Download 7th Grade Units - Alton School District

Illinois Model Curriculum Scope & Sequence
Grade/Course: _7th Grade Math_________
Unit
Core Standards
Supporting Standards
Approximate
Time Frame
4-5 weeks
1) Ratios &
Proportionality
7.RP. 1
7.RP.2.
7.NS.1
7.NS.2
7.NS.3
7.EE.2
2) Ratio &
Proportion
Applications
7.RP.3
7.G.1
7.NS.2
7.NS.3
7.EE.2
7.EE.3
2-3 weeks
3) Rational
Number
Operations
4) Expressions
7.NS.1
7.NS.2
7.NS.3
7.EE.1
7.EE.2
7.EE.2
5-6 weeks
7.NS.1
7.NS.2
2-3 weeks
5) Equations
7.EE.3
7.EE.4
7.G.4
7.SP.1
7.SP.2
7.SP.3
7.SP.4
7.NS.1
7.NS.2
4-5 weeks
7.NS.1
7.NS.2
7.NS.3
7.EE.2
7.EE.3
7.NS.1
7.NS.2
7.NS.3
7.EE.2
7.EE.3
7.EE.2
7.EE.3
7.EE.4
3-4 weeks
6) Data
Distributions
7) Probability
7.SP.5
7.SP.6
7.SP.7
7.SP.8
8) Geometric
Measurement
7.G.2
7.G.3
7.G.5
7.G.6
2-3 weeks
4-5 weeks
Grade 7:
Unit 1: Ratios & Proportionality
Approximate time frame: 4 – 5 weeks
Connections to Previous Learning:
Students in Grade 6 learn the concepts of ratio and unit rate as well as the precise mathematical language used to describe these relationships. They learn to
solve problems using ratio and rate reasoning using a variety of tools such as tables, tape diagrams, double number lines and equations.
Focus of this Unit:
Students apply concepts of ratio and unit rate learned in grade 6 to fluently compute unit rates, represent proportional relationships between quantities, and
compare and contrast proportional relationships in real world contexts.
Connections to Subsequent Learning:
Students will apply their understanding of ratios and proportionality to situations involving multi-step ratio and percent problems as well as scale drawings.
From the 6-7 Ratios and Proportional Reasoning Progression Document, pp. 8-10:
In Grade 7, students extend their reasoning about ratios and proportional relationships in several ways. Students use ratios in cases that involve pairs of rational
number entries, and they compute associated unit rates. They identify these unit rates in representations of proportional relationships. They work with
equations in two variables to represent and analyze proportional relationships. They also solve multi-step ratio and percent problems, such as problems
involving percent increase and decrease.
At this grade, students will also work with ratios specified by rational numbers, such as Students continue to use ratio tables, extending this use to finding unit
rates.
4/20/2013 5:14:52 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 1
Additional Standards = Approximately 10%
Grade 7:
Unit 1: Ratios & Proportionality
Recognizing proportional relationships Students examine situations carefully, to determine if they describe a proportional relationship. For example, if Josh is 10
and Reina is 7, how old will Reina be when Josh is 20? We cannot solve this problem with the proportion 10/7 = 20/R because it is not the case that for every 10
years that Josh ages, Reina ages 7 years. Instead, when Josh has aged 10 another years, Reina will as well, and so she will be 17 when Josh is 20.
For example, if it takes 2 people 5 hours to paint a fence, how long will it take 4 people to paint a fence of the same size (assuming all the people work at the
same steady rate)? We cannot solve this problem with the proportion 2/5 = 4/h because it is not the case that for every 2 people, 5 hours of work are needed to
paint the fence. When more people work, it will take fewer hours. With twice as many people working, it will take half as long, so it will take only 2.5 hours for 4
people to paint a fence. Students must understand the structure of the problem, which includes looking for and understand the roles of “for every,” “for each,”
and “per.”
Students recognize that graphs that are not lines through the origin and tables in which there is not a constant ratio in the entries do not represent proportional
relationships. For example, consider circular patios that could be made with a range of diameters. For such patios, the area (and therefore the number of pavers
it takes to make the patio) is not proportionally related to the diameter, although the circumference (and therefore the length of stone border it takes to
encircle the patio) is proportionally related to the diameter. Note that in the case of the circumference, C, of a circle of diameter D, the constant of
proportionality in C = pi D is the number pi, which is not a rational number.
Equations for proportional relationships As students work with proportional relationships, they write equations of the form y = cx, where c is a constant of
proportionality, i.e., a unit rate. They see this unit rate as the amount of increase in y as x increases by 1 unit in a ratio table and they recognize the unit rate as
the vertical increase in a “unit rate triangle” or “slope triangle” with horizontal side
of length 1 for a graph of a proportional relationship.
Students connect their work with equations to their work with tables and
diagrams. For example, if Seth runs 5 meters every 2 seconds, then how long will it
take Seth to run 100 meters at that rate? The traditional method is to formulate an
equation, 5/2 = 100/T, cross-multiply, and solve the resulting equation to solve the
problem. If 5/2 and 100/T are viewed as unit rates obtained from the equivalent
ratios 5 : 2 and 100 : T , then they must be equivalent fractions because equivalent
ratios have the same unit rate. To see the rationale for cross-multiplying, note that
when the fractions are given the common denominator 2 •T, then the numerators
become 5 • T and 2 • 100 respectively. Once the denominators are equal, the
fractions are equal exactly when their numerators are equal, so 5•Tmust equal 2
•100 for the unit rates to be equal. This is why we can solve the equation 5 •T = 2
•100 to find the amount of time it will take for Seth to run 100 meters.
4/20/2013 5:14:52 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 2
Additional Standards = Approximately 10%
Grade 7:
Unit 1: Ratios & Proportionality
A common error in setting up proportions is placing numbers in incorrect locations. This is especially easy to do when the order in which quantities are stated in
the problem is switched within the problem statement. For example, the second of the following two problem statements is more difficult than the first because
of the reversal.
“If a factory produces 5 cans of dog food for every 3 cans of cat food, then when the company produces 600 cans of dog food, how many cans of cat food will it
produce?”
“If a factory produces 5 cans of dog food for every 3 cans of cat food, then how many cans of cat food will the company produce when it produces 600 cans of
dog food?”
Such problems can be framed in terms of proportional relationships and the constant of proportionality or unit rate, which is obscured by the traditional method
of setting up proportions. For example, if Seth runs 5 meters every 2 seconds, he runs at a rate of 2.5 meters per second, so distance d (in meters) and time t (in
seconds) are related by d = 2:5t. If d=100 then t = 100/ 2:5 =40, so he takes 40 seconds to run 100 meters.
Desired Outcomes
Standard(s):
Analyze proportional relationships and use them to solve real-world and mathematical problems
7.RP.1 Compute unit rates associates with rations of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. For
example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.
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.
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
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.
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.
4/20/2013 5:14:52 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 3
Additional Standards = Approximately 10%
Grade 7:
Unit 1: Ratios & Proportionality
Supporting Standards:
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.
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.
c) Understand subtraction of rational numbers as adding 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.
d) Apply properties of operations as strategies to add and subtract rational numbers.
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.
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.
c) Apply properties of operations as strategies 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.
7. NS.3 Solve real-world and mathematical problems involving the four operations with rational number.
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.”
WIDA Standard: (English Language Learners)
English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.
English Language Learners benefit from:
• visual representations such as double number lines and bar models for representing and solving problems involving proportional relationships.
• attention to the language of problems to determine ratio and rate representations appropriate to given contexts.
Understandings: Students will understand that …
•
•
•
Rates, ratios, percentages and proportional relationships express how quantities change in relationship to each other.
Rates, ratios, percentages and proportional relationships can be represented in multiple ways.
Rates, ratios, percentages and proportional relationships can be applied to problem solving situations.
4/20/2013 5:14:52 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 4
Additional Standards = Approximately 10%
Grade 7:
Unit 1: Ratios & Proportionality
Essential Questions:
•
•
•
•
How do rates, ratios, percentages and proportional relationships apply to our world?
When and why do I use proportional comparisons?
How does comparing quantities describe the relationship between them?
How do graphs illustrate proportional relationships?
Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)
*1. Make sense of problems and persevere in solving them. Students make sense of ratio and unit rates in real-world contexts. They persevere by selecting
and using appropriate representations for the given contexts.
*2. Reason abstractly and quantitatively. Students will reason about the value of the rational number in relation the models that are created to represent
3.
*4.
5.
*6.
7.
8.
them.
Construct viable arguments and critique the reasoning of others.
Model with mathematics. Students create models using tape diagrams, double number lines, manipulatives, tables and graphs to represent real-world and
mathematical situations involving ratios and proportions. For example, students will examine the relationships between slopes of lines and ratio tables in
the context of given situations.
Use appropriate tools strategically.
Attend to precision. Students attend to the ratio and rate language studied in grade 6 to represent and solve problems involving rates and ratios.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
4/20/2013 5:14:52 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 5
Additional Standards = Approximately 10%
Grade 7:
Unit 1: Ratios & Proportionality
Advanced Skills/Concepts:
Prerequisite Skills/Concepts:
Students should already be able to:
•
Understand ratio concepts and use ratio reasoning to solve
problems. (6.RP 1-3)
Some students may be ready to:
•
•
•
•
Knowledge: Students will know…
Skills: Students will be able to …
•
All standards in this unit go beyond the knowledge level.
•
•
•
•
•
•
•
4/20/2013 5:14:52 PM
Priority Standards = Approximately 70%
Analyze proportional relationships and use them to solve real-world and
mathematical problems. (Compute unit rates. Recognize, represent and
explain proportional relationships using tables, graphs, equations, diagrams
and verbal descriptions. Use proportional relationships to solve multi-step
ratio and percent problems.) (7.RP.1-3)
Apply and extend previous understandings of operations with fractions to add,
subtract, multiply, and divide rational numbers. (7.NS.1-3)
Apply properties of operations as strategies to add, subtract, factor and
expand linear expressions with rational coefficients. (7.EE.1)
Solve real-life and mathematical problems using numerical and algebraic
expressions and equations. (7.EE.3-4)
Compute unit rates involving rational numbers, fractions, and complex fractions.
(7.RP.1)
Compute ratios of length in like or different units. (7.RP.1)
Compute ratios of area and other measurements in like or different units. (7.RP.1)
Determine whether two quantities are in a proportional relationship by using a
table and or graph. (7.RP.2)
Identify the constant of proportionality (unit rate) in tables, graphs, diagrams, and
verbal descriptions. (7.RP.2)
Create and solve equations to represent proportional relationships. (7.RP.2)
Use words to describe the location of a point on a graph and its relationship to the
origin. (7.RP.2)
Explain what a point on a graph of a proportional relationship means in terms of
the situation. (how does the one quantity relate to the other) (7.RP.2)
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 6
Additional Standards = Approximately 10%
Grade 7:
Unit 1: Ratios & Proportionality
Academic Vocabulary:
Critical Terms:
Supplemental Terms:
Simple Interest
Percent increase
Percent decrease
Commission
Percent error
Rate of change
Gratuity
Tax
Tip
Ratio
Rate
Proportion
Percent
Unit rate
Equivalency
Greatest Common Factor (GCF)
Least Common Multiple (LCM)
4/20/2013 5:14:52 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 7
Additional Standards = Approximately 10%
Grade 7:
Unit 2: Ratio & Proportion Applications
Time Frame: Approximately 2-3 Weeks
Connections to previous learning:
In Grade 6, students develop an understanding of ratio and proportion using ratio tables, tape diagrams, and double number lines.
Focus of this unit:
Students extend their understanding of ratios and develop understanding of proportionality to solve single and multi-step problems involving such real world
contexts as percent of increase or decrease and scale drawing.
Connections to Subsequent Learning:
Students will use this understanding of proportionality to find scale factors between geometric figures and develop understandings of congruence and similarity.
They will use ratio tables to study statistics and probability.
From the 6-7, Ratios and Proportional Relationships Progression Document, pp.10-11:
Multistep problems Students extend their work to solving multistep ratio and percent problems.7.RP.3 Problems involving percent increase or percent decrease
require careful attention to the referent whole. For example, consider the difference in these two percent decrease and percent increase problems:
Skateboard problem 1. After a 20% discount, the price of a SuperSick skateboard is $140. What was the price before the discount?
Skateboard problem 2. A Super Sick skateboard costs $140 now, but its price will go up by 20%. What will the new price be after the increase?
The solutions to these two
problems are different
because the 20% refers to
different wholes or 100%
amounts. In the first
problem, the 20% is 20% of
the larger pre-discount
amount, whereas in the
second problem, the 20% is
20% of the smaller preincrease amount.
Notice that the distributive
property is implicitly involved
in working with percent
decrease and increase. For example, in the first problem, if x is the original price of the skateboard (in dollars), then after the 20% discount, the new price is x20% x. The distributive property shows that the new price is 80% x:
x- 20% x = 100% x - 20% x = (100% - 20%) x = 80% x
4/20/2013 5:14:54 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 1
Additional Standards = Approximately 10%
Grade 7:
Unit 2: Ratio & Proportion Applications
Percentages can also be used in making comparisons between two quantities. Students must attend closely to the wording of such problems to determine what
the whole or 100% amount a percentage refers to.
Connection to Geometry: One new context for proportions at Grade 7 is scale drawings. To compute unknown lengths from known lengths, students can set up
proportions in tables or equations, or they can reason about how lengths compare multiplicatively. Students can use two kinds of multiplicative comparisons.
They can apply a scale factor that relates lengths in two different figures, or they can consider the ratio of two lengths within one figure, find a multiplicative
relationship between those lengths, and apply that relationship to the ratio of the corresponding lengths in the other figure. When working with areas, students
should be aware that areas do not scale by the same factor that relates lengths. (Areas scale by the square of the scale factor that relates lengths, if area is
measured in the unit of measurement derived from that used for length.)
Desired Outcomes
Standard(s):
Analyze proportional relationships and use them to solve real-world and mathematical problems.
7. RP.3 Use proportional relationships to solve multi-step ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities
and commissions, fees, percent increase and decrease, percent error.
Draw, construct, and describe geometrical figures and describe the relationships between them.
7.G.1 Solve problems involving scale drawing of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a
scale drawing in a different scale.
4/20/2013 5:14:54 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 2
Additional Standards = Approximately 10%
Grade 7:
Unit 2: Ratio & Proportion Applications
Supporting Standards:
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.
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/1) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real-world contexts.
c) Apply properties of operations as strategies 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.
7.NS.3 Solve real-world and mathematical problems involving the four operations with rational number.
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.”
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 ¾ inches long in the center of a door
that is 27 ½ 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.
WIDA Standard: (English Language Learners)
English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.
English language learners will benefit from:
• hands-on and virtual manipulative experiences using geometric figures and fraction, decimal, percent equivalence tools.
• explicit vocabulary instruction with regard to ratio and percent situations.
Understandings: Students will understand that …
•
•
•
•
Rates, ratios, percentages and proportional relationships can be applied to problem solving situations such as interest, tax, discount, etc.
Rates, ratios, percentages and proportional relationships can be applied to solve multi-step ratio and percent problems.
Scale drawings can be applied to problem solving situations involving geometric figures.
Geometrical figures can be used to reproduce a drawing at a different scale.
4/20/2013 5:14:54 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 3
Additional Standards = Approximately 10%
Grade 7:
Unit 2: Ratio & Proportion Applications
Essential Questions:
•
•
•
How can I use proportional relationships to solve ratio and percent problems?
How can I use scale drawings to compute actual lengths and area?
How can I use geometric figures to reproduce a drawing at a different scale?
Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)
*1. Make sense of problems and persevere in solving them. Students exhibit this standard when they represent and interpret proportional relationships to
*2.
3.
*4.
5.
6.
7.
8.
solve ratio and percent problems using visual models, proportions and other equations. They also make sense of proportional situations that involve scale
drawings using diagrams and equations.
Reason abstractly and quantitatively. Students will reason about the value of rational numbers in real-world contexts when representing and solving
problems. They will apply proportional reasoning to scale drawings and determine if calculations are appropriate to the contexts.
Construct viable arguments and critique the reasoning of others. Students will be expected to articulate their problem solving processes and explain the
connection between the various representations (visual, tabular, algebraic, real-life) used to solve problems involving scale drawings and other proportional
relationships.
Model with mathematics. Students will use double number lines, tape diagrams and ratio charts to represent real-world situations involving proportional
relationships.
Use appropriate tools strategically. Students are expected to select appropriate measurement and construction tools when reproducing scale drawings.
Attend to precision. Students will attend to the units of measurement when solving and representing problems with scale drawings. They will also attend
to precise mathematical language when interpreting and communicating about problem solving with ratio and percent situations.
Look for and make use of structure. Students look for patterns when solving proportional relationships and interpreting scale drawings.
Look for and express regularity in repeated reasoning. Students use repeated reasoning when they replicate drawings at different scales.
Prerequisite Skills/Concepts:
Advanced Skills/Concepts:
Students should already be able to:
Some students may be ready to:
•
•
•
Understand ratio concepts and use ratio reasoning to solve
problems. (6.RP.1-3)
Compute unit rates associated with rations of fractions,
including ratios of lengths, areas, and other quantities
measured in like or different units. (7.RP.1)
Recognize and represent proportional relationships between
quantities. (7.RP.2)
4/20/2013 5:14:54 PM
Priority Standards = Approximately 70%
•
•
Understand the connections between proportional relationships to interpret
unit rate as the slope of the graph (8.EE.5)
Proportional relationships can be applied to solve congruence and similarity
applications. (8.G.2)
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 4
Additional Standards = Approximately 10%
Grade 7:
Unit 2: Ratio & Proportion Applications
Knowledge: Students will know…
Skills: Students will be able to…
All standards for this unit go beyond the knowledge level.
•
•
•
•
•
•
•
Solve multi-step ratio and percent problems. (7.RP.3)
Solve problems involving simple interest and tax. (7.RP.3)
Solve problems involving markups and markdowns, gratuities and commissions,
and fees. (7.RP.3)
Solve problems involving percent increase, percent decrease, and percent (margin
of) error. (7.RP.3)
Solve problems involving scale drawings of geometric figures. (7.G.1)
Compute actual lengths and areas from a scale drawing. (7.G.1)
Reproduce a scale drawing at a different scale. (7.G.1)
Academic Vocabulary:
Critical Terms:
Supplemental Terms:
Ratio
Proportion
Percent increase
Percent decrease
Percent error
Markdowns
Markups
Scale
Tax
Gratuity
Area
Volume
Simple interest
equivalent
4/20/2013 5:14:54 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 5
Additional Standards = Approximately 10%
Grade 7:
Unit 3: Rational Number Operations
Time Frame: Approximately 5-6 Weeks
Connections to Previous Learning: In Grade 6, students develop an understanding of rational numbers by using vertical and horizontal number lines and by
applying their previous knowledge of whole numbers and integers to rational number systems.
Students have a great deal of background in operations from K-6 that will lead to achievement of the expectations in this unit. In the primary grades, students
build understanding of whole number addition and subtraction. In the intermediate grades, students build understanding of multiplication and division with
whole numbers and addition, subtraction, multiplication and division with fractions and decimals. 6th grade students learn to divide fractions by fractions with
understanding and work to build an understanding of the value of integers. All of these concepts are crucial for students to be successful in this unit on
operations with rational numbers. In the past, students have used number line models with whole numbers, fractions and decimals. 6th graders have
represented integers on vertical and horizontal number line diagrams, but this is their first experience in adding and subtracting positive and negative rational
numbers.
Focus of this Unit: Students will be able to add, subtract, multiply, and divide rational numbers fluently as well as solve real-world and mathematical problems.
In this unit, students will represent their calculations using number line models, equations or expressions, and real world applications.
Connections to Subsequent Learning: Students will use their understanding of rational numbers and apply them to linear functions.
4/20/2013 5:14:55 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 1
Additional Standards = Approximately 10%
Grade 7:
Unit 3: Rational Number Operations
Desired Outcomes
Standard(s):
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
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.
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.
c) Understand subtraction of rational numbers as adding 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.
d) Apply properties of operations as strategies to add and subtract rational numbers.
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.
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/1) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real-world contexts.
c) Apply properties of operations as strategies 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.
7. NS.3 Solve real-world and mathematical problems involving the four operations with rational number.
Supporting 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.”
WIDA Standard: (English Language Learners)
English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.
English language learners benefit from:
• The use of manipulatives for developing understanding of operations with positive and negative numbers.
• Attention to mathematical vocabulary when solving real-world and mathematical problems involving positive and negative numbers.
4/20/2013 5:14:55 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 2
Additional Standards = Approximately 10%
Grade 7:
Unit 3: Rational Number Operations
Understandings: Students will understand that …
•
•
•
•
•
Rational numbers use the same properties as whole numbers.
Rational numbers can be used to represent and solve real-life situation problems.
Rational numbers can be represented with visuals (including distance models), language, and real-life contexts.
A number line model can be used to represent the unique placement of any number in relation to other numbers.
There are precise terms and sequence to describe operations with rational numbers.
Essential Questions:
•
•
How are rational numbers used and applied in real-life and mathematical situations?
What is the relationship between properties of operations and types of numbers?
Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)
*1. Make sense of problems and persevere in solving them. Students explain and demonstrate rational number operations by using symbols, visuals, words,
and real life contexts. Students demonstrate perseverance while using a variety of strategies. (number lines, manipulatives, drawings, etc.)
2.
* Reason abstractly and quantitatively. Students demonstrate quantitative reasoning by representing and solving real world situations using visuals,
numbers, and symbols. They demonstrate abstract reasoning by translating numerical sentences into real world situations.
*3. Construct viable arguments and critique the reasoning of others. Students will discuss rules for operations with rational numbers using appropriate
terminology and tools/visuals. Students apply properties to support their arguments and constructively critique the reasoning of others while supporting
their own position.
4. Model with mathematics. Students model understanding of rational number operations using tools such as algebra tiles, counters, visuals, and number
lines and connect these models to solve problems involving real-world situations.
5. Use appropriate tools strategically. Students demonstrate their ability to select and use the most appropriate tool (paper/pencil, manipulatives, and
calculators) while solving problems with rational numbers.
6. Attend to precision. Students demonstrate precision by using correct terminology and symbols and labeling units correctly. Students use precision in
calculation by checking the reasonableness of their answers and making adjustments accordingly.
7. Look for and make use of structure. Students look for structure in positive and negative rational numbers when they place them appropriately on the
number line. They use this structure in calculation when considering the position of numbers on the number line. In addition, students recognize the
problem solving structures of word problems and use this awareness to aid in solving them.
8. Look for and express regularity in repeated reasoning. Students will use manipulatives to explore the patterns of operations with rational numbers.
Students will use these patterns to develop algorithms. They can use these algorithms to solve problems with a variety of problem solving structures.
4/20/2013 5:14:55 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 3
Additional Standards = Approximately 10%
Grade 7:
Unit 3: Rational Number Operations
Advanced Skills/Concepts:
Prerequisite Skills/Concepts:
Some students may be ready to:
Students should already be able to:
•
Develop a conceptual understanding of positive and negative
numbers. (6.NS.5-6)
•
•
•
•
•
•
Know that there are numbers that are not rational, and approximate them by
rational numbers. (8.NS.1-2)
Interpret and apply positive and negative slopes of lines and positive and negative
coefficients in equations; develop understanding of square roots and irrational
numbers.
Understand relationships between positive and negative coefficients or values for
variables; use positive and negative integers to communicate directions in two
dimensions.
Evaluate algebraic expressions involving positive and negative coefficients or values
for variables; interpret isometrics in the plane given in symbolic form.
Graph equations on coordinate grids; locate square roots on the number line.
Use the properties and order of operations to write equivalent expressions and
solve equations.
Knowledge: Students will know…
Skills: Students will be able to …
All standards in this unit go beyond the knowledge level.
•
•
•
•
•
•
•
•
•
•
•
4/20/2013 5:14:55 PM
Priority Standards = Approximately 70%
Add and subtract rational numbers. (7.NS.1)
Represent addition and subtraction on a horizontal or vertical number line
diagram. (7.NS.1)
Use words, visuals and symbols to describe situations in which opposite quantities
combine to make 0. (7.NS.1)
Represent addition of quantities with symbols, visuals and words by showing
positive or negative direction from one quantity to the other. (7.NS.1)
Show that a number and its opposite have a sum of 0 using visuals, symbols, words
and real-world contexts. (7.NS.1)
Use the term “additive inverse” to describe 2 numbers whose sum is zero. (7.NS.1)
Use commutative, distributive, associative, identity, and inverse properties to add
and subtract rational numbers. (7.NS.1)
Use the term “absolute value” to describe the distance from zero on number line
diagram and with symbols. (7.NS.1)
Multiply and divide rational numbers. (7.NS.2)
Use the distributive property to multiply positive and negative rational numbers
using symbols, visuals, words and real-life contexts. (7.NS.2)
Interpret products of rational numbers by describing real-world contexts. (7.NS.2)
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 4
Additional Standards = Approximately 10%
Grade 7:
Unit 3: Rational Number Operations
•
•
•
•
•
•
4/20/2013 5:14:55 PM
Priority Standards = Approximately 70%
Identify situations when integers can and cannot be divided. (7.NS.2)
Use words and real-world contexts to explain why the quotient of two integers is a
rational number. (7.NS.2)
Identify and apply properties used when multiplying and dividing rational numbers.
(7.NS.2)
Convert a rational number to a decimal using long division. (7.NS.2)
Identify terminating or repeating decimal representations of rational numbers.
(7.NS.2)
Solve real world and mathematical problems involving the four operations with
rational numbers. (7.NS.3)
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 5
Additional Standards = Approximately 10%
Grade 7:
Unit 3: Rational Number Operations
Academic Vocabulary:
Critical Terms:
Supplemental Terms:
Commutative Property
Distributive Property
Integers
Negative Numbers
Opposites
Positive Numbers
Income/Profit
Absolute Value
Additive Inverse
Associative Property
Expanding
Factoring
Quadrant I, II, III, IV
Order of Operations
Rational Numbers
Area
Coordinate Grid
Decimals
Expressions
Fact Family
Fractions
Mathematical Sentence
Number Line
Number Sentence
Operations
Ordered Pair
Variable
4/20/2013 5:14:55 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 6
Additional Standards = Approximately 10%
Grade 7:
Unit 4: Expressions
Time Frame: Approximately 2-3 weeks
Connections to Previous Learning:
In 6th grade, students read, write and evaluate numerical expressions involving variables and whole number exponents. They apply properties of operations
using the appropriate order of operations to generate equivalent expressions.
Focus of this unit:
Students simplify general linear expressions that involve rational coefficients and distribute negative numbers to solve real world and mathematical problems.
Connections to Subsequent Learning:
A more complete understanding of order of operations and their properties will lay the foundation for the extensive study of functions next year.
From the 6-8, Expressions and Equations Progression Document pp. 8-9:
Use properties of operations to generate equivalent expressions In Grade 7 students start to simplify general linear expressions with rational coefficients.
Building on work in Grade 6, where students used conventions about the order of operations to parse, and properties of operations to transform, simple
expressions such as 2(3 + 8x) or 10 – 2p, students now encounter linear expressions with more operations and whose transformation may require an
understanding of the rules for multiplying negative numbers, such as 7 – 2(3 – 8x).
In simplifying this expression students might come up with answers such as:
• 5(3 – 8x), mistakenly detaching the 2 from the indicated multiplication.
• 7 – 2(-5x), through a determination to perform the computation in parentheses first, even though no simplification is possible.
• 7 – 6 – 16x, through an imperfect understanding of the way the distributive law works or of
the rules for multiplying negative numbers.
In contrast with the simple linear expressions they see in Grade 6, the more complex expressions
student seen in Grade 7 afford shifts of perspective, particularly because of their experience with
negative numbers: for example, students might see 7 – 2(3 – 8x) as 7 – (2(3 – 8x)) or as
7 + (-2)(3 + (-8)x) (MP7).
As students gain experience with multiple ways of writing an expression, they also learn that
different ways of writing expressions can serve different purposes and provide different ways of
seeing a problem. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as
“multiply by 1.05.” In the example on the right, the connection between the expressions and the
figure emphasize that they all represent the same number, and the connection between the
structure of each expression and a method of calculation emphasize the fact that expressions are
built up from operations on numbers.
4/20/2013 5:14:56 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 1
Additional Standards = Approximately 10%
Grade 7:
Unit 4: Expressions
Desired Outcomes
Standard(s):
Use properties of operations to generate equivalent expressions
7.EE.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
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.”
Supporting Standards:
7.NS1 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.
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.
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.
d) Apply properties of operations as strategies to add and subtract rational numbers.
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.
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.
c) Apply properties of operations as strategies 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
WIDA Standard: (English Language Learners)
English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.
English language learners benefit from:
• explicit vocabulary instruction with regard to the components of an algebraic expression.
4/20/2013 5:14:56 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 2
Additional Standards = Approximately 10%
Grade 7:
Unit 4: Expressions
Understandings: Students will understand that …
•
•
•
•
•
•
Variables can be used to represent numbers in any type of mathematical problem.
Understand the difference between an expression and an equation.
Expressions you simplify and equations you solve for the variable’s value.
Write and solve multi-step equations including all rational numbers.
Some equations may have more than one solution and understand inequalities.
Properties of operations allow us to add, subtract, factor, and expand linear expressions.
Essential Questions:
•
•
When and how are expressions, equations, inequalities and graphs applied to real world situations?
How can the order of operations be applied to evaluating expressions, and solving from one-step to multi-step equations?
Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)
1. Make sense of problems and persevere in solving them.
*2. Reason abstractly and quantitatively. Students demonstrate quantitative reasoning by representing and solving real world situations using visuals,
*3.
*4.
*5.
*6.
7.
8.
numbers, and symbols. They demonstrate abstract reasoning by translating expressions, equations, inequalities and linear relationships into real world
situations.
Construct viable arguments and critique the reasoning of others. Students will discuss the differences among expressions, equations and inequalities
using appropriate terminology and tools/visuals. Students will apply their knowledge of equations and inequalities to support their arguments and critique
the reasoning of others while supporting their own position.
Model with mathematics. Students will model an understanding of expressions, equations, inequalities, and graphs using tools such as algebra tiles/blocks,
counters, protractors, compasses, and visuals to represent real world situations.
Use appropriate tools strategically. Students demonstrate their ability to select and use the most appropriate tool (pencil/paper, manipulatives,
calculators, protractors, etc.) while simplifying/evaluating/analyzing expressions, solving equations and representing and analyzing linear relationships.
Attend to precision. Students demonstrate precision by correctly using numbers, variables and symbols to represent expressions, equations and linear
relationships, and correctly label units. Students use precision in calculation by checking the reasonableness of their answers and making adjustments
accordingly. Students will use appropriate geometric language to describe and label figures.
Look for and make use of structure. to solve for the variable. Students will also examine the relationship between the structure of a circle and the
formulas for area and circumference.
Look for and express regularity in repeated reasoning.
4/20/2013 5:14:56 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 3
Additional Standards = Approximately 10%
Grade 7:
Unit 4: Expressions
Advanced Skills/Concepts:
Prerequisite Skills/Concepts:
Some students may be ready to:
Students should already be able to:
•
•
Apply and extend previous understandings of arithmetic to
algebraic expressions. (6.EE.1-4)
Add, subtract multiply and divide positive and negative numbers.
(7.NS.1-2)
Knowledge: Students will know…
Solve real-life and mathematical problems using numerical and algebraic
expressions and equations. (7.EE.3-4)
•
Skills: Students will be able to …
•
All standards in this unit go beyond the knowledge level.
•
•
•
•
Use Commutative, Associative, Distributive, Identity, and Inverse Properties to
add and subtract linear expressions with rational coefficients. (7.EE.1)
Use Commutative, Associative, Distributive, Identity, and Inverse Properties to
factor and expand linear expressions with rational coefficients. (7.EE.1)
Rewrite an expression in a different form. (7.EE.2)
Choose the form of an expression that works best to solve a problem. (7.EE.2)
Explain your reasoning for the choice of expression used to solve a problem.
(7.EE.2)
Academic Vocabulary:
Critical Terms:
Supplemental Terms:
Distributive Property
Commutative Property
Associative Property
Multiplicative Property of Zero
Variable
Numerical expression
Algebraic expression
Term
Coefficient
Constant
Equation
Inequality
Algebra
Property
Order of operations
Evaluate
Simplest form
4/20/2013 5:14:56 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 4
Additional Standards = Approximately 10%
Grade 7:
Unit 5: Equations
Approximate Time Frame: 4-5 weeks
Connections to Previous Learning:
In grade 6, students learned to read, write, interpret and solve one-variable equations and inequalities in real-life and mathematical situations.
Focus of this Unit:
Students will read, write, interpret and solve multi-step real-life and mathematical problems using algebraic and numerical expressions, equations and
inequalities.
Connections to Subsequent Learning:
Understanding that equations can have multiple solutions will lay a foundation for the study of solving systems of simultaneous linear equations.
From the 6-8, Expressions and Equations Progression Document, pp. 8-10:
Use properties of operations to generate equivalent expressions: In Grade 7 students start to simplify general linear expressions with rational coefficients.
Building on work in Grade 6, where students used conventions about the order of operations to parse, and
properties of operations to transform, simple expressions such as 2(3+8x) or 10-2p, students now encounter
linear expressions with more operations and whose transformation may require an understanding of the rules
for multiplying negative numbers, such as 7 − 2 3 − 8 .
In simplifying this expression students might come up with answers such as
• 5(3 − 8 ), mistakenly detaching the 2 from the indicated multiplication
• 7 − 2(−5 ), through a determination to perform the computation in parentheses first, even though no
simplification is possible
• 7 − 6 − 16 , through an imperfect understanding of the way the distributive law works or of the rules
for multiplying negative numbers.
In contrast with the simple linear expressions they see in Grade 6, the more complex expressions students seen
in Grade 7 afford shifts of perspective, particularly because of their experience with negative numbers: for
example, students might see 7 − 2(3 − 8 ) as 7 − 2(3 − 8 ) or as 7 + (−2)(3 + (−8) )
As students gain experience with multiple ways of writing an expression, they also learn that different ways of writing expressions can serve different
purposes and provide different ways of seeing a problem. For example, + 0.05 − 1.05 means that “increase by 5%” is the same as “multiply by 1:05.” In the
example on the right, the connection between the expressions and the figure emphasize that they all represent the same number, and the connection between
the structure of each expression and a method of calculation emphasize the fact that expressions are built up from operations on numbers.
Solve real-life and mathematical problems using numerical and algebraic expressions and equations: By Grade 7 students start to see whole numbers,
integers, and positive and negative fractions as belonging to a single system of rational numbers, and they solve multi-step problems involving rational numbers
presented in various forms. Students use mental computation and estimation to assess the reasonableness of their solutions. For example, the following
statement appeared in an article about the annual migration of the Bartailed Godwit from Alaska to New Zealand:
4/20/2013 5:14:58 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 1
Additional Standards = Approximately 10%
Grade 7:
Unit 5: Equations
She had flown for eight days – nonstop – covering approximately 7,250 miles at an average speed of
nearly 35 miles per hour.
Students can make the rough mental estimate: 8 × 24 × 35 − 8 × 12 × 70 < 100 × 70 − 700 to
recognize that although this astonishing statement is in the right ballpark, the average speed is in fact
greater than 35 miles per hours, suggesting that one of the numbers in the article must be wrong.
As they build a systematic approach to solving equations in one variable, students continue to
compare arithmetical and algebraic solutions to work problems. For example they solve the problem.
The perimeter of a rectangle is 54cm and dividing by 2, and also by setting up the equation.
2 + 2 ∙ 6 = 54
The steps in solving the equation mirror the steps in the numerical solution. As problems get more
complex, algebraic methods become more valuable. For example, in the cyclist problem in the margin, the
numerical solution requires some insight in order to keep the cognitive load of the calculations in check.
By contrast, choosing the letter s to stand for the unknown speed, students build an equation by adding the distances travelled in three hours (3 3 ∙ 12.5)
and setting them equal to 63 to get
3 + 3 ∙ 12.5 − 63
It is worthwhile exploring two different possible next steps in the solution of this equation:
3 + 37.5 − 64and3( + 12.5) − 63
The first is suggested by a standard approach to solving linear equations; the second is suggested by a comparison with the numerical solution described earlier.
Students also set up and solve inequalities, recognizing the ways in which the process of solving them is similar to the process of solving linear
equations:
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 solution.
Students also recognize one important new consideration in solving inequalities: multiplying or dividing both sides of an inequality by a negative number
reverses the order of the comparison it represents.
It is useful to present a context that allows students to make sense of this. For example,
If the price of a ticket to a school concert is p dollars then the attendance is 1000 − 50p. What range of prices ensures that at least 600 people attend?
Students recognize that the requirement of at least 600 people leads to inequality1000 − 50 ≥ 600. Before solving the inequality they use common sense to
anticipate that that answer will be of form ≤?, since higher prices result in lower attendance. Note that inequalities using ≤
≥ are included in this
standard in addition to>
<.
4/20/2013 5:14:58 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 2
Additional Standards = Approximately 10%
Grade 7:
Unit 5: Equations
Desired Outcomes
Standard(s):
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
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 ¾ inches long in the center of a door
that is 27 ½ 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.
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?
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.
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
7.G.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between
the circumference and area of a circle.
4/20/2013 5:14:58 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 3
Additional Standards = Approximately 10%
Grade 7:
Unit 5: Equations
Supporting Standards:
7.NS1 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.
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.
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.
d) Apply properties of operations as strategies to add and subtract rational numbers.
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.
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.
c) Apply properties of operations as strategies 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
WIDA Standard: (English Language Learners)
English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.
English language learners benefit from:
• Explicit vocabulary instruction regarding the structure and elements of equations.
• The use of area models for working with multiplicative relationships.
Understandings: Students will understand that …
•
•
•
•
Expressions can be manipulated to suit a particular purpose to solve problems efficiently.
Mathematical expressions, equations, inequalities and graphs are used to represent and solve real-world and mathematical problems.
Properties, order of operations, and inverse operations are used to simplify expressions and solve equations efficiently.
Algebra is applied when solving geometric problems (i.e. circumference and area of a circle).
Essential Questions:
• When and how are expressions, equations, inequalities and graphs applied to real world situations?
• What are some possible real-life situations to which there may be more than one solution?
• How does the ongoing use of fractions and decimals apply to real-life situations?
4/20/2013 5:14:58 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 4
Additional Standards = Approximately 10%
Grade 7:
Unit 5: Equations
Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)
*1. Make sense of problems and persevere in solving them. Students explain and demonstrate an understanding of expressions, equations, and linear
*2.
3.
*4.
*5.
*6.
*7.
8.
relationships by using symbols, visuals, words, and real-life contexts. Students demonstrate perseverance by drawing from their own experiences, prior
knowledge, and repertoire of strategies until they find the solution for the given situation.
Reason abstractly and quantitatively. Students demonstrate quantitative reasoning by representing and solving real-world situations using visuals,
numbers, and symbols. They demonstrate abstract reasoning by translating expressions, equations, inequalities and linear relationships into real-world
situations.
Construct viable arguments and critique the reasoning of others.
Model with mathematics. Students will model an understanding of expressions, equations, inequalities, and graphs using tools such as algebra tiles/blocks,
counters, protractors, compasses, and visuals to represent real-world situations.
Use appropriate tools strategically. Students demonstrate their ability to select and use the most appropriate tool (pencil/paper, manipulatives,
calculators, protractors, etc.) while simplifying/evaluating/analyzing expressions, solving equations and representing and analyzing linear relationships.
Attend to precision. Students demonstrate precision by correctly using numbers, variables and symbols to represent expressions, equations and linear
relationships, and correctly label units. Students use precision in calculation by checking the reasonableness of their answers and making adjustments
accordingly. Students will use appropriate geometric language to describe and label figures.
Look for and make use of structure. Students will apply structure by defining the variable and choosing an appropriate mode of representation to aid in
solving the problem. (i.e. diagram, table, graph, number line). Students will model with an equation to solve for the variable. Students will also examine the
relationship between the structure of a circle and the formulas for area and circumference.
Look for and express regularity in repeated reasoning.
Prerequisite Skills/Concepts:
Students should already be able to:
•
•
•
Advanced Skills/Concepts:
Some students may be ready to:
Computation with all positive and negative rational numbers
(7.NS.1-2)
Solve real-world and mathematical problems with rational
numbers (7.NS.3)
Apply properties of operations to add, subtract, factor and expand
linear expressions (7.EE.1)
4/20/2013 5:14:58 PM
Priority Standards = Approximately 70%
•
Graph linear relationships and interpret slope (8.EE.5)
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 5
Additional Standards = Approximately 10%
Grade 7:
Knowledge: Students will know…
Unit 5: Equations
Skills: Students will be able to do…
•
All standards in this unit go beyond the knowledge level.
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
4/20/2013 5:14:58 PM
Priority Standards = Approximately 70%
Use commutative, associative, distributive, identity, and inverse properties to
calculate with numbers in any form (whole numbers, fractions and decimals).
(7.EE.3)
Convert between rational number forms (whole numbers, fractions and
decimals) to solve problems as appropriate. (7.EE.3)
Solve multi-step mathematical problems posed with positive and negative
rational numbers in any form (whole numbers, fractions, and decimals), using
tools strategically. (7.EE.3)
Solve multi-step real-life problems posed with positive and negative rational
numbers in any form (whole numbers, fractions, and decimals), using tools
strategically. (7.EE.3)
Use mental computation and estimation strategies to assess the reasonableness
of the answer. (7.EE.3)
Translate words or real-life situations into variable equations. (7.EE.4)
Translate words or real-life situations into variable inequalities. (7.EE.4)
Solve one- or two-step equations with rational numbers fluently. (7.EE.4)
Solve word problems leading to one- or two-step equations with rational
numbers. (7.EE.4)
Construct simple equations and inequalities with rational numbers to solve
problems. (7.EE.4)
Compare an algebraic solution to an arithmetic solution, identifying the
sequence of the operations used in each approach. (7.EE.4)
Solve word problems leading to one- or two-step inequalities with rational
numbers. (7.EE.4)
Graph the solution set of inequalities and interpret it in the context of the
problem. (7.EE.4)
Know the formulas for the area and circumference of a circle. (7.G.4)
Use the formulas for area and circumference of a circle to solve problems.
(7.G.4)
Informally, derive the area formula for a circle based on circumference. (7.G.4)
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 6
Additional Standards = Approximately 10%
Grade 7:
Unit 5: Equations
Academic Vocabulary:
Critical Terms:
Supplemental Terms:
Linear
Coefficient
Factored form
Circumference
Combining like terms
Inverse operation
Rate of change
Evaluate
Expression
Equivalent
Rational number
Commutative property
Associative property
Distributive property
Identity properties
Expanded form
Equation
Inequality
Circle
Term
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Adapted from UbD framework
Supporting Standards = Approximately 20%
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Additional Standards = Approximately 10%
Grade 7:
Unit 6: Data Distributions
Time Frame: Approximately 2-4 Weeks
Connections to Previous Learning:
Students in Grade 6 learn the concepts of ratio and unit rate as well as the precise mathematical language used to describe these relationships. They learn to
solve problems using ratio and rate reasoning using a variety of tools such as tables, tape diagrams, double number lines and equations.
Focus of this Unit: Students build upon their understanding of statistics by examining how selected data can be used to draw conclusions, make predictions, and
compare populations.
Connections to Subsequent Learning: Students will apply their experiences with data, coordinate planes and linear functions to the study of variables related to
a question of interest. They will analyze bivariate data through linear models and frequency tables.
From the 6-8 Statistics and Probability Progression document, pp. 8-10:
Random sampling: In earlier grades students have been using data, both categorical and
measurement, to answer simple statistical questions, but have paid little attention to how the data
were selected. A primary focus for Grade 7 is the process of selecting a random sample, and the
value of doing so. If three students are to be selected from the class for a special project, students
recognize that a fair way to make the selection is to put all the student names in a box, mix them
up, and draw out three names “at random.” Individual students realize that they may not get
selected, but that each student has the same chance of being selected. In other words, random
sampling is a fair way to select a subset (a sample) of the set of interest (the population). A statistic
computed from a random sample, such as the mean of the sample, can be used as an estimate of
that same characteristic of the population from which the sample was selected. This estimate must
be viewed with some degree of caution because of the variability in both the population and
sample data. A basic tenet of statistical reasoning, then, is that random sampling allows results
from a sample to be generalized to a much larger body of data, namely, the population from which
the sample was selected.
“What proportion of students in the seventh grade of your school chooses football as their favorite sport?” Students realize that they do not have the time and
energy to interview all seventh graders, so the next best way to get an answer is to select a random sample of seventh graders and interview them on this issue.
The sample proportion is the best estimate of the population proportion, but students realize that the two are not the same and a different sample will give a
slightly different estimate. In short, students realize that conclusions drawn from random samples generalize beyond the sample to the population from which
the sample was selected, but a sample statistic is only an estimate of a corresponding population parameter and there will be some discrepancy between the
two. Understanding variability in sampling allows the investigator to gauge the expected size of that discrepancy.
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Grade 7:
Unit 6: Data Distributions
The variability in samples can be studied by means of simulation. Students are to take a random
sample of 50 seventh graders from a large population of seventh graders to estimate the proportion
having football as their favorite sport. Suppose, for the moment, that the true proportion is 60%, or
0.60. How much variation can be expected among the sample proportions? The scenario of selecting
samples from this population can be simulated by constructing a “population” that has 60% red
chips and 40% blue chips, taking a sample of 50 chips from that population, recording the number of
red chips, replacing the sample in the population, and repeating the sampling process. (This can be
done by hand or with the aid of technology, or by a combination of the two.) Record the proportion
of red chips in each sample and plot the results.
The dot plots show results for 200 such random samples of size 50 each. Note that the sample
proportions pile up around 0.60, but it is not too rare to see a sample proportion down around 0.45
or up around .0.75. Thus, we might expect a variation of close to 15 percentage points in either
direction. Interestingly, about that same amount of variation persists for true proportions of 50%
and 40%, as shown in the dot plots. Students can now reason that random samples of size 50 are
likely to produce sample proportions that are within about 15 percentage points of the true
population value. They should now conjecture as to what will happen if the sample size is doubled or
halved, and then check out the conjectures with further simulations. Why are sample sizes in public
opinion polls generally around 1000 or more, rather than as small as 50?
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Supporting Standards = Approximately 20%
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Grade 7:
Unit 6: Data Distributions
Informal comparative inference: To estimate a population mean or median, the best practice is to select a random sample from that population and use the
sample mean or median as the estimate, just as with proportions. But, many of the practical problems dealing with measures of center are comparative in
nature, as in comparing average scores on the first and second exam or comparing average salaries between female and male employees of a firm. Such
comparisons may involve making conjectures about population parameters and constructing arguments based on data to support the conjectures (MP3).
If all measurements in a population are known, no sampling is necessary and data comparisons involve
the calculated measures of center. Even then, students should consider variability figures in the margin
show the female life expectancies for countries of Africa and Europe. It is clear that Europe tends to
have the higher life expectancies and a much higher median, but some African countries are
comparable to some of those in Europe. The mean and MAD for Africa are 53.6 and 9.5 years,
respectively, whereas those for Europe are 79.3 and 2.8 years. In Africa, it would not be rare to see a
country in which female life expectancy is about ten years away from the mean for the continent, but
in Europe the life expectancy in most countries is within three years of the mean.
For random samples, students should understand that medians and means computed from samples will
vary from sample to sample and that making informed decisions based on such sample statistics
requires some knowledge of the amount of variation to expect. Just as for proportions, a good way to
gain this knowledge is through simulation, beginning with a population of known structure.
The following examples are based on data compiled from nearly 200 middle school students in the
Washington, DC area participating in the Census at Schools Project. Responses to the question, “How
many hours per week do you usually spend on homework?,” from a random sample of 10 female
students and another of 10 male students from this population gave the results plotted to the left.
Females have a slightly higher median, but students houl drealize that there is too much variation in
the sample data to conclude that, in this population, females have a higher median homework time.
An idea of how much variation to expect in samples of size 10 is needed.
Simulation to the rescue! Students can take multiple samples of size 10 from the Census of Schools
data to see how much the sample medians themselves tend to vary. The sample medians for 100
random samples of size 10 each, with 100 samples of males and 100 samples of females, is shown to left. This plot shows that the sample medians vary much
less than the homework hours themselves and provides more convincing evidence that the female median homework hours is larger than that for males. Half of
the female sample medians are within one hour of 4 while half of the male sample medians are within half hour of 3, although there is still overlap between the
two groups.
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Supporting Standards = Approximately 20%
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Grade 7:
Unit 6: Data Distributions
A similar analysis based on sample means gave the results seen in the margin. Here, the overlap of the two distributions is more severe and the evidence weaker
for declaring that the females have higher mean study hours than males.
Desired Outcomes
Standard(s):
Use random sampling to draw inferences about a population.
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.
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.
Draw informal comparative inferences about two populations
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.
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.
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Priority Standards = Approximately 70%
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Supporting Standards = Approximately 20%
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Grade 7:
Unit 6: Data Distributions
Supporting Standards:
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.
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.
c) Understand subtraction of rational numbers as adding 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.
d) Apply properties of operations as strategies to add and subtract rational numbers.
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.
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/1) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real-world contexts.
c) Apply properties of operations as strategies 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.
7. NS.3 Solve real-world and mathematical problems involving the four operations with rational number.
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.”
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 ¾ inches long in the center of a door
that is 27 ½ 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.
WIDA Standard: (English Language Learners)
English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.
English language learners benefit from:
• Explicit instruction regarding statistical language with regard to visual data representations.
• Clear examples of bias expressed using simple language and supported with visual representations.
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Supporting Standards = Approximately 20%
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Additional Standards = Approximately 10%
Grade 7:
Unit 6: Data Distributions
Understandings: Students will 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.
Random sampling tends to produce representative samples and support valid inferences.
Two data distributions can be compared using visual and numerical representations based upon measures of center and measures of variability to draw
conclusions.
Essential Questions:
•
•
•
How can two data distributions be compared?
How can statistics be used to gain information about a sample population?
How can a random sample of a larger population be used to draw inferences?
Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)
1. Make sense of problems and persevere in solving them. Students make sense of information by connecting visual, tabular, and symbolic representations
of sample populations in real-life contexts.
2.
* Reason abstractly and quantitatively. Students’ reason about the values in data representations based upon their relationship to the real number line.
* 3. Construct viable arguments and critique the reasoning of others. Students use data to make inferences from sample sets. They construct viable
arguments by referring to representations as evidence of their inferences and question each other regarding these inferences.
*4. Model with mathematics. Students generate representative samples in real-world contexts and represent these visually, in tables, and symbolically to gain
information from sample sets.
5. Use appropriate tools strategically. Students choose appropriate mathematical and visual representations, including technology-based tools, to represent
the data distributions.
6. Attend to precision. Students use precision to collect accurate measurement information from sample populations and precise language when generating
and interpreting data.
7. Look for and make use of structure. Students interpret data representations in tables, histograms, box plots and scatter plots by examining the features of
those representations.
8. Look for and express regularity in repeated reasoning. Students use their prior experiences with visual representations of data to interpret new data
sample sets.
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Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
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Additional Standards = Approximately 10%
Grade 7:
Unit 6: Data Distributions
Prerequisite Skills/Concepts:
Advanced Skills/Concepts:
Students should already be able to:
Some students may be ready to:
•
•
Develop understanding of statistical variability. (6.SP.1-3)
Summarize and describe distributions. (6.SP.4-5)
Knowledge: Students will know…
•
Skills: Students will be able to …
•
All standards in this unit go beyond the knowledge level.
•
•
•
•
•
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Priority Standards = Approximately 70%
Investigate patterns of association in bivariate data (8.SP.1-4)
Recognize and identify that different sampling techniques must be used in real life
situations, because it is very difficult to survey an entire population. (7.SP.1)
Select appropriate sample sizes based on a population in real-life situations and
explain why generalizations about a population from a sample are valid only if the
sample is random and representative of that population. (7.SP.1)
Collect data from a sample population in order to predict information about a
population. (7.SP.1)
Interpret data from a random sample to draw inferences about a population with
an unknown characteristic of interest. (7.SP.2)
Generate multiple samples (or simulated samples) of the same size to determine
the variation in estimates or predictions by comparing the samples. (7.SP.2)
Identify the degree of overlap between two numerical sets of data. (7.SP.3)
Visually compare two numerical data distributions with like ranges. (7.SP.3)
Measure the difference between the centers of two different data distributions
and express this difference as a multiple of a measure of variability. (7.SP.3)
Use measures of center and measures of variability for numerical data from
random samples to draw informal comparative inferences about two populations.
(7.SP.4)
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 7
Additional Standards = Approximately 10%
Grade 7:
Unit 6: Data Distributions
Academic Vocabulary:
Critical Terms:
Supplemental Terms:
Random sample
Biased sample
Unbiased sample
Histogram
Box plot
Dot plot
Double box plot
Double dot plot
Statistics
Mean
Median
Mode
4/20/2013 5:15:00 PM
Priority Standards = Approximately 70%
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Supporting Standards = Approximately 20%
Page 8
Additional Standards = Approximately 10%
Grade 7:
Unit 7: Probability
Approximate Time Frame: 2-3 weeks
Connections to Previous Learning:
Students in Grade 6 learn the concepts of ratio and unit rate as well as the precise mathematical language used to describe these relationships. They learn to
solve problems using ratio and rate reasoning using a variety of tools such as tables, tape diagrams, double number lines and equations.
Focus of this Unit:
In Grade 7 students develop a general understanding of the likelihood of events occurring by realizing that probabilities fall between 0 and 1. They gather data
from simulations to estimate theoretical probability using the experimental probability. Students will make predictions about the relative frequency of an event
by using simulations to collect, record, organize and analyze data. They will develop probability models to be used to find the probability of simple and
compound events. Students will determine from each sample space the probability or fraction of each possible outcome.
Connections to Subsequent Learning:
The concepts developed at Grade 7 extend to the high school Conditional Probability and Rules of Probability Standards.
From the 6-8 Statistics and Probability Progression Document pp.7-8:
Chance processes and probability models In Grade 7, students build their understanding of probability on a relative frequency view of the subject, examining
the proportion of “successes” in a chance process—one involving repeated observations of random outcomes of a given event, such as a series of coin tosses.
“What is my chance of getting the correct answer to the next multiple choice question?” is not a probability question in the relative frequency sense. “What is
my chance of getting the correct answer to the next multiple choice question if I make a random guess among the four choices?” is a probability question,
because the student could set up an experiment of multiple trials to approximate the relative frequency of the outcome. And two students doing the same
experiment will get nearly the same approximation. These important points are often overlooked in discussions of probability.
Students begin by relating probability to the long-run (more than five or ten trials) relative frequency
of a chance event, using coins, number cubes, cards, spinners, bead bags, and so on. Hands-on
activities with students collecting the data on probability experiments are critically important, but
once the connection between observed relative frequency and theoretical probability is clear, they
can move to simulating probability experiments via technology (graphing calculators or computers).
It must be understood that the connection between relative frequency and probability goes two
ways. If you know the structure of the generating mechanism (e.g., a bag with known numbers of red
and white chips), you can anticipate the relative frequencies of a series of random selections (with
replacement) from the bag. If you do not know the structure (e.g., the bag has unknown numbers of
red and white chips), you can approximate it by making a series of random selections and recording
the relative frequencies. This simple idea, obvious to the experienced, is essential and not obvious at all to the novice. The first type of situation, in which the
structure is known, leads to “probability”; the second, in which the structure is unknown, leads to “statistics.”
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Supporting Standards = Approximately 20%
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Grade 7:
Unit 7: Probability
A probability model provides a probability for each possible non-overlapping outcome for a chance process so that the total probability over all such outcomes is
unity. The collection of all possible individual outcomes is known as the sample space for the model. For example, the sample space for the toss of two coins (fair
or not) is often written as {TT, HT, TH, HH}. The probabilities of the model can be either theoretical (based on the structure of the process and its outcomes) or
empirical (based on observed data generated by the process). In the toss of two balanced coins, the four outcomes of the sample space are given equal
theoretical probabilities of ¼ because of the symmetry of the process—because the coins are balanced, an outcome of heads is just as likely as an outcome of
tails. Randomly selecting a name from a list of ten students also leads to equally likely outcomes with probability 0.10 that a given student’s name will be
selected. If there are exactly four seventh graders on the list, the chance of selecting a seventh grader’s name is 0.40. On the other hand, the probability of a
tossed thumbtack landing point up is not necessarily ½ just because there are two possible outcomes; these outcomes may not be equally likely and an empirical
answer must be found by tossing the tack and collecting data.
The product rule for counting outcomes for chance events should be used in finite situations like tossing two or three coins or rolling two number cubes. There is
no need to go to more formal rules for permutations and combinations at this level. Students should gain experience in the use of diagrams, especially trees and
tables, as the basis for organized counting of possible outcomes from chance processes. For example, the 36 equally likely (theoretical probability) outcomes
from the toss of a pair of number cubes are most easily listed on a two-way table. An archived table of census data can be used to approximate the (empirical)
probability that a randomly selected Florida resident will be Hispanic.
After the basics of probability are understood, students should experience setting up a model and using simulation (by hand or with technology) to collect data
and estimate probabilities for a real situation that is sufficiently complex that the theoretical probabilities are not obvious. For example, suppose, over many
years of records, a river generates a spring flood about 40% of the time. Based on these records, what is the chance that it will flood for at least three years in a
row sometime during the next five years?
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Priority Standards = Approximately 70%
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Supporting Standards = Approximately 20%
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Additional Standards = Approximately 10%
Grade 7:
Unit 7: Probability
Desired Outcomes
Standard(s):
Investigate chance processes and develop, use and evaluate probability models.
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 in neither unlikely nor likely, and a
probability near 1 indicates a likely event.
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.
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.
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
7.SP.8 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.
b) Represent sample spaces 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.
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?
4/20/2013 5:15:01 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
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Additional Standards = Approximately 10%
Grade 7:
Unit 7: Probability
Supporting Standards:
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.
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.
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.
d) Apply properties of operations as strategies to add and subtract rational numbers .
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.
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.
c) Apply properties of operations as strategies 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
7. NS.3 Solve real-world and mathematical problems involving the four operations with rational number.
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.”
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 ¾ inches long in the center of a door
that is 27 ½ 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.
WIDA Standard: (English Language Learners)
English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.
English language learners will benefit from:
• Explicit instruction with regard to understanding the contexts for probability models.
• Explicit vocabulary instruction with regard to probability language.
4/20/2013 5:15:01 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 4
Additional Standards = Approximately 10%
Grade 7:
Unit 7: Probability
Understandings: Students will understand that …
• The probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring.
• The probability of a chance event is approximated by collecting data on the chance process that produces it, observing its long-run relative frequency,
•
•
and predicting the approximate relative frequency given the probability.
A probability model, which may or may not be uniform, is used to find probabilities of events.
Various tools are used to find probabilities of compound events. (Including organized lists, tables, tree diagrams, and simulations.)
Essential Questions:
•
•
•
•
How are probability and the likelihood of an occurrence related and represented?
How is probability approximated?
How is a probability model used?
How are probabilities of compound events determined?
Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)
1. Make sense of problems and persevere in solving them. Students make sense of probability situations by creating visual, tabular and symbolic models to
represent the situations. They persevere through approximating probabilities and refining approximations based upon data.
*2. Reason abstractly and quantitatively. Students’ reason about the numerical values used to represent probabilities as a value between 0 and 1.
* 3. Construct viable arguments and critique the reasoning of others. Students approximate probabilities and create probability models and explain reasoning
for their approximations. They also question each other about the representations they create to represent probabilities.
*4. Model with mathematics. Students model real world populations using mathematical probability representations that are algebraic, tabular or graphic.
5. Use appropriate tools strategically. Students select and use technological, graphic or real-world contexts to model probabilities.
6. Attend to precision. Students use precise language and calculations to represent probabilities in mathematical and real-world contexts.
7. Look for and make use of structure. Students recognize that probability can be represented in tables, visual models, or as a rational number.
8. Look for express regularity in repeated reasoning. Students use repeated reasoning when approximating probabilities. They refine their approximations
based upon experiences with data.
Prerequisite Skills/Concepts:
Advanced Skills/Concepts:
Students should already be able to:
Some students may be ready to:
•
•
•
Understand statistical variability (6.SP.2)
Display data in various ways (6.SP.4)
Interpret and summarize data as a numerical set in relation to
its context (6.SP.5)
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Priority Standards = Approximately 70%
•
•
Understand independence and conditional probability and use them to
interpret data. (S-CP.1-5)
Use the rules of probability to compute probabilities of compound events in a
uniform probability model. (S-CP.6-7)
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 5
Additional Standards = Approximately 10%
Grade 7:
Knowledge: Students will know…
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0 represents an event that is impossible.
1 represents an event is certain.
The closer to 1, the more likely an event is to occur.
The closer to 0, the less likely an event is to occur.
All standards in this unit go beyond the knowledge level.
Unit 7: Probability
Skills: Students will be able to…
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4/20/2013 5:15:01 PM
Priority Standards = Approximately 70%
Represent the probability of a chance event as a number between 0 and 1. (7.SP.5)
Use the terms “likely”, “unlikely,” to describe the probability represented by the
fractions used. (7.SP.5)
Approximate the probability of a chance event by collecting data on the chance
process that produces it and observing its long-run relative frequency. (7.SP.6)
Predict the approximate relative frequency of a chance event given the probability.
(7.SP.6)
Develop a uniform probability model by assigning equal probability to all
outcomes, and use the model to determine probabilities of events. (7.SP.7)
Develop a probability model (which may not be uniform) by observing frequencies
in data generated from a chance process. (7.SP.7)
Compare probabilities from a model to observed frequencies. (7.SP.7)
If the agreement between a model and observed frequencies is not good, explain
possible sources of the discrepancy. (7.SP.7)
Find probabilities of compound events using organized lists, tables, tree diagrams,
and simulation. (7.SP.8)
Represent the probability of a compound event as the fraction of outcomes in the
sample space for which the compound event occurs. (7.SP.8)
Represent sample spaces for compound events using methods such as organized
lists, tables and tree diagrams. (7.SP.8)
For an event described in everyday language (e.g., “rolling double sixes”), identify
the outcomes in the sample space which compose the event. (7.SP.8)
Design and use a simulation to generate frequencies for compound events. (7.SP.8)
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 6
Additional Standards = Approximately 10%
Grade 7:
Unit 7: Probability
Academic Vocabulary:
Critical Terms:
Supplemental Terms:
Simulation
Compound event
Probability
Sample space
Random sample
Outcome
Theoretical probability
Experimental probability
Relative Frequency
Tree diagram
Likelihood
Counting Principle
Uniform probability model
Empirical probability
Equally likely
More likely
Less likely
Fair
Unfair
Simple event
Fraction
Decimal
Percent
Combination
Permutation
Dependent Event
Independent Event
Complementary Event
Relative frequency
4/20/2013 5:15:01 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 7
Additional Standards = Approximately 10%
Grade 7:
Unit 8: Geometric Measurement
Approximate time frame: 4-5 weeks
Connections to previous learning:
Students should have had experiences with problems involving areas and volumes extend previous work and provide a context for developing and using
equations. Students’ competencies in shape composition and decomposition should be highly developed. These competencies form the foundation for
understanding multiplication, formulas for area and volume, and the coordinate plane. Students also analyze, compose and decompose polyhedral solids.
Students extend their understanding of properties of two-dimensional shapes to use the coordinate systems. As a precursor for learning to describe crosssections of three-dimensional figures, students use drawings and physical models to learn to identify parallel lines in three-dimensional shapes, as well as lines
perpendicular to a plane, lines parallel to a plane, the plane passing through three given points, and the plane perpendicular to a given line at a given point.
Focus of this unit:
In grade 7, students begin to reason about relationships among two-dimensional figures using scale and informal geometric constructions, and 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, volume of two- and three-dimensional objects
composed of triangles, quadrilaterals, polygons, cubes and right prisms.
Connections to subsequent learning:
The skills and concepts attained in this unit will prepare students for work on congruence and similarity in grade 8. This will lead to readiness to understand and
apply the Pythagorean Theorem.
From the K-6, Geometry Progression Document, pg. 20:
Composition and decomposition of shapes is used throughout geometry from Grade 6 to high school and beyond. Compositions and decompositions of regions
continue to be important for solving a wide variety of area problems, including justifications of formulas and solving real world problems that involve complex
shapes. Decompositions are often indicated in geometric diagrams by an auxiliary line, and using the strategy of drawing an auxiliary line to solve a problem are
part of looking for and making use of structure (MP7). Recognizing the significance of an existing line in a figure is also part of looking for and making use of
structure. This may involve identifying the length of an associated line segment, which in turn may rely on students’ abilities to identify relationships of line
segments and angles in the figure. These abilities become more sophisticated as students gain more experience in geometry. In Grade 7, this experience
includes making scale drawings of geometric figures and solving problems involving angle measure, surface area, and volume (which build on understandings
described in the Geometric Measurement Progression as well as the ability to compose and decompose figures).
4/20/2013 5:15:02 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 1
Additional Standards = Approximately 10%
Grade 7:
Unit 8: Geometric Measurement
Desired Outcomes
Standard(s):
Draw, Construct, and describe geometrical figures and describe the relationships between them.
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.
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.
Solve real-life and mathematical problems in involving angle measure, area, surface area, and volume
7.G.5 Use facts about supplementary, complementary, vertical, and adjacent angles in multi-step problem to write and solve simple equations for an
unknown angle in a figure.
7.G.6 Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles,
quadrilaterals, polygons, cubes, and right prisms.
Supporting Standards:
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.”
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 ¾ inches long in the center of a door
that is 27 ½ 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.
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?
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.
WIDA Standard: (English Language Learners)
English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.
English language learners benefit from:
• tactile and virtual manipulatives for the study of geometric figures and their components.
• explicit vocabulary instruction with regard to geometric and measurement terminology.
4/20/2013 5:15:02 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 2
Additional Standards = Approximately 10%
Grade 7:
Unit 8: Geometric Measurement
Understandings: Students will understand that …
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Real world and geometric structures are composed of shapes and spaces with specific properties.
Shapes are defined by their properties.
Shapes have a purpose for designing structures.
Three-dimensional figures have relationships to specific two-dimensional figures.
Planes that cut polyhedra create related two-dimensional figures.
Essential Questions:
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How are forms and objects created or represented?
How are two-dimensional and three-dimensional space related?
How are specific characteristics and a classification system useful in analyzing and designing structures?
How does our understanding of geometry help us to describe real-world objects?
Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)
*1. Make sense of problems and persevere in solving them. Students make sense of the problems involving geometric measurements (area, volume, surface
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*7.
*8.
area, etc.) through their understanding of the relationships between these measurements. They demonstrate this by choosing appropriate strategies for
solving problems involving real-world and mathematical situations.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others. Students are able to construct arguments using concrete referents such as objects,
drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades.
Model with mathematics. Students are able to apply the geometry concepts they know to solve problems arising in everyday life, society and the
workplace. This may include applying area and surface of 2-dimensional figures to solve interior design problems or surface area and volume of 3dimensional figures to solve architectural problems.
Use appropriate tools strategically. Mathematically proficient students consider available tools that might include concrete models, a ruler, a protractor,
or dynamic geometry software such as virtual manipulatives and simulations. When making mathematical models, they know that technology can enable
them to visualize the results of varying assumptions, explore consequences, and compare predictions with data.
Attend to precision.
Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. They recognize the significance of an
existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They can see complicated things as single
objects or as being composed of several objects.
Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general
methods and for shortcuts.
4/20/2013 5:15:02 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 3
Additional Standards = Approximately 10%
Grade 7:
Unit 8: Geometric Measurement
Prerequisite Skills/Concepts:
Advanced Skills/Concepts:
Some students may be ready to:
Students should already be able to:
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Solve problems involving surface area of figures composed of
triangles and rectangles.
Solve problems involving volume of right rectangular prisms.
Represent three-dimensional figures using nets.
Find area of quadrilaterals and triangles, and of other polygons
through decomposition strategies.
4/20/2013 5:15:02 PM
Priority Standards = Approximately 70%
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Understand congruence and similarity using physical models, transparencies, or
geometry software. (8.G.1-5)
Solve real-world and mathematical problems involving volume of cylinders, cones,
and spheres. (8.G.9)
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 4
Additional Standards = Approximately 10%
Grade 7:
Unit 8: Geometric Measurement
Skills: Students will be able to …
Knowledge: Students will know…
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Properties and vocabulary associated with angles, polygons, prisms,
pyramids, cross sections
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Facts about supplementary, complementary, vertical, and adjacent
angles.
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4/20/2013 5:15:02 PM
Priority Standards = Approximately 70%
Use freehand, ruler, protractor and technology to draw geometric shapes with given
conditions. (7.G.2)
Construct triangles from 3 measures of angles or sides. (7.G.2)
Given conditions, determine what and how many type(s) of triangles are possible to
construct. (7.G.2)
Describe the two-dimensional figures that result from slicing three-dimensional figures
(right rectangular prisms and right rectangular pyramids). (7.G.3)
Identify and describe supplementary, complementary, vertical, and adjacent angles. (7.G.5)
Use understandings of supplementary, complementary, vertical and adjacent angles to
write and solve equations. (7.G.5)
Explain (verbally and in writing) the relationships between the angles formed by two
intersecting lines. (7.G.5)
Solve mathematical problems involving area, volume and surface area of two- and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right
prisms. (7.G.6)
Solve real-world problems involving area, volume and surface area of two- and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right
prisms. (7.G.6)
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 5
Additional Standards = Approximately 10%
Grade 7:
Unit 8: Geometric Measurement
Academic Vocabulary:
Critical Terms:
Supplemental Terms:
Three-dimensional
Two-dimensional
Surface area
volume
Intersecting lines
Vertex
Complementary angles
Supplementary angles
Cross-sections
Right rectangular prism
Right rectangular pyramid
Constructions
Virtual manipulative
Cube
Planar section
Compose
Decompose
Nets
Volume
Area
Polygon
Pyramid
Prism
Triangle
Angle
Right angle
Obtuse angle
Degrees
Acute angle
Angle measure
Line segment
Prism
Pyramid
Plane
4/20/2013 5:15:02 PM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 6
Additional Standards = Approximately 10%