Grade 7 Overview
Ratios and Proportional Relationships (RP)
 Analyze proportional relationships and use them to solve real-world and
mathematical problems. (1, 2, 3)
The Number System (NS)
 Apply and extend previous understandings of operations with fractions
to add, subtract, multiply, and divide rational numbers. (1, 2, 3)
Expressions and Equations (EE)
 Use properties of operations to generate equivalent expressions. (1, 2)
 Solve real-life and mathematical problems using numerical and
algebraic expressions and equations. (3, 4)
Mathematical Practices (MP)
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.
Geometry (G)
 Draw, construct and describe geometrical figures and describe the
relationships between them. (1, 2, 3)
 Solve real-life and mathematical problems involving angle measure,
area, surface area, and volume. (4, 5, 6)
Statistics and Probability (SP)
 Use random sampling to draw inferences about a population. (1, 2)
 Draw informal comparative inferences about two populations. (3, 4)
 Investigate chance processes and develop, use, and evaluate
probability models. (5, 6, 7,8)
*Numbers following cluster heading indicate standards included in the cluster.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
1
In Grade 7, instructional time should focus on four critical areas: (1) developing understanding of and applying proportional relationships; (2)
developing understanding of operations with rational numbers and working with expressions and linear equations; (3) solving problems
involving scale drawings and informal geometric constructions, and working with two- and three-dimensional shapes to solve problems
involving area, surface area, and volume; and (4) drawing inferences about populations based on samples.
(1) Students extend their understanding of ratios and develop understanding of proportionality to solve single- and multi-step problems. Students use
their understanding of ratios and proportionality to solve a wide variety of percent problems, including those involving discounts, interest, taxes, tips,
and percent increase or decrease. Students solve problems about scale drawings by relating corresponding lengths between the objects or by using
the fact that relationships of lengths within an object are preserved in similar objects. Students graph proportional relationships and understand the
unit rate informally as a measure of the steepness of the related line, called the slope. They distinguish proportional relationships from other
relationships.
(2) Students develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal representation),
and percents as different representations of rational numbers. Students extend addition, subtraction, multiplication, and division to all rational
numbers, maintaining the properties of operations and the relationships between addition and subtraction, and multiplication and division. By applying
these properties, and by viewing negative numbers in terms of everyday contexts (e.g., amounts owed or temperatures below zero), students explain
and interpret the rules for adding, subtracting, multiplying, and dividing with negative numbers. They use the arithmetic of rational numbers as they
formulate expressions and equations in one variable and use these equations to solve problems.
(3) Students continue their work with area from Grade 6, solving problems involving the area and circumference of a circle and surface area of threedimensional objects. In preparation for work on congruence and similarity in Grade 8 they reason about relationships among two-dimensional figures
using scale drawings and informal geometric constructions, and they gain familiarity with the relationships between angles formed by intersecting
lines. Students work with three-dimensional figures, relating them to two-dimensional figures by examining cross-sections. They solve real-world and
mathematical problems involving area, surface area, and volume of two- and three-dimensional objects composed of triangles, quadrilaterals,
polygons, cubes and right prisms.
(4) Students build on their previous work with single data distributions to compare two data distributions and address questions about differences
between populations. They begin informal work with random sampling to generate data sets and learn about the importance of representative
samples for drawing inferences.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
2
Standards for Mathematical Practice
Standards
Mathematical Practices are
Students are expected to:
listed throughout the grade
level document in the 2nd
column to reflect the need to
connect the mathematical
practices to mathematical
content in instruction.
7.MP.1. Make sense of
problems and persevere in
solving them.
7.MP.2. Reason abstractly
and quantitatively.
7.MP.3. Construct viable
arguments and critique the
reasoning of others.
7.MP.4. Model with
mathematics.
Explanations and Examples
In grade 7, students solve problems involving ratios and rates and discuss how they solved
them. Students solve real world problems through the application of algebraic and geometric
concepts. Students seek the meaning of a problem and look for efficient ways to represent and
solve it. They may check their thinking by asking themselves, “What is the most efficient way to
solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?”.
In grade 7, students represent a wide variety of real world contexts through the use of real
numbers and variables in mathematical expressions, equations, and inequalities. Students
contextualize to understand the meaning of the number or variable as related to the problem and
decontextualize to manipulate symbolic representations by applying properties of operations.
In grade 7, students construct arguments using verbal or written explanations accompanied by
expressions, equations, inequalities, models, and graphs, tables, and other data displays (i.e.
box plots, dot plots, histograms, etc.). They further refine their mathematical communication
skills through mathematical discussions in which they critically evaluate their own thinking and
the thinking of other students. They pose questions like “How did you get that?”, “Why is that
true?” “Does that always work?”. They explain their thinking to others and respond to others’
thinking.
In grade 7, students model problem situations symbolically, graphically, tabularly, and
contextually. Students form expressions, equations, or inequalities from real world contexts and
connect symbolic and graphical representations. Students explore covariance and represent two
quantities simultaneously. They use measures of center and variability and data displays (i.e.
box plots and histograms) to draw inferences, make comparisons and formulate predictions.
Students use experiments or simulations to generate data sets and create probability models.
Students need many opportunities to connect and explain the connections between the different
representations. They should be able to use all of these representations as appropriate to a
problem context.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
3
Standards for Mathematical Practice
Standards
Mathematical Practices are
Students are expected to:
listed throughout the grade
level document in the 2nd
column to reflect the need to
connect the mathematical
practices to mathematical
content in instruction.
7.MP.5. Use appropriate
tools strategically.
Standards for Mathematical Practice
Standards
Mathematical Practices are
Students are expected to:
listed throughout the grade
level document in the 2nd
column to reflect the need to
connect the mathematical
practices to mathematical
content in instruction.
7.MP.6. Attend to
precision.
7.MP.7. Look for and make
use of structure.
Explanations and Examples
Students consider available tools (including estimation and technology) when solving a
mathematical problem and decide when certain tools might be helpful. For instance, students in
grade 7 may decide to represent similar data sets using dot plots with the same scale to visually
compare the center and variability of the data. Students might use physical objects or applets to
generate probability data and use graphing calculators or spreadsheets to manage and
represent data in different forms.
Explanations and Examples
In grade 7, students continue to refine their mathematical communication skills by using clear
and precise language in their discussions with others and in their own reasoning. Students
define variables, specify units of measure, and label axes accurately. Students use appropriate
terminology when referring to rates, ratios, probability models, geometric figures, data displays,
and components of expressions, equations or inequalities.
Students routinely seek patterns or structures to model and solve problems. For instance,
students recognize patterns that exist in ratio tables making connections between the constant of
proportionality in a table with the slope of a graph. Students apply properties to generate
equivalent expressions (i.e. 6 + 2x = 2 (3 + x) by distributive property) and solve equations
(i.e. 2c + 3 = 15, 2c = 12 by subtraction property of equality; c=6 by division property of equality).
Students compose and decompose two- and three-dimensional figures to solve real world
problems involving scale drawings, surface area, and volume. Students examine tree diagrams
or systematic lists to determine the sample space for compound events and verify that they have
listed all possibilities.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
4
7.MP.8. Look for and
express regularity in
repeated reasoning.
In grade 7, students use repeated reasoning to understand algorithms and make generalizations
about patterns. During multiple opportunities to solve and model problems, they may notice that
a/b ÷ c/d = ad/bc and construct other examples and models that confirm their generalization.
They extend their thinking to include complex fractions and rational numbers. Students formally
begin to make connections between covariance, rates, and representations showing the
relationships between quantities. They create, explain, evaluate, and modify probability models
to describe simple and compound events.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
5
Grade 7
Ratio and Proportional Relationships (RP) (1 Cluster)
7.RP Analyze proportional relationships and use them to solve real-world and mathematical problems (Cluster 1Standards 1, 2, and 3)
Essential Concepts
Essential Questions
 A ratio is a comparison of two quantities (by division) and usually
represents a part-to-part comparison.
 A fraction is usually a part to whole comparison or represents a division
problem.
 A quotient of a ratio is a unit rate.
 Proportionality can be determined by equivalent ratios, a constant of
proportionality, or a unit rate.
 Proportionality can be determined from a graph, table, or equation by
finding a constant of proportionality (unit rate).
 Ratio can be extended into solving single and multi-step proportionality
problems and percent problems.
 Unit rate is the slope of a proportional relationship that, when graphed,
is a linear equation that goes through the origin.
 Linear equations when graphed are straight lines.
 Every point on a graph of a proportional relationship has a meaning in
terms of the situation.
7.RP1
 Standard 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. Example,
if a person walks ½ mile in
each ¼ hour, compute the
unit rate as the complex
fraction ½/¼ miles per
hour, equivalently 2 miles
per hour.
Mathematical
Practices
7.MP.2. Reason
abstractly and
quantitatively.




How are fractions and ratios alike? How are they different?
How do you compute a unit rate given a comparison of two quantities?
How do you determine if two ratios are proportional?
Given a table, graph, equation, diagram or verbal description, how do
you determine if it is showing a proportional relationship?
 How are proportional relationships represented on a graph, a table and
an equation?
 How does an understanding of proportionality connect to solving
problems involving interest, taxes, tips and discounts?
 What are the key points on a proportional graph that best describe the
proportional relationship?
Examples & Explanations
 Example, if a person walks ½ mile in each ¼ hour, compute the unit rate as the complex fraction
½/¼ miles per hour, equivalently 2 miles per hour.
7.MP.6. Attend to
precision.
Connections: 6-8.RST.7;
SC07-S1C2-04;
ET07-S1C1-01
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
6
7.RP Analyze proportional relationships and use them to solve real-world and mathematical problems (Cluster 1Standards 1, 2, and 3)
7.RP.2
Standard 7.RP.2
Mathematical
Examples & Explanations
Recognize and represent
Students may use a content web site and/or interactive white board to create tables and graphs of
Practices
proportional relationships
between quantities.
a. Decide whether two
quantities are in a
proportional relationship,
e.g., by testing for
equivalent ratios in a
table or graphing on a
coordinate plane and
observing whether the
graph is a straight line
through the origin.
b. Identify the constant of
proportionality (unit rate)
in tables, graphs,
equations, diagrams, and
verbal descriptions of
proportional relationships.
c. Represent proportional
relationships by
equations. For example, if
total cost t is proportional
to the number n of items
purchased at a constant
price p, the relationship
between the total cost
and the number of items
can be expressed as t =
pn.
d. Explain what a point (x, y)
on the graph of a
proportional relationship
means in terms of the
situation, with special
attention to the points (0,
7.MP.1. Make
sense of problems
and persevere in
solving them.
proportional or non-proportional relationships. Graphing proportional relationships represented in a
table helps students recognize that the graph is a line through the origin (0,0) with a constant of
proportionality equal to the slope of the line.
Examples:
7.MP.2. Reason
abstractly and
quantitatively.

7.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
A student is making trail mix. Create a graph to determine if the quantities of nuts and fruit
are proportional for each serving size listed in the table. If the quantities are proportional,
what is the constant of proportionality or unit rate that defines the relationship? Explain how
you determined the constant of proportionality and how it relates to both the table and
graph.
Serving Size
Cups of Nuts (x)
Cups of Fruit (y)
7.MP.4. Model
with mathematics.
1
1
2
2
2
4
3
3
6
4
4
8
7.MP.5. Use
appropriate tools
strategically.
7.MP.6. Attend to
precision.
7.MP.7. Look for
and make use of
structure.
7.MP.8. Look for
and express
regularity in
repeated
reasoning.
The relationship is proportional. For each of the other serving sizes there are 2 cups of fruit for
every 1 cup of nuts (2:1).
The constant of proportionality is shown in the first column of the table and by the slope of the line
on the graph.

The graph below represents the cost of gum packs as a unit rate of $2 dollars for every
pack of gum. The unit rate is represented as $2/pack. Represent the relationship using a
table and an equation.
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
7
0) and (1, r) where r is the
unit rate.
Connections: 6-8.WHST.2c-f;
6-8.WHST.1c;
6-8.RST.7; 6-8.RST.4; ET07S6C2-03;
ET07-S1C1-01; SC07-S1C401;
SC07-S2C2-03
Table:
Number of Packs of Gum (g)
0
1
2
3
4
Cost in Dollars (d)
0
2
4
6
8
Equation: d = 2g, where d is the cost in dollars and g is the packs of gum

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.
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).
7.RP.3
Standard 7.RP.3
Mathematical
Use proportional relationships Practices
to solve multistep ratio and
percent problems. Examples:
simple interest, tax, markups
7.MP.1. Make
sense of problems
and persevere in
solving them.
Examples & Explanations
Examples of multistep ratio ad percent problems include, simple interest, tax, markups and
markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
8
and markdowns, gratuities
and commissions, fees,
percent increase and
decrease, percent error.
Connections: 6-8.RST.3;
SS07-S5C3-01; SC07-S4C304; SC07-S4C3-05
7.MP.2. Reason
abstractly and
quantitatively.
7.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
Students should be able to explain or show their work using a representation (numbers, words,
pictures, physical objects, or equations) and verify that their answer is reasonable. Models help
students to identify the parts of the problem and how the values are related. For percent increase
and decrease, students identify the starting value, determine the difference, and compare the
difference in the two values to the starting value.
Examples:
 Gas prices are projected to increase 124% by April 2015. A gallon of gas currently costs
$4.17. What is the projected cost of a gallon of gas for April 2015?
A student might say: “The original cost of a gallon of gas is $4.17. An increase of 100%
means that the cost will double. I will also need to add another 24% to figure out the final
projected cost of a gallon of gas. Since 25% of $4.17 is about $1.04, the projected cost of a
gallon of gas should be around $9.40.”
7.MP.4. Model
with mathematics.
7.MP.5. Use
appropriate tools
strategically.
$4.17 + 4.17 + (0.24  4.17) = 2.24 x 4.17
7.MP.6. Attend to
precision.
7.MP.7. Look for
and make use of
structure.
7.MP.8. Look for
and express
regularity in
repeated
reasoning.

100%
100%
24%
$4.17
$4.17
?
A sweater is marked down 33%. Its original price was $37.50. What is the price of the
sweater before sales tax?
37.50
Original Price of Sweater
33% of
37.50
Discount
67% of 37.50
Sale price of sweater
The discount is 33% times 37.50. The sale price of the sweater is the original price minus
the discount or 67% of the original price of the sweater, or Sale Price = 0.67 x Original
Price.
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
9

A shirt is on sale for 40% off. The sale price is $12. What was the original price? What was
the amount of the discount?
Discount
40% of original price
Sale Price - $12
60% of original price
0.60p = 12
Original Price (p)

At a certain store, 48 television sets were sold in April. The manager at the store wants to
encourage the sales team to sell more TVs and is going to give all the sales team
members a bonus if the number of TVs sold increases by 30% in May. How many TVs
must the sales team sell in May to receive the bonus? Justify your solution.

A salesperson set a goal to earn $2,000 in May. He receives a base salary of $500 as well
as a 10% commission for all sales. How much merchandise will he have to sell to meet his
goal?
After eating at a restaurant, your bill before tax is $52.60 The sales tax rate is 8%. You
decide to leave a 20% tip for the waiter based on the pre-tax amount. How much is the tip
you leave for the waiter? How much will the total bill be, including tax and tip? Express
your solution as a multiple of the bill.
The amount paid = 0.20 x $52.50 + 0.08 x $52.50 = 0.28 x $52.50
Additional Domain Information – Ratio and Proportional Relationships (RP)
Key Vocabulary (BOLD means most important)





Axis
Compare
Data
Decimal
Denominator





Dependent
Difference
Equation
Equivalence
Equivalent ratios





Fraction
Graph/coordinate
graph
Independent
Percent
Proportion





Rate
Ratio
Scale/Scale factor
Unit Rate
Variables
Example Resources

Books
 Holt – Chapter 5-1
 CMP2 – Variables ad patterns (7th grade unit)

Technology
 Illustrative Mathematics
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
10





My.hrw.com
http://www.phschool.com/cmp2/
nlvm.usu.edu/
www.quia.com
Exemplary Lessons
 Bagel Proportions- A real-life example—taken from a bagel shop, of all places—is used to get students to think about solving a problem
symbolically. Students must decipher a series of equations and interpret results to understand the point that the bagel shop’s owner is
trying to make.

Highway Robbery The National Bank of Illuminations has been robbed! Students apply their knowledge of rations, unit rates, and
proportions to sort through the clues and deduce which suspect is the true culprit.


Travel in the Solar System
This mini unit affords students the opportunity to think about two aspects of the time required to complete space travel within the solar
system. First, students consider the amount of time that space travelers must spend on the journey. Second, students think about what
kinds of events might occur on Earth while the space travelers are on their journey. Thinking about both situations improves students’
concept of time and distance as well as improves their understanding of the solar system.


Students will:
1. Practice computation of rations, unit rates, and proportions.
2. Apply skills to an authentic context, develop problem solving and deductive skills.
Students will:
3. Apply measurement and computation to gain insight into the large numbers associated with distances in space.
4. Calculate the amount of time needed to travel to each of the planets.
5. Reflect upon events that would happen on earth while they are travelling through space.
Understanding Rational Numbers and Proportions
Students use real-world models to develop an understanding of fractions, decimals proportions, and problem solving.
The three activities in this investigation center on situations involving rational numbers and proportions that students encounter at a
bakery. These activities involve several important concepts of rational numbers and proportions, including partitioning a unit into equal
parts, the quotients interpretation of fractions, the area model of fractions, determining fractional parts of a unit not cut into equal-sized
pieces, equivalence, unit prices, and multiplication of fractions.

Student will:
1. Represent parts of a whole using an area interpretation of fraction
2. Determine the fractional part of a whole when parts are not cut into equal sized pieces
3. Develop an understanding of the quotient interpretation of fractions
4. Find the unit cost of items that are part of a set
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
11
5. Determine the relationship among parts of a whole that are unequal-sized pieces
6. Express fractional parts of a whole as decimal equivalents
Assessments
All assessments for this domain must focus on assessing the degree, to which seventh grade students can, within the parameters specified in the 3
standards included in this domain,
 analyze proportional relationships and use them to solve real-world and mathematical problems.
Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments, (e.g., pre-assessments,
daily checks for understanding, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation;
summative assessments (e.g., unit assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All
district-adopted resources contain multiple assessment tools and include online resources that can be used for the purposes delineated above.
PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills,
concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The
performance measure will be administered as close to the end of the school year as possible. The second, an end of the year machine-scorable
summative assessment will be administered after approximately 90% of the school year. These assessments are to begin during the SY 2014-2015.
Common Student Misconceptions
Students have difficulty determining which quantities should be the numerator and denominator in a unit rate, such as $2.85 per gallon. Point
out that “per” means “for each” or “apiece”. The quantity after “per” is the denominator, for example, $2.85/gallon.
Students often think, because of notation, that fractions and ratios are the same thing. For example they think that because we can write the ratio
of boys to girls in the class as
2
, students think that this is fraction as opposed to the ratio. Using ratios in context is key to understanding the difference
3
between a fraction and ratio.
Students often have a difficult time understanding that fractions and ratios may represent different comparisons. It is very important when
teaching that we label
 all ratios and fractions.
Fractions always express a part-to-whole comparison, but ratios can express a part-to-whole comparison or a part-to-part comparison. For
example, the ratio of boys to girls in a class is 7:10, or it could be the ratio of boys to the total class 7:17 making it a fraction.
Students often misinterpret a ratio table as an INPUT/OUTPUT table. Students may try to create a rule rather than see the equivalency and
multiplicative relationship among ratios.
Students have difficulty understanding that every ratio can create 2 unit rates. For example, if you buy 3 oranges for $2, then you have two unit
rates; 1.5 oranges per dollar or $0.66 per orange.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
12
When graphing points on a coordinate plane, student commonly go up and down first and then left or right. Students often graph the y-value
before the x. Another way to think about it is student often plot the x-value as the y and visa versa. For example, given the point (3, -2), student will go up
three and then left 2 as opposed to going right 3 and then down 2.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
13
The Number System (NS) (1 Cluster)
7.NS Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide
rational numbers (Cluster 1- Standards 1, 2, and 3)
Essential Concepts
Essential Questions
 When opposites are combined the sum is always 0.
 A positive number is represented by moving right on a number line, a
negative number is represented moving left on a number line.
 Absolute value is the distance that number is away from zero on a
number line, also referred to as the magnitude of a number in real-world
contexts.
 Addition of rational numbers (p + q) is the number located a distance |q|
from p, in the positive or negative direction depending on whether q is
positive or negative.
 Subtraction of rational numbers is the same as adding the additive
inverse, p – q = p + (–q).
 Distance between two rational numbers on the number line is the
absolute value of their difference.
 Patterns and properties of operations are used to generate rules for
multiplying and dividing positive and negative rational numbers.
 Any rational number can be written as a fraction, decimal, percent or
quotient of integers with a non-0 divisor.
 Rational numbers can be converted to a decimal that either ends in 0 or
repeat.
 Properties of operations are used as strategies to compute real- world
problems with rational numbers.
7.NS.1
Standard 7.NS.1
Apply and extend previous
understandings of addition
and subtraction to add and
subtract rational numbers;
represent addition and
subtraction on a horizontal or
vertical number line diagram.
a. Describe situations in
which opposite quantities
combine to make 0. For
example, a hydrogen
atom has 0 charge
Mathematical
Practices
7.MP.2. Reason
abstractly and
quantitatively.
7.MP.4. Model
with mathematics.
 How do you model operations with integers using a vertical or horizontal
number line?
 How is subtraction of integers related to addition of integers?
 How is absolute value related to finding the distance between two
numbers on a number line?
 What is an example of a real-world problem that is solved by doing
addition or subtraction with a positive and negative number?
 What is an example of a real-world problem that is solved by doing
multiplication with a positive and negative number?
 What is an example of a real-world problem that is solved by doing
division with a positive and negative number?
 Why does multiplying or dividing two negatives equal a positive?
 Given a decimal, how do you determine if the decimal represents a
rational numbers?
Examples & Explanations
Visual representations may be helpful as students begin this work; they become less necessary as
students become more fluent with the operations.
Examples:
 Use a number line to illustrate:
o p-q
o p + (- q)
o Is this equation true p – q = p + (-q)
7.MP.7. Look for
and make use of
structure.
Continued on next page
Tucson Unified School District
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Grade 7
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14
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.

-3 and 3 are shown to be opposites on the number line because they are equal distance
from zero and therefore have the same absolute value and the sum of the number and it’s
opposite is zero.

A hydrogen atom has 0 charge because its two constituents are oppositely charged.

You have $4 and you need to pay a friend $3. What will you have after paying your friend?
4 + (-3) = 1 or (-3) + 4 = 1
Connections: 6-8.WHST.2f;
6-8.WHST.2b;
6-8.RST.3; 6-8.RST.7; ET07S1C1-01;
SS07-S4C5-04
Tucson Unified School District
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Grade 7
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15
7.NS Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide
rational numbers (Cluster 1- Standards 1, 2, and 3)
Essential Concepts
Essential Questions
 When opposites are combined the sum is always 0.
 A positive number is represented by moving right on a number line, a
negative number is represented moving left on a number line.
 Absolute value is the distance that number is away from zero on a
number line, also referred to as the magnitude of a number in real-world
contexts.
 Addition of rational numbers (p + q) is the number located a distance |q|
from p, in the positive or negative direction depending on whether q is
positive or negative.
 Subtraction of rational numbers is the same as adding the additive
inverse, p – q = p + (–q).
 Distance between two rational numbers on the number line is the
absolute value of their difference.
 Patterns and properties of operations are used to generate rules for
multiplying and dividing positive and negative rational numbers.
 Any rational number can be written as a fraction, decimal, percent or
quotient of integers with a non-0 divisor.
 Rational numbers can be converted to a decimal that either ends in 0 or
repeat.
 Properties of operations are used as strategies to compute real- world
problems with rational numbers.
7.NS.2
Standard 7.NS.2
Apply and extend previous
understandings of
multiplication and division and
of fractions to multiply and
divide rational numbers.
a. Understand that
multiplication is extended
from fractions to rational
numbers by requiring that
operations continue to
satisfy the properties of
operations, particularly
the distributive property,
Mathematical
Practices
7.MP.2. Reason
abstractly and
quantitatively.
7.MP.4. Model
with mathematics.
 How do you model operations with integers using a vertical or horizontal
number line?
 How is subtraction of integers related to addition of integers?
 How is absolute value related to finding the distance between two
numbers on a number line?
 What is an example of a real-world problem that is solved by doing
addition or subtraction with a positive and negative number?
 What is an example of a real-world problem that is solved by doing
multiplication with a positive and negative number?
 What is an example of a real-world problem that is solved by doing
division with a positive and negative number?
 Why does multiplying or dividing two negatives equal a positive?
 Given a decimal, how do you determine if the decimal represents a
rational numbers?
Examples & Explanations
Multiplication and division of integers is an extension of multiplication and division of whole
numbers.
Examples:
 Examine the family of equations. What patterns do you see? Create a model and context
for each of the products. Write and model the family of equations related to 3 x 4 = 12.
7.MP.7. Look for
and make use of
structure.
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Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
16
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.
Equation
Number Line Model
Context
2x3=6
Selling two packages
of apples at $3.00
per pack
2 x -3 = -6
Spending 3 dollars
each on 2 packages
of apples
-2 x 3 = -6
Owing 2 dollars to
each of your three
friends
-2 x -3 = 6
Forgiving 3 debts of
$2.00 each
Connections: 6-8.RST.4; 68.RST.5;
SC07-S1C3-01; SS07-S5C304
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Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
17
7.NS Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide
rational numbers (Cluster 1- Standards 1, 2, and 3)
Essential Concepts
Essential Questions
 When opposites are combined the sum is always 0.
 A positive number is represented by moving right on a number line, a
negative number is represented moving left on a number line.
 Absolute value is the distance that number is away from zero on a
number line, also referred to as the magnitude of a number in real-world
contexts.
 Addition of rational numbers (p + q) is the number located a distance |q|
from p, in the positive or negative direction depending on whether q is
positive or negative.
 Subtraction of rational numbers is the same as adding the additive
inverse, p – q = p + (–q).
 Distance between two rational numbers on the number line is the
absolute value of their difference.
 Patterns and properties of operations are used to generate rules for
multiplying and dividing positive and negative rational numbers.
 Any rational number can be written as a fraction, decimal, percent or
quotient of integers with a non-0 divisor.
 Rational numbers can be converted to a decimal that either ends in 0 or
repeat.
 Properties of operations are used as strategies to compute real- world
problems with rational numbers.
7.NS.3
Standard 7.NS.3
Solve real-world and
mathematical problems
involving the four operations
with rational numbers.
(Computations with rational
numbers extend the rules for
manipulating fractions to
complex fractions.)
Mathematical
Practices
7.MP.1. Make
sense of problems
and persevere in
solving them.
7.MP.2. Reason
abstractly and
quantitatively.
 How do you model operations with integers using a vertical or horizontal
number line?
 How is subtraction of integers related to addition of integers?
 How is absolute value related to finding the distance between two
numbers on a number line?
 What is an example of a real-world problem that is solved by doing
addition or subtraction with a positive and negative number?
 What is an example of a real-world problem that is solved by doing
multiplication with a positive and negative number?
 What is an example of a real-world problem that is solved by doing
division with a positive and negative number?
 Why does multiplying or dividing two negatives equal a positive?
 Given a decimal, how do you determine if the decimal represents a
rational numbers?
Examples & Explanations
Examples:
 Your cell phone bill is automatically deducting $32 from your bank account every month.
How much will the deductions total for the year?
-32 + -32 + -32 + -32 + -32 + -32 + -32 + -32 + -32 + -32 + -32 + -32 = 12 (-32)

It took a submarine 20 seconds to drop to 100 feet below sea level from the surface. What
was the rate of the descent?
Connection: 6-8.RST.3
7.MP.5. Use
appropriate tools
strategically.
Continued on next page
Tucson Unified School District
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Grade 7
Board Approved 03/27/2012
18
 100 feet
 5 feet

 -5 ft/sec
20 seconds 1 second
7.MP.6. Attend to
precision.
7.MP.7. Look for
and make use of
structure.
7.MP.8. Look for
and express
regularity in
repeated
reasoning.
Additional Domain Information – The Number System (NS)
Key Vocabulary (Bold means most important)




Absolute value
Additive inverse
Algorithm
Commutative
property



Decimals
Distributive
property
Expanding
Factoring



Fractions
Integers
Mathematical
sentence




Negative numbers
Opposites
Order of
operations
Ordered pair



Positive numbers
Quadrant I, II, III, IV
Rational numbers
Example Resources

Books
 Holt Mathematics – Chapter Chapter 2
 Connected Math – Accentuate the negative (7th grade unit)
 Math in Context – Operations

Technology
 Illuminations
 National Library of Virtual Manipulatives
 NCTM
 Quia
 pbs learning media

Exemplary Lessons
 Sharing Prize Money
(click Number Sense; grade 7; click descriptor; scroll to bottom and click see illustration.)
Alignment 1: 7.NS.3
Grade 7
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
19
Domain NS: The Number System
Cluster
 Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
Standard
 Solve real-world and mathematical problems involving the four operations with rational numbers.Computations with rational
numbers extend the rules for manipulating fractions to complex fractions.
 The three seventh grade classes at Sunview Middle School collected the most boxtops for a school fundraiser, and so
they won a $600 prize to share between them. Mr. Aceves’ class collected 3,760 box tops, Mrs. Baca’s class collected
2,301, and Mr. Canyon’s class collected 1,855. How should they divide the money so that each class gets the same
fraction of the prize money as the fraction of the box tops that they collected?

Football Finances
 In this activity, students analyze pictures of football stands to make estimates related to the attendance at the Super Bowl. The
students will realize that estimates must, at times, be made with little background information and that a range of answers might
be correct. Students also make estimates about the television audience. (Can easily include negative integers.)

Exploring Krypto
 The rules of Krypto are amazingly simple — combine five numbers using the standard arithmetic operations to create a target
number. Finding a solution to one of the more than 3 million possible combinations can be quite a challenge, but students love it.
And you’ll love that the game helps to develop number sense, computational skill, and an understanding of the order of
operations. (Can use red and black cards to include negative integers.)
Learning Objectives


Students will:
1. Investigate the game of Krypto and develop strategies for finding solutions efficiently.
2. Explore the order of operations using Krypto challenges.
Number and Operations
• This is a large pdf, that contains multiple lessons from fractions through proportional reasoning
Assessments
All assessments for this domain must focus on assessing the degree, to which seventh grade students can, within the parameters specified in the 3
standards included in this domain,
 apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments, (e.g., pre-assessments,
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
20
daily checks for understanding, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation;
summative assessments (e.g., unit assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All
district-adopted resources contain multiple assessment tools and include online resources that can be used for the purposes delineated above.
PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills,
concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The
performance measure will be administered as close to the end of the school year as possible. The second, an end of the year machine-scorable
summative assessment will be administered after approximately 90% of the school year. These assessments are to begin during the SY 2014-2015.
Common Student Misconceptions
Student errors suggest that students interpret and treat multi-digit numbers as single-digit numbers placed adjacent to each other, rather than
using place value meanings for the digits in different positions. For example, they see 762 as a 7, a 6 and a 2 as opposed to 700+60+2 = 762.
The rational number system requires that students reformulate their concept of number in a major way. For example, students must go beyond
whole-number ideas, in which a number expresses a fixed quantity, to understand numbers that are expressed in relationship to other numbers.
Students believe that estimation is guessing and not a useful strategy. Students need to understand and be exposed to the process of estimation
as a way of quickly checking the reasonableness of a solution.
Students need to work with +/- integer chips and number lines, patterns and real world examples to develop an understanding of the
operations with integers. The more time you spend with models the better. Students have difficulty grasping operations with integers and need multiple
representations to help them. Money, points for games and sports are great real world examples that help students understanding.
When students merely memorize procedures, they fail to understand the deeper ideas that could make it easier to remember-and apply-what they learn.
Students commonly think that an exponent means to multiply the base times the exponent. For example, students think that 32 = 3x2 as opposed
3
to 3x3. Students need to understand the fact that, x , means x  x  x , means x times x times x, not 3x or 3 times x.
When subtracting integers, students subtract the smaller number from the larger number in each column no matter where it is. The idea that
subtraction is the additive inverse is crucial for students. Students need a great deal of work with addition of integers. Show them that subtraction is the
same, with the added step of translatingthe subtraction
 to the additive inverse. This will make it more likely that students will retain the information. It is
very important that students have firm understanding of opposites in order to help them understand the additive inverse is the opposite.
Students commonly don’t label numbers in real world problems and then misinterpret the meaning of their answers. Be sure that students use
labels throughout their work when they are solving a problem in context.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
21
Expressions and Equations (EE) (2 Clusters)
7.EE Use properties of operations to generate equivalent expressions (Cluster 1- Standards 1 and 2)
Essential Concepts
Essential Questions
 Properties of operations can be used to form equivalent forms of linear
expressions.
 Different forms of equivalent expression show different aspects of a
problem.
 Expressions and equations can be written in different forms depending
on the context of the problem and how the quantities within it are
related.
7.EE.1
Standard 7.EE.1
Apply properties of operations
as strategies to add, subtract,
factor, and expand linear
expressions with rational
coefficients.
Connection: 6-8.RST.5
Mathematical
Practices
7.MP.2. Reason
abstractly and
quantitatively.
 How are properties of operations used to create equivalent forms of
linear expressions?
 Given the expression 2x -6, what is a story problem that would relate to
this expression?
 Why would you want to look at different forms of equivalent expression
in terms of a real-world problem?
 How can you write an expression in multiple ways, including a visual
representation of the problem, and determine which representation is
most useful in solving a problem?
Examples & Explanations
Examples:
 Write an equivalent expression for

3x  5  2 .
Suzanne thinks the two expressions 23a  2  4a and 10a  2 are equivalent? Is she
correct? Explain why or why not?
7.MP.6. Attend to
precision.

 for:
Write equivalent expressions
3a  12 .


Possible solutions might include factoring as in 3(a  4) , or other expressions such as
7.MP.7. Look for
and make use of
structure.
a  2a  7  5 .

A rectangle is twice as long as wide. One way to write an expression to find the perimeter
would be w  w  2w  2w . Write the expression in two other ways.
Solution:

6w
OR 2(w)  2(2 w) .
An equilateral triangle has a perimeter of 6x  15 . What is the length of each of the sides
of the triangle?
Solution: 3(2 x  5) , therefore each side is 2 x  5 units long.
Tucson Unified School District
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Grade 7
Board Approved 03/27/2012
22
7.EE Use properties of operations to generate equivalent expressions (Cluster 1- Standards 1 and 2)
7.EE.2
Mathematical
Examples & Explanations
Standard 7.EE.2
Examples:
Practices
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.”
Connections: 6-8.WHST.1b,c;
6-8.WHST.2b-c; 6-8.RST.3;
6-8.RST.7; SS07-S5C2-09;
SC07-S2C2-03
7.MP.2. Reason
abstractly and
quantitatively.
7.MP.6. Attend to
precision.
7.MP.7. Look for
and make use of
structure.
7.MP.8. Look for
and express
regularity in
repeated
reasoning

Jamie and Ted both get paid an equal hourly wage of $9 per hour. This week, Ted made
an additional $27 dollars in overtime. Write an expression that represents the weekly
wages of both if J = the number of hours that Jamie worked this week and T = the number
of hours Ted worked this week? Can you write the expression in another way?
 Example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”
Students may create several different expressions depending upon how they group the quantities
in the problem.
One student might say: To find the total wage, I would first multiply the number of hours Jamie
worked by 9. Then I would multiply the number of hours Ted worked by 9. I would add these two
values with the $27 overtime to find the total wages for the week. The student would write the
expression 9 J  9T  27 .
Another student might say: To find the total wages, I would add the number of hours that Ted and
Jamie worked. I would multiply the total number of hours worked by 9. I would then add the
overtime to that value to get the total wages for the week. The student would write the expression
9( J  T )  27
A third student might say: To find the total wages, I would need to figure out how much Jamie
made and add that to how much Ted made for the week. To figure out Jamie’s wages, I would
multiply the number of hours she worked by 9. To figure out Ted’s wages, I would multiply the
number of hours he worked by 9 and then add the $27 he earned in overtime. My final step would
be to add Jamie and Ted wages for the week to find their combined total wages. The student would
write the expression (9J )  (9T  27)

Given a square pool as shown in the picture, write four different expressions to find the
total number of tiles in the border. Explain how each of the expressions relates to the
diagram and demonstrate that the expressions are equivalent. Which expression do you
think is most useful? Explain your thinking.
Tucson Unified School District
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Grade 7
Board Approved 03/27/2012
23
7.EE Solve real-life and mathematical problems using numerical and algebraic expressions and equations (Cluster 1Standards 3 and 4)
Essential Concepts
Essential Questions
 Mental math and estimation strategies for calculations in problem
solving contexts extend from students’ work with whole number
operations and are used to check reasonableness of answers.
 Students can fluently move between fractions, decimals and percents in
order to solve multi-step real world and mathematical problems.
 Real-world problems can be represented and solved using visual
models, equations or inequalities.
 Real-world situations can be represented and solved using linear
equations with rational numbers of the form px+q = r and p(x+q) =r.
 Real-world situations can be represented and solved using linear
inequalities with rational numbers of the form px+q < r and p(x+q) > r.
 Solutions sets for inequalities are graphed on number lines.
7.EE.3
Standard 7.EE.3 .
Solve multi-step real-life and
mathematical problems
posed with positive and
negative rational numbers in
any form (whole numbers,
fractions, and decimals),
using tools strategically.
Apply properties of operations
to calculate with numbers in
any form; convert between
forms as appropriate; and
assess the reasonableness of
answers using mental
computation and estimation
strategies. For example: If a
woman making $25 an hour
gets a 10% raise, she will
make an additional 1/10 of
her salary an hour, or $2.50,
for a new salary of $27.50. If
you want to place a towel bar
Mathematical
Practices
7.MP.1. Make
sense of problems
and persevere in
solving them.
7.MP.2. Reason
abstractly and
quantitatively.
7.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
 What estimation strategies can you use to determine the
reasonableness of an answer?
 How does being able to move fluently between fractions, decimals and
percent help to simplify solving and check reasonableness of real world
problems?
 How do you solve a linear equation of the form px+q = r or p(x+q) =r?
 How do you solve a linear inequalities of the form px+q < r or
 p(x+q) > r?
 How do you determine if a real-world situation will be represented using
an equation or and inequality?
Examples & Explanations
Estimation strategies for calculations with fractions and decimals extend from students’ work with
whole number operations. Estimation strategies include, but are not limited to:
 Front-end estimation with adjusting (using the highest place value and estimating from the
front end making adjustments to the estimate by taking into account the remaining
amounts),
 Clustering around an average (when the values are close together an average value is
selected and multiplied by the number of values to determine an estimate),
 Rounding and adjusting (students round down or round up and then adjust their estimate
depending on how much the rounding affected the original values),
 Using friendly or compatible numbers such as factors (students seek to fit numbers
together - i.e., rounding to factors and grouping numbers together that have round sums
like 100 or 1000), and
 Using benchmark numbers that are easy to compute (students select close whole numbers
for fractions or decimals to determine an estimate).
7.MP.4. Model
with mathematics.
7.MP.5. Use
appropriate tools
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
24
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.
Connections: 6-8.WHST.1b,c;
6-8.WHST2b;
6-8.RST.7; ET07-S6C2-03
7.EE.4
Standard 7.EE.4
Use variables to represent
quantities in a real-world or
mathematical problem, and
construct simple equations
and inequalities to solve
problems by reasoning about
the quantities.
a. Solve word problems
leading to equations of
the form px+q=r and
p(x+q)=r, where p, q, and
r are specific rational
numbers. Solve
equations of these forms
fluently. Compare an
algebraic solution to an
arithmetic solution,
identifying the sequence
of the operations used in
each approach. For
example, the perimeter of
a rectangle is 54 cm. Its
strategically.
7.MP.6. Attend to
precision.
Example:
 The youth group is going on a trip to the state fair. The trip costs $52. Included in that price
is $11 for a concert ticket and the cost of 2 passes, one for the rides and one for the game
booths. Each of the passes cost the same price. Write an equation representing the cost of
the trip and determine the price of one pass.
7.MP.7. Look for
and make use of
structure.
7.MP.8. Look for
and express
regularity in
repeated
reasoning.
Mathematical
Practices
7.MP.1. Make
sense of problems
and persevere in
solving them.
x
x
52

11
2x + 11 = 52
2x = 41
x = $20.5
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.
Examples & Explanations
Examples:

Amie had $26 dollars to spend on school supplies. After buying 10 pens, she had $14.30
left. How much did each pen cost?

The sum of three consecutive even numbers is 48. What is the smallest of these numbers?

Solve:

Florencia has at most $60 to spend on clothes. She wants to buy a pair of jeans for $22
dollars and spend the rest on t-shirts. Each t-shirt costs $8. Write an inequality for the
number of t-shirts she can purchase.

7.MP.4. Model
with mathematics.
Steven has $25 dollars. He spent $10.81, including tax, to buy a new DVD. He needs to
set aside $10.00 to pay for his lunch next week. If peanuts cost $0.38 per package
including tax, what is the maximum number of packages that Steven can buy?
7.MP.5. Use
appropriate tools
strategically.
Write an equation or inequality to model the situation. Explain how you determined whether
to write an equation or inequality and the properties of the real number system that you
used to find a solution.
7.MP.2. Reason
abstractly and
quantitatively.
7.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
5
n  5  20
4
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
25
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.
7.MP.6. Attend to
precision.
7.MP.7. Look for
and make use of
structure.

The perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

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
7.MP.8. Look for
and express
regularity in
repeated
reasoning.
1
x  3  2 and graph your solution on a number line.
2
Connections: 6-8.SRT.3; 68.RST.4
Additional Domain Information – Expressions and Equations (EE)
Key Vocabulary (BOLD means most important)






Break-even point
Coefficient
Constant rate of change
Constant term
Coordinate pair
Dependent variable






Equation of a line
Function
Graph/coordinate
graph
Independent variable
Intersecting lines
Linear function






Linear relationship
Origin
Pattern of change
Point of intersection
Ratio/rate
Rise






Run
Scale
Slope
Solution to an
equation
Solving an equation
Steepness





Symbolic method
Table
Variable
x-intercept
y-intercept
Example Resources

Books
 HOLT TEXT ONLINE
 Chapter 12 Equations/Inequalities
 Connected Math Project II (7th grade Units: Moving Straight Ahead & Variables and Patterns)
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
26

Math In Context (Algebra- Building Formulas)

Technology
 National Library of Virtual Manipulatives Algebra 6-8
 My.hrw.com
 http://www.phschool.com/cmp2/

Exemplary Lessons (from Illuminations)
 Barbie Bungie
 Bouncing Tennis Balls
 Amazing Profit
 Illustrative Mathematics -This website has a wide variety of lessons and activities that support the Common Core.
Assessments
All assessments for this domain must focus on assessing the degree, to which seventh grade students can, within the parameters specified in the 4
standards included in this domain,
 use properties of operations to generate equivalent expressions, and
 solve real-life and mathematical problems using numerical and algebraic expressions and equations.
Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments, (e.g., pre-assessments,
daily checks for understanding, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation;
summative assessments (e.g., unit assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All
district-adopted resources contain multiple assessment tools and include online resources that can be used for the purposes delineated above.
PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills,
concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The
performance measure will be administered as close to the end of the school year as possible. The second, an end of the year machine-scorable
summative assessment will be administered after approximately 90% of the school year. These assessments are to begin during the SY 2014-2015.
Common Student Misconceptions
Students may think that the notation 4x means forty-something. For example, when evaluating 4x when x = 7, substitution does not result in the
expression meaning 47, students need to understand that 4x means x multiplied by 4 or x+x+x+x. Using parenthesis when substituting in the value can
prevent some of this confusion. For example, 4(7) = 28.
Students commonly combine unlike terms such as 5x+ 2y. Students may answer the question as 7xy. Here they do not understand the concept of
like terms and the order of operations. Another common error is to say that 4x+6 = 10x. Again this stems from not understanding that the multiplication
needs to be done before the addition.
When students are simplifying expression with the distributive property, students commonly don’t multiply and show x x  x . For example,
2
Tucson Unified School District
Mathematics Curriculum
27
Grade 7
Board Approved 03/27/2012

given the problem 2x(5x+3) student commonly give the answer 10x +6x as opposed to 10x 2 + 6x.
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?
When graphing inequalities, students assume that the inequality sign is showing which way to shade on the number line. Students need to be
exposed to inequalities where the variable is either on the left or the right. (x> 5 and 5< x). Students learn how to read the meaning of the symbols and
thus giving them a greater understanding of what to graph on a number line. Most students are only exposed to the variable on the left, which would
explain why the inequality signs does tell then which way to shade on a number line.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
28
Geometry (G) (2 Clusters)
7.G Draw, construct, and describe geometrical figures and describe the relationships between them (Cluster 1Standards 1, 2, and 3)
Essential Concepts
Essential Questions
 Scale drawings are images that are proportional to the original object by
a multiplicative relationship.
 Scale drawings can be reproduced using different ratios of side lengths,
making them larger or smaller through multiplication.
 Actual side length and areas of figures can be found from scale
drawings.
 Given constraints for triangles will determine whether a unique triangle,
more than one triangle, or no triangle can be constructed.
7.G.1
Standard 7.G.1
Solve problems involving
scale drawings of geometric
figures, such as computing
actual lengths and areas from
a scale drawing and
reproducing a scale drawing
at a different scale.
Connections: 6-8.RST.7;
SC07-S1C2-04; SS07-S4C603; SS07-S4C1-01; SS07S4C1-02; ET07-S1C1-01
Mathematical
Practices
7.MP.1. Make
sense of problems
and persevere in
solving them.
 How are ratios of side lengths used to compute side lengths from one
scale drawing to another?
 How do you use a scale drawing to find an actual length or area for a
figure?
 Given certain constraints, how can you determine if you have a unique
triangle, more than one triangle, or no triangle at all?
 What shapes are created when planar cuts are made diagonally,
perpendicularly, and parallel to the base in a rectangular prism or
rectangular pyramid?
Examples & Explanations
Example:
 Julie showed you the scale drawing of her room. If each 2 cm on the scale drawing equals
5 ft, what are the actual dimensions of Julie’s room? Reproduce the drawing at 3 times its
current size.
7.MP.2. Reason
abstractly and
quantitatively.
7.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
7.MP.4. Model
with mathematics.
7.MP.5. Use
appropriate tools
strategically.
7.MP.6. Attend to
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
29
precision.
7.MP.7. Look for
and make use of
structure.
7.MP.8. Look for
and express
regularity in
repeated
reasoning.
7.G.2
Standard 7.G.2
Draw (freehand, with ruler
and protractor, and with
technology) geometric
shapes with given conditions.
Focus on constructing
triangles from three measures
of angles or sides, noticing
when the conditions
determine a unique triangle,
more than one triangle, or no
triangle.
Connections: 6-8.RST.4; 68.RST.7;
6-8.WHST.2b,2f; SC07S1C2-04;
ET07-S1C2-01; ET07-S6C103
Mathematical
Practices
7.MP.4. Model
with mathematics.
7.MP.5. Use
appropriate tools
strategically.
7.MP.6. Attend to
precision.
Examples & Explanations
Conditions may involve points, line segments, angles, parallelism, congruence, angles, and
perpendicularity.
Examples:
Is it possible to draw a triangle with a 90˚ angle and one leg that is 4 inches long and one leg
that is 3 inches long? If so, draw one. Is there more than one such triangle?
 Draw a triangle with angles that are 60 degrees. Is this a unique triangle? Why or why not?

Draw an isosceles triangle with only one 80 degree angle. Is this the only possibility or can
you draw another triangle that will also meet these conditions?

Can you draw a triangle with sides that are 13 cm, 5 cm and 6cm?

Draw a quadrilateral with one set of parallel sides and no right angles.
7.MP.7. Look for
and make use of
structure.
7.MP.8. Look for
and express
regularity in
repeated
reasoning.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
30
7.G Draw, construct, and describe geometrical figures and describe the relationships between them (Cluster 1Standards 1, 2, and 3)
7.G.3
Standard 7.G.3
Mathematical
Examples & Explanations
Describe the two-dimensional Practices
Example:
figures that result from slicing
three-dimensional figures, as
in plane sections of right
rectangular prisms and right
rectangular pyramids.
Connections: 6-8.WHST.1b;
6-8.WHST.2b
7.MP.2. Reason
abstractly and
quantitatively.

Using a clay model of a rectangular prism, describe the shapes that are created when planar
cuts are made diagonally, perpendicularly, and parallel to the base.
7.MP.4. Model
with mathematics.
7.MP.5. Use
appropriate tools
strategically.
7.MP.7. Look for
and make use of
structure.
7.G Solve real-life and mathematical problems involving angle measure, area, surface area, and volume (Cluster 1Standards 4, 5, and 6)
Essential Concepts
Essential Questions
 Pi is derived by finding the ratio of the circumference to the diameter for
any circle.
 The circumference of a circle is 2r or d .
 Using the fact that circumference of a circle is 2r ; the area formula of
2
a circle can be derived to be r .
 Equations can be written and used to find the value of missing angles in
multi-step problems involving
complementary, supplementary, adjacent


and vertical angles.
 The volume of a prism
 is calculated by taking the area of the base times
the height (V = B x h).
 Surface area of prisms and pyramids is calculated by finding the sum of
the area of each of its faces.
 How do you derive the value of pi given any circle?
 How do you derive the area formula for a circle from the formula for the
circumference of a circle?
 What is a real-world situation in which you use the area and/or
circumference of a circle?
 What is a real-world situation in which you would need to calculate the
volume and surface area of a three dimensional figure?
 How can you use what we know about areas of triangles, quadrilaterals
and polygons to find the surface area of three-dimensional figures?
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
31
7.G.4
Standard 7.G.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.
Connections: 6-8.WHST.1d;
SC07-S2C2-03; ET07-S6C203; ET07-S1C4-01
Mathematical
Practices
7.MP.1. Make
sense of problems
and persevere in
solving them.
7.MP.2. Reason
abstractly and
quantitatively.
7.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
7.MP.4. Model
with mathematics.
Examples & Explanations
Examples:
 The seventh grade class is building a mini golf game for the school carnival. The end of the
putting green will be a circle. If the circle is 10 feet in diameter, how many square feet of grass
carpet will they need to buy to cover the circle? How might you communicate this information to
the salesperson to make sure you receive a piece of carpet that is the correct size?

Students measure the circumference and diameter of several circular objects in the room
(clock, trash can, door knob, wheel, etc.). Students organize their information and discover the
relationship between circumference and diameter by noticing the pattern in the ratio of the
measures. Students write an expression that could be used to find the circumference of a circle
with any diameter and check their expression on other circles.

Students will use a circle as a model to make several equal parts as you would in a pie model.
The greater number the cuts, the better. The pie pieces are laid out to form a shape similar to a
parallelogram. Students will then write an expression for the area of the parallelogram related
to the radius (note: the length of the base of the parallelogram is half the circumference, or πr,
and the height is r, resulting in an area of πr2. Extension: If students are given the
circumference of a circle, could they write a formula to determine the circle’s area or given the
area of a circle, could they write the formula for the circumference?
7.MP.5. Use
appropriate tools
strategically.
7.MP.6. Attend to
precision.
7.MP.7. Look for
and make use of
structure.
7.MP.8. Look for
and express
regularity in
repeated
reasoning.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
32
7.G Solve real-life and mathematical problems involving angle measure, area, surface area, and volume (Cluster 1Standards 4, 5, and 6)
Essential Concepts
Essential Questions
 Pi is derived by finding the ratio of the circumference to the diameter for
any circle.
 The circumference of a circle is 2r or d .
 Using the fact that circumference of a circle is 2r ; the area formula of
2
a circle can be derived to be r .
 Equations can be written and used to find the value of missing angles in
multi-step problems involving
complementary, supplementary, adjacent


and vertical angles.
 The volume of a prism
 is calculated by taking the area of the base times
the height (V = B x h).
 Surface area of prisms and pyramids is calculated by finding the sum of
the area of each of its faces.
 How do you derive the value of pi given any circle?
 How do you derive the area formula for a circle from the formula for the
circumference of a circle?
 What is a real-world situation in which you use the area and/or
circumference of a circle?
 What is a real-world situation in which you would need to calculate the
volume and surface area of a three dimensional figure?
 How can you use what we know about areas of triangles, quadrilaterals
and polygons to find the surface area of three-dimensional figures?
7.G.5
Standard 7.G.5
Use facts about
supplementary,
complementary, vertical, and
adjacent angles in a multistep problem to write and
solve simple equations for an
unknown angle in a figure.
Connection: ET07-S1C4-01
Mathematical
Practices
Examples & Explanations
Set and solve an equation to find the value of angle x.
7.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
7.MP.4. Model
with mathematics.
7.MP.5. Use
appropriate tools
strategically.
x + 35o = 180o so m  145
o
Set and solve equations to find the missing angle measures.

7.MP.6. Attend to
precision.
7.MP.7. Look for
and make use of
structure.
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
33
Possible solutions:
x  45 o  90 o
mx  45 o
mx  my
45 o  my
y  z  45 o  180 o
45 o  z  45 o  180 o mz90 o
7.G Solve real-life and mathematical problems involving angle measure, area, surface area, and volume (Cluster 1Standards 4, 5, and 6)
Essential Concepts
Essential Questions

 Pi is derived by finding the ratio of the circumference
to the diameter for  How do you derive the value of pi given any circle?
any circle.
 The circumference of a circle is 2r or d .
 Using the fact that circumference of a circle is 2r ; the area formula of
2
a circle can be derived to be r .
 Equations can be written and used to find the value of missing angles in
multi-step problems involving
complementary, supplementary, adjacent


and vertical angles.
 The volume of a prism
 is calculated by taking the area of the base times
the height (V = B x h).
 Surface area of prisms and pyramids is calculated by finding the sum of
the area of each of its faces.
7.G.6
Standard 7.G.6
Solve real-world and
mathematical problems
involving area, volume and
surface area of two- and
three-dimensional objects
Mathematical
Practices
7.MP.1. Make
sense of problems
and persevere in
 How do you derive the area formula for a circle from the formula for the
circumference of a circle?
 What is a real-world situation in which you use the area and/or
circumference of a circle?
 What is a real-world situation in which you would need to calculate the
volume and surface area of a three dimensional figure?
 How can you use what we know about areas of triangles, quadrilaterals
and polygons to find the surface area of three-dimensional figures?
Examples & Explanations
Students understanding of volume can be supported by focusing on the area of base times the
height to calculate volume. Students understanding of surface area can be supported by focusing
on the sum of the area of the faces. Nets can be used to evaluate surface area calculations.
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
34
composed of triangles,
quadrilaterals, polygons,
cubes, and right prisms.
Connections: 6-8.WHST.2a;
ET07-S1C4-01
solving them.
7.MP.2. Reason
abstractly and
quantitatively.
Examples:
 Choose one of the figures shown below and write a step by step procedure for determining
the area. Find another person that chose the same figure as you did. How are your
procedures the same and different? Do they yield the same result?
7.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
7.MP.4. Model
with mathematics.
7.MP.5. Use
appropriate tools
strategically.

A cereal box is a rectangular prism. What is the volume of the cereal box? What is the
surface area of the cereal box? (Hint: Create a net of the cereal box and use the net to
calculate the surface area.) Make a poster explaining your work to share with the class.


Find the area of a triangle with a base length of three units and a height of four units.
Find the area of the trapezoid shown below using the formulas for rectangles and triangles.
7.MP.6. Attend to
precision.
12
3
7.MP.7. Look for
and make use of
structure.
7
7.MP.8. Look for
and express
regularity in
repeated
reasoning.
Additional Domain Information – Geometry (G)
Key Vocabulary (BOLD means most important)





Area
Base
Circumference
Cone
Congruent





Cylinder
Diameter
Dimensions
Edge
Face





Height
Length
Net
Perimeter
Pi
Example Resources
Tucson Unified School District
Mathematics Curriculum





Prism
Pyramid
Radius
Rectangular prism
Right prism





Sphere
Surface area
Unit cube
Volume
Width
Grade 7
Board Approved 03/27/2012
35

BOOKS
 HOLT ONLINE TEXT
o HOLT Chapter 5 Lesson 5-7-5-9 Similar Figures, Scale Drawings and Scale Models

CMP2
o
2 Dimensional Covering & Surrounding-Investigation #4 for Circles
o
Shapes & Designs (6th grade)
o
2 and 3 Dimensional- Filling and Wrapping (7th Grade)

Math in Context: Geometry- Packages and Polygons

A Collection of Math Lessons 6-8 Marilyn Burns

Technology
 My.hrw.com
 http://www.phschool.com/cmp2/
 nlvm.usu.edu/
 http://connectedmath.msu.edu/CD/Grade7/BoxFill/index.html

Exemplary Lessons
 Illuminations: Paper Pool The interactive Paper Pool game provides an opportunity for students to develop their understanding of ratio,
proportion, greatest common factor and least common multiple.
 In this investigation, students are asked to play a game called Paper Pool. The game is played on rectangular grids made of congruent
squares, such as the one shown here.
 Illuminations: Polygon Capture In this lesson, students classify polygons according to more than one property at a time. In the context of
a game, students move from a simple description of shapes to an analysis of how properties are related. This lesson was adapted from
an article which appeared in the October, 1998 edition of Mathematics Teaching in the Middle School.
 Illuminations: Cubed Cans In this lesson, students will use formulas they have explored for the volume of a cylinder and convert them
into the same volume for rectangular prisms while trying to minimize the surface area. Various real world cylindrical objects will be
measured and converted into a prism to hold the same volume. As an extension, students may design and create a rectangular prism
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
36
container according to their dimensions to compare and contrast with the cylinder.
Assessments
All assessments for this domain must focus on assessing the degree, to which seventh grade students can, within the parameters specified in the 6
standards included in this domain,
 draw, construct and describe geometrical figures and describe the relationships between them, and
 solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments, (e.g., pre-assessments,
daily checks for understanding, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation;
summative assessments (e.g., unit assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All
district-adopted resources contain multiple assessment tools and include online resources that can be used for the purposes delineated above.
PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills,
concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The
performance measure will be administered as close to the end of the school year as possible. The second, an end of the year machine-scorable
summative assessment will be administered after approximately 90% of the school year. These assessments are to begin during the SY 2014-2015.
Common Student Misconceptions
Students do not understand that doubling side measures does not double the area. For example, students have a hard time recognizing that when
you double the side lengths to scale a figure that the perimeter will also double, but the area will quadruple. Are area will be the scale factor squared. So
if you triple the side lengths then the area will be 32 = 9 times larger.
It is important that students properly read the ruler. For example: cm vs in or mm vs cm
Some students might have difficulty recalling that if the scale factor is greater than 1, then the new figure is bigger than the original. To remind
students, write the following ratio on the board. A scale factor is model: original.
Students commonly have difficulty understanding that a scale factor smaller than 1 will create a new figure that is smaller then the original.
For example if your scale factor is
1
or .25, then the image will have side lengths that are a quarter of the size of the original. When you multiply by a
4
fraction you are really dividing, thus creating a smaller image.
Students commonly use the diagonal length of triangles and parallelograms as the height. Students need to make sure they understand that the
height an object is always
 perpendicular to the base of the object.
Students believe that Pi is a rational number because we tell them to use 3.14 in calculations. Students need to understand that 3.14 is an
approximation of Pi not the exact value.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
37
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). Students need to understand that circumference
is measured is 1 dimension (in) where as area is measured in 2 (in2).
Students confuse the meanings of acute and obtuse when classifying triangles. Consider this mnemonic device for students who still confuse
acute and obtuse angles. Have those students use the phrase “ a cute little angle” to remember that acute angles are < 90 degrees.
Some students assume that lines are perpendicular if they appear perpendicular in a drawing or picture. Remind students that perpendicular
lines must be labeled at their intersection with a small square to show that they form right angles, or must be labeled as perpendicular before they can be
identified as such.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
38
Statistics and Probability (SP) (3 Clusters)
7.SP Use random sampling to draw inferences about a population (Cluster 1- Standards 1 and 2)
Essential Concepts
Essential Questions
 Random samplings create sample populations, which mimic the
demographics of a larger population, that are used to collect and
generalize information.
 Data from random samplings can be used to create valid inferences
about an unknown characteristic of interest.
7.SP.1
Standard 7.SP 1
Understand that statistics can
be used to gain information
about a population by
examining a sample of the
population; generalizations
about a population from a
sample are valid only if the
sample is representative of
that population. Understand
that random sampling tends
to produce representative
samples and support valid
inferences.
Mathematical
Practices
7.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
7.MP.6. Attend to
precision.
 How do you create a random sampling?
 How do you determine if a sampling is a random sampling?
 What is the purpose for collecting data using a random sampling and
why is it important?
Examples & Explanations
Example:
 The school food service wants to increase the number of students who eat hot lunch in the
cafeteria. The student council has been asked to conduct a survey of the student body to
determine the students’ preferences for hot lunch. They have determined two ways to do
the survey. The two methods are listed below. Identify the type of sampling used in each
survey option. Which survey option should the student council use and why?
1. Write all of the students’ names on cards and pull them out in a draw to determine who
will complete the survey.
2. Survey the first 20 students that enter the lunchroom.
Connections: SS07-S4C4-04;
SS07-S4C4-05;
SC07-S3C1-02; SC07-S4C304;
ET07-S4C2-01; ET07-S4C202;
ET07-S6C2-03;
7.SP.2
Standard 7.SP.2
Use data from a random
sample to draw inferences
about a population with an
unknown characteristic of
interest. Generate multiple
samples (or simulated
Mathematical
Practices
7.MP.1. Make
sense of problems
and persevere in
solving them.
Examples & Explanations
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.
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
39
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.
Connections: 6-8.WHST.1b;
SC07-S1C3-04; SC07-S1C305; SC07-S1C3-06;
SC07-S1C4-05; SC07-S2C203;
ET07-S1C3-01; ET07-S1C302;
ET07-S4C2-02; ET07-S6C203
7.MP.2. Reason
abstractly and
quantitatively.

Below is the data collected from two random samples of 100 students regarding student’s
school lunch preference. Make at least two inferences based on the results.
7.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
7.MP.5. Use
appropriate tools
strategically.
7.MP.6. Attend to
precision.
7.MP.7. Look for
and make use of
structure.
7.SP Draw informal comparative inferences about two populations (Cluster 1- Standards 3 and 4)
Essential Concepts
Essential Questions
 Measures of center and measures of variability are used to determine
informal (generalizations) or formal (numerical) differences between two
data sets with similar variability.
7.SP.3
Standard 7.SP.3
Informally assess the degree
of visual overlap of two
numerical data distributions
with similar variabilities,
measuring the difference
between the centers by
expressing it as a multiple of
a measure of variability. For
example, the mean height of
Mathematical
Practices
7.MP.1. Make
sense of problems
and persevere in
solving them.
7.MP.2. Reason
abstractly and
quantitatively.
 Why would you want to compare two different data sets?
 What measure of center or measure of variability would you use to
compare two data sets and why?
Examples & Explanations
Students can readily find data as described in the example on sports team or college websites.
Other sources for data include American Fact Finder (Census Bureau), Fed Stats, Ecology
Explorers, USGS, or CIA World Factbook. Researching data sets provides opportunities to connect
mathematics to their interests and other academic subjects. Students can utilize statistic functions
in graphing calculators or spreadsheets for calculations with larger data sets or to check their
computations. Students calculate mean absolute deviations in preparation for later work with
standard deviations.
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
40
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.
Connections: 6-8.WHST.1b;
SC07-S1C4-01; SC07-S1C402;
SC07-S1C4-03; SS07-S4C101;
SS07-S4C1-02; SS07-S4C105;
SS07-S4C4-06; SS07-S4C603;
ET07-S1C3-01; ET07-S1C302;
ET07-S4C2-01; ET07-S4C202;
ET07-S6C2-03
7.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
7.MP.4. Model
with mathematics.
7.MP.5. Use
appropriate tools
strategically.
7.MP.6. Attend to
precision.
Example:
 Jason wanted to compare the mean height of the players on his favorite basketball and
soccer teams. He thinks the mean height of the players on the basketball team will be
greater but doesn’t know how much greater. He also wonders if the variability of heights of
the athletes is related to the sport they play. He thinks that there will be a greater variability
in the heights of soccer players as compared to basketball players. He used the rosters
and player statistics from the team websites to generate the following lists.
Basketball Team – Height of Players in inches for 2010-2011 Season
75, 73, 76, 78, 79, 78, 79, 81, 80, 82, 81, 84, 82, 84, 80, 84
Soccer Team – Height of Players in inches for 2010
73, 73, 73, 72, 69, 76, 72, 73, 74, 70, 65, 71, 74, 76, 70, 72, 71, 74, 71, 74, 73, 67,
70, 72, 69, 78, 73, 76, 69
To compare the data sets, Jason creates a two dot plots on the same scale. The
shortest player is 65 inches and the tallest players are 84 inches.
7.MP.7. Look for
and make use of
structure.
In looking at the distribution of the data, Jason observes that there is some overlap
between the two data sets. Some players on both teams have players between 73
and 78 inches tall. Jason decides to use the mean and mean absolute deviation to
compare the data sets. Jason sets up a table for each data set to help him with the
calculations.
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
41
The mean height of the basketball players is 79.75 inches as compared to the mean height
of the soccer players at 72.07 inches, a difference of 7.68 inches.
The mean absolute deviation (MAD) is calculated by taking the mean of the absolute
deviations for each data point. The difference between each data point and the mean is
recorded in the second column of the table. Jason used rounded values (80 inches for the
mean height of basketball players and 72 inches for the mean height of soccer players) to
find the differences. The absolute deviation, absolute value of the deviation, is recorded in
the third column. The absolute deviations are summed and divided by the number of data
points in the set.
The mean absolute deviation is 2.14 inches for the basketball players and 2.53 for the
soccer players. These values indicate moderate variation in both data sets. There is
slightly more variability in the height of the soccer players. The difference between the
heights of the teams is approximately 3 times the variability of the data sets (7.68 ÷ 2.53 =
3.04).
Soccer Players (n = 29)
Height (in) Deviation from
Mean (in)
65
67
69
69
69
70
70
70
71
71
71
72
72
72
72
73
73
73
-7
-5
-3
-3
-3
-2
-2
-2
-1
-1
-1
0
0
0
0
+1
+1
+1
Tucson Unified School District
Mathematics Curriculum
Absolute
Deviation
(in)
7
5
3
3
3
2
2
2
1
1
1
0
0
0
0
1
1
1
Basketball Players (n = 16)
Height (in)
Deviation from
Mean (in)
Absolute
Deviation (in)
73
75
76
78
78
79
79
80
80
81
81
82
82
84
84
84
7
5
4
2
2
1
1
0
0
1
1
2
2
4
4
4
-7
-5
-4
-2
-2
-1
-1
0
0
1
1
2
2
4
4
4
Grade 7
Board Approved 03/27/2012
42
73
73
73
74
74
74
74
76
76
76
78
Σ = 2090
+1
+1
+1
+2
+2
+2
+2
+4
+4
+4
+6
1
1
1
2
2
2
2
4
4
4
6
Σ = 62
Mean = 2090 ÷ 29 =72 inches
MAD = 62 ÷ 29 = 2.13 inches
7.SP.4
Standard 7.SP.4
Use measures of center and
measures of variability for
numerical data from random
samples to draw informal
comparative inferences about
two populations. For
example, decide whether the
words in a chapter of a
seventh-grade science book
are generally longer than the
words in a chapter of a fourthgrade science book.
Connections: 6-8.WHST.1b;
ET07-S1C3-01; ET07-S1C302;
ET07-S4C2-01; ET07-S4C202;
ET07-S6C2-03; SC07-S1C301;
SC07-S1C3-05; SC07-S1C403;
Mathematical
Practices
7.MP.1. Make
sense of problems
and persevere in
solving them.
7.MP.2. Reason
abstractly and
quantitatively.
7.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
Σ = 1276
Σ = 40
Mean = 1276 ÷ 16 =80 inches
MAD = 40 ÷ 16 = 2.5 inches
Examples & Explanations
Measures of center include mean, median, and mode. The measures of variability include range,
mean absolute deviation, and interquartile range.
Example:
 The two data sets below depict random samples of the housing prices sold in the King
River and Toby Ranch areas of Arizona. Based on the prices below which measure of
center will provide the most accurate estimation of housing prices in Arizona? Explain your
reasoning.
o King River area {1.2 million, 242000, 265500, 140000, 281000, 265000, 211000}
o Toby Ranch homes {5million, 154000, 250000, 250000, 200000, 160000, 190000}
 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.
7.MP.4. Model
with mathematics.
7.MP.5. Use
appropriate tools
strategically.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
43
SC07-S2C2-03; SC07-S4C304;
SS07-S4C2-01; SS07-S4C406;
SS07-S4C4-09
7.MP.6. Attend to
precision.
7.MP.7. Look for
and make use of
structure.
7.SP Investigate chance processes and develop, use, and evaluate probability models (Cluster 1- Standards 5, 6, 7,
and 8)
Essential Concepts
Essential Questions
 A number between 0 and 1 represents the probability of the likelihood of
an event occurring, where 0 is impossible and 1 is certain the event will
occur.
 Sample spaces for compound events are represented using organized
lists, tables and tree diagrams.
 Actual probabilities, simple or compound, are the fraction of outcomes in
the sample space for which the event or compound event occurs.
 Experiments and simulations are used to collect data to determine the
chance probability.
 The more times an experiment or simulation is done the closer the
chance probability should be to the actual probability, simple or
compound.
7.SP.5
Standard 7.SP.5
Understand that the
probability of a chance event
is a number between 0 and 1
that expresses the likelihood
of the event occurring. Larger
numbers indicate greater
likelihood. A probability near
0 indicates an unlikely event,
a probability around ½
indicates an event that is
neither unlikely nor likely, and
a probability near 1 indicates
a likely event.
Mathematical
Practices
 What is the range of probabilities of an event occurring and what do the
numbers mean?
 What types of experiments represent situations in which the
probabilities of each outcome are equally likely?
 What is an example of a compound event and what makes it a
compound event?
 What is a sample space and why do you need it for calculating
probabilities?
 How can you represent the sample space of compound events?
 How do you design and use a simulation to generate frequencies for
compound events?
Examples & Explanations
7.MP.4. Model
with mathematics.
Probability can be expressed in terms such as impossible, unlikely, likely, or certain or as a number
between 0 and 1 as illustrated on the number line. Students can use simulations such as Marble
Mania on AAAS or the Random Drawing Tool on NCTM’s Illuminations to generate data and
examine patterns.
7.MP.5. Use
appropriate tools
strategically.
Marble Mania http://www.sciencenetlinks.com/interactives/marble/marblemania.html
Random Drawing Tool - http://illuminations.nctm.org/activitydetail.aspx?id=67
7.MP.6. Attend to
precision.
7.MP.7. Look for
and make use of
structure.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
44
Connections: 6-8.WHST.1b;
SS07-S5C1-04; ET07-S1C301; ET07-S1C3-02
Example:
 The container below contains 2 gray, 1 white, and 4 black marbles. Without looking, if you
choose a marble from the container, will the probability be closer to 0 or to 1 that you will
select a white marble? A gray marble? A black marble? Justify each of your predictions.
7.SP.6
Standard 7.SP.6
Approximate the probability of
a chance event by collecting
data on the chance process
that produces it and
observing its long-run relative
frequency, and predict the
approximate relative
frequency given the
Mathematical
Practices
7.MP.1. Make
sense of problems
and persevere in
solving them.
7.MP.2. Reason
Examples & Explanations
Students can collect data using physical objects or graphing calculator or web-based simulations.
Students can perform experiments multiple times, pool data with other groups, or increase the
number of trials in a simulation to look at the long-run relative frequencies.
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
45
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.
Connections: 6-8.WHST.1a;
ET07-S1C2-01; ET07-S1C202; ET07-S1C2-03;
ET07-S1C3-01; ET07-S1C302
ET07-S4C2-01; ET07-S6C103;
ET07-S6C2-03; SC07-S1C203;
SC07-S1C2-05; SC07-S1C305;
SC07-S1C4-03; SC07-S1C405;
SC07-S2C2-03
7.SP.7
Standard 7.SP.7
Develop a probability model
and use it to find probabilities
of events. Compare
probabilities from a model to
observed frequencies; if the
agreement is not good,
explain possible sources of
the discrepancy.
a. Develop a uniform
probability model by
assigning equal
probability to all
outcomes, and use the
model to determine
probabilities of events.
For example, if a student
abstractly and
quantitatively.
7.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
Example:
 Each group receives a bag that contains 4 green marbles, 6 red marbles, and 10 blue
marbles. Each group performs 50 pulls, recording the color of marble drawn and replacing
the marble into the bag before the next draw. Students compile their data as a group and
then as a class. They summarize their data as experimental probabilities and make
conjectures about theoretical probabilities (How many green draws would you expect if you
were to conduct 1000 pulls? 10,000 pulls?).

Students create another scenario with a different ratio of marbles in the bag and make a
conjecture about the outcome of 50 marble pulls with replacement. (An example would be
3 green marbles, 6 blue marbles, 3 blue marbles.)

Students try the experiment and compare their predictions to the experimental outcomes to
continue to explore and refine conjectures about theoretical probability.

When rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200
times, but probably not exactly 200 times.
7.MP.4. Model
with mathematics.
7.MP.5. Use
appropriate tools
strategically.
Mathematical
Practices
7.MP.1. Make
sense of problems
and persevere in
solving them.
7.MP.2. Reason
abstractly and
quantitatively.
7.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
Examples & Explanations
Students need multiple opportunities to perform probability experiments and compare these results
to theoretical probabilities. Critical components of the experiment process are making predictions
about the outcomes by applying the principles of theoretical probability, comparing the predictions
to the outcomes of the experiments, and replicating the experiment to compare results.
Experiments can be replicated by the same group or by compiling class data. Experiments can be
conducted using various random generation devices including, but not limited to, bag pulls,
spinners, number cubes, coin toss, and colored chips. Students can collect data using physical
objects or graphing calculator or web-based simulations. Students can also develop models for
geometric probability (i.e. a target).
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.
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
46
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.MP.4. Model
with mathematics.

If you choose a point in the square, what is the probability that it is not in the circle?

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.MP.5. Use
appropriate tools
strategically.
7.MP.6. Attend to
precision.
7.MP.7. Look for
and make use of
structure.
7.MP.8. Look for
and express
regularity in
repeated
reasoning.
Connections: 6-8.WHST.2d;
SC07-S1C2-02;
ET07-S1C2-01; ET07-S1C202;
ET07-S1C2-03; ET07-S1C301;
ET07-S1C3-02; ET07-S4C201;
ET07-S4C2-02; ET07-S6C103;
ET07-S6C2-03
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
47
7.SP Investigate chance processes and develop, use, and evaluate probability models (Cluster 1- Standards 5, 6, 7,
and 8)
7.SP.8
Standard 7.SP.8
Mathematical
Examples & Explanations
Find probabilities of
In our state (AZ.4.OA.3.1) 4th grade students have previously worked with counting problems of
Practices
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?
7.MP.1. Make
sense of problems
and persevere in
solving them.
7.MP.2. Reason
abstractly and
quantitatively.
7.MP.4. Model
with mathematics.
7.MP.5. Use
appropriate tools
strategically.
7.MP.7. Look for
and make use of
structure.
possibilities using systematic listing, arrays, charts and tree diagrams.
Examples:
 Students conduct a bag pull experiment. A bag contains 5 marbles. There is one red
marble, two blue marbles and two purple marbles. Students will draw one marble without
replacement and then draw another. What is the sample space for this situation? Explain
how you determined the sample space and how you will use it to find the probability of
drawing one blue marble followed by another blue marble.

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?

Show all possible arrangements of the letters in the word FRED using a tree diagram. If
each of the letters is on a tile and drawn at random, what is the probability that you will
draw the letters F-R-E-D in that order? What is the probability that your “word” will have an
F as the first letter?
7.MP.8. Look for
and express
regularity in
repeated
reasoning.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
48
Connections: 6-8.WHST.2d;
ET07-S1C2-01; ET07-S1C202; ET07-S1C2-03;
SC07-S1C4-03; SC07-S1C405;
SC07-S1C2-02; SC07-S1C203
Additional Domain Information – Statistics and Probability (SP)
Key Vocabulary (BOLD means most important)
Area model
Binomial probability
Categorical data
Distribution
Equally likely
Expected value
Experimental probability
Fair game
Law of large numbers
Line plot
Mean
Measures
Measures of center
Median
Mode
Numerical data
Outcome
Outlier
Payoff
Probability
Random
Sample space
Scatter plot
Stem-and- leaf plot
Theoretical probability
Tree diagram
Variability of a set of
numerical data
Example Resources

Books
 HOLT ONLINE TEXT
o Holt- Chapter 7-Collecting, Displaying and Analyzing Data
o Holt-Chapter 12- Probability
 Math In Context Probability and Data Analysis
o Dealing with Data
o Second Chance
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
49

CMP2
o How Likely Is it? (6th)
o What Do You Expect? (7th)

Technology/Web Resources
 Design a Dart Board CMP2
 Adjustable Spinner
 My.hrw.com
 http://www.phschool.com/cmp2/
 nlvm.usu.edu/

Exemplary Lessons
 Using NBA Stats for Box and Whisker Plots In this lesson, students use information from NBA statistics to make and compare box and
whisker plots. The data provided in the lesson come from the NBA, but you could apply the lesson to data from the WNBA or any other
sports teams or leagues for which player statistics are available.



Where is Everybody?- Using two online activities (State Data Map and Canada Data Map), students use ratios and percents to
compare population density and explore various statistical measures.
Probability Explorations- 2 lessons- Students will explore theoretical and experimental probability and the relationship between them.
Students will also graph an experiment to further explore the relationship according to the law of large numbers.
Individual Lessons
 Lesson 1 - What Are My Chances?
Students will conduct five experiments through stations to compare theoretical and experimental probability. The class data will be
combined to compare with previously established theoretical probability.
 Lesson 2 - Probably Graphing
Student will conduct a coin tossing experiment for 30 trials. Their results will be graphed and shows a line graph that progresses toward
the theoretical probability. The graph will also allow for a representation of heads or tails throughout the experiment.
 Probability Unit 6-8
Throughout this unit of seven lessons, students use hands-on learning activities to explore statistics and probability. Designed for
mentoring situations at middle school level, the lessons focus on the essentials. For example, the probability activities introduce fairness
in games and the computation of probability. Lessons include teaching guidelines as well as all handouts. MSP full record
Assessments
All assessments for this domain must focus on assessing the degree to which seventh grade students can, within the parameters specified in the 8cs
standards included in this domain,
 use random sampling to draw inferences about a population,
 draw informal comparative inferences about two populations, and
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
50

investigate chance processes and develop, use, and evaluate probability models.
Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments, (e.g., pre-assessments,
daily checks for understanding, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation;
summative assessments (e.g., unit assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All
district-adopted resources contain multiple assessment tools and include online resources that can be used for the purposes delineated above.
PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills,
concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The
performance measure will be administered as close to the end of the school year as possible. The second, an end of the year machine-scorable
summative assessment will be administered after approximately 90% of the school year. These assessments are to begin during the SY 2014-2015.
Common Student Misconceptions
Students believe that one random sample is representative of the entire population. Many samples must be taken in order to make an
inference that is valid. By comparing the results of one random sample with the results of multiple random samples, students can correct this
misconception.
Students are confused when to use mean, median or mode as an appropriate measure of center. The more opportunity students have to answer
the question, “which measure of center best represent the data?” the less confusion they will have.
Students often expect the theoretical and experimental probabilities of the same data to match. Providing multiple opportunities for students to
experience simulations of situations in order to find and compare the experimental probability to the theoretical probability helps students to 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.
Students often struggle making organized lists or trees for a situation in order to determine the theoretical probability. Having students start
with simpler situations that have fewer elements enables them to have successful experiences with organizing lists and trees diagrams. Ask guiding
questions to help students create methods for creating organized lists and trees for situations with more elements.
Students often struggle when converting forms of probability from fractions to percents and vice versa. To help students with the discussion of
probability, don’t allow the symbol manipulation/conversions to detract from the conversations. By having students use technology such as a graphing
calculator or computer software to simulate a situation and graph the results, the focus is on the interpretation of the data. Students then make
predictions about the general population based on these probabilities.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
51
COMPARATIVE ANALYSIS: GRADE 7 MATHEMATICS
PLANNING FOR CONTENT SHIFTS
2010 STANDARD
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 ½ mile in
each ¼ hour, compute the unit rate as the complex fraction ½/¼
miles per hour, equivalently 2 miles per hour.
2008 PO
PLAN
M05-S1C1-05 Use ratios and unit rates to
model, describe and extend problems in
context.
M07-S1C1-01 Recognize and convert
between expressions for positive and
negative rational numbers, including
fractions, decimals, percents, and ratios.
M07-S1C2-03 Solve problems involving
percentages, ratio and proportion, including
tax, discount, tips, and part/whole
relationships. (no complex ratios)
M08-S3C4-02 Solve problems involving
simple rates.
7.RP.2 Recognize and represent proportional relationships
between quantities.
a. Decide whether two quantities are in a proportional
relationship, e.g., by testing for equivalent ratios in a table
or graphing on a coordinate plane and observing whether the
graph is a straight line through the origin.
b. Identify the constant of proportionality (unit rate) in tables,
graphs, equations, diagrams, and verbal descriptions of
proportional relationships.
M07-S3C2-01 Use a table of values to graph an
equation or proportional relationship; describe the
graph’s characteristics.
M05-S1C1-05 Use ratios and unit rates to model,
describe and extend problems in context.
M07-S3C1-01 Recognize, describe, create, and
analyze numerical and geometric sequences using
tables or graphs; make conjectures about these
sequences.
M07-S3C2-01 Use a table of values to graph an
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
52
2010 STANDARD
2008 PO
equation or proportional relationship; describe the
PLAN
graph’s characteristics.
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.
M07-S3C4-01 Use graphs and tables to model and
analyze change.
M08-S3C2-03 Write the rule for a simple function
using algebraic notation.
M08-S3C2-05 Demonstrate that proportional
relationships are linear using equations, graphs, or
tables.
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.
NEW
M07-S1C2-03 Solve problems involving
percentages, ratio and proportion, including tax,
discount, tips, and part/whole relationships.
M08-S1C2-03 Solve problems involving percent
increase, percent decrease, and simple interest
rates.
7.NS.1 Apply and extend previous
understandings of addition and
subtraction to add and subtract rational
numbers; represent addition and
M06-S1C2-01 Apply and interpret the concepts of
addition and subtraction with integers using models.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
53
2010 STANDARD
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 realworld 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.
2008 PO
PLAN
NEW
M06-S1C1-05 Express that a number’s
distance from zero on the number line is its
absolute value.
M07-S1C1-04 Model and solve simple problems
involving absolute value.
M08-S1C1-04 Model and solve problems
involving absolute values.
MHS-S1C1-03 Express that the distance
between two numbers is the absolute value
of their difference.
M07-S1C2-01 Add, subtract, multiply, and
divide integers.
M07-S1C2-05 Simplify numerical
expressions using the order of operations
and appropriate mathematical properties.
7.NS.2 Apply and extend previous
understandings of multiplication and
division and of fractions to multiply and
divide rational numbers.
Tucson Unified School District
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Grade 7
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2010 STANDARD
2008 PO
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 realworld 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.
M07-S1C2-01 Add, subtract, multiply, and
divide integers.
7.NS.3 Solve real-world and mathematical
problems involving the four operations with
rational numbers. (Computations with rational
numbers extend the rules for manipulating
M07-S1C1-01 Recognize and convert
between expressions for positive and
negative rational numbers, including
fractions, decimals, percents, and ratios.
PLAN
M07-S1C2-01 Add, subtract, multiply, and
divide integers.
M07-S1C2-02 Solve problems with rational
numbers and appropriate operations using
exact answers or estimates.
M07-S1C1-01 Recognize and convert between
expressions for positive and negative rational
numbers, including fractions, decimals, percents,
and ratios.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
55
2010 STANDARD
fractions to complex fractions.)
7.EE.1 Apply properties of operations as
strategies to add, subtract, factor, and expand
linear expressions with rational coefficients.
7.EE.2 Understand that rewriting an expression
in different forms in a problem context can shed
light on the problem and how the quantities in it
are related. For example, a + 0.05a = 1.05a
means that “increase by 5%” is the same as
“multiply by 1.05.”
2008 PO
PLAN
M07-S1C2-02 Solve problems with rational
numbers and appropriate operations using
exact answers or estimates.
M06-S1C2-06 Apply the commutative,
associative, distributive, and identity
properties to evaluate numerical
expressions involving whole numbers.
M06-S1C2-07 Simplify numerical
expressions (involving fractions, decimals,
and exponents) using the order of
operations with or without grouping
symbols.
M07-S1C2-05 Simplify numerical
expressions using the order of operations
and appropriate mathematical properties.
M08-S3C3-03 Analyze situations, simplify,
and solve problems involving linear
equations and inequalities using the
properties of the real number system. (not
factoring and expansion)
MHS-S3C3-08 Simplify and evaluate
polynomials, rational expressions,
expressions containing absolute value, and
radicals.
M07-S3C3-01 Write a single variable
algebraic expression or one-step equation
given a contextual situation.
M08-S3C3-01 Write or identify algebraic
expressions, equations, or inequalities that
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
56
2010 STANDARD
2008 PO
PLAN
represent a situation.
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.
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?
M06-S3C3-02 Create and solve two-step
equations that can be solved using inverse
properties with fractions and decimals.
M07-S1C3-02 Make estimates appropriate
to a given situation.
M07-S3C3-03 Solve multi-step equations
using inverse properties with rational
numbers.
M07-S3C3-03 Solve multi-step equations
using inverse properties with rational
numbers.
M07-S3C3-05 Create and solve two-step
equations that can be solved using inverse
operations with rational numbers.
M08-S3C3-01 Write or identify algebraic
expressions, equations, or inequalities that
represent a situation.
M08-S3C3-03 Analyze situations, simplify,
and solve problems involving linear
Tucson Unified School District
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Grade 7
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2010 STANDARD
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.
7.G.1 Solve problems involving scale drawings
of geometric figures, such as computing actual
lengths and areas from a scale drawing and
reproducing a scale drawing at a different scale.
7.G.2 Draw (freehand, with ruler and protractor,
and with technology) geometric shapes with
given conditions. Focus on constructing triangles
from three measures of angles or sides, noticing
when the conditions determine a unique triangle,
more than one triangle, or no triangle.
7.G.3 Describe the two-dimensional
figures that result from slicing three-
2008 PO
PLAN
equations and inequalities using the
properties of the real number system.
M07-S3C3-06 Create and solve one-step
inequalities with whole numbers. (one step
only)
M08-S3C3-01 Write or identify algebraic
expressions, equations, or inequalities that
represent a situation.
M08-S3C3-03 Analyze situations, simplify,
and solve problems involving linear
equations and inequalities using the
properties of the real number system.
M06-S4C4-03 Estimate the measure of objects
using a scale drawing or map.
M07-S4C4-04 Determine actual lengths
based on scale drawings or maps. (only
length)
M08-S4C4-02 Solve geometric problems
using ratios and proportions.
M05-S4C1-01 Draw and label 2dimensional figures given specific attributes
including angle measure and side length.
(no specific focus on triangles)
M07-S4C4-07 Measure to the appropriate degree
of accuracy and justify reasoning.
M08-S4C1-02 Predict results of combining,
subdividing, and changing shapes of plane figures
and solids.
Tucson Unified School District
Mathematics Curriculum
Grade 7
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2010 STANDARD
dimensional figures, as in plane sections
of right rectangular prisms and right
rectangular pyramids.
7.G.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.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.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.
2008 PO
PLAN
M06-S4C1-01 Define  (pi) as the ratio
between the circumference and diameter of
a circle and explain the relationship among
the diameter, radius, and circumference.
M07-S4C4-01 Solve problems involving the
circumference and area of a circle by calculating
and estimating.
M07-S5C2-11 Use manipulatives and other
modeling techniques to defend π (pi) as a ratio of
circumference to diameter.
M06-S4C1-02 Solve problems using properties of
supplementary, complementary, and vertical angles.
M07-S4C1-02 Analyze and determine
relationships between angles created by
parallel lines cut by a transversal.
M07-S3C3-05 Create and solve two-step
equations that can be solved using inverse
operations with rational numbers
M07-S4C4-03 Calculate the area and
perimeter of composite 2-dimensional
figures. (areas)
M07-S4C4-05 Create a net to calculate the
surface area of a given solid. (nets)
M07-S4C4-06 Identify the appropriate unit
of measure to compute the volume of an
object and justify reasoning. (volume)
M07-S4C1-03 Draw and classify 3dimensional figures with appropriate labels
showing specified attributes of parallelism,
congruence, perpendicularity, and
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
59
2010 STANDARD
2008 PO
PLAN
symmetry.
M08-S4C4-03 Calculate the surface area
and volume of rectangular prisms, right
triangular prisms, and cylinders. (surface
area and volume)
7.SP.1 Understand that statistics can be
used to gain information about a
population by examining a sample of the
population; generalizations about a
population from a sample are valid only if
the sample is representative of that
population. Understand that random
sampling tends to produce
representative samples and support valid
inferences.
7.SP.2 Use data from a random sample
to draw inferences about a population
with an unknown characteristic of
interest. Generate multiple samples (or
simulated samples) of the same size to
gauge the variation in estimates or
predictions. For example, estimate the
mean word length in a book by randomly
sampling words from the book; predict
the winner of a school election based on
randomly sampled survey data. Gauge
how far off the estimate or prediction
might be.
M07-S2C1-04 Distinguish between a simple
random and non-random sample.
7.SP.3 Informally assess the degree of visual
overlap of two numerical data distributions with
similar variabilities, measuring the difference
between the centers by expressing it as a
M08-S2C1-01 Solve problems by selecting,
constructing, interpreting, and calculating with
displays of data, including box and whisker plots
and scatterplots.
M08-S2C1-02 Make inferences by comparing the
same summary statistic for two or more data sets.
M08-S2C1-05 Evaluate the design of an
experiment.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
60
2010 STANDARD
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.
2008 PO
M08-S2C1-02 Make inferences by comparing the
same summary statistic for two or more data sets.
7.SP.4 Use measures of center and measures
of variability for numerical data from random
samples to draw informal comparative inferences
about two populations. For example, decide
whether the words in a chapter of a seventhgrade science book are generally longer than the
words in a chapter of a fourth-grade science
book.
M06-S2C1-03 Use extreme values, mean, median,
mode, and range to analyze and describe the
distribution of a given data set. (measures of center
and variability)
M08-S2C1-02 Make inferences by comparing the
same summary statistic for two or more data sets.
MHS-S2C1-04 Make inferences by comparing data
sets using one or more summary statistics.
MHS-S2C1-05 Determine which measure of center
is most appropriate in a given situation and explain
why.
7.SP.5 Understand that the probability of a
chance event is a number between 0 and 1 that
expresses the likelihood of the event occurring.
Larger numbers indicate greater likelihood. A
probability near 0 indicates an unlikely event, a
probability around ½ indicates an event that is
neither unlikely nor likely, and a probability near
1 indicates a likely event.
7.SP.6 Approximate the probability of a chance
event by collecting data on the chance process
that produces it and observing its long-run
relative frequency, and predict the approximate
relative frequency given the probability. For
example, when rolling a number cube 600 times,
predict that a 3 or 6 would be rolled roughly 200
times, but probably not exactly 200 times.
PLAN
M08-S2C1-03 Describe how summary statistics
relate to the shape of the distribution.
M04-S2C2-01 Describe elements of theoretical
probability by listing or drawing all possible
outcomes of a given event and predicting the
outcome using word and number benchmarks.
M05-S2C2-01 Describe the theoretical probability of
events and represent the probability as a fraction,
decimal, or percent.
M05-S2C2-02 Explore probability when performing
experiments by




predicting the outcome,
recording the data,
comparing outcomes of the experiment to
predictions, and
comparing the results of multiple repetitions
of the experiment.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
61
2010 STANDARD
2008 PO
M07-S2C2-03 Compare the results of multiple
repetitions of the same probability experiment to the
theoretical probability.
PLAN
7.SP.7 Develop a probability model and
use it to find probabilities of events.
Compare probabilities from a model to
observed frequencies; if the agreement
is not good, explain possible sources of
the discrepancy.
a. Develop a uniform probability model
by assigning equal probability to all
outcomes, and use the model to
determine probabilities of events.
For example, if a student is selected
at random from a class, find the
probability that Jane will be selected
and the probability that a girl will be
selected.
b. Develop a probability model (which
may not be uniform) by observing
frequencies in data generated from a
chance process. For example, find
the approximate probability that a
spinning penny will land heads up or
that a tossed paper cup will land
open-end down. Do the outcomes
M06-S2C2-01 USE DATA COLLECTED FROM
MULTIPLE TRIALS OF A SINGLE EVENT TO FORM
A CONJECTURE ABOUT THE THEORETICAL
PROBABILITY.
M06-S2C2-02 Use theoretical probability to
 predict experimental outcomes,
 compare the outcome of the experiment to
the prediction, and
 replicate the experiment and compare
results.
M05-S2C2-02 Explore probability when performing
experiments by




predicting the outcome,
recording the data,
comparing outcomes of the experiment to
predictions, and
comparing the results of multiple repetitions
of the experiment.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
62
2010 STANDARD
for the spinning penny appear to be
equally likely based on the observed
frequencies?
7.SP.8 Find probabilities of compound
events using organized lists, tables, tree
diagrams, and simulation.
a. Understand that, just as with simple
events, the probability of a
compound event is the fraction of
outcomes in the sample space for
which the compound event occurs.
b. Represent sample spaces for
compound events using methods
such as organized lists, tables and
tree diagrams. For an event
described in everyday language
2008 PO
M06-S2C2-02 Use theoretical probability to
 predict experimental outcomes,
 compare the outcome of the experiment to
the prediction, and
 replicate the experiment and compare
results.
M07-S2C2-02 Experiment with two different events
to determine whether the two events are dependent
or independent of each other.
M07-S2S2-03 Compare the results of multiple
repetitions of the same probability experiment to the
theoretical probability.
M07-S2C2-04 Compare probabilities to determine
fairness in experimental situations.
M08-S2C2-02 Interpret probabilities within a given
context and compare the outcome of an experiment
to predictions made prior to performing the
experiment.
MHS-S2C2-05 Use concepts and formulas of area
to calculate geometric probabilities.
PLAN
M07-S2C2-01 Determine conditional probabilities
(experimental) in compound probability experiments.
M07-S2C3-01 Analyze relationships among the tree
diagrams where items repeat and do not repeat;
make numerical connections to the multiplication
principle of counting.
M08-S2C2-01 Determine theoretical and
experimental conditional probabilities in compound
probability experiments.
M06-S2C2-03 Determine all possible outcomes
(sample space) of a given situation using a
systematic approach.
M07-S2C2-01 Determine conditional probabilities
(experimental) in compound probability experiments.
M07-S2C3-01 Analyze relationships among the tree
diagrams where items repeat and do not repeat;
Tucson Unified School District
Mathematics Curriculum
Grade 7
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63
2010 STANDARD
(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?
7.MP.1 Make sense of problems and
persevere in solving them.
2008 PO
make numerical connections to the multiplication
principle of counting.
M08-S2C2-03 Use all possible outcomes (sample
space) to determine the probability of dependent
and independent events.
M07-S2C2-01 Determine conditional probabilities
(experimental) in compound probability experiments.
PLAN
M07-S5C2-01 Analyze a problem situation to
determine the question(s) to be answered.
M07-S5C2-02 Analyze and compare mathematical
strategies for efficient problem solving; select and
use one or more strategies to solve a problem.
M07-S5C2-03 Identify relevant, missing, and
extraneous information related to the solution to a
problem
M07-S5C2-04 Represent a problem situation using
multiple representations, describe the process used
to solve the problem, and verify the reasonableness
of the solution.
M07-S5C2-05 Apply a previously used problemsolving strategy in a new context.
M07-S5C2-06 Communicate the answer(s) to the
question(s) in a problem using appropriate
representations, including symbols and informal and
formal mathematical language.
M07-S5C2-07 Isolate and organize mathematical
information taken from symbols, diagrams, and
graphs to make inferences, draw conclusions, and
justify reasoning.
M07-S5C2-08 Make and test conjectures based on
information collected from explorations and
Tucson Unified School District
Mathematics Curriculum
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64
2010 STANDARD
7.MP.2 Reason abstractly and quantitatively.
7.MP.3 Construct viable arguments and
critique the reasoning of others.
7.MP.4 Model with mathematics.
7.MP.4 Model with mathematics.
7.MP.5 Use appropriate tools strategically.
2008 PO
experiments.
M07-S5C2-07 Isolate and organize mathematical
information taken from symbols, diagrams, and
graphs to make inferences, draw conclusions, and
justify reasoning.
PLAN
M07-S5C2-04 Represent a problem situation using
multiple representations, describe the process used
to solve the problem, and verify the reasonableness
of the solution.
M07-S5C2-07 Isolate and organize mathematical
information taken from symbols, diagrams, and
graphs to make inferences, draw conclusions, and
justify reasoning.
M07-S5C2-08 Make and test conjectures based on
information collected from explorations and
experiments.
M07-S5C2-02 Analyze and compare mathematical
strategies for efficient problem solving; select and
use one or more strategies to solve a problem.
M07-S5C2-04 Represent a problem situation using
multiple representations, describe the process used
to solve the problem, and verify the reasonableness
of the solution.
M07-S5C2-05 Apply a previously used problemsolving strategy in a new context.
M07-S5C2-06 Communicate the answer(s) to the
question(s) in a problem using appropriate
representations, including symbols and informal and
formal mathematical language.
M07-S5C2-07 Isolate and organize mathematical
information taken from symbols, diagrams, and
graphs to make inferences, draw conclusions, and
justify reasoning.
M07-S5C2-02 Analyze and compare mathematical
strategies for efficient problem solving; select and
use one or more strategies to solve a problem.
M07-S5C2-08 Make and test conjectures based on
information collected from explorations and
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
65
2010 STANDARD
7.MP.6 Attend to precision.
7.MP.7 Look for and make use of structure.
7.MP.8 Look for and express regularity in
repeated reasoning.
MOVEMENT
MOVED TO GRADE 6
MOVED TO GRADE 6
MOVED TO GRADE 8
MOVED TO GRADE 8
MOVED TO GRADE 8
REMOVED
MOVED TO GRADE 8
MOVED TO GRADE 8
2008 PO
experiments.
M07-S5C2-06 Communicate the answer(s) to the
question(s) in a problem using appropriate
representations, including symbols and informal and
formal mathematical language.
M07-S5C2-07 Isolate and organize mathematical
information taken from symbols, diagrams, and
graphs to make inferences, draw conclusions, and
justify reasoning.
M07-S5C2-08 Make and test conjectures based on
information collected from explorations and
experiments.
M07-S5C2-04 Represent a problem situation using
multiple representations, describe the process used
to solve the problem, and verify the reasonableness
of the solution.
2008 PO
M07-S1C1-02 Find or use factors, multiples, or prime
factorization within a set of numbers.
M07-S1C1-03 Compare and order rational numbers
using various models and representations.
M07-S1C2-04 Represent and interpret numbers
using scientific notation (positive exponents only).
M07-S1C3-01 Estimate and apply benchmarks for
rational numbers and common irrational numbers.
M07-S1C3-03 Estimate square roots of numbers less
than 1000 by locating them between two consecutive
whole numbers.
M07-S1C3-04 Estimate the measure of an object in
one system of units given the measure of that object
in another system and the approximate conversion
factor.
M07-S2C1-01 Solve problems by selecting,
constructing, and interpreting displays of data
including multi-line graphs and scatterplots.
M07-S2C1-02 Interpret trends in a data set, estimate
values for missing data, and predict values for points
Tucson Unified School District
Mathematics Curriculum
PLAN
PLAN
Grade 7
Board Approved 03/27/2012
66
beyond the range of the data set.
MOVED TO GRADE 8
REMOVED
REMOVED
MOVED TO GRADE 6
MOVED TO GRADE 6
MOVED TO HIGH SCHOOL
MOVED TO HIGH SCHOOL
MOVED TO GRADE 8
MOVED TO GRADE 8
MOVED TO GRADE 6
MOVED TO GRADE 6
REMOVED
REMOVED
M07-S2C1-03 Identify outliers and determine their
effect on mean, median, mode, and range.
M07-S2C3-02 Solve counting problems using Venn
diagrams and represent the answer algebraically.
M07-S2C4-01 Use vertex-edge graphs and
algorithmic thinking to represent and find solutions to
practical problems related to Euler/Hamilton paths
and circuits.
M07-S3C3-02 Evaluate an expression containing one
or two variables by substituting numbers for the
variables.
M07-S3C3-04 Translate between graphs and tables
that represent a linear equation.
M07-S4C1-01 Recognize the relationship between
central angles and intercepted arcs; identify arcs and
chords of a circle.
M07-S4C1-04 Describe the relationship between the
number of sides in a regular polygon and the sum of
its interior angles.
M07-S4C1-05 Identify corresponding parts of
congruent figures.
M07-S4C2-01 Model the result of a double
transformation (translations or reflections) of a 2dimensional figure on a coordinate plane using all
four quadrants.
M07-S4C4-02 Identify polygons having the same
perimeter or area.
M07-S5C1-01 Create an algorithm to determine the
area of a given composite figure.
M07-S5C2-09 Solve logic problems using multiple
variables and multiple conditional statements using
words, pictures, and charts.
M07-S5C2-10 Demonstrate and explain that the
process of solving equations is a deductive proof.
Tucson Unified School District
Mathematics Curriculum
Grade 7
Board Approved 03/27/2012
67