Download Greeley-Evans School District 6- 7th Grade: 2015

Greeley-Evans School District 6- 7th Grade: 2015-2016
Mathematics
Content Area
Grade Level
7th Grade
Standard
Grade Level Expectations (GLE)
GLE Code
1. Number Sense,
Properties, and
Operations
1.
Proportional reasoning involves comparisons and multiplicative relationships among
ratios
MA10-GR.7-S.1-GLE.1
2.
Formulate, represent, and use algorithms with rational numbers flexibly, accurately, and
efficiently
MA10-GR.7-S.1-GLE.2
2. Patterns, Functions, and
Algebraic Structures
1. Properties of arithmetic can be used to generate equivalent expressions
MA10-GR.7-S.2-GLE.1
2. Equations and expressions model quantitative relationships and phenomena
MA10-GR.7-S.2-GLE.2
3. Data Analysis, Statistics,
and Probability
1. Statistics can be used to gain information about populations by examining samples
MA10-GR.7-S.3-GLE.1
2. Mathematical models are used to determine probability
MA10-GR.7-S.3-GLE.2
4. Shape, Dimension, and
Geometric Relationships
1. Modeling geometric figures and relationships leads to informal spatial reasoning and
proof
MA10-GR.7-S.4-GLE.1
2. Linear measure, angle measure, area, and volume are fundamentally different and
require different units of measure
MA10-GR.7-S.4-GLE.2
Colorado 21st Century Skills
Invention
Critical Thinking and Reasoning: Thinking
Deeply, Thinking Differently
Information Literacy: Untangling the Web
Collaboration: Working Together, Learning
Together
Self-Direction: Own Your Learning
Invention: Creating Solutions
Mathematical Practices:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Module Titles
Ratios and Proportional Relationships
Rational Numbers
Length of Unit
Dates
31 Days
22 Days
August 27-October 12
October 13-November 12
Expressions and Equations
Percent and Proportional Relationships
Statistics and Probability
27 Days
25 Days
30 Days
November 13-January 11
January 12-February 18
February 19-April 7
Geometry
31 Days
April 8-May 20
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Greeley-Evans School District 6- 7th Grade: 2015-2016
MATHEMATICAL PRACTICES
MATHEMATICAL PRACTICE 1: Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints,
relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt.
They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their
progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on
their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and
graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or
pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask
themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
MATHEMATICAL PRACTICE 2: Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving
quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a
life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into
the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved;
attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
MATHEMATICAL PRACTICE 3: Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures
and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use
counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible
arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible
arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can 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. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others,
decide whether they make sense, and ask useful questions to clarify or improve the arguments.
MATHEMATICAL PRACTICE 4: Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as
simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the
community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another.
Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that
these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables,
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of
the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
MATHEMATICAL PRACTICE 5: Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a
protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools
appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations.
For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by
strategically using estimation and other mathematical knowledge. 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. Mathematically proficient students at various grade levels are able to identify relevant external
mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen
their understanding of concepts.
MATHEMATICAL PRACTICE 6: Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the
meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to
clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the
problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine
claims and make explicit use of definitions.
MATHEMATICAL PRACTICE 7: Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as
seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well-remembered 7 × 5 + 7
× 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. 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 also can step back for an overview and shift
perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3(x y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.
MATHEMATICAL PRACTICE 8: 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. Upper elementary students might notice when
dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope
as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y - 2)/(x - 1) = 3. Noticing the regularity
in the way terms cancel when expanding (x - 1)(x + 1), (x - 1)(x2 + x + 1), and (x - 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As
they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness
of their intermediate results.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
Suggested Big Idea
Content Emphasis Cluster
Mathematical Practices
Common Assessment
Graduate Competency
CCSS Priority Standards
CCSS.MATH.CONTENT.7.EE.B.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.
CCSS.MATH.CONTENT.7.RP.A.2
Recognize and represent
proportional relationships between
quantities.
CCSS.MATH.CONTENT.7.RP.A.2.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.
CCSS.MATH.CONTENT.7.RP.A.2.B
Identify the constant of
proportionality (unit rate) in tables,
Module 1: Ratios and Proportional Relationships
Analyze proportional relationships and use them to solve real-world and mathematical problems.
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
Draw, constructs, and describes geometrical figures and describes the relationships
between them.
MP.1. Make sense of problems and persevere in solving them.
MP.2. Reason abstractly and quantitatively.
MP.3. Construct viable arguments and critique the reasoning of others.
MP.4. Model with mathematics.
MP.5. Use appropriate tools strategically.
MP.6. Attend to precision.
MP.7. Look for and make use of structure.
MP.8. Look for and express regularity in repeated reasoning.
End of Module Assessment
Prepared graduates make both relative (multiplicative) and absolute (arithmetic) comparisons between quantities. Multiplicative
thinking underlies proportional reasoning.
Prepared graduates use critical thinking to recognize problematic aspects of situations, create mathematical models, and present
and defend solutions
Prepared graduates apply transformation to numbers, shapes, functional representations, and data
Cross-Content
Writing Focus
Language/Vocabulary Misconceptions
Connections
Literacy Connections
RST.6-8.4
Determine the meaning
of symbols, key terms,
and other domainspecific words and
phrases as they are used
in a specific scientific
or technical context
relevant to grades 6-8
texts and topics.
RST.6-8.5
Analyze the structure an
author uses to organize
a text, including how
the major sections
contribute to the whole
and to an understanding
of the topic.
RST.6-8.7
Integrate quantitative or
technical information
Writing Connection
WHST.6-8.2
Write
informative/explanatory
texts, including the
narration of historical
events, scientific
procedures/
experiments, or
technical processes.
a. Introduce a topic
clearly, previewing
what is to follow;
organize ideas,
concepts, and
information into
broader categories as
appropriate to
achieving purpose;
include formatting (e.g.,
headings), graphics
(e.g., charts, tables),
and multimedia when
Academic VocabularyCross discipline
language- Compute,
Identify, Represent,
Explain, Estimate,
Solve
Technical VocabularyDiscipline-specific
languageProportional To,
Proportional
Relationship,
Constant of
Proportionality, Oneto One
Correspondence,
Scale Drawing and
Scale Factor , Ratio,
Rate, Unit Rate ,
Equivalent Ratio ,
Ratio Table
A common error is to reverse the position of
the variables when writing equations.
Students may find it useful to use variables
specifically related to the quantities rather
than using x and y.
Constructing verbal models can also be
helpful. A student might describe the
situation as “the number of packs of gum
times the cost for each pack is the total cost
in dollars”. They can use this verbal model
to construct the equation. Students can
check their equation by substituting values
and comparing their results to the table. The
checking process helps student revise and
recheck their model as necessary. The
number of packs of gum times the cost for
each pack is the total cost (g x 2 = d).
Student’s may have misconceptions about
correctly setting up proportions, how to read a
ruler, doubling side measures, and does not
double perimeter.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
graphs, equations, diagrams, and
verbal descriptions of proportional
relationships.
CCSS.MATH.CONTENT.7.RP.A.2.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.
CCSS.MATH.CONTENT.7.RP.A.2.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.
CCSS.MATH.CONTENT.7.G.1
Solve problems involving scale
drawings of geometric figures,
including computing actual lengths
and areas from a scale drawing and
reproducing a scale drawing at a
different scale.
expressed in words in a
text with a version of
that information
expressed visually (e.g.,
in a flowchart, diagram,
model, graph, or table).
RST.6-8.8
Distinguish among facts,
reasoned judgment
based on research
findings, and
speculation in a text.
useful to aiding
comprehension.
b.Develop the topic with
relevant, well-chosen
facts, definitions,
concrete details,
quotations, or other
information and
examples.
c. Use appropriate and
varied transitions to
create cohesion and
clarify the relationships
among ideas and
concepts.
d.Use precise language
and domain-specific
vocabulary to inform
about or explain the
topic.
e. Establish and maintain
a formal style and
objective tone.
f. Provide a concluding
statement or section that
follows from and
supports the
information or
explanation presented.
L.6-8.6
Acquire and use
accurately gradeappropriate general
academic and
domain-specific
words and phrases;
gather vocabulary
knowledge when
considering a word or
phrase important to
comprehension or
expression.
L.6-8.4
Determine or clarify
the meaning of
unknown and
multiple-meaning
words and phrases
choosing flexibly
from a range of
strategies.
WHST.6-8.4
Produce clear and
coherent writing in
which the development,
organization, and style
are appropriate to task,
purpose, and audience.
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Greeley-Evans School District 6- 7th Grade: 2015-2016
Ratios and Proportional Relationships


Module 1


Topic A: Proportional Relationships (7.RP.2a)
Topic B: Unit Rate and the Constant of Proportionality (7.RP.2b, 7.RP.2c,
7.RP.2d, 7.EE.4a)
Topic C: Ratios and Rates Involving Fractions (7.RP.1, 7.RP.3, 7.EE.4a)
Topic D: Ratios of Scale Drawings (7.RP.2b, 7.G.1)
Length of Unit
31 Days
August 27-October 12
Analyze proportional relationships and use them to solve real-world and mathematical problems.
7.RP.1 Compute unit rates associated with ratios 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 ½ / ¼ miles per hour,
equivalently 2 miles per hour.
7.RP.2
Content Standards
(Priority Standards)
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 at 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.
7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns,
gratuities and commissions, fees, percent increase and decrease, percent error.
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
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?
Draw, construct, and describe geometrical figures and describe the relationships between them.
7.G.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and
reproducing a scale drawing at a different scale.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
Inquiry Questions
See Knowledge Packets for specific questions.
What is a proportion?
How is the constant of proportionality calculated?
What are the different ways the constant of proportionality is represented?
How can you determine if two ratios are in a proportional relationship by using a ratio table?
How is the constant of proportionality visible in the graph of a proportional relationship?
How can you represent proportional relationships by equations?
Why is the point (1, r) a point on the equation y = xr and how does that relate to r being the unit rate?
Why is it important to keep track of the order of proportionality when analyzing relationships?
Why can the constant of proportionality be expressed in two ways?
Why is the unit ratio equivalent to the constant of proportionality?
Key Knowledge and Skills (Procedural Skill and Application)
My students will be able to (Do)…











Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.
Compare and contrast proportional and non-proportional linear relationships.
Use ratios to make predictions in proportional situations.
Analyze proportional relationships and use them to solve real-world and mathematical problems.
Decide whether two quantities are in a proportional relationship, including 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.
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
Represent proportional relationships by equations.
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.
Use proportional relationships to solve multistep ratio and percent problems.
Solve problems involving percent of a number, discounts, taxes, simple interest, percent increase, and percent decrease.
Use multiplication by a constant factor (unit rate) to represent proportional relationships.
Resources
Technology
Technology links that provide ways for students to deepen their understanding of the mathematics in the unit and can be used to differentiate
student learning.-Vertical teaming – calculators, video links;
Materials
Ratio Tables, Coordinate Grid.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
End of Unit Common Assessment on Schoolcity:


Performance/Learning
Tasks (Assessments)
Scanned into School City or students take the assessment online
Should be in addition to individually developed formative assessments
Pre Assessment Module 1 Topic A
Post Assessment Module 1 Topic A
Pre Assessment Module 1 Topic B and C
Post Assessment Module 1 Topic B and C
Pre Assessment Module 1 Topic D
Post Assessment Module 1 Topic D
***Please see the Knowledge Packets for specific lesson information.
Topic A: Proportional Relationships (7.RP.2a)
Lesson 1: An Experience in Relationships as Measuring Rate.
**Exit Ticket from EngageNY. http://www.youtube.com/watch?feature=player_embedded&v=tCKstDXMslQ
Lesson 2: Proportional Relationships
Lessons 3–4: Identifying Proportional and Non-Proportional Relationships in Tables.
Lessons 5–6: Identifying Proportional and Non-Proportional Relationships in Graphs
Topic B: Unit Rate and the Constant of Proportionality (7.RP.2b, 7.RP.2c, 7.RP.2d, 7.EE.4a) (Combined with Topic C)
Lesson 7: Unit Rate as the Constant of Proportionality
Lessons 8–9: Representing Proportional Relationships with Equations
Lesson 10: Interpreting Graphs of Proportional Relationships
Instructional Notes
Topic C: Ratios and Rates Involving Fractions (7.RP.1, 7.RP.3, 7.EE.4a) (Combined with Topic B)
Lessons 11–12: Ratios of Fractions and Their Unit Rates
Lesson 13: Finding Equivalent Ratios Given the Total Quantity. Students may need a calculator for the first activity to convert from mixed
numbers to improper fractions. Also there are great scaffolding questions to ask students for each exercise.
Lesson 14: Multistep Ratio Problems.
Lesson 15: Equations of Graphs of Proportional Relationships Involving Fractions
Topic D: Ratios of Scale Drawings (7.RP.2b, 7.G.1)
Lesson 16: Relating Scale Drawings to Ratios and Rates: you will need to have the intro activity pages displayed on the board for students.
Lesson 18: Computing Actual Lengths from a Scale Drawing
Lesson 19: Computing Actual Areas from a Scale Drawing
Lesson 20: An Exercise in Creating a Scale Drawing
Lessons 21–22: An Exercise in Changing Scales
End-of-Module Assessment and Rubric Topics A through D (assessment 1 day, return 1 day, remediation or further applications 2 days)
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio
of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received,
candidate C received nearly three votes.”
6.RP.2
Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio
relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.”
“We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” i
6.RP.3
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape
diagrams, double number line diagrams, or equations.
a. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the
pairs of values on the coordinate plane. Use tables to compare ratios.
b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns,
then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving
finding the whole, given a part and the percent.
d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing
quantities.
Foundational
Standards
Solve real-world and mathematical problems involving area, surface area, and volume.
6.G.1
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into
triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
6.G.3
Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with
the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical
problems.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
Suggested Big Idea
Content Emphasis Cluster
Mathematical Practices
Common Assessment
Graduate Competency
CCSS Priority Standards
CCSS.MATH.CONTENT.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.
CCSS.MATH.CONTENT.7.NS.1.D
Apply properties of operations as
strategies to add and subtract
rational numbers.
CCSS.MATH.CONTENT.7.NS.A.3
Solve real-world and mathematical
problems involving the four
operations with rational numbers.
Computations with rational numbers
Module 2: Rational Numbers
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
CCSS.MATH.CONTENT.7.EE.4
CCSS.MATH.CONTENT.7.NS.1.D
CCSS.MATH.CONTENT.7.NS.A.3
MP.1. Make sense of problems and
MP.2. Reason abstractly and
MP.1. Make sense of problems and
persevere in solving them.
quantitatively.
persevere in solving them.
MP.2. Reason abstractly and
MP.4. Model with mathematics.
MP.2. Reason abstractly and
quantitatively.
MP.7. Look for and make use of
quantitatively.
MP.3. Construct viable arguments and
structure.
MP.5. Use appropriate tools strategically.
critique the reasoning of others.
MP.6. Attend to precision.
MP.4. Model with mathematics.
MP.7. Look for and make use of
MP.5. Use appropriate tools strategically.
structure.
MP.6. Attend to precision.
MP.8. Look for and express regularity in
MP.7. Look for and make use of
repeated reasoning.
structure.
MP.8. Look for and express regularity in
repeated reasoning.
End of Module Assessment
Prepared graduates use critical thinking to recognize problematic aspects of situations, create mathematical models, and present
and defend solutions.
Prepared graduates are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate
(mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and
transparency.
Cross-Content
Writing Focus
Language/Vocabulary Misconceptions
Connections
Literacy Connections
RST.6-8.4
Determine the meaning of
symbols, key terms, and
other domain-specific
words and phrases as
they are used in a
specific scientific or
technical context
relevant to grades 6-8
texts and topics.
RST.6-8.5
Analyze the structure an
author uses to organize
a text, including how
the major sections
contribute to the whole
and to an understanding
of the topic.
Writing Connection
WHST.6-8.2
Write
informative/explanatory
texts, including the
narration of historical
events, scientific
procedures/ experiments,
or technical processes.
f. Introduce a topic clearly,
previewing what is to
follow; organize ideas,
concepts, and
information into broader
categories as appropriate
to achieving purpose;
include formatting (e.g.,
headings), graphics (e.g.,
charts, tables), and
multimedia when useful
to aiding comprehension.
Academic VocabularyCross discipline
language- apply,
represent, interpret,
convert, construct,
reasoning
Technical VocabularyAdditive Identity,
Inverse, Break-Even
Point, Distance
Formula, Loss,
Multiplicative
Identity, Profit,
Repeating
Decimal, Terminating
Decimal, Absolute
Value, Associative
Property (of
Multiplication and
Addition),
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
extend the rules for manipulating
fractions to complex fractions.
RST.6-8.7
Integrate quantitative or
technical information
expressed in words in a
text with a version of
that information
expressed visually (e.g.,
in a flowchart, diagram,
model, graph, or table).
RST.6-8.8
Distinguish among facts,
reasoned judgment
based on research
findings, and
speculation in a text.
g. Develop the topic with
relevant, well-chosen
facts, definitions,
concrete details,
quotations, or other
information and
examples.
h. Use appropriate and
varied transitions to
create cohesion and
clarify the relationships
among ideas and
concepts.
i. Use precise language and
domain-specific
vocabulary to inform
about or explain the
topic.
j. Establish and maintain a
formal style and
objective tone.
f. Provide a concluding
statement or section that
follows from and
supports the information
or explanation presented.
WHST.6-8.4
Produce clear and coherent
writing in which the
development,
organization, and style
are appropriate to task,
purpose, and audience.
Commutative Property
(of Multiplication and
Addition), Credit, Debit,
Deposit, Distributive
Property (of
Multiplication Over
Addition), Expression,
Equation, Integer,
Inverse, Multiplicative
Inverse, Opposites,
Overdraft, Positives,
Negatives, Rational
Numbers, Withdraw
L.6-8.6
Acquire and use
accurately gradeappropriate general
academic and domainspecific words and
phrases; gather
vocabulary knowledge
when considering a
word or phrase
important to
comprehension or
expression.
L.6-8.4
Determine or clarify
the meaning of
unknown and
multiple-meaning
words and phrases
choosing flexibly from
a range of strategies.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
Rational Numbers

Module 2


Content Standards
(Priority Standards)
Topic A: Addition and Subtraction of Integers and Rational Numbers
(7.NS.A.1)
Topic B: Multiplication and Division of Integers and Rational Numbers
(7.NS.A.2)
Topic C: Applying Operations with Rational Numbers to Expressions and
Equations (7.NS.A.3, 7.EE.A.2, 7.EE.B.4a)
Length of Unit
22 Days
October 13-November 12
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
7.NS.A.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.A.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.A.3 Solve real‐world and mathematical problems involving the four operations with rational numbers. ii
Use properties of operations to generate equivalent expressions.
7.EE.A.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.”
Solve real‐life and mathematical problems using numerical and algebraic expressions and equations.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
7.EE.B.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?
See Knowledge Packets for specific questions.
Inquiry Questions
What does it mean in real life to have a negative value or quantity of something?
What happens when you multiply positive or negative numbers?
How could you represent an integer such as -5 as an "arrow" on the number line?
Are there multiple ways of representing integers?
Key Knowledge and Skills (Procedural Skill and Application)
My students will be able to (Do)…













Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
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)
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 realworld contexts.
Apply properties of operations as strategies to add and subtract rational numbers.
Understand 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
Understand integers can be divided, provided 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 sums, products and quotients of rational numbers by describing real-world contexts
Apply properties of operations as strategies to multiply and divide rational numbers.
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
Solve real-world and mathematical problems involving the four operations with rational numbers.
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.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
Resources
Technology
Materials
Technology links that provide ways for students to deepen their understanding of the mathematics in the unit and can be used to differentiate
student learning.-Vertical teaming – calculators, video links;
Integer game in lesson 1
End of Unit Common Assessment on Schoolcity:
Performance/Learning
Tasks (Assessments)


Scanned into School City or students take the assessment online
Should be in addition to individually developed formative assessments
Pre Assessment Module 2 Topic A and B
Post Assessment Module 2 Topic A and B
***Please see the Knowledge Packets for specific lesson information.
Topic A: Addition and Subtraction of Integers and Rational Numbers (7.NS.A.1)
**There is some good fraction practice in this topic.
Lesson 1: Opposite Quantities Combine to Make Zero
**The Integer Game is used for several lessons. There is a lot of cutting required, so plan ahead.
Lesson 2: Using the Number Line to Model the Addition of Integers
Lesson 3: Understanding Addition of Integers
Lesson 4: Efficiently Adding Integers and Other Rational Numbers
Lesson 5: Understanding Subtraction of Integers and Other Rational Numbers
Lesson 6: The Distance Between Two Rational Numbers
Lesson 7: Addition and Subtraction of Rational Numbers
Lessons 8–9: Applying the Properties of Operations to Add and Subtract Rational Numbers
Instructional Notes
Topic B: Multiplication and Division of Integers and Rational Numbers (7.NS.A.2)
Lesson 10: Understanding Multiplication of Integers
Lesson 11: Develop Rules for Multiplying Signed Numbers
Lesson 12: Division of Integers
**The Exit Ticket for Lesson 12 is a MUST!
**Use the Fact Fluency Sheet
Lesson 13: Converting Between Fractions and Decimals Using Equivalent Fractions
Lesson 14: Converting Rational Numbers to Decimals Using Long Division
Lesson 15: Multiplication and Division of Rational Numbers
Lesson 16: Applying the Properties of Operations to Multiply and Divide Rational Numbers
Topic C: Applying Operations with Rational Numbers to Expressions and Equations (7.NS.A.3, 7.EE.A.2, 7.EE.B.4a) (Moved to Module 3)
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
Use equivalent fractions as a strategy to add and subtract fractions.
5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions
in such a way as to produce an equivalent sum or difference of fractions with like denominators.
For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
5.NF.B.3 Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole
numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent
the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes
are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by
weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
5.NF.B.4
Foundational
Standards
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations
a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same
with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
6.NS.A.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual
fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction
model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of
8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally?
How many 3/4‐cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2
square mi?
Compute fluently with multi-digit numbers and find common factors and multiples.
6.NS.B.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
Apply and extend previous understandings of numbers to the system of rational numbers.
6.NS.C.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g.,
temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and
negative numbers to represent quantities in real‐world contexts, explaining the meaning of 0 in each situation.
6.NS.C.6
Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous
grades to represent points on the line and in the plane with negative number coordinates.
a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of
the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.
6.NS.C.7
Understand ordering and absolute value of rational numbers.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude
for a positive or negative quantity in a real‐world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to
describe the size of the debt in dollars.
Apply and extend previous understandings of arithmetic to algebraic expressions.
6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers.
a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation
“Subtract y from 5” as 5 – y.
b. Identify parts of an expression using mathematical terms (sum, term, product, factor quotient, coefficient); view one or more parts of
an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single
entity and a sum of two terms.
c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real‐world problems.
Perform arithmetic operations, including those involving whole‐number exponents, in the conventional order when there are no
parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6 s2 to find the volume
and surface area of a cube with sides of length s = 1/2.
6.EE.A.3
Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3
(2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the
equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
6.EE.A.4
Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is
substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of
which number y stands for.
Reason about and solve one‐variable equations and inequalities.
6.EE.B.6 Use variables to represent numbers and write expressions when solving a real‐world or mathematical problem; understand that a
variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
6.EE.B.7
Solve real‐world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q
and x are all nonnegative rational numbers.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
Suggested Big Idea
Content Emphasis Cluster
Mathematical Practices
Common Assessment
Graduate Competency
CCSS Priority Standards
CCSS.MATH.CONTENT.7.EE.B.3
Solve multi-step real-life and
mathematical problems posed with
positive and negative rational
numbers in any form (whole
numbers, fractions, and decimals),
using tools strategically. Apply
properties of operations to calculate
with numbers in any form; convert
between forms as appropriate; and
assess the reasonableness of
answers using mental computation
and estimation strategies. For
example: If a woman making $25 an
hour gets a 10% raise, she will make
an additional 1/10 of her salary an
hour, or $2.50, for a new salary of
$27.50. If you want to place a towel
bar 9 3/4 inches long in the center of
a door that is 27 1/2 inches wide, you
will need to place the bar about 9
inches from each edge; this estimate
can be used as a check on the exact
computation.
Module 3: Expressions and Equations
Use properties of operations to generate equivalent expressions
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
Solve real-life and mathematical problems involving angle measure, area, surface area, and
volume
MP.1. Make sense of problems and persevere in solving them.
MP.2. Reason abstractly and quantitatively.
MP.3. Construct viable arguments and critique the reasoning of others.
MP.4. Model with mathematics.
MP.5. Use appropriate tools strategically.
MP.6. Attend to precision.
MP.7. Look for and make use of structure.
MP.8. Look for and express regularity in repeated reasoning.
End of Module Assessment
Prepared graduates use critical thinking to recognize problematic aspects of situations, create mathematical models, and present
and defend solutions
Prepared graduates understand quantity through estimation, precision, order of magnitude, and comparison. The reasonableness of
answers relies on the ability to judge appropriateness, compare, estimate, and analyze error
Cross-Content
Writing Focus
Language/Vocabulary Misconceptions
Connections
Literacy Connections
RST.6-8.4
Determine the meaning of
symbols, key terms, and
other domain-specific
words and phrases as
they are used in a
specific scientific or
technical context
relevant to grades 6-8
texts and topics.
RST.6-8.5
Analyze the structure an
author uses to organize
a text, including how
the major sections
contribute to the whole
and to an understanding
of the topic.
RST.6-8.7
Integrate quantitative or
technical information
expressed in words in a
Writing Connection
WHST.6-8.2
Write
informative/explanatory
texts, including the
narration of historical
events, scientific
procedures/
experiments, or
technical processes.
k. Introduce a topic
clearly, previewing
what is to follow;
organize ideas,
concepts, and
information into
broader categories as
appropriate to
achieving purpose;
include formatting (e.g.,
headings), graphics
(e.g., charts, tables),
and multimedia when
useful to aiding
comprehension.
Academic Vocabularyapply, convert,
represent, identify,
interpret,
expanded form ,
standard form
factored form,
coefficient, circle,
diameter,
circumference,
radius, pi, circular
region or disk
Technical VocabularyEquivalent,
expressions,
inequalities,
properties of
operations, addition,
subtraction,
multiplication,
division, factoring,
expansion, arithmetic
solution strategy,
algebraic solution
As students begin to build and work with
expressions containing more than two
operations, students tend to set aside the
order of operations. For example having a
student simplify an expression like 8 + 4(2x
- 5) + 3x can bring to light several
misconceptions. Do the students
immediately add the 8 and 4 before
distributing the 4? Do they only multiply the
4 and the 2x and not distribute the 4 to both
terms in the parenthesis? Do they collect all
like terms 8 + 4 – 5, and 2x + 3x? Each of
these show gaps in students’ understanding
of how to simplify numerical expressions
with multiple operations.
Students may believe:
Pi is an exact number rather than
understanding that 3.14 is just an
approximation of pi. Many students are
confused when dealing with circumference
(linear measurement) and area. This
confusion is about an attribute that is
measured using linear units (surrounding)
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
CCSS.MATH.CONTENT.7.EE.B.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.
CCSS.MATH.CONTENT.7.G.B.6
Solve real-world and mathematical
problems involving area, volume and
surface area of two- and threedimensional objects composed of
triangles, quadrilaterals, polygons,
cubes, and right prisms.
text with a version of
that information
expressed visually (e.g.,
in a flowchart, diagram,
model, graph, or table).
RST.6-8.8
Distinguish among facts,
reasoned judgment
based on research
findings, and
speculation in a text.
l. Develop the topic with
relevant, well-chosen
facts, definitions,
concrete details,
quotations, or other
information and
examples.
m.
Use appropriate
and varied transitions to
create cohesion and
clarify the relationships
among ideas and
concepts.
n. Use precise language
and domain-specific
vocabulary to inform
about or explain the
topic.
o.Establish and maintain
a formal style and
objective tone.
f. Provide a concluding
statement or section that
follows from and
supports the
information or
explanation presented.
WHST.6-8.4
Produce clear and
coherent writing in
which the development,
organization, and style
are appropriate to task,
purpose, and audience.
strategy, arithmetic
operations, algebraic
equations,
correctness, algebraic
manipulations,
operation, both sides,
negative number,
reverse
vs. an attribute that is measured using area
units (covering).
L.6-8.6
Acquire and use
accurately gradeappropriate general
academic and
domain-specific
words and phrases;
gather vocabulary
knowledge when
considering a word or
phrase important to
comprehension or
expression.
L.6-8.4
Determine or clarify
the meaning of
unknown and
multiple-meaning
words and phrases
choosing flexibly
from a range of
strategies.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
Expressions and Equations

Module 3


Topic A: Use Properties of Operations to Generate Equivalent Expressions
(7.EE.A.1, 7.EE.A.2)
Topic B: Solve Problems Using Expressions, Equations, and Inequalities
(7.EE.B.3, 7.EE.B.4, 7.G.B.5)
Topic C: Use Equations and Inequalities to Solve Geometry Problems (7.G.B.4,
7.G.B.6)
Length of Unit
27 Days
November 13-January 11
Use properties of operations to generate equivalent expressions.
7.EE.A.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
7.EE.A.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.”
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
7.EE.B.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers,
fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert
between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For
example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new
salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 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.
Content Standards
(Priority Standards)
7.EE.B.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.B.4
Know the formulas for the area and circumference of a circle and solve problems; give an informal derivation of the relationship between
the circumference and area of a circle.
7.G.B.5
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and use them to solve
simple equations for an unknown angle in a figure.
7.G.B.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.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
Inquiry Questions
See Knowledge Packets for specific questions.
Why are there different ways to solve equations?
What does it mean to solve an equation arithmetically?
How can variables help us understand relationships in the world?
What relationships exist between area and volume?
Would you rather have a rectangular trampoline or a circular trampoline with an area of 190 sq. ft?
Key Knowledge and Skills (Procedural Skill and Application)
My students will be able to (Do)






Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
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
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.
Fluently solve equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers.
Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.
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.
Resources
Technology
Technology links that provide ways for students to deepen their understanding of the mathematics in the unit and can be used to differentiate
student learning.-Vertical teaming – calculators, video links;
Materials
Coordinate Plane (included in lessons), protractor, nets for three-dimensional figures (included in lessons)
End of Unit Common Assessment on Schoolcity:


Performance/Learning
Tasks (Assessments)
Scanned into School City or students take the assessment online
Should be in addition to individually developed formative assessments
Pre Assessment Module 3 Topic A and B
Post Assessment Module 3 Topic A and B
Pre Assessment Module 3 Topic C
Post Assessment Module 3 Topic C
***Please see the Knowledge Packets for specific lesson information.
Instructional Notes
Module 2 Topic C: Applying Operations with Rational Numbers to Expressions and Equations (7.NS.A.3, 7.EE.A.2, 7.EE.B.4a)
Lesson 17: Comparing Tape Diagram Solutions to Algebraic Solutions
Lessons 18–19: Writing, Evaluating, and Finding Equivalent Expressions with Rational Numbers
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
Lesson 20: Investments—Performing Operations with Rational Numbers
Lesson 21: If-Then Moves with Integer Number Cards
Lessons 22–23: Solving Equations Using Algebra
Topic A: Use Properties of Operations to Generate Equivalent Expressions (7.EE.A.1, 7.EE.A.2)
Lessons 1–2: Generating Equivalent Expressions: The any order, any grouping property is referenced in the Progressions for the Common
Core State Standards in Mathematics: Grades 6–8, Expressions and Equations.
Lessons 3–4: Writing Products as Sums and Sums as Products
Lesson 5: Using the Identity and Inverse to Write Equivalent Expressions
Lesson 6: Collecting Rational Number Like Terms
Topic B: Solve Problems Using Expressions, Equations, and Inequalities (7.EE.B.3, 7.EE.B.4, 7.G.B.5)
Lesson 7: Understanding Equations: read the entire “lesson notes” section and pay close attention to the key vocabulary that is included.
Lessons 8–9: Using the If-Then Moves in Solving Equations: The FAQs on solving equations in the lesson are designed to help teachers
understand the structure of the next set of lessons. Before reading the FAQ, it may be helpful to review the properties of operations and
the properties of equality listed in Table 3 and Table 4 of the Common Core State Standards (CCSS). For Lesson 9, the materials for the
game are at the very end of the lesson.
Lessons 10–11: Angle Problems and Solving Equations: for lesson 11, students will need protractors.
Lesson 12: Properties of Inequalities: for the sprint exercise in the beginning, make sure to provide solutions to problems that
were not solved by the students. You will also need dice (enough to split the class up into four groups) for the second activity.
Lesson 13: Inequalities
Lesson 14: Solving Inequalities
Lesson 15: Graphing Solutions to Inequalities
Topic C: Use Equations and Inequalities to Solve Geometry Problems (7.G.B.4, 7.G.B.6)
Lesson 16: The Most Famous Ratio of All: students will need a compass in this lesson
Lesson 17: The Area of a Circle
Lesson 18: More Problems on Area and Circumference
Lesson 19: Unknown Area Problems on the Coordinate Plane: for example 1, circulate the room to check student progress and to ensure
that students are drawing the figures correctly.
Lesson 20: Composite Area Problems
Lessons 21–22: Surface Area. Use a transparency to show students how the nets overlap where the lateral faces together form a longer
rectangular region and the bases are represented by "wings" on either side of that triangle. You may want to use student work for this if
you see a good example.
Lessons 23–24: The Volume of a Right Prism. The Opening Exercise for Lesson 24 requires a small amount of water. Have an absorbent
towel available to soak up the water at the completion of the exercise.
Lessons 25–26: Volume and Surface Area
Foundational
Standards
Understand and apply properties of operations and the relationship between addition and subtraction
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
1.OA.3
Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known.
(Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12.
(Associative property of addition.)
Understand properties of multiplication and the relationship between multiplication and division.
3.OA.5
Apply properties of operations as strategies to multiply and divide.2 Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known.
(Commutative property of multiplication.) 3 x 5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30.
(Associative property of multiplication.) Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 +
16 = 56. (Distributive property.)
Geometric measurement: understand concepts of angle and measure angles.
4.MD.C.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of
angle measurement:
a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of
the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a
“one degree angle,” and can be used to measure angles.
b. An angle that turns through 𝑛 one-degree angles is said to have an angle measure of 𝑛 degrees.
4.MD.C.6
Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
4.MD.C.7
Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the
sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world
and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.
Apply and extend previous understandings of arithmetic to algebraic expressions.
6.EE.A.3
Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression
3(2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the
equivalent expression 6(4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
6.EE.A.4
Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is
substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of
which number y stands for.
Reason about and solve one-variable equations and inequalities.
6.EE.B.6
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a
variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
6.EE.B.7
Solve real-world and mathematical problems by writing and solving equations in the form x + p = q and px = q for cases in which p, q
and x are all nonnegative rational numbers.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
6.EE.B.8
Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world mathematical problem. Recognize
that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line
diagrams.
Solve real-world and mathematical problems involving area, surface area, and volume.
6.G.A.1
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into
triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
6.G.A.2
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction
edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the
formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems.
6.G.A.4
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these
figures. Apply these techniques in the context of solving real-world and mathematical problems.
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
7.NS.A.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.A.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 realworld 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.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
Suggested Big Idea
Content Emphasis Cluster
Mathematical Practices
Common Assessment
Graduate Competency
CCSS Priority Standards
CCSS.MATH.CONTENT.7.EE.3
Solve multi-step real-life and mathematical
problems posed with positive and negative
rational numbers in any form (whole
numbers, fractions, and decimals), using
tools strategically. Apply properties of
operations to calculate with numbers in any
form; convert between forms as appropriate;
and assess the reasonableness of answers
using mental computation and estimation
strategies. For example: If a woman making
$25 an hour gets a 10% raise, she will make
an additional 1/10 of her salary an hour, or
$2.50, for a new salary of $27.50. If you
want to place a towel bar 9 3/4 inches long
in the center of a door that is 27 1/2 inches
wide, you will need to place the bar about 9
inches from each edge; this estimate can be
used as a check on the exact computation.
CCSS.MATH.CONTENT.7.RP.2
Recognize and represent proportional
Module 4: Percent and Proportional Relationships
Use properties of operations to generate equivalent expressions.
Analyze proportional relationships and use them to solve real-world and mathematical problems.
Draw, constructs, and describes geometrical figures and describes the relationships between them.
MP.1. Make sense of problems and persevere in solving them.
MP.2. Reason abstractly and quantitatively.
MP.3. Construct viable arguments and critique the reasoning of others.
MP.4. Model with mathematics.
MP.5. Use appropriate tools strategically.
MP.6. Attend to precision.
MP.7. Look for and make use of structure.
MP.8. Look for and express regularity in repeated reasoning.
End of Module Assessment
Prepared graduates use critical thinking to recognize problematic aspects of situations, create mathematical models, and present
and defend solutions
Prepared graduates make both relative (multiplicative) and absolute (arithmetic) comparisons between quantities. Multiplicative
thinking underlies proportional reasoning
Prepared graduates apply transformation to numbers, shapes, functional representations, and data
Cross-Content
Writing Focus
Language/Vocabulary Misconceptions
Connections
Literacy Connections
RST.6-8.4
Determine the meaning
of symbols, key terms,
and other domainspecific words and
phrases as they are used
in a specific scientific
or technical context
relevant to grades 6-8
texts and topics.
RST.6-8.5
Analyze the structure an
author uses to organize
a text, including how
the major sections
contribute to the whole
and to an understanding
of the topic.
RST.6-8.7
Writing Connection
WHST.6-8.2
Write
informative/explanatory
texts, including the
narration of historical
events, scientific
procedures/
experiments, or
technical processes.
p.Introduce a topic
clearly, previewing
what is to follow;
organize ideas,
concepts, and
information into
broader categories as
appropriate to
achieving purpose;
include formatting (e.g.,
headings), graphics
(e.g., charts, tables),
Academic VocabularyCross discipline
language- Compute,
discount, estimate,
identify, percent,
table, tax, unit cost
Technical VocabularyAbsolute Value,
Percent Error,
Complex Fraction,
Constant of
Proportionality,
Discount Price,
Expression, Equation,
Equivalent Ratios,
Fee, Fraction,
Greatest Common
Factor, Length of a
Segment, One-toOne
As students begin to build and work with
expressions containing more than two
operations, students tend to set aside the
order of operations. For example having a
student simplify an expression like 8 + 4(2x
- 5) + 3x can bring to light several
misconceptions. Do the students
immediately add the 8 and 4 before
distributing the 4? Do they only multiply the
4 and the 2x and not distribute the 4 to both
terms in the parenthesis? Do they collect all
like terms
8 + 4 – 5, and 2x + 3x? Each of these show
gaps in students’ understanding of how to
simplify numerical expressions with
multiple operations.
A common error is to reverse the position of
the variables when writing equations.
Students may find it useful to use variables
specifically related to the quantities rather
than using x and y.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
relationships between quantities.
CCSS.MATH.CONTENT.7.RP.A.2.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.
CCSS.MATH.CONTENT.7.RP.A.2.B
Identify the constant of proportionality (unit
rate) in tables, graphs, equations, diagrams,
and verbal descriptions of proportional
relationships.
CCSS.MATH.CONTENT.7.RP.A.2.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.
CCSS.MATH.CONTENT.7.RP.A.2.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.
CCSS.MATH.CONTENT.7.G.1
Solve problems involving scale drawings of
geometric figures, including computing
actual lengths and areas from a scale
drawing and reproducing a scale drawing at
a different scale.
Integrate quantitative or
technical information
expressed in words in a
text with a version of
that information
expressed visually (e.g.,
in a flowchart, diagram,
model, graph, or table).
RST.6-8.8
Distinguish among facts,
reasoned judgment
based on research
findings, and
speculation in a text.
and multimedia when
useful to aiding
comprehension.
q.Develop the topic with
relevant, well-chosen
facts, definitions,
concrete details,
quotations, or other
information and
examples.
r. Use appropriate and
varied transitions to
create cohesion and
clarify the relationships
among ideas and
concepts.
s. Use precise language
and domain-specific
vocabulary to inform
about or explain the
topic.
t. Establish and maintain
a formal style and
objective tone.
f. Provide a concluding
statement or section that
follows from and
supports the
information or
explanation presented.
WHST.6-8.4
Produce clear and
coherent writing in
which the development,
organization, and style
are appropriate to task,
purpose, and audience.
Correspondence,
Original Price,
Percent, Proportional
To, Proportional
Relationship, Rate,
Ratio, Rational
Number, Sales Price,
Scale Drawing, Scale
Factor, Unit Rate
L.6-8.6
Acquire and use
accurately gradeappropriate general
academic and
domain-specific
words and phrases;
gather vocabulary
knowledge when
considering a word or
phrase important to
comprehension or
expression.
Constructing verbal models can also be
helpful. A student might describe the
situation as “the number of packs of gum
times the cost for each pack is the total cost
in dollars”. They can use this verbal model
to construct the equation. Students can
check their equation by substituting values
and comparing their results to the table. The
checking process helps student revise and
recheck their model as necessary. The
number of packs of gum times the cost for
each pack is the total cost (g x 2 = d).
Student’s may have misconceptions about
correctly setting up proportions, how to read
a ruler, doubling side measures, and does
not double perimeter.
L.6-8.4
Determine or clarify
the meaning of
unknown and
multiple-meaning
words and phrases
choosing flexibly
from a range of
strategies.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
Percent and Proportional Relationships


Module 4


Topic A: Finding the Whole (7.RP.A.1, 7.RP.A.2c, 7.RP.A.3)
Topic B: Percent Problems Including More than One Whole (7.RP.A.1, 7.RP.A.2,
7.RP.A.3, 7.EE.B.3)
Topic C: Scale Drawings (7.RP.A.2b, 7.G.A.1)
Topic D: Population, Mixture, and Counting Problems Involving Percents
(7.RP.A.2c, 7.RP.A.3, 7.EE.B.3)
Length of Unit
25 Days
January 12-Februrary 18
Analyze proportional relationships and use them to solve real-world and mathematical problems.
7.RP.A.1
Compute unit rates associated with ratios 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 ½ / ¼ miles per hour,
equivalently 2 miles per hour.
Content
Standards
(Priority
Standards)
7.RP.A.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 at 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.
7.RP.A.3
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns,
gratuities and commissions, fees, percent increase and decrease, percent error.
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
7.EE.B.3
Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers,
fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert
between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For
example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new
salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 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.
Draw, construct, and describe geometrical figures and describe the relationships between them.
7.G.A.1
Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and
reproducing a scale drawing at a different scale.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
See Knowledge Packets for specific questions.
Inquiry
Questions
How do we calculate percent of increase or decrease using proportions?
Why isn't a discount of 25% followed by a discount of 10% equivalent to a discount of 35%?
How could you compare 15% of 100 to 100% of 15?
How so we use ratios and percentages in the real world?
Key Knowledge and Skills (Procedural Skill and Application)
My students will be able to (Do)…


Solve problems involving percents of numbers, discounts, taxes, simple interest, and percent increase or decrease.
Use proportional relationships to solve multistep ratio and percent problems.
Resources
Technology
Technology links that provide ways for students to deepen their understanding of the mathematics in the unit and can be used to differentiate
student learning.-Vertical teaming – calculators, video links;
Materials
End of Unit Common Assessment on Schoolcity:


Performance/Learning
Tasks (Assessments)
Scanned into School City or students take the assessment online
Should be in addition to individually developed formative assessments
Pre Assessment Module 4 Topic B
Post Assessment Module 4 Topic B
Pre Assessment Module 4 Topic C and D
Post Assessment Module 4 Topic C and D
***Please see the Knowledge Packets for specific lesson information.
Instructional Notes
Topic A: Finding the Whole (7.RP.A.1, 7.RP.A.2c, 7.RP.A.3)
Lesson 1: Percent
Lesson 2: Part of a Whole as a Percent
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
Lesson 3: Comparing Quantities with Percent
Lesson 4: Percent Increase and Decrease
Lesson 5: Finding One-Hundred Percent Given Another Percent
Lesson 6: Fluency with Percents
Topic B: Percent Problems Including More than One Whole (7.RP.A.1, 7.RP.A.2, 7.RP.A.3, 7.EE.B.3)
Lesson 7: Markup and Markdown Problems
Lesson 10: Simple Interest
Lesson 11: Tax, Commissions, Fees, and Other Real-World Percent Problems
Topic C: Scale Drawings (7.RP.A.2b, 7.G.A.1)
Lesson 12: The Scale Factor as a Percent for a Scale Drawing
Lesson 13: Changing Scales
Lesson 14: Computing Actual Lengths from a Scale Drawing
Lesson 15: Solving Area Problems Using Scale Drawings
**Look at MathScapes Getting In Shape lesson 4 to see if it would be an addition or in place of lesson for Topic C. Also, look at MathScapes
From the Ground Up lessons 1-3 so see if these could be used in place of some lessons in Topic C.
Topic D: Population, Mixture, and Counting Problems Involving Percents (7.RP.A.2c, 7.RP.A.3, 7.EE.B.3)
Lesson 16: Population Problems
Lesson 17: Mixture Problems
Lesson 18: Counting Problems
Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.A.1
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The
ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” Or, “For every vote
candidate A received, candidate C received nearly three votes.”
6.RP.A.2
Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio
relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of
sugar.” Or, “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”4 4 Expectations for unit rates in this grade are
limited to non‐complex fractions.
6.RP.A.3
Use ratio and rate reasoning to solve real‐world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape
diagrams, double number line diagrams, or equations.
a. Make tables of equivalent ratios relating quantities with whole‐number measurements, find missing values in the tables, and plot
the pairs of values on the coordinate plane. Use tables to compare ratios.
Foundational
Standards
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns,
then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving
finding the whole, given a part and the percent.
d. Use ratio reasoning to convert measurement units; manipulate and transform units.
Solve real‐world and mathematical problems involving area, surface area, and volume.
6.G.A.1
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing
into triangles and other shapes; apply these techniques in the context of solving real‐world and mathematical problems.
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
7.NS.A.1b
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and
subtraction on a horizontal or vertical number line diagram.
b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is
positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers
by describing real‐world contexts.
7.NS.A.3 Solve real-world and mathematical problems involving the four operations with rational numbers.5
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
7.EE.B.4a
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?
4 Expectations
for unit rates in this grade are limited to non-complex fractions.
with rational numbers extend the rules for manipulating fractions to complex fractions.
5 Computations
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
Suggested Big Idea
Content Emphasis Cluster
Mathematical Practices
Common Assessment
Graduate Competency
CCSS Priority Standards
CCSS.MATH.CONTENT.7.SP.B.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.
CCSS.MATH.CONTENT.7.SP.C.8
Find probabilities of compound events
using organized lists, tables, tree diagrams,
and simulation.
CCSS.MATH.CONTENT.7.SP.C.8.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.
CCSS.MATH.CONTENT.7.SP.C.8.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
Module 5: Statistics and Probability
Draw informal comparative inferences about two populations.
Investigate chance processes and develop, use, and evaluate probability models.
CCSS.MATH.CONTENT.7.SP.B.4
CCSS.MATH.CONTENT.7.SP.C.8
MP.1. Make sense of problems and persevere in solving them.
MP.1. Make sense of problems and persevere in solving
MP.2. Reason abstractly and quantitatively.
them.
MP.3. Construct viable arguments and critique the reasoning of
MP.2. Reason abstractly and quantitatively.
others.
MP.4. Model with mathematics.
MP.4. Model with mathematics.
MP.5. Use appropriate tools strategically.
MP.5. Use appropriate tools strategically.
MP.7. Look for and make use of structure.
MP.6. Attend to precision.
MP.8. Look for and express regularity in repeated reasoning
MP.7. Look for and make use of structure.
End of Module Assessment
Prepared graduates use critical thinking to recognize problematic aspects of situations, create mathematical models, and present
and defend solutions
Cross-Content
Writing Focus
Language/Vocabulary Misconceptions
Connections
Literacy Connections
RST.6-8.4
Determine the meaning of
symbols, key terms, and
other domain-specific
words and phrases as
they are used in a
specific scientific or
technical context
relevant to grades 6-8
texts and topics.
RST.6-8.5
Analyze the structure an
author uses to organize
a text, including how
the major sections
contribute to the whole
and to an understanding
of the topic.
RST.6-8.7
Integrate quantitative or
technical information
expressed in words in a
text with a version of
that information
expressed visually (e.g.,
Writing Connection
WHST.6-8.2
Write
informative/explanatory
texts, including the
narration of historical
events, scientific
procedures/
experiments, or
technical processes.
u. Introduce a topic
clearly, previewing
what is to follow;
organize ideas,
concepts, and
information into
broader categories as
appropriate to
achieving purpose;
include formatting (e.g.,
headings), graphics
(e.g., charts, tables),
and multimedia when
useful to aiding
comprehension.
v. Develop the topic with
relevant, well-chosen
Academic Vocabularydraw, construct,
describe, recognize
Technical VocabularyProbability, Probability
Model, Uniform
Probability Model,
Compound Event,
Tree Diagram,
Simulation, Random
Sample, Inference,
Measures of Center,
Measures of
Variability, Mean
Absolute Deviation,
Shape
Students often expect the theoretical and
experimental probabilities of the same data
to match.
By providing multiple opportunities for
students to experience simulations of
situations in order to find and compare the
experimental probability to the theoretical
probability, students discover that rarely are
those probabilities the same.
Students often expect that simulations will
result in all of the possibilities. All
possibilities may occur in a simulation, but
not necessarily. Theoretical probability does
use all possibilities. Note examples in
simulations when some possibilities are not
shown.
L.6-8.6
Acquire and use
accurately gradeappropriate general
academic and
domain-specific
words and phrases;
gather vocabulary
knowledge when
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
compose the event.
CCSS.MATH.CONTENT.7.SP.C.8.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?
in a flowchart, diagram,
model, graph, or table).
RST.6-8.8
Distinguish among facts,
reasoned judgment
based on research
findings, and
speculation in a text.
facts, definitions,
concrete details,
quotations, or other
information and
examples.
w.
Use appropriate
and varied transitions to
create cohesion and
clarify the relationships
among ideas and
concepts.
x. Use precise language
and domain-specific
vocabulary to inform
about or explain the
topic.
y. Establish and maintain
a formal style and
objective tone.
f. Provide a concluding
statement or section that
follows from and
supports the
information or
explanation presented.
considering a word or
phrase important to
comprehension or
expression.
L.6-8.4
Determine or clarify
the meaning of
unknown and
multiple-meaning
words and phrases
choosing flexibly
from a range of
strategies.
WHST.6-8.4
Produce clear and
coherent writing in
which the development,
organization, and style
are appropriate to task,
purpose, and audience.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
Statistics and Probability

Module 5



Topic A: Calculating and Interpreting Probabilities (7.SP.C.5, 7.SP.C.6, 7.SP.C.7,
7.SP.C.8a, 7.SP.C.8b)
Topic B: Estimating Probabilities (7.SP.C.6, 7.SP.C.7, 7.SP.C.8c)
Topic C: Random Sampling and Estimated Population Characteristics (7.SP.A.1,
7.SP.A.2)
Topic D: Comparing Populations (7.SP.B.3, 7.SP.B.4)
Length of Unit
30 Days
February 19-April 7
Using random sampling to draw inferences about a population.
7.SP.A.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.A.2
Content Standards
(Priority Standards)
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.B.3
Informally assess the degree of visual overlap of two numerical data distributions with similar variability, 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.B.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.
Investigate chance processes and develop, use, and evaluate probability models.
7.SP.C.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 1/2 indicates an event that is neither unlikely nor likely,
and a probability near 1 indicates a likely event.
7.SP.C.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.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
7.SP.C.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 frequencies?
7.SP.C.8
Inquiry Questions
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?
See Knowledge Packets for specific questions.
Why do we investigate samples rather entire populations?
Describe a game where chance was involved?
If you stand next to a door of a shopping store and ask every person who walks in about their favorite show brand, describe your
prediction of the results of the survey.
Is it possible to use probability to predict the future? Why or why not?
What does a probability near 0 indicate? Around 1/2 indicate? Around 1 indicate?
What is a compound event?
What are models that can be used to represent probability outcomes for compound events?
Key Knowledge and Skills (Procedural Skill and Application)
My students will be able to (Do)...




Understand 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 random sampling tends to produce representative samples and support valid inferences
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
Assess the degree of visual overlap Informally of two numerical data distributions with similar variables, measuring the difference between the
centers by expressing it as a multiple of a measure of variability
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016











Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two
populations
Understand probability of a chance event as a number between 0 and 1 that expresses the likelihood of the event occurring; larger numbers indicate
greater likelihood
Recognize a probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a
probability near 1 indicates a likely event
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
Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy
Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events
Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process
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
Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams
Identify the outcomes in a sample space for an event described in everyday language
Design and use a simulation to generate frequencies for compound events
Resources
Technology
Technology links that provide ways for students to deepen their understanding of the mathematics in the unit and can be used to differentiate
student learning.-Vertical teaming – calculators, video links;
Materials
Histograms, Graphing Calculator, Dot Plots, Spinners, coins,
End of Unit Common Assessment on Schoolcity:


Performance/Learning
Tasks (Assessments)
Scanned into School City or students take the assessment online
Should be in addition to individually developed formative assessments
Pre Assessment Module 5 Topic A and B
Post Assessment Module 5 Topic A and B
Pre Assessment Module 5 Topic D
Post Assessment Module 5 Topic D
***Please see the Knowledge Packets for specific lesson information.
Instructional Notes
Topic A: Calculating and Interpreting Probabilities (7.SP.C.5, 7.SP.C.6, 7.SP.C.7, 7.SP.C.8a, 7.SP.C.8b)
Lesson 1: Chance Experiments
*Paperclip spinner needed, Spinners, probability Line
Lesson 2: Estimating Probabilities by Collecting Data
*A box of animal crackers for Example 2
*Paperclip spinner needed, Spinners
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
*See Teachers Resources in Engage for Line Plots, spinners, and charts
Lesson 3: Chance Experiments with Equally Likely Outcomes
*Possible need to explain Sample Spaces
Lesson 4: Calculating Probabilities for Chance Experiments with Equally Likely Outcomes
*Explaining Theoretical Probability with replacement
Lesson 5: Chance Experiments with Outcomes that Are Not Equally Likely
*Adding fractions with unlike denominators
Lesson 6: Using Tree Diagrams to Represent a Sample Space and to Calculate Probabilities
Lesson 7: Calculating Probabilities of Compound Events
*Adding and Multiplying decimals
Topic B: Estimating Probabilities (7.SP.C.6, 7.SP.C.7, 7.SP.C.8c)
Lesson 8: The Difference Between Theoretical Probabilities and Estimated Probabilities
*Uses Coin and Results Charts
Lesson 9: Comparing Estimated Probabilities to Probabilities Predicted by a Model
*2 Bags with chips in them
Lessons 10–11: Using Simulation to Estimate a Probability
*Dice and coin
Lesson 12: Using Probability to Make Decisions
*Multiple devices for finding probability
*cups/bags and different colored items
Topic C: Random Sampling and Estimated Population Characteristics (7.SP.A.1, 7.SP.A.2)
Lesson 13: Populations, Samples, and Generalizing from a Sample to a Population
*Copy of the Census from Teachers Guide
Lesson 14: Selecting a Sample
Lesson 15: Random Sampling
*Collection of pennies
*Grocery store price lists from teachers guide
Lesson 16: Methods for Selecting a Random Sample
*Random number generator
www.rossmanchance.com/applets/RandomGen/GenRandom01.htm
Lesson 17: Sampling Variability
Lesson 18: Estimating a Population Mean
Lesson 19: Understanding Variability when Estimating a Population Proportion
Lesson 20: Estimating a Population Proportion
Topic D: Comparing Populations (7.SP.B.3, 7.SP.B.4)
for these lessons, use the generated tables and charts in the Copy Ready section from EngageNY
Lesson 21: Why Worry About Sampling Variability?
Lessons 22–23: Using Sample Data to Decide if Two Population Means Are Different
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
Foundational
Standards
Summarize and describe distributions.
6.SP.B.5
Summarize numerical data sets in relation to their context, such as by:
a. Reporting the number of observations.
b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as
well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the
data were gathered.
d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were
gathered.
Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.A.3c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving
finding the whole, given a part and the percent.
Analyze proportional relationships and use them to solve real‐world and mathematical problems.
7.RP.A.2 Recognize and represent proportional relationships between quantities.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
Suggested Big Idea
Content Emphasis Cluster
Mathematical Practices
Common Assessment
Graduate Competency
CCSS Priority Standards
CCSS.MATH.CONTENT.7.G.B.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.
Module 6: Geometry
Solve real-life and mathematical problems involving angle measure, area, surface area, and
volume.
MP.1. Make sense of problems and persevere in solving them.
MP.2. Reason abstractly and quantitatively.
MP.3. Construct viable arguments and critique the reasoning of others.
MP.4. Model with mathematics.
MP.5. Use appropriate tools strategically.
MP.6. Attend to precision.
MP.7. Look for and make use of structure.
MP.8. Look for and express regularity in repeated reasoning.
End of Module Assessment
Prepared graduates understand quantity through estimation, precision, order of magnitude, and comparison. The reasonableness of
answers relies on the ability to judge appropriateness, compare, estimate, and analyze error
Cross-Content
Writing Focus
Language/Vocabulary Misconceptions
Connections
Literacy Connections
RST.6-8.4
Determine the meaning of
symbols, key terms, and
other domain-specific
words and phrases as
they are used in a
specific scientific or
technical context
relevant to grades 6-8
texts and topics.
RST.6-8.5
Analyze the structure an
author uses to organize
a text, including how
the major sections
contribute to the whole
and to an understanding
of the topic.
RST.6-8.7
Integrate quantitative or
technical information
expressed in words in a
text with a version of
that information
expressed visually (e.g.,
Writing Connection
WHST.6-8.2
Write
informative/explanatory
texts, including the
narration of historical
events, scientific
procedures/
experiments, or
technical processes.
z. Introduce a topic
clearly, previewing
what is to follow;
organize ideas,
concepts, and
information into
broader categories as
appropriate to
achieving purpose;
include formatting (e.g.,
headings), graphics
(e.g., charts, tables),
and multimedia when
useful to aiding
comprehension.
aa.
Develop the
topic with relevant,
well-chosen facts,
definitions, concrete
Academic Vocabulary-
Composition, draw,
protractor, ruler,
slice, threedimensional, twodimensional
Technical VocabularyCorrespondence,
Identical Triangles,
Unique Triangle,
Right Rectangular
Pyramid, Surface of
a Pyramid, Vertical
Angles, Adjacent
Angles,
Complementary
Angles,
Supplementary
Angles, Angles on a
Line, Angles on a
Point, Right
rectangular Prism
Students may believe:
Pi is an exact number rather than
understanding that 3.14 is just an
approximation of pi.
Many students are confused when dealing
with circumference (linear measurement)
and area. This confusion is about an
attribute that is measured using linear units
(surrounding) vs. an attribute that is
measured using area units (covering).
L.6-8.6
Acquire and use
accurately grade-
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
in a flowchart, diagram,
model, graph, or table).
RST.6-8.8
Distinguish among facts,
reasoned judgment
based on research
findings, and
speculation in a text.
details, quotations, or
other information and
examples.
bb.
Use appropriate
and varied transitions to
create cohesion and
clarify the relationships
among ideas and
concepts.
cc.
Use precise
language and domainspecific vocabulary to
inform about or explain
the topic.
dd.
Establish and
maintain a formal style
and objective tone.
f. Provide a concluding
statement or section that
follows from and
supports the
information or
explanation presented.
appropriate general
academic and
domain-specific
words and phrases;
gather vocabulary
knowledge when
considering a word or
phrase important to
comprehension or
expression.
L.6-8.4
Determine or clarify
the meaning of
unknown and
multiple-meaning
words and phrases
choosing flexibly
from a range of
strategies.
WHST.6-8.4
Produce clear and
coherent writing in
which the development,
organization, and style
are appropriate to task,
purpose, and audience.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
Geometry
Module 6





Topic A: Unknown Angles (7.G.B.5)
Topic B: Constructing Triangles (7.G.A.2)
Topic C: Slicing Solids (7.G.A.3)
Topic D: Problems Involving Area and Surface Area (7.G.B.6)
Topic E: Problems Involving Volume (7.G.B.6)
Length of Unit
31 Days
April 8-May 20
Draw, construct, and describe geometrical figures and describe the relationships between them.
7.G.A.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.A.3
Content Standards
(Priority Standard)
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations
for an unknown angle in a figure.
7.G.B.6
Inquiry Questions
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-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.
See Knowledge Packets for specific questions.
 Why is pi an important number?
 How many two-dimensional shapes can you make by slicing a three-dimensional object?
 How are the circumference and diameter of a circle related?
 How does the derivation of the formula for the area of a circle rely on both the circumference and radius of the circle?
 Why is the scale factor for side lengths and perimeters different from the one for areas?
 What are the formulas to find circumference of circles?
 What is the formula to find area of circles?
 How is the volume of pyramids related to the rectangular prism they fit into?
 How do nets help us find three-dimensional objects’ surface areas?
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
Key Knowledge and Skills (Procedural Skill and Application)
My students will be able to (Do)…






Draw (freehand, with ruler and protractor, with technology) geometric shapes with given condition, such as triangles from three measures of angles
or sides, noting when conditions determine unique triangles, more than one triangle, or no triangle.
Describe two-dimensional figures that result from slicing three-dimensional figures.
Know formulas for area and circumference of circles and use them to solve problems.
Use facts about supplementary, complementary, vertical, and adjacent angles in multi-step problems to write and solve simple equations for
unknown angles in figures.
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.
Know formulas for volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
Resources
Technology
Materials
Technology links that provide ways for students to deepen their understanding of the mathematics in the unit and can be used to differentiate
student learning.-Vertical teaming – calculators, video links;
Rulers, Protractors, Compass, Grid Paper
End of Unit Common Assessment on Schoolcity:
Performance/Learning
Tasks (Assessments)


Scanned into School City or students take the assessment online
Should be in addition to individually developed formative assessments
Pre Assessment Module 6 Topic D and E
Post Assessment Module 6 Topic D and E
***Please see the Knowledge Packets for specific lesson information.
Instructional Notes
**Module 6 is less focused on 2D geometry and formulas and is heavy on constructions and cross sections. Practice of 2D Geometry is woven
throughout other modules but can be revisited here as desired.
Topic A: Unknown Angles (7.G.B.5)
Lesson 1: Complementary and Supplementary Angles
Lessons 2–4: Solve for Unknown Angles using Equations
Topic B: Constructing Triangles (7.G.A.2)
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
Lesson 5: Unique Triangles
Lesson 6: Drawing Geometric Shapes
Lesson 7: Drawing Parallelograms
Lesson 8: Drawing Triangles
Lesson 9: Conditions for a Unique Triangle―Three Sides and Two Sides and the Included Angle
Lesson 10: Conditions for a Unique Triangle—Two Angles and a Given Side
Lesson 11: Conditions on Measurements that Determine a Triangle
Lesson 12: Unique Triangles―Two Sides and a Non-Included Angle
Lessons 13–14: Checking for Identical Triangles
Lesson 15: Using Unique Triangles to Solve Real-World and Mathematical Problems
Topic C: Slicing Solids (7.G.A.3)
Lessons 16: Slicing a Right Rectangular Prism with a Plane
Lesson 17: Slicing a Right Rectangular Pyramid with a Plane
Lesson 18: Slicing on an Angle
Lesson 19: Understanding Three-Dimensional Figures
Topic D: Problems Involving Area and Surface Area (7.G.B.6) (Combine with Topic E)
Lesson 20: Real-World Area Problems
Lesson 21: Mathematical Area Problems
Lesson 22: Area Problems with Circular Regions
Lessons 23–24: Surface Area
Topic E: Problems Involving Volume (7.G.B.6) (Combine with Topic D)
Lesson 25: Volume of Right Prisms
Lesson 26: Volume of Composite Three-Dimensional Objects
Lesson 27: Real-World Volume Problems
End-of-Module Assessment and Rubric
 For question 5, area of circles was taught in unit 3.
Foundational
Standards
Geometric measurement: understand concepts of angle and measure angles.
4.MD.C.7 Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the
sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real-world
and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.
Solve real-world and mathematical problems involving area, surface area, and volume.
6.G.A.1
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into
triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
Revised 6/16/2015
Greeley-Evans School District 6- 7th Grade: 2015-2016
6.G.A.2
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction
edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the
formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems.
6.G.A.4
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these
figures. Apply these techniques in the context of solving real-world and mathematical problems.
Solve real-life and mathematical problems involving area, surface area, and volume.
7.G.B.4
Know the formulas for 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.
i
Expectations for unit rates in this grade are limited to non-complex fractions.
ii
Computations with rational numbers extend the rules for manipulating fractions to complex fractions.
Revised 6/16/2015