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This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: www.njctl.org New Jersey Center for Teaching and Learning Progressive Mathematics Initiative

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7th Grade Math Number System www.njctl.org 2013-01-28

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Use Normal View for the Interactive Elements To use the interactive elements in this presentation, do not select the Slide Show view. Instead, select Normal view and follow these steps to set the view as large as possible: On the View menu, select Normal. Close the Slides tab on the left. In the upper right corner next to the Help button, click the ^ to minimize the ribbon at the top of the screen. On the View menu, confirm that Ruler is deselected. On the View tab, click Fit to Window. Use Slide Show View to Administer Assessment Items To administer the numbered assessment items in this presentation, use the Slide Show view. (See Slide 6 for an example.) Setting the PowerPoint View

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Number System Unit Topics Number System, Opposites & Absolute Value Comparing and Ordering Rational Numbers Adding Rational Numbers Turning Subtraction Into Addition Adding and Subtracting Rational Numbers Review Multiplying Rational Numbers Dividing Rational Numbers Operations with Rational Numbers Click on the topic to go to that section Converting Rational Numbers to Decimals Common Core Standards: 7.NS.1, 7.NS.2, 7.NS.3

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Number System, Opposites & Absolute Value Return to Table of Contents

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1 Do you know what an integer is? Yes No

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Number System 0.22 Natural 1,2,3... Whole 0 Integer...-4, -3, -2, -1 Rational 1/5 5/2 8.3 -2.756 -3/4 1/3 -1/11 Real Irrational

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{...-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7...} Definition of Integer: The set of whole numbers, their opposites and zero. Define Integer Examples of Integer: X

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Definition of Rational: A number that can be written as a simple fraction (Set of integers and decimals that repeat or terminate) Define Rational 0, -5, 8, 0.44, -0.23, Examples of rational numbers: 9, ½ X

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Definition of Irrational: A real number that cannot be written as a simple fraction Define Irrational Examples of irrational numbers: X

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integer rationalirrational Classify each number as specific as possible: Integer, Rational or Irrational 5 -6 0 -21 -65 1 3.2 -6.32 9 2.34437 x 10 3 ½ ¾ 3¾ π 5

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0 -2-3-4 -5 12 345 Rational Numbers on a Number Line Negative Numbers Positive Numbers Numbers to the left of zero are less than zero Numbers to the right of zero are greater than zero Zero is neither positive or negative Zero

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2Which of the following are examples of integers? A B C D E -5 0 -3.2 12 1 2

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3Which of the following are examples of rational numbers? A B C D E 1313 -3 10 0.25 75%

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Numbers In Our World

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You might hear "And the quarterback is sacked for a loss of 5 yards." This can be represented as an integer: -5 Or, "The total snow fall this year has been 6 inches more than normal." This can be represented as an integer: +6 or 6 Numbers can represent everyday situations

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1.Spending $6.75 2.Gain of 11 pounds 3.Depositing $700 4.10 degrees below zero 5.8 strokes under par (par = 0) 6. feet above sea level Write a number to represent each situation:

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4Which of the following numbers best represents the following scenario: The effect on your wallet when you spend $10.25. A B C D -10.25 10.25 0 +/- 10.25

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5Which of the following integers best represents the following scenario: Earning $50 shoveling snow. A B C D -50 50 0 +/- 50

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6 Which of the following numbers best represents the following scenario: You dive feet to explore a sunken ship. A B C D 0

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10234567 89 10 -2-3-4-5-6-7-8-9-10 The numbers -4 and 4 are shown on the number line. Both numbers are 4 units from 0, but 4 is to the right of 0 and -4 is to the left of zero. The numbers -4 and 4 are opposites. Opposites are two numbers which are the same distance from zero. Opposites

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7 What is the opposite of -7?

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8What is the opposite of 18.2?

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What happens when you add two opposites? Try it and see... A number and its opposite have a sum of zero. Numbers and their opposites are called additive inverses. Click to Reveal

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Integers are used in game shows. In the game of Jeopardy you: gain points for a correct response lose points for an incorrect response can have a positive or negative score Jeopardy

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When a contestant gets a $100 question correct: Score = $100 Then a $50 question incorrect: Score = $50 Then a $200 question incorrect: Score = -$150 How did the score become negative? Let's take a look...

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When a contestant gets a $100 question correct Then a $50 question incorrect Then a $200 question incorrect Question Answered Integer Representation New Score 100 Correct 50 Incorrect 200 Incorrect -50 100 50 -150 -200 Let's organize our thoughts...

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Question Answered Integer Representation New Score 150 Incorrect 50 Incorrect 200 Correct -50 -150 -200 0 200 When a contestant gets a $150 question incorrect Then a $50 question incorrect Then a $200 question correct Now you try...

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When a contestant gets a $50 question incorrect Then a $150 question correct Then a $200 question incorrect Question Answered Integer Representation New Score Now you try...

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9After the following 3 responses what would the contestants score be? $100 incorrect $200 correct $50 incorrect

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10After the following 3 responses what would the contestants score be? $200 correct $50 correct $300 incorrect

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11After the following 3 responses what would the contestants score be? $150 incorrect $50 correct $100 correct

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12After the following 3 responses what would the contestants score be? $50 incorrect $100 incorrect

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13After the following 3 responses what would the contestants score be? $200 correct $50 correct $100 incorrect

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An integer is a whole number, zero or its opposite. A rational number is a number that can be written as a simple fraction. An irrational number is a number that cannot be written as a simple fraction. Number lines have negative numbers to the left of zero and then positive numbers to the right. Zero is neither positive nor negative. Numbers can represent real life situations. To Review X

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Absolute Value of Numbers The absolute value is the distance a number is from zero on the number line, regardless of direction. Distance and absolute value are always non-negative (positive or zero). 10234567 89 10 -2-3-4-5-6-7-8-9-10 What is the distance from 0 to 5? 10234567 89 10 -2 -3-4-5-6-7-8-9 -10 What is the distance from 0 to -5?

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10234567 89 10 -2-3-4-5-6-7-8-9 -10 Absolute value is symbolized by two vertical bars |4||4| What is the | 4 | ? This is read, "the absolute value of 4"

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|-4| = 4 |-9| = 9 = 9.6 |9.6| Use the number line to find absolute value. 10234567 89 10 -2-3-4-5-6-7-8-9-10 Move to check Move to check Move to check

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14Find

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15Find |-8|

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16What is ?

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17What is ?

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18Find

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19What is the absolute value of the number shown in the generator? (Click for web site)

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A B C D E -30 -15 0 15 30 20Which numbers have 15 as their absolute value?

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A B C D E -100 -50 0 50 100 21 Which numbers have 100 as their absolute value?

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Comparing and Ordering Rational Numbers Return to Table of Contents

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To compare rational numbers, plot points on the number line. The numbers farther to the right are larger. The numbers farther to the left are smaller. Use the Number Line 1 0 2 3 4 5 6 7 8 9 10 -2-3-4-5-6-7-8-9 -10

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Place the number tiles in the correct places on the number line.

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Now, can you see: Which integer is largest? Which is smallest?

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Where do rational numbers go on the number line? Go to the board and write in the following numbers:

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Put these numbers on the number line. Which number is the largest? The smallest?

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Comparing Positive Numbers Numbers can be equal to; less than; or more than another number. The symbols that we use are: Equals "=" Less than " " For example: 4 = 4 4 2 When using, remember that the smaller side points at the smaller number.

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A B C = < > 22 10.5 is ______ 15.2.

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23 7.5 is ______ 7.5 A B C = < >

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24 3.2 is ______ 5.7 A B C = < >

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Comparing Negative Numbers The larger the absolute value of a negative number, the smaller the number. That's because it is farther from zero, but in the negative direction. For example: -4 = -4 -4 > -6 -4 < -2 1 0 2 3 4 5 6 7 8 9 10 -2-3-4 -5 -6 -7-8-9 -10 Remember, the number farther to the right on a number line is larger.

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Comparing Negative Numbers One way to think of this is in terms of money. You'd rather have $20 than $10. But you'd rather owe someone $10 than $20. Owing money can be thought of as having a negative amount of money, since you need to get that much money back just to get to zero. So owing $10 can be thought of as -$10.

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25 -4.75 ______ -4.75 A B C = < >

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1 0 2 3 4 5 6 7 8 9 10 -2-3-4 -5 -6 -7-8-9 -10 26 -4 ______ -5 A B C = < >

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27 A B C = < >

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28-14.75 is ______ -6.2 A B C = < >

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29-14.2 is ______ -14.3 A B C = < >

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Comparing All Numbers Any negative number is less than zero or any positive number. Any positive number is greater than zero or any negative number. 1 0 2 3 4 5 6 7 8 9 10 -2-3-4 -5 -6 -7-8-9 -10

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Drag the appropriate inequality symbol between the following pairs of numbers: 1)-3.2 5.8 3)63 36 5)-6.7 -3.9 7)-24 -17 9)-8.75 -8.25 2)-237 -259 4)-10.2 -15.4 6)127 172 8) 10) -10 -7 < >>>>>>>>>>>>>>>>>>>>< < < < < < < < < < < < < < < < < < < < <

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30 A B C = < >

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31 A B C = < >

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32 A B C = < >

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33 A B C = < >

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34 A B C = < >

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35 A B C = < >

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A thermometer can be thought of as a vertical number line. Positive numbers are above zero and negative numbers are below zero.

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36If the temperature reading on a thermometer is 10 ℃, what will the new reading be if the temperature: falls 3 degrees?

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37If the temperature reading on a thermometer is 10 ℃, what will the new reading be if the temperature: rises 5 degrees?

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38 If the temperature reading on a thermometer is 10 ℃, what will the new reading be if the temperature: falls 12 degrees?

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39If the temperature reading on a thermometer is -3 ℃, what will the new reading be if the temperature: falls 3 degrees?

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40If the temperature reading on a thermometer is -3 ℃, what will the new reading be if the temperature: rises 5 degrees?

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41If the temperature reading on a thermometer is -3 ℃, what will the new reading be if the temperature: falls 12 degrees?

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Adding Rational Numbers Return to Table of Contents

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Symbols We will use "+" to indicate addition and "-" for subtraction. Parentheses will also be used to show things more clearly. For instance, if we want to add -3 to 4 we will write: 4 + (-3), which is clearer than 4 + -3. Or if we want to subtract -4 from -5 we will write: -5 - (-4), which is clearer than -5 - -4.

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While this section is titled "Addition" we're going to learn here how to both add and subtract using the number line. Addition and subtraction are inverse operations (they have the opposite effect). If you add a number and then subtract the same number you haven't changed anything. Addition undoes subtraction, and vice versa. Addition: A walk on the number line.

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1.Start at zero 2.Walk the number of steps indicated by the first number. 3.Walk the number of steps given by the second number. 4.Look down, you're standing on the answer. Addition: A walk on the number line. Rules for directions Go to the right for positive numbers Go to the left for negative numbers Go in the opposite direction when subtracting, rather than adding, the second number Subtracting a negative number means you move to the right: since that's the opposite of moving to the left Here's how it works.

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Let's do 3 + 4 on the number line. 1.Start at zero 2.Walk the number of steps indicated by the first number. 3.Take the number of steps given by the second number. 4.Look down, you're standing on the answer.

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Let's do 3 + 4 on the number line. 1.Start at zero 2.Walk the number of steps indicated by the first number. 3.Take the number of steps given by the second number. 4.Look down, you're standing on the answer. Go to the right for positive numbers

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Let's do 3 + 4 on the number line. 1.Start at zero 2.Walk the number of steps indicated by the first number. 3.Take the number of steps given by the second number. 4.Look down, you're standing on the answer. Go to the right for positive numbers

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Let's do -4 + (-5) on the number line. 1.Start at zero 2.Walk the number of steps indicated by the first number. 3.Take the number of steps given by the second number. 4.Look down, you're standing on the answer.

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Go to the left for negative numbers Let's do -4 + (-5) on the number line. 1.Start at zero 2.Walk the number of steps indicated by the first number. 3.Take the number of steps given by the second number. 4.Look down, you're standing on the answer.

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Go to the left for negative numbers Let's do -4 + (-5) on the number line. 1.Start at zero 2.Walk the number of steps indicated by the first number. 3.Take the number of steps given by the second number. 4.Look down, you're standing on the answer.

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Let's do 5 + -7 on the number line. 1.Start at zero 2.Walk the number of steps indicated by the first number. 3.Take the number of steps given by the second number. 4.Look down, you're standing on the answer. 1 023 4 5 6 7 8 9 10 -2-3-4-5-6-7-8-9-10

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1 023 4 5 6 7 8 9 10 -2-3-4-5-6-7-8-9-10 Let's do 5 + -7 on the number line. 1.Start at zero 2.Walk the number of steps indicated by the first number. 3.Take the number of steps given by the second number. 4.Look down, you're standing on the answer. Go to the right for positive numbers

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1 023 4 5 6 7 8 9 10 -2-3-4-5-6-7-8-9-10 Let's do 5 + -7 on the number line. 1.Start at zero 2.Walk the number of steps indicated by the first number. 3.Take the number of steps given by the second number. 4.Look down, you're standing on the answer. Go to the left for negative numbers

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Let's do -4 + 9.5 on the number line. 1.Start at zero 2.Walk the number of steps indicated by the first number. 3.Take the number of steps given by the second number. 4.Look down, you're standing on the answer. 1 023 4 5 6 7 8 9 10 -2-3-4-5-6-7-8-9-10

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1 023 4 5 6 7 8 9 10 -2-3-4-5-6-7-8-9-10 Go to the left for negative numbers Let's do -4 + 9.5 on the number line. 1.Start at zero 2.Walk the number of steps indicated by the first number. 3.Take the number of steps given by the second number. 4.Look down, you're standing on the answer.

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1 023 4 5 6 7 8 9 10 -2-3-4-5-6-7-8-9-10 Let's do -4 + 9.5 on the number line. 1.Start at zero 2.Walk the number of steps indicated by the first number. 3.Take the number of steps given by the second number. 4.Look down, you're standing on the answer. Go to the right for positive numbers

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Addition: Using Absolute Values You can always add using the number line. But if we study our results, we can see how to get the same answers without having to draw the number line. We'll get the same answers, but more easily.

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1023456789 10 -2-3-4-5-6-7-8-9-10 10234567 8910 -2-3-4-5-6-7-8-9-10 10234567 8910 -2-3-4-5-6-7-8-9-10 3 + 4 = 7 -4 + 9.5 = 5.5 5 + (-7) = -2 -4 + (-5) = -9 102345678910-2-3-4-5-6-7-8-9-10 We can see some patterns here that allow us to create rules to get these answers without drawing. Addition: Using Absolute Values

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To add integers with the same sign 1. Add the absolute value of the integers. 2. The sign stays the same. (Same sign, find the sum) 3 + 4 = 7 3 + 4 = 7; both signs are positive; so 3 + 4 = 7 4 + 5 = 9; both signs are negative; so -4 + (-5) = -9 Addition: Using Absolute Values -4 + (-5) = -9 102345678910-2-3-4-5-6-7-8-9-10 10234567 8910 -2-3-4-5-6-7-8-9-10

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Interpreting the Absolute Value Approach The reason the absolute value approach works, if the signs of the integers are the same, is: The absolute value is the distance you travel in a direction, positive or negative. If both numbers have the same sign, the distances will add together, since they're both asking you to travel in the same direction. If you walk one mile west and then two miles west, you'll be three miles west of where you started.

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To add integers with different signs 1. Find the difference of the absolute values of the integers. 2. Keep the sign of the integer with the greater absolute value. (Different signs, find the difference) -4 + 9.5 = 5.5 9.5 - 4 = 5.5; 9.5 > 4, and 9.5 is positive; so -4 + 9.5 = 5.5 7 - 5 = 2; 7 > 5 and 9 is negative; so 5 + (-7) = -2 Addition: Using Absolute Values 1023456789 10 -2-3-4-5-6-7-8-9-10 10234567 8910 -2-3-4-5-6-7-8-9-10 5 + (-7) = -2

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Interpreting the Absolute Value Approach If the signs of the integers are the different: For the 2 nd leg of your trip you're traveling in the opposite direction of the 1 st leg, undoing some of your original travel. The total distance you are from your starting point will be the difference between the two distances. The sign of the answer must be the same as that of the larger number, since that's the direction you traveled farther. If you walk one mile west and then two miles east, you'll be one mile east of where you started.

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Adding Rational Numbers: To add integers with the same sign 1. Add the absolute value of the integers. 2. The sign stays the same. (Same sign, find the sum) To add integers with different signs 1. Find the difference of the absolute values of the integers. 2. Keep the sign of the integer with the greater absolute value. (Different signs, find the difference)

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4211 + (-4) =

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43-4 + (-4) =

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4417 + (-20) =

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45-15 + (-30) =

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46-5 + 10 =

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47-9 + (-2.3) =

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485.3 + (-8.4) =

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494.8 + (12.6) =

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50-14.3 + 7.93 =

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Turning Subtraction Into Addition Return to Table of Contents

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Subtracting Numbers Subtracting a number is the same as adding it's opposite. (Add a line, change the sign of the second number)

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Subtracting Numbers Subtracting a number is the same as adding it's opposite. We can see this from the number line, remembering our rules for directions. Compare these two problems: 8 - 5 and 8 + (-5). For "8 - 5" we move 8 steps to the right and then move 5 steps to the left, since the negative sign tells us to move in the opposite direction that we would for +5. For "8 + (-5)" we move 8 steps to the right, and then 5 steps to the left since we're adding -5. Either way, we end up at +3. 10234567 8910 -2-3-4-5-6-7-8-9-1010234567 8910 -2-3-4-5-6-7-8-9-10

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Subtracting Negative Numbers Compare these two problems: 8 - (-2) and 8 + 2. For "8 - (-2)" we move 8 steps to the right and then move 2 steps to the right, since the negative sign tells us to move in the opposite direction that we would for -2. For "8 + 2" we move 8 steps to the right, and then 2 steps to the right since we're adding 2. Either way, we end up at +10. 10234567 8910 -2-3-4-5-6-7-8-9-1010234567 8910 -2-3-4-5-6-7-8-9-10

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Subtracting Numbers Any subtraction can be turned into addition by: Changing the subtraction sign to addition. Changing the integer after the subtraction sign to its opposite. EXAMPLES: 5 - (-3) is the same as 5 + 3 -12 - 17 is the same as -12 + (-17)

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54 Convert the subtraction problem into an addition problem. 8 – 4 A B C D -8 + 4 8 + (-4) -8 + (-4) 8 + 4

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55Convert the subtraction problem into an addition problem. -3.7 - (-10.1) A B C D -3.7 + 10.1 3.7 + (-10.1) -3.7 + (-10.1) 3.7 + 10.1

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A B C D 56Convert the subtraction problem into an addition problem.

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57Convert the subtraction problem into an addition problem. A B C D

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58Convert the subtraction problem into an addition problem. 1 - 9 A B C D -1 + 9 1 + (-9) -1 + (-9) 1 + 9

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Adding and Subtracting Rational Numbers Review Return to Table of Contents

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59Calculate the sum or difference. -6 – 2

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60Calculate the sum or difference. 5 + (-5)

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61Calculate the sum or difference. -10.5 + 6.2

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62Calculate the sum or difference. 7.3 – (-4)

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63Calculate the sum or difference.

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64Calculate the sum or difference. 9.27 + (-8.38)

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65Calculate the sum or difference. -4.2 + (-5.9)

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66Calculate the sum or difference. -2 – (-3.95)

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67Calculate the sum or difference. 5 - 6 + (-7)

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68Calculate the sum or difference. 19 + (-12) - 11

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69Calculate the sum or difference. -2.3 + 4.1 + (-12.7)

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70Calculate the sum or difference. -8.3 - (-3.7) + 5.2

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71Calculate the sum or difference.

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Multiplying Rational Numbers Return to Table of Contents

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Symbols In the past, you may have used "x" to indicate multiplication. For example "3 times 4" would have been written as 3 x 4. However, that will be a problem in the future since the letter "x" is used in algebra as a variable. There are two ways we will indicate multiplication: 3 times 4 will be written as either 3∙4 or 3(4).

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Parentheses The second method of showing multiplication, 3(4), is to put the second number in parentheses. Parentheses have also been used for other purposes. When we want to add -3 to 4 we will write that as 4 + (-3), which is clearer than 4 + -3. Also, whatever operation is in parentheses is done first. The way to write that we want to subtract 4 from 6 and then divide by 2 would be (6 - 4) ÷ 2 = 1. Removing the parentheses would yield 6 - 4 ÷ 2 = 4, since we work from left to right.

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Multiplying Rational Numbers Multiplication is just a quick way of writing repeated additions. These are all equivalent: 3 ∙ 4 3 +3 + 3 + 3 4 + 4 + 4 they all equal 12.

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We know how to add with a number line. Let's just do the same thing with multiplication by just doing repeated addition. To do that, we'll start at zero and then just keep adding: either 3+3+3+3 or 4+4+4. We should get the same result either way, 12. Multiplying Rational Numbers

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10234567 89 10 -2-3 Let's do 4 x 3 on the number line. 11 13 121416 1715 We'll do it as 3+3+3+3 and as 4+4+4

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1 0 234567 89 10 -2-3 11 13 121416 17 15 Try 5 x 2 on the number line. Try it as 5+5 and as 2+2+2+2+2

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Multiplying Negative Numbers Let's use the same approach to determine rules for multiplying negative numbers. If we have 4 x (-3) we know we can think of that as (-3) added to itself 4 times. But we don't know how to think of adding 4 to itself -3 times, so let's just get our answer this way: 4 x (-3) = (-3)+(-3)+(-3)+(-3)

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-13 -14 -12-11-10-9-8-7 -6-5 -4 -15-16-17 -3 -202 31 4 x (-3) On the Number Line 4 x (-3) = (-3)(-3)(-3)(-3)(-3) So we can see that 4 x (-3) = -12

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4 ∙ 3 4 + 4 + 4 12 4(-3) (-3) + (-3) + (-3) -12 Multiplying positive numbers has a positive value. Multiplying a negative number and a positive number has a negative value. What about multiplying together two negative numbers: what is the sign of (-4)(-3) Sign Rules for Multiplying Rational Numbers ?

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Multiplying Negative Numbers We can't add something to itself a negative number of time; we don't know what that means. But we can think of our rule from earlier, where a (-) sign tells us to reverse direction. So if we think of (-4)(-3) as -(4)(-3) we can then see that the answer will be the opposite of (-12):12 Each negative sign makes us reverse direction once, so two multiplied together gets us back to the positive direction.

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4 ∙ 3 4 + 4 + 4 12 4(-3) (-4) + (-4) + (-4) -12 Multiplying positive numbers yields a positive result. Multiplying a negative number and a positive number yields a negative result. Multiplying two negative numbers together yields a positive result. Sign Rules for Multiplying Rational Numbers (-4)(-3) -((-4) + (-4) + (-4)) -(-12) 12

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Every time you multiply by a negative number you change the sign. Multiplying with one negative number makes the answer negative. Multiplying with a second negative change the answer back to positive. 1(-3) = -3 -3(-4) = 12 Multiplying Rational Numbers

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When multiplying two numbers with the same sign (+ or -), the product is positive. When multiplying two numbers with different signs, the product is negative. When multiplying several numbers with different signs, count the number of negatives. An even amount of negatives = positive product An odd amount of negatives = negative product Multiplying Rational Numbers

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We can also see these rules when we look at the patterns below: 3(3) = 9-5(3) = -15 3(2) = 6-5(2) = -10 3(1) = 3-5(1) = -5 3(0) = 0-5(0) = 0 3(-1) = -3 -5(-1) = 5 3(-2) = -6 -5(-2) = 10 3(-3) = -9 -5(-3) = 15 2.5(3) = 7.5 -3.1(3)(-2) = 18.3 2.5(2) = 5 -3.1(2)(-2) = 12.4 2.5(1) = 2.5 -3.1(1)(-2) = 6.2 2.5(0) = 0 -3.1(0)(-2) = 0 2.5(-1) = -2.5 -3.1(-1)(-2) = -6.2 2.5(-2) = -5 -3.1(-2)(-2) = -12.4 2.5(-3) = -7.5 -3.1(-3)(-2) = -18.3 Multiplying Rational Numbers

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72What is the value of (-3)(-9)?

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73What is the value of 3.1 ∙ 7?

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74What is the value of 5(-4.82)?

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75What is the value of (-3.2)(-6.4)?

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76What is the value of -8 ∙ 7.6?

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77What is the value of (-5.12)(-9)?

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78What is the value of -2(-7)(-4)?

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79What is the value of:

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80What is the value of:

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81What is the value of:

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82What is the value of:

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Dividing Rational Numbers Return to Table of Contents

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Division Symbols You may have mostly used the "÷" symbol to show division. We will also represent division as a fraction. Remember: 99 ÷ 3 = 3 3 are both ways to show division. = 3

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Dividing Rational Numbers Division is the opposite of multiplication, just like subtraction is the opposite of addition. When you divide a number, by another number, you are finding out how many of that second number would have to add together to get the first number. For instance, since 5∙2 = 10, that means that I could divide 10 into 5 groups of 2's, or 2 groups of 5's. This is just what we did on the number line for multiplication, but backwards. Let's try 10 ÷ 2

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Try 10 ÷ 2 on the number line This means how many lengths of 2 would be needed to add up to 10. The answer is 5: the number of red arrows of length 2 that end to end give a total length of 10. 1 0 234567 89 10 -2-3 11 13 121416 1715

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1 0 234567 89 10 -2-3 11 13 121416 1715 Try 10 ÷ 5 on the number line This means how many lengths of 5 would be needed to add up to 10. The answer is 2: the number of green arrows of length 5 that, end to end, give a total length of 10.

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-12 ÷ 3 On the Number Line This can be read as how many steps of 3 would it take to get to -12. Each red arrow represents a step of 3, so we can see that -12 ÷ 3 = -4 (The answer is negative because the steps are to the left.) -13 -14 -12-11-10-9-8-7 -6-5 -4 -15-16-17 -3 -202 31

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-15 ÷ 3 = -5 We know that -5(3) = -15, so it makes sense that -15 ÷ 3 = -5. We also know 3(-5) = -15. So, what is the value of -15 ÷ -5 The value must be positive 3, because 3(-5) = -15 -15 3 = -5 Dividing Rational Numbers

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The quotient of two positive numbers is positive. The quotient of a positive and negative number is negative. The quotient of two negative numbers is positive. When dividing several numbers with different signs, count the number of negatives. An even amount of negatives = positive quotient An odd amount of negatives = negative quotient Dividing Rational Numbers

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83Find the value of 32 ÷ 4

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84 Find the value of:

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85 Find the value of:

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86Find the value of:

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87Find the value of:

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88Find the value of:

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89Find the value of:

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93Find the value of:

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Operations with Rational Numbers Return to Table of Contents

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When simplifying expressions with rational numbers, you must follow the order of operations while remembering your rules for positive and negative numbers!

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Order of Operations Parentheses Exponents Multiplication Division Addition Subtraction Complete at the same time...whichever comes first...from left to right (ALL Grouping Symbols)

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Let's simplify this step by step... What should you do first? 5 - (-2) = 5 + 2 = 7 What should you do next? (-3)(7) = -21 What is your last step? -7 + (-21) = -28 -7 + (-3)[5 - (-2)]

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Let's simplify this step by step... What should you do first? What should you do second? What should you do third?What should you do last? Click to Reveal Click to Reveal Click to Reveal Click to Reveal

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-12÷3(-4) 94Simplify the expression.

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95 [-1 - (-5)] + [7(3 - 8)]

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96 40 - (-5)(-9)(2)

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975.8 - 6.3 + 2.5

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-3(-4.7)(5-3.2) Simplify the expression.98

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99

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100

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[3.2 + (-15.6)] - 6[4.1 - (-5.3)] 101Complete the first step of simplifying. What is your answer?

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[3.2 + (-15.6)] - 6[4.1 - (-5.3)] 102 Complete the next step of simplifying. What is your answer?

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[3.2 + (-15.6)] - 6[4.1 - (-5.3)] 103 Complete the next step of simplifying. What is your answer?

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[3.2 + (-15.6)] - 6[4.1 - (-5.3)] 104 Complete the next step of simplifying. What is your answer?

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105 Simplify the expression.

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106Simplify the expression.

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107

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(-4.75)(3) - (-8.3) 109

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Solve this one in your groups.

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How about this one?

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110

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111 [(-3.2)(2) + (-5)(4)][4.5 + (-1.2)]

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Converting Rational Numbers to Decimals Return to Table of Contents

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Definition of Rational: A number that can be written as a simple fraction (Set of integers and decimals that repeat or terminate) In order for a number to be rational, you should be able to divide the fraction and have the decimal either terminate or repeat. Do you recall the definition of a Rational Number? x

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Use long division! Divide the numerator by the denominator. If the decimal terminates or repeats, then you have a rational number. If the decimal continues forever, then you have an irrational number. How can you convert Rational Numbers into Decimals?

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Long Division Review

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116 Convert to a decimal (if the number is repeating, use bar notation in your notebook but enter the repeating number(s) 3 times on your responder)

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117 Convert to a decimal (if the number is repeating, use bar notation in your notebook but enter the repeating number(s) 3 times on your responder)

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118 Convert to a decimal (if the number is repeating, use bar notation in your notebook but enter the repeating number(s) 3 times on your responder)

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119 Convert to a decimal (if the number is repeating, use bar notation in your notebook but enter the repeating number(s) 3 times on your responder)

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120 Convert to a decimal (if the number is repeating, use bar notation in your notebook but enter the repeating number(s) 3 times on your responder)

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121 Convert to a decimal (if the number is repeating, use bar notation in your notebook but enter the repeating number(s) 3 times on your responder)

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122 Convert to a decimal (if the number is repeating, use bar notation in your notebook but enter the repeating number(s) 3 times on your responder)

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123 Convert to a decimal (if the number is repeating, use bar notation in your notebook but enter the repeating number(s) 3 times on your responder)

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124 Convert to a decimal (if the number is repeating, use bar notation in your notebook but enter the repeating number(s) 3 times on your responder)

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125 Convert to a decimal (if the number is repeating, use bar notation in your notebook but enter the repeating number(s) 3 times on your responder)

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126 Convert to a decimal (if the number is repeating, use bar notation in your notebook but enter the repeating number(s) 3 times on your responder)

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127 Convert to a decimal (if the number is repeating, use bar notation in your notebook but enter the repeating number(s) 3 times on your responder)

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