Download Standards - Joliet Public Schools District 86

Joliet Public Schools District 86
Mathematics Curriculum
Aligned with
The New Illinois State Standards Incorporating the Common Core
Grades 6-8
Charles E. Coleman, Ed.D.
Superintendent
June 2013
Joliet Public Schools District 86
Mathematics Curriculum
Aligned with
The New Illinois State Standards Incorporating the Common Core
Grades 6-8
Charles E. Coleman, Ed.D.
Superintendent
June 2013
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Joliet Public Schools District 86
Mathematics Mission Statement
The mission of the Joliet Public Schools District 86 Mathematics Curriculum is to
develop a program of study that supports focused instruction of the concepts and key
skills at each grade level. This curriculum provides structured opportunities for students
to become mathematical thinkers in a global society.
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Acknowledgements
Thank you to the members of the Grades 6-8 Mathematics Common Core Curriculum Team
for their collaborative work in creating this guide.
Linda Anton, Dirksen Junior High School
Kristin Gedmin, Hufford Junior High School
Susan Groves, Washington Junior High School
Mary Pickens, Gompers Junior High School
Christine Reed, Gompers Junior High School
Jaculin Taylor-Nowak, Dirksen Junior High School
Lisa Trilli-Mayfield, Hufford Junior High School
Misael Vargas, Gompers Junior High School
Annie Walker, Washington Junior High School
Erin Buteau, Technology Department, John F. Kennedy Administrative Center
Caiti Dominick, Special Services Department, John F. Kennedy Administrative Center
Jan Taylor, Curriculum & Instruction, John F. Kennedy Administrative Center
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Table of Contents
Section
Page
The Standards for Mathematical Practice
The 8 Mathematical Practice Standards
How to Read the Content Standards
Grade 6 Standards
Grade 7 Standards
Grade 8 Standards
Common Core References
5
6
7
8
112
183
247
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THE STANDARDS FOR MATHEMATICAL PRACTICE
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators
at all levels should seek to develop in their students. These practices rest on important ―processes
and proficiencies‖ with longstanding importance in mathematics education. The first of these are the
NCTM process standards of problem solving, reasoning and proof, communication, representation,
and connections. The second are the strands of mathematical proficiency specified in the National
Research Council‘s report Adding It Up: adaptive reasoning, strategic competence, conceptual
understanding (comprehension of mathematical concepts, operations and relations), procedural
fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and
productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile,
coupled with a belief in diligence and one‘s own efficacy).
The Common Core State Standards, Standards for Mathematical Practice.
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MATHEMATICAL PRACTICE STANDARDS
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity and reasoning.
The Common Core State Standards, Standards for Mathematical Practice.
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CONTENT STANDARDS
HOW TO READ THE GRADE LEVEL STANDARDS
Standards define what students should understand and be able to do.
Clusters summarize groups of related standards. Note that standards from different clusters may
sometimes be closely related, because mathematics is a connected subject.
Domains are larger groups of related standards. Standards from different domains may sometimes
be closely related.
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Mathematics | Grade 6 – Critical Areas
In Grade 6, instructional time should focus on four critical areas: (1) connecting ratio and rate to whole number multiplication
and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and
extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting,
and using expressions and equations; and (4) developing understanding of statistical thinking.
(1) Students use reasoning about multiplication and division to solve ratio and rate problems about quantities. By viewing equivalent
ratios and rates as deriving from, and extending, pairs of rows (or columns) in the multiplication table, and by analyzing simple
drawings that indicate the relative size of quantities, students connect their understanding of multiplication and division with ratios
and rates. Thus students expand the scope of problems for which they can use multiplication and division to solve problems, and they
connect ratios and fractions. Students solve a wide variety of problems involving ratios and rates.
(2) Students use the meaning of fractions, the meanings of multiplication and division, and the relationship between multiplication and
division to understand and explain why the procedures for dividing fractions make sense. Students use these operations to solve
problems. Students extend their previous understandings of number and the ordering of numbers to the full system of rational
numbers, which includes negative rational numbers, and in particular negative integers. They reason about the order and absolute
value of rational numbers and about the location of points in all four quadrants of the coordinate plane.
(3) Students understand the use of variables in mathematical expressions. They write expressions and equations that correspond to
given situations, evaluate expressions, and use expressions and formulas to solve problems. Students understand that expressions in
different forms can be equivalent, and they use the properties of operations to rewrite expressions in equivalent forms. Students know
that the solutions of an equation are the values of the variables that make the equation true. Students use properties of operations and
the idea of maintaining the equality of both sides of an equation to solve simple one-step equations. Students construct and analyze
tables, such as tables of quantities that are in equivalent ratios, and they use equations (such as 3x = y) to describe relationships
between quantities.
(4) Building on and reinforcing their understanding of number, students begin to develop their ability to think statistically. Students
recognize that a data distribution may not have a definite center and that different ways to measure center yield different values. The
median measures center in the sense that it is roughly the middle value. The mean measures center in the sense that it is the value that
each data point would take on if the total of the data values were redistributed equally, and also in the sense that it is a balance point.
Students recognize that a measure of variability (interquartile range or mean absolute deviation) can also be useful for summarizing
data because two very different sets of data can have the same mean and median yet be distinguished by their variability. Students
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learn to describe and summarize numerical data sets, identifying clusters, peaks, gaps, and symmetry, considering the context in which
the data were collected.
Students in Grade 6 also build on their work with area in elementary school by reasoning about relationships among shapes to
determine area, surface area, and volume. They find areas of right triangles, other triangles, and special quadrilaterals by decomposing
these shapes, rearranging or removing pieces, and relating the shapes to rectangles. Using these methods, students discuss, develop,
and justify formulas for areas of triangles and parallelograms. Students find areas of polygons and surface areas of prisms and
pyramids by decomposing them into pieces whose area they can determine. They reason about right rectangular prisms with fractional
side lengths to extend formulas for the volume of a right rectangular prism to fractional side lengths. They prepare for work on scale
drawings and constructions in Grade 7 by drawing polygons in the coordinate plane.
The Common Core State Standards
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Grade 6 Overview
Ratios and Proportional Relationships
• Understand ratio concepts and use ratio reasoning to solve problems.
The Number System
• Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
• Compute fluently with multi-digit numbers and find common factors and multiples.
• Apply and extend previous understandings of numbers to the system of rational numbers.
Expressions and Equations
• Apply and extend previous understandings of arithmetic to algebraic expressions.
• Reason about and solve one-variable equations and inequalities.
• Represent and analyze quantitative relationships between dependent and independent variables.
Geometry
• Solve real-world and mathematical problems involving area, surface area, and volume.
Statistics and Probability
• Develop understanding of statistical variability.
• Summarize and describe distributions.
The Common Core State Standards
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Mathematical Content Standards
Grade 6
Ratios and Proportional Relationships 6.RP
Enduring Understandings
Essential Questions
Ratios and proportional relationships are used to express how
How can ratios and proportional relationships be used to
quantities are related and how quantities change in relation to
determine unknown quantities.
each other.
Major Cluster: Understand ratio concepts and use ratio reasoning to solve problems.
Standard: 6.RP.1.
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example,
“The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote
candidate A received, candidate C received nearly three votes.”
Explanation:
 Students will know that a ratio is the comparison of two quantities or measures. The comparison can be part-to-whole
(ratio of guppies to all fish in an aquarium) or part-to-part (ratio of guppies to goldfish).
 Students need to understand each of these ratios when expressed in the following forms: 6 to 15 or 6:15. These values can
2
be reduced to , 2 to 5 or 2:5
5
Learning Targets:
 I can define the term ratio and demonstrate my understanding by giving various examples.
 I can write a ratio that describes a relationship between two quantities.
 I can explain the relationship that a ratio represents.
Vocabulary:
Ratio
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Sample Problem(s):
Sample 1
A comparison of 6 guppies and 9 goldfish could be expressed in any of the following forms:
6
, 6 or 6:9. If the number of
9
guppies is represented by black circles and the number of goldfish is represented by white circles,
These values can be regrouped into 2 black circles (goldfish) to 2 white circles (guppies), which would reduce the ratio to
2
, 2 to 3 or 2:3.
3
Students should be able to identify and describe any ratio using ―For every _____, there are _____‖. In the examples above, the
ratio could be expressed saying, ―For every 2 goldfish, there are 3 guppies‖.
Resources:
 Learning About Ratios: A Sandwich Study
http://tinyurl.com/LearningAboutRatios
Students make peanut butter sandwiches using different ratios of ingredients in this activity.
 Proportional Reasoning
http://www.learner.org/courses/learningmath/algebra/session4/
In this lesson, students explore proportions by examining proportional relationships, absolute and relative comparisons,
and looking at graphs to learn about these relationships.
 Ratios and Proportions
http://tinyurl.com/RatiosProportions
This is an interactive slideshow that explains ratios and proportions and provides examples and sample problems that the
student can work through.
 What's My Ratio?
http://tinyurl.com/MyRatios
In this lesson, students investigate proportionality by using linear measurement. They also find the ratio between similar
figures.
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Major Cluster: Understand ratio concepts and use ratio reasoning to solve problems.
Standard: 6.RP.2
Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio
relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is ¾ cup of flour for each cup of
sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”(Expectations for unit rates in this grade are
limited to non-complex fractions).
Explanation:
Expectations for unit rates in this grade are limited to non-complex fractions. Both the numerator and denominator of the
original ratio will be whole numbers.
 Students will know a unit rate expresses a ratio as part-to-one or one unit of another quantity. For example, if there are 2
cookies for 3 students, each student receives 2/3of a cookie, so the unit rate is 2/3:1. If a car travels 240 miles in 4 hours,
the car travels 60 miles per hour (60:1).
 Students understand the unit rate from various contextual situations.
 Students will understand rate language (for each, for every, per…)
Learning Targets:
 I can define the term ―unit rate.‖
 I can demonstrate my understanding by giving various examples.
 I can recognize a ratio written as a unit rate
 I can explain a unit rate
 I can describe the ratio relationship represented by a unit rate
 I can convert a given ratio to a unit rate
Vocabulary:
ratio, rate, unit rate
Sample Problem(s):
Sample 1
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Sample 2
Resources:
 Learning About Ratios: A Sandwich Study
http://tinyurl.com/LearningAboutRatios
Students make peanut butter sandwiches using different ratios of ingredients in this activity.
 Ratios and Proportions
http://tinyurl.com/RatiosProportions
This is an interactive slideshow that explains ratios and proportions and provides examples and sample problems that the
student can work through.
 What's My Ratio?
http://tinyurl.com/MyRatios
In this lesson, students investigate proportionality by using linear measurement. They also find the ratio between similar
figures.
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Major Cluster: Understand ratio concepts and use ratio reasoning to solve problems.
Standard: 6.RP.3
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios,
tape diagrams, double number line diagrams, or equations.
a. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and
plot the pairs of values on the coordinate plane. Use tables to compare ratios.
Explanation:
 Students will know that ratios and rates can be used in ratio tables and graphs to solve problems. Previously, students have
used additive reasoning in tables to solve problems.
 Students need to begin using multiplicative reasoning. To aid in the development of proportional reasoning the crossproduct algorithm is not expected at this level.
 When working with ratio tables and graphs, whole number measurements are the expectation for this standard.
Learning Targets:
 I can create a table of equivalent ratios
 I can solve real world problems involving proportional reasoning by using various diagrams
 I can use the proportional relationship to find missing values in a table of equivalent ratios.
 I can compare ratios presented in various tables
 I can plot corresponding values from an equivalent ratio table on a coordinate grid.
Vocabulary:
ratio, rate, coordinate plane, tape diagram, proportional relationship, equivalent ratio
Sample Problem(s):
Sample 1
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Sample 2
Sample 3
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Sample 4
Resources:
 Grid and Percent It
http://tinyurl.com/GridPercent
This lesson plans provides a 10 x 10 model so that students can understand how to solve percent problems.
 IXL Game: Ratios, proportions, and percents
http://tinyurl.com/IXLRatioGame
This game helps understand ratios, proportions, and percents, specifically percents of numbers and money amounts.
 Modeling Linear Relationships
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


http://tinyurl.com/LinearModeling
Students compare arm span and height in people to learn about proportion. They also use a scatter plot graph to analyze
their data.
Proportional Reasoning
http://www.learner.org/courses/learningmath/algebra/session4/
In this lesson, students explore proportions by examining proportional relationships, absolute and relative comparisons,
and looking at graphs to learn about these relationships.
Ratio Word Problems
http://www.mathplayground.com/MTV/pbratio1.html
Students view videos explaining how to solve word problems dealing with ratios and then get a chance to solve them on
their own.
Ratios and Proportions
http://tinyurl.com/RatiosProportions
This is an interactive slideshow that explains ratios and proportions and provides examples and sample problems that the
student can work through.
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Major Cluster: Understand ratio concepts and use ratio reasoning to solve problems.
Standard: 6.RP.3
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios,
tape diagrams, double number line diagrams, or equations.
b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4
lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
Explanation:
 Students recognize the use of ratios, unit rate and multiplication in solving problems, which could allow for the use of
fractions and decimals.
Learning Targets:
 I can solve real-world problems involving unit pricing by using various diagrams.
 I can solve real-world problems involving constant speed by using various diagrams.
Vocabulary:
ratio, rate, unit rate, coordinate plane, constant speed, unit pricing
Sample Problem(s):
Sample 1
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Sample 2
Resources:
 Grid and Percent It
http://tinyurl.com/GridPercent
This lesson plans provides a 10 x 10 model so that students can understand how to solve percent problems.
 IXL Game: Ratios, proportions, and percents
http://tinyurl.com/IXLRatioGame
This game helps sixth graders understand ratios, proportions, and percents, specifically percents of numbers and money
amounts.
 Modeling Linear Relationships
http://tinyurl.com/LinearModeling
Students compare arm span and height in people to learn about proportion. They also use a scatter plot graph to analyze
their data.
 Proportional Reasoning
http://www.learner.org/courses/learningmath/algebra/session4/
In this lesson, students explore proportions by examining proportional relationships, absolute and relative comparisons,
and looking at graphs to learn about these relationships.
 Ratio Word Problems
http://www.mathplayground.com/MTV/pbratio1.html
Students view videos explaining how to solve word problems dealing with.
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Major Cluster: Understand ratio concepts and use ratio reasoning to solve problems.
Standard: 6.RP.3
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape
diagrams, double number line diagrams, or equations.
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving
finding the whole, given a part and the percent.
Explanation:
 This is the students‘ first introduction to percents. Percentages are a rate per 100. Models, such as percent bars or 10 x 10 grids
should be used to model percents.
 Students use percentages to find the part when given the percent, by recognizing that the whole is being divided into 100 parts
and then taking a part of them (the percent).
 Students also find the whole, given a part and the percent.
Learning Targets:
 I can use visual representations (e.g., strip diagrams, percent bars, one-hundred grids) to model percents
 I can write a percent as a rate per one-hundred
 I can use proportional reasoning to find the percent of a given number
 I can use proportional reasoning to find the whole when given both the part and the percent
Vocabulary:
equivalent ratio, rate, unit rate, percent, coordinate plane, proportional reasoning, strip diagram, percent bar
Sample Problem(s):
Sample 1
Sample 2
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Sample 3
Sample 4
Resources:
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



Grid and Percent It
http://tinyurl.com/GridPercent
This lesson plans provides a 10 x 10 model so that students can understand how to solve percent problems.
Modeling Linear Relationships
http://tinyurl.com/LinearModeling
Students compare arm span and height in people to learn about proportion. They also use a scatter plot graph to analyze their
data.
Ratio Word Problems
http://www.mathplayground.com/MTV/pbratio1.html
Students view videos explaining how to solve word problems dealing with ratios and then get a chance to solve them on their
own.
What's My Ratio?
http://tinyurl.com/MyRatios
In this lesson, students investigate proportionality by using linear measurement. They also find the ratio between similar
figures.
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Major Cluster: Understand ratio concepts and use ratio reasoning to solve problems.
Standard: 6.RP.3
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios,
tape diagrams, double number line diagrams, or equations.
d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing
quantities.
Explanation:
 A ratio can be used to compare measures of two different types, such as inches per foot, milliliters per liter and
centimeters per inch.
 Students recognize that a conversion factor is a fraction equal to 1 since the quantity described in the numerator and
denominator is the same. For example 12 inches is a conversion 1 foot factor since the numerator and denominator name
the same amount.
 Students use ratios as conversion factors and the identity property for multiplication to convert ratio units. For example,
how many centimeters are in 7 feet, given that 1 inch = 2.54 cm.
7 feet x 12 inches x 2.54 cm = 7 feet x 12 inches x 2.54 cm = 7 x 12 x 2.54 cm = 213.36 cm
1 foot
1 inch
1 foot
1 inch
Note: Conversion factors will be given. Conversions can occur both between and across the metric and
English system. Estimates are not expected.
Learning Targets:
 I can explain that a conversion factor is a fraction equal to 1.
 I can convert measurement units using ratio reasoning.
 I can convert measurement units between Metric and English using ratio reasoning.
Vocabulary:
ratio, rate, unit rate, conversion factor
Sample Problem(s):
Sample 1
How many centimeters are in 7 feet, given that 1 inch ≈ 2.54 cm?
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Resources:
 IXL Game: Ratios, proportions, and percents
http://tinyurl.com/IXLRatioGame
This game helps sixth graders understand ratios, proportions, and percents, specifically percents of numbers and money
amounts.
 Modeling Linear Relationships
http://tinyurl.com/LinearModeling
Students compare arm span and height in people to learn about proportion. They also use a scatter plot graph to analyze
their data.
 Pie Chart
http://www.shodor.org/interactivate/activities/PieChart/
Students enter values in this applet and create pie charts in which they can vary the number or size of sections and
display as fractions or percentages.
 Proportional Reasoning
http://www.learner.org/courses/learningmath/algebra/session4/
In this lesson, students explore proportions by examining proportional relationships, absolute and relative comparisons,
and looking at graphs to learn about these relationships.
 Ratio Word Problems
http://www.mathplayground.com/MTV/pbratio1.html
Students view videos explaining how to solve word problems dealing with ratios and then get a chance to solve them on
their own.
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The Number System 6.NS
Enduring Understandings
Essential Questions
Rational numbers can be represented in multiple ways and are
In what ways can rational numbers be useful?
useful when examining situations involving numbers that are
not whole.
Major Cluster: Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
Standard: 6.NS.1
Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using
visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷(3/4) = (8/9 )
because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share ½ lb of
chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with
length 3/4 mile and area1/2 square mile?
Explanation:
 In 5th grade students divided whole numbers by unit fractions. Students continue this understanding by using visual
models and equations to divide whole numbers by fractions and fractions by fractions to solve word problems. Students
understand that a division problem such as 3 ’ 2/5 is asking, ―how many 2/5 are in 3?‖
Learning Targets:
 I can use a visual model to represent the division of a fraction by a fraction
 I can divide fractions by fractions using an algorithm or mathematical reasoning
 I can use mathematical reasoning to justify the standard algorithm for fraction division
 I can solve real world problems involving the division of fractions
 I can interpret (explain) the quotient in the context of the problem
 I can create story contexts for the problems involving the division of a fraction by a fraction
Vocabulary:
quotient, interpret, reciprocal, justify
Sample Problem(s):
Sample 1
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Sample 2
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Sample 3
Sample 4
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Resources:
 Division Practice
http://learnzillion.com/lessons/204-divide-fractions-by-fractions-using-models
This lesson uses a model to show division of fractions
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Additional Cluster: Compute fluently with multi-digit numbers and find common factors and multiples.
Standard: 6.NS.2
Fluently divide multi-digit numbers using the standard algorithm.
Explanation:
 Procedural fluency is defined by the Common Core as ―skill in carrying out procedures flexibly, accurately, efficiently
and appropriately‖. In 5th grade, students were introduced to division through concrete models and various strategies to
develop an understanding of this mathematical operation (limited to 4-digit numbers divided by 2-digit numbers). In 6th
grade, students become fluent in the use of the standard division algorithm. This understanding is foundational for work
with fractions and decimals in 7th grade.
 Students are expected to fluently and accurately divide multi-digit whole numbers. Divisors can be any number of digits
at this grade level.
Learning Targets:
 I can use the standard algorithm to fluently divide multi-digit numbers.
Vocabulary:
fluent, dividend, divisor, quotient, remainder
Sample Problem(s):
Sample 1
Sample 2
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Resources:
 Division Practice (partial quotients)
http://www.youtube.com/watch?v=qWstA8EZr2w&safe=active
Video shows how to use partial quotients when dividing multi-digit numbers.
 Partial Quotients
http://everydaymath.uchicago.edu/teaching-topics/computation/div-part-quot/
A video showing how to use partial quotients when dividing multi-digit numbers.
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Additional Cluster: Compute fluently with multi-digit numbers and find common factors and multiples.
Standard: 6.NS.3
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
Explanation:
 Multiplication and division of decimals was introduced in 5th grade (decimals to the hundredth place).
 Students become fluent in the use of the standard algorithms of each of these operations.
Learning Targets:
 I can fluently add multi-digit decimals using the standard algorithm.
 I can fluently subtract multi-digit decimals using the standard algorithm.
 I can fluently multiply multi-digit decimals using the standard algorithm.
 I can fluently divide multi-digit decimals using the standard algorithm.
Vocabulary:
fluently, decimal
Sample Problem(s):
Sample 1
First estimate the sum of 12.3 and 9.75.
Solution: An estimate of the sum would be 12 + 10 or 22. Students could also state if their estimate is high or low.
Answers of 230.5 or 2.305 indicate the students are not considering place value when adding.
Resources:
 IXL – Add and Subtract Money
http://www.ixl.com/math/grade-6/add-and-subtract-money-amounts
This Game helps students understand adding and subtracting decimals using money.
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Additional Cluster: Compute fluently with multi-digit numbers and find common factors and multiples.
Standard: 6.NS.4
Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole
numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common
factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2). Apply and
extend previous understandings of numbers to the system of rational numbers.
Explanation:
GCF
 Students will find the greatest common factor of two whole numbers less than or equal to 100. For example, the greatest
common factor of 40 and 16 can be found by: listing the factors of 40 (1, 2, 4, 5, 8, 10, 20, 40) and 16 (1, 2, 4, 8, 16), then
taking the greatest common factor (8) or listing the prime factors of 40 (2 • 2 • 2 • 5) and 16 (2 • 2 • 2 • 2) and then
multiplying the common factors (2 • 2 • 2 = 8).
 Students also understand that the greatest common factor of two prime numbers will be 1.
 Students use the greatest common factor and the distributive property to find the sum of two whole numbers. For example,
36 + 8 can be expressed as 4 (9 + 2) = 4 (11) = 44
LCM
 Students find the least common multiple of two whole numbers less than or equal to twelve. For example, the least
common multiple of 6 and 8 can be found by: listing the multiplies of 6 (6, 12, 18, 24, 30, …) and 8 (8, 26, 24, 32, 40…),
then taking the least in common from the list (24); or using the prime factorization.
Learning Targets:
 I can find all factors of any given number, less than or equal to 100
 I can find the greatest common factor of any two numbers less than or equal to 100.
 I can create a list of all multiples for any number less than or equal to 12
 I can find the least common multiple of any two numbers, less than or equal to 12.
 I can use the distributive property to rewrite a simple addition problem when addends have a common factor
Vocabulary:
factor, multiple, greatest common factor, least common multiple, distributive property, addends
Sample Problem(s):
Sample 1
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Sample 2
Sample 3
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Sample 4
Resources:
 Factors
http://www.shodor.org/interactivate/lessons/Factors/
This lesson is designed to help students understand factors of whole numbers.
 Finding Factors
http://www.shodor.org/interactivate/lessons/FindingFactors/
This lesson plan's activities give students practice in finding the factors of whole numbers.
 Patterns in Pascal's Triangle
http://www.shodor.org/interactivate/lessons/PatternsInPascal
This lesson plan was designed to help students understand Pascal's Triangle and its patterns.
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
Sets and the Venn Diagram
http://www.shodor.org/interactivate/lessons/SetsTheVennDiagram/
This lesson is designed to help students understand the ideas surrounding sets and Venn diagrams.
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Major Cluster: Apply and extend previous understandings of numbers to the system of rational numbers.
Standard: 6.NS.5
Understand that positive and negative numbers are used together to describe quantities having opposite directions or values
(e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use
positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
Explanation:
 Students use rational numbers (fractions, decimals, and integers) to represent real-world contexts and understand the
meaning of 0 in each situation.
Learning Targets:
 I can give examples of how positive numbers are used to describe real world situations.
 I can give examples of how negative numbers are used to describe real world situations.
 I can recognize that positive signs represent opposite values and/or directions.
 I can recognize that negative signs represent opposite values and/or directions.
 I can explain that the number zero is the point at which direction or value will change.
Vocabulary:
positive, negative, opposite value
Sample Problem(s):
Sample1
Resources:
 OPUS
http://tinyurl.com/num3p5k
Sample problems for whole class instruction
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Major Cluster: Apply and extend previous understandings of numbers to the system of rational numbers.
Standard: 6.NS.6
Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from
previous grades to represent points on the line and in the plane with negative number coordinates.
a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the
opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite.
Explanation:
 Students extend the number line to represent all rational numbers.
 Students recognize that a number and its opposite are equidistance from zero (reflections about the zero). The
opposite sign (–) shifts the number to the opposite side of 0. For example, – 4 could be read as ―the opposite of 4‖
which would be negative 4.
Learning Targets:
 I can model rational numbers as a point on a number line.
 I can explain why every rational number can be represented by a point on a number line.
 I can plot a number and its opposite on a number line and recognize that they are equidistant from zero.
 I can find the opposite of any given number including zero.
Vocabulary:
rational number, integer, opposite, coordinate plane, ordered pair, quadrant, reflection
Sample Problem(s):
Sample 1
Resources:
 Sample Assessment Task: Cake Weighing
http://tinyurl.com/CakeWeigh
This sample assessment task provides a situation that students can understand, even if it is not in their everyday
experience. Use the navigation at the upper right of this page to access the task.
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Major Cluster: Apply and extend previous understandings of numbers to the system of rational numbers.
Standard: 6.NS.6
Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from
previous grades to represent points on the line and in the plane with negative number coordinates.
b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that
when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
Explanation:
 Students extend the number line to represent all rational numbers.
 Students recognize that number lines may be either horizontal or vertical (i.e. thermometer).
 Students recognize the point where the x-axis and y-axis intersect as the origin.
 Students identify the four quadrants.
 Students are able to identify the quadrant for an ordered pair based on the signs of the coordinates. For example,
students recognize that in Quadrant II, the signs of all ordered pairs would be (–, +).
 Students understand the relationship between two ordered pairs differing only by signs as reflections across one or both
axes.
Learning Targets:
 I can use the signs of the coordinates to determine the location of an ordered pair in the coordinate plane
 I can understand the relationship between two ordered pairs differing only by signs as reflections across one or both
axes.
Vocabulary:
rational number, integer, opposite, coordinate plane, ordered pair, quadrant, reflection
Sample Problem(s):
Sample 1
Graph the following points in the correct quadrant of the coordinate plane. If the point is reflected across the x-axis, what are
the coordinates of the reflected points? What similarities are between coordinates of the original point and the reflected point?
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Resources:
 Sample Assessment Task: Cake Weighing
http://tinyurl.com/CakeWeigh
This four-part sample assessment task provides a novel situation in an authentic context that all students can
understand, even though it is not likely to be in their everyday experience. Use the navigation at the upper right of this
page to access the task.
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Major Cluster: Apply and extend previous understandings of numbers to the system of rational numbers.
Standard: 6.NS.6
Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from
previous grades to represent points on the line and in the plane with negative number coordinates.
c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position
pairs of integers and other rational numbers on a coordinated plane.
Explanation:
 Students are able to plot all rational numbers on a number line (either vertical or horizontal) or identify the values of
given points on a number line. For example, students are able to identify where the following numbers would be on a
1
number line: –4.5, 2, 3.2, –3 3/5, 0.2, –2, 1 .
2
Learning Targets:
 I can find a point on a number line.
 I can find a point on a coordinate plane.
 I can position a point from a number line.
 I can position a point from a coordinate plane.
Vocabulary:
rational number, integer, opposite, coordinate plane, ordered pair, quadrant, reflection
Sample Problem(s):
Sample 1
Resources:
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
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Sample Assessment Task: Cake Weighing
http://tinyurl.com/CakeWeigh
This four-part sample assessment task provides a novel situation in an authentic context that all students can
understand, even though it is not likely to be in their everyday experience. Use the navigation at the upper right of
this page to access the task.
Major Cluster: Apply and extend previous understandings of numbers to the system of rational numbers.
Standard: 6.NS.7
Understand ordering and absolute value of rational numbers.
a. Interpret statements of inequality as statements about the relative position of two numbers on a number line. For example,
interpret -3>-7 as a statement that -3 is located to the right of -7 on a number line oriented from left to right.
Explanation:
 Students identify the absolute value of a number as the distance from zero but understand that although the value of 7 is less than -3, the absolute value (distance) of -7 is greater than the absolute value (distance) of -3.
 Students use inequalities to express the relationship between two rational numbers, understanding that the value of
numbers is smaller moving to the left on a number line. For example, –4 ½ < –2 because –4 ½ is located to the left of
–2 on the number line.
Learning Targets:
 I can describe the relative position of two numbers on a number line when given an inequality.
 I can describe the value of a number in relation to 0.
Vocabulary:
absolute value, positive, negative, inequality
Sample Problem(s):
Sample 1
Resources:
 Sample Assessment Task: Cake Weighing
http://tinyurl.com/CakeWeigh
This four-part sample assessment task provides a novel situation in an authentic context that all students can
understand, even though it is not likely to be in their everyday experience. Use the navigation at the upper right of
this page to access the task.
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Major Cluster: Apply and extend previous understandings of numbers to the system of rational numbers.
Standard: 6.NS.7
Understand ordering and absolute value of rational numbers.
b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write -3°C>7°C to express the fact than -3°C is warmer then -7°C.
Explanation:
 Students write statements using < or > to compare rational number in context. However, explanations should
reference the context rather than ―less than‖ or ―greater than‖. For example, the balance in Sue‘s checkbook was –
12.55. The balance in Ron‘s checkbook was –10.45. Since –12.55 < –10.45, Sue owes more than Ron. The
interpretation could also be ―Ron owes less than Sue‖.
Learning Targets:
 I can interpret a given inequality in terms of a real-world situation.
 I can write statements comparing rational numbers in real-world contexts.
Vocabulary:
rational number, integer
Sample Problem(s):
Sample 1
Sample 2
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Sample 3
Resources:
 Comparing Temperatures
http://www.illustrativemathematics.org/illustrations/285
The purpose of the task is for students to compare signed numbers in a real-world context. It could be used for
either assessment or instruction if the teacher were to use it to generate classroom discussion.
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Major Cluster: Apply and extend previous understandings of numbers to the system of rational numbers.
Standard: 6.NS.7
Understand ordering and absolute value of rational numbers.
c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute as
magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of -30 dollars,
write | –30|= 30 to describe the size of the debt in dollars.
Explanation:
 Students understand absolute value as the distance from zero and recognize the symbols | | as representing absolute
value. For example, | –7 | can be interpreted as the distance –7 is from 0 which would be 7. Likewise | 7 | can be
interpreted as the distance 7 is from 0 which would also be 7. In real-world contexts, the absolute value can be used
to describe size or magnitude. For example, for an ocean depth of –900 feet, write | –900| = 900 to describe the
distance below sea level.
Learning Targets:
 I can understand absolute value as the distance from zero.
 I can describe absolute value as the magnitude (size, enormity) of the number in a real-world situation.
Vocabulary:
rational number, integer, opposite, magnitude, absolute value
Sample Problem(s):
Sample 1
Sample 2
Resources:
 Sample Assessment Task: Cake Weighing
http://tinyurl.com/CakeWeigh
This sample assessment task provides a novel situation in an authentic context that students can understand. Use the
navigation at the upper right of this page to access the task.
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Major Cluster: Apply and extend previous understandings of numbers to the system of rational numbers.
Standard: 6.NS.7
Understand ordering and absolute value of rational numbers.
d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance
less than -30 dollars represents a debt greater than 30 dollars.
Explanation:
 When working with positive numbers, the absolute value (distance from zero) of the number and the value of the
number is the same; therefore, ordering is not problematic. However, negative numbers have a distinction that
students need to understand. As the negative number increases (moves to the left on a number line), the value of the
number decreases.
Learning Targets:
 I can make a comparison statement referring to a real-world situation given a signed number.
 I can make a comparison statement referring to a real-world situation given the absolute value of a signed number.
Vocabulary:
rational number, integer, opposite, absolute value
Sample Problem(s):
Sample 1
1. Display:
The Earth‘s temperature range spans from 136°F in Libya to -129°C at the Vostok Station in Antarctica.
2. Ask students to turn and talk to a partner and make two subsequent statements about the original statement, such as: ―It is
much colder in Antarctica than Libya,‖ ―We live in a much warmer place than Antarctica,‖ or ―Those temperatures are
extremes.‖
3. Ask some students to share their statements with the whole group and discuss them, eliciting follow-up questions as
appropriate.
Resources:
 Sample Assessment Task: Cake Weighing
http://tinyurl.com/CakeWeigh
This four-part sample assessment task provides a novel situation in an authentic context that all students can
understand, even though it is not likely to be in their everyday experience. Use the navigation at the upper right of
this page to access the task.
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Major Cluster – Apply and extend previous understandings of numbers to the system of rational numbers.
Standard: 6.NS.8.
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of
coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
Explanation:
 Students find the distance between points whose ordered pairs have the same x-coordinate (vertical) or same y-coordinate
(horizontal). For example, the distance between (–5, 2) and (–9, 2) would be 4 units. This would be a horizontal line since
the y-coordinates are the same. In this scenario, both coordinates are in the same quadrant. The distance can be found by
using a number line to find the distance between –5 and –9. Students could also recognize that –5 is 5 units from 0
(absolute value) and that –9 is 9 units from 0 (absolute value). Since both of these are in the same quadrant, the distance
can be found by finding the difference between 9 and 5. (| 9 | - | 5 |).
 Students will know that coordinates could also be in two quadrants. For example, the distance between (3, –5) and (3, 7)
would be 12 units. This would be a vertical line since the x-coordinates are the same. The distance can be found by using
a number line to count from –5 to 7 or by recognizing that the distance (absolute value) from –5 to 0 is 5 units and the
distance (absolute value) from 0 to 7 is 7 units so the total distance would be 5 + 7 or 12 units.
Learning Targets:
 I can graph points in all four quadrants of the coordinate plane to solve real-world mathematical problems.
 I can use absolute values to find the distance between two points with the same x-coordinates.
 I can use absolute values to find the distance between two points with the same y-coordinates.
Vocabulary:
coordinate plane, quadrant, coordinates, x-coordinate, y- coordinate, absolute value
Sample Problem(s):
Sample 1
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Sample 2
Resources:
 IXL Game: Coordinate graphing
http://tinyurl.com/coordinategraph
This game will help sixth graders learn to graph points on a coordinate plane. Note: The IXL site requires subscription
for unlimited use.
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Expressions and Equations 6.EE
Enduring Understandings
Essential Questions
Algebraic expressions and equations are used to model real-life How can algebraic expressions and equations be used to model,
problems and represent quantitative relationships, so that the
analyze, and solve mathematical situations?
numbers and symbols can be mindfully manipulated to reach a
solution or make sense of the quantitative relationships
Major Cluster – Apply and extend previous understandings of arithmetic to algebraic expressions.
Standard: 6.EE.1
Write and evaluate numerical expressions involving whole-number exponents.
Explanation:
 Students demonstrate the meaning of exponents to write and evaluate numerical expressions with whole number
exponents. The base can be a whole number, positive decimal or a positive fraction.
 Students recognize that an expression with a variable represents the same mathematics (i.e. x4 can be written as x • x • x •
x)
Learning Targets:
 I can explain the meaning of a number raised to a power/exponent through repeated multiplication.
 I can write numerical expressions involving whole-number exponents
 I can evaluate numerical expressions involving whole-number exponents.
Vocabulary:
base, exponent, evaluate, power
Sample Problem(s):
Sample 1
Sample 2
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Sample 3
Resources:
 Algebra Four
http://www.shodor.org/interactivate/activities/AlgebraFour/
This lesson contains a game activity designed to help students practice solving algebraic equations.
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Major Cluster – Apply and extend previous understandings of arithmetic to algebraic expressions.
Standard: 6.EE.2
Write, read, and evaluate expressions in which letters stand for numbers.
a. Write expressions that record operations with numbers and with letters standing for numbers.
For example, express the calculation “Subtract y from 5” as 5 – y.
Explanation:
 Students write expressions from verbal descriptions using letters and numbers.
 Students understand order is important in writing subtraction and division problems.
 Students understand that the expression ―5 times any number, n‖ could be represented with 5n and that a number and
letter written together means to multiply.
Learning Targets:
 I can identify parts of an algebraic expression by using correct mathematical terms.
 I can write an algebraic expression representing a sum given a verbal expression.
 I can write an algebraic expression representing a difference given a verbal expression.
 I can write an algebraic expression representing a product given a verbal expression.
 I can write an algebraic expression representing a quotient given a verbal expression.
Vocabulary:
sum, difference, term, product, factor, quotient, coefficient, algebraic expression, substitute, evaluate, verbal expression
Sample Problem(s):
Sample 1
Resources:
 Algebra Balance Scales : Negatives
http://tinyurl.com/NegativesScale
This applet presents balance scales students can use to understand how to balance the scales and equations using
negative numbers.
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

IXL Game: Algebra: Evaluate expressions
http://tinyurl.com/Evaluateexpressions
This game is designed to help sixth graders understand how to evaluate expressions involving integers. This is just one
of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use.
Linear Function Machine
http://www.shodor.org/interactivate/activities/LinearFunctMachine/
By putting different values into the linear function machine students will explore simple linear functions.
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Major Cluster – Apply and extend previous understandings of arithmetic to algebraic expressions.
Standard: 6.EE.2
Write, read, and evaluate expressions in which letters stand for numbers.
b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more
parts of an expression as a single entity.
For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two
terms.
Explanation:
 Students use appropriate mathematical language to write verbal expressions from algebraic expressions.
Students can describe expressions such as 3 (2 + 6) as the product of two factors: 3 and (2 + 6). The quantity (2 + 6) is viewed as
one factor consisting of two terms.
Learning Targets:
 I can recognize an expression as both a single value and as two or more terms on which an operation is performed.
 I can use the correct mathematical language to identify parts of a verbal expression
 I can use the correct mathematical language to identify parts of an algebraic expression.
Vocabulary:
sum, difference, term, product, factor, quotient, coefficient, arithmetic expression, algebraic expression
Sample Problem(s):
Sample 1
Students write algebraic expressions:
 7 less than 3 time a number
Solution: 3x – 7
 3 times the sum of a number and 5
Solutions: 3 (x+5)
 7 less than the product of 2 and a number
Solution: 2x – 7
 Twice the difference between a number and 5
Solution: 2(z - 5)
 The quotient of the sum of x plus 4 and 2
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Solution:
x4
2
Resources:
 Algebra Balance Scales : Negatives
http://tinyurl.com/NegativesScale
This applet presents balance scales students can use to understand how to balance the scales and equations using
negative numbers.
 IXL Game: Algebra: Evaluate expressions
http://tinyurl.com/Evaluateexpressions
This game is designed to help sixth graders understand how to evaluate expressions involving integers. This is just one
of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use.
 Linear Function Machine
http://www.shodor.org/interactivate/activities/LinearFunctMachine/
By putting different values into the linear function machine students will explore simple linear functions.
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Major Cluster – Apply and extend previous understandings of arithmetic to algebraic expressions.
Standard: 6.EE.2
Write, read, and evaluate expressions in which letters stand for numbers.
c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world
problems. Perform arithmetic operations, including those involving whole- number exponents, in the conventional order when
there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6s2 to
find the volume and surface area of a cube with sides of length s = ½.
Explanation:
 Students evaluate algebraic expressions, using order of operations as needed.
 Given an expression such as 3x + 2y, find the value of the expression when x is equal to 4 and y is equal to 2.4. This
problem requires students to understand that multiplication is understood when numbers and variables are written together
and to use the order of operations to evaluate.
 Given a context and the formula arising from the context, students could write an expression and then evaluate for any
number.
Learning Targets:
 I can evaluate an algebraic expression for a given value.
 I can substitute values in formulas to solve real-world problems.
 I can apply the order of operations when evaluating arithmetic expressions.
 I can apply the order of operations when evaluating algebraic expressions.
Vocabulary:
sum, difference, term, product, factor, quotient, coefficient, arithmetic expression, algebraic expression, substitute, evaluate
Sample Problem(s):
Sample 1
Sample 2
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Sample 3
Sample 4
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Sample 5
Sample 6
Resources:
 Algebra Balance Scales : Negatives
http://tinyurl.com/NegativesScale
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

Students can use this applet understand how to balance the scales and equations using negative numbers.
IXL Game: Algebra: Evaluate expressions
http://tinyurl.com/Evaluateexpressions
This game is designed to help sixth graders understand how to evaluate expressions involving integers. This is just one
of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use.
Linear Function Machine
http://www.shodor.org/interactivate/activities/LinearFunctMachine/
By putting different values into the linear function machine students will explore simple linear functions.
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Major Cluster – Apply and extend previous understandings of arithmetic to algebraic expressions.
Standard: 6.EE.3
Apply the properties of operations to generate equivalent expressions.
For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the
distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations
to y + y + y to produce the equivalent expression 3y.
Explanation:
Students use the following properties of operations to write equivalent expressions:.
Learning Targets:
 I can apply the associative property of addition to generate equivalent expressions.
 I can apply the commutative property of addition to generate equivalent expressions
 I can apply the additive property of 0 to generate equivalent expressions
 I can apply the existence of additive inverses to generate equivalent expressions
 I can apply the associative property of multiplication to generate equivalent expressions
 I can apply the commutative property of multiplication to generate equivalent expressions
 I can apply the multiplicative identity property of 1 to generate equivalent expressions
 I can apply the existence of multiplicative inverses to generate equivalent expressions
 I can apply the distributive property of multiplication over addition to generate equivalent expressions.
Vocabulary:
equivalent expressions, distributive property, commutative property of addition, commutative property of multiplication,
associative property of addition, associative property of multiplication, additive identity property of 0, additive inverse,
multiplicative identity property of 1, multiplicative inverse
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Sample Problem(s):
Sample 1
Sample 2
Sample 3
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Sample 4
Resources:
 Algebra Four
http://www.shodor.org/interactivate/activities/AlgebraFour/
This lesson contains a game activity designed to help students practice solving algebraic equations.
64 | P a g e
Major Cluster – Apply and extend previous understandings of arithmetic to algebraic expressions.
Standard: 6.EE.4
Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is
substituted into them).
For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y
stands for.
Explanation:
 Students demonstrate an understanding of like terms as quantities being added or subtracted with the same variables and
exponents.
 Students connect their experiences with finding and identifying equivalent forms of whole numbers and can write
expressions in various forms.
 Students can prove that the expressions are equivalent by simplifying each expression into the same form.
Learning Targets:
 I can identify whether two expressions are equivalent.
 I can justify two expressions are equivalent.
Vocabulary:
equivalent expression
Sample Problem(s):
Sample 1
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Resources:
 What are Equivalent Expressions?
http://tinyurl.com/equivalentexpressions
A great tutorial step by step video to help explain equivalent expressions
 Equivalent Expression Calculator
http://tinyurl.com/equivalentexpressioncalculator
This is a calculator tool that finds equivalent expressions.
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Major Cluster – Reason about and solve one-variable equations and inequalities.
Standard: 6.EE.5
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any,
make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation
or inequality true.
Explanation:
 Beginning experiences in solving equations should require students to understand the meaning of the equation as well as
the question being asked.
 Solving equations using reasoning and prior knowledge should be required of students to allow them to develop effective
strategies such as using reasoning, fact families, and inverse operations.
 Students may use balance models in representing and solving equations and inequalities.
Learning Targets:
 I can explain that solving an equation or inequality leads to finding the value or values of the variable that will make a
true mathematical statement.
 I can substitute a given value into an algebraic equations or inequality to determine whether it is part of the solution set.
Vocabulary:
equation, inequality, substitute, solve, solution
Sample Problem(s):
Sample 1
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Sample 2
Sample 3
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Resources:
 Algebra Four
http://www.shodor.org/interactivate/activities/AlgebraFour/
This lesson contains a game activity designed to help students practice solving algebraic equations.
69 | P a g e
Major Cluster – Reason about and solve one-variable equations and inequalities.
Standard: 6.EE.6
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a
variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
Explanation:
 Students write expressions to represent various real-world situations.
 Given a contextual situation, students define variables and write an expression to represent the situation.
 Connecting writing expressions with story problems and/or drawing pictures will give students a context for this work. It
is important for students to read algebraic expressions in a manner that reinforces that the variable represents a number.
Learning Targets:
 I can use a variable to write an algebraic expression that represents a real-world situation when a specific number is
unknown.
 I can explain and give example of how a variable can represent a single unknown number or can represent any number in
a specified set.
 I can use a variable to write an expression that represents a consistent relationship in a particular pattern.
Vocabulary:
variable, constant, algebraic expression
Sample Problem(s):
Sample 1
Sample 2
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Sample 3
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Resources:
 What is a Variable?
http://tinyurl.com/whatisvariable
Understanding that a variable is just a symbol that can represent different values.
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Major Cluster – Reason about and solve one-variable equations and inequalities.
Standard: 6.EE.7
Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in
which p, q and x are all nonnegative rational numbers.
Explanation:
 Students write equations from real-world problems and then use inverse operations to solve one-step equations. Equations
may include fractions and decimals with non-negative solutions.
 Students create and solve equations that are based on real-world situations. It may be beneficial for students to draw
pictures that illustrate the equation in problem situations. Solving equations using reasoning and prior knowledge should
be required of students to allow them to develop effective strategies.
Learning Targets:
 I can write equations that represent real- world problems.
 I can solve real-world problems using equations in the form x + p = q where p and q are given numbers.
 I can solve real-world problems using equations in the form px = q where p and q are given numbers.
Vocabulary:
variable, constant, algebraic expression
Sample Problem(s):
Sample 1
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Sample 2
Resources:
 6th Grade Equations
http://www.youtube.com/playlist?list=PLnIkFmW0ticNQCGwCgisYNtfdFx6f9OsH
This will take you to a Learn Zillion lesson pertaining to this standard.
 Problem Solving Using Equations
http://tinyurl.com/oft5hoe
Practice real-world and quantitative problems by solving equations in the x + p = q form and the px = q form.
74 | P a g e
Major Cluster – Reason about and solve one-variable equations and inequalities.
Standard: 6.EE.8
Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem.
Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on
number line diagrams.
Explanation:
 Students write an inequality and represent solutions on a number line for various contextual situations.
Learning Targets:
 I can write a simple inequality to represent the constraints or condition of numerical values in a real-world or
mathematical problem
 I can explain what the solution set of an inequality represents.
 I can show the solution set of an inequality by graphing it on a number line.
Vocabulary:
inequality
Sample Problem(s):
Sample 1
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Sample 2
Sample 3
Resources:
 Math Genie
http://www.math-play.com/Inequality-Game.html
In this inequality game, Genie will be there to help you solve inequalities and word problems involving inequalities.
There are great hints if a student gets the answer wrong.
 Inequality Game
http://mrnussbaum.com/geniusboxing/
A fun inequality game for students.
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Major Cluster – Represent and analyze quantitative relationships between dependent and independent variables.
Standard: 6.EE.9
Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to
express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable.
Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the
equation.
For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the
equation d = 65t to represent the relationship between distance and time.
Explanation:
 Students recognize that the independent variable is graphed on the x-axis; the dependent variable is graphed on the y-axis.
 Students recognize that not all data should be graphed with a line. Data that is discrete would be graphed with coordinates
only.
 Students are expected to recognize and explain the impact on the dependent variable when the independent variable
changes (As the x variable increases, how does the y variable change?).
 Additionally, students should be able to write an equation from a word problem and understand how the coefficient of the
dependent variable is related to the graph and /or a table of values.
 Students can use many forms to represent relationships between quantities. Multiple representations include describing
the relationship using language, a table, an equation, or a graph. Translating between multiple representations helps
students understand that each form represents the same relationship and provides a different perspective on the function.
Learning Targets:
 I can create a table of two variables that represents a real-world situation in which one quantity will change in relation to
the other.
 I can explain the difference between the independent variable and the dependent variable.
 I can determine the independent and dependent variable in a relationship
 I can write an algebraic equation that represents the relationship between the two variables.
 I can create a graph by plotting the dependent variable on the x-axis and the independent variable on the y-axis of a
coordinate plane.
 I can analyze the relationship between the dependent and independent variables by comparing the table, graph, and
equation.
Vocabulary:
independent variables, dependent variables, coordinate plane
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Sample Problem(s):
Sample 1
Resources:
 Algebra Four
http://www.shodor.org/interactivate/activities/AlgebraFour/
This lesson contains a game activity designed to help students practice solving algebraic equations.
 Linear Function Machine
http://www.shodor.org/interactivate/activities/LinearFunctMachine/
By putting different values into the linear function machine students will explore simple linear functions.
 Proportional Reasoning
http://www.learner.org/courses/learningmath/algebra/session4/
In this lesson, students explore proportions by examining proportional relationships, absolute and relative comparisons,
and looking at graphs to learn about these relationships.
 Simple Plot
http://www.shodor.org/interactivate/activities/SimplePlot/
The applet in this lesson plan allows the student to plot ordered pairs and understand functions.
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Geometry 6.G
Enduring Understandings
Essential questions
Geometric Attributes (such as shapes, lines, angles, figures,
How does geometry better describe objects?
and planes) provide descriptive information about an object‘s
properties and position in space and support visualization and
problem solving
Supporting Cluster: Solve real-world and mathematical problems involving area, surface area, and volume.
Standard: 6.G.1
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing
into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
Explanation:
 Students continue to understand that area is the number of squares needed to cover a plane figure.
 Finding the area of triangles is introduced in relationship to the area of rectangles – a rectangle can be decomposed into
two congruent triangles. Therefore, the area of the triangle is ½ the area of the rectangle. The area of a rectangle can be
found by multiplying base x height; therefore, the area of the triangle is ½ bh or (b x h)/2.
 Students decompose shapes into rectangles and triangles to determine the area.
 Students should know the formulas for rectangles and triangles. ―Knowing the formula‖ does not mean memorization of
the formula. To ―know‖ means to have an understanding of why the formula works and how the formula relates to the
measure (area) and the figure. This understanding should be for all students.
Learning Targets:
 I can model how to find the area of a parallelogram by decomposing it and recomposing the parts to form a rectangle.
 I can model how to find the area of a right triangle by composing two right triangles into a rectangle.
 I can model how to find the area of a triangle by composing or decomposing two congruent triangles into a parallelogram.
 I can model how to find the area of a trapezoid by composing or decomposing two congruent trapezoids into a rectangle
and one or more triangles.
 I can model how to find the area of other polygons by decomposing them into simpler shapes such as triangles, rectangles,
and parallelograms and combing the areas of those simple shapes.
 I can explain the relationship between the formulas for the area of rectangle, parallelograms, triangles, and trapezoids.
 I can apply these techniques in the context of solving real-world and mathematical problems.
Vocabulary:
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polygon, right triangle, quadrilateral, parallelogram, area, square unit, right trapezoid, composing, decomposing, congruent
Sample Problem(s):
Sample 1
Sample 2
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Sample 3
Sample 4
Sample 5
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Sample 6
Resources:
 Area
http://www.shodor.org/interactivate/lessons/Area/
This lesson is designed to help students be able to calculate the area of a random shape on a grid. It also explains the
correlation between the size of the perimeter and the number of different possible areas that can be contained within that
perimeter.
 Area Explorations
http://www.shodor.org/interactivate/lessons/AreaExplorations/
In this lesson, students will explore the area of irregular shapes to find multiple different methods for calculating area
 Pentagon Puzzles
http://tinyurl.com/PentagonPuzzles
By deconstructing pentagons into triangles, students in this activity learn how to calculate the area of pentagons.
 Surface Area and Volume
http://www.shodor.org/interactivate/lessons/SurfaceAreaAndVolume/
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




An online activity is the focus of this lesson plan to help students understand the concepts of surface area and volume.
Surface Area of Prisms
http://www.shodor.org/interactivate/lessons/SurfaceAreaPrisms/
In this lesson, students will understand surface area and how solve for the surface area of triangular prisms.
Table for 22: A Real-World Geometry Project
https://www.teachingchannel.org/videos/real-world-geometry-lesson
This Teaching Channel video has students apply knowledge of area and perimeter to solve real-world problems. This
site provides a lesson plan and student handouts. (13 minutes)
Triangle Area
http://www.shodor.org/interactivate/lessons/TriangleArea/
This interactive lesson plan will help students understand how to find the area of a right triangle.
Triangle Explorer
http://www.shodor.org/interactivate/activities/TriangleExplorer
The applet in this lesson allows students to draw triangles and calculate their area.
What's Fun About Surface Area?
https://www.teachingchannel.org/videos/teaching-surface-area
In this Teaching Channel video, an educator helps students construct an understanding of surface area. (7 minutes)
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Supporting Cluster: Solve real-world and mathematical problems involving area, surface area, and volume.
Standard: 6.G.2
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction
edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the
formulas V = lwh and V = bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems.
Explanation:
 Previously students calculated the volume of right rectangular prisms (boxes) using whole number edges. The unit cube was 1
x 1 x 1. In 6th grade the unit cube will have fractional edge lengths. (i.e. ½ • ½ • ½ ) Students find the volume of the right
rectangular prism with these unit cubes.
 Students need multiple opportunities to measure volume by filling rectangular prisms with blocks and looking at the
relationship between the total volume and the area of the base. Through these experiences, students derive the volume formula
(volume equals the area of the base times the height).
 In addition to filling boxes, students can draw diagrams to represent fractional side lengths, connecting with multiplication of
fractions. This process is similar to composing and decomposing two dimensional shapes.
Learning Targets:
 I can find the volume of a right rectangular prism by reasoning about the number of unit cubes it takes to cover the first layer
of the prism and the number of layers needed to fill the entire prism.
 I can generalize finding the volume of a right rectangular prism to the equation V=lwh or V=Bh.
 I can solve real-world problems that involve finding the volume of right rectangular prisms.
Vocabulary:
right rectangular prism, base, height, area, volume, cubic unit
Sample Problem(s):
Sample 1
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Sample 2
Sample 3
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Resources:
 Boxed In and Wrapped Up
http://tinyurl.com/BoxedInWrappedUp
This lesson asks students to find the volume and surface area of a rectangular box and then convert it into a cubical box with
the same volume.
 Surface Area and Volume
http://www.shodor.org/interactivate/lessons/SurfaceAreaAndVolume
An online activity is the focus of this lesson plan to help students understand the concepts of surface area and volume.
 Surface Area of Rectangular Prisms
http://www.shodor.org/interactivate/lessons/SurfaceAreaRectangular/
This lesson is designed to help students understand the concept of surface area by specifically finding the surface area of a
rectangular prism.
 Volume of Prisms
http://www.shodor.org/interactivate/lessons/VolumePrisms/
This is a lesson designed to help students understand how to solve problems for the volume of triangular prisms.
 Volume of Rectangular Prisms
http://www.shodor.org/interactivate/lessons/VolumeRectangular/
This lesson is designed to help students understand how to solve for the volume of rectangular prisms.
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Supporting Cluster: Solve real-world and mathematical problems involving area, surface area, and volume.
Standard 6.G.3
Draw polygons in the coordinate plane given coordinates for the vertices; use to find the length of a side joining points with the same
first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical
problems.
Explanation:
 Students are given the coordinates of polygons to draw in the coordinate plane. If both x-coordinates are the same (2, -1) and
(2, 4), then students recognize that a vertical line has been created and the distance between these coordinates is the distance
between -1 and 4, or 5. If both the y-coordinates are the same (-5, 4) and (2, 4), then students recognize that a horizontal line
has been created and the distance between these coordinates is the distance between -5 and 2, or 7. Using this understanding,
student solve real-world and mathematical problems, including finding the area and perimeter of geometric figures drawn on a
coordinate plane.
Learning Targets:
 I can plot vertices in the coordinate plane to draw specific polygons.
 I can use the coordinates of the vertices of a polygon to find the length of a specific side.
 I can apply the techniques of plotting points, drawing figures, and finding lengths on the coordinate plane to solve real-world
problems.
Vocabulary:
vertex/vertices, coordinate, polygon
Sample Problem(s):
Sample 1
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Sample 2
Resources:
 Cartesian Coordinate System
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http://www.shodor.org/interactivate/lessons/CartesianCoordinate/
This lesson is designed to help students understand the Cartesian plane, specifically how to plot points, read coordinates and
find the ratio of the rise over run for slope.
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Supporting Cluster: Solve real-world and mathematical problems involving area, surface area, and volume.
Standard: 6.G.4
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of
these figures. Apply these techniques in the context of solving real-world and mathematical problems.
Explanation:
 Students represent three-dimensional figures whose nets are composed of rectangles and triangles.
 Students recognize that parallel lines on a net are congruent.
 Using the dimensions of the individual faces, students calculate the area of each rectangle and/or triangle and add these
sums together to find the surface area of the figure.
 Students construct models and nets of three dimensional figures, and describe them by the number of edges, vertices, and
faces. Solids include rectangular and triangular prisms. Students are expected to use the net to calculate the surface area.
 Students also describe the types of faces needed to create a three dimensional figure. Students make and test conjectures
by determining what is needed to create a specific three-dimensional figure.
Learning Targets:
 I can match a net to the correct three-dimensional figure.
 I can draw a net for a given three-dimensional figure.
 I can use a net to find the surface area of a given three-dimensional figure.
 I can solve real-world problems that involve finding the surface are of a three-dimensional figure.
Vocabulary:
rectangular prism, triangular prism, tetrahedron, net, surface area, rectangular pyramid
Sample Problem(s):
Sample 1
Sample 2
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Resources:
 Boxed In and Wrapped Up
http://tinyurl.com/BoxedInWrappedUp
This lesson asks students to find the volume and surface area of a rectangular box and then convert it into a cubical box
with the same volume.
 Surface Area and Volume
http://www.shodor.org/interactivate/lessons/SurfaceAreaAndVolume
An online activity is the focus of this lesson plan to help students understand the concepts of surface area and volume.
 What's Fun About Surface Area?
https://www.teachingchannel.org/videos/teaching-surface-area
In this Teaching Channel video an educator helps students construct an understanding of surface area. (7 minutes)
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Statistics and Probability-6.SP
Enduring Understandings
Essential Questions
The rules of probability can lead to more valid and reliable
How is probability used to make informed decisions about
predictions about the likelihood of an event occurring.
uncertain events?
Additional Cluster – Develop understanding of statistical variability.
Standard: 6.SP.1
Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the
answers.
For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question
because one anticipates variability in students’ ages.
Explanation:
 Students differentiate between statistical questions and those that are not.
Learning Targets:
 I can explain what makes a good statistical question.
 I can develop a question that can be used to collect statistical information.
Vocabulary:
variability
Sample Problem(s):
Sample 1
Agree & Disagree Statements
Teacher Notes
Using the set of questions below, you must decide if each question is a statistical question by circling your choice. After selecting
either agree or disagree, justify your thinking about why you agree or disagree.
1) How old am I?
Agree
Disagree
Justify your answer:
The data produced would not have variability therefore would not be a statistical question.
2) How old are the students in my school
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Agree
Disagree
Justify your answer:
The data produced would have variability. All students in a school would not be the same age. Therefore, the question is a
statistical question.
3) How old are the members of my family?
Agree
Disagree
Justify your answer:
The data produced would have variability. All family members are not the same age. Therefore, the question is a statistical
question.
Resources:
 Math Task: The Missing Words
http://www.uen.org/core/math/downloads/missing_words.pdf
Math Task Overview: Students should be able to explain a reasonable strategy for determining the number of missing
words. They should accurately compute the mean and range, and select an appropriate graph for displaying the data.
Students will explore the concepts of variability and distribution of a data set.
 Modeling Linear Relationships
http://www.learner.org/courses/learningmath/data/session7/part_c
Students compare arm span and height in people to learn about proportion. They also use a scatter plot graph to analyze
their data.
 What's My Ratio?
http://tinyurl.com/MyRatios
In this lesson, students investigate proportionality by using linear measurement. They also find the ratio between similar
figures.
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Additional Cluster – Develop understanding of statistical variability.
Standard: 6.SP.2
Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center,
spread, and overall shape.
Explanation:
 Students examine the distribution of a data set and discuss the center, spread and overall shape with dot plots, histograms
and box plots.
 Students know that distribution can be described using center (median or mean), and spread.
Learning Targets:
 I can explain that there are three ways that the distribution of a set of data can be described: by its center, spread, and
overall shape.
 I can describe the center of a set of statistical data.
 I can describe the spread of a set of statistical data.
 I can describe the overall shape of the set of data.
Vocabulary:
distribution, center, spread, mean, median
Sample Problem(s):
Sample 1
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Sample 2
1. What is the median number of children for the 16 households?
2. Explain how to find the median.
3. What does the median tell you?
4. Describe the distribution of data by its spread and overall shape.
Resources:
 Box Plots
http://www.shodor.org/interactivate/lessons/BoxPlots
By completing this lesson, students will understand the concept of median, quartiles, and how to build a box plot.
 Math Task: The Missing Words
http://www.uen.org/core/math/downloads/missing_words.pdf
Students should be able to explain a reasonable strategy for determining the number of missing words. They should
accurately compute the mean and range, and select an appropriate graph for displaying the data. Students will explore
the concepts of variability and distribution of a data set.
 Modeling Linear Relationships
http://www.learner.org/courses/learningmath/data/session7/part_c/
Students compare arm span and height in people to learn about proportion. They also use a scatter plot graph to analyze
their data.
 The Bell Curve
http://www.shodor.org/interactivate/lessons/TheBellCurve/
This lesson and activity introduces the student to the concept of the Bell Curve and distribution.
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Additional Cluster – Develop understanding of statistical variability.
Standard: 6.SP.3
Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of
variation describes how its values vary with a single number.
Explanation:
 Students understand that data sets contain many numerical values that can be summarized by one number such as a
measure of center. The measure of center gives a numerical value to represent the center of the data (i.e. midpoint of an
ordered list or the balancing point). Another characteristic of a data set is the variability (or spread) of the values.
Learning Targets:
 I can define a measure of center as a single value that summarizes a data set.
 I can define a measure of variation by how its values vary with a single number.
Vocabulary:
measure of center, measure of variation, mean, median
Sample Problem(s):
Sample 1
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Resources:
 Box Plots
http://www.shodor.org/interactivate/lessons/BoxPlots/
By completing this lesson, students will understand the concept of median, quartiles, and how to build a box plot.
 IXL Game: Statistics
http://tinyurl.com/IXLGameStatistics
This game is designed to help sixth graders understand how to calculate mean, median, mode, and range.
 Spinner
http://www.shodor.org/interactivate/activities/BasicSpinner/
By manipulating a spinner and its pointer students will learn about probability in this activity.
 What's My Ratio?
http://tinyurl.com/MyRatios
In this lesson, students investigate proportionality by using linear measurement. They also find the ratio between similar
figures.
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Additional Cluster – Summarize and describe distributions.
Standard: 6.SP.4
Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
Explanation:
 Students display data sets using number lines. Dot plots, histograms and box plots are three graphs to be used.
Learning Targets:
 I can organize and display data as a line plot or dot plot.
 I can organize and display data in a histogram.
 I can organize and display data in a box plot.
 I can determine the upper and lower extremes, median, and upper and lower quartiles of a set of data and use this information
to display the data in a box plot.
Vocabulary:
line plot, dot plot, histogram, median, lower extreme, lower quartile, upper quartile, upper extreme, box plot, outlier
Sample Problem(s):
Sample 1
Sample 2
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Sample 3
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Resources:
 Pie Chart
http://www.shodor.org/interactivate/activities/PieChart/
Students enter values in this applet and create pie charts in which they can vary the number or size of sections and display as
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



fractions or percentages.
Box Plots
http://www.shodor.org/interactivate/lessons/BoxPlots/
By completing this lesson, students will understand the concept of median, quartiles, and how to build a box plot.
Create a Graph
http://nces.ed.gov/nceskids/createagraph/default.aspx
This applet allows students to create bar, line, area, pie, and XY graphs.
Math Task: The Missing Words
http://www.uen.org/core/math/downloads/missing_words.pdf
Math Task Overview: Students should be able to explain a reasonable strategy for determining the number of missing words.
They should accurately compute the mean and range, and select an appropriate graph for displaying the data. Students will
explore the concepts of variability and distribution of a data set.
The Bell Curve
http://www.shodor.org/interactivate/lessons/TheBellCurve/
This lesson and activity introduces the student to the concept of the Bell Curve and distribution.
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Additional Cluster – Summarize and describe distribution.
Standard: 6.SP.5
Summarize numerical data sets in relation to their context, such as by:
a. Reporting the number of observations.
Explanation:
 Students record the number of observations.
 Using histograms, students determine the number of values between specified intervals.
 Given a box plot and the total number of data values, students identify the number of data points that are represented by
the box. Reporting of the number of observations must consider the attribute of the data sets, including units (when
applicable).
Learning Targets:
 I can report the number of observations in a data set or display.
Vocabulary:
observations, data
Sample Problem(s):
Sample 1
The bar chart represents the outcome of a penalty shoot-out competition. Each person in the competition was allowed six shots at
the goal. The graph shows, for example, that four people only scored one goal with their six shots.
(a) How many people were involved in the shoot-out?
(b) Find the values for the Mean and Median, and explain how you calculated each answer.
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Resources:
 Create a Graph
http://nces.ed.gov/nceskids/createagraph/default.aspx
This applet allows students to create bar, line, area, pie, and XY graphs.
 The Bell Curve
http://www.shodor.org/interactivate/lessons/TheBellCurve/
This lesson and activity introduces the student to the concept of the Bell Curve and distribution.
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Additional Cluster – Summarize and describe distribution.
Standard: 6.SP.5
Summarize numerical data sets in relation to their context, such as by:
b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
Explanation:
 Given a box plot and the total number of data values, students identify the number of data points that are represented by
the box. Reporting of the number of observations must consider the attribute of the data sets, including units (when
applicable). Consideration may need to be given to how the data was collected (i.e. Random sampling)
Learning Targets:
 I can describe how data was measured using the correct units of measurement.
 I can describe the attribute under investigation.
Vocabulary:
attribute
Sample Problem(s):
Sample 1
Lil Wayne keeps two pens of chickens in his back yard (for the eggs).
(a) How many chickens are in each pen?
(b) What units are used to measure the chickens' weights?
Resources:
 Create a Graph
http://nces.ed.gov/nceskids/createagraph/default.aspx
This applet allows students to create bar, line, area, pie, and XY graphs.
 The Bell Curve
http://www.shodor.org/interactivate/lessons/TheBellCurve/
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This lesson and activity introduces the student to the concept of the Bell Curve and distribution.
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Additional Cluster – Summarize and describe distribution.
Standard: 6.SP.5
Summarize numerical data sets in relation to their context, such as by:
c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute
deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the
context in which the data were gathered.
Explanation:
 Given a set of data values, students summarize the measure of center with the median or mean.
 Given a set of data values students can describe measures of variation using the interquartile range or the Mean Absolute
Deviation.
Learning Targets:
 I can determine the mean of the collected data.
 I can determine the median of the collected data.
 I can determine the measures of variance using range of collected data
 I can determine the measures of variance using interquartile range of collected data.
 I can determine the measures of variance using mean absolute deviation of collected data
 I can describe overall patterns in the data and how they relate to the context of the problem.
 I can describe any deviations for the overall pattern and how they relate to the context of the problem.
Vocabulary:
measure of center, mean, median, measure of variability, range, interquartile range, mean absolute deviation., outlier.
Sample Problem(s):
Sample 1
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1. What is the median of this set of data? What process did you use to find your solution?
2. What is the mean of this set of data? What process did you use to find your solution?
Solution
1. What is the median of this set of data?
I put the numbers in order from least to greatest(2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7) The number
in the middle is the median.
2. What is the mean of this set of data? Add the numbers together and divide by how many.
4.193548387
Resources:
 Create a Graph
http://nces.ed.gov/nceskids/createagraph/default.aspx
This applet allows students to create bar, line, area, pie, and XY graphs.
 The Bell Curve
http://www.shodor.org/interactivate/lessons/TheBellCurve/
This lesson and activity introduces the student to the concept of the Bell Curve and distribution.
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Additional Cluster – Summarize and describe distribution.
Standard: 6.SP.5
Summarize numerical data sets in relation to their context, such as by:
d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data
were gathered.
Explanation:
 Students understand how the measures of center and measures of variability are represented by the graphical display.
 Students describe the context of the data, using the shape of the data and are able to use this information to determine an
appropriate measure of center and measure of variability.
 Students summarize numerical data by providing background information about the attribute being measured, methods
and unit of measurement, the context of data collection activities, the number of observations, and summary statistics.
Summary statistics include quantitative measures of center, spread, and variability including extreme values (minimum
and maximum), mean, median, mode, range, quartiles, interquartile ranges, and mean absolute deviation.
Learning Targets:
 I can justify the use of a particular measure of center or measure of variability based on the shape of the data.
 I can use a measure of center to draw inferences about the shape of the data distribution.
 I can use a measure of variation to draw inferences about the shape of the data distribution.
Vocabulary:
measure of center, mean, median, mode, measure of variability, range, interquartile range, mean absolute deviation.
Sample Problem(s):
Sample 1
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Sample 2
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Sample 3
The following data represents the number of TV shows watched per week, by a group of ten students; 3, 1, 2, 6, 1, 1, 3, 9, 0, 1
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What measure of center best describe the data shown above? Explain your reasoning.
What measure of variability best describes the data shown above? Explain your reasoning.
Resources:
 Create a Graph
http://nces.ed.gov/nceskids/createagraph/default.aspx
This applet allows students to create bar, line, area, pie, and XY graphs.
 The Bell Curve
http://www.shodor.org/interactivate/lessons/TheBellCurve/
This lesson and activity introduces the student to the concept of the Bell Curve and distribution.
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Mathematics | Grade 7 – Critical Areas
In Grade 7, instructional time should focus on four critical areas: (1) developing understanding of and applying proportional
relationships; (2) developing understanding of operations with rational numbers and working with expressions and linear
equations; (3) solving problems involving scale drawings and informal geometric constructions, and working with two- and
three-dimensional shapes to solve problems involving area, surface area, and volume; and (4) drawing inferences about
populations based on samples.
(1) Students extend their understanding of ratios and develop understanding of proportionality to solve single- and multi-step
problems. Students use their understanding of ratios and proportionality to solve a wide variety of percent problems, including those
involving discounts, interest, taxes, tips, and percent increase or decrease. Students solve problems about scale drawings by relating
corresponding lengths between the objects or by using the fact that relationships of lengths within an object are preserved in similar
objects. Students graph proportional relationships and understand the unit rate informally as a measure of the steepness of the related
line, called the slope. They distinguish proportional relationships from other relationships.
(2) Students develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal
representation), and percents as different representations of rational numbers. Students extend addition, subtraction, multiplication,
and division to all rational numbers, maintaining the properties of operations and the relationships between addition and subtraction,
and multiplication and division. By applying these properties, and by viewing negative numbers in terms of everyday contexts (e.g.,
amounts owed or temperatures below zero), students explain and interpret the rules for adding, subtracting, multiplying, and dividing
with negative numbers. They use the arithmetic of rational numbers as they formulate expressions and equations in one
variable and use these equations to solve problems.
(3) Students continue their work with area from Grade 6, solving problems involving the area and circumference of a circle and
surface area of three dimensional objects. In preparation for work on congruence and similarity in Grade 8 they reason about
relationships among two-dimensional figures using scale drawings and informal geometric constructions, and they gain
familiarity with the relationships between angles formed by intersecting lines. Students work with three-dimensional figures, relating
them to two dimensional figures by examining cross-sections. They solve real-world and mathematical problems involving area,
surface area, and volume of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right
prisms.
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(4) Students build on their previous work with single data distributions to compare two data distributions and address questions about
differences between populations. They begin informal work with random sampling to generate data sets and learn about the
importance of representative samples for drawing inferences.
The Common Core State Standards
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Grade 7 Overview
Ratios and Proportional Relationships
• Analyze proportional relationships and use them to solve real-world and mathematical problems.
The Number System
• Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational
numbers.
Expressions and Equations
• Use properties of operations to generate equivalent expressions.
• Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
Geometry
• Draw, construct and describe geometrical figures and describe the relationships between them.
• Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
Statistics and Probability
• Use random sampling to draw inferences about a population.
• Draw informal comparative inferences about two populations.
• Investigate chance processes and develop, use, and evaluate probability models.
The Common Core State Standards
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Mathematical Content Standards
Grade 7
Ratios and Proportional Relationships 7.RP
Enduring Understandings
Essential Question
Ratios and proportional relationships are used to express how
How can ratios and proportional relationships be used to
quantities are related and how quantities change in relation to
determine unknown quantities?
each other.
Major Cluster: Analyze proportional relationships and use them to solve real-world and mathematical problems.
Standard 7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities
1
1
measured in like or different units. For example, if a person walks
mile in each hour, compute the unit rate as the complex
2
4
1 1
fraction ÷ miles per hour, equivalently 2 miles per hour.
2 4
Explanation:
 Students will be able to compute unit rates when a fraction is in the numerator and denominator.
 Students will convert units of measure to make comparisons.
Learning Targets:
 I can compute a unit rate by multiplying both quantities by the same factor.
 I can compute a unit rate by dividing both quantities by the same factor.
Vocabulary:
Ratio, Rate, Unit Rate
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Sample Problem(s):
Sample 1
1
1
degree each
hour, what is the increase in temperature expressed as a unit rate?
5
2
1
1
2
Solution: degree ÷ of an hour, equivalently of a degree per hour.
5
2
5
Sample 2
1
If Monica reads 7 pages in 9 minutes, what is her average reading rate in pages per minute, and in pages per hour?
2
1
15
Solution: 7 pages ÷ 9 minutes, equivalently
of a page per minute.
2
18
1
3
Solution: 7 pages ÷
of an hour, equivalently 50 pages per hour.
2
20
Sample 3
1
1
John mows of a lawn in 10 minutes. Marica mows
of a lawn in 6 minutes. A student claims that Marica is mowing faster
3
4
If the temperature is rising
because she only worked for 6 minutes, while John worked for 10. Is the student‘s reasoning correct? Why or why not?
Solution: Yes, because in 1 minute Marica mows
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1
3
of the lawn and John mows
in one minute.
25
100
Resources:
 Leaky Faucets
http://tinyurl.com/Leaky-Faucets
Three week unit on ratio and proportional understanding.
 Proportional Reasoning Performance Tasks
http://tinyurl.com/proportional-reasoning
This is a 3-4 week unit that focuses on identifying and using unit rates.
 Resources for Ratios and Proportional Relationships
http://tinyurl.com/c3x2fwf
Link provides additional links that include resources, lessons, and units
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Major Cluster: Analyze proportional relationships and use them to solve real-world and mathematical problems.
Standard 7.RP.2 Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a
coordinate plane and observing whether the graph is a straight line through the origin.
Explanation:
 Students will be able to determine if quantities are proportional using graphs or tables.
Learning Targets:
 I can represent proportional relationships between quantities.
 I can use tables or graphs to determine if ratios are proportionate.
Vocabulary:
proportional relationships, unit rate, equivalent ratios, origin
Sample Problem(s):
Sample 1
The table below gives the price for different numbers of books. Do the numbers in the table represent a proportional relationship?
Number of
Books
1
3
4
7
Price
$3
$9
$12
$18
Solution: No, if it was a proportional relationship 7 books would have a price of $21
Sample 2
Look at the graph below. Do the two quantities represent a proportional relationship?
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Solution: Yes, the two quantities represent a proportional relationship as depicted by the constant rate of change of the line in the
graph above.
Resources:
 Leaky Faucets
http://tinyurl.com/Leaky-Faucets
Three week unit on ratio and proportional understanding.
 Proportional Reasoning Performance Tasks
http://tinyurl.com/proportional-reasoning
This is a 3-4 week unit that focuses on identifying and using unit rates.
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Major Cluster: Analyze proportional relationships and use them to solve real-world and mathematical problems.
Standard 7.RP.2 Recognize and represent proportional relationships between quantities.
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of
proportional relationships.
Explanation:
 Students will be able to identify the rate of change in tables, graphs, equations, diagrams, and verbal descriptions of
proportional relationships.
Learning Targets:
 I can identify the constant of proportionality when numerical relationships are provided in a table.
 I can identify the constant of proportionality when numerical relationships are provided in a graph.
 I can identify the constant of proportionality when numerical relationships are provided in an equation.
 I can identify the constant of proportionality when numerical relationships are provided in a diagram.
 I can identify the constant of proportionality when numerical relationships are provided in a description.
Vocabulary:
proportional relationships, constant of proportionality, unit rate, equivalent ratios, origin
Sample Problem(s):
Sample 1:
Does the following table show a proportional relationship?
Year of Service
Income
1
$22,000
2
$44,000
4
$88,000
5
$110,000
Solution: Yes, for every 1 year of service, the income increases $22,000
Sample 2:
The following graph represents the cost of bananas at a store. What is the constant of proportionality?
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Solution: From the graph it can be determined that 4 pounds of bananas are $1.00; So, 1 pound of bananas is $0.25, which is the
constant proportionality for the graph.
Resources:
 Leaky Faucets
http://tinyurl.com/Leaky-Faucets
Three week unit on ratio and proportional understanding.
 Proportional Reasoning Performance Tasks
http://tinyurl.com/proportional-reasoning
This is a 3-4 week unit that focuses on identifying and using unit rates.
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Major Cluster: Analyze proportional relationships and use them to solve real-world and mathematical problems.
Standard 7.RP.2 Recognize and represent proportional relationships between quantities.
c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items
purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
Explanation:
 Students will be able to write equations that represent proportional relationships.
Learning Targets:
 I can write an equation to represent a proportional relationship.
Vocabulary:
proportional relationships, constant of proportionality, unit rate, equivalent
Sample Problem(s):
Sample 1
Gas is selling at the pump at $3.98 per gallon. Represent this as an equation.
Solution: $3.98g = c where g = gallons pumped and c = cost.
Resources:
 Leaky Faucets
http://tinyurl.com/Leaky-Faucets
Three week unit on ratio and proportional understanding.
 Proportional Reasoning Performance Tasks
http://tinyurl.com/proportional-reasoning
This is a 3-4 week unit that focuses on identifying and using unit rates.
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Major Cluster: Analyze proportional relationships and use them to solve real-world and mathematical problems.
Standard 7.RP.2 Recognize and represent proportional relationships between quantities.
d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to
the points (0, 0) and (1, r) where r is the unit rate.
Explanation:
 Students will be able to explain how each ordered pair along the graph of a proportional line relates to the items being
compared.
 Students will identify the origin as the initial starting point that lacks any value.
 Students will recognize the unit rate as the y-value when the x-coordinate equals 1.
Learning Targets:
 I can explain what the ordered pairs on a proportional graph mean.
 I can explain what the origin is on a proportional graph.
 I can determine the unit rate from a proportional graph.
Vocabulary:
Proportional relationships, constant of proportionality, unit rate, equivalent ratios, origin
Sample Problem(s):
Sample 1
The following graph represents the cost of bananas at a store. Describe what point (2,50) means. Describe what the point (0,0)
means in given situation. What is the unit rate?
Solution: Point (2,50) means that when you purchase 2 lbs of bananas, the cost if $0.50. Point (0,0) means that there is no cost if
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no bananas have been purchased. The unit rate is $0.25 per pound of bananas.
Resources:
 Leaky Faucets
http://tinyurl.com/Leaky-Faucets
Three week unit on ratio and proportional understanding.
 Proportional Reasoning Performance Tasks
http://tinyurl.com/proportional-reasoning
This is a 3-4 week unit that focuses on identifying and using unit rates.
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Major Cluster: Analyze proportional relationships and use them to solve real-world and mathematical problems.
Standard 7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax,
markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error
Explanation:
 Students will be able to solve multistep ratio and percent problems using proportional relationships.
Learning Targets:
 I can compute a multistep ratio problem using proportional relationships.
 I can compute a multistep percent problem using proportional relationships.
Vocabulary:
Percent, simple interest, percent increase, percent decrease, percent error, tax, markup, markdown, discount, gratuity,
commission, fees
Sample Problem(s):
Sample 1
Find the selling price of a $60 video game with a 28% markup and 6% tax
Solution: $81.40
Sample 2
You and four friends go to a restaurant for dinner. All together you ordered $96 worth of food. There is 7.5% tax on your meal.
You will include a 20% gratuity after tax. Your friend contributes a coupon that takes $10 off your meal (before tax and
gratuity). Find the final cost for this dinner. If they divide the total equally among them, how much does each of the friends pay?
Solution:
$96-10 = $86.
$86 * .075 = $6.45
$6.45 + 86 = $92.45
$92.45 * .20 = $18.49
$18.49 + $92.45 = $110.94
$110.94 / 5 is approximately $22.19 per person
Sample 3
In a sale, the store reduces all prices by 25% each week.
Does this mean that, after 4 weeks, everything in the store will cost $0? If not, why not?
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Solution: No because there is a new starting value at the beginning of each week after taking 25% off of that starting price.
Therefore, prices would get lower and lower but never reach zero.
Resources:
 Leaky Faucets
http://tinyurl.com/Leaky-Faucets
Three week unit on ratio and proportional understanding.
 Proportional Reasoning Performance Tasks
http://tinyurl.com/proportional-reasoning
This is a 3-4 week unit that focuses on identifying and using unit rates.
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The Number System 7.NS
Enduring Understandings:
Essential Question:
Rational numbers can be represented in multiple ways and are
In what ways can rational numbers be useful?
useful when examining situations involving numbers that are
not whole.
Major Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and
divide rational numbers.
Standard: 7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers;
represent addition and subtraction on a horizontal or vertical number line diagram.
a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its
two constituents are oppositely charged
Explanation:
 Students will use number lines to model addition and subtraction.
 Students understand that a number and its opposite equals zero.
Learning Targets:
 I can use a number line to model additive inverse.
 I can describe real-world situations where opposite quantities have a sum of zero.
Vocabulary:
positive, negative, opposite, additive inverse, zero pair, integer, rational number
Sample Problem(s):
Sample 1
Model with a number line -3 + 3.
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Solution:
+3
Resources:
 Rational Numbers
http://tinyurl.com/blpamyl
The unit provides students with the opportunity to deepen their understanding of rational numbers in various forms.
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Major Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide
rational numbers.
Standard: 7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers;
represent addition and subtraction on a horizontal or vertical number line diagram.
b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is
positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers
by describing real-world contexts.
Explanation:
 Students will be able to add integers using a number line.
 Students will be able to comprehend addition of integers in real-world contexts.
Learning Targets:
 I can use a number line to model addition of rational numbers.
 I can describe real world situations that apply to the sum of rational numbers.
Vocabulary:
positive, negative, opposite, additive inverse, absolute value, integer, rational number
Sample Problem(s):
Sample 1
Ashley and Payton were given the following problem to model on a number line:
Tony has -$18.75 in his bank account. If he deposits $19.00 into his account, what is his new account balance?
Ashley answered -$0.25
Peyton answered $0.25
Whose answer is correct? Why?
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Sample 2
Write a real world situation that describes $25 + $1.25 = $26.25
Solution
For Example, A sweater costs $25, there was $1.25 tax, my total was $26.25.
Resources:
 Rational Numbers
http://tinyurl.com/blpamyl
The unit provides students with the opportunity to deepen their understanding of rational numbers in various forms.
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Major Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and
divide rational numbers.
Standard: 7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers;
represent addition and subtraction on a horizontal or vertical number line diagram.
c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between
two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
Explanation:
 Students will rewrite subtraction problems as addition problems using the opposite of the second term.
 Students will be able to model subtraction of integers on the number line and recognize the difference between the two
numbers is the absolute value of their difference.
 Students will be able to comprehend the subtraction of integers in real-world contexts.
Learning Targets:
 I can rewrite a subtraction problem as an addition problem using the additive inverse.
 I can use a number line to model subtraction of rational numbers.
 I can apply the difference of rational numbers to real world contexts.
Vocabulary:
positive, negative, opposite, additive inverse, absolute value, integer, rational number
Sample Problem(s):
Sample 1
Use a number line to model the following problem.
If the temperature reading on a thermometer is 10 degrees, what will the new reading be if the temperature falls 3 degrees?
Resources:
 Rational Numbers
http://tinyurl.com/blpamyl
The unit provides students with the opportunity to deepen their understanding of rational numbers in various forms.
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Major Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide
rational numbers.
Standard: 7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers;
represent addition and subtraction on a horizontal or vertical number line diagram.
d. Apply properties of operations as strategies to add and subtract rational numbers.
Explanation:
 Students will be able to understand the integer algorithms and apply them to properties of operations.
Learning Targets:
 I can apply the Associative property to add rational numbers.
 I can apply the Commutative property to add rational numbers.
 I can apply the Additive identity property of 0 to add rational numbers.
 I can apply the Additive inverse to add rational numbers.
 I can apply the Associative property to subtract rational numbers.
 I can apply the Commutative property to subtract rational numbers.
 I can apply the Additive identity property of 0 to subtract rational numbers.
 I can apply the Additive inverse to subtract rational numbers.
Vocabulary:
positive, negative, opposite, additive inverse, absolute value, integer, rational number
Sample Problem(s):
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Sample 1
Which of the following are possible? If yes, give an example. If no, explain why not.
1. positive + positive = positive
2. positive + positive = negative
4. negative + positive = negative
5. positive - positive = positive
7. negative - positive = positive
8. negative - positive = negative
3. negative + positive = positive
6. positive - positive = negative
Resources:
 Rational Numbers
http://tinyurl.com/blpamyl
The unit provides students with the opportunity to deepen their understanding of rational numbers in various forms.
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Major Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and
divide rational numbers.
Standard: 7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and
divide rational numbers.
a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy
the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for
multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
Explanation:
 Students will apply rules for multiplying r, particularly involving distributing a negative number.
Learning Targets:
 I can use the Associative property to multiply integers.
 I can use the Commutative property to multiply integers.
 I can use the Multiplication identity property of 1 to multiply integers.
 I can use the Existence of multiplication inverses to multiply integers.
 I can use the Distributive property of multiplication over addition.
 I can compose real world problems that apply to multiplying integers.
Vocabulary:
integer
Sample Problem(s):
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Sample 1
iTunes sells 4 iPhone apps at the cost of $2 per app (4 x 2 = 8).
Sample 2
You spend $3 each on 4 bottles of Gatorade. (4 x -3 = -12)
Sample 3
Your brother owes $6 to each of 4 friends, (-6 x 4 = -24).
Sample 4
You tell 3 of your friends not to worry about paying you the $6 each that they owe you. (-3 x -6 = 18).
Sample 5
Distribute the red/yellow counter chips (Algebra Tiles or Blocks can also be used) to the students. Using the counter ships, ask
them to model: -3(4) = -12. Also, prompt them to think of -3(4) = 12 as ―4 groups of -3.‖
Sample 6
Have the students determine the fact family for the multiplication sentence -3(4) = -12.
-3(4) = -12
-12  -3 = 4
4(-3) = -12
-12  4 = -3
Resources:
 Rational Numbers
http://tinyurl.com/blpamyl


The unit provides students with the opportunity to deepen their understanding of rational numbers in various forms.
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Major Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and
divide rational numbers.
Standard: 7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and
divide rational numbers.
b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero
divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by
describing real-world contexts.
Explanation:
 Students will be able to apply the rules for dividing integers.
 -(p/q) = (-p)/q = p/(-q) for example –(3/4) = (-3)/4 = 3/(-4)
 Students will understand that you can not divide by zero.
Learning Targets:
 I can model/explain that a fraction is a division problem.
 I understand that if p and q are integers, then –(p/q) = (–p)/q = p/(–q).
 I can interpret quotients of rational numbers by describing real world context that apply to dividing integers.
Vocabulary:
integer, quotient, rational numbers
Sample Problem(s):
Sample 1
1. Present the students with the following problem:
You decide to open a savings account with the Columbia Bank. For this particular account, you must maintain an *average
monthly balance of $10, or they will charge you a $15 penalty fee that is deducted from your account. However, if you do
maintain a balance of $30, then you will be rewarded with a deposit in your account of $5. A “deposit” is the same as a
positive number and a “withdrawal” is the same as a negative number. Your first monthly bank statement is as follows:
*“Average” may be a skill that needs to be reviewed with the students.
Date
May 5
May 6
May 8
May 11
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Deposit
Withdrawal
$10
$15
$3
$5
Balance
May 15
May 19
May 21
May 22
May 24
May 27
May 28
May 30
$6
$8
$12
$10
$5
$7
$10
$25
Procedures:
A. Complete the balance column, based on the deposits and withdrawals.
B. Calculate the approximate average balance for the month of May in order to determine if you will be charged a penalty
fee. Show how you calculated your work.
C. Based on whether you received a $15 penalty fee or a $5 reward in your account, calculate your new
average monthly balance.
Answers – Monthly balance = -$24  12 deposits/withdrawals = average monthly balance of -$2.00. Based on this, you will
be charged a penalty fee of $15.00. Therefore, the monthly balance of -$24 plus -$15 = new monthly balance of -$39.00.
Based on this, your new average monthly balance will be -$3.00.

Resources:
 Rational Numbers
http://tinyurl.com/blpamyl
The unit provides students with the opportunity to deepen their understanding of rational numbers in various forms.
137 | P a g e
Major Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and
divide rational numbers.
Standard: 7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and
divide rational numbers.
c. Apply properties of operations as strategies to multiply and divide rational numbers.
Explanation:
Students will be able to determine and apply the appropriate properties of operations to multiply and divide rational numbers.
Learning Targets:
 I can apply a property of operation to multiply rational numbers.
 I can apply a property of operation to divide rational numbers.
Vocabulary:
integer, rational number, terminating decimal, repeating decimal
Sample Problem(s):
Solution: A
Resources:
 Rational Numbers
http://tinyurl.com/blpamyl
The unit provides students with the opportunity to deepen their understanding of rational numbers in various forms.
138 | P a g e
Major Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and
divide rational numbers.
Standard: 7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and
divide rational numbers.
d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s
or eventually repeats
Explanation:
 Students will understand that a fraction represents division and apply that understanding to convert a fraction to a decimal
using long division.
 Students will understand that if a number is considered rational, it will eventually terminate or repeat.
Learning Targets:
 I can convert a fraction to a decimal by dividing the numerator by the denominator.
 I know that the decimal of the rational number either terminates or repeats.
Vocabulary:
integer, rational number, terminating decimal, repeating decimal, numerator, denominator
Sample Problem(s):
Sample 1
15
Convert the following fractions into a decimal using long division.
9
Solution: 1.66
Resources:

Rational Numbers
http://tinyurl.com/blpamyl
The unit provides students with the opportunity to deepen their understanding of rational numbers in various forms.
139 | P a g e
Major Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and
divide rational numbers.
Standard: 7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers.
(Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
Explanation:
 Students solve problems with all rational numbers including complex fractions (fractions in the numerator and/or in the
denominator).
Learning Targets:
 I can solve real world problems using the four operations with rational numbers.
Vocabulary:
rational number, complex fraction
Sample Problem(s):
Sample 1
Jessica and Eric are helping their teacher buy supplies for a project. Every student will get a bag with 2 pencils and 30 index
cards. The teacher gave Jessica $17 to buy pencils from the school store. The pencils come in boxes of 12 and cost $1.69 per
box. Eric was given $19 to buy index cards at an office supply store. Index cards are sold in packs of 150 and cost $2.99 per
pack. Jane buys as many boxes of pencils as she can afford. Eric buys as many packages of index cards as he can afford. How
many complete bags of supplies can they make?
A) fewer than 10
B) between 10 and 24
C) between 25 and 40
D) more than 40
Solution: C
Resources:
 Rational Numbers
http://tinyurl.com/blpamyl
The unit provides students with the opportunity to deepen their understanding of rational numbers in various forms.
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Expressions and Equations 7.EE
Essential Question: How can algebraic expressions and
Enduring Understandings:
Algebraic expressions and equations are used to model real-life
equations be used to model, analyze, and solve mathematical
problems and represent quantitative relationships, so that the
situations?
numbers and symbols can be mindfully manipulated to reach a
solution or make sense of the quantitative relationships.
Major Cluster: Use properties of operations to generate equivalent expressions.
Standard 7.EE.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational
coefficients.
Explanation:
 Students apply concepts from properties of operations to combine like terms.
 Students apply properties of operations and work with rational numbers (integers and positive / negative fractions and
decimals) to write equivalent expressions.
Learning Targets:
 I can apply the commutative property to add linear expressions with rational coefficients.
 I can apply the associative property to add linear expressions with rational coefficients.
 I can apply the distributive property to add and/or subtract linear expressions with rational coefficients.
 I can apply the distributive property to factor a linear expression with rational coefficients.
 I can apply the distributive property to expand a linear expression with rational coefficients.
Vocabulary:
commutative property, associative property, distributive property, linear expression, coefficient, like terms, equivalent expressions,
expanded expression, rational numbers
Sample Problem(s):
Sample 1
Write an equivalent expression for 7(x+5) – 4
Sample 2
Sharon thinks the two expressions 2(5a – 2) + 4a and 14a – 2 is equivalent? Is she correct?
Explain why or why not?
Sample 3
Write equivalent expressions for: 9b + 20
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Resources:
 MSED
http://mdk12.org/instruction/academies/resources_2012/Mathematics/pdf/Math_Academy_Presentations/M.S._EEA_to_mdk1
2.org/Day_1_MS_Afternoon_Lesson_Seeds/7_Lesson_Seed.docx
This lesson allows students to expand and factor expressions (distributive property)
 HCPSS Secondary Mathematics Office
https://grade7commoncoremath.wikispaces.hcpss.org/file/view/7.EE.1%20Lesson%20Distributive%20Property.doc/34984323
8/7.EE.1%20Lesson%20Distributive%20Property.doc
This lesson allows students to explain how to find the product of expressions.
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Major Cluster: Use properties of operations to generate equivalent expressions.
Standard 7.EE.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem
and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply
by 1.05.”
Explanation:
 Students understand that an expression can be represented in multiple ways.
Learning Targets:
 I can use equivalent expressions to represent the same quantity.
 I can demonstrate the relationship between equivalent expressions and the quantity they represent.
Vocabulary
linear expression, equivalent expressions
Sample Problem(s):
Patricia, Hugo and Sun work at a music store. Each week, Patricia works three more than twice the number of hours that Hugo
works. Sun works 2 less than Hugo.
(a) Let x represent the number of hours that Hugo works each week. The number of hours that Hugo, Patricia, and Sun work can
be modeled is shown below. Write an expression that represents each person's number of hours.
Hugo ___________
Patricia ___________
Sun ___________
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(b) Model the total number of hours that Patricia and Sun work together. Draw the result below. Then write an expression for the
drawing.
(c) Like tiles are tiles that have the same shape. Using your model, group like tiles together and remove the zero pairs. Draw the
result below. Then write an expression for your drawing.
Patricia, Hugo and Sun work at a music store. Each week, Patricia works three more than twice the number of hours that Hugo
works. Sun works 2 less than Hugo.
(a) Let x represent the number of hours that Hugo works each week. The number of hours that Hugo, Patricia, and Sun work can
be modeled is shown below. Write an expression that represents each person's number of hours.
Hugo ___________
Patricia ___________
Sun ___________
(b) Model the total number of hours that Patricia and Sun work together. Draw the result below. Then write an expression for the
drawing.
(c) Like tiles are tiles that have the same shape. Using your model, group like tiles together and remove the zero pairs. Draw the
result below. Then write an expression for your drawing.
Sample 2:
Which expression is not equivalent to the other three? Justify your reasoning.
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5n + 10n -5
5n (1 + 2) – 5
15 – 5
5(n-1)
—Utah Common Core Academy
Resources:
 http://www.ode.state.or.us/wma/teachlearn/commoncore/mat.07.pt.4.sfund.a.401_v1.pdf
The student will use the content for the domains of expressions and equations and ratio and proportional relationships to
explore the profit for three different fundraising plans. The student will use the content for the domain of statistics and
probability and expressions and equations to analyze the validity of claims about the fundraising project.
 http://eucc2011.wikispaces.com/file/view/Expressions+Learning+Cycle+7EE2+(1).docx
Students will be able to recognize component parts of an expression and identify equivalency.
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Major Cluster: Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
Standard 7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any
form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers
in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and
estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her
salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door
that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on
the exact computation.
Explanation:
 Students will be able to solve contextual problems using rational numbers (positive and negative) in a variety of forms.
 Students convert between fractions, decimals, and percents as needed to solve the problem.
 Students will use estimation to justify the reasonableness of their answers.
Learning Targets:
 I can solve multi-step real world problems using positive and negative rational numbers (fractions, decimals, and
percents).
 I can solve multi-step real world problems by applying the properties of operations.
 I can use mental computation and estimation strategies to determine if my answer makes sense.
Vocabulary:
equivalent expressions, rational numbers
Sample Problem(s):
Sample 1
A framed picture 24 inches wide and 28 inches high will be hung on a wall. The picture will be hung where the distance from the
floor to ceiling is 8 feet. The center of the picture must be 5 ¼ feet from the floor. Determine the distance from the ceiling to the
top of the picture frame.
Solution: 1ft 7in
Sample 2
The Glee Club is going on a trip to Navy Pier. The trip costs $56. Included in that price is $10 for a boat ride and the cost of 2
passes to ride the Ferris Wheel. Each of the passes cost the same price. Write an equation representing the cost of the trip and
determine the price of one pass.
2x + 10=56 ; one ticket cost $23.00
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Resources:

http://www.yummymath.com/wp-content/uploads/black-friday.pdf
Students will be able to add some understanding and analysis of their Black Friday shopping trip. They will calculate
savings in dollars and percents with this investigation.
 http://www.learner.org/courses/learningmath/number/session4/part_c/index.html
Students will use manipulative to model addition, subtraction, multiplication, and division with positive and negative
integers
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Major Cluster: Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
Standard: 7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations
and inequalities to solve problems by reasoning about the quantities.
a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational
numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the
sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is
its width?
Explanation:
 Students will be able to create equations to model word problems.
 Students will be able to solve linear equations that model word problems.
 Students will connect arithmetic solution processes to algebraic solution processes.
Learning Targets:
 I can analyze a word problem to determine the quantities and operations necessary to write an equation.
 I can define a variable to represent an unknown quantity given a real-world problem.
 I can write an algebraic equation by defining a variable and using the values given to represent a word problem.
 I can compare an arithmetic solution to an algebraic solution.
 I can solve equations in the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers.
Vocabulary:
rational number,
Sample Problem(s):
Sample 1
Larry cleans 3 offices per night for 5 nights. It takes 25 minutes per office. He is paid $750. How much does he make per hour?
Solution: $120.00
Sample 2
Mr. Williamson is paid a weekly salary of $325 plus 7% of his weekly sales. At the end of one week, he earned $500. How much
were his sales for the week?
Solution: $2,500
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Sample 3
The sum of three consecutive even numbers is 36. What is the smallest of these values?
Solution: 11
Resources:
 http://www.yummymath.com/wp-content/uploads/Remembering_Robinson.pdf
In this activity, students will consider how many hits and homeruns Jackie might have had if he had begun his MLB career
at the age of 20.
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Major Cluster: Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
Standard: 7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations
and inequalities to solve problems by reasoning about the quantities.
b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational
numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson,
you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number
of sales you need to make, and describe the solutions.
Explanation:
 Students will be able to create inequalities to model word problems.
 Students will be able to solve linear inequalities that model word problems.
 Students will be able to graph the solution of the inequality.
 Students will connect arithmetic solution processes to algebraic solution processes.
Learning Targets:
 I can analyze a word problem to determine the quantities and operations necessary to write an inequality.
 I can write an algebraic inequality by defining a variable and using the values given to represent a real-world problem.
 I can solve a simple inequality and graph the solution on a number line.
 I can describe the solution to an inequality in relation to the problem.
Vocabulary:
Inequality
Sample Problem(s):
Sample 1
Solve the following inequality.
3
- y – 10 > 2
4
Solution: y < -16
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Sample 2
Elizabeth purchased an iphone this morning. She also purchased an automatic application update service plan. This service is
$14.00 annually. There is a $3.50 charge for each application update service session. Elizabeth budgeted $120 for the first year of
having this service plan. What is the maximum number of application updates Elizabeth can purchase in her first year of having
this plan?
Solution: u < 30
Sample 3
Sarah's monthly earnings include a fixed salary of $4,000 and 7% commission on all her monthly sales. To cover her vacation
expenses, Sarah needs to earn an income of at least $9,600 this month. Write an inequality that represents the amount of sales
Sarah needs to cover her vacation expenses. Solve the inequality and then graph its solution on a number line.
Solution: s > $ 80,000
Resources:
 Fantasy Football
http://www.yummymath.com/2011/fantasy-football/
Students first calculate football points given touchdowns, yardage gains and fantasy pass interceptions. They are then
challenged to create and use equations to explain their method of calculation. Finally, students solve equations as they try
to find the number of passes, touchdowns, or interceptions that yield given point totals.
 Toy Trains
http://insidemathematics.org/common-core-math-tasks/7th-grade/7-2009%20Toy%20Trains.pdf
This problem gives students a chance to find and use a number pattern and find an algebraic expression for the number
pattern using the concept of trains.
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Geometry 7.G
Enduring Understandings:
Essential Question:
Geometric attributes (such as shapes, lines, angles, figures, and How does our understanding of geometry help us to describe
planes) provide descriptive information about an object‘s
real-world objects?
properties and position in space and support visualization and
problem solving.
Additional Cluster: Draw, construct, and describe geometrical figures and describe the relationships between them.
Standard: 7.G.1- Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas
from a scale drawing and reproducing a scale drawing at a different scale.
Explanation:
 Students determine the dimensions of figures when given a scale and identify the affect of a scale on an actual length and
area.
 Students identify the scale factor given two figures.
 Using a given scale drawing, students reproduce the drawing at a different scale.
 Students understand the lengths will change by a factor equal to the product of the magnitude of the two size
transformations.
Learning Targets:
 I can use a scale drawing to determine the actual dimensions of a geometric figure.
 I can use a scale drawing to determine the actual area of a geometric figure.
 I can change scales on similar drawings.
Vocabulary:
Scale factor, scale drawing, similar, geometric figures
Sample Problem(s):
Sample1
Julie shows the scale drawing of her room below. If each 2 cm on the scale drawing equals 5 ft, what are the actual dimensions of
Julie‘s room? Reproduce the drawing at 3 times its current size.
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Solution:
5.6 cm = 4 ft
1.2cm= 3ft 2.8 cm = 7 ft
4.4 cm = 11 ft
4 cm =10 ft
Resources:
 Glowing Rectangles
http://www.yummymath.com/2011/glowing-rectangles
Kids spend hours each day using television, computers, mobile devices and so on. Use that interest to motivate your
student‘s analysis of screen ratios.
 Malawian House
http://www.wfu.edu/education/ncctm08/houses.pdf
Students will use the concepts of similarity, scale factors, and conversion factors to make scale drawings of different
Malawian houses
 Linking Length, Perimeter, Area and Volume
http://illuminations.nctm.org/LessonDetail.aspx?id=U98
 Learning about Length, Perimeter, Area, and Volume of Similar Objects Using Interactive Figures
http://www.nctm.org/standards/content.aspx?id=26884
This two-part example illustrates how students can learn about the length, perimeter, area, and volume of similar objects
using dynamic figures.
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Additional Cluster: Draw, construct, and describe geometrical figures and describe the relationships between them.
Standard: 7.G.2- Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus
on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more
than one triangle, or no triangle.
Explanation:
 Students will be able to draw geometric shapes freehand and using appropriate tools.
 Students understand the characteristics of angles that create triangles (for example, can a triangle have more than one
obtuse angle? Will three sides of any length create a triangle?)
Learning Targets:
 I can draw a geometric shape with specific conditions.
 I can construct a triangle when given three side lengths.
 I can construct a triangle when given three angle measurements.
 I can construct a triangle when given a combination of side and angle measurements.
 I can determine the conditions that will result in unique triangles, multiple triangles, or no triangle.
Vocabulary:
obtuse angle, acute angle, right angle, acute triangle, obtuse triangle, and right triangle.
Sample Problem (s):
Sample1
Draw a quadrilateral with one set of parallel sides and no right angles.
Students understand the characteristics of angles and side lengths that create a unique triangle, more than one triangle or no
triangle.
Sample 2
Can a triangle have more than one obtuse angle? Explain your reasoning.
Sample 3
Will three sides of any length create a triangle? Explain how you know which will work.
Possibilities to examine are:
a. 13 cm, 5 cm, and 6 cm
b. 3 cm, 3cm, and 3 cm
c. 2 cm, 7 cm, 6 cm
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Solution:
―A‖ above will not work; ―B‖ and ―C‖ will work. Students recognize that the sum of the two smaller sides must be larger than the
third side.
Sample 4
Is it possible to draw a triangle with a 90° angle and one leg that is 4 inches long and one leg that is 3 inches long?
If so, draw one. Is there more than one such triangle?
(NOTE: Pythagorean Theorem is NOT expected – this is an exploration activity only)
Sample 5
Draw a triangle with angles that are 60 degrees. Is this a unique triangle? Why or why not?
Sample 6
Draw an isosceles triangle with only one 80 degree angle. Is this the only possibility or can another triangle be drawn
that will meet these conditions?
Through exploration, students recognize that the sum of the angles of any triangle will be 180⁰.
Resources:
 Making Triangles (applet)
http://www.nctm.org/standards/content.aspx?id=25007
Focuses attention on the concept of triangle, helping students understand the mathematical meaning of a triangle and the
idea of congruence, or sameness, in geometry.
 Triangle Explorer
http://shodor.org/interactivate/activities/TriangleExplorer/
Students learn about areas of triangles and about the Cartesian coordinate system through experimenting with triangles
drawn on a grid.
 Triangle Calculator
http://ostermiller.org/calc/triangle.html
Enter values three of the six sides and angles of the triangle and the other three values will be computed. The number of
significant values entered will determine the number of significant figures in the results.
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Additional Cluster: Draw construct, and describe geometrical figures and describe the relationships between them.
Standard: 7.G.3- Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of
right rectangular prisms and right rectangular pyramids.
Explanation:
 Students need to describe the resulting face shape from cuts made parallel and perpendicular to the bases of right
rectangular prisms and pyramids. Cuts made parallel will take the shape of the base; cuts made perpendicular will take the
shape of the lateral (side) face. Cuts made at an angle through the right rectangular prism will produce a parallelogram.
Learning Targets:
 I can describe and model the different ways to slice a 3-dimensional figure (vertical, horizontal, angle).
 I can describe the different 2-dimensional cross sections that will result depending on how you slice the 3-dimensional
figures.
Vocabulary:
polygons, right rectangular prism, right rectangular pyramid, cross section
Sample Problem(s):
Sample 1
Students need to describe the resulting face shape from cuts made parallel and perpendicular to the bases of right rectangular
prisms and pyramids. Cuts made parallel will take the shape of the base; cuts made perpendicular will take the shape of the lateral
(side) face. Cuts made at an angle through the right rectangular prism will produce a parallelogram;
If the pyramid is cut with a plane (green) parallel to the base, the intersection of the pyramid and the plane is a square cross
section (red).
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If the pyramid is cut with a plane (green) passing through the top vertex and perpendicular to the base, the intersection of the
pyramid and the plane is a triangular cross section (red).
If the pyramid is cut with a plane (green) perpendicular to the base, but not through the top vertex, the intersection of the pyramid
and the plane is a trapezoidal cross section (red).
Resources:
 Plasticine Geometry
http://www.cemc.uwaterloo.ca/resources/emmyfiles/2009-10/by_strand/EN_GE0910C3_6.pdf
This activity becomes extremely involved but the activity of trying to find all possible cross-sections is a nice challenge.
Not all students will be able to find them all, but the process of trying is good mathematical exercise, bearing not only the
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specific content standards but also on the mathematical practice of perseverance (MP1).
 Learn Zillion Lesson
http://learnzillion.com/lessonsets/200-describe-the-twodimensional-figures-that-result-from-slicing-threedimensionalfigures
In this lesson you will learn how to describe the cross sections of a right rectangular prism by slicing at different angles.
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Additional Cluster: Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
Standard: 7.G.4- Know the formulas for the area and circumference of circle and use them to solve problems; give an informal
derivation of the relationship between the circumference and the area of a circle.
Explanation:
 ―Know the formula‖ does not mean memorization of the formula. To ―know‖ means to have an understanding of why the
formula works and how the formula relates to the measure (area and circumference) and the figure. This understanding
should be for all students.
 Students understand the relationship between radius and diameter and their relationships to the center of a circle. Students
also understand the ratio of circumference to diameter can be expressed as Pi. Building on these understandings, students
generate the formulas for circumference and area.
Learning Targets:
 I can identify the parts of a circle.
 I know that circumference of a circle is the distance around the circle.
 I can explain/show how the formulas for area and circumference of circles are derived.
 I can state the formula for finding the area of a circle.
 I can state the formula for finding the circumference of a circle.
 I can use formulas to find the area and circumference of a circle.
 I can determine the diameter and radius of a circle when the circumference is given.
 I can use a ratio and algebraic reasoning to compare the area and circumference of a circle.
Vocabulary:
radius, diameter, center, area, pi, circumference
Sample Problem(s):
Sample 1
The seventh grade class is building a mini-golf game for the school carnival. The end of the putting green will be a circle. If the
circle is 10 feet in diameter, how many square feet of grass carpet will they need to buy to cover the circle? How might someone
communicate this information to the salesperson to make sure he receives a piece of carpet that is the correct size? Use 3.14 for
pi.
Solution: Area = Πr2
Area = 3.14 (5)2
Area = 78.5 ft2
To communicate this information, ask for a 9 ft by 9 ft square of carpet.
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Sample 2
The center of the circle is at (5, -5). What is the area of the circle?
Solution:
The radius of the circle is 4. Using the formula, Area = Πr2, the area of the circle is approximately 50.24 units2. Students build
on their understanding of area from 6th grade to find the area of left-over materials when circles are cut from squares and
triangles or when squares and triangles are cut from circles.
Sample 3
What is the perimeter of the inside of the track.
Solution:
The ends of the track are two semicircles, which would form one circle with a diameter of 62m. The circumference of this part
would be 194.68 m. Add this to the two lengths of the rectangle and the perimeter is 2194.68 m
Resources:
 Which is Bigger
http://insidemathematics.org/common-core-math-tasks/7th-grade/7-2004%20Which%20is%20Bigger.pdf
Performance Task in which students must be able to analyze characteristics and properties of three-dimensional geometric
shapes and apply the appropriate techniques, tools, and formulas to determine measurements, such as circumference and
height.
 Illuminations Circle Tool
http://illuminations.nctm.org/ActivityDetail.aspx?ID=116
How do the area and circumference of a circle compare to its radius and diameter? This activity allows you to investigate
these relationships in the Intro and Investigation sections and then hone your skills in the Problems section.
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Additional Cluster: Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
Standard: 7.G.5- Use the facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to
write and solve simple equations for an unknown angle in a figure.
Explanation:
 Students will be able to define and understand properties of supplementary, complementary, vertical and adjacent angles.
 Students will be able to use properties of supplementary, complementary, vertical and adjacent angles to solve for
unknown angles in a figure.
 Students will be able to write and solve equations based on a diagram of intersecting lines with some known angle
measures.
Learning Targets:
 I can state the relationship between supplementary, complementary and vertical angles.
 I can use angle relationships to write algebraic equations for unknown angles.
 I can use algebraic reasoning and angle relationships to solve multi-step problems.
Vocabulary:
supplementary angles, complementary angles, vertical angles, adjacent angles
Sample Problem(s):
Sample 1
Write and solve an equation to find the measure of angle x.
Solution:
Find the measure of the missing angle inside the triangle (180 – 90 – 40), or 50°.
The measure of angle x is supplementary to 50°, so subtract 50 from 180 to get a measure of 130° for x.
Sample 2
Find the measure of angle x.
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Solution:
First, find the missing angle measure of the bottom triangle (180 – 30 – 30 = 120). Since the 120 is a vertical angle
to x, the measure of x is also 120°.
Resources:
 What is Angles?
http://www.shodor.org/interactivate/activities/Angles/
This activity allows the user to practice important angle vocabulary.
 XP Math
http://www.xpmath.com/forums/arcade.php?do=play&gameid=103
Game of complementary and supplementary angle pairs practice.
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Additional Cluster: Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
Standard 7.G.6 Solve real-world and mathematical problems involving area, volume, and surface area of two-dimensional and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
Explanation:
 Students continue work from 5th and 6th grade to work with area, volume and surface area of two- dimensional and threedimensional objects. (composite shapes)
 Surface area formulas are not the expectation with this standard. Building on work with nets in the 6th grade, students should
recognize that finding the area of each face of a three-dimensional figure and adding the areas will give the surface area.
 Students understanding of volume can be supported by focusing on the area of base times the height to calculate volume.
 Students understanding of surface area can be supported by focusing on the sum of the area of the faces. Nets can be used to
evaluate surface area calculations.
Learning Targets:
 I can determine the area of two-dimensional figures including those found in real-world contexts.
 I can determine the surface area of three-dimensional figures found in real-world contexts.
 I can determine the volume of three-dimensional figures found in real-world contexts.
Vocabulary:
length, width, base, height, altitude, surface area, and volume
Sample Problem(s):
Sample 1
The surface area of a cube is 96 in2. What is the volume of the cube?
Solution:
The area of each face of the cube is equal. Dividing 96 by 6 gives an area of 16 in2 for each face. Because each
face is a square, the length of the edge would be 4 in. The volume could then be found by multiplying 4 x 4 x 4 or
64 in3.
Resources:
 LFS Digital Unit – Three Dimensional Objects
http://agi.seaford.k12.de.us/sites/LFSdigital/units/Math%20Unit%20Topics/Three%20dimensional%20object.aspx
Unit focusing on the concept that the volume and surface area of prisms and cylinders can be calculated and applied to real
world problems. This unit contains a number of links to Discovery Education resources.
 Measuring Henry‘s Cabin
http://www.cyberbee.com/henrybuilds/extensions.html
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
In this lesson and activity students will determine the surface area and volume of a house and also reconstruct it on a smaller
scale.
Boxed In and Wrapped Up
http://www.teachengineering.org/view_lesson.php?url=http%3A%2F%2Fwww.teachengineering.org%2Fcollection%2Fduk_
%2Flessons%2Fduk_boxes_mary_less%2Fduk_boxes_mary_less.xml
Students find the volume and surface area of a rectangular box (e.g., a cereal box), and then figure out how to convert that box into a new, cubical box having the same
volume as the original.
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Stats & Probability 7.SP
Essential Question: How is probability used to make informed
Enduring Understandings:
The rules of probability can lead to more valid and reliable
decisions about uncertain events?
predictions about the likelihood of an event occurring.
Supporting Cluster: Use random sampling to draw inferences about a population.
Standard 7.SP.1 Understand that statistics can be used to gain information about a population by examining a sample of the
population; generalizations about a population from a sample are valid only if the sample is representative of that population.
Understand that random sampling tends to produce representative samples and support valid inferences.
Explanation:
 Students understand that representative samples can be used to make valid inferences about a population.
 Students understand that a random sample increases the likelihood of obtaining a representative sample of a population.
Learning Targets:
 I can make inferences about a population by examining a sample.
 I can explain how the sample is valid in relation to the population it is representing.
 I can explain that random sampling tends to produce representative samples.
Vocabulary:
sample, population, random sample, representative sample, inference
Sample Problem(s):
Sample 1
The school food service wants to increase the number of students who eat hot lunch in the cafeteria. The student council has been
asked to conduct a survey of the student body to determine the students‘ preferences for hot lunch. They have determined three ways
to do the survey. The three methods are listed below. Determine if each survey option would produce a random sample. Which survey
option should the student council use and why?
1. Write all of the students‘ names on cards and pull them randomly to determine who will complete the survey.
2. Survey the first 20 students that enter the lunchroom.
3. Survey every 3rd student who gets off a bus.
Resources:
 Polling
http://www.explorelearning.com/index.cfm?method=cResource.dspDetail&ResourceID=507
Link provides teachers and students with interactive activities that address stats and probability.
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
Manga High
http://www.mangahigh.com/en_us/curriculum##curriculum
This site includes challenges that address this standard.
 Learnzillion
http://learnzillion.com/lessons?utf8=%E2%9C%93&filters%5Bsubject%5D=math&query=&filters%5Bgrade%5D%5B%5D
=7&filters%5Bdomain%5D=SP%3A+Statistics+and+Probability&filters%5Bstandard%5D=7.SP.2%3A+Use+data+from+
a+random+sample+to+draw
This link provides effective lessons that are related to this standard as well as Extension Activities at the end of the lesson.
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Supporting Cluster: Use random sampling to draw inferences about a population.
Standard: 7.SP.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest.
Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example,
estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on
randomly sampled survey data. Gauge how far off the estimate or prediction might be.
Explanation:
 Students collect and use multiple samples of data to answer question(s) about a population.
 Students explore the variation in estimates or predictions based on multiple samples of the same data.
Learning Targets:
 I can make inferences about a population based on data taken from a random sample.
 I can compare and contrast multiple samples of the same data.
 I can gauge the variation in estimates or predictions.
Vocabulary:
variation, population, sample, random sample, prediction, inference, simulation, gauge
Sample Problem(s):
Sample 1
th
Given the first name of all students in your grade, predict the most common name in the U.S. for 7 graders. How good of an estimate
do you think your sample provides? Explain your reasoning.
Sample 2
Design a method of gathering a random sample from the student body to determine their favorite NFL team.
Resources:
 Learnzillion
http://learnzillion.com/lessons?utf8=%E2%9C%93&filters%5Bsubject%5D=math&query=&filters%5Bgrade%5D%5B%5D
=7&filters%5Bdomain%5D=SP%3A+Statistics+and+Probability&filters%5Bstandard%5D=7.SP.2%3A+Use+data+from+
a+random+sample+to+draw...
This link provides effective lessons that are related to this standard as well as Extension Activities at the end of the lesson.
 Manga High
http://www.mangahigh.com/en_us/curriculum##curriculum
This site includes challenges that address this standard.
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Additional Cluster: Draw informal comparative inferences about two populations.
Standard: 7.SP.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities,
measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height
of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability
(mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.
Explanation:
 Students can readily find data as described in the example on sports team or college websites.
 Students calculate mean absolute deviations in preparation for later work with standard deviations.
 Researching data sets provides opportunities to connect mathematics to their interests and other academic subjects.
Learning Targets:
 I can informally compare inferences about two populations.
 I can determine the difference between the centers by expressing it as a multiple of a measure of variability.
Vocabulary:
centers, variabilities, mean, median, mean absolute deviation, interquartile range, measure of variability
Sample Problem(s):
Sample1
Jason wanted to compare the mean height of the players on his favorite basketball and soccer teams. He thinks the mean height of the
players on the basketball team will be greater but doesn‘t know how much greater. He also wonders if the variability of heights of the
athletes is related to the sport they play. He thinks that there will be a greater variability in the heights of soccer players as compared
to basketball players. He used the rosters and player statistics from the team websites to generate the following lists.
Basketball Team – Height of Players in inches for 2010 Season
75, 73, 76, 78, 79, 78, 79, 81, 80, 82, 81, 84, 82, 84, 80, 84
Soccer Team – Height of Players in inches for 2010 Season
73, 73, 73, 72, 69, 76, 72, 73, 74, 70, 65, 71, 74, 76, 70, 72, 71, 74, 71, 74, 73, 67, 70, 72, 69, 78, 73, 76, 69
To compare the data sets, Jason creates two dot plots on the same scale. The shortest player is 65 inches and the tallest players are 84
inches.
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In looking at the distribution of the data, Jason observes that there is some overlap between the two data sets. Some players on both
teams have players between 73 and 78 inches tall. Jason decides to use the mean and mean absolute deviation to compare the data sets.
The mean height of the basketball players is 79.75 inches as compared to the mean height of the soccer players at
72.07 inches, a difference of 7.68 inches. The mean absolute deviation (MAD) is calculated by taking the mean of the absolute
deviations for each data point. The difference between each data point and the mean is recorded in the second column of the table. The
difference between each data point and the mean is recorded in the second column of the table. Jason used rounded values (80 inches
for the mean height of basketball players and 72 inches for the mean height of soccer players) to find the differences. The absolute
deviation, absolute value of the deviation, is recorded in the third column. The absolute deviations are summed and divided by the
number of data points in the set. The mean absolute deviation is 2.14 inches for the basketball players and 2.53 for the soccer players.
These values indicate moderate variation in both data sets.
Solution:
There is slightly more variability in the height of the soccer players. The difference between the heights of the
teams (7.68) is approximately 3 times the variability of the data sets (7.68 ÷ 2.53 = 3.04; 7.68 ÷ 2.14 = 3.59)
Resources:
 Learnzillion
http://learnzillion.com/lessons?utf8=%E2%9C%93&filters%5Bsubject%5D=math&query=&filters%5Bgrade%5D%5B%5D
=7&filters%5Bdomain%5D=SP%3A+Statistics+and+Probability&filters%5Bstandard%5D=7.SP.2%3A+Use+data+from+
a+random+sample+to+draw...
This link provides effective lessons that are related to this standard as well as Extension Activities at the end of the lesson.
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Additional Cluster: Draw informal comparative inferences about two populations.
Standard: 7.SP.4 Use measures of center and measures of variability for numerical data from random samples to draw informal
comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science
book are generally longer than the words in a chapter of a fourth-grade science book.
Explanation:
 Students compare two sets of data using measures of center (mean and median) and variability mean absolute deviation
and interquartile range. Showing two graphs vertically rather than side by side helps students make comparisons.
Learning Targets:
 I can compare two populations by using the means of data collected from random samples.
 I can compare two populations by using the medians of data collected from random samples.
 I can compare two populations by using the mean absolute deviations of data from random samples.
 I can compare two populations by using the interquartile ranges of data from random samples.
Vocabulary:
measures of variability, measures of center, mean, median, mean absolute deviation, interquartile range, population, random
sample.
Sample Problem(s):
Sample 1
The two data sets below depict random samples of the management salaries in two companies. Based on the
salaries below, which measure of center will provide the most accurate estimation of the salaries for each company?
 Company A: 1.2 million; 242,000; 265,500; 140,000; 281,000; 265,000; 211,000

190,000
Solution:
The median would be the most accurate measure since both companies have one value in the million that is far from the other
values and would affect the mean.
Resources:
 Manga High
http://www.mangahigh.com/en_us/curriculum##curriculum
This site includes challenges that address this standard.
 Learnzillion
http://learnzillion.com/lessonsets/300-using-measures-of-center-and-variability-to-draw-informal-comparativeinferences-1
This lesson is specific to this standard and has additional extension activities linked to the ―I Can‖ statements above.
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Supporting Cluster: Investigate chance processes and develop, use, and evaluate probability models.
Standard: 7.SP.5
Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event
occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ½
indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
Explanation:
 Students recognize that the probability of any single event can be expressed in terms such as impossible, unlikely, likely,
or certain
 Students recognize that the probability of any single event can be expressed as a number between 0 and 1, inclusive.
Learning Targets:
 I can recognize and explain that probabilities are expressed as a number between 0 and 1.
 I can interpret a probability of 0 as impossible.
 I can interpret a probability near 0 as unlikely to occur.
 I can interpret a probability near ½ as being as equally to occur as to not occur.
 I can interpret a probability near 1 as likely to occur.
 I can interpret a probability of 1 as certain.
Vocabulary:
likely, unlikely, ratio, percent, decimal, probability
Sample Problem(s):
Sample 1
The container below contains 2 gray, 1 white, and 4 black marbles. Without looking, if Eric chooses a marble from the container,
will the probability be closer to 0 or to 1 that Eric will select a white marble? A gray marble? A black marble? Justify each of
your predictions.
Solution:
White marble: Closer to 0
Gray marble: Closer to 0
Black marble: Closer to 1
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Resources:
 How I Roll
http://tinyurl.com/bbsjdtw
This task asks students to identify the number of outcomes possible from rolling two number cubes, to consider the
likelihood of one of the cubes ―winning‖ on any given roll by showing the sample space, and to determine the probability
of a certain outcome.
 The Probability Of Having An Event Using A Number Line
http://learnzillion.com/lessons/1238-describe-the-probability-of-an-event-using-a-number-line
Students will use a number line to find the probability of an event.
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Supporting Cluster: Investigate chance processes and develop, use, and evaluate probability models.
Standard: 7.SP.6 Approximate the probability of a chance event by collecting data on the chance process that produces it and
observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example,
when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200
times.
Explanation:
 Students collect data from a probability experiment, recognizing that as the number of trials increase, the experimental
probability approaches the theoretical probability. The focus of this standard is relative frequency -- The relative
frequency is the observed number of successful events for a finite sample of trials. Relative frequency is the observed
proportion of a successful event, expressed as the value calculated by dividing the number of times an event occurs by the
total number of times an experiment is carried out.
Learning Targets:
 I can use probability to predict the number of times a particular event will occur given a specific number of trials.
 I can collect data on a chance process to approximate its probability.
 I can use my results from the experimental probability to explain why they are not equivalent to the outcomes in the
theoretical probability.
 I can define probability as a ratio.
 I can represent experimental probability as a number between zero and one.
Vocabulary:
likely, unlikely, probability, experimental probability, relative frequency
Sample Problem(s):
Sample1
Suppose we toss a coin 50 times and have 27 heads and 23 tails. We define a head as a success. The relative frequency of heads
is: 27/50 = 54%.The probability of a head is 50%. The difference between the relative frequency of 54% and the probability of
50% is due to small sample size. The probability of an event can be thought of as its long-run relative frequency when the
experiment is carried out many times. Students can collect data using physical objects or technology simulations. Students can
perform experiments multiple times, pool data with other groups, or increase the number of trials in a simulation to look at the
long-run relative frequencies.
Sample2
Each group receives a bag that contains 4 green marbles, 6 red marbles, and 10 blue marbles. Each group performs
50 pulls, recording the color of marble drawn and replacing the marble into the bag before the next draw. Students compile their
data as a group and then as a class. They summarize their data as experimental probabilities and make conjectures about
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theoretical probabilities (How many green draws would are expected if 1000 pulls are conducted? 10,000 pulls?). Students create
another scenario with a different ratio of marbles in the bag and make a conjecture about the outcome of 50 marble pulls with
replacement. (An example would be 3 green marbles, 6 blue marbles, 3 blue marbles.) Students try the experiment and compare
their predictions to the experimental outcomes to continue to explore and refine conjectures about theoretical probability.
Resources:
 Yellow Starburst
http://ccssmath.org/?page_id=667

Students will use starburst to find the probability of a chance event by collecting data. They will investigate how many of
each color there is in a pack of starburst and compare their assumptions to the actual and find their percent of error.
Fair Game
http://insidemathematics.org/common-core-math-tasks/7th-grade/7-2003%20Fair%20Game.pdf
This task challenges a student to use understanding of probabilities to represent the sample space for simple and
compound events. A student must use information about probabilities to estimate probability of future events and
construct an argument about the fairness of a game.
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Supporting Cluster: Investigate chance processes and develop, use, and evaluate probability models.
Standard: 7.SP.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to
observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine
probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be
selected and the probability that a girl will be selected.
Explanation:
 Probabilities are useful for predicting what will happen over the long run. Using theoretical probability, students predict
frequencies of outcomes.
 Students recognize an appropriate design to conduct an experiment with simple probability events, understanding that the
experimental data give realistic estimates of the probability of an event but are affected by sample size.
 Students need multiple opportunities to perform probability experiments and compare these results to theoretical
probabilities. Critical components of the experiment process are making predictions about the outcomes by applying the
principles of theoretical probability, comparing the predictions to the outcomes of the experiments, and replicating the
experiment to compare results. Experiments can be replicated by the same group or by compiling class data. Experiments
can be conducted using various random generation devices including, but not limited to, bag pulls, spinners, number
cubes, coin tosses, and colored chips. Students can collect data using physical objects or technology simulations. Students
can also develop models for geometric probability (i.e. a target).
Learning Targets:
 I can develop a simulation to model probability.
 I can utilize the simulation to determine the probability of specific events.
Vocabulary:
likely, unlikely, probability model, simulation, experimental probability, sources of discrepancy
Sample Problem(s):
Sample 1
If Mary chooses a point in the square, what is the probability that it is not in the circle?
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Solution:
The area of the square would be 12 x 12 or 144 units squared.
The area of the circle would be 113.04 units squared. The probability that
a point is not in the circle would be 21.5%
Resources:
 How I Roll
http://tinyurl.com/bbsjdtw
This task asks students to identify the number of outcomes possible from rolling two number cubes, to consider the
likelihood of one of the cubes ―winning‖ on any given roll by showing the sample space, and to determine the probability
of a certain outcome.
 How Many Buttons
http://www.illustrativemathematics.org/illustrations/1022
This task uses student generated data to assess standard 7.SP.7. This task could also be extended to address Standard
7.SP.1 by adding a small or whole class discussion of whether the class could be considered as a representative sample of
all students at your school.
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Supporting Cluster: Investigate chance processes and develop, use, and evaluate probability models.
Standard: 7.SP.7b Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to
observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.
For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land
open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
Explanation:
 Probabilities are useful for predicting what will happen over the long run. Using theoretical probability, students predict
frequencies of outcomes.
 Students recognize an appropriate design to conduct an experiment with simple probability events, understanding that the
experimental data give realistic estimates of the probability of an event but are affected by sample size.
 Students need multiple opportunities to perform probability experiments and compare these results to theoretical
probabilities. Critical components of the experiment process are making predictions about the outcomes by applying the
principles of theoretical probability, comparing the predictions to the outcomes of the experiments, and replicating the
experiment to compare results. Experiments can be replicated by the same group or by compiling class data. Experiments
can be conducted using various random generation devices including, but not limited to, bag pulls, spinners, number
cubes, coin tosses, and colored chips. Students can collect data using physical objects or technology simulations.
 Students can also develop models for geometric probability (i.e. a target).
Learning Targets:
 I can develop a simulation to model probability.
 I can determine the probability of all events is equally likely to occur.
 I can determine the probability of events that may not be equally likely to occur.
 I can utilize the simulation to determine the probability of specific events.
Vocabulary:
likely, unlikely, probability model, simulation, experimental probability
Sample Problem(s):
Sample 1 Jason is tossing a fair coin. He tosses the coin ten times and it lands on heads eight times. If Jason tosses the coin an
eleventh time, what is the probability that it will land on heads?
Solution:
The probability would be ½. The result of the eleventh toss does not depend on the previous results.
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Sample 2
Conduct an experiment using a Styrofoam cup by tossing the cup and recording how it lands.
 How many trials were conducted?
 How many times did it land right side up?
 How many times did it land upside down?
 How many times did it land on its side?
 Determine the probability for each of the above results according to the collected data.
Resources:
 How I Roll
http://tinyurl.com/bbsjdtw
This task asks students to identify the number of outcomes possible from rolling two number cubes, to consider the
likelihood of one of the cubes ―winning‖ on any given roll by showing the sample space, and to determine the probability
of a certain outcome.
 Comparison of theoretical and experimental
http://learnzillion.com/lessons/1589-explain-discrepancies-in-results-from-a-probability-model-by-comparing-theexperimental-and-theoretical-probabilities
In this lesson you will learn how to explain discrepancies in results from a probability model by comparing experimental
and theoretical probabilities.
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Supporting Cluster: Investigate chance processes and develop, use, and evaluate probability models.
Standard: 7.SP.8a Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample
space for which the compound event occurs.
Explanation:
 Students use tree diagrams, frequency tables, organized lists, and simulations to determine the probability of compound
events paying close attention to writing as a fraction based off of the sample space.
Learning Targets:
 I can create a sample space of all possible outcomes for a compound event by using an organized list.
 I can create a sample space of all possible outcomes for a compound event by using a table
 I can create a sample space of all possible outcomes for a compound event by using a tree diagram.
 I can create a sample space of all possible outcomes for a compound event by using simulation.
Vocabulary:
likely, unlikely, compound event, sample space, probability, tree diagram, outcomes, favorable outcomes
Sample Problem(s):
Sample 1
For your ice cream sundae you can choose from chocolate, strawberry, or vanilla ice cream, and red, blue, or green sprinkles. It
can be served in small or large cups. A sundae gets one flavor of ice cream, one sprinkle, and one size. List the different kinds of
sundaes you can order.
Solution:
Making an organized list will identify that there are 18 different types of sundaes.
Resources:
 How I Roll
http://tinyurl.com/bbsjdtw
This task asks students to identify the number of outcomes possible from rolling two number cubes, to consider the
likelihood of one of the cubes ―winning‖ on any given roll by showing the sample space, and to determine the probability
of a certain outcome.
 Magna High
 http://www.magnahigh.com
This site includes challenges that address this standard.
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Supporting Cluster: Investigate chance processes and develop, use, and evaluate probability models.
Standard: 7.SP.8b Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
b. Represent for compound events using methods such as organized lists, tables, and tree diagrams. For an event described in
everyday language (e.g., ―rolling double sixes‖), identify the outcomes in the sample space which compose the event.
Explanation:
 Students use tree diagrams, frequency tables, and organized lists, and simulations to determine the probability of
compound events to find probabilities of real-world situations.
Learning Targets:
 I can represent a sample space of all possible outcomes for a compound event by using an organized list.
 I can represent a sample space of all possible outcomes for a compound event by using a table.
 I can represent a sample space of all possible outcomes for a compound event by using a tree diagram.
 I can represent a sample space of all possible outcomes for a compound event by using simulation.
 I can identify the outcome in my sample space described in everyday language.
Vocabulary:
likely, unlikely, compound event, sample space, probability, tree diagram, outcomes, favorable outcomes, simulation
Sample Problem(s):
Sample1
Show all possible arrangements of the letters in the word FRED using a tree diagram. If each of the letters is on a tile and drawn
at random, what is the probability of drawing the letters F-R-E-D in that order? What is the probability that a ―word‖ will have an
F as the first letter?
Solution:
There are 24 possible arrangements (4 choices • 3 choices • 2 choices • 1 choice)
The probability of drawing F-R-E-D in that order is 1/24
The probability that a ―word‖ will have an F as the first letter is 6/24 or 1/4
Resources:
 How I Roll
http://tinyurl.com/bbsjdtw
This task asks students to identify the number of outcomes possible from rolling two number cubes, to consider the
likelihood of one of the cubes ―winning‖ on any given roll by showing the sample space, and to determine the probability
of a certain outcome.
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Supporting Cluster: Investigate chance processes and develop, use, and evaluate probability models.
Standard: 7.SP.8c Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation
tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at
least 4 donors to find one with type A blood?
Explanation:
 Students can create a simulation that represents a given situation using materials to model certain events.
Learning Targets:
 I can create a sample space of all possible outcomes for a compound event bys using an organized list, a table or a tree
diagram.
 I can use the sample space to compare the number of favorable outcomes the total number of outcomes and determine the
probability of the compound event.
 I can design and utilize a simulation to predict the probability of a compound event.
Vocabulary:
likely, unlikely, compound event, sample space, probability, tree diagram, outcomes, favorable outcomes, simulation.
Sample Problem(s):
Sample 1
Suppose you are a prisoner in a far away land. The king takes pity on you and gives you a chance to leave. He shows you the
maze below. At the start and at each fork in the path, you must spin the spinner and follow the path that it points to. You may
request that the key to freedom be placed in one of the two rooms. In which room should you place the key to have the best
chance of freedom?
Notice that the probability of ending the maze in any one room is dependent on the result of the first spin.
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Solution: Place the key in Room B for the best chance of freedom. Accept multiple strategies for solving and multiple
7
9
representations of the solution. (7 ways to get to room A; 9 ways to get to room B or
of the time you get in Room A and
16
16
of the time you get in Room B)
Resources:
 How I Roll


http://tinyurl.com/bbsjdtw
This task asks students to identify the number of outcomes possible from rolling two number cubes, to consider the
likelihood of one of the cubes ―winning‖ on any given roll by showing the sample space, and to determine the probability
of a certain outcome.
 http://www.magnahigh.com
This site includes challenges that address this standard.
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Mathematics | Grade 8 – Critical Areas
In Grade 8, instructional time should focus on three critical areas: (1) formulating and reasoning about expressions and
equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems
of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; (3)
analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding
and applying the Pythagorean Theorem.
(1) Students use linear equations and systems of linear equations to represent, analyze, and solve a variety of problems. Students
recognize equations for proportions (y/x = m or y = mx) as special linear equations (y = mx + b), understanding that the constant of
proportionality (m) is the slope, and the graphs are lines through the origin. They understand that the slope (m) of a line is a constant
rate of change, so that if the input or x-coordinate changes by an amount A, the output or y-coordinate changes by the amount m·A.
Students also use a linear equation to describe the association between two quantities in bivariate data (such as arm span vs. height for
students in a classroom). At this grade, fitting the model, and assessing its fit to the data are done informally. Interpreting the model in
the context of the data requires students to express a relationship between the two quantities in question and to interpret components of
the relationship (such as slope and y-intercept) in terms of the situation. Students strategically choose and efficiently implement
procedures to solve linear equations in one variable, understanding that when they use the properties of equality and the concept of
logical equivalence, they maintain the solutions of the original equation. Students solve systems of two linear equations in two
variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the same line. Students use linear
equations, systems of linear equations, linear functions, and their understanding of slope of a line to analyze situations and solve
problems.
(2) Students grasp the concept of a function as a rule that assigns to each input exactly one output. They understand that functions
describe situations where one quantity determines another. They can translate among representations and partial representations of
functions (noting that tabular and graphical representations may be partial representations), and they describe how aspects of the
function are reflected in the different representations. (3) Students use ideas about distance and angles, how they behave under
translations, rotations, reflections, and dilations, and ideas about congruence and similarity to describe and analyze two-dimensional
figures and to solve problems. Students show that the sum of the angles in a triangle is the angle formed by a straight line, and that
various configurations of lines give rise to similar triangles because of the angles created when a transversal cuts parallel lines.
Students understand the statement of the Pythagorean Theorem and its converse, and can explain why the Pythagorean Theorem holds,
for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points
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on the coordinate plane, to find lengths, and to analyze polygons. Students complete their work on volume by solving problems
involving cones, cylinders, and spheres.
(3) Students use ideas about distance and angles, how they behave under translations, rotations, reflections, and dilations, and ideas
about congruence and similarity to describe and analyze two-dimensional figures and to solve problems. Students show that the sum
of the angles in a triangle is the angle formed by a straight line, and that various configurations of lines give rise to similar triangles
because of the angles created when a transversal cuts parallel lines. Students understand the statement of the Pythagorean Theorem
and its converse, and can explain why the Pythagorean Theorem holds, for example, by decomposing a square in two different ways.
They apply the Pythagorean Theorem to find distances between points on the coordinate plane, to find lengths, and to analyze
polygons. Students complete their work on volume by solving problems involving cones, cylinders, and spheres.
The Common Core State Standards
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Grade 8 Overview
The Number System
• Know that there are numbers that are not rational, and approximate them by rational numbers.
Expressions and Equations
• Work with radicals and integer exponents.
• Understand the connections between proportional relationships, lines, and linear equations.
• Analyze and solve linear equations and pairs of simultaneous linear equations.
Functions
• Define, evaluate, and compare functions.
• Use functions to model relationships between quantities.
Geometry
• Understand congruence and similarity using physical models, transparencies, or geometry software.
• Understand and apply the Pythagorean Theorem.
• Solve real-world and mathematical problems involving volume of cylinders, cones and spheres.
Statistics and Probability
• Investigate patterns of association in bivariate data.
The Common Core State Standards
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Mathematical Content Standards
Grade 8
The Number System 8.NS
Enduring Understandings
Essential Questions
Rational numbers can be represented in multiple ways and are
In what ways can rational numbers be useful?
useful when examining situations involving numbers that are
not whole.
Supporting Cluster: Know there are numbers that are not rational, and approximate them by rational numbers.
Standard: 8.NS.1
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion;
for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats
eventually into a rational number.
Explanation:
 Students distinguish between rational and irrational numbers.
 Students know that any number that can be expressed as a fraction, a whole number, a terminating or repeating decimal is
a rational number.
 Students know that any number that does terminate or repeat is irrational.
 Students recognize that the decimal equivalent of a fraction will either terminate or repeat.
Learning Targets:
 I can classify a number as rational based on its decimal expansion.
 I can classify a number as irrational based on its decimal expansion.
 I can convert a repeating decimal into a fraction
Vocabulary:
Rational number, Irrational number, decimal expansion
Sample Problem(s):
Sample 1
Classify the following numbers as rational or irrational based on its decimal expansion.
 0.75
0.23
1.72416...
0.0003
9.99
-5.34
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8.004
142.034…
4.1
3.14…
6.120120012…
-8.0
Solutions:
Rational numbers: 0.23,  0.75 , 0.0003, 9.99, -5.34, 8.004, 4.1 ,
Irrational numbers: 1.72416..., 142.034…, 6.120120012…
-8.0
Sample 2
Note: This practice can be easily made into a Smart Board activity.
Task: Students will convert a repeating decimal into a fraction.
Game: In partners or groups of 3 students will play a game of tic-tac-toe. Students will be presented with 10 decimals and 2
tic-tac-toe boards. They will express the decimals as fractions and look for the fraction form on the tic-tac-toe boards.
1. Present the information below for students to copy down
1) 0.3
6) 0.6
1
9
8
11
1
13
25
99
1
6
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2) 0.27
3) 0.1
7) 0.83
8) 0.5
3
11
23
99
1
10
2
11
2
3
4) 0.16
9) 0.23
5) 0.1
10) 0.36
6
7
1
2
1
7
5
6
1
3
3
22
2
3
2
7
2) Instruct students to convert each decimal to a rational number. If the rational number is on the tic-tac-toe board, students
should ―X‖ it out. The first person in each group to get three ―X‘s‖ in a row wins.
Solution: The second column of the first board should have three ―X‘s‖ in a row, and the last row of the second board.
1
1) 3
3
2) 11
1
3) 9
2
6) 3
5
7) 6
1
8) 2
1
4) 6
9)
1
5) 10
23
99
4
10) 11
Resources:
 Line Up Cards Activity
http://www.regentsprep.org/Regents/math/ALGEBRA/AOP1/Tcards.htm
Give each student a card and ask the class to arrange themselves around the room in numerical order.
 Rational and Irrational Number Game
http://www.math-play.com/rational-and-irrational-numbers-game/rational-and-irrational-numbers-game.html
Students will drag each number in the correct bin; and classify each number as rational or irrational number before time
runs out.
 Rational and Irrational Numbers
http://www.regentsprep.org/Regents/math/ALGEBRA/AOP1/Lrat.htm
This is a lesson that provides a diagram and table of rational and irrational numbers
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Supporting Cluster: Know there are numbers that are not rational, and approximate them by rational numbers.
Standard: 8.NS.2
Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a
number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2,
show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
Explanation:
 Students locate rational and irrational numbers on the number line.
 Students compare and order rational and irrational numbers.
 Students understand that the value of a square root can be approximated between integers
 Students understand that non-perfect square roots are irrational.
Learning Targets:
 I can plot the estimated value of an irrational number on a number line.
 I can use estimated values to compare two or more irrational numbers.
 I can compare rational and irrational numbers.
 I can order rational and irrational numbers.
Vocabulary: rational number, irrational number, non-perfect square root
Sample Problem(s):
Sample 1
Plot the estimated value of the square roots on a number line.
Square root
1
2
3
4
5
6
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Sample 2
Give students a set of 10 cards (shown below). Have students work with partners to sort the cards into 2 piles
- Rational and irrational.
- Plot estimated values on the number line.
0,
3
,
5
2,
0. 8 ,
4,
π,
16 , -2,
5,
4.173
Resources:
 Rational and Irrational Numbers
http://www.mathsisfun.com/irrational-numbers.html
This lesson clearly defines rational and irrational numbers. It will assist students better understand on any square root of a
prime is an irrational number. Also, there are a few list of problems that students will be able to work on.
 Plot Real Numbers on a Number line
http://www.illustrativemathematics.org/illustrations/337
When students plot irrational numbers on the number line, it helps reinforce the idea that they fit into a number system that
includes the more familiar integer and rational numbers.
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Expressions and Equations 8.EE
Enduring Understandings Essential Questions Algebraic expressions and equations are used to model real-life How can algebraic expressions and equations be used to model,
problems and to represent quantitative relationships, so that the analyze and solve mathematical situations?
numbers and symbols can be mindfully manipulated to reach a
solution or make sense of the quantitative relationship.
Major Cluster: Work with radicals and integer exponents.
Standard: 8.EE.1
Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example,
32 × 3-5= 3-3= 1/33= 1/27.
Explanation:
Students will use numerical bases and the laws of exponents, students generate equivalent expressions.
Exponents rules and properties
Rule name
Rule
Example
a n · a m = a n+m
23 · 24 = 23+4 = 128
a n · b n = (a · b) n
32 · 42 = (3·4)2 = 144
a n / a m = a n-m
25 / 23 = 25-3 = 4
a n / b n = (a / b) n
43 / 23 = (4/2)3 = 8
(a n) m = a n·m
(23)2 = 23·2 = 64
Product rules
Quotient rules
Power rules
m
an
Negative exponents
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m
= a (n
)
a-n = 1 / a n
2
2
2 3 = 2 (3
) = 512
2-3 = 1/23 = 0.125
Zero rules
a0 = 1
50 = 1
a1 = a
51 = 5
1n = 1
15 = 1
One rules
Learning Targets:
 I can use the property of integer exponents to simplify an exponent expression raised to first power.
 I can use the property of integer exponents to simplify an exponent expression raised to zero power.
 I can use the property of integer exponents to simplify an expression raised to a negative power.
 I can use the property of integer exponents to find the product of exponent expressions with the same base.
 I can use the property of integer exponents to find the quotient of exponent expressions with the same base.
Vocabulary:
integer, exponent
Sample Problem(s):
Sample 1
Discovery Lessons:
Finding Rules of Exponents Activity
Process:
Use this table to complete the first table with the students on the handout.
Exponent
Number
Repeated Multiplication
3
27
3 3  3
3
3
2
1
3
0
3
3
3
3
1
2
3
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What‘s the pattern?
÷3
Rules:
Any number raised to the power of 1 is __________________________________.
Any number raised to the power of 0 is __________________________________.
When the exponent is negative the number is a ____________________________.
Answers to tables:
Products
Expression
Expression written as repeated
multiplication
4 3
2 *2
(2*2*2*2)*(2*2*2)
Number of
factors
7
Product as a power
27
31*34
52*54
x2 *x 4
Rules:
Quotients
Expression
28
3
2
35
33
57
56
X8
X2
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Expression written as repeated
multiplication
2*2*2*2*2*2*2*2
2*2*2
Simplified
expression
Number of
factors
Quotient as a
power
2*2*2*2*2
5
25
Rule:
Power to a Power
Expression
Expression written as repeated
multiplication
4 3
(2 )
(2*2*2*2) (2*2*2*2) (2*2*2*2)
(32)4
(52)2
(x5) 4
Rule:
Number of
factors
12
Product as a power
212
Question: How can you use patterns to discover rules for multiplying and dividing integers?
Products
Expression
Expression written as repeated
Number of
Product as a power
multiplication
factors
24*23
31*34
52*54
x2 *x 4
Rule:
Quotients
Expression
28
23
35
33
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Expression written as repeated
multiplication
Simplified
expression
Number of
factors
Quotient as a
power
57
56
X8
X2
Rule:
Power to a Power
Expression
Expression written as repeated
multiplication
Number of
factors
Product as a power
(24)3
(31)4
(52)4
(x2) 4
Rule:
Activity Type: Group & Label
Note: This can be used as an overall review activity using all properties, or on a smaller basis that focuses on only a few properties.
Procedure:
1. The students will be placed into small groups. A teacher should use any grouping strategy that fits the needs of his or her
students. They will be given a packet containing a wide variety of expressions that need to be simplified using the various
properties of exponents.
2. First, the students must identify which property they must use in order to simply the expression. They will group each
expression into the proper property category. Once the students are finished grouping, they will have the instructor check
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that the grouping is correct. The instructor should give feedback for any misplaced expressions. A small example:
Product Property of Exponents
Power of a Power
Quotient Property of Exponents
3. Upon successful grouping from step 2, the students must then simplify the problems. The students can work individually
on each expression and compare work to help each other if a problem arises. Once the students agree to a final solution,
they will submit their groups work to the teacher.
Extension:
 Have students fix any answers where they have a negative exponent using the negative exponent property.
Resources:
 Slides from Exponents (Tutorial 20)
http://www.ohiorc.org/orc_documents/orc/for_mathematics/tutorials/40_handout.pdf
This is a summary that teacher can print out for the students after the lessons are being taught. Or students can make flash
cards or a little booklet as an activity in class to generate and summarize the exponent rules as they go along with the
lesson.
 Self Check Practice: Exponents (Meaning and Laws)
http://www.ohiorc.org/orc_documents/orc/for_mathematics/tutorials/40_selfcheck.pdf
This is an activity that students will be able to work on the computer or on hard copies. Students can self check the
answers right way.
 Exponent Jeopardy
http://www.math-play.com/Exponents-Jeopardy/Exponents-Jeopardy.html
Students can be organized in four teams (blue, red, green and yellow). The team must be selected before they provide the
answer. The game will track the time and points for each team.
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Major Cluster: Work with radicals and integer exponents.
Standard: 8.EE.2
Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational
number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
Explanation:
 Students recognize that squaring a number and taking the square root √ of a number are inverse operations; likewise, cubing a
number and taking the cube root are inverse operations.
 Students recognize perfect squares and cubes, understanding that non-perfect squares and non-perfect cubes are irrational.
Learning Targets:
 I can evaluate the square root of a perfect square
 I can evaluate the cube root of a perfect cube.
 I can explain that the square root of a non-perfect square will be irrational.
Vocabulary:
cube, square, cube root, radicand, radical, perfect square, perfect cube, irrational, non-perfect square roots
Sample Problem(s):
Sample 1
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Sample 2
Sample 3
Resources:
 Smart Board Radical Activity
http://mathequalslove.blogspot.com/2012/10/best-activity-ever-for-reviewing-square.html
Teacher can follow the instructions and make this activity on the smart board. Simply, the radical sign is already placed on the
smart board and two number cubes. Students are organized into two groups such as girls vs. boys. One female and one male
student roll a number cube and all students must evaluate the radical. The game is scored by tally mark. Prize may be optional
depends on individual teachers.
 Non-Perfect Square Root Activity
http://www.learnalberta.ca/content/mejhm/html/object_interactives/square_roots/flashHelp/pdf/SquareRootsPrintActivity.pdf
This is a printable activity. The activity starts with evaluate perfect square root and progresses to evaluate non-perfect square
root. Students need to do estimation and graph square root of a non-perfect square on a number line.
 Root Jeopardy
http://teachers.sduhsd.k12.ca.us/abrown/activities/jeopardy/rootjeopardy.htm
This activity is categorized as vocabulary, in betweens, perfect squares, exact roots and cubes. The game can be played in
various ways such as girls vs. boys, grouped in different colors or by Group Generator on the Smart Board. Also, there are
vocabulary review activity is on the end of the jeopardy activity.
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Major Cluster: Work with radicals and integer exponents.
Standard: 8.EE.3
Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities,
and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 ×
108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.
Explanation:
 This is the first time scientific notation appears. Students will not have prior knowledge.
 Students express numbers in scientific notation.
 Students compare and interpret scientific notation quantities in the context of the situation. For example, if the exponent
increases by one, the value increases 10 times.
Learning Targets:
 I understand that scientific notation can be used to express very large and very small numbers.
 I can convert very large quantities between standard form and scientific notation.
 I can convert very small quantities between standard form and scientific notation.
 I can compare quantities written in scientific notation.
Vocabulary:
scientific notation, power of ten, standard form, factors, ―how many times as big‖
Sample Problem(s):
Sample 1
To set a purpose: have students solve addition and subtraction problems with very large numbers and then very small numbers
without a calculator (choose values where results would be in scientific notation in the calculator)
Then have the students use the calculator to solve the same problems.
Have the students compare the answers using a calculator to those without a calculator, and look for a pattern.
Lead to using scientific notation.
 Have students generate a set of cards written in scientific notation.
 Have students put the cards into order least to greatest
 Then write in standard form
Resources:
 Scientific Cards Match Up Activity
http://www.regentsprep.org/Regents/math/ALGEBRA/AO2/TScicard2.htm
This activity requires two colors of 3x5 note cards. One color is written in scientific notation, and other color is written in
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standard notation. Students need to match up the scientific notation with correct standard notation.
 Scientific Cards Line Up Activity
http://www.regentsprep.org/Regents/math/ALGEBRA/AO2/TScicards.htm
Teacher prepares 3x5 cards with a variety of numbers written in scientific notation. Students who are participating will
receive a card. Students will line up with their card valuates in the front of the classroom from least to greatest. The
audience will critique the results.
 Converting Numbers
http://janus.astro.umd.edu/cgi-bin/astro/scinote.pl
This activity will permit students to practice conversion in scientific notation to standard notation and vice versa. There is
a ―Help me!‖ button with a ―?‖ next to it just in case you need a review of how to do conversion by multiplying in
scientific notation step by step.
 Scientific Notation-Activity
http://www.thatquiz.org/tq/previewtest?L/E/M/S/54231139454157
This activity is written in multiple choice question form. It can be just a quick check assessment or can be used as a quiz
to formally assess students‘ understanding.
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Major Cluster: Work with radicals and integer exponents.
Standard: 8.EE.4
Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation
are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities
(e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
Explanation:
 Students use laws of exponents to add or subtract numbers written in scientific notation.
 Students use laws of exponents to multiply or divide numbers written in scientific notation.
 Students understand scientific notation as generated on various calculators or other technology.
Learning Targets:
 I can add numbers written in scientific notation.
 I can subtract numbers written in scientific notation.
 I can multiply numbers written in scientific notation.
 I can divide numbers written in scientific notation.
 I can select the appropriate units for measurements.
 I can identify the various ways scientific notation is displayed on calculators and computer software.
 I can interpret the various ways scientific notation is displayed on calculators and computer software.
Vocabulary:
scientific notation, power of ten
Sample Problem(s):
Use the cards of populations written in scientific notation, have students perform operations (addition, subtraction, multiplication,
and division) to answer questions.
Sample1
What is the combined population of Illinois & Indiana?
How many times larger is the population of Texas compared to Wyoming?
Resources:
 Multiplying and Dividing Scientific Notation
http://go.hrw.com/resources/go_sc/hst/HSTMW261.PDF
This activity consists of two parts: multiplying and dividing scientific notation. It clearly explains how to multiply and
divide step-by-step. Students will be able to use the explanation and complete the tables.
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
Youtube-Multiplying and Dividing Scientific Notations
http://www.youtube.com/watch?v=nfTbTFJhbgo&safe=active
This video will provide a lesson on how to multiplying and dividing scientific notations. It can be used as a regular lesson
or for the students who are absent for the lesson.
 Computer Activity for Multiplying and Dividing Scientific Notation
http://www.edinformatics.com/math_science/scinot_mult_div.htm
 This activity is self-guided and students can work on their own paces. Students can summit answers and it will
immediately give them feedback with the correct answer.
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Major Cluster: Understand the connections between proportional relationships, lines, and linear equations.
Standard: 8.EE.5
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional
relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to
determine which of two moving objects has greater speed.
Explanation:
 This is the first time the students are introduced to slope.
 Students identify the unit rate (or slope) in graphs, tables and equations to compare two or more proportional relationships.
 Students should be able to compare rates of change between different representations (graph, table, verbal descriptions, and
equations)
Learning Targets:
 I can graph a proportional relationship in the coordinate plane.
 I can interpret the unit rate of a proportional relationship as the slope of a graph.
 I can use a graph, a table, or an equation to determine the unit rate of a proportional relationship
 I can use the rate of change to make comparisons between various proportional relationships represented in a graph, a table,
or an equation.
Vocabulary:
proportional relationship, unit rate, slope
Sample Problem(s):
Sample 1- Person 1‘s trip is represented graph below. Person 2‘s trip is represented by the equation below.
Distance
400
300
200
100
Time
0
1
2
3
4
5
6
7
8
Person 2‘s trip is represented by the equation y = 65x, where x is the time in hours and y is the distance in miles.
Question: what is the speed of person 1? What is the speed of person 2?
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Solution to the graph: In the above graph, person 1‘s car is traveling 50 miles per hour.
Person 2 is traveling 65 mi/hr. We want students to understand that the coefficient of x is the speed which is another way of
saying slope.
Sample 3
In Class Activity for students:
Separate students into small groups.
Give each group a string long enough to go from one corner of the room to the other.
Give each group a different distance/time problem involving square tiles on the floor.
Group A- slope of ½ For every 1 tile forward, they go sideways two. Repeat until they reach the other side of the classroom
Group B – Slope of 1. Step forward one tile and then sideways one tile.
Group C – Slope of 4
Group D – slope of 2/3 …….. Etc…
After reaching the far side of the room, have students use the string to make a line from where they started to where they end.
Have the groups make observations about each line and how and why they are different.
Resources:
 Proportional Relationship and Slope
http://www.cpm.org/pdfs/state_supplements/Proportional_Relationships_Slope.pdf
This activity consists of lesson, practice problems and answers to the questions. Students will explore various proportional
relationships that are compared with two different quantities.
 Hoop Shoot-Slope Game
http://www.crctlessons.com/slope-game.html
This slope game is an excellent way to test your algebra skills and also have fun. Students can play the game on the
computer and they can choose one or two players to compete. Whoever solves slope questions correctly, he/she will have a
three-point shoot. Whoever answers questions incorrectly, he/she will miss the three-point shoot. Whoever scores the
most points wins!
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Cluster: Understand the connections between proportional relationships, lines, and linear equations.
Standard: 8.EE.6
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate
plane; derive the equation y = mx + b for a line intercepting the vertical axis at b.
Explanation:
 Students will understand that triangles are similar when there is a constant rate of proportion between them.
 Using a graph, students construct triangles between two points on a line and compare the sides to understand that the slope (ratio of
rise to run) is the same between any two points on a line.
Learning Targets:
 I can use similar triangles to prove slope is constant.
 I can derive the y = mx + b equation from the graph of a line.
Vocabulary:
Right triangle, leg, hypotenuse, similar triangles, ratio, slope, proportional relationship, y-intercepts
Sample Problem(s):
Activity: https://grade8commoncoremath.wikispaces.hcpss.org/file/view/8.EE.6+Lesson+Similar+Triangles+and+Slope.doc
The student can use the ideas learned in the previous lesson to explore the similar triangles on the same line and on the same graph are
similar.
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Resources:
 Interactive Slope
http://www.mathopenref.com/coordslope.html
Students can adjust the line below by dragging an orange dot at point A or B. The slope of the line is continuously recalculated.
Students can also drag the origin point at (0,0). It has a summary and explanations of slopes: positive, negative, zero and
undefined.
 Interactive Slope Lesson
http://www.virtualnerd.com/algebra-1/linear-equation-analysis/slope-rate-of-change/slope-examples/slope-from-two-points
This electronic lesson is focused on finding a slope by using two points. Students can use it as reinforcement. Or it can be used
for the students who are absent for the lesson.
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Major Cluster: Analyze and solve linear equations and pairs of simultaneous linear equations.
Standard: 8.EE.7
Solve linear equations in one variable.
a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of
these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of
the form x = a, a = a, or a = b results (where a and b are different numbers).
b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions
using the distributive property and collecting like terms.
Explanation:
 Students will create and solve one-variable equations with one, infinite, or no solutions.
 Students will solve on variable equations using rational numbers, distributive property, and combining like terms.
 Students recognize that the solution to the equation is the value(s) of the variable, which make a true equality when
substituted back into the equation.
Learning Targets:
 I can solve linear equations by using the distributive property and/or collecting like terms.
 I can give examples of linear equations with one solution.
 I can give examples of linear equations with infinite solutions.
 I can give examples of linear equations with no solution.
Vocabulary:
Linear equation, equivalent equations, rational number, coefficient, collecting like terms, like terms, solution
Sample Problem(s):
Sample 1
Solve for x.
1. -3(x + 7) + 3x = 4
2. 3x – 8 = 4x – 8
3. 3(x + 1) – 5 = 3x – 2
Sample 2
Solve the following.
1. 7(m – 3) = 7
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2.
1 2
3 1
 y  y
4 3
4 3
Resources:
 Solving Multi-Step Equations
http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Solving%20Multi-Step%20Equations.pdf
This is Kuta Software printable worksheet. It consists of 20 questions, answer document and one critical thinking
question.
 Interactive Solving Multi-Step Equations
http://www.purplemath.com/modules/solvelin3.htm
This is interactive activity. Students can be working on each problem individually. They can choose what variable

they want to solve, number of steps, and graph of the equation.
Hot Math Practice Problems
http://hotmath.com/help/gt/genericalg1/section_1_3.html
This activity provides a list of problems. Students can work on the problems individually on the computer. It will
provide step-by-step solving multi-step equation when students encounter difficulties.
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Major Cluster: Analyze and solve linear equations and pairs of simultaneous linear equations.
Standard: 8.EE.8
Analyze and solve pairs of simultaneous linear equations.
a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their
graphs, because points of intersection satisfy both equations simultaneously.
b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve
simple cases by inspection. For example, 3x + 2y = 5 and 3x +2y = 6 have no solution because 3x + 2y cannot simultaneously be
5 and 6.
c. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates
for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
Explanation:
 Students graph two linear equations, recognizing that the ordered pair for the point of intersection is the x-value that will
generate the given y-value for both equations.
 Students recognize that graphed lines with one point of intersection (different slopes) will have one solution, parallel lines
(same slope, different y-intercepts) have no solutions, and lines that are the same (same slope, same y-intercept) will have
infinitely many solutions.
 Students should be exposed to the various methods of solving systems of equations.
 Students will solve real world problems utilizing systems of equations.
Learning Targets:
 I can explain how the point of intersection of two lines on a graph corresponds to the solution.
 I can solve a system of linear equations using algebraic and mathematical reasoning.
 I can estimate the solution of the system using the graphs of two linear equations.
 I can write a system of linear equations to represent a real-world problem.
 I can solve real world problems dealing with systems with linear equations.
 I can interpret the solution in the context of the problem.
Vocabulary:
Linear equation, equivalent equations, rational number, coefficient, like terms, solution
Sample Problem(s):
Sample 1
Solve the following system of equations
3x + 4y = 7
-2x + 8y = 10
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Plant A and Plant B are on different watering schedules. This affects their rate of growth. Compare the growth of the two plants
to determine when their heights will be the same.
Let W = number of weeks
Let H = height of the plant after W weeks
W
0
1
2
3
Plant A
H
4 (0,4)
6 (1,6)
8 (2,8)
10 (3,10)
Plant B
W
H
0 2
(0,1)
1 6
(1,6)
2 10 (2,10)
3 14 (3,14)
Given each set of coordinates, graph their corresponding lines.
Solution:
Resources:
 Solving System by Graphing
http://www.ixl.com/math/algebra-1/solve-a-system-of-equations-by-graphing
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This is interactive activity online. Students will graph a system in a coordinate and summit the solution in ordered

pair. It will keep track of scores and the time spending on solving the system and will provide explanation.
Solving System by Substitution
http://www.ixl.com/math/algebra-1/solve-a-system-of-equations-using-substitution
This is interactive activity online. Students will solve the system by substitution and summit the solution in ordered
pair. It will keep track of scores and the time spending on solving the system and will provide explanation.
 Solving System by Elimination
http://www.ixl.com/math/algebra-1/solve-a-system-of-equations-using-elimination
This is interactive activity online. Students will solve the system by elimination and summit the solution in ordered
pair. It will keep track of scores and the time spending on solving the system and will provide explanation.
 Solving Systems of Linear Equations Jeopardy Style Review
http://www.superteachertools.com/jeopardyx/jeopardy-review-game.php?gamefile=1297688635
This jeopardy game consists of rewrite in y = mx + b, solving the system by graphing, substitution, elimination and
order pair is the solution to the system.
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Functions 8.F
Enduring Understandings –
Essential Questions –
The characteristics of functions and their representations are
How are functions useful?
useful in making sense of patterns and solving problems
involving quantitative relationships.
Major Cluster: Define, evaluate and compare functions.
Standard: 8.F.1
Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered
pairs consisting of an input and the corresponding output.1
1
Function notation is not required in Grade 8.
Explanation:
 Students understand that functions occur when there is exactly one y value that is associated with any x value.
 Students can identify functions from equations, graphs, and tables/ordered pairs.
Learning Targets:
 I can identify a function that is a rule that assigns each input exactly one output.
Vocabulary:
function, input, output, rate of change.
Sample Problem(s):
Sample 1
Graphs
Students recognize graphs such as the one below is a function using the vertical line test, showing that each x-value has only one
y-value;
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Tables or Ordered Pairs
Students read tables or look at a set of ordered pairs to determine functions and identify equations where there is only one output
(y-value) for each input (x-value).
Functions Not A Function
{(0, 2), (1, 3), (2, 5), (3, 6)}
Equations
Students recognize equations such as y = x or y = x2 + 3x + 4 as functions; whereas, equations such as x2 + y2 = 25 are not
functions.
FUNCTION
NOT A FUNCTION
Resources:
 Identify Function Activity
http://www.ixl.com/math/algebra-1/identify-linear-functions
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This is a multiple choice interactive activity online. It will keep track of scores and the time spending on solving the
problem and will provide explanation.
Mathopolis Quiz
http://www.mathopolis.com/questions/q.php?id=536&site=1&ref=/sets/domain-rangecodomain.html&qs=536_537_538_539_1174_1175_1176_2341_2342_2343
Students will be able to take a ten-question quiz online. It will provide feedback, explanations and a ―help‖ bottom.
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Major Cluster: Define, evaluate and compare functions.
Standard: 8.F.2
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal
descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic
expression, determine which function has the greater rate of change.
Explanation:
 Students compare functions from different representations. e.g. increasing, decreasing, ranges, slopes, domains,
Learning Targets:
 I can compare two functions that are represented differently (e.g., algebraically, in a table, graphically or a verbal representation).
Vocabulary:
function, input, output, linear, non-linear, rate of change.
Sample Problem(s):
Of the four linear functions represented below, which has the greatest rate of change?
Verbally
A number, y, is three less than twice a number, x.
Graphically
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Table of Values
x
f(x)
-3
-9
3
3
6
9
Algebraically
2y + 3 = 3x
Resources:
 Compare Function Activity
http://www.mytestbook.com/worksheet.aspx?test_id=1419&grade=8&subject=Math&topics=Algebra%20Compare%20functi
ons,%20graphs,%20tables
This is a printable worksheet which also provides answers keys.
 Function Machine
http://www.mathplayground.com/functionmachine.html
Students choose inputs and the machine will produce the outputs. Students will generate a function rule.
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Major Cluster: Define, evaluate and compare functions.
Standard: 8.F.3
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are
not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its
graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
Explanation:
 Students use equations, graphs and tables to categorize functions as linear or non-linear.
 Students recognize that points on a straight line will have the same rate of change between any two of the points.
Learning Targets:
 I can explain why the equation y = mx + b represents a linear function
 I can interpret the slope in relation to the function.
 I can interpret the y- intercept in relation to the function.
 I can give examples of functions that are non-linear.
Vocabulary:
slope, linear function, y-intercept, non-linear function, y = mx+b
Sample Problem(s):
Sample 1
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Resources:
 Slope and Y-intercept Activity
http://dlc.k12.ar.us/Resources/Mathematics/Algebra_II/SlopeInterceptActivity.pdf
Students will find the coordinates of a given linear equations, graph those points, list the slope and y-intercept of the
resulting line, and use those to evaluate the original equation.
 Rag to Riches Slope and Y-intercept Activity
http://www.quia.com/rr/379720.html
This game asks students to identify slope and y-intercept in multiple choice format. It provides hits and total amount of
money as motivation.
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Supporting Cluster: Use functions to model relationships between quantities.
Standard: 8.F.4
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the
function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph.
Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a
table of values.
Explanation:
 Students use the slope and y-intercepts to write a linear function in the form y = mx +b. Situations may be given as a verbal
description, two ordered pairs, a table, a graph, or rate of change and another point on the line. Students interpret slope and yintercept in the context of the given situation. Students recognize that in a table the y-intercept is the y-value when x is equal
to 0.
 In contextual situations, the y-intercept is generally the starting value or the value in the situation when the independent
variable is 0. The slope is the rate of change that occurs in the problem.
Learning Targets:
 I can translate between a table, graph, equation and/or verbal description of a linear function.
 I can identify and interpret the slope and y-intercept from a graph of a linear function.
 I can identify and interpret the slope and y-intercept from a table of values.
 I can identify and interpret the slope and y-intercept from a verbal description.
Vocabulary:
function, linear, rate of change, slope, y-intercept, initial value
Sample Problem(s):
A company charges $35 a day for the car as well as charging a one-time $25 fee for the car‗s navigation system (GPS).Write an
expression for the cost in dollars, c, as a function of the number of days, d.
Resources:
 Writing y = mx + b Game
http://hotmath.com/hotmath_help/games/kp/Karappan_Poochi_Sound.swf
Students will find the slope and y-intercept by following along the cockroaches on the coordinate plane. The cockroaches
will move along a straight line. Your mission: find the equation of the line in slope-intercept form. Students score if they
get the correct slope. Otherwise, the cockroaches will cumulates.

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
Hoop Shoot Game
http://www.math-play.com/slope-intercept-game.html
This slope game is an excellent way to test your algebra skills and also have fun. Students can play the game on the
computer and they can choose one or two players to compete. Whoever solves slope questions correctly, he/she will have a
three-point shoot. Whoever answers questions incorrectly, he/she will miss the three-point shoot. Whoever scores the
most points wins!
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Supporting Cluster: Use functions to model relationships between quantities.
Standard: 8.F.5
Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is
increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been
described verbally.
Explanation:
 Given a verbal description of a situation, students sketch a graph to model that situation.
 Given a graph of a situation, students provide a verbal description of the situation.
Learning Targets:
 I can create a graph from a qualitative verbal description.
 I can qualitatively describe the functional relationship of a graph.
Vocabulary:
function, qualitatively, qualitative .
Sample Problem(s):
Sample 1
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Sample 2
Resources:
 Slope-Intercept Form Review Jeopardy
http://www.superteachertools.com/jeopardyx/jeopardy-review-game.php?gamefile=1340250564
This game is grouped in 10 teams. It consists of vocabulary, slope, y-intercept, expression of slope-intercept form and
graphs.
 Scattering Interpretation Slope and y-Intercept Game
http://quizlet.com/20643749/scatter/
This is a matching game. Students drag each description to the line of best fit on a scatter plot. The game is timed.
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Geometry 8.G
Enduring Understandings –
Essential Questions –
Geometric attributes ( such as shapes, lines, angles, figures, and How does geometry better describe objects?
planes) provide descriptive information about an object‘s
properties and position in space and support visualization and
problem solving.
Major Cluster: Understand congruence and similarity using physical models, transparencies, or geometry software.
Standard: 8.G.1
Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
Explanation:
 The properties of rotations, reflections, and translations for congruent figures are:
o Lines are taken to lines, and line segments to line segments of the same length.
o Angles are taken to angles of the same measure.
o Parallel lines are taken to parallel lines.
 Students understand that translations, rotations, and reflections produce images of exactly the same size and shape as the preimage.
Learning Targets:
 I can verify the properties of rotation for congruent figures.
 I can verify the properties of reflections for congruent figures.
 I can verify the properties of translations for congruent figures.
Vocabulary:
transformation, translation, reflection, rotation, parallel line, congruent, line segments, line
Sample Problem(s):
Activity: Students will draw angles, intersecting lines and parallel lines to use as pre-images. Next, students will
place a piece of patty paper or transparency over the image and trace the figures (once traced they can be transformed)
Materials: graph paper, patty paper or transparency sheet, ruler, protractors
Transformation 1: Rotation.
 Have students draw a pair of intersecting lines (pre-image)on graph paper
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 Next, student will place a piece of patty paper or transparency over the pre-image and trace it (image).
 Have students rotate the image on the transparency by placing the tip of the pencil on the transparency paper.
 Have the students compare the images (length, angle measures, parallel lines, and non-parallel lines)
 Conclusion: same size and same shape as pre- image but in different location.
Transformation 2: Reflection
 Have students draw an angle on graph paper in quadrant 2 or 3 of a coordinate
 Next, student will place a piece of patty paper or transparency over the pre-image.
 Flip (reflect) the image on the transparency parallel with the y-axis.
 Have the students compare the images (length, angle measure, parallel lines, and non-parallel lines)
 Conclusion: same size and same shape as pre- image but in different location.
Transformation 3: Translation
 Have students create a set of parallel lines on graph paper.
 Place the image transparency over the pre-image and trace the it
 Have students move the image on the transparency up, down, left right, combination of moves.
 Have the students compare the images (length, angle measure, parallel lines, and non-parallel lines)
 Conclusion: same size and same shape as pre- image but in different location.
Resources:
 Matching Congruent Figures
http://www.narragansett.k12.ri.us/resources/necap%20support/gle_support/Math/resources_geometry/math_congr.htm
Matching congruent figures using reflections, translations, or rotations means to identify if figures are congruent by
determining if the figures only differ in location.
 Rotation Activity
http://www.mathsisfun.com/geometry/rotation.html
Students will choose a specific shape and rotate in a specific angle in the coordinate plane.
 Reflection Activity
http://www.mathsisfun.com/geometry/reflection.html
Students will choose a specific shape and reflect over a specific line in the coordinate plane.
 Translation Activity
http://www.mathsisfun.com/geometry/translation.html
Students will choose a specific shape and translate in a specific distance in the coordinate plane.
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Major Cluster: Understand congruence and similarity using physical models, transparencies, or geometry software.
Standard: 8.G.2
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of
rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between
them.
Explanation:
 Students examine two figures to determine congruency by identifying the rigid transformation(s) that produced the figures.
 Students understand that translation that occurred between the congruent figures.
Learning Targets:
 I can examine two figures to determine congruency.
 I can describe a sequence of transformation between two congruent figures.
Vocabulary
congruent, transformation, reflection, rotation, translation
Sample Problem(s):
Sample 1
Have students graph the trapezoid A (1, 3) B (3, 7) C(8, 7) D(10, 3) as pre-image, follow the steps from the activity in 8.G.1.
Conclusion should be the same. Same size and shape different location.
Sample 2
Students will describe a sequence of transformations to ―move‖ a pre-image to an image
Materials: Graph paper, pattern blocks (polygons). Students will work individually at first, then switch to partners.
Individually,
1. Have students trace a polygon(pre-image) onto graph paper.
2. They should then rotate, reflect and translate the figure, tracing the image after each transformation. They should write
down the steps (being very specific as to angle rotation, line of symmetry, and direction) after each move.
3. Trace the pre-image on a NEW sheet of graph paper in its ORIGINAL position and then exchange the pre-image with
another student.
Partners or groups of 3
1. One student will read the written instructions for his/her transformation to her partner(s), who will follow the directions
and draw the image.
2. Compare the pre-image with the image. If the pre-image and image don not match, they should work together and review
each step to identify any errors.
3. Switch roles and repeat.
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Resources:
 Congruency and Similar Shape Lesson
http://www.brainpopjr.com/math/geometry/congruentandsimilarshapes/grownups.weml
This is a Smart Board lesson on Brain Pop Jr. The lesson helps children explore geometry and understand and identify
relationships between different figures. This movie will introduce congruent and similar shapes.
 Congruency and Similar Shape
http://www.sheppardsoftware.com/mathgames/geometry/shapeshoot/SSCongruentSimilar.htm
It is a computer game that will reinforce students‘ knowledge on congruency and similarity. Students will use mouse to
click on congruent or similar shapes.
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Major Cluster: Understand congruence and similarity using physical models, transparencies, or geometry software.
Standard: 8.G.3
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
Explanation:
 Students identify resulting coordinates from translations, reflections, dilations and rotations recognizing the relationship
between the coordinates and the transformation.
Learning Targets:
 I can identify resulting coordinates from translations.
 I can identify resulting coordinates from reflections.
 I can identify resulting coordinates from dilations.
 I can identify resulting coordinates from rotations.
Vocabulary:
transformation, translation, reflection, rotation, dilation, similar
Sample Problem(s):
Triangle 2 is __________ times the size of triangle 1. Describe the change in dilation from triangle 1 to triangle 2.
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Resources:
 Triangle Congruency with Coordinates
http://www.illustrativemathematics.org/standards/k8
This task gives students a chance to explore several issues relating to rigid motions of the plane and triangle congruence.
 Tutorial Transformation Lesson
http://www.virtualnerd.com/middle-math/integers-coordinate-plane/transformations
This is a popular tutorial website that provides all the transformation lessons. It can reinforce students‘ understanding or
simply for the students who missed lessons.
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Major Cluster: Understand congruence and similarity using physical models, transparencies, or geometry software.
Standard: 8.G.4
Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of
rotations, reflections, translations, and dilations; given two similar two- dimensional figures, describe a sequence that exhibits the
similarity between them.
Explanation:
 Students describe the sequence that would produce similar figures.
 Given a pre-image and the required sequence, students can create a similar figure.
Learning Targets:
 I can examine two figures to determine similarity.
 I can describe a sequence of transformation between two similar figures.
Vocabulary:
transformation, translation, reflection, rotation, dilation, similar
Sample Problem(s):
Sample 1Use the website/lesson in small group or individual to facilitate lesson. http://learnzillion.com/lessonsets/289
Resources:
 Identify Similarity
http://www.ixl.com/math/geometry/identify-similar-figures
Students will identify two figures. The activity will keep track of scores and time of spending on the solving the problems.
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Major Cluster: Understand congruence and similarity using physical models, transparencies, or geometry software.
Standard: 8.G.5
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when
parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies
of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why
this is so.
Explanation:
 Students use exploration and deductive reasoning to determine relationships that exist between:
o Angle sums and exterior angle sums of triangles
o Angles created when parallel lines are cut by a transversal
o The angle-angle criterion for similarity of triangles
 Students will construct various triangles and find the measures of the interior and exterior angles.
Learning Targets:
 I can informally argue that the sum of any triangle‘s interior angles is 180°
 I can informally argue that the sum of the two opposite interior angles is equal to the exterior angle.
 I can informally argue that corresponding angles of parallel lines cut by a transversal are congruent.
 I can informally argue that alternate interior angles of parallel lines cut by a transversal are congruent.
 I can informally argue that alternate exterior angles of parallel lines cut by a transversal are congruent.
 I can informally argue that same side interior angles of parallel lines cut by a transversal are supplementary.
 I can informally argue that same side exterior angles of parallel lines cut by a transversal are supplementary.
 I can informally argue the angle/angle rule to the similarity between triangles.
Vocabulary:
interior angle, alternate interior angle, corresponding angles, exterior angle, same side interior angles, same side exterior angles,
parallel lines, transversal, similar
Sample Problem(s):
Sample 1: Angle sum of triangles
 Provide the students with a paper with various triangles (several examples of each: right triangle, obtuse triangle and acute
triangle).
For each triangle:
 Have students measure the interior angle of each triangle and record in the table.
 Have students sum the angles and round the sum to the nearest ten.
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Triangle
Type of
triangle
Angle A
Angle B
Angle C
Sum of
Angles
1
2
3
4
5
6
7
8
9
After all triangles have been measured:
 Have students write a statement from the relationship found regarding the sum of the angles.
Sample 2: Angle sum of triangles
Students will draw and then cut a triangle and rearrange the angles to form a line, showing the sum of the angles in the
triangle is 180
 Have students draw a large triangle on graph paper, labeling the vertices INSIDE and cut it out.
 Have students cut the triangle so that the original triangle is now divided into three pieces, each piece with an angle
labeled.
 Have students rearrange the vertices so that a common point is the vertex of all the three angles and the sides touch but do
NOT overlap.
 Students will find out that no matter how they arrange them it will create a straight line (also known as a straight angle).
 Ask students what this proves about the sum of the measures of the angles of a triangle?
Sample 3: Angle sum and exterior angles of triangles
Give student handout with image shown drawn on it.
 Have students measure All angles (interior and exterior)
 Have students answer questions:
1. Which angles are exterior angles?
2. Which are interior angles?
3. What remote interior angles are paired with each exterior angle?
4. What is the sum of the exterior angles?
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f
b
a
e
c
d
Additional Activity:
Go to http://www.mathwarehouse.com/geometry/triangles/angles/remote-exterior-and-interior-angles-of-a-triangle.php.
Scroll down to ―Make Angle‖ to create an exterior angle of a triangle. Notice the exterior angle of a triangle is equal to the sum
of the 2 remote interior angles.
Sample 4: Angles created by parallel lines cut by a transversal
 Have students draw two parallel lines on graph paper and label the angles with letters
 Have students draw a transversal lines intersecting the parallel lines
 Have students measure the resulting angles and draw conclusions
A
B
C D
E F
G H
Angle
Measure in degrees
Angle
A
E
B
F
C
G
D
H
Examples of questions (but not all inclusive)
What angles are congruent to angle A?
What angles are congruent to angle B?
List all congruent angles.
Name of supplementary angles that are not touching.
Measure in degrees
Follow up with identifying angle types and relationships (alternate interior & exterior, vertical, corresponding, supplementary)
Resources:
 A + B + C = 180°
http://www.mathsisfun.com/proof180deg.html
Students can drag a point on a triangle and observe the angle sum of any triangle is 180°.
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
Interactive Demonstration of Interior Angle Sum
http://www.mathwarehouse.com/geometry/triangles/
This is an interactive activity. Besides students can drag a point on a triangle and observe the angle sum of any triangle is
180°, it also provides practice problems to solve a missing angle.
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Major Cluster: Understand and apply the Pythagorean Theorem.
Standard: 8.G.6
Explain a proof of the Pythagorean Theorem and its converse.
Explanation:
 Students explain the Pythagorean Theorem as it relates to the area of squares coming off of all sides of a right triangle.
 Students also understand that given three side lengths with this relationship forms a right triangle.
Learning Targets:
 I can explain algebraic reasoning to relate the visual model to the Pythagorean Theorem.
 I can explain how to use the Pythagorean Theorem to determine if a given triangle is a right triangle.
Vocabulary:
Pythagorean Theorem, leg, hypotenuse, converse
Sample Problem(s):
Sample 1
Materials: grid papers
Calculators
Procedures:
1. Students are going to draw triangles from given measures, representing and computing the areas of the squares of each
side.
2. Students record the data in the chart (the one below).
3. Students will make a conjecture about the relationship among the areas within each triangle.
Triangle Measure of leg 1 Measure of leg 2 Area of square on
Area of square Area of square
leg 1
on leg 2
on hypotenuse
1
2
4. Students will test out their conjectures, then explain and discuss their findings
5. Students are introduced the Pythagorean Theorem and explain the pattern they have explored.
Resources:
 Discovering the Pythagorean Theorem
http://www.regentsprep.org/Regents/math/ALGEBRA/AT1/TActive.htm
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
This website provides step-by-step activity.
Pythagorean Theorem Jeopardy
http://www.math-play.com/Pythagorean-Theorem-Jeopardy/Pythagorean-Theorem-Jeopardy.html
This game can be grouped into four teams (blue, red, green and yellow). The game consists of finding hypotenuse,
unknown leg, and Pythagorean Theorem conversion.
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Major Cluster: Understand and apply the Pythagorean Theorem.
Standard: 8.G.7
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in
two and three dimensions.
Explanation:
 Students will be able to apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world
and mathematical problems in two and three dimensions.
Learning Targets:
 I can apply the Pythagorean Theorem to find an unknown side length of a right triangle.
 I can draw a diagram and use the Pythagorean Theorem to solve real-world and mathematical problems involving right
triangles in two and three dimensional figures.
Vocabulary:
Pythagorean Theorem, leg, hypotenuse, converse
Sample Problem(s):
Sample 1
Painters use ladders to paint on high buildings and often use the help of the Pythagorean Theorem to complete their work. Take
for example a painter who has to paint a wall which is about 8 m high. The painter has to put the ladder 6 m away to avoid a rack
in between. What will be the length of the ladder required by the painter to complete his work?
Sample 2
Mark and Jose are arguing. Mark says it is impossible to draw a right triangle with sides measuring 8 inches, 12 inches, and 17
inches. Jose says it is possible. Who is correct?
Resources:
 Pythagorean Theorem Game
http://www.kidsnumbers.com/pythagorean-theorem-game.php
This is interactive online game can be played individually. Students will find each leg and hypotenuse.
 Pythagorean Theorem Jeopardy
http://www.math-play.com/Pythagorean-Theorem-Jeopardy/Pythagorean-Theorem-Jeopardy.html
The game consists of finding hypotenuse, unknown leg, and Pythagorean Theorem conversion.
 Finding Missing Leg
http://www.ixl.com/math/algebra-1/pythagorean-theorem
Students will find the length of the missing leg. They will keep track of scores and time of spending on the solving the
problems.
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Major Cluster: Understand and apply the Pythagorean Theorem.
Standard: 8.G.8
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Explanation:
 Students will apply the Pythagorean Theorem to find the distance between two points on the coordinate plane.
Learning Targets:
 I can find the distance between two points on a coordinate plane using the Pythagorean Theorem.
Vocabulary:
Pythagorean Theorem, leg, hypotenuse, converse
Sample Problem(s):
Sample 1:
Materials: graphing paper
Calculators
Procedures:
1. Students will choose any two set of coordinates at a time.
2. Students will graph the points on the coordinate plane (as shown below diagram).
3. Students will create right triangles and apply Pythagorean Theorem to find the distance
between the two given points.
(1, -6), (4, 8), (9, -2), (0, -7), (-3, -8), (-2, 5), (3, 2), (-10, -3), (5, 4), (-9, -1)
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Resources:
 Pythagorean Distance between Two Points
http://gamedev.michaeljameswilliams.com/2009/05/08/pythagorean-distance-between-two-points/
This website explains on how to use Pythagorean Theorem to find the distance of a line on a graphing paper.
 Distance between Two points Activity
http://www.ixl.com/math/grade-8/distance-between-two-points
Students will find the distance between two points on a coordinate plane. The activity will keep track of scores and time
of spending on the solving the problems.
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Additional Cluster: Solve real-world and mathematical problems involving volume of cylinders, cones and spheres.
Standard: 8.G.9
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical
problems.
Explanation:
 Students will find the volume of cylinders, cones and spheres by using the correct formula.
 Students understand the relationship between the volume of:
o Cylinders and cones
o Cylinders and spheres to the corresponding formulas.
Learning Targets:
 I can solve real-world and mathematical problems involving the volume of cylinders by using the formulas.
 I can solve real-world and mathematical problems involving the volume of cones by using the formulas.
 I can solve real-world and mathematical problems involving the volume of spheres by using the formulas.
Vocabulary:
Volume, cylinder, sphere, cone
Sample Problem(s):
James wanted to plant pansies in his new planter. He wondered how much potting soil he should buy to fill it. Use the
measurements in the diagram below to determine the planter‗s volume.
Resources:
 Finding Volumes of Prism and Cylinder Activity
http://www.ixl.com/math/geometry/volume-of-prisms-and-cylinders
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Students will find the volumes of prisms and cylinders. The activity will keep track of scores and time of spending on the
solving the problems.
 Finding Volumes of Pyramid and Cone Activity
http://www.ixl.com/math/geometry/volume-of-pyramids-and-cones
Students will find the volumes of prisms and cylinders. The activity will keep track of scores and time of spending on the
solving the problems.
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STATISTICS AND PROBABILITY – 8.SP
Enduring Understandings –
Essential Questions –
The rules of probability can lead to more valid and reliable
How is probability used to make informed decisions about
predictions about the likelihood of an event occurring.
certain events?
Supporting Cluster: Investigate patterns of association in bivariate data.
Standard: 8.SP.1
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities.
Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
Explanation:
 Students represent measurement (numerical) data on a scatter plot, recognizing patterns of association.
 Students will identify patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear
association.
Learning Targets:
 I can plot data to create a scatter plot.
 I can interpret the relationship between two variables.
 I can identify the patterns of association in the context of the data sample.
Vocabulary:
scatter plot, bivariate, clustering, outliers, positive association, negative association, linear association, non-linear association,
correlation
Sample Problem(s):
 Please go to the links in the resource section of this document. There are some interesting ideas for data collection and
analysis.
 http://www.waterbury.k12.ct.us/wmd/site/files/gr8stations.pdf
 Watch a class use the Barbie Bungee jumping activity
https://www.teachingchannel.org/videos/stem-lesson-ideas-bungee-jump?fd=1
Resources:
 Making a Scatter Plot
http://www.target.k12.mt.us/cms/lib7/MT01000812/Centricity/Domain/68/SP_Scatter_Plot_Notes_and_Practice_0.pdf
This is a printable activity. Students need to graph the points and analyze the data.
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Supporting Cluster: Investigate patterns of association in bivariate data.
Standard: 8.SP.2
Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a
linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to
the line
Explanation:
 Students understand that a straight line (trend line or line of best fit) can represent a scatter plot with linear association.
Learning Targets:
 I can recognize whether or not data plotted on a scatter plot has a linear association.
 I can draw a straight trend line to approximate the linear relationship between the points of two data sets.
 I can make inferences regarding the reliability of the trend line by noting the closeness of the data points to the line.
Vocabulary:
scatter plot, nonlinear association, trend line, line of best fit
Sample Problem(s):
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Resources:
 Birds Eggs
http://illustrativemathematics.org/illustrations/41
This task has students to identify a correlation and use it to make interpolative predictions, and reason about the properties
of bird eggs from a graph of the data.
 Scatter Plots and Lines of Best Fit Worksheet
http://shamokinmath.wikispaces.com/file/view/line%20of%20best%20fit.pdf/404034640/line%20of%20best%20fit.pdf
This is a printable activity. It includes various examples that involve everyday life. Students need to make a graph, find
the line of best fit, and make predictions.
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Supporting Cluster: Investigate patterns of association in bivariate data.
Standard:
8.SP.3
Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and
intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional
hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.
Explanation:
 Students interpret the slope and y-intercept of the line in the context of the problem.
Learning Targets:
 I can use the equation of the trend line that approximates the linear relationship between the plotted points of two data sets.
 I can interpret the y-intercept and slope of the equation in context of the situation.
Vocabulary:
linear model, bivariate, slope, y-intercept, trend line, line of best fit.
Sample Problem(s):
The capacity of the fuel tank in a car is 13.5 gallons. The table below shows the number of miles traveled and how many gallons
of gas are left in the tank. Describe the relationship between the variables. If the data is linear, determine a line of best fit. Do you
think the line represents a good fit for the data set? Why or why not? What is the average fuel efficiency of the car in miles per
gallon?
Miles Traveled
0 75 120 160 250 300
Gallons Used
0 2.3 4.5 5.7 9.7 10.7
Resources:
 Exploring Linear Data
http://illuminations.nctm.org/LessonDetail.aspx?id=L298
Lesson showing construct scatterplots of two-variable data, interpret individual data points and make conclusions about
trends in data, especially linear relationships, estimate and write equations of lines of best fit
 Barbie Bungee
http://illuminations.nctm.org/LessonDetail.aspx?id=L646
Collect data using a rubber band bungee cord and a Barbie. Use the data to construct a scatterplot and line of best fit.
 In a Heartbeat
http://www.pbs.org/teachers/connect/resources/4384/preview/
Students will gather data, create a scatter plots and analyze the data with the purpose of determining if a correlation exits.
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Supporting Cluster: Investigate patterns of association in bivariate data.
Standard: 8.SP.4
Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative
frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected
from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two
variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and
whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
Explanation:
 Students recognize that categorical data can also be described numerically through the use of a two-way table.
Learning Targets:
 I understand the patterns of association are seen in bivariate data.
 I can create a two-way table to record frequencies of bivariate categorical values.
 I can determine the relative frequencies for rows and/or columns of a two-way table.
 I can use relative frequencies and context for the problem to describe possible associations between two sets of data.
Vocabulary:
bivariate, categorical data, two-way table, frequency, relative frequency
Sample Problem(s):
For example, a survey was conducted to determine if boys eat breakfast more often than girls. The following table shows the
results:
Eat breakfast on a regular basis
Do not eat breakfast on a regular basis
Male
190
130
Female
110
165
Students can use the information from the table to compare the probabilities of males eating breakfast (190 of the 320 males is
59%) and females eating breakfast (110 of the 375 females is 29%) to answer the question. From this data, it can be determined
that males do eat breakfast more regularly than females.
The table illustrates the results when 100 students were asked the survey questions: Do you have a curfew? Do you have
assigned chores? Is there evidence that those who have a curfew also tend to have chores?
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Solution: Of the students, who answered that they had a curfew, 40 had chores and 10 did not. Of the students who answered they
did not have a curfew, 10 had chores and 40 did not. From this sample, there appears to be a positive correlation between having
a curfew and having chores.
Resources:
 Two-Way tables lesson
http://stattrek.com/statistics/two-way-table.aspx
Lesson on bivariate data and video
 What‘s Your favorite Subject
http://illustrativemathematics.org/illustrations/973
Data that can be used for students to practice using relative frequencies calculated for rows or columns to describe
possible association between the two variables.
 Music & Sports
http://illustrativemathematics.org/illustrations/1098
This activity can be used for students to gather bivariate data, make a two-way chart and analyze the meaning
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Common Core State Documents Referenced
In addition to the resources listed within the curriculum guide, the following state websites were utilized:
Arizona Department of Education – www.azed.gov/azcommoncore
Delaware Department of Education – http://www.doe.k12.de.us
Illinois State Board of Education – http://www.isbe.net
New York Department of Education – www.engageny.org
South Carolina Department of Education – http://ed.sc.gove/core
Utah Department of Education – www.schools.utah.gov/core
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