Grade 7 Unit 1: Add, Subtract, Multiply and Divide Rational Numbers (6 Weeks)

Stage 1 – Desired Results

Established Goals Unit Description Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers and use this understanding to solve real world and mathematical problems involving the four operations. The Mathematical Practices should be evident throughout instruction and connected to the content addressed in this unit. Students should engage in mathematical tasks that provide an opportunity to connect content and practices. Common Core Learning Standards 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 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.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.3 Solve real-world and mathematical problems involving the four operations with rational numbers. 7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

Common Core Standards of 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. 8. Look for and express regularity in repeated reasoning. ESL Language Standards Standard 1 1. Identify and use reading and listening strategies to make text comprehensible and meaningful. 7. Present information clearly in a variety of oral and written forms for different audiences and purposes related to all academic content areas. 12. Convey information and ideas through spoken and written language, using conventions and features of American English appropriate to audience and purpose. 15. Apply self-monitoring and self-correcting strategies for accurate language production and oral and written presentation, using established criteria for effective presentation of information. 16. Apply learning strategies to acquire information and make texts comprehensible and meaningful. 9. Apply learning strategies to examine and interpret a variety of materials. Standard4 5. Explain actions, choices, and decisions in social and academic situations.

Big Ideas 1. The set of real numbers is infinite and each whole number can be associated with a unique point on the number line.

Essential Questions 1. How do properties of real numbers relate to properties of rational numbers? 1. How are integers and real numbers related? 1. It is possible to use "less than" or "greater than" concepts with integers? 1. How can a meaning be found for operations on negative numbers? 1. How can we use models to help us understand operations with integer numbers? 1. How can we use number properties to help us understand the meaning of computing with integers?

Content (Students will know….) A. Properties of operations with whole numbers hold for addition and subtraction of rational numbers B. Addition and subtraction can be represented on a vertical or horizontal number line C. Adding a number and its opposite equal zero D. Subtracting a rational number is the same as adding the additive inverse

Skills (Students will be able to…) A1. Add and subtract positive and negative rational numbers using properties of operations on whole numbers A2. Apply properties of operations as strategies to add and subtract rational numbers B1. Use the number line as a tool to add and subtract

rational numbers (for example model using thermometer or

hot air balloon model)

C1.Describe real life situations in which quantities combine to make zero C2. Show that a number and its opposite have a sum of zero (are additive inverses) D1. Subtract rational numbers by using the additive inverse, p – q = p + (-q)

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

F. p + q is a number located a distance |p| from q, in the

positive or negative direction depending on whether p is

positive or negative G. Models, diagrams and manipulatives are useful in developing computational understanding of multiplying and dividing integers H. The strategies we use to multiply and divide fractions can be applied to multiplying and dividing rational numbers I. Multiplication is extended from fractions to rational numbers by requiring operations continue to satisfy the properties of operations and the rules for multiplying signed numbers J. Integers can be divided provided that the divisor is not zero and the quotient is a rational number K. The decimal form of a rational number terminates eventually or ends in zero L. Real world problems can be solved using operations with rational numbers

D2. Apply this principle in a real world context

E1. Subtract two rational numbers by finding the absolute

value of their difference

E2. Apply this principle to a real world context

F1. Add and subtract integers with absolute value

G. Use models to develop the meaning of algorithms for

integer operations

The two color counters model

The stack or row model

The hot air balloon model (vertical number line)

The number line model (horizontal number line)

The charged particle model for multiplying integers

H1. Multiply and divide rational numbers

H2. Use strategies from multiplying and dividing fractions to

multiplying and dividing rational numbers

I1. Use the distributive property to multiply rational

numbers, for example: (-5)(-4) = (-1)(-1) + (5)(4)

I2. Interpret products of rational numbers by describing real

world contexts

J1. Divide integers such that if p and q are integers:

)(

)(

q

p

q

p

q

p.

J2. Interpret quotients of rational numbers by describing real

world contexts

K1. Recognize what makes a rational number a rational

number

K2. Convert a rational number to a decimal using long

division

L1. Solve real-world and mathematical problems involving the four operations with positive and negative rational numbers L2. Apply properties of operations to calculate with numbers in any form L3. Covert between different forms of rational numbers as appropriate L4. Assess the reasonableness of answers using mental calculations and estimation strategies

Terms/ Vocabulary Additive Inverse, Multiplicative Inverse, Absolute Value, Integer, Long Division, Natural Number, Negative Number, Opposite Number, Positive Number, Rational Number, Repeating Decimal, Terminating Decimal, Zero Pair, Distributive Property

Stage 2 – Assessment Evidence

Performance Task Initial Assessment: Diagnostic Assessment Final Assessment: Stock Market Savvy

Other Evidence Weekly assessments, projects, journal prompts, exit slips, etc

Stage 3 – Learning Plan

Impact Mathematics CCLS Aligned Lessons:

The following lessons will support some of the essential questions aligned in this unit map.

7.NS.1 Impact Lesson 3.1 – Add and Subtract with Negative Numbers pgs 126-152 (Omit pages 131-133) 7.NS.2 Impact Lesson 3.2 – Multiply and Divide with Negative Numbers pgs 154-168 Impact Lesson 7.2 – Irrational Numbers pgs 336-339 Impact Lesson 2.3 – Fraction and Decimal Equivalents (*In Course 1) pgs 94-97 7.NS.3 Impact Lesson 3.1 – Add and Subtract with Negative Numbers pg 142

Impact Lesson 3.2 – Multiply and Divide with Negative Numbers pgs 162-164, 166-168 7.NS.3 Sharing Price Money:www.illustrativemathematics.org Simplifying Expressions Using Order of Operations: http://www.uen.org/Lessonplan/preview.cgi?LPid=23376 www.nymathstandards.pbworks.com www.mathgoodies.com/lessons/vol5/challenge_vol.5.html Teaching Resources: Impact 7th Grade Teacher Guide-Chapter 3 Resources pg126B (Adding and subtracting of integers) Impact 7th Grade Teacher Guide-Chapter 3 Resources pg154B (Multiplication Division of integers) Impact 7th Grade Teacher Guide-Chapter7 Resources pg320B (Rational numbers) At the end of this unit-Sample activities

Task Resources:

https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_7_7thGrade_Unit1SE.

pdf

-What’s Your Sign? pg 14 -Helicopters and Submarines pg 17 - Hot Air Balloon pg 18 - Debits and Credits pg 21 -Multiplying Integers pg 24 -Multiplying Rational Numbers pg 27 -Patterns of Multiplication and Division pg 29 -The Repeater vs. The Terminator pg 33

SAMPLE ACTIVITIES: Models for Teaching Operations of Integers These models have been adapted from http://teachers.henrico.k12.va.us/math/hcpsalgebra1/. The following are some everyday events that can be used to help students develop a conceptual understanding of addition and subtraction of integers. • Getting rid of a negative is a positive. For example: Johnny used to cheat, fight and swear. Then he stopped cheating and fighting. Now he only has 1 negative trait so… (3 negative traits) - (2 negative traits) = (1 negative trait) or (-3) - (-2) = (-1) • Using a credit card example can make this subtraction concept clearer. If you have spent money you don't have (-5) and paid off part of it (+3), you still have a negative balance (-2) as a debt, or (-5) + 3 = (-2). • Draw a picture of a mountain, the shore (sea level) and the bottom of the ocean. Label sea level as 0. Any of the following models can be used to help students understand the process of adding or subtracting integers. If students have trouble understanding and using one model you can show students how to use another model. 1. The Charged Particles Model (same as using two-color counters)

When using charged particles to subtract, 3 – (-4) for example, you begin with a picture of 3 positive

particles.

Since there are no negative values to “take away”, you must use the Identity Property of Addition to rename positive 3 as 3 + 0. This is represented by 4 pairs of positive and negative particles that are equivalent to 4 zeros.

Now that there are negative particles, you can “take away” 4 negative particles.

The modeled problem shows that the result of subtracting 4 negative particles is actually like adding 4 positive particles. The result is 7 positive particles. This is a great way to show why 3 – (-4) = 3 + 4 = 7

Two-Color Counters Method

When using two-colored counters you would use the yellow side to represent positive integers and the red side to represent negative numbers. The problem represented is -3 – 5.

2. The Stack or Row Model

To model positive and negative integers, use colored linking cubes and graph paper. Graph paper and colored pencils will allow students to record problems and results. Students should also write the problems and answers numerically.

Create stacks or rows of numbers with the colored linking cubes and combine/compare the cubes. If the numbers have the same sign, then the cubes will be the same color. Stress that adding is like combining, so make a stack or row to show this.

(-3) + (-4) = (-7)

If the numbers are not the same sign (color), for example -3 + 5, you compare the stacks of different colors. Using the concept of zero pairs, the result

is the difference between the stacks or the result is based

on the higher stack. This is easy to see and understand.

For subtraction you create zeros by pairing one of each color. Then add as many zeros to the first number as needed so that you can take away what the problem calls for. Now physically take away the indicated amount and see what is left. The example problem shown is 3 – (-4).

3. The Hot Air Balloon Model

Sand bags (negative integers) and Hot Air bags (positive integers) can be used to illustrate operations with integers. Bags can be put on (added to) the balloon or taken off (subtracted).

Here is an example: -3 - (-4) = ? • The balloon starts at -3 (think of the balloon being 3 feet below sea level or 3 feet below the level of a canyon) and you take off 4 sand bags. • Now, think about what happens to a balloon if you remove sand bags, the balloon gets lighter. So, the balloon would go up 4 units.

If you think in terms of a vertical number line, it would start at -3 and end up at 1, so -3 - (-4) = 1. To help students make the connection between -3 - (-4) and -3 + (+4), present the addition and subtraction questions using the same numbers.

Another example would include the first addition question as 9 + (-5) and the first subtraction question would then be 9 - (+5). The students see that putting on 5 sand bags (negative) produces the same result as taking off 5 hot air bags (positive).

4. The Number Line Model

You can describe addition and subtraction of integers with a number line and a toy car. The car faces forward (to the right) to represent a positive direction. The car is moved forward to represent a positive integer. The car flips around backward (facing left) to represent a negative direction or subtraction. The car is moved backward (reverse) to represent a negative integer.

Example 1: 4 + 4 = 8

Example 2: 4 + (- 8) = -4

Example 3: 4 – (-4) = 8

5. Charged Particle Model for Multiplication

The charged particle method can be used to illustrate multiplication of integers.

To begin, a model with a 0 charge is illustrated. The 0 charge model will allow us to work with positive and negative integers.

Example 1: In this problem, 3 x (-2), three groups of two

negative charges is added to the 0 charged field. The result

is (-6).

Example 2: (-3) x (-2) = ?

For more resources of multiplication and division of integers, see pages 144 – 146 of Van de Walle, J., & Lovin, L.H. (2006). Teaching Student-centered Mathematics: Grades 5-8. Boston, MA: Pearson Education and Allyn & Bacon.

Grade 7 Unit 1: Initial Diagnostic Assessment

1. Model and solve the following expressions using the number line.

a. −2 + (−5) =

b. 4 + (−9) =

c. -6 – 2 =

d. -4 - (-1) =

2. Every month, Larry’s cell phone bill of $23.75 is deducted automatically from his checking account. How much money was deducted from September to December? Write an equation that represents this situation and then solve.

3. Mary owes her father $360 she borrowed to pay for a new bike. She is to pay him back over

12 months. Assuming that each month’s payment is the same, how much does she owe him

each month? Write an equation that represents this situation and solve.

4. Use the ledger to record the information and answer the questions. On May 5, your beginning balance is $8.00 o On May 6, you spent $4.38 on a gallon of ice cream at Marty’s Ice Cream Parlor. o On May 7, you spent $3.37 on crackers, a candy bar, and a coke from the corner store. o On May 8, you received $10 for cutting the neighbor’s grass. o On May 8, you spent $14.80 on a downloaded book for your Kindle.

Date Transaction Payment Deposit Balance

a. What is your balance after four transactions? Show all of your calculations.

b. Did you have enough to purchase the book for your Kindle? If not, how much money do you need to earn to have an account balance of $0?

5. Petra earned scores of 50

49,

60

54

,40

30and on her last three English quizzes.

a. Find each score as a decimal.

b. Arrange the decimal scores in order from least to greatest

6. A roller coaster begins at 4

390 feet above ground level. Then it ascends

8

115 feet.

a. Write an addition equation to find the height of the coaster after its first ascent. Then solve

it.

b. What subtraction equation could you use to find the height of the coaster after its first

ascent?

7. Paul and Karen went on a hot air balloon ride. The table shows the relationship between the change in altitude and the change in air temperature for 45 minutes of their flight.

Change in Altitude (Feet)

Change in Air Temperature (ºF)

0 0˚

500 -1.75˚

1,000 -3.5˚

-500 1.75˚

-1,000 3.5˚

a. What is the range in temperature during their flight? Show your calculation. b. What was the mean temperature change during Paul and Karen’s flight? Show your

calculations. c. What was the median temperature change?

7th Grade Unit 1: Initial Diagnostic Assessment

Scoring Guide

Diagnostic Assessment Rubric

Points Section

Points

1. Gives correct answer and has correct model on the number line

a. -7 b. -5 c. -8 d. -3

1

1

1

1

4

2. Gives the correct answer of $-95 or $95 deducted from his checking account

Gives Correct Equation: $-23.75 (4) = $-95

1

1

2

3. Gives correct answer of: Mary owes her father $30 a month.

Gives correct equation of: -360 ÷ 12 = -30

1

1

2

4. Fills out Ledger Correctly and Completely:

Date Transaction Payment Deposit Balance

5/6 Ice Cream $4.38 $8.00-$4.38 =

$3.62

5/7 Corner Store $3.37 $0.25

5/8 Cut Grass $10.00 $10.25

5/8 Kindle Book $14.80 $-4.55

a. Gives correct answer of $-4.55 Shows calculations:

- 22.55

+18.00

-4.55

10.00

+8.00

18.00

-4.38

-3.37

-14.80

-22.55

b. Gives correct answer of: No, I didn’t. I need another $4.55 to bring the balance to zero.

2

2

1

5

5. a. Gives correct equation with correct answer: 90 3/4 = 90 6/8 15 1/8 = 15 1/8 (fraction conversion)

90 6/8 + 15 1/8 = 105 7/8 feet (correct equation & answer)

b. Gives correct equation

X – 15 1/8 = 19 3/4

1

1

2

6. a. Givers correct answer with calculation shown Range = 3.5 – (-3.5) = 7 degrees

b. Gives correct answer with calculation shown

0 + (-1.75) + (-3.5) + 1.75 + 3.5 = 0

0 ÷ 5 = 0

Answer: Zero Degrees

1

1

3

c. Gives correct answer of: -3.5 degrees

1

7. a. Gives correct decimal for each score 30/40 = .75 54/60= .9 49/50= .98

b. Arranges the decimal in the correct order

.75, .90, .98

1

1

2

Total Points 20

Novice Apprentice Practitioner Expert

0-4 points 5-10 points 11-16 points 17-20 points

Grade 7 Unit 1 Final Assessment: Stock Market Savvy

Mr. Greene’s 7th grade class has been studying financial markets in their “math-economics” class. Each student in his class keeps a portfolio of stocks that they “buy” and “sell” according to their price in the actual stock market.

1. In December, 11 students had portfolios that increased in value and 18 students had portfolios that decreased in value. How many more students decreased than increased the value? Use a number line to model the situation, and then solve.

2. The Dow Jones Industrial Average closed at 13,233 points on Monday and closed at 13,145 points on Tuesday.

a. What was the change in points from Monday to Tuesday? Write an equation and solve.

b. What was the percentage change in the index from Monday to Tuesday? Show your work.

3. The prices of some popular stocks are listed below. Rewrite all three prices in the same format (fraction, decimal, or percent), and then place them on the number line diagram provided.

Stock Price ($)

Calculation Equivalent Form

Office Depot

2-3/8

Staples 225%

Office Max 2-7/14

2 3

4. Monica purchased 8 shares of Abercrombie and Fitch stock at $23.70. She wanted to buy

another stock so, she sold half the shares at $22.95 and the other half at $20.56. What was the result of these transactions? Did she make money or lose money? How much? Show all of your mathematical thinking in the space below.

5. Every month that he makes a profit on his stock portfolio, Jose withdraws 25% of his earnings

and puts it into his savings account. Below is a table of his earnings/losses:

Month Opening Balance of Portfolio

Closing Balance of Portfolio

September $500 $519

October $519 $604

November $604 $577

December $577 $599

a. In which month did he make the greatest profit? How much profit?

b. How much did Jose deposit into his savings account from September to December? Show all of your mathematical thinking in the space below:

6. The New York Times reports these changes in the price of Apple’s stock over four days: - 1/8,

-5/8, 3/8, and -9/8. What is the average daily change?

7th Grade Unit 1: Final Assessment

Stock Market Savvy Scoring Guide

Stock Market Savvy Rubric

Points Section

Points

1. Gives correct answer and shows correct model on the number line

7 more students decreased than increased

-7 0 11

+11 – 18 = -7

1

1

2

2. a. Gives correct equation with the correct answer

13,233 – 13,145 = 88

b. X 100 = 67%

2

1

3

3. Fills out chart correctly with either fractions, decimals or percents (1 point for each row)

Stock Price ($) Calculation Equivalent Form

Office

Depot

A

2 3/8 2 3/8 = 2 3/8 2 3/8

2.375

2.375%

Staples

B

225% 2.25 = 2 25/100

= 2 ¼

= 2 2/8

2 ¼

2.25

225%

Office Max

C

2 7/14 2 7/14 = 2 ½

2 ½ = 2 4/8

2 ½

2.50

250%

Correctly places prices on the number line and labels number line correctly

B A C

I I I I I I I I I

2 2-1/8 2-2/8 2-3/8 2 ½ 2-5/8 2-6/8 2-7/8 3

1

1

1

2

5

4. Shows all work and provides final answer

(-23.70) (80) = $-189.60

(22.95) (4) = $+91.80

(20.56) (4) = $+82.24

-189.60 + 174.04 = -15.56

1

1

1

1

4

5. a. Gives the correct month and amount of profit

He made the greatest profit in October

Her made $85.00

b. Gives correct answer of $31.50 and shows work

$19 + $85 + $22 = $126

126 (.25) = $31.50 or 126 ÷ 4 = $31.50

1

1

2

6. a. Givers correct answer of -3/8 with work shown -1/8 + (-5/8) + 3/8 + (-9/8)

-15/8 + 3/8 = -12/8

-12/8 ÷ 4 = -3/8

2

2

Total Points 18 18

Novice Apprentice Practitioner Expert

0-4 points 5-7 points 8-14 points 15-18 points