SCUSD Curriculum Map
Curriculum Map
DRAFT Last Updated August 1, 2014
Grade 7/8 Compacted Mathematics
Mathematics
Grade 7/8
Compacted
Sacramento City Unified School District
1
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Table of Contents
Compacted 7th/8th Year-at-a-Glance........................................................................................................................................................................................................................................................................................3
Unit #1: Proportional Reasoning and Relationships ...............................................................................................................................................................................................................................................................4
Unit #2: Applying Proportional Reasoning to Problems with Percents ..................................................................................................................................................................................................................................8
Unit #3: Operations with Rational Numbers – Addition and Subtraction ........................................................................................................................................................................................................................... 11
Unit #4: Operations with Rational Numbers – Multiplication and Division ........................................................................................................................................................................................................................ 15
Unit #5: Equivalent Expressions ........................................................................................................................................................................................................................................................................................... 19
Unit #6: Problem Solving with Equations and Inequalities .................................................................................................................................................................................................................................................. 22
Unit 7: Linear Relationships ................................................................................................................................................................................................................................................................................................. 28
Unit #8: Data Analysis .......................................................................................................................................................................................................................................................................................................... 32
Unit #9: Probability .............................................................................................................................................................................................................................................................................................................. 36
Unit #10: 2-Dimensional and 3-Dimensional Geometric Figures......................................................................................................................................................................................................................................... 40
Unit 11: Irrational Numbers and the Pythagorean Theorem............................................................................................................................................................................................................................................... 44
Unit 12: Exponents ............................................................................................................................................................................................................................................................................................................... 51
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SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Compacted 7th/8th Year-at-a-Glance
Month
Unit
Content Standards
September
Unit #1
Proportional Reasoning and Relationships
7.RP.1, 2
7.G.1
Unit #2
Applying Proportional Reasoning to Problems with Percents
7.RP.3
District Benchmark 1
October
District Benchmark 2
November
December/January
February
March
District Benchmark 3
CAASPP
(Smarter Balanced Summative Test)
Unit #3
Operations with Rational Numbers –Addition and Subtraction
7.NS.1, 3
Unit #4
Operations with Rational Numbers –Multiplication and Division
7.NS.2, 3
Unit #5
Equivalent Expressions
Unit #6
Problem Solving with Equations and Inequalities
Unit #7
Linear Relationships
7.EE.1, 2
Unit #8
Data Analysis
Unit #9
Probability
April
Unit #10
2-Dimensional and 3-Dimensional Geometric Figures
April/May
Unit #11
Irrational Numbers and The Pythagorean Theorem
May/June
Unit #12
Exponents
7.EE.3, 4
8.EE.7
8.EE.5, 6
8.F.2
7.SP.1, 2, 3, 4
7.SP.5, 6, 7, 8
7.G.1, 2, 3, 4, 5, 6
8.NS.1, 2
8.EE.2
8.G.6, 7, 8, 9
8.EE.1, 3, 4
3
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #1: Proportional Reasoning and Relationships
(Approx. # of Days ___)
Content Standards: 7.RP.1,2 and 7.G.1
In this unit, students will be able to use ratios and proportions appropriately
Common Core State Standards-Mathematics:
Ratios and Proportional Relationships 7.RP
Analyze proportional relationships and use them to solve real-world and mathematical problems.
1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour,
compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.
2. Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight
line through the origin.
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost
and the number of items can be expressed as t = pn.
d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate
Geometry 7.G
Draw, construct, and describe geometrical figures and describe the relationships between them.
1. Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
Standards for Mathematical Practice of Emphasis:
SMP.1 - Make Sense of Problems and Persevere in Solving Them
SMP.2 - Reason Abstractly and Quantitatively
SMP.4 - Model with Mathematics
ELD Standards to Support Unit:
Part I: Interacting in Meaningful Ways:
A. Collaborative:
2. Interacting with others in written English in various communicative forms
4
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
4. Adapting language choices to various contexts
B. Interpretive:
5. Listening actively to spoken English in a range of social and academic contexts.
C. Productive:
11. Supporting own opinions and evaluating others’ opinions in speaking and writing.
Part II: Learning About How English Works
A. Expanding and Enriching Ideas
5. Modifying to add details.
B. Connecting and Condensing Ideas
6. Connecting Ideas
7. Condensing Ideas
SEL Competencies:
Self-awareness
Self-management
Social awareness
Relationship skills
Responsible decision making
Unit #1 Proportional Reasoning and Relationships
Essential Questions
Assessments for
Learning
Assessments/Tasks
aligned to learning
outcomes.
Note: These Assessments
are suggested, not
required.
Sequence of Learning Outcomes
7.RP.1, 7.RP.2, 7.G.1
Students will be able to….
Strategies for Teaching and
Learning
Differentiation
(EL/SpEd/GATE)
Resources
CA Mathematics Framework Gr. 7
p. 6 – 14
Progressions for the Common Core – Ratios
and Proportional Relationships Gr. 6-7
North Carolina
7th Grade Math Unpacked Content: pgs. 6- 9,
25-26
5
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #1 Proportional Reasoning and Relationships
Essential Questions

Assessments for
Learning
Sequence of Learning Outcomes
7.RP.1, 7.RP.2, 7.G.1
What is the role of
unit rate in solving
problems?
How do you know
which of the two
unit rates is
important for a
problem?
What makes a
relationship
proportional?
http://www.illustrativem 1) Identify and utilize unit rates to solve
athematics.org/illustratio
real-world problems with proportional
ns/82
relationships containing whole numbers,
fractions and decimals by using visual
representations. (Framework p.12)
For Learning Outcomes
1-5:
7.RP.1
http://www.engageny.or
g
(this link is to a module
that has a variety of
tasks that relate to the
learning outcomes)
How do you know
which of the two
unit rates is
important for a
problem?
http://www.illustrativem 1) Use their understanding of unit rates
athematics.org/illustratio
and proportionality to create equations,
ns/101
a d
both in the form  and y = kx, to

What makes a
relationship
proportional?

What makes a
http://www.illustrativem 2) Identify, utilize and write equations with
athematics.org/illustratio
unit rates developed from scale
ns/107
drawings to solve problems and
reproduce a scale drawing at a different
scale. (Framework p.35, 36)
7.G.1
http://www.illustrativem 3) Use unit rate or constant of



b
Strategies for Teaching and
Learning
Teaching Ratios through Tape
Diagrams and Double Number
Lines:
http://math.kennesaw.edu/~twata
nab/DeKalb%20Title%20I%20Sum
mit%202012.pdf
Video on Teaching Unit Rate with
Tape Diagrams:
https://learnzillion.com/lessons/84
1-create-unit-rate-using-tape-dia
gram
Differentiation
(EL/SpEd/GATE)
Resources
Big Ideas
Section 5.1:
p.152-163 – exploratory activities
p. 164 – definition
p. 167-169 – practice finding the unit rate
p. 171 – activity 3 – examples of when to use
the unit rate
http://www.virtualnerd.com/middle-math/all/
(See Ratios, Proportions, and Percent)
Section 5.3
p. 178 – 184
Section 5.6
p. 199 – Activity 2 & 3 – Real World Examples
of Direct Variation
p. 200 – Identifying Direct Variation
p. 201 – Real Life Application
c
solve real-world problems. (Framework
p.12)
7.RP.2
Video on Scale Drawings:
http://www.virtualnerd.com/middl
e-math/ratios-proportions-percent
/scale-drawings-models/scale-draw
ing-definition
Section 7.5
p. 300 – Definition of Scale Models & Drawings
p. 301 – Definition of Scale Factor/Unit Rate
p. 303 – 304 – Examples of Scale Drawings
Video on Understanding Unit Rates:
Section 5.2
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SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #1 Proportional Reasoning and Relationships
Essential Questions
relationship
proportional?

Assessments for
Learning
athematics.org/illustratio
ns/1527
Sequence of Learning Outcomes
7.RP.1, 7.RP.2, 7.G.1
Strategies for Teaching and
Learning
proportionality to determine if a
https://learnzillion.com/lessons/24
relationship is proportional. Students
10
should explore a variety of
non-examples including: no relationship,
linear but not proportional, inverse
relationships, non-similar figures.
(Framework p.8,9)
7.RP.2
http://map.mathshell.org 4) Given a real-world example, work
simultaneously with a graph, table and
How is the constant /materials/download.php
?fileid=1070
equation. Determine if there is a
of proportionality
constant of proportionality in each
represented in a
representation. If so, identify the
graph, table and
constant of proportionality in each
equation?
representation, giving careful attention
to the point (1, r) on a graph.
7.G.1
Differentiation
(EL/SpEd/GATE)
Resources
p. T-164 – Example 2 Discusses a common
error. This can be used as an entry point to
discuss which unit rate is important for the
problem.
p. 170 & 171 – Determining Proportions
p. 172 – Comparing Ratios by simplifying to
simplest form
p. 173 – Example 3 Comparing Unit Rates to
Determine Proportionality
p. 176 – Graphing Proportions
Section 5.6
p. 201 – Example 3
p. 203 & T-203 – Practice with proportions
using multiple representations
Proportional Reasoning Sample Lesson:
http://math.serpmedia.org/dragonfly/dragonfl
y.pdf
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SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #2: Applying Proportional Reasoning to Problems with Percents
(Approx. # of Days ___)
Content Standards: 7.RP.3
In this unit, students will be able to apply proportions
Common Core State Standards-Mathematics:
Ratios and Proportional Relationships 7.RP
Analyze proportional relationships and use them to solve real-world and mathematical problems.
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.
Standards for Mathematical Practice:
SMP.1 - Make Sense of Problems and Persevere in Solving Them
SMP.3 - Construct Viable Arguments and Critique the Reasoning of Others
SMP.4 - Model with Mathematics
SMP.5 - Use Appropriate Tools Strategically
SEL Competencies:
Self-awareness
Self-management
Social awareness
ELD Standards to Support Unit:
Part I: Interacting in Meaningful Ways:
5) Collaborative:
3. Interacting with others in written English in various communicative forms
6. Adapting language choices to various contexts
6) Interpretive:
7. Listening actively to spoken English in a range of social and academic contexts.
7) Productive:
11. Supporting own opinions and evaluating others’ opinions in speaking and writing.
Part II: Learning About How English Works
C. Expanding and Enriching Ideas
5. Modifying to add details.
D. Connecting and Condensing Ideas
6. Connecting Ideas
7. Condensing Ideas
Relationship skills
Responsible decision making
8
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #2 Applying Proportional Reasoning to Problems with Percents
Essential Questions
Assessments for
Learning
Assessments/Tasks
aligned to learning
outcomes:
Sequence of Learning Outcomes
7.RP.3
Strategies for Teaching and
Learning
Students will be able to….
Differentiation
(EL/SpEd/GATE)
Resources
CA Mathematics Framework Gr. 7
p. 14 – 16
Progressions for the Common Core – Ratios
and Proportional Relationships Gr. 6-7
Note: These Assessments
are suggested, not
required.
North Carolina
7th Grade Math Unpacked Content: p. 10- 13
7th Grade Common Core State Standards Flip
Book



How can you
check for
reasonableness as
you solve a
problem and in
your answer?
How do you round
percentages
strategically to
estimate?
What are the
http://www.illustrativem 1) Estimate and calculate tips, simple
athematics.org/illustratio
interest, tax, fees and mark ups using
ns/106
bar modeling, double number lines and
algorithmic procedures. (Framework
http://map.mathshell.org
p.14, 15)
/materials/download.php
7.RP.3
?fileid=1042
Videos (click on links below):
 Estimating Percents Using Bar
Modeling
 Finding Cost After Tax
 Percent Increase/Decrease
Building background knowledge: Lessons
6.1-6.4 (for changing % to decimals,
reinforce the concept of why we move the
decimal twice)
Lessons 6.4, 6.6 and 6.7
For tax: Need p. 250 problem #3 and p. 251
problems #24 & #25
For tip: see framework p. 15
Relationships between decimals, fractions,
9
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #2 Applying Proportional Reasoning to Problems with Percents
Essential Questions
Assessments for
Learning
Sequence of Learning Outcomes
7.RP.3
Strategies for Teaching and
Learning
connections
between bar
modeling, double
number lines and
the algorithmic
procedure?
Resources
and percentages:
http://www.mathsisfun.com/decimal-fractionpercentage.html

When bar
modeling, how do
you decide how to
“chunk” the
percents in the
model? (Ex: How
is 15%
represented? Is it
10% + 5% or 10%
+ 1% +1% +1%
+1% +1% or….)

Which quantity
http://map.mathshell.org 3) Estimate and calculate percent change
represents the
/materials/download.p
including identifying the original value
whole (or 100%)?
hp?fileid=794
and comparing the difference in two
How can you check
values to the starting price. (Framework
for reasonableness
p.15, 16)
as you solve a
7.RP.3
problem and in your
answer?

Differentiation
(EL/SpEd/GATE)
http://www.illustrativem 2) Estimate and calculate discounts,
Videos (click on links below):
athematics.org/illustratio
markdowns and sales using bar
 Estimate Percent Using Bar
ns/105
modeling, double number lines and
Modeling
algorithmic procedures.
 Tax, Tips, and Discounts
http://map.mathshell.org
7.RP.3
/materials/download.php
?fileid=1524
Lessons 6.4 and 6.6
Lessons 6.5
Refer to 6.1 for Fraction, decimal, percent
conversions
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SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #3: Operations with Rational Numbers – Addition and Subtraction
(Approx. # of Days ____)
Content Standards: 7.NS.1, 3
In his unit, students will be able to add & subtract integers
Common Core State Standards-Mathematics:
The Number System 7.NS
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
1. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.
b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a
sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of
their difference, and apply this principle in real-world contexts.
d. Apply properties of operations as strategies to add and subtract rational numbers.
3. Solve real-world and mathematical problems involving the four operations with rational numbers.
Standards for Mathematical Practice:
SMP.1 - Make Sense of Problems and Persevere in Solving Them
SMP.2 - Reason Abstractly and Quantitatively
SMP.3 - Construct Viable Arguments and Critique the Reasoning of Others
SMP.4 - Model with Mathematics
SMP.5 - Use Appropriate Tools Strategically
SMP.6 - Attend to Precision
SMP.7 - Look For and Make Use of Structure
SMP.8 - Look For and Express Regularity in Repeated Reasoning
ELD Standards to Support Unit:
Part I: Interacting in Meaningful Ways:
8) Collaborative:
4. Interacting with others in written English in various communicative forms
8. Adapting language choices to various contexts
9) Interpretive:
11
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
9. Listening actively to spoken English in a range of social and academic contexts.
10) Productive:
11. Supporting own opinions and evaluating others’ opinions in speaking and writing.
Part II: Learning About How English Works
E. Expanding and Enriching Ideas
5. Modifying to add details.
F. Connecting and Condensing Ideas
6. Connecting Ideas
7. Condensing Ideas
Essential Questions
Assessments for
Learning
Assessments/Tasks
aligned to learning
outcomes:
SEL Competencies:
Self-awareness
Self-management
Social awareness
Relationship skills
Responsible decision making
Unit #3 Operations with Rational Numbers – Addition and Subtraction
Sequence of Learning Outcomes
Strategies for Teaching and
Differentiation
7.NS.1, 7.NS.3
Learning
(EL/SpEd/GATE)
Students will be able to…
Note: These Assessments
are suggested, not
required.
Resources
CA Mathematics Framework Gr. 7
p. 18 – 28
http://www.cde.ca.gov/ci/ma/cf/documents/a
ug2013gradeseven.pdf
Progressions for the Common Core – The
Number System gr. 6-8
North Carolina
7th Grade Math Unpacked Content: p. 14 – 1 7



What is a zero
pair?
How can you use
zero pairs to solve
problems?
How do you know
For learning outcomes 1 1) Understand and develop fluency adding Strategies for adding and
&2
rational numbers (integers, fractions and subtracting positive and negative
http://map.mathshell.org
decimals) by creating zero pairs using
numbers (click on links):
/materials/lessons.php?t
counting chips, the number line,
 Counting Chips
askid=453#task453
decomposition and mental math. Apply
 T-Charts
understanding to solve real-world
7th Grade Common Core State Standards Flip
Book
Big Ideas 7th grade TE:
Lesson 1.2
p. 8 and 9
(focuses only on integers)
The remainder of this lesson is more applicable
12
SCUSD Curriculum Map
Essential Questions





which number to
decompose when
creating zero
pairs?
How do zero pairs
and number lines
compare and
contrast?
What is the most
efficient method
to use for any
given problem?
How can you
subtract
something that
isn’t there? (Ex: -3
– 2, how can you
subtract 2
positives from 3
negatives?)
How do you know
how many zero
pairs to add to a
problem in order
to subtract (take
away)?
Grade 7/8 Compacted Mathematics
Assessments for
Learning
http://www.illustrativem
athematics.org/illustratio
ns/46
Unit #3 Operations with Rational Numbers – Addition and Subtraction
Sequence of Learning Outcomes
Strategies for Teaching and
Differentiation
7.NS.1, 7.NS.3
Learning
(EL/SpEd/GATE)
problems. (Framework p.20-21)
7.NS.1, 7.NS.3
http://www.illustrativem
athematics.org/illustratio
ns/998
http://www.illustrativem
athematics.org/illustratio
ns/317




Decomposition
Number Line
Reading the problem aloud
Making connections to
real-world problems (i.e.
problems involving money,
debt, etc.)
Stay away from rote memorization
techniques or mnemonic devices
(e.g., “keep-change-change”,
etc.)
2) Understand and develop fluency
Strategies for adding and
subtracting rational numbers (integers,
subtracting positive and negative
fractions and decimals) by creating zero
numbers:
pairs using counting chips and the
 Counting Chips
number line, with an emphasis on
 “Taking Away”
“taking away” and by seeing subtraction
as the inverse of addition (c – b = a
means a + b = c). Apply understanding to
solve real-world problems. (Framework
p.22, 23)
7.NS.1, 7.NS.3
Why is subtracting http://www.illustrativem 3) Compare and contrast work with
athematics.org/illustratio
addition and subtraction of rational
a negative
ns/310
numbers to build the understanding that
equivalent to
p – q = p + (-q) for the purpose of
adding a positive?
Resources
to learning outcome 4
Lesson 2.2
p. 50
(includes all rational numbers) Students have
experience with adding and subtracting all
positive rational numbers in earlier grades so
these two lessons can be taught
simultaneously.
Big Ideas 7th grade TE:
Lesson 1.3
p.14 only
This lesson focuses primarily on learning
outcome 3 and does not sufficiently meet
the expectations of this outcome
Big Ideas 7th grade TE:
Lesson 1.3
p.14 (integers only)
13
SCUSD Curriculum Map
Essential Questions

How can you
transition from
using a particular
method to solve a
problem to just
knowing the
answer?
Grade 7/8 Compacted Mathematics
Assessments for
Learning
Unit #3 Operations with Rational Numbers – Addition and Subtraction
Sequence of Learning Outcomes
Strategies for Teaching and
Differentiation
7.NS.1, 7.NS.3
Learning
(EL/SpEd/GATE)
thinking of any subtraction problem as
an addition problem with a negative
quantity. Apply understanding to solve
real-world problems.
7.NS.1, 7.NS.3
4) Synthesize the work they have done
with addition and adding the opposite to
create an algorithm around comparing
quantities of rational numbers and
either adding or subtracting.
(Framework p.21)
7.NS.1
Resources
lesson 2.3
p. 58 (all rational numbers)
*see note in outcome 1 about teaching the
lessons simultaneously.
Big Ideas 7th grade TE:
Lesson 1.2
Starting on p. 10
14
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #4: Operations with Rational Numbers – Multiplication and Division
(Approx. # of days ____)
Content Standards: 7.NS.2, 3
In this unit, students will be able to multiply and divide integers
Common Core State Standards-Mathmetics:
The Number System 7.NS
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property,
leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–
p)/q = p/(–q). Interpret quotients of rational numbers by describing real world contexts.
c. Apply properties of operations as strategies to multiply and divide rational numbers.
d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
3. Solve real-world and mathematical problems involving the four operations with rational numbers.
Standards for Mathematical Practice:
SMP.1 - Make Sense of Problems and Persevere in Solving Them
SMP.2 - Reason Abstractly and Quantitatively
SMP.3 - Construct Viable Arguments and Critique the Reasoning of Others
SMP.4 - Model with Mathematics
SMP.5 - Use Appropriate Tools Strategically
SMP.6 - Attend to Precision
SMP.7 - Look For and Make Use of Structure
SMP.8 - Look For and Express Regularity in Repeated Reasoning
ELD Standards to Support Unit:
Part I: Interacting in Meaningful Ways:
11) Collaborative:
5. Interacting with others in written English in various communicative forms
10. Adapting language choices to various contexts
12) Interpretive:
15
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
11. Listening actively to spoken English in a range of social and academic contexts.
13) Productive:
11. Supporting own opinions and evaluating others’ opinions in speaking and writing.
Part II: Learning About How English Works
G. Expanding and Enriching Ideas
5. Modifying to add details.
H. Connecting and Condensing Ideas
6. Connecting Ideas
7. Condensing Ideas
Essential Questions
Assessments for
Learning
Assessments/Tasks
aligned to learning
outcomes:
SEL Competencies:
Self-awareness
Self-management
Social awareness
Relationship skills
Responsible decision making
Unit #4 Operations with Rational Numbers – Multiplication and Division
Sequence of Learning Outcomes
Strategies for Teaching and
Differentiation
7.NS.2, 7.NS.3
Learning
(EL/SpEd/GATE)
Students will be able to…
Note: These Assessments
are suggested, not
required.


Where do the
rules of signed
numbers come
from?
Why is the
product of two
negative numbers
a positive
Resources
CA Mathematics Framework Gr. 7
p. 18 – 28
http://www.cde.ca.gov/ci/ma/cf/documents/a
ug2013gradeseven.pdf
Progressions for the Common Core – The
Number System gr. 6-8
1) Understand and develop fluency of
Definition of multiplication for
multiplication of integers through
integers, for example:
definition of integers and multiplication
3(-4) is three groups of negative
as repeated addition. Additional
four or -4 + -4 + -4 = -12 and
methods that should be explored
-3(-4) is the opposite of three
include using patterns in products of
groups of negative four or –(-4 +
integers and the proof of why (-1)(-1) =
-4 + -4) = -(-12) = 12.
1. (Framework p. 25, 26)
7.NS.2 Videos:
North Carolina
7th Grade Math Unpacked Content: p. 14 – 1 7
Big Ideas 7th grade TE:
Lesson 1.4
p. 22
It is essential here that the conceptual
understanding is built and this leads to the
discovery of the rules mentioned in outcome 2.
Proof for (-1)(-1) = 1
16
SCUSD Curriculum Map
Essential Questions
Grade 7/8 Compacted Mathematics
Assessments for
Learning
Unit #4 Operations with Rational Numbers – Multiplication and Division
Sequence of Learning Outcomes
Strategies for Teaching and
Differentiation
7.NS.2, 7.NS.3
Learning
(EL/SpEd/GATE)
number?
Understanding multiplication using
number lines:
http://www.youtube.com/watch
?v=K_Jqdw3NpEw
Resources
p.64
Multiplication using
decomposition:
http://www.youtube.com/watch?v
=Q_V1brQtxT0



What are
examples of
multiplying and
dividing signed
rational numbers
in real life?
How can you
extend the rules
for integers to all
rational numbers?
How do the rules
for multiplying
signed numbers
help you know the
rules for dividing
signed numbers?
2) Develop the rules for multiplying
integers and extend that understanding
to all rational numbers for the purpose
of fluency. Apply rules of signed
numbers to real-world contexts.
7.NS.2
3) Extend the rules of multiplication to
division of integers using the inverse
relationship between multiplication and
division. Apply division of integers to
real-world contexts.
7.NS.2
Proof of (-1)(-1) = 1
https://www.khanacademy.org/ma
th/arithmetic/absolute-value/mul
t_div_negatives/v/why-a-negativ
e-times-a-negative-is-a-positive
Students look for patterns of
products in integers.
Big Ideas 7th grade TE:
Lesson 1.5
p. 28
17
SCUSD Curriculum Map
Essential Questions

How do you know
if a number is
rational?
Grade 7/8 Compacted Mathematics
Assessments for
Learning
Unit #4 Operations with Rational Numbers – Multiplication and Division
Sequence of Learning Outcomes
Strategies for Teaching and
Differentiation
7.NS.2, 7.NS.3
Learning
(EL/SpEd/GATE)
4) Apply rules of multiplication and division Multiplying fractions:
to all rational numbers. Solve real-world
3 5
 1  5 
  3  
problems involving both operations.
4 7
 4  7 
7.NS.2
Resources
Big Ideas 7th grade TE:
Lesson 2.4
p. 66
http://www.illustrativem 5) Convert rational numbers to decimals
athematics.org/illustratio
using long division; know that the
ns/604
decimal form of a rational number
terminates in 0’s or repeats.
http://www.illustrativem
7.NS.2
athematics.org/illustrat
ions/593
Big Ideas 7th grade TE:
Lesson 2.1
p.46
http://www.illustrativem 6) Solve real-world and mathematical
athematics.org/illustratio
problems involving the four operations
ns/298
with rational numbers. (Framework
p.28)
7.NS.2
Embedded into the practice of each of the
above mentioned lessons.
18
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #5: Equivalent Expressions
(Approx. # of Days ____)
Content Standards: 7.EE.1,2
In this unit, students will be able to identify and generate equivalent expressions.
Common Core State Standards-Mathematics:
Expressions and Equations 7.EE
Use properties of operations to generate equivalent expressions.
1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
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.”
Standards for Mathematical Practice:
SMP.3 - Construct Viable Arguments and Critique the Reasoning of Others
SMP.7 - Look For and Make Use of Structure
SMP.8 - Look For and Express Regularity in Repeated Reasoning
SEL Competencies:
Self-awareness
Self-management
Social awareness
ELD Standards to Support Unit:
Part I: Interacting in Meaningful Ways:
14) Collaborative:
6. Interacting with others in written English in various communicative forms
12. Adapting language choices to various contexts
15) Interpretive:
13. Listening actively to spoken English in a range of social and academic contexts.
16) Productive:
11. Supporting own opinions and evaluating others’ opinions in speaking and writing.
Part II: Learning About How English Works
I. Expanding and Enriching Ideas
5. Modifying to add details.
J. Connecting and Condensing Ideas
6. Connecting Ideas
7. Condensing Ideas
Relationship skills
Responsible decision making
19
SCUSD Curriculum Map
Essential Questions
Grade 7/8 Compacted Mathematics
Suggested
Assessments for
Learning
Assessments/Tasks
aligned to learning
outcomes:
Unit #5 Equivalent Expressions
Sequence of Learning Outcomes
Strategies for Teaching and
7.EE.1, 7.EE.2
Learning
Students will be able to…
Note: These Assessments
are suggested, not
required.
Differentiation
(EL/SpEd/GATE)
Resources
CA Mathematics Framework Gr. 7
p. 28 – 31
Progressions for the Common Core –
Expressions and Equations Gr. 6 – 8
North Carolina
7th Grade Math Unpacked Content: p. 18 – 2 0
7th Grade Common Core State Standards Flip
Book

Of the many
possible
equivalent
expressions, how
does each
represent the
meaning of a
given situation?
http://www.illustrativem 1) Generate equivalent expressions
Use pattern problems like the “Pool
athematics.org/illustratio
containing rational numbers by
Border Problem” (Framework p.
ns/541
combining like terms in mathematical
31).
and real-world problems. Compare the
meaning of each equivalent expression Possible use of manipulatives:
in the context of real-world problems.
Integer tiles
(Framework p.29)
7.EE.2 Other real-world problems could
include:
 Perimeter/Area Problems
 Cell Phone Plans

Of the many
possible
equivalent
expressions,
which of them
best represents
the meaning of
http://www.illustrativem 2) Generate equivalent expressions
athematics.org/illustratio
containing rational numbers using the
ns/543
distributive property, both expanding
http://www.illustrativem
and factoring, in mathematical and
athematics.org/illustratio
real-world problems. Compare the
ns/1450
meaning of each equivalent expression
in the context of real-world problems.
20
SCUSD Curriculum Map
Essential Questions
the situation?

Grade 7/8 Compacted Mathematics
Suggested
Assessments for
Learning
Unit #5 Equivalent Expressions
Sequence of Learning Outcomes
Strategies for Teaching and
7.EE.1, 7.EE.2
Learning
Differentiation
(EL/SpEd/GATE)
Resources
(Framework p.29)
7.EE.2
http://www.illustrativem
3)
Generate
equivalent
expressions
How does
containing rational numbers using the
changing one term athematics.org/illustratio
ns/433
distributive property, addition and
of an expression
subtraction, i.e. 8 – 2(0.5x + 1) in
change the
mathematical and real-world problems.
meaning of the
Compare the meaning of each
context?
equivalent expression in the context of
real-world problems.
7.EE.1
21
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #6: Problem Solving with Equations and Inequalities
(Approx. # Days)
Content Standards: 7.EE.3,4 and 8.EE.7
Math Common Core Content Standards:
Domain: Expressions and Equations 7.EE
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
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.
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?
Expressions and Equations 8.EE
Analyze and solve linear equations and pairs of simultaneous linear equations.
7. Solve linear equations in one variable.
a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given
equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
b.
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
Standards for Mathematical Practice:
SMP-1. Make Sense of Problems and Persevere in Solving Them
SMP-2. Reason Abstractly and Quantitatively
SMP-3. Construct Viable Arguments and Critique the Reasoning of Others
SMP-6. Attend to Precision
SEL Competencies:
Self-awareness
Self-management
Social awareness
Relationship skills
Responsible decision making
ELD Standards to Support Unit:
Part I: Interacting in Meaningful Ways:
D. Collaborative:
7. Interacting with others in written English in various communicative forms
14. Adapting language choices to various contexts
22
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
E. Interpretive:
15. Listening actively to spoken English in a range of social and academic contexts.
F. Productive:
11. Supporting own opinions and evaluating others’ opinions in speaking and writing.
Part II: Learning About How English Works
K. Expanding and Enriching Ideas
5. Modifying to add details.
L. Connecting and Condensing Ideas
6. Connecting Ideas
7. Condensing Ideas
Essential Questions
Assessments for
Learning
Assessments/Tasks
aligned to learning
outcomes:
Note: These Assessments
are suggested, not
required.
Unit #6 Problem Solving with Equations and Inequalities
Sequence of Learning Outcomes
Strategies for Teaching and
7.EE.3, 7.EE.4, 8.EE.7
Learning
Students will be able to…
Differentiation
(EL/SpEd/GATE)
Resources
CA Mathematics Framework Gr. 7
p. 31 – 33
Progressions for the Common Core –
Expressions and Equations Gr. 6 – 8
North Carolina
7th Grade Math Unpacked Content: p. 21 – 24
7th Grade Common Core State Standards Flip
Book
CA Mathematics Framework Gr. 8
p. 17-18
North Carolina
8th Grade Math Unpacked Content:
p. 15 – 16
23
SCUSD Curriculum Map
Essential Questions
Grade 7/8 Compacted Mathematics
Assessments for
Learning
Unit #6 Problem Solving with Equations and Inequalities
Sequence of Learning Outcomes
Strategies for Teaching and
7.EE.3, 7.EE.4, 8.EE.7
Learning
Differentiation
(EL/SpEd/GATE)
Resources
8th Grade Common Core State Standards Flip
Book



http://www.illustrativem 1) Solve multi-step, real-life and
What are some
athematics.org/illustratio
mathematical problems by using
arithmetic tools
ns/997
arithmetic methods such as bar
you can use to
modeling, Guess and Check, drawing a
solve real-life
picture or other tools instead of creating
problems?
an equation. Use estimation to assess
When is it
the reasonableness of answers.
appropriate to use
7.EE.3
arithmetic tools
and when is it
appropriate to
solve equations
algebraically?
For equations
such as
Problem solving strategies for
real-world context problems
(click on links, where available):
 Bar Modeling (video)
 Drawing a picture
 Make a table
 Guess and Check
 Estimation (3/7 of $105 is
about ½ of $100)
 Integer Tiles
 Side-by-side instruction
 Multiple Representations
Using multiple representations to solve
problems:
http://www.acoe.org/acoe/files/EdServices/M
ath/OneStepEquationsMultipleApproachesV3.
pdf
5x  10  25
and
2
9 x  6  10 ,
3

what are different
methods for
solving
algebraically?
When solving a
problem using
both methods
(arithmetic tools
and algebraically),
where do you see
24
SCUSD Curriculum Map
Essential Questions




Grade 7/8 Compacted Mathematics
Assessments for
Learning
Unit #6 Problem Solving with Equations and Inequalities
Sequence of Learning Outcomes
Strategies for Teaching and
7.EE.3, 7.EE.4, 8.EE.7
Learning
Differentiation
(EL/SpEd/GATE)
Resources
relationships in
your work?
How do you
interpret the
graph of an
inequality in terms
of the context of
the problem?
http://www.illustrativem 2) Generate equations equivalent to px + q
What is the
= r with rational coefficients and solve
purpose of using athematics.org/illustratio
ns/108
mathematical and real-life situations
inverse
using inverse operations.
operations?
7.EE.4
http://www.illustrativem 3) Generate equations equivalent to p(x +
What does your
q) = r with rational coefficients and solve
solution mean in athematics.org/illustratio
mathematical and real-life situations
the context of the ns/478
using inverse operations.
problem?
7.EE.4
http://www.illustrativem 4) Compare and contrast the use of
athematics.org/illustratio
arithmetic (see 1) versus algebraic
ns/712
methods (see 2 and 3) of solving
equations equivalent to px + q = r and
p(x + q) = r in mathematical and real-life
situations. *
7.EE.4
Use inverse operations to solve
algebraic equations (i.e. creating
zeroes and ones). Warn against the
language of “cancel out.”
Learning Outcome 4 can be
embedded in Outcomes 2 and 3
25
SCUSD Curriculum Map
Essential Questions





Grade 7/8 Compacted Mathematics
Assessments for
Learning
Unit #6 Problem Solving with Equations and Inequalities
Sequence of Learning Outcomes
Strategies for Teaching and
7.EE.3, 7.EE.4, 8.EE.7
Learning
http://www.illustrativem 5) Generate and solve inequalities with
Use investigation to help students
Why and when
rational numbers, in the form of px + q <
understand the reason for
would you reverse athematics.org/illustratio
ns/643
r and px + q > r (including < and >) that
reversing inequality symbols
an inequality
arise from real world problems. Graph
when multiplying or dividing by
symbol?
the
solution
region
and
interpret
the
negative numbers.
When solving
meaning of solutions in the context of
https://www.youtube.com
equations and
the problem.
inequalities, using
7.EE.4
inverse
operations, how
do you know
whether to create
a zero or a one?
How do you
interpret the
graph of an
inequality in terms
of the context of
the problem?
What is a
solution?
Differentiation
(EL/SpEd/GATE)
Resources
Why the inequality sign changes when
multiplying or dividing by a negative number:
http://www.algebra.com/algebra/homework/I
nequalities/Inequalities.faq.question.203735.
html
Learning Outcomes 6-8: 6) Use inverse operations to solve linear
Inverse operations (creating zeroes
http://www.illustrativem
equations in one variable with rational
and ones)
athematics.org/illustrat
coefficients, including equations that
“Geometric situations”: For
ions/550
have variables and constants on both
example, writing a linear
sides of the equal sign, arising in
equation to solve for the
http://www.illustrativem
algebraic, geometric, and real-world
measure of a missing angle of a
athematics.org/illustrat
situations.
triangle.
ions/392
8.EE.7
http://www.illustrativem
athematics.org/illustrat 7) Analyze a given equation to determine
How can you
whether it has one solution (x = a),
make assumptions ions/999
infinite solutions (a = a), or zero
or predictions
*Embed in Learning Outcomes 1 &
3 (See Framework p. 18).
26
SCUSD Curriculum Map
Essential Questions

Grade 7/8 Compacted Mathematics
Assessments for
Learning
about the number http://map.mathshell.org
of solutions at
/materials/download.p
multiple points
hp?fileid=1154
throughout the
process of solving http://map.mathshell.org
linear equations?
/materials/lessons.php
?taskid=442#task442
What does it
mean for a linear
equation to have http://map.mathshell.org
/materials/lessons.php
one solution, no
?taskid=487#task487
solutions, or
infinite solutions?
Unit #6 Problem Solving with Equations and Inequalities
Sequence of Learning Outcomes
Strategies for Teaching and
7.EE.3, 7.EE.4, 8.EE.7
Learning
Differentiation
(EL/SpEd/GATE)
Resources
solutions (a = b); explain their reasoning Throughout the process of solving
using the definition of solution.*
an equation, students should
8.EE.7 make assumptions or predictions
about the number of solutions by
comparing each side of the
equation. For example, students
should reason that the equation
5x + 2 = 5x + 2 must have infinite
solutions without having to
simplify further.
8) Solve linear equations with rational
number coefficients, including equations
whose solutions require expanding
expressions using the distributive
property and collecting like terms,
arising in algebraic, geometric, and
real-world situations.
8.EE.7
27
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit 7: Linear Relationships
(Approx. 24 Days)
Content Standards: 8.EE.5,6 and 8.F.2
In this unit students will be able to identify, compare, and graph linear relationships.
Common Core State Standards-Mathematics:
Expressions and Equations
8.EE
Understand the connections between proportional relationships, lines, and linear equations.
5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a
distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
6. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the
equation y = mx + b for a line intercepting the vertical axis at b.
Functions
8.F
Define, evaluate, and compare functions.
2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a
table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
Standards for Mathematical Practice:
SMP-1. Make Sense of Problems and Persevere in Solving Them
SMP-2. Reason Abstractly and Quantitatively
SMP-4. Model with Mathematics
SMP-5. Use Appropriate Tools Strategically
SMP-8. Look For and Express Regularity in Repeated Reasoning
ELD Standards to Support Unit:
Part I: Interacting in Meaningful Ways:
G. Collaborative:
8. Interacting with others in written English in various communicative forms
16. Adapting language choices to various contexts
H. Interpretive:
SEL Competencies:
Self-awareness
Self-management
Social awareness
Relationship skills
Responsible decision making
28
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
17. Listening actively to spoken English in a range of social and academic contexts.
Productive:
11. Supporting own opinions and evaluating others’ opinions in speaking and writing.
Part II: Learning About How English Works
M. Expanding and Enriching Ideas
5. Modifying to add details.
N. Connecting and Condensing Ideas
6. Connecting Ideas
7. Condensing Ideas
I.
Unit #7 Linear Relationships
Essential Questions
Assessments for
Learning
Assessments/Tasks
aligned to learning
outcomes:
Note: These Assessments
are suggested, not
required.
Sequence of Learning Outcomes
8.EE.5, 8.EE.6, 8.F.2
Students will be able to…
Strategies for Teaching and
Learning
Differentiation
e.g., EL, SpEd, GATE
Differentiation Support
for Unit:
Use of math journals for
differentiation and
formative assessment
(use link below)
https://www.teaching
channel.org/videos/m
ath-journals
Flexible grouping:
 Content
 Interest
 Project/product
Resources
CCSS Support for Unit:
CA Mathematics Framework Gr. 8
p. 11 – 17, 23
Progressions for the Common Core –
Expressions and Equations Gr. 6-8
North Carolina
8th Grade Math Unpacked Content:
p. 13 – 14 and 20 – 21
f
8th Grade Common Core State Standards Flip
Book
29
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #7 Linear Relationships
Essential Questions




Assessments for
Learning
Sequence of Learning Outcomes
8.EE.5, 8.EE.6, 8.F.2
Strategies for Teaching and
Learning
Differentiation
e.g., EL, SpEd, GATE
Resources
For Learning Outcomes
1) Graph proportional relationships given a Using similar triangles to prove
 Level
1 - 2:
real-world context and interpret the unit slope formula (videos and
(Heterogeneous/
http://www.illustrative
rate as the slope of the graph.
practice problems, click on links
Homogeneous)
mathematics.org/illustr
8.EE.5 below):
Tiered:
ations/129
http://www.youtube.com/
 Independent
https://www.khanacademy
Management Plan
http://www.illustrativem
http://www.illustrativemathematic
(Must Do/May Do)
athematics.org/illustrat
s.org/illustrations/1537
 Grouping
ions/55
o Content
When given a context, pay
o Rigor w/in the
http://www.illustrativem
attention to the units involved
concept
athematics.org/illustrat
throughout problem solving
o Project-based
ions/184
process.
learning
o Homework
2) Compare two different proportional
Use side-by-side instruction with
What are some
o Grouping
relationships
represented
in
different
graphs,
tables,
equations,
and
examples of linear
o Formative
ways, for example, in a graph, a table, an verbal descriptions for a given
relationships that
Assessment
equation, and a verbal description.
real-world context (for example,
are/are not
Anchor
Activities:
8.EE.5,
8.F.2
use
a
graphic
organizer
for
proportional?
 Content-related
student work).
How do you
tasks for early
know?
finishers
o Game
How can you use http://www.illustrativem 3) Use similar triangles to explain why the Emphasize the similarities and
o Investigation
athematics.org/illustrat
slope m is the same between any two
differences between proportional
the slope of a line
o Partner
ions/1537
distinct points on a non-vertical line in
and non-proportional
to find additional
Activity
the coordinate plane. (Framework p. 16)
relationships.
points on the line?
o
Stations
8.EE.6
Depth
and
Complexity
For Learning Outcomes
4) Derive and understand slope/rate of
Derivation (in experiences 4-6)
4 – 7:
change given a real-world context by
should be studied in Framework, Prompts/Icons:
 Depth
http://www.illustrativem
using graphs, tables, equations (y=mx)
p. 17
o Language of
athematics.org/illustrat
and verbal descriptions in the first
What does the
slope of a
proportional
relationship mean
in the context of a
problem?
Where do you see
the slope in the
problem? In the
table? In the
graph? In the
equation?
30
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #7 Linear Relationships
Essential Questions
Assessments for
Learning
ions/641
Sequence of Learning Outcomes
8.EE.5, 8.EE.6, 8.F.2
quadrant.
8.EE.6


http://www.illustrativem
athematics.org/illustrat 5) Derive and understand slope/rate of
ions/352
change with a y-intercept given a
real-world context by using graphs,
http://www.illustrativem
tables, equations (y=mx + b) and verbal
athematics.org/illustrat
descriptions in the first quadrant.
ions/86
8.EE.6
6) Derive and understand slope/rate of
http://www.illustrativem
change and y-intercept in context in all
athematics.org/illustrat
quadrants.
ions/1552
8.EE.5
7) Model real-world problems with the
Given a context,
relationships y=mx and y=mx + b.
which quadrants
Determine what parts of the graph make
are reasonable for
sense in context of the situation.
your graph? Why?
8.EE.5
What is
similar/different
about the
equations y=mx
and y=mx + b?
Strategies for Teaching and
Learning
Differentiation
e.g., EL, SpEd, GATE
Resources
the Discipline
Patterns
Unanswered
Questions
o Rules
o Trends
o Big Ideas
 Complexity
See Differentiation
Resources at:
http://scusd-math.wik
ispaces.com/home
o
o
31
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #8: Data Analysis
(Approx. # of Days ____)
Content Standards: 7.SP.1,2,3,4
In this unit, students will be able to gather, sort, and appropriately interpret data
Common Core State Standards-Mathematics:
Statistics and Probability 7.SP
Use random sampling to draw inferences about a population.
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.
2. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the
variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled
survey data. Gauge how far off the estimate or prediction might be.
Draw informal comparative inferences about two populations.
3. Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of
variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on
either team; on a dot plot, the separation between the two distributions of heights is noticeable.
4. Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a
chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.
Standards for Mathematical Practice:
SMP.1 - Make Sense of Problems and Persevere in Solving Them
SMP.2 - Reason Abstractly and Quantitatively
SMP.3 - Construct Viable Arguments and Critique the Reasoning of Others
SMP.4 - Model with Mathematics
SMP.5 - Use Appropriate Tools Strategically
SMP.6 - Attend to Precision
SMP.7 - Look For and Make Use of Structure
32
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
ELD Standards to Support Unit:
Part I: Interacting in Meaningful Ways:
17) Collaborative:
9. Interacting with others in written English in various communicative forms
18. Adapting language choices to various contexts
18) Interpretive:
19. Listening actively to spoken English in a range of social and academic contexts.
19) Productive:
11. Supporting own opinions and evaluating others’ opinions in speaking and writing.
Part II: Learning About How English Works
O. Expanding and Enriching Ideas
5. Modifying to add details.
P. Connecting and Condensing Ideas
6. Connecting Ideas
7. Condensing Ideas
Essential Questions
Assessments for
Learning
Assessments/Tasks
aligned to learning
outcomes:
Note: These Assessments
are suggested, not
required.
SEL Competencies:
Self-awareness
Self-management
Social awareness
Relationship skills
Responsible decision making
Unit #8 Data Analysis
Sequence of Learning Outcomes
Strategies for Teaching and
7.SP.1, 7.SP.2, 7.SP.3, 7.SP.4
Learning
Students will be able to…
Differentiation
(EL/SpEd/GATE)
Resources
CA Mathematics Framework Gr. 7
p. 38 – 42
Progressions for the Common Core – Statistics
and Probability Gr. 6-8
North Carolina
7th Grade Math Unpacked Content: p. 34 – 38
7th Grade Common Core State Standards Flip
Book
33
SCUSD Curriculum Map
Essential Questions





Grade 7/8 Compacted Mathematics
Assessments for
Learning
Unit #8 Data Analysis
Sequence of Learning Outcomes
Strategies for Teaching and
7.SP.1, 7.SP.2, 7.SP.3, 7.SP.4
Learning
For Learning Outcomes 1 1) Determine if a given random sample is
How do you
– 4:
representative of a population, and
conduct a random
http://www.engageny.or
make generalizations about the
sample to most
g
population based on characteristics of
accurately reflect
(pg: 82 – 157)
the sample.
a population?
7.SP.2
How do you know (This module contains a
variety of tasks that
if a random
relate to multiple
sample is
learning outcomes.)
representative of
a population?
Possible Unit Project:
http://www.ciese.org/cur
riculum/tempproj/
2) Make predictions about a population
How do you know
given data from a random sample, and
if your inferences
then generate and analyze data from
and predictions
additional random samples representing
about a
the same population to determine the
population are
validity of the predictions. (Framework,
valid?
p. 39, 40)
Why might you
7.SP.1
conduct more
than one random
sample of the
same population?
What kinds of
inferences or
predictions can
you make from
looking at visual
representations of
given data sets
3) Make inferences, predictions, and
comparisons from visual representations
(for example, dot plots and box plots) of
given data sets.
7.SP.3
Differentiation
(EL/SpEd/GATE)
Resources
Random Sampling of a Population:
http://www.glencoe.com (from
Glencoe textbook)
Videos (Random Sampling):
http://learnzillion.com/lessons/271
6-take-a-simple-random-sample
http://learnzillion.com/lessons/320
6-generate-survey-data-throughsimulations
Videos (click on links below):
Dot Plots
Box Plots
Interquartile Range
34
SCUSD Curriculum Map
Essential Questions
Grade 7/8 Compacted Mathematics
Assessments for
Learning
Unit #8 Data Analysis
Sequence of Learning Outcomes
Strategies for Teaching and
7.SP.1, 7.SP.2, 7.SP.3, 7.SP.4
Learning
Differentiation
(EL/SpEd/GATE)
Resources
(for example, dot
plots)?


How can you use
the mean absolute
deviation (MAD)
of a given data
set?
When is it
appropriate to use
the different
measures of
center (mean and
median) and when
is it appropriate to
use the different
measures of
variability (MAD
and inter-quartile
range)?
4) Determine if the averages (mean or
Video (click on link below):
median) of two or more given data sets Mean Absolute Deviation
serve as a valuable reference for
comparison based on the variance
(mean absolute deviation or
inter-quartile range) of the data set.
(Framework, p. 41, 42).
7.SP.4
35
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #9: Probability
(Approx. # of Days ____)
Content Standards: 7.SP.5,6,7,8
In this unit, students will be able to identify, develop, and use probability to measure outcomes.
Common Core State Standards- Mathematics:
Statistics and Probability 7.SP
Investigate chance processes and develop, use, and evaluate probability models.
5. Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0
indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
6. Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given
the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
7. Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the
discrepancy.
a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from
a class, find the probability that Jane will be selected and the probability that a girl will be selected.
b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny
will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
8. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify
the outcomes in the sample space which compose the event.
c.
Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have
type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
Standards for Mathematical Practice:
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
36
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
ELD Standards to Support Unit:
Part I: Interacting in Meaningful Ways:
20) Collaborative:
10. Interacting with others in written English in various communicative forms
20. Adapting language choices to various contexts
21) Interpretive:
21. Listening actively to spoken English in a range of social and academic contexts.
22) Productive:
11. Supporting own opinions and evaluating others’ opinions in speaking and writing.
Part II: Learning About How English Works
Q. Expanding and Enriching Ideas
5. Modifying to add details.
R. Connecting and Condensing Ideas
6. Connecting Ideas
7. Condensing Ideas
Essential Questions
Assessments for
Learning
Assessments/Tasks
aligned to learning
outcomes:
Sequence of Learning Outcomes
7.SP.5, 7.SP.6, 7.SP.7, 7.SP.8
SEL Competencies:
Self-awareness
Self-management
Social awareness
Relationship skills
Responsible decision making
Unit #9 Probability
Strategies for Teaching and
Learning
Students will be able to…
Note: These Assessments
are suggested, not
required.

Conduct class discussions about
Why does it make http://www.engageny.or 1) Determine the probability of a chance
g
event and represent it as a number
observed data (e.g. flipping a
sense that the
(pp. 1 - 82)
between 0 and 1 (for example, the
coin), paying attention to
probability of a
(This module contains a
probability of flipping heads on a quarter similarities and differences
chance event is
variety of tasks that
is ½), and understand that a probability
between students’ observations,
represented as a
Differentiation
(EL/SpEd/GATE)
Resources
Technology for random CA Mathematics Framework Gr. 7
sampling:
p. 42 – 45
http://www.randomizer.
org/
Progressions for the Common Core – Statistics
and Probability Gr. 6-8
http://stattrek.com/stati
stics/random-number- North Carolina
generator.aspx
7th Grade Math Unpacked Content: p. 39 – 43
Videos (click on links below)
Using organized lists
Using tables
Using tree diagrams
37
SCUSD Curriculum Map
Essential Questions


Grade 7/8 Compacted Mathematics
Assessments for
Learning
number between
relate to multiple
0 and 1?
learning experiences.)
How do you know
if your predictions http://www.illustrativem
athematics.org/illustrat
based on
ions/1581
observed
frequencies are
valid?
What is a
reasonable
number of data
points to collect in
order to make a
prediction about
the probability of
a chance event?
Unit #9 Probability
Sequence of Learning Outcomes
Strategies for Teaching and
7.SP.5, 7.SP.6, 7.SP.7, 7.SP.8
Learning
near zero is an unlikely event, while a
probability near 1 is a likely event.
7.SP.5
Differentiation
(EL/SpEd/GATE)
Resources
and focusing on any predictions
that can be made.
Possible chance events:
 Rolling dice
 Flipping coins
 Choosing cards from a deck
 Choosing colored objects
 Spinner (video)
http://www.illustrativem 2) Collect data from a chance event (for
athematics.org/illustrat
example, rolling a die), calculate the
ions/1216
probability based on the observed
frequencies, and use proportional
reasoning to make predictions.
7.SP.6

What are some
http://map.mathshell.org 3) Compare the theoretical probability of a
/materials/tasks.php?ta
chance event to the probability based on
skid=367#task367
observed frequencies, and explain any
http://map.mathshell.org
possible sources of discrepancies.
/materials/lessons.php
7.SP.7
?taskid=225&subpage=
concept
http://www.illustrativem 4) Find probabilities of compound events
38
SCUSD Curriculum Map
Essential Questions
similarities and
differences when
using an organized
list, a table, and a
tree diagram to
find probabilities
of compound
events?
Grade 7/8 Compacted Mathematics
Assessments for
Learning
athematics.org/illustrat
ions/885
Unit #9 Probability
Sequence of Learning Outcomes
Strategies for Teaching and
7.SP.5, 7.SP.6, 7.SP.7, 7.SP.8
Learning
Differentiation
(EL/SpEd/GATE)
Resources
using organized lists, tables, and tree
diagrams.
7.SP.8

What are some
similarities and
differences
between simple
events and
compound
events?
5) Make connections between finding the
probability of a simple event and finding
the probability of a compound event.
7.SP.8

How do you
design a
simulation that
represents a
compound event?
6) Design and use a simulation from a
compound event (for example, rolling
two dice) to generate frequencies.
7.SP.8
39
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #10: 2-Dimensional and 3-Dimensional Geometric Figures
(Approx. # of Days ____)
Content Standards: 7.G.1,2,3,4,5,6
In this unit, students will be able to identify, draw, classify, and understand common geometric figures.
Common Core State Standards- Mathematics:
Geometry 7.G
Draw, construct, and describe geometrical figures and describe the relationships between them.
1. Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
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.
3. Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
4. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
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.
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.
Standards for Mathematical Practice:
3. Construct Viable Arguments and Critique the Reasoning of Others
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
SEL Competencies:
Self-awareness
Self-management
ELD Standards to Support Unit:
Social awareness
Part I: Interacting in Meaningful Ways:
23) Collaborative:
11. Interacting with others in written English in various communicative forms
22. Adapting language choices to various contexts
24) Interpretive:
23. Listening actively to spoken English in a range of social and academic contexts.
Relationship skills
Responsible decision making
40
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
25) Productive:
11. Supporting own opinions and evaluating others’ opinions in speaking and writing.
Part II: Learning About How English Works
S. Expanding and Enriching Ideas
5. Modifying to add details.
T. Connecting and Condensing Ideas
6. Connecting Ideas
7. Condensing Ideas
Essential Questions
Assessments for
Learning
Assessments/Tasks
aligned to learning
outcomes:
Unit #10 2-Dimensional and 3-Dimensional Geometric Figures
Sequence of Learning Outcomes
Strategies for Teaching and
Differentiation
7.G.1, 7.G.2, 7.G.3, 7.G.4, 7.G.5, 7.G.6
Learning
(EL/SpEd/GATE)
Students will be able to…
GeoGebra

http://map.mathshell.org 1) Draw triangles (freehand, with ruler and
What are the
/materials/lessons.php
protractor and with technology) given
criteria for 3 side
?taskid=581&subpage=
three out of six possible criteria, for
lengths to form a
concept
example two side lengths and an angle.
triangle?
Determine if the triangle exists, is
What is an
unique, or determines more than one
example of a
triangle.
situation where
7.G.2
you could be given
three pieces of
information about
a triangle and
CA Mathematics Framework Gr. 7
p. 33 – 38
North Carolina
7th Grade Math Unpacked Content: p. 25 – 33
Note: These Assessments
are suggested, not
required.

Resources
7th Grade Common Core State Standards Flip
Book
Informal introduction to triangle
inequality theorem.
http://www.mathopenref.com/tria
ngleinequality.html
Informal introduction to triangle
congruence theorems: SSS, SSA,
AAS, SAS, AAA.
http://www.regentsprep.org/Rege
nts/math/geometry/GP4/BegTriPrf.
htm
41
SCUSD Curriculum Map
Essential Questions
Grade 7/8 Compacted Mathematics
Assessments for
Learning
Unit #10 2-Dimensional and 3-Dimensional Geometric Figures
Sequence of Learning Outcomes
Strategies for Teaching and
Differentiation
7.G.1, 7.G.2, 7.G.3, 7.G.4, 7.G.5, 7.G.6
Learning
(EL/SpEd/GATE)
Resources
have more than
one possible
drawing that fit
the given criteria?

What is  (pi)?
Why is it an
important number
and how is it
used?
2) Write and solve equations for unknown
angles in figures involving
supplementary, complementary, vertical
and adjacent angles.*
7.G.2, 7.G.5
Use circles to explore supplementary,
complementary, vertical and adjacent angles.
http://www.mathsisfun.com/geometry/circle-t
heorems.html
3) Explore the relationship between the
circumference and diameter of circles to
discover .
7.G.4
Exploration of Pi (lesson):
https://www.teachervision.com/math/lessonplan/3430.html
http://www.illustrativem 4) Build on understanding of
athematics.org/illustratio
circumference, diameter and  to
ns/1553
generate formulas for circumference
and area of circles and use them to solve
http://www.illustrativem
mathematical and real-world problems.
athematics.org/illustratio
7.G.4
ns/34

How do you
subdivide a
composite figure
to find its area?
Videos (click on links below)
Explore and generate formulas for
circumference and area of a circle.
Finding Circumference
Finding the Area of a Circle
5) Find the area of triangles, quadrilaterals, Compare the use of addition and
and other polygons, including composite subtraction when finding the area
figures composed of triangles,
of composite figures, for example:
quadrilaterals, and polygons, in the
Sample Lesson:
http://cc.betterlesson.com/lesson/441863/are
a-of-composite-shapes-using-a-grid
42
SCUSD Curriculum Map
Essential Questions

What is the
relationship
between the
ratios of side
lengths and areas
of geometric
figures in scale
drawings?
Grade 7/8 Compacted Mathematics
Assessments for
Learning
Unit #10 2-Dimensional and 3-Dimensional Geometric Figures
Sequence of Learning Outcomes
Strategies for Teaching and
Differentiation
7.G.1, 7.G.2, 7.G.3, 7.G.4, 7.G.5, 7.G.6
Learning
(EL/SpEd/GATE)
Resources
context of real-world and mathematical
problems.
7.G.6
http://www.illustrativem 6) Investigate relationships between side
athematics.org/illustratio
lengths and areas in scale drawings of
ns/107
geometric figures, and reproduce a scale
drawing at a diffe6ent scale.
http://map.mathshell.org
7.G.1
/materials/lessons.php?t
askid=494&subpage=pro
blem
7) Identify the two-dimensional figure that
results from slicing a plane section of a
three-dimensional figure.

What is the
relationship
between area,
surface area, and
volume?
7.G.3
http://www.illustrativem 8) Solve real-world and mathematical
athematics.org/illustratio
problems involving the surface area and
ns/266
volume of cubes and right prisms, and
explore the relationship between
surface area and volume.
7.G.6
43
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit 11: Irrational Numbers and the Pythagorean Theorem
(Approx. 20 Days)
Content Standards:
8.NS.1-2, 8.EE.2, 8.G.6 – 9
In this unit, students will understand distinctions between rational and irrational numbers. Students will work with radical and exponential equations in various geometric applications.
Common Core State Standards-Mathematics:
The Number System
8.NS
Know that there are numbers that are not rational, and approximate them by rational numbers.
1. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually,
and convert a decimal expansion which repeats eventually into a rational number.
2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For
example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
Expressions and Equations
8.EE
Work with radicals and integer exponents.
2. Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube
roots of small perfect cubes. Know that √2 is irrational.
Geometry
8.G
Understand and apply the Pythagorean Theorem.
6. Explain a proof of the Pythagorean Theorem and its converse.
7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
44
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Standards for Mathematical Practice:
SMP-1. Make Sense of Problems and Persevere in Solving Them
SMP-3. Construct Viable Arguments and Critique the Reasoning of Others
SMP-4. Model with Mathematics
SMP-8. Look For and Express Regularity in Repeated Reasoning
ELD Standards to Support Unit:
Part I: Interacting in Meaningful Ways:
J. Collaborative:
12. Interacting with others in written English in various communicative forms
24. Adapting language choices to various contexts
K. Interpretive:
25. Listening actively to spoken English in a range of social and academic contexts.
L. Productive:
11. Supporting own opinions and evaluating others’ opinions in speaking and writing.
Part II: Learning About How English Works
U. Expanding and Enriching Ideas
5. Modifying to add details.
V. Connecting and Condensing Ideas
6. Connecting Ideas
7. Condensing Ideas
SEL Competencies:
Self-awareness
Self-management
Social awareness
Relationship skills
Responsible decision making
45
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #11 Irrational Numbers and The Pythagorean Theorem
Essential Questions
Assessments for
Learning
Assessments/Tasks
aligned to learning
outcomes:
Note: These Assessments
are suggested, not
required.
Sequence of Learning Outcomes
8.NS.1, 2 , 8.EE.2 , 8.G.6, 7, 8, 9
Students will be able to…
Strategies for Teaching and
Learning
Differentiation
e.g., EL, SpEd, GATE
Differentiation Support
for Unit:
Use of math journals for
differentiation and
formative assessment
(use link below)
https://www.teaching
channel.org/videos/m
ath-journals
Flexible grouping:
 Content
 Interest
 Project/product
 Level
(Heterogeneous/
Homogeneous)
Tiered:
 Independent
Management Plan
(Must Do/May Do)
 Grouping
o Content
o Rigor w/in the
concept
o Project-based
learning
o Homework
o Grouping
o Formative
Assessment
Resources
CCSS Support for Unit:
CA Mathematics Framework Gr. 8
p. 6 – 11, 30 – 32
Progressions for the Common Core – The
Number System Gr. 6-8
North Carolina
8th Grade Math Unpacked Content:
p. 6 – 7, 9 – 10, 33 – 38
8th Grade Common Core State Standards Flip
Book
46
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #11 Irrational Numbers and The Pythagorean Theorem
Essential Questions




Assessments for
Learning
Sequence of Learning Outcomes
8.NS.1, 2 , 8.EE.2 , 8.G.6, 7, 8, 9
Strategies for Teaching and
Learning
Differentiation
e.g., EL, SpEd, GATE
Resources
For Learning Outcomes 1 1) Investigate how to prove that
For Outcomes 1-2, all values should Anchor Activities:
Converting repeating decimals into fractions
– 3:
terminating decimals are rational
be explored, not only numbers
 Content-related
http://map.mathshell.org
because they can be written in the form
between 0 and 1 (e.g., 2.8, -3.6,
tasks for early
𝑝
/materials/lessons.php
finishers
and
√9).
using place value.
𝑞
?taskid=421#task421
o Game
8.NS.1 Students should use a calculator to
o Investigation
verify their decimal-to-fraction
http://www.illustrativem
o Partner
conversion.
athematics.org/illustrat
Activity
ions/334
o Stations
Video (click on link):
Depth and Complexity
Distingush between rational and
http://www.illustrativem
Prompts/Icons:
irrational numbers
athematics.org/illustrat
 Depth
ions/1538
o Language of
For Outcomes 1-3, it may be
the Discipline
valuable for students to place the
o Patterns
numbers on a number line.
o Unanswered
Questions
o Rules
o Trends
2) Investigate non-terminating, repeating
What is the
o Big Ideas
decimals are rational because they can
appropriate level
 Complexity
𝑝
of precision in
be written in the form 𝑞 using the
See Differentiation
estimating an
conversion method (Framework p. 7).
Resources at:
irrational number
8.NS.1
http://scusd-math.wikis
in a given
paces.com/home
real-world
context?
How do you know
if a number is
rational?
How do you
determine what to
multiply the
equation by when
converting
non-terminating
repeating
decimals into
fractions?
What strategies
do you have to
turn a decimal
into a fraction and
vice-versa?
3) Use a calculator to explore the expanded
decimal values of π and non-perfect
squares to notice that they are
47
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #11 Irrational Numbers and The Pythagorean Theorem
Essential Questions



Assessments for
Learning
Sequence of Learning Outcomes
8.NS.1, 2 , 8.EE.2 , 8.G.6, 7, 8, 9
Strategies for Teaching and
Learning
Differentiation
e.g., EL, SpEd, GATE
Resources
non-terminating and non-repeating.
Students will use this understanding to
conclude why they cannot be written as
fractions using the conversion method.
8.NS.2
For Learning Outcomes 4 4) Estimate the values of irrational
– 6:
numbers using a method of squaring
http://www.illustrativem
rational numbers (For example estimate
athematics.org/illustratio
the decimal expansion of √2 by
ns/336
showing that √2 is between 1 and 2,
then between 1.4 and 1.5, and explain
http://www.illustrativem
how to continue on to get better
athematics.org/illustratio
approximations - Framework p. 8).
ns/337
Represent this number on the number
line.
http://www.illustrativem
8.NS.2
athematics.org/illustratio 5) Evaluate expressions for square roots of Video (click on link):
ns/1221
small perfect squares and cube roots of Find the square root of a perfect
small perfect cubes, using the concept of square
http://illuminations.nctm
repeated multiplication.
.org/Lesson.aspx?id=40
8.EE.2
82
What do square
root and cube
root actually
mean?
Why are squares
and square roots
and cubes and
cube roots inverse Unit Assessment
operations?
(Exponents and Radicals)
Are all square
roots irrational?
Why or why not?
48
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #11 Irrational Numbers and The Pythagorean Theorem
Essential Questions


In the
Pythagorean
Theorem, does it
matter which side
is labeled a?
Which two lengths
are able to be
interchanged
within the
Pythagorean
Theorem?
What conditions
need to be met in
order to prove a
triangle is a right
triangle?
Assessments for
Learning
For Learning Outcomes 7
– 9:
http://map.mathshell.org
/materials/download.p
hp?fileid=804
http://map.mathshell.org
/materials/lessons.php
?taskid=408#task408
http://map.mathshell.org
/materials/tasks.php?ta
skid=280#task280
http://map.mathshell.org
/materials/tasks.php?ta
skid=276#task276
http://map.mathshell.org
/materials/download.p
hp?fileid=1098
Sequence of Learning Outcomes
8.NS.1, 2 , 8.EE.2 , 8.G.6, 7, 8, 9
Strategies for Teaching and
Learning
6) Solve equations in the form of x2 = p and
x3 = p using inverse operations (where p
is a positive rational number).
8.EE.2
7) Understand the Pythagorean Theory
Videos on Pythagorean Theorem:
using a proof and explore the proof with  Pythagorean Theorem
multiple right triangles. Using the
construction
same proof, students will explore
 Pythagorean Theorem
whether or not the Pythagorean
discovery and real-world
Theorem applies to non-right triangles.
problem to solve
8.G.6
Differentiation
e.g., EL, SpEd, GATE
Resources
Sample Lesson:
Review of right triangles and the relationships
of their sides.
8) Use the Pythagorean Theorem to solve Use Outcome #4 to approximate
for unknown side lengths (rational and
irrational answers appropriately
irrational) in right triangles in real-world
for real-world Pythagorean
and mathematical problems in two and
Theorem problems
three dimensions.
8.G.7
49
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #11 Irrational Numbers and The Pythagorean Theorem
Essential Questions
Assessments for
Learning
Sequence of Learning Outcomes
8.NS.1, 2 , 8.EE.2 , 8.G.6, 7, 8, 9
Strategies for Teaching and
Learning
Differentiation
e.g., EL, SpEd, GATE
Resources
9) Given two coordinates, students will
Video:
draw a right triangle and use the
Find distance between two points
Pythagorean Theorem to find the
on the coordinate plane using
distance between the two coordinates
Pythagorean
(i.e. the length of the hypotenuse of the
right triangle).
8.G.8
http://www.illustrativem 10) Solve real-life and mathematical
athematics.org/illustrat
problems using the formulas for volume
ions/520
of cylinders, cones, and spheres.
8.G.9
http://www.illustrativem
athematics.org/illustrat
ions/521
http://www.illustrativem
athematics.org/illustrat
ions/112
http://www.illustrativem
athematics.org/illustrat
ions/517
http://map.mathshell.org
/materials/lessons.php
?taskid=410#task410
50
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit 12: Exponents
(Approx. 15 Days)
Content Standards: 8.EE.1,3,4
In this unit, students will be able to use exponents and scientific notation appropriately.
Common Core State Standards-Mathematics:
Expressions and Equations
8.EE
Work with radicals and integer exponents.
1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.
3. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For
example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.
4. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for
measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
Standards for Mathematical Practice:
SMP-1. Make Sense of Problems and Persevere in Solving Them
SMP-3. Construct Viable Arguments and Critique the Reasoning of Others
SMP-5. Use Appropriate Tools Strategically
SMP-6. Attend to Precision
SMP-7. Look For and Make Use of Structure
SMP-8. Look For and Express Regularity in Repeated Reasoning
ELD Standards to Support Unit:
Part I: Interacting in Meaningful Ways:
M. Collaborative:
13. Interacting with others in written English in various communicative forms
26. Adapting language choices to various contexts
N. Interpretive:
27. Listening actively to spoken English in a range of social and academic contexts.
O. Productive:
11. Supporting own opinions and evaluating others’ opinions in speaking and writing.
SEL Competencies:
Self-awareness
Self-management
Social awareness
Relationship skills,
Responsible decision making
51
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Part II: Learning About How English Works
W. Expanding and Enriching Ideas
5. Modifying to add details.
X. Connecting and Condensing Ideas
6. Connecting Ideas
7. Condensing Ideas
Unit #12 Exponents
Essential Questions
Assessments for
Learning
Assessments/Tasks
aligned to learning
outcomes:
Note: These Assessments
are suggested, not
required.
Sequence of Learning Outcomes
8.EE.1, 8.EE.3, 8.EE.4
Students will be able to…
Strategies for Teaching and
Learning
Differentiation
e.g., EL, SpEd, GATE
Differentiation Support
for Unit:
Use of math journals for
differentiation and
formative assessment
(use link below)
https://www.teaching
channel.org/videos/m
ath-journals
Flexible grouping:
 Content
 Interest
 Project/product
 Level
(Heterogeneous/
Homogeneous)
Tiered:
 Independent
Management Plan
Resources
CCSS Support for Unit:
CA Mathematics Framework Gr. 8
p. 8 – 11
Progressions for the Common Core –
Expressions and Equations Gr. 6-8
North Carolina
8th Grade Math Unpacked Content:
p. 8 – 12
8th Grade Common Core State Standards Flip
Book
52
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #12 Exponents
Essential Questions


Where do the
rules for
exponents come
from?
Why do we use
scientific
notation?

Assessments for
Learning
Sequence of Learning Outcomes
8.EE.1, 8.EE.3, 8.EE.4
Strategies for Teaching and
Learning
For Learning Outcomes 1 1) Use the definition of an exponent to
Use the definition of an exponent
– 5:
expand and simplify expressions (with
to expand and simplify
http://www.illustrativem
positive integer exponents only) in order expressions in order to general
athematics.org/illustrat
to generate the rule for multiplying
rules, for example:
ions/395
powers with the same base:
23  2 4  2  2  2  2  2  2  2  27 Use
vocabulary like “How many
a m  a n  a mn
8.EE.1 factors of 2 do you see?”
For example,
2   (2 )(2 )  (2  2  2)  (2  2  2)  2
3 2
3
3
6
2) Expand and simplify expressions (with
positive integer exponents only) in order
to generate the rule for raising a power
to a power: (a m )n  a mn
8.EE.1
3) Expand and simplify expressions by
“finding ones” (with positive integer
exponents only) in order to generate the
rule dividing powers with the same base:
am
 a mn
n
a

How can you
prove that a0 = 1?
8.EE.1
0
4) Make predictions about powers raised
Why is a  1 ? (See proofs…)
to zero exponents. Generate and prove http://www.homeschoolmath.
0
the rule for zero exponents: a  1
Differentiation
e.g., EL, SpEd, GATE
Resources
(Must Do/May Do)
Grouping
o Content
o Rigor w/in the
concept
o Project-based
learning
o Homework
o Grouping
o Formative
Assessment
Anchor Activities:
 Content-related
tasks for early
finishers
o Game
o Investigation
o Partner
Activity
o Stations
Depth and Complexity
Prompts/Icons:
 Depth
o Language of
the Discipline
o Patterns
o Unanswered
Questions
o Rules
o Trends

53
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #12 Exponents
Essential Questions
Assessments for
Learning
Sequence of Learning Outcomes
8.EE.1, 8.EE.3, 8.EE.4
Strategies for Teaching and
Learning
8.EE.1 https://www.khanacademy.org

Why isn’t a-n = –
an?
5) Make predictions about powers raised
to negative exponents. Generate and
prove the rule for negative exponents:
a
n
1
 n
a
Why is a
n

1
?
an
(See proofs…)
http://www.homeschoolmath.
Differentiation
e.g., EL, SpEd, GATE
Resources
o Big Ideas
 Complexity
See Differentiation
Resources at:
http://scusd-math.wik
ispaces.com/home
8.EE.1 https://www.khanacademy.org


Why do we use 10 For Learning Outcomes 6 6) Given large or small numbers in
– 9:
standard form, express them in scientific
as a base for
http://www.illustrativem
notation; Given large or small numbers
numbers
athematics.org/illustrat
in scientific notation, express them in
expressed in
ions/823
standard form.
scientific
8.EE.3
notation?
http://www.illustrativem
athematics.org/illustrat 7) Use numbers expressed in scientific
How can you use
ions/476
notation (in the form of a single digit
estimation to
times an integer power of 10) to
compare two
http://www.illustrativem
determine which one has a greater value
numbers
athematics.org/illustrat
(for example, determine which value is
expressed in
ions/1291
greater and explain how you know: 8 x
scientific
105 and 9 x 104).
notation?
http://www.illustrativem
8.EE.3
athematics.org/illustrat 8) Using numbers expressed in scientific
For Outcomes 8 & 9, students will
ions/1593
notation (in the form of a single digit
choose units of appropriate size
times an integer power of 10), estimate
for a given situation. Students
http://www.illustrativem
how many times greater one number is
should be able to interpret and
athematics.org/illustrat
than the other number (for example, the understand scientific notation as
54
SCUSD Curriculum Map
Grade 7/8 Compacted Mathematics
Unit #12 Exponents
Essential Questions
Assessments for
Learning
ions/113
Sequence of Learning Outcomes
8.EE.1, 8.EE.3, 8.EE.4
Strategies for Teaching and
Learning
Differentiation
e.g., EL, SpEd, GATE
Resources
population of the U.S. is 3 x 108 and the
it has been generated by a
population of the world is 7 x 109, so the
calculator.
population of the world is more than 20
times larger).
About how much greater is the
8.EE.3 world population than the U.S.
population?
7×109
3×108
7×10×108
70
=
= ≈ 23
3×108
3
The world population is about 23
times greater.
How much greater is 6 x 10-8 than 9
x 10-9?
6×10−8
9×10−9

How do the rules
of exponents
apply to
performing
operations with
numbers
expressed in
scientific
notation?
=
6×10×10−9
9×10−9
=
60
9
≈ 6.7
9) Given a mathematical or real-world
Use the rules for integer exponents
problem, perform operations with
(see Outcomes 1 – 5) to perform
numbers expressed in scientific
operations with numbers
notation, including problems where both expressed in scientific notation.
decimal and scientific notation are used.
8.EE.4
55