7th  Grade  Mathematics                                                            

Unit  #2:  Computing  with  Rational  Numbers  Pacing:  25  Days  Unit Overview

In Grade 6, students formed a conceptual understanding of integers through the use of the number line, absolute value, and opposites and extended their understanding to include the ordering and comparing of rational numbers. In this unit, students build on their understanding of rational numbers to add, subtract, multiply, and divide signed numbers. Previous work in computing the sums, differences, products, and quotients of fractions serves as a significant foundation to access the content of this unit.

Prerequisite Skills Vocabulary Mathematical Practices 1) Conduct all four operations with multi-digit numbers and

decimals with fluency 2) Add and subtract fractions with like and unlike denominators 3) Convert mixed numbers to improper fractions and improper

fractions to mixed numbers 4) Simplify fractions 5) Construct a basic number line

Fractions Natural Numbers Absolute value Repeating Decimal Integer Positive integer Negative integer Rational numbers Whole numbers Opposites Assets Debt Additive inverse Graph Bar notation Terminating Decimal

MP.1: Make sense of problems and persevere in solving them

MP.2: Reason abstractly and quantitatively

MP.3: Construct viable arguments and critique the reasoning of others

MP.4: Model with mathematics

MP.5: Use appropriate tools strategically

MP.6: Attend to precision

MP.7: Look for and make use of structure

MP.8: Look for and express regularity in repeated reasoning

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Common Core State Standards Progression of Skills                                                                   According to the PARCC Model Content Framework, Standard 3.NF.2 should serve as an opportunity for in- depth focus:                              

According to the PARCC Model Content Framework, Standard 7.NS.3 should serve as opportunity for in-depth focus:

“When students work toward meeting this standard (which is closely connected to 7.NS.1 and 7.NS.2), they consolidate their skill and understanding of addition, subtraction, multiplication, and division of rational numbers.”

According to the PARCC Model Content Framework, The key advance in addition and subtraction of rational numbers between

seventh and eighth grade is: “By working with equations such as x2=2 and in geometric contexts such as Pythagorean theorem, students enlarge their concept of number beyond the system of rationals to include irrational numbers. They represent these numbers with radical expressions and approximate these numbers with rationals.”  

6th Grade 7th Grade 8th Grade 6.NS.5: Understand that positive and negative numbers are used together to describe quantities having opposite directions or values.

7.NS.1a: Describe situations in which opposite quantities combine to make 0

N/A  

6.NS.5: Use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.  

7.NS.1b: 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.

N/A  

6.NS.7c: Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.

7.NS.1c: 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.

8.NS.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).

N/A   7.NS.1d: Apply properties of operations as strategies to add and subtract rational numbers.

N/A  

  7.NS.3: Solve real-world and mathematical problems involving the four operations with rational numbers.

N/A  

 

Major  Standards  (70%)  

Additional  Standards  (10%)  

   

7.NS.1: Add and Subtract Rational Numbers

7.NS.2: Multiply and Divide Rational Numbers

7.NS.3: Solve Real World and Mathematical

Problems Involving Rational Numbers

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Big Ideas Students Will… • How can knowledge about additive

inverse with addition and subtraction and the Associative Property be applied to rational numbers?

• What does the common phrase "add the opposite" mean in terms of subtraction of rational numbers?

• How do I represent positive and negative numbers on a number line? How can one extend the coordinate grid to represent positive and negative coordinates?

• What is the difference between an

integer and a rational number? • What does a remainder in a quotient

represent? How does this compare to a quotient that is a decimal?

• What rules can we find to generalize patterns when operating with positive and negative numbers?

• What is a repeating decimal? When converting a rational number to a decimal how can you tell if the quotient is a repeating decimal?

Know/Understand Be Skilled At… • The concept of absolute value, as well as the

symbolic representation of it in equations (ex: |q|).

• How absolute value applies in the real world. • The definition of rational numbers, integers,

and the additive inverse. • 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

• That subtraction of rational numbers as

adding the additive inverse, p – q = p + (–q). • The distance between two rational numbers

on the number line is the absolute value of their difference.

• That a terminating decimal ends in 0

• That a decimal can repeat

• Adding and subtracting rational numbers.

• Representing addition and subtraction on a horizontal or vertical number line diagram.

• Applying properties of operations as strategies

to add and subtract rational numbers.

• Explaining why and when the product or difference should be in the absolute value form.

• Writing the product or difference in the form of

an absolute value when necessary.

• Adding, subtracting, multiplying, and dividing rational numbers, including complex fractions.

• Explaining how they used the properties of

operations to solve the problem.

• Convert fractions, decimals and percents to equivalent forms; specifically, converting rational numbers to decimals using division

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Unit Sequence Student Friendly Objective

SWBAT… Key Points/

Teaching Tips Sample Assessment Item

from Exit Ticket Instructional

Resources 1 Use integers to describe

everyday situations. Justify that an integer plus its opposite add to zero.

• Use “Hot Air Balloons” to design a conceptual and inquiry-based lesson before engaging students with My Math and Engage NY Resources

• Note: resources will need to be combined and modified significantly to meet this objective

At 8:00 a.m. the temperature outside was -5°C. At 6:00 p.m., the temperature was 25°C. Display the integers on a number line.

“Hot Air Balloons” (Appendix C) Engage NY Module 2 Lesson 1 (Appendix C) My Math Chapter 3 Lesson 1

2 Model adding positive and negative integers using number lines. Find the sums of positive and negative integers. Observe patterns and deduce/justify the rule for adding integers.

• Pacing: 2 days • Combine/sequence resources as

needed to meet these objectives

1) Look at the number line below.

What equation is modeled on the number line? A) -3 + 3 = 6 B) -3 + 6 = 3* C) -3 + (-6) = 0 D) -3 – 6 = 3

Engage NY Module 2 Lessons 2 – 4 (Appendix C) My Math Inquiry Lab (Pages 199 – 202) My Math Chapter 3 Lesson 2

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4 Model subtracting positive and negative integers using number lines. Find the difference of positive and negative integers. Observe patterns and deduce/justify the rule for subtracting integers.

• Pacing: 2 days • Combine/sequence resources as

needed to meet these objectives

Ojos del Salado is the highest mountain in Chile, with a peak at about 6900 meters above sea level. The Atacama Trench, just off the coast of Peru and Chile, is about 8100 meters below sea level (at its lowest point).

• What is the difference in elevations between Mount Ojos del Salado and the Atacama Trench?

• Is the elevation halfway between the peak of Mount Ojos del Salado and the Atacama Trench above sea level or below sea level? Explain without calculating the exact value.

What elevation is halfway between the peak of Mount Ojos del Salado and the Atacama Trench?

Engage NY Module 2 Lesson 5 (Appendix C) My Math Inquiry Lab (Pages 211 - 214) My Math Chapter 3 Lesson 3

5

6 Determine and explain how the difference between two rational numbers is related to their difference

  Engage NY Module 2 Lesson 6 (Appendix C) My Math Chapter 3 Inquiry Lab (Pages 223 – 224)

7 Flex Day Recommended Resources:

“Sign Your Name” (Appendix C) “Deep Freeze Task” (Appendix C)

My Math Chapter 4 Are You Ready? And Inquiry Lab (Pages 260 – 262) – pre-requisite skills necessary for next set of objectives

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8 Use a number line to demonstrate how the rules for adding and subtracting integers apply to rational numbers

•

  Engage NY Module 2 Lesson 7 (Appendix C) My Math Chapter 4 Inquiry Lab (Pages 279 – 282)

9 Apply properties of operations to add and subtract rational numbers

• Pacing: 2 days • Combine/sequence resources as

needed to meet these objectives

  Engage NY Module 2 Lessons 8 -9 (Appendix C) My Math Chapter 4 Lesson 5 Resource for Remediation: My Math Chapter 4 Lessons 3 & 4

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11 Flex Day Recommended Resources:

“Distances Between Houses” (Appendix C) 12 Determine when the product

of two integers is positive and when it is negative. Justify and explain rules for multiplying integers.

My Math Chapter 3 Inquiry Lab (Pages 229 – 232) Engage NY Module 2 Lesson 10 (Appendix C)

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13 Multiply integers • Students understand the rules for multiplication of integers and that multiplying the absolute values of integers results in the absolute value of the product. The sign, or absolute value, of the product is positive if the factors have the same sign and negative if they have opposite signs.

• Students realize that (-1)(-1) = (1) and see that it can be proven to be true mathematically through the use of the distributive property and the additive inverse.

Engage NY Module 2 Lesson 11 (Appendix C) My Math Chapter 3 Lesson 4

14 Divide integers • Students recognize that division is the reverse process of multiplication, and that integers can be divided provided the divisor is not zero.

• Students understand that every quotient of integers (with a non-zero divisor) is a rational number and divide signed numbers by dividing their absolute values to get the absolute value of the quotient. The quotient is positive if the divisor and dividend have the same signs and negative if they have opposite signs.

Engage NY Module 2 Lesson 12 (Appendix C) My Math Chapter 3 Lesson 5

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15 Flex Day Small group instruction based on data from lessons 13 and 14

16 Convert between fractions

and decimals • Pacing: 2 days • understand that the context of

a real-life situation often determines whether a rational number should be represented as a fraction or decimal.

Malia found a "short cut" to find the decimal representation of the fraction below: 117 250 Rather than use long division she noticed that because 250×4=1000, then: 117 x 4 = 468 = .468 250 x 4 = 1000

a. For which of the following fractions does Malia's strategy work to find the decimal representation?

b. For each one for which the strategy does work, use it to find the decimal representation. For the fractions for which you can apply this strategy, use another strategy to find their decimal equivalent.

c. For which denominators can Malia's strategy work?

Engage NY Module 2 Lessons 13 & 14 (Appendix C) My Math Chapter 4 Lesson 1

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18 Compare and order rational numbers

You, your cousin, and a friend each take the same number of free throws at a basketball hoop. Who made the most free throws?  

My Math Chapter 4 Lesson 2

19 Demonstrate how the rules for multiplying and dividing integers apply to rational numbers. Interpret products and quotients of rational numbers in real world contexts

• Engage NY Module 2 Lesson 15 (Appendix C) “Why is a Negative x a Negative Always a Positive?” (Appendix C)

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20 Apply properties of operations to multiply and divide rational numbers

• Pacing: UP to two days A water well drilling rig has dug to a height of -60 feet after one full day of continuous use.

• Assuming the rig drilled at a constant rate, what was the height of the drill after 15 hours?

If the rig has been running constantly and is currently at a height of -143.6 feet, for how long has the rig been running?

Engage NY Module 2 Lesson 16 (Appendix C) My Math Chapter 4 Lessons 6 & 8

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22 - 24 3 Flex Days (Instruction Based on Data) Recommended Resources:

“Repeating Decimal as Approximation” (Appendix C) “Field Trip Funding” (Appendix C) “What Does it Cost?” (Appendix C)

“Pizzeria Profits” (Appendix C) “Sharing Prize Money” (Appendix C)

“The Repeater vs. The Terminator” (Appendix C) Engage NY Mid-Module Assessment (Appendix C)

My Math 21st Century Career in Astronomy (Pages 251 – 252) My Math Chapter 3 Review (Pages 253 – 256)

My Math 21st Century Career in Fashion Design (Pages 335 – 336) My Math Chapter 4 Review (Pages 337 – 340)

25 End of Unit Assessment MCLASS Beacon Assessment 7.2

Appendix B *This assessment will be administered online

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Appendix A: Unpacked Standards Guide

Source: Public Schools of North Carolina NCDPI Collaborative Workspace  

Standard Unpacking What do these standards mean a child will know and be able to do?

7.NS. 1. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. a. Describe situations in which

opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

c. Understand subtraction of rational numbers as adding the

7.NS. 1. Visual representations may be helpful as students begin this work; they become less necessary as students become more fluent with the operations. Examples: • Use a number line to illustrate:

o p - q o p + (- q) o Is this equation true p – q = p + (-q)

• -3 and 3 are shown to be opposites on the number line because they are equal distance from zero and therefore have the same absolute value and the sum of the number and it’s opposite is zero.

You have $4 and you need to pay a friend $3. What will you have after paying your friend? 4 + (-3) = 1 or (-3) + 4 = 1

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additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

d. Apply properties of operations as strategies to add and subtract rational numbers.

7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)

7.NS.3. Examples: • Your cell phone bill is automatically deducting $32 from your bank account every month. How much will the deductions total for the year? -32 + -32 + -32 + -32 + -32 + -32 + -32 + -32 + -32 + -32 + -32 + -32 = 12 (-32) • It took a submarine 20 seconds to drop to 100 feet below sea level from the surface. What was the rate of the descent?

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7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. a. Understand that multiplication is

extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.

c. Apply properties of operations as strategies to multiply and divide rational numbers.

d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.

7.NS.2 Students understand that multiplication and division of integers is an extension of multiplication and division of whole numbers. Students recognize that when division of rational numbers is represented with a fraction bar, each number can have a negative sign. Example 1: Which of the following fractions is equivalent to

€

−45

? Explain your reasoning.

a. b. c.

Example Set 2: Examine the family of equations in the table below. What patterns are evident? Create a model and context for each of the products. Write and model the family of equations related to 3 x 4 = 12.

Equation Number Line Model Context

2 • 3 = 6

Selling two packages of apples at $3.00 per pack

2 • -3 = -6

Spending 3 dollars each on 2 packages of apples

-2 • 3 = -6

Owing 2 dollars to each of your three friends

-2 • -3 = 6

Forgiving 3 debts of $2.00 each

€

4−5

€

−1620

€

−4−5

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Using long division from elementary school, students understand the difference between terminating and repeating decimals. This understanding is foundational for the work with rational and irrational numbers in 8th grade. Example 3: Using long division, express the following fractions as decimals. Which of the following fractions will result in terminating decimals; which will result in repeating decimals? Identify which fractions will terminate (the denominator of the fraction in reduced form only has factors of 2 and/or 5)