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 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

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

Natural Numbers Repeating Decimal

Mathematical Practices 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

Common Core State Standards                                                                

6th Grade

  Additional   Standards   (10%)    

            Major     Standards     (70%)            

   

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

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.”  

Progression of Skills 7th Grade

8th Grade

N/A   6.NS.5: Understand that 7.NS.1a: Describe According PARCC Model Content Framework, positive and negative to the situations in which Standard 3.NF.2 should serve as an opportunity for innumbers are used opposite quantities together todepth describefocus: combine to make 0 quantities having opposite directions or values.   7.NS.1b: N/A   6.NS.5: Use positive and Understand p + q as the negative numbers to number located a represent quantities in distance |q| from p, in the real-world contexts, positive or negative explaining the meaning direction depending on of 0 in each situation.   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. 6.NS.7c: Understand the 7.NS.1c: Understand 8.NS.2: Use rational absolute value of a subtraction of rational approximations of rational number as its numbers as adding the irrational numbers to distance from 0 on the additive inverse, p compare the size of number line; interpret q= p + (-q). Show that the irrational numbers, absolute value as distance between two locate them approximately magnitude for a positive rational numbers on the on a number line diagram, or negative quantity in a number line is the and estimate the value of real-world situation. absolute value of their expressions (e.g., π2). difference, and apply this principle in real-world contexts. N/A   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.  

2  |  P a g e    

Big Ideas •

•

•

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?

Students Will… •

•

Know/Understand The concept of absolute value, as well as the symbolic representation of it in equations (ex: |q|).

Be Skilled At… • Adding and subtracting rational numbers. • Representing addition and subtraction on a horizontal or vertical number line diagram.

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

• 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.

•

What is the difference between an integer and a rational number?

•

That subtraction of rational numbers as adding the additive inverse, p – q = p + (–q).

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

•

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

•

The distance between two rational numbers on the number line is the absolute value of their difference.

• Explaining how they used the properties of operations to solve the problem.

•

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

•

That a terminating decimal ends in 0

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

•

That a decimal can repeat

•

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

3  |  P a g e    

Unit Sequence 1

Student Friendly Objective SWBAT… Use integers to describe everyday situations.

•

Justify that an integer plus its opposite add to zero. •

2

3

Model adding positive and negative integers using number lines.

• •

Key Points/ Teaching Tips 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

Sample Assessment Item from Exit Ticket 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.

Pacing: 2 days Combine/sequence resources as needed to meet these objectives

1) Look at the number line below.

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

Find the sums of positive and negative integers. Observe patterns and deduce/justify the rule for adding integers.

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

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

My Math Chapter 3 Lesson 2

4  |  P a g e    

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.

5

• •

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.

6

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

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

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

5  |  P a g e    

8

9

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

•

Apply properties of operations to add and subtract rational numbers

• •

 

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

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

10

Resource for Remediation: My Math Chapter 4 Lessons 3 & 4 11

12

Flex Day Recommended Resources: “Distances Between Houses” (Appendix C) 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)

6  |  P a g e    

13

Multiply integers

•

•

14

Divide 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. 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 11 (Appendix C) My Math Chapter 3 Lesson 4

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

7  |  P a g e    

15

16

Flex Day Small group instruction based on data from lessons 13 and 14 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 = 468 250 x 4 = 1000

17

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

= .468

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? 8  |  P a g e    

18

Compare and order rational numbers

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

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

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

9  |  P a g e    

20

               

21

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.

Engage NY Module 2 Lesson 16 (Appendix C)

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

My Math Chapter 4 Lessons 6 & 8

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?

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

10  |  P a g e    

   

Appendix A: Unpacked Standards Guide Source: Public Schools of North Carolina NCDPI Collaborative Workspace   Unpacking Standard 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 7.NS. 1. Visual representations may be helpful as students begin this work; they become less necessary as subtraction to add and subtract students become more fluent with the operations. rational numbers; represent addition Examples: and subtraction on a horizontal or • Use a number line to illustrate: vertical number line diagram. op-q a. Describe situations in which o p + (- q) o Is this equation true p – q = p + (-q) opposite quantities combine to make 0. For example, a • -3 and 3 are shown to be opposites on the number line because they are equal distance from zero and hydrogen atom has 0 charge therefore have the same absolute value and the sum of the number and it’s opposite is zero. because its two constituents are oppositely charged. b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.

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

c. Understand subtraction of rational numbers as adding the 11  |  P a g e    

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?

12  |  P a g e    

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 a.

4 −5

b.

−16 20

c.

−4 −5

−4 ? Explain your reasoning. 5

€

€ € 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 2•3=6

2 • -3 = -6

-2 • 3 = -6

-2 • -3 = 6

Number Line Model

Context Selling two packages of apples at $3.00 per pack Spending 3 dollars each on 2 packages of apples Owing 2 dollars to each of your three friends Forgiving 3 debts of $2.00 each

13  |  P a g e    

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)  

14  |  P a g e