Mathematics Grade 7/8 Curriculum Map CompactedMathematics Grade 7/8 Compacted DRAFT Last Updated October 30, ... CA Mathematics Framework Gr. 7 p. 14 – 16 Progressions for the Common

SCUSD Curriculum Map – Last Updated 10/30/14 Grade 7/8 Compacted Mathematics

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Curriculum Map

Mathematics Grade 7/8 Compacted

DRAFT Last Updated October 30, 2014 Sacramento City Unified School District


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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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Compacted 7th/8th Year-at-a-Glance

Month Unit Content Standards “Big Ideas MATH” Courses 2 and 3

Textbook Chapters

September

Unit #1 Proportional Reasoning and Relationships

7.RP.1, 2 Course 2: Ch. 5

7.G.1 Course 2: Ch. 7.5

Unit #2 Applying Proportional Reasoning to Problems with Percents

7.RP.3 Course 2: Ch. 5.1 and 5.3

Ch. 6.3 – 6.7

October

Unit #3 Operations with Rational Numbers –Addition and Subtraction

7.NS.1, 3 Course 2: Ch. 1 and 2

Unit #4 Operations with Rational Numbers –Multiplication and Division

7.NS.2, 3 Course 2: Ch. 1 and 2

November Unit #5

Equivalent Expressions 7.EE.1, 2 Course 2: Ch. 3.1 and 3.2

December/January Unit #6

Problem Solving with Equations and Inequalities

7.EE.3 Course 2: Ch. 6 (6.1, 6.2, 6.4)

7.EE.4 Course 2: Ch. 3 and 4

8.EE.7 Course 3: Ch. 1

February Unit #7

Linear Relationships 8.EE.5, 6

8.F.2 Course 3: Ch. 4

March

Unit #8 Data Analysis

7.SP.1 – 4 Course 2: Ch. 10.6 – 10.7

Unit #9 Probability

7.SP.5 – 8 Course 2: Ch. 10.1 – 10.5

April Unit #10

2-Dimensional and 3-Dimensional Geometric Figures 7.G.1 – 6 Course 2: Ch. 7, 8, 9

April/May Unit #11

Irrational Numbers and The Pythagorean Theorem

8.NS.1, 2 8.EE.2

8.G.6 – 9 Course 3: Ch. 7 and 8

May/June Unit #12

Exponents 8.EE.1, 3, 4 Course 3: Ch. 9


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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


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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


Essential Questions Assessments for Learning

Sequence of Learning Outcomes 7.RP.1, 7.RP.2, 7.G.1

Strategies for Teaching and Learning

Differentiation (EL/SpEd/GATE)

Resources

Assessments/Tasks aligned to learning outcomes. Note: These Assessments

are suggested, not required.

Students will be able to….

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

SEL Competencies: Self-awareness

Self-management

Social awareness

Relationship skills

Responsible decision making

http://www.cde.ca.gov/ci/ma/cf/documents/aug2013gradeseven.pdf


http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf


http://www.ncpublicschools.org/docs/acre/standards/common-core-tools/unpacking/math/7th.pdf




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Resources

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.illustrativemathematics.org/illustrations/82 For Learning Outcomes

1-5: http://www.engageny.or

g (this link is to a module

that has a variety of tasks that relate to the learning outcomes)

1) Identify and utilize unit rates to solve real-world problems with proportional relationships containing whole numbers, fractions and decimals by using visual representations. (Framework p.12)

7.RP.1

Teaching Ratios through Tape

Diagrams and Double Number

Lines:

http://math.kennesaw.edu/~twatanab/DeKalb%20Title%20I%20Summit%202012.pdf

Video on Teaching Unit Rate with

Tape Diagrams:

https://learnzillion.com/lessons/841-create-unit-rate-using-tape-diagram

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)

How do you know which of the two unit rates is important for a problem?

http://www.illustrativemathematics.org/illustrations/101

1) Use their understanding of unit rates and proportionality to create equations,

both in the formc

d

b

a and y = kx, to

solve real-world problems. (Framework p.12)

7.RP.2

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

What makes a relationship proportional?


2) Identify, utilize and write equations with unit rates developed from scale drawings to solve problems and reproduce a scale drawing at a different scale. (Framework p.35, 36)

7.G.1

Video on Scale Drawings: http://www.virtualnerd.com/middle-math/ratios-proportions-percent/scale-drawings-models/scale-drawing-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

What makes a http://www.illustrativem 3) Use unit rate or constant of Video on Understanding Unit Rates: Section 5.2




http://www.engageny.org/sites/default/files/resource/attachments/g7-m1-student-materials.pdf

http://www.engageny.org/sites/default/files/resource/attachments/g7-m1-student-materials.pdf







http://www.virtualnerd.com/middle-math/all/







http://www.virtualnerd.com/middle-math/ratios-proportions-percent/scale-drawings-models/scale-drawing-definition






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Resources

relationship proportional?

athematics.org/illustrations/1527

proportionality to determine if a relationship is proportional. Students 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

https://learnzillion.com/lessons/2410

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

How is the constant of proportionality represented in a graph, table and equation?

http://map.mathshell.org/materials/download.php?fileid=1070

4) Given a real-world example, work simultaneously with a graph, table and equation. Determine if there is a constant of proportionality in each representation. If so, identify the constant of proportionality in each representation, giving careful attention to the point (1, r) on a graph.

7.G.1

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/dragonfly.pdf





https://learnzillion.com/

https://learnzillion.com/







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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


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



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.


C. Expanding and Enriching Ideas


D. Connecting and Condensing Ideas

6. Connecting Ideas

7. Condensing Ideas


Self-management

Social awareness

Relationship skills



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Sequence of Learning Outcomes 7.RP.3



Resources

Assessments/Tasks aligned to learning outcomes:

Note: These Assessments


Students will be able to….

CA Mathematics Framework Gr. 7 p. 14 – 16 Progressions for the Common Core – Ratios

and Proportional Relationships Gr. 6-7 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.illustrativemathematics.org/illustrations/106 http://map.mathshell.org/materials/download.php?fileid=1042

1) Estimate and calculate tips, simple interest, tax, fees and mark ups using bar modeling, double number lines and algorithmic procedures. (Framework p.14, 15)

7.RP.3

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,







http://katm.org/wp/wp-content/uploads/flipbooks/7th_FlipBookEdited21.pdf








https://learnzillion.com/lessons/3556-estimate-a-percent-value-using-a-bar-model


https://learnzillion.com/lessons/3441-write-an-expression-to-find-the-cost-of-an-item-with-tax

https://learnzillion.com/lessons/3581-calculate-percent-increase-and-decrease-in-context


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Sequence of Learning Outcomes 7.RP.3



Resources

connections between bar modeling, double number lines and the algorithmic procedure?

and percentages: http://www.mathsisfun.com/decimal-fraction-percentage.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….)

http://www.illustrativemathematics.org/illustrations/105 http://map.mathshell.org/materials/download.php?fileid=1524

2) Estimate and calculate discounts, markdowns and sales using bar modeling, double number lines and algorithmic procedures.

7.RP.3

Videos (click on links below):

Estimate Percent Using Bar Modeling

Tax, Tips, and Discounts

Lessons 6.4 and 6.6

Which quantity represents the whole (or 100%)?

How can you check for reasonableness as you solve a problem and in your answer?


3) Estimate and calculate percent change including identifying the original value and comparing the difference in two values to the starting price. (Framework p.15, 16)

7.RP.3

Lessons 6.5 Refer to 6.1 for Fraction, decimal, percent conversions

http://www.mathsisfun.com/decimal-fraction-percentage.html

http://www.mathsisfun.com/decimal-fraction-percentage.html









https://learnzillion.com/lessons/3507-apply-taxes-tips-and-discounts-using-a-proportion-and-scale-factor





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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


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




9) Interpretive:


12

9. Listening actively to spoken English in a range of social and academic contexts. 10) Productive:



E. Expanding and Enriching Ideas


F. Connecting and Condensing Ideas

6. Connecting Ideas

7. Condensing Ideas

Unit #3 Operations with Rational Numbers – Addition and Subtraction


Sequence of Learning Outcomes 7.NS.1, 7.NS.3



Resources




Students will be able to…

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 7th Grade Common Core State Standards Flip Book

What is a zero pair?

How can you use zero pairs to solve problems?

How do you know

For learning outcomes 1 & 2 http://map.mathshell.org/materials/lessons.php?taskid=453#task453

1) Understand and develop fluency adding rational numbers (integers, fractions and decimals) by creating zero pairs using counting chips, the number line, decomposition and mental math. Apply understanding to solve real-world

Strategies for adding and subtracting positive and negative numbers (click on links):

Counting Chips

T-Charts

Big Ideas 7th grade TE: Lesson 1.2 p. 8 and 9 (focuses only on integers) The remainder of this lesson is more applicable


Self-management

Social awareness

Relationship skills






http://commoncoretools.me/wp-content/uploads/2013/07/ccssm_progression_NS+Number_2013-07-09.pdf






http://map.mathshell.org/materials/lessons.php?taskid=453#task453



http://learnzillion.com/lessons/2621-add-integers-using-chips

http://www.mathfox.com/adding-integers-using-t-chart/


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Resources

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?

http://www.illustrativemathematics.org/illustrations/46 http://www.illustrativemathematics.org/illustrations/998 http://www.illustrativemathematics.org/illustrations/317

problems. (Framework p.20-21) 7.NS.1, 7.NS.3

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.)

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.

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)?

2) Understand and develop fluency subtracting rational numbers (integers, fractions and decimals) by creating zero pairs using counting chips and the number line, with an emphasis on “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

Strategies for adding and subtracting positive and negative numbers:

Counting Chips

“Taking Away”

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

Why is subtracting a negative equivalent to adding a positive?


3) Compare and contrast work with addition and subtraction of rational numbers to build the understanding that p – q = p + (-q) for the purpose of

Big Ideas 7th grade TE: Lesson 1.3 p.14 (integers only)










http://www.youtube.com/watch?v=fQX74Eeo4Tw

http://learnzillion.com/lessons/3007-add-rational-numbers-using-algorithms-and-number-lines

http://learnzillion.com/lessons/2621-add-integers-using-chips





14






Resources

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

lesson 2.3 p. 58 (all rational numbers) *see note in outcome 1 about teaching the

lessons simultaneously.

How can you transition from using a particular method to solve a problem to just knowing the answer?

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

Big Ideas 7th grade TE: Lesson 1.2 Starting on p. 10


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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:


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




12) Interpretive:


16

11. Listening actively to spoken English in a range of social and academic contexts. 13) Productive:



G. Expanding and Enriching Ideas


H. Connecting and Condensing Ideas

6. Connecting Ideas

7. Condensing Ideas

Unit #4 Operations with Rational Numbers – Multiplication and Division





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

Where do the rules of signed numbers come from?

Why is the product of two negative numbers a positive

1) Understand and develop fluency of multiplication of integers through definition of integers and multiplication as repeated addition. Additional methods that should be explored include using patterns in products of integers and the proof of why (-1)(-1) = 1. (Framework p. 25, 26)

7.NS.2

Definition of multiplication for integers, for example:

3(-4) is three groups of negative four or -4 + -4 + -4 = -12 and -3(-4) is the opposite of three groups of negative four or –(-4 + -4 + -4) = -(-12) = 12.

Videos:

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


Self-management

Social awareness

Relationship skills











17






Resources

number?

Understanding multiplication using number lines: http://www.youtube.com/watch?v=K_Jqdw3NpEw

Multiplication using

decomposition: http://www.youtube.com/watch?v

=Q_V1brQtxT0 Proof of (-1)(-1) = 1 https://www.khanacademy.org/ma

th/arithmetic/absolute-value/mult_div_negatives/v/why-a-negative-times-a-negative-is-a-positive

p.64

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?

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

Students look for patterns of products in integers.

How do the rules for multiplying signed numbers help you know the rules for dividing signed numbers?

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

Big Ideas 7th grade TE: Lesson 1.5 p. 28

http://www.youtube.com/watch?v=K_Jqdw3NpEw

http://www.youtube.com/watch?v=K_Jqdw3NpEw

http://www.youtube.com/watch?v=Q_V1brQtxT0

http://www.youtube.com/watch?v=Q_V1brQtxT0

https://www.khanacademy.org/math/arithmetic/absolute-value/mult_div_negatives/v/why-a-negative-times-a-negative-is-a-positive





18






Resources

How do you know if a number is rational?

4) Apply rules of multiplication and division to all rational numbers. Solve real-world problems involving both operations.

7.NS.2

Multiplying fractions:

7

5

4

13

7

5

4

3

Big Ideas 7th grade TE: Lesson 2.4 p. 66

http://www.illustrativemathematics.org/illustrations/604 http://www.illustrativem


5) Convert rational numbers to decimals using long division; know that the decimal form of a rational number terminates in 0’s or repeats.

7.NS.2

Big Ideas 7th grade TE: Lesson 2.1 p.46


6) Solve real-world and mathematical problems involving the four operations with rational numbers. (Framework p.28)

7.NS.2

Embedded into the practice of each of the above mentioned lessons.











19

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.


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







I. Expanding and Enriching Ideas


J. Connecting and Condensing Ideas

6. Connecting Ideas 7. Condensing Ideas


Self-management

Social awareness

Relationship skills



20

Unit #5 Equivalent Expressions

Essential Questions Assessments for Learning Sequence of Learning Outcomes 7.EE.1, 7.EE.2

Strategies for Teaching and Learning Differentiation e.g. EL, SpEd, GATE

Resources


Note: These Assessments are

suggested, not required.


Differentiation Support for Unit:

Use of math journals for differentiation and formative assessment (use link below) https://www.teachingchannel.org/videos/math-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-base

d learning o Homework o Grouping o Formative

Assessment

CCSS Support for Unit: 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?


1) Generate equivalent expressions containing rational numbers by combining like terms in mathematical and real-world problems. Compare the meaning of each equivalent expression in the context of real-world problems. (Framework p.29)

7.EE.2

Use pattern problems like the “Pool Border Problem” (Framework p. 31).

Possible use of manipulatives: Integer

tiles Other real-world problems could include:

Perimeter/Area Problems

Cell Phone Plans

Big Ideas 7th grade TE: Lesson 3.1 p. 82 Example 4 and problems 26 and 29 do pretty good job of incorporating a real-life context. It is important for students to make sense of what the parts of the expression represent based on the context both before and after simplification. Problem 24 is a pretty cool error analysis problem that addresses common mistakes. Lesson3.2 p.86: uses manipulatives and some alternative structure to support the understanding of combining like terms.

Of the many possible equivalent expressions, which of them best represents the

http://www.illustrativemathematics.org/illustrations/543 http://www.illustrativemathematics.org/illustrations/1450

2) Generate equivalent expressions containing rational numbers using the distributive property, both expanding and factoring, in mathematical and real-world

Example of expanding using the distributive property:

2(3x + 5) = (3x + 5) + (3x + 5) = 6x + 10

Big Ideas 7th grade TE: Lesson 3.2 “Extension” p. 92 This addresses factoring. Expansion is not

addressed explicitly, but there are plenty of problems that provide the opportunity for

https://www.teachingchannel.org/videos/math-journals





http://commoncoretools.files.wordpress.com/2011/04/ccss_progression_ee_2011_04_25.pdf













21

Unit #5 Equivalent Expressions

Essential Questions Assessments for Learning Sequence of Learning Outcomes 7.EE.1, 7.EE.2

Strategies for Teaching and Learning Differentiation e.g. EL, SpEd, GATE

Resources

meaning of the situation?

problems. Compare the meaning of each equivalent expression in the context of real-world problems. (CA Framework p.29)

7.EE.2

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 o Big Ideas

Complexity See Differentiation

Resources at: http://scusd-math.wikispaces.com/home

teaching this approach.

How does changing one term of an expression change the meaning of the context?


3) Generate equivalent expressions containing rational numbers using the distributive property, addition and subtraction, i.e. 8 – 2(0.5x + 1) in mathematical and real-world problems. Compare the meaning of each equivalent expression in the context of real-world problems.

7.EE.1 & 2

Big Ideas 7th grade TE: Embedded within lessons 3.1 and 3.2

http://scusd-math.wikispaces.com/home





22

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?


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



D. Collaborative: 7. Interacting with others in written English in various communicative forms 14. Adapting language choices to various contexts

SEL Competencies: Self-awareness Self-management Social awareness Relationship skills Responsible decision making


23

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.


K. Expanding and Enriching Ideas


L. Connecting and Condensing Ideas

6. Connecting Ideas

7. Condensing Ideas

Unit #6 Problem Solving with Equations and Inequalities


Sequence of Learning Outcomes 7.EE.3, 7.EE.4, 8.EE.7



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









http://www.cde.ca.gov/ci/ma/cf/documents/aug2013gradeeight.pdf






24






Resources

8th Grade Common Core State Standards Flip Book

What are some arithmetic tools you can use to solve real-life problems?

When is it appropriate to use arithmetic tools and when is it appropriate to solve equations algebraically?

For equations such as

25105 x

and

10693

2x ,

what are different methods for solving algebraically?

When solving a problem using both methods (arithmetic tools and algebraically), where do you see


1) Solve multi-step, real-life and mathematical problems posed with positive and negative rational numbers by using arithmetic methods, such as: bar modeling, Guess and Check, drawing a picture or other tools instead of creating an equation. Use estimation to assess the reasonableness of answers.

7.EE.3

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/Math/OneStepEquationsMultipleApproachesV3.pdf The textbook only focuses on percent

problems for 7.EE.3, but the standard calls for problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals). The lessons below will only address part of this standard:

Big Ideas MATH 7th grade TE: Lesson 6.1 shows examples of how to represent percent numbers on a grid. Lesson 6.2 shows examples of using a number line to order positive fractions, decimals, and percents. Lesson 6.3 shows examples of estimating and using a bar model to solve percent problems. Lesson 6.4 shows examples of using a bar model to solve percent problems, and how to make and use a table of values to solve percent problems.

http://katm.org/wp/wp-content/uploads/flipbooks/8thFlipFinaledited.pdf





http://www.showme.com/sh/?h=uAslN8C

http://www.wccusd.net/cms/lib03/CA01001466/Centricity/domain/60/lessons/Grade%206%20Lessons/SolvingEquationsMultipleMethodsV4.pdf

http://www.acoe.org/acoe/files/EdServices/Math/OneStepEquationsMultipleApproachesV3.pdf




25






Resources

relationships in your work?

How do you interpret the graph of an inequality in terms of the context of the problem?

What is the purpose of using inverse operations?


2) Generate equations equivalent to px + q = r with rational coefficients and solve mathematical and real-life situations using inverse operations.

7.EE.4

Students will be able to solve (procedurally) two-step equations. Students will also be able to read a word problem, write an equation (in the form px + q = r or p(x + q) = r) related to that word problem, solve it, and interpret their result in terms of the context of the word problem. Use inverse operations to solve algebraic equations (i.e. creating zeroes and ones). Warn against the language of “cancel out.” For equations in the form p(x + q) = r, it may be reasonable to use the distributive property first, and it may also be reasonable to divide by the coefficient (p) first. Students should be able to see both methods as appropriate ways to solve an equation in this form.

Big Ideas MATH 7th grade TE: Lessons 3.3 – 3.5

What does your solution mean in the context of the problem?


3) Generate equations equivalent to p(x + q) = r with rational coefficients and solve mathematical and real-life situations using inverse operations.

7.EE.4

This Learning Outcome is not specifically called out in the textbook lessons, though there are some problems in the homework in Lesson 3.5 (#32-34) in which the equations are in the form p(x + q) = r.








26






Resources


4) Compare and contrast the use of arithmetic (see 1) versus algebraic 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

Learning Outcome 4 can be embedded in Outcomes 2 and 3

(This is not a specified lesson in the textbook)

Why and when would you reverse an inequality symbol?

When solving equations and inequalities, using 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?


5) Generate and solve inequalities with rational numbers, in the form of px + q < r and px + q > r (including < and >) that arise from real world problems. Graph the solution region on a number line and interpret the meaning of solutions in the context of the problem.

7.EE.4

Use investigation to help students understand the reason for reversing inequality symbols when multiplying or dividing by negative numbers.

https://www.youtube.com

Why the inequality sign changes when multiplying or dividing by a negative number:

http://www.algebra.com/algebra/homework/Inequalities/Inequalities.faq.question.203735.html

Big Ideas MATH 7th grade TE: Lessons 4.1 – 4.3

What is a solution?

Learning Outcomes 6-8: http://www.illustrativem


6) Use inverse operations to solve linear equations in one variable with rational coefficients, including equations that have variables and constants on both sides of the equal sign, arising in

Inverse operations (creating zeroes and ones)

“Geometric situations”: For example, writing a linear equation to solve for the

Big Ideas Math 8th Grade TE: Lessons 1.1 – 1.3 Note: The text teaches the idea of “undoing” operations. This map calls for teaching what we are “doing” to each side of the equation in







https://www.youtube.com/watch?v=y5vx0oXVyY0








27






Resources


http://www.illustrativem


http://map.mathshell.org

/materials/download.php?fileid=1154


/materials/lessons.php?taskid=442#task442


/materials/lessons.php?taskid=487#task487

algebraic, geometric, and real-world situations.

8.EE.7

measure of a missing angle of a triangle.

order to isolate the variable and maintain equality (creating zeroes and ones)

How can you make assumptions or predictions about the number of solutions at multiple points throughout the process of solving linear equations?

What does it mean for a linear equation to have one solution, no solutions, or infinite solutions?

7) Analyze a given equation to determine whether it has one solution (x = a), infinite solutions (a = a), or zero solutions (a = b); explain their reasoning using the definition of solution.*

8.EE.7

*Embed in Learning Outcomes 1 & 3 (See Framework p. 18).

Throughout the process of solving

an equation, students should 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.

In Chapter 1, this concept is briefly mentioned in the teacher notes and is present in some practice problems, but the text does not sufficiently address this learning outcome which is absolutely critical in order to meet the expectations of the standard.

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

Big Ideas Math 8th Grade TE: Lesson 1.3 p. 20 The distributive property is addressed, but

combining like terms is not.

















28

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.



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



G. Collaborative: 8. Interacting with others in written English in various communicative forms 16. Adapting language choices to various contexts

H. Interpretive:



29

17. Listening actively to spoken English in a range of social and academic contexts. I. Productive:



M. Expanding and Enriching Ideas


N. Connecting and Condensing Ideas

6. Connecting Ideas

7. Condensing Ideas

Unit #7 Linear Relationships


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







Flexible grouping: Content Interest Project/product

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















30






Resources

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?

For Learning Outcomes 1 - 2:


http://www.illustrativemath

ematics.org/illustrations/55



1) Graph proportional relationships given a real-world context and interpret the unit rate as the slope of the graph.

8.EE.5

When given a context, pay attention to the units involved throughout problem solving process.

Level (Heterogeneous/ Homogeneous)

Tiered: Independent


Grouping o Content o Rigor w/in the

concept o Project-based

learning o Homework o Grouping o Formative

Assessment Anchor Activities: Content-related


Activity o Stations


o Language of

Big Ideas Math 8th Grade TE: Lesson 4.3

What are some examples of linear relationships that are/are not proportional? How do you know?

2) Compare two different proportional relationships represented in different ways, for example, in a graph, a table, an equation, and a verbal description.

8.EE.5, 8.F.2

Use side-by-side instruction with graphs, tables, equations, and verbal descriptions for a given real-world context (for example, use a graphic organizer for student work).

Big Ideas Math 8th Grade TE: Lesson 4.3 p. 161 Example 3 p. 162 #9 p. 163 #10

How can you use the slope of a line to find additional points on the line?


3) 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. (Framework p. 16)

8.EE.6

Using similar triangles to prove slope formula (videos and practice problems, click on links below):

http://www.youtube.com/ https://www.khanacademy http://www.illustrativemathematics.org/illustrations/1537 Emphasize the similarities and differences

Big Ideas Math TE: Lesson 4.2 p. 149 Activity 2 This learning outcome builds on their

previous knowledge of similar triangles.

Students should be able to explain that two right triangles whose hypotenuses













http://www.youtube.com/watch?v=TqpT0xsiMGY

http://www.youtube.com/watch?v=TqpT0xsiMGY

https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-relationships-functions/cc-8th-similarity-slope/v/similar-triangles-to-prove-that-the-slope-is-constant-for-a-line




31






Resources

between proportional and non-proportional relationships.

the Discipline o Patterns o Unanswered




are on the same line, are similar. Students should use their understanding

of similar triangles to conclude that the slope between any two points on a line is the same.

For Learning Outcomes 4 – 7: http://www.illustrativemath








4) Derive and understand slope/rate of change given a real-world context by using graphs, tables, equations (y=mx) and verbal descriptions in the first quadrant.

8.EE.6

Derivation (in experiences 4-6) should be studied in Framework, p. 17

Big Ideas Math 8th Grade TE: Lesson 4.2 This lesson introduces slope and the

slope formula. Students should derive the slope formula on their own from their understandings from Learning Outcome 3, in which they understand slope through similar triangles.

What is similar/different about the equations y=mx and y=mx + b?

5) Derive and understand slope/rate of change with a y-intercept given a real-world context by using graphs, tables, equations (y=mx + b) and verbal descriptions in the first quadrant.

8.EE.6

Big Ideas Math 8th Grade TE: Lesson 4.3 p. 159 Activity 3: Deriving an Equation Lesson 4.4

6) Derive and understand slope/rate of change and y-intercept in context in all quadrants.

8.EE.6


Given a context, which quadrants are reasonable for your graph? Why?

7) Model real-world problems with the relationships y=mx and y=mx + b. Determine what parts of the graph make sense in context of the situation.

8.EE.6

Big Ideas Math TE: Lesson 4.4 p. 169 Activity 3: Real-Life Application p. 170-171 #17, 24, 26
















32

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


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


33







O. Expanding and Enriching Ideas 5. Modifying to add details.

P. Connecting and Condensing Ideas 6. Connecting Ideas

7. Condensing Ideas



Sequence of Learning Outcomes 7.SP.1, 7.SP.2, 7.SP.3, 7.SP.4



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


Self-management

Social awareness

Relationship skills




http://commoncoretools.files.wordpress.com/2011/12/ccss_progression_sp_68_2011_12_26_bis.pdf







34






Resources

How do you conduct a random sample to most accurately reflect a population?

How do you know if a random sample is representative of a population?

For Learning Outcomes 1 – 4:

http://www.engageny.org

(pg: 82 – 157) (This module contains a

variety of tasks that relate to multiple learning outcomes.)

Possible Unit Project: http://www.ciese.org/cur

riculum/tempproj/

1) Determine if a given random sample is representative of a population, and make generalizations about the population based on characteristics of the sample.

7.SP.2

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-through-simulations

Big Ideas MATH 7th grade TE: Lesson 10.6

How do you know if your inferences and predictions about a population are valid?

Why might you conduct more than one random sample of the same population?

2) Make predictions about a population given data from a random sample, and then generate and analyze data from additional random samples representing the same population to determine the validity of the predictions. (Framework, p. 39, 40)

7.SP.1

Big Ideas MATH 7th grade TE: Lesson 10.6 Extension Lesson 10.6 (p. 446)

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

Videos (click on links below): Dot Plots Box Plots Interquartile Range


http://www.engageny.org/sites/default/files/resource/attachments/math-g7-m5-student-materials.pdf


http://www.ciese.org/curriculum/tempproj/

http://www.ciese.org/curriculum/tempproj/

http://www.glencoe.com/sec/math/prealg/prealg04/add_lesson/using_sampling_pa1.pdf

http://learnzillion.com/lessons/2716-take-a-simple-random-sample

http://learnzillion.com/lessons/2716-take-a-simple-random-sample

http://learnzillion.com/lessons/3206-generate-survey-data-through-simulations



http://learnzillion.com/lessons/2842-create-a-dot-plot

http://www.khanacademy.org/math/probability/descriptive-statistics/Box-and-whisker%20plots/v/box-and-whisker-plots

http://learnzillion.com/lessons/3596-compare-iqr-using-box-plots


35






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 median) of two or more given data sets 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

Video (click on link below): Mean Absolute Deviation


http://learnzillion.com/lessons/3578-compare-two-populations-using-mean-absolute-deviation


36

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


37







Q. Expanding and Enriching Ideas 5. Modifying to add details.

R. Connecting and Condensing Ideas 6. Connecting Ideas

7. Condensing Ideas

Unit #9 Probability





Resources





Technology for random sampling:

http://www.randomizer.org/

http://stattrek.com/stati

stics/random-number-generator.aspx

CA Mathematics Framework Gr. 7 p. 42 – 45 Progressions for the Common Core – Statistics

and Probability Gr. 6-8 North Carolina 7th Grade Math Unpacked Content: p. 39 – 43

Why does it make sense that the probability of a chance event is represented as a

http://www.engageny.org (pp. 1 - 82) (This module contains a

variety of tasks that

1) Determine the probability of a chance event and represent it as a number between 0 and 1 (for example, the probability of flipping heads on a quarter is ½), and understand that a probability

Conduct class discussions about observed data (e.g. flipping a coin), paying attention to similarities and differences between students’ observations,

Videos (click on links below) Using organized lists Using tables Using tree diagrams


Self-management

Social awareness

Relationship skills




http://stattrek.com/statistics/random-number-generator.aspx











http://www.youtube.com/watch?v=tc6F54fbLRU

http://learnzillion.com/lessons/1862-find-the-probability-of-a-compound-event-by-creating-a-table

http://learnzillion.com/lessons/1861-find-the-probability-of-a-compound-event-by-creating-a-tree-diagram


38

Unit #9 Probability





Resources

number between 0 and 1?

How do you know if your predictions based on 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?

relate to multiple learning experiences.)

http://www.illustrativem


near zero is an unlikely event, while a probability near 1 is a likely event.

7.SP.5

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)

Big Ideas MATH 7th grade TE: Lesson 10.1 and 10.2


2) Collect data from a chance event (for example, rolling a die), calculate the probability based on the observed frequencies, and use proportional reasoning to make predictions.

7.SP.6

Have students collect their own data.

Big Ideas MATH 7th grade TE: Lesson 10.2 and 10.3

http://map.mathshell.org/materials/tasks.php?taskid=367#task367

http://map.mathshell.org/materials/lessons.php?taskid=225&subpage=concept

3) Compare the theoretical probability of a chance event to the probability based on observed frequencies, and explain any possible sources of discrepancies.

7.SP.7

Example Students can compare the

theoretical probability of flipping tails on a quarter (0.5) to an observed probability (they may flip a quarter 20 times and get tails 8 times).

See p.412, Activity 1





http://learnzillion.com/lessons/1206-calculate-the-probability-of-an-event-by-creating-a-ratio












39

Unit #9 Probability





Resources

What are some similarities and differences when using an organized list, a table, and a tree diagram to find probabilities of compound events?


4) Find probabilities of compound events 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

Big Ideas MATH 7th grade TE: Lesson 10.5 The textbook compares independent events to

dependent events.

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

Big Ideas MATH 7th grade TE: Extension Lesson 10.5, p.436 –437





40

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






Self-management

Social awareness

Relationship skills



41



S. Expanding and Enriching Ideas 5. Modifying to add details.

T. Connecting and Condensing Ideas 6. Connecting Ideas

7. Condensing Ideas

Unit #10 2-Dimensional and 3-Dimensional Geometric Figures


Sequence of Learning Outcomes 7.G.1, 7.G.2, 7.G.3, 7.G.4, 7.G.5, 7.G.6



Resources





GeoGebra

CA Mathematics Framework Gr. 7 p. 33 – 38 North Carolina 7th Grade Math Unpacked Content: p. 25 – 33 7th Grade Common Core State Standards Flip Book

What are the criteria for 3 side lengths to form a triangle?

What is an example of a situation where you could be given three pieces of information about a triangle and


1) Draw triangles (freehand, with ruler and protractor, and with technology) given three out of six possible criteria, for example two side lengths and an angle. Determine if the triangle exists, is unique, or determines more than one triangle.

7.G.2

Informal introduction to triangle inequality theorem. http://www.mathopenref.com/triangleinequality.html Informal introduction to triangle congruence theorems: SSS, SSA, AAS, SAS, AAA. http://www.regentsprep.org/Regents/math/geometry/GP4/BegTriPrf.htm












http://www.mathopenref.com/triangleinequality.html

http://www.mathopenref.com/triangleinequality.html

http://www.regentsprep.org/Regents/math/geometry/GP4/BegTriPrf.htm




42






Resources

have more than one possible drawing that fit the given criteria?

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-theorems.html Big Ideas MATH 7th grade TE: Lesson 7.1 Lesson 7.2 Extension Lesson 7.3 p.288-289

What is (pi)? Why is it an important number and how is it used?

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/lesson-

plan/3430.html Big Ideas MATH 7th grade TE: Lesson 8.1

http://www.illustrativemathematics.org/illustrations/1553 http://www.illustrativemathematics.org/illustrations/34

4) Build on understanding of

circumference, diameter and to generate formulas for circumference and area of circles and use them to solve mathematical and real-world problems.

7.G.4

Videos (click on links below) Explore and generate formulas for circumference and area of a circle. Finding Circumference

Finding the Area of a Circle


How do you subdivide a composite figure to find its area?

5) Find the area of triangles, quadrilaterals, and other polygons, including composite figures composed of triangles, quadrilaterals, and polygons, in the

Compare the use of addition and subtraction when finding the area of composite figures, for example:

Sample Lesson: http://cc.betterlesson.com/lesson/441863/are

a-of-composite-shapes-using-a-grid

http://www.mathsisfun.com/geometry/circle-theorems.html

http://www.mathsisfun.com/geometry/circle-theorems.html

https://www.teachervision.com/math/lesson-plan/3430.html

https://www.teachervision.com/math/lesson-plan/3430.html







http://learnzillion.com/lessons/818-find-the-circumference-of-a-circle

http://learnzillion.com/lessons/819-find-the-area-of-a-circle

http://cc.betterlesson.com/lesson/441863/area-of-composite-shapes-using-a-grid

http://cc.betterlesson.com/lesson/441863/area-of-composite-shapes-using-a-grid


43






Resources

context of real-world and mathematical problems.

7.G.6

Big Ideas MATH 7th grade TE: Lesson 8.2 Lesson 8.3 Lesson 8.4

What is the relationship between the ratios of side lengths and areas of geometric figures in scale drawings?

http://www.illustrativemathematics.org/illustrations/107 http://map.mathshell.org/materials/lessons.php?taskid=494&subpage=problem

6) Investigate relationships between side lengths and areas in scale drawings of geometric figures, and reproduce a scale drawing at a different scale.

7.G.1

Big Ideas MATH 7th grade TE: Lesson 7.5 (Refer back to what students learned in Unit 1

about scale drawings).

7) Identify the two-dimensional figure that results from slicing a plane section of a three-dimensional figure.

7.G.3

Big Ideas MATH 7th grade TE: Extension Lesson 9.5 p.388-389

What is the relationship between area, surface area, and volume?


8) Solve real-world and mathematical problems involving the surface area and volume of cubes and right prisms, and explore the relationship between surface area and volume.

7.G.6

Big Ideas MATH 7th grade TE: Lesson 9.1 Lesson 9.2 Lesson 9.3 Lesson 9.4 Lesson 9.5




http://map.mathshell.org/materials/lessons.php?taskid=494&subpage=problem








44

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.



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.


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.


45

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



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.


U. Expanding and Enriching Ideas 5. Modifying to add details.

V. Connecting and Condensing Ideas 6. Connecting Ideas

7. Condensing Ideas



46

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



Resources









Tiered: Independent


Grouping o Content o Rigor w/in the

concept o Project-based

learning o Homework o Grouping o Formative

Assessment


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














47





Resources

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?

For Learning Outcomes 1 – 3: http://map.mathshell.org/mat

erials/lessons.php?taskid=421#task421

http://www.illustrativemathe

matics.org/illustrations/334 http://www.illustrativemathe

matics.org/illustrations/1538

1) Investigate how to prove that terminating decimals are rational because they can be written in the

form 𝑝

𝑞 using place value.

8.NS.1

For Outcomes 1-2, all values should be explored, not only numbers between 0

and 1 (e.g., 2.8, -3.6, and √9).

Students should use a calculator to verify their decimal-to-fraction conversion.

Video (click on link): Distingush between rational and

irrational numbers For Outcomes 1-3, it may be valuable for

students to place the numbers on a number line.

Anchor Activities: Content-related


Activity o Stations



o Patterns o Unanswered


Complexity See Differentiation Resources at: http://scusd-math.wikispaces.com/home

Video (click on link): Distingush between rational and

irrational numbers

What is the appropriate level of precision in estimating an irrational number in a given real-world context?

2) Investigate and discover that non-terminating, repeating decimals are rational because they can be

written in the form 𝑝

𝑞 using the

conversion method (Framework p. 7). 8.NS.1

Converting repeating decimals into fractions Big Ideas Math 8th Grade TE: Extension Lesson 7.4 p. 316 – 317 “Repeating Decimals”

3) Use a calculator to explore the expanded decimal values of π and non-perfect squares to notice that they are non-terminating and










http://learnzillion.com/lessons/221-distinguish-between-rational-and-irrational-numbers






http://www.instructables.com/id/Decimal-to-Fraction-1/step2/Converting-repeating-decimals-to-fractions/

http://www.instructables.com/id/Decimal-to-Fraction-1/step2/Converting-repeating-decimals-to-fractions/


48





Resources

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 – 6: http://www.illustrativemathematics.org/illustrations/336 http://www.illustrativemathematics.org/illustrations/337 http://www.illustrativemathematics.org/illustrations/1221 http://illuminations.nctm.org/

Lesson.aspx?id=4082

Unit Assessment (Exponents

and Radicals)

4) Estimate the values of irrational numbers using a method of squaring rational numbers (For example

estimate the decimal expansion of √2

by showing that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations - Framework p. 8). Represent this number on the number line. 8.NS.2


What do square root and cube root actually mean?

Why are squares and square roots and cubes and cube roots inverse operations?

Are all square roots irrational? Why or why not?

5) Evaluate expressions for square roots of small perfect squares and cube roots of small perfect cubes, using the concept of repeated multiplication. 8.EE.2

Video (click on link):

Find the square root of a perfect square

Big Ideas Math 8th Grade TE: Lessons 7.1 and 7.2

6) Solve equations in the form of x2 = p and x3 = p using inverse operations (where p is a positive rational number).

Big Ideas Math 8th Grade TE: Lesson 7.1 p292 – 293 #28, 33, 35, 37

(students write equations in the form x2 = p and solve using square roots).







http://illuminations.nctm.org/Lesson.aspx?id=4082

http://illuminations.nctm.org/Lesson.aspx?id=4082

http://cc.betterlesson.com/lesson/443160/exponents-and-radicals-unit-assessment-day-1-of-2

http://cc.betterlesson.com/lesson/443160/exponents-and-radicals-unit-assessment-day-1-of-2

http://learnzillion.com/lessons/1810-find-the-square-root-of-a-perfect-square


49





Resources

8.EE.2

Lesson 7.2 p.297 Example 4, p.298 - 299 #21 – 22, 29, 31 – 33 (students will write equations in the form x3 = p and solve).

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?

For Learning Outcomes 7 – 9: http://map.mathshell.org/mat

erials/download.php?fileid=804





1) Understand the Pythagorean Theory using a proof and explore the proof with multiple right triangles. Using the same proof, students will explore whether or not the Pythagorean Theorem applies to non-right triangles.

8.G.6

Videos on Pythagorean Theorem:

Pythagorean Theorem construction

Pythagorean Theorem discovery and real-world problem to solve

Sample Lesson: Review of right triangles and the

relationships of their sides. Big Ideas Math 8th Grade TE: Lesson 7.3 p. 300 – 301, Activities 1 – 3

2) Use the Pythagorean Theorem to solve for unknown side lengths (rational and irrational) in right triangles in real-world and mathematical problems in two and three dimensions.

8.G.7

Use Outcome #4 to approximate irrational answers appropriately for real-world Pythagorean Theorem problems

Big Ideas Math 8th Grade TE: Lesson 7.3 p. 302 – 303, Examples 1 – 3

3) Given two coordinates, students will draw a right triangle and use the Pythagorean Theorem to find the

Video: Find distance between two points on the

coordinate plane using Pythagorean

















http://learnzillion.com/lessons/2704-use-the-area-of-squares-proof-to-relate-side-lengths-of-a-right-triangle

https://www.teachingchannel.org/videos/teaching-pythagorean-theorem

https://www.teachingchannel.org/videos/teaching-pythagorean-theorem

http://cc.betterlesson.com/lesson/586142/playing-around-with-pythagoras-day-2

http://cc.betterlesson.com/lesson/586142/playing-around-with-pythagoras-day-2

http://learnzillion.com/lessons/3406-find-distance-between-two-points-on-the-coordinate-plane-using-the-pythagorean-theorem

http://learnzillion.com/lessons/3406-find-distance-between-two-points-on-the-coordinate-plane-using-the-pythagorean-theorem


50





Resources

distance between the two coordinates (i.e. the length of the hypotenuse of the right triangle).

8.G.8





matics.org/illustrations/517 http://map.mathshell.org/mat

erials/lessons.php?taskid=410#task410

4) Solve real-life and mathematical problems using the formulas for volume of cylinders, cones, and spheres.

8.G.9

Big Ideas Math 8th Grade TE: Chapter 8, lessons 8.1 – 8.3 (Lesson 8.4 includes Surface Area which

is not addressed in the standards).













51

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.



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



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


52


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 Sequence of Learning Outcomes 8.EE.1, 8.EE.3, 8.EE.4



Resources









Tiered: Independent


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














53

Unit #12 Exponents




Resources

Where do the rules for exponents come from?

Why do we use scientific notation?

For Learning Outcomes 1 – 5: http://www.illustrativemathe


1) Use the definition of an exponent to expand and simplify expressions (with positive integer exponents only) in order to generate the rule for multiplying powers with the

same base: nmnm aaa 8.EE.1

Use the definition of an exponent to expand and simplify expressions in order to general rules, for example:

743 2222222222 Use vocabulary like “How many factors of 2 do you see?”

For example,

63323 2)222()222()2)(2(2


Grouping o Content o Rigor w/in

the concept o Project-base

d learning o Homework o Grouping o Formative

Assessment Anchor Activities: Content-related


Activity o Stations



o Patterns


Lesson 10.2 (start with nmnm aaa rule only)

2) Expand and simplify expressions (with positive integer exponents only) in order to generate the rule for raising a power to a power:

mnnm aa )(

8.EE.1

Big Ideas Math 8th Grade TE:

Lesson 10.2 (include mnnm aa )( rule)

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: nm

n

m

aa

a

8.EE.1





54

Unit #12 Exponents




Resources

How can you prove that a0 = 1?

4) Make predictions about powers raised to zero exponents. Generate and prove the rule for zero

exponents: 10 a 8.EE.1

Why is 10 a ? (See proofs…) http://www.homeschoolmath. https://www.khanacademy.org

o Unanswered Questions

o Rules o Trends o Big Ideas



Big Ideas Math 8th Grade TE: Lesson 10.4 (start with zero exponents only)

Why isn’t a-n = –an?

5) Make predictions about powers raised to negative exponents. Generate and prove the rule for

negative exponents: n

n

aa

1

8.EE.1

Why isn

n

aa

1 ?

(See proofs…) http://www.homeschoolmath. https://www.khanacademy.org

Big Ideas Math 8th Grade TE: Lesson 10.4 (include negative exponents)

Why do we use 10 as a base for numbers expressed in scientific notation?

For Learning Outcomes 6 – 9: http://www.illustrativemathe







6) Given large or small numbers in standard form, express them in scientific notation; Given large or small numbers in scientific notation, express them in standard form.

8.EE.3


How can you use estimation to compare two numbers expressed in scientific notation?

7) Use numbers expressed in scientific notation (in the form of a single digit times an integer power of 10) to determine which one has a greater value (for example, determine which value is greater and explain how you know: 8 x 105

and 9 x 104).


http://www.homeschoolmath.net/teaching/zero-exponent-proof.php

https://www.khanacademy.org/math/arithmetic/exponents-radicals/world-of-exponents/v/raising-a-number-to-the-0th-and-1st-power



http://www.homeschoolmath.net/teaching/negative_zero_exponents.php

https://www.khanacademy.org/math/arithmetic/exponents-radicals/negative-exponents-tutorial/v/negative-exponent-intuition













55

Unit #12 Exponents




Resources


8.EE.3

8) Using numbers expressed in scientific notation (in the form of a single digit times an integer power of 10), estimate how many times greater one number is than the other number (for example, the population of the U.S. is 3 x 108 and the population of the world is 7 x 109, so the population of the world is more than 20 times larger).

8.EE.3

Example: About how much greater is the world

population than the U.S. population? 7×109

3×108=

7×10×108

3×108=

70

3≈ 23

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=

6×10×10−9

9×10−9=

60

9≈ 6.7

Big Ideas Math 8th Grade TE: Lesson 10.5 p.439 #3 (this example starts to

develop this Learning Outcome; for more detail see Strategies for Teaching and Learning).

How do the rules of exponents apply to performing operations with numbers expressed in scientific notation?

9) Given a mathematical or real-world problem, perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used.

8.EE.4

Use the rules for integer exponents (see Learning Experiences 1 – 5) to perform operations with numbers expressed in scientific notation.

Students will choose units of appropriate

size for a given situation. Students should be able to interpret and understand scientific notation as it has been generated by a calculator.



Documents

Mathematics Grade 7/8 Curriculum Map CompactedMathematics Grade 7/8 Compacted DRAFT Last Updated October 30, ... CA Mathematics Framework Gr. 7 p. 14 – 16 Progressions for the Common