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Common Core Learning StandardsGRADE 7 MathematicsTHE NUMBER SYSTEM

Common Core Learning Standards Concepts Embedded Skills Vocabulary

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Additive Inverse

Solve problems where two quantities add to make a sum of 0 (additive inverse).

additive inverse

rational numbers

7.NS.1a. 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.

Describe real world situations where two quantities add to make a sum of zero.

SAMPLE TASKSI. Charlie owes his friend ten dollars. After one week he pays him seven dollars. After the second week he pays him 3 dollars. How much money does Charlie owe him after the second week?

II. During a football game, Freddie Jackson loses 6 yards on the first down and then gains four yards during the second down. Explain what Freddie Jackson needs to do on the third down to make his team be back where they were when they started.

Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Alignment Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.

III. Prove that -5 + 5 = 0.

IV. The following table shows the expenses and sales for a gift store.

Month Expenses SalesJanuary $150 $75

February $200 $150March $100 ?

What does the store need make in sales during March to break even? Explain your reasoning.

V. Kendrick adds 34 cup chicken stock to a pot. Then he takes

34 cup of chicken stock out of the pot. What is the overall increase or decrease

in the amount of chicken stock in the pot?




Sum of rational

numbers as a Distance (absolute

value)

Define the sum of two rational numbers as the distance one addend is away from the total by the absolute value of the other addend. Example: -4 + -3 = -7 …so the distance -4 from -7 is |-3| or left 3.

Distance Addend Sum rational

numbersDefine the direction of the distance on a number line based on the sign of the addend. Negative is left/down and positive is right/up.

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.

Define additive inverse as a rational number added to its negative which results in a sum of zero. Solve real world problems involving adding rational numbers.

SAMPLE TASKS

See BelowI. The temperature at 8am is -4⁰F. By 4pm the temperature had increased 9⁰F. Use the thermometer below to demonstrate what the temperature is at 4pm.


II. Joe has $428.56 in his bank account. He makes a withdrawal of $140 and a deposit of $46.71. What is Joe’s new balance?

III. Sarah and Lindsay are tugging on a rope that is 7 38 meters long. The rope breaks into two pieces. Sarah’s new piece is 3.75 meters. Who

has the longer piece? Show work to prove whose piece is longer.

IV. Jacob is trying to add -8 and -3. Use the number line to show the answer.


V. The table below shows the golf scores for a group of people after nine holes.

Round of 9-Hole GolfGolfer Jan Sergei Sue Yanni Raz Leon FelicityScore -6 6 3 -4 2 -3 -5

Which of the golfers have scores that are additive inverses? Explain your answer.




Difference of rational

numbers as adding the

additive inverse.

Compare subtracting rational numbers to adding the additive inverse. Example: -4 -( -3) = -4 + 3

additive inverse

absolute value distance

Prove that the distance between two rational numbers is equal to the absolute value of their difference. Example: the distance between -4 and -1 on a number line equals |-4 - (-1)| = 3

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.

Solve real world problems describing the skill above. Example: How many degrees cooler is -4°F to -1°F

SAMPLE TASKSI. The temperature in Buffalo is -4°F. It is 5°F colder in Grand Island. To find the temperature in Grand Island Kelly wants to do -4 – 5. Matthew wants to do -4 + (-5) to find Grand Island’s temperature. Whose process is correct, or will both work? Explain.


II. A cave explorer climbed from an elevation of -15 meters to an elevation of -8 meters.

Part A: What vertical distance did the explorer climb? Show your work.

Part B: Does it matter which way you subtract the values when finding the distance? Explain.

III. Allison ran 5 14 miles. Jimmy ran 3 23 miles. How much further did Allison run than Jimmy?


IV. Buffalo’s temperature is -15°F. Albany’s temperature is 5°F. How many degrees cooler is Buffalo? Justify your answer.




Properties of operations

with rational numbers

Apply commutative, associative, additive inverse, and distributive properties to solve addition and subtraction of rational numbers. Example: -3 + 9 = 9 + -3

commutative property

associative property

distributive property

additive inverse

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

SAMPLE TASKSI. Charles was given the following problem to complete:

−12

+ 38+12

Charles thinks it will be easier to add −12 and

12 first. Explain whether or not Charles is allowed to solve the problem this way.

II. A football player had a loss of 3 yards on his first play and a gain of 9 yards on his second play. To find his total yardage Leslie wants to add -3 + 9. Show another way to solve this problem.


III. Amanda is making two different cookie recipes. One recipe calls for 12 a cup of sugar. The other recipe calls for

34 of a cup of sugar.

Amanda wants to triple both of the recipes, so how much total sugar will she need?

IV. Alex and Greg were trying to solve the following problem: -5 + -3 + 3. Their work is shown below:

Which property allows both of their work to be correct?


Alex’s Work

(-5 + -3) + 3

-8 + 3

-5

Greg’s Work

-5 + (-3 + 3)

-5 + 0

-5



multiplication of rational numbers

Apply and extend the commutative, associative, and distributive property of multiplication from fractions to rational numbers. Focus on the distributive property.




rational numbers

fractions signed

numbers

Prove the rules for multiplying signed numbers by applying the distributive property.

7.NS.2a. 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.

Solve real-world problems involving signed numbers. Example: A car has 5 gallons of gas left. During our 5 day trip, we used 3 gallons per day. How much gas did we start out with?

SAMPLE TASKSI. Flora made 6 withdrawals of $24.50 each from her bank account. By how much did her account balance change?


II. Prove that the distributive property works by showing the work for the distributive property on the left hand side and comparing it to the answer on the right hand side.

Distributive Property

−13

(3+18)

Order of Operations

−13

(3+18)

−13

(21)

-7

III. Mrs. Smith gave the following problem to the class. Half the class solved the problem using the order of operations and half the class used the distributive property to solve the problem.

−12

(6−4)

Justify that either method will give you a correct answer.


IV. Greg has $80 left in his bank account. He made 6 withdrawals of $5 each. How much money did he start with in his account? Show work to support your answer.




Division of rational numbers

Define the quotient of two integers (divisor not = 0) as a rational number.

Division rational

numbers negative

symbol integer numerator denominator quotient divisor

Solve real word problems involving division of rational numbers.

7.NS.2b. 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.

Explain that a negative symbol can be written in the numerator, denominator, or next to the fraction without changing the value of the fraction.

SAMPLE TASKS

I. Which one of the following fractions is not equivalent to the other fractions? Explain.

−254

−254

−25−4 -6.25

25−4


II. A recipe calls for 2 34 cups of flour. You only have a 14 cup measuring cup. How many times will you need to fill the

14 measuring cup?

III. Divide five by four. Is your answer a rational number? Explain.

IV. A trail is 13.5 miles long. There are markers every 0.25 mile along the trail, including at the end of the trail. How many markers are there in all? Show your work.




Multiply and divide rational numbers by applying commutative, associative, and distributive properties.




rational numbers

7.NS.2c. Apply properties of operations as strategies to multiply and divide rational numbers.

SAMPLE TASKSI. Andrew is trying to solve the following problem: -0.5 • 6.8 • -2

Andrew’s first step is to rewrite the problem as shown below.

-0.5 • -2 • 6.8

Part A: Identify which property Andrew had in mind when he rewrote the problem.

Part B: Explain why applying this property makes the problem easier to solve.


II. Prove that the distributive property does NOT work for the following problem:

12(300÷50)

III. Write an equivalent expression for 54 ÷ 13 .

IV. Part A: Solve the following problem: (-0.5 • 6) • -4

Part B: Apply the associative property to the problem above and prove that the associative property works for multiplication.




fraction to decimal

conversion

Divide the numerator of a fraction by its denominator using long division.

long division rational

numbers fractions numerator denominator divide terminates repeats

Define a rational number as a decimal that terminates or eventually repeats.

7.NS.2d. 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.

SAMPLE TASKS

I. Use long division to prove that 45 is a rational number.

II. Sharon thinks that the decimal 0.3 is not a rational number. Explain to her why she is incorrect.


III. Part A: Put the following complex fraction in simplest form:

5649

.

Part B: Is your answer in Part A a rational number. Explain why or why not.

IV. Determine if 711 is a rational number. Explain your answer.




real world applications of rational numbers

Solve real world problems involving all four operations with rational numbers. (Keep in mind order of operations.)

add subtract multiply divide rational

numbers order of

operations

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

SAMPLE TASKSI. A group of golfers received the following scores after 9 holes of golf.

Round of 9-Hole GolfGolfer Jan Sergei Sue Yanni Raz Leon FelicityScore -6 6 3 -4 2 -3 -5

Part A: What is the average score for the group of golfers? Show your work.


Part B: Twice the sum of 4 and Yanni’s score is Bob’s score. What was Bob’s score? Show your work.

II. The temperature at 8 am was 5°F. It decreased 2°F per hour for the next 6 hours. What was the temperature at 2 pm?

III. Herbert’s checking account balance was $233. His account has overdraft protection if he withdraws more than his balance, but the bank charges $12 covering the overdraft. Herbert made the transactions shown in the table below:

Withdrawals Deposits$120 $75$120 $50.24$120 $18.01$200 $150.50

Part A: Does Herbert’s account require overdraft protection? Why or why not.


Part B: What is Herbert’s final balance? Show your work.

IV. A chef adds 12 cup of stock to a stew 5 times while it is cooking. During this time, he also removes

34 cup of stew 4 times to serve to

guests. Write and simplify an expression to find by how much the overall amount of stew increased or decreased. Show your work.


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