Operations with Integers

Transcription

1 Operations with Integers Integers and Absolute Values 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 as strategies to calculate with numbers in any form; convert between. CC.7.EE.3 Lesson objectives Teachers' notes 1

2 positive 4 ft 4 ft/sec 4 ft/sec 2

3 120 ft 120 ft/sec negative -120 ft/sec ft/sec and 16 ft/sec 3 The object with the velocity of 4 feet per second, because 4 > 3. 3

4 Speed tells how fast an object is moving. Velocity tells how fast it is moving and in what direction. DEFINITION: The number of units, or distance, between an integer and zero is called the absolute value. The absolute value of a number a is written as a. 10 units Answer How many units away from zero is -10? } } units Answer How many units away from zero is +10? 4

5 The absolute value of an integer is never negative because it asks, "How far away is it from zero?" Click on the arrow to help you decide if negative 15 and positive 15 are the same distance from zero Yes! No! Finding Absolute Value The absolute value of an integer is its distance from 0 on a number line. Use a number line to find each absolute value. 3) Touch above the number line for help. 4) 5

6 What is the absolute value of each number. Drag it through the tunnel and see! Match pairs of EQUIVALENT absolute values. 6

7 Adding Integers 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 as strategies to calculate with numbers in any form; convert between. CC.7.EE.3 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. CC.7.NS.1b Apply properties of operations as strategies to add and subtract rational numbers. CC.7.NS.1d Lesson objectives Teachers' notes 7

8 Modeling Integer Addition using a Number Line. Add. Use a number line. 1) Step 2 Step 1 Start at 0. Move left 7 spaces. Then move left 4 more spaces. 2) Step 2 Step 1 Start at 0. Move left 12 spaces. Then move right 19 spaces. 3) = 22 Step 1 Step 2 Start at 0. Move right 9 spaces. Then move right 13 spaces. 4) 2 + (-5) = -3 Plus a negative 5 Plus a negative 5 Step 2 Positive 2 Step Start at 0. Move right 2 spaces. Then move 5 spots in the negative direction. 5) 4 + (-4) = 0 negative 4 Step 1 Step 2 positive 4 Step 1 Step Start at 0. Move right 4 spaces. Then move 4 spots in the negative direction. 8

9 The Positive, the Negative, the SUM obj: to add positive and negative integers 1 Place 1 or 1 to represent the first number All 1 1 numbers added together represent 0. 3 Add or remove chips 1 1 as indicated by each expression to verify that they are equivalent The Positive, the Negative, the SUM Add positive and negative integers to make 0. 5) =

10 The Positive, the Negative, the SUM Add positive and negative integers to make 0. 6) = The Positive, the Negative, the SUM Add positive and negative integers to make 0. 7) =

11 Two numbers that are the same distance from 0, but on opposite sides of 0, are called opposites. For example, 3 and 3 are opposites. Adding Integers Using Absolute Values To add integers, first determine the sign of each number. If the signs are the same, then add the numbers and keep the sign. If the signs are different, then subtract the absolute value of the numbers and keep the sign of the number with the greatest absolute value. 9) The signs are different. Find the difference of the absolute values. Think: 8 4 = 4. Use the sign of the integer with the greater absolute value (positive). 10) 23 + (-35) -12 The signs are different. Find the difference of the absolute values. Think: = 12. Use the sign of the integer with the greater absolute value (positive). 11

12 Another way to think of it Lose $7 Lose $ = 16 Lose $4 Lose $ = 6 Lose $16 overall Lose $6 overall Win $5 Win $ = 8 Win $8 overall The list shows four bank account transactions in July. Find the change C in the account balance. 12

13 Subtracting Integers 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 as strategies to calculate with numbers in any form; convert between. CC.7.EE.3 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. CC.7.NS.1c Apply properties of operations as strategies to add and subtract rational numbers. CC.7.NS.1d Lesson objectives Teachers' notes 3-4 Subtracting Integers Subtracting Integers Add the Opposite 13

14 Modeling Integer Subtraction using a number line. Use a number line to find each difference. 1) 4-1 = 3 Step 2 Step 1 Start at 0. Move right 4 spaces. To subtract 1, move to the left. 2) -3-1 = -4 Step 2 Step 1 Start at 0. Move left 3 spaces. To subtract 1, move to the left. 3) -2 - (-4) = 2 Step 2 Step 1 Start at 0. Move left 2 spaces. To subtract -4, move to the right. 14

15 Using Tiles to Add & Subtract Integers We can use Algebra Tiles to represent positive and negative numbers. Yellow = Positive, Red = Negative + What do we get when we have one positive and one negative? + HINT What is 3 5? -8 + Need a Zero Pair? + 15

16 What is 6 ( 8)? 14 + Need a Zero Pair? + What is 4 7? Need a Zero Pair? + 16

17 What is 5 ( 1)? 6 + Need a Zero Pair? + 17

18 IN YOUR OWN WORDS How are adding integers and subtracting integers related? Write a general rule for subtracting integers. Answer The Rule is... Steps to solve: SAME CHANGE OPPOSITE 1. Leave the first number alone. 2. Change the minus sign to a plus 3. Use the OPPOSITE of the second number ( 20) 18

19 What's the Rule? 4-5 = (- 5) = (-3) = = = (-5) = (-3) = = 15 19

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21 3-5 Multiplying and Dividing Integers 21

22 Multiplying Integers 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 as strategies to calculate with numbers in any form; convert between. CC.7.EE.3 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. CC. 7.NS.2a Apply properties of operations as strategies to multiply and divide rational numbers. CC. 7.NS.2c Lesson objectives Teachers' notes Lesson 22

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24 Multiplying Integers by Skip Counting Find each product. Use a number line to model repeated addition and skip counting. 1) 2) 24

25 Check Point 25

26 The bar graph shows the number of taxis a company has in service. The number of taxis decreases by the same amount each year for four years. Find the total change in the number of taxis. The bar graph shows that the number of taxis in service decreases by 50 each year. Use a model to solve the problem. 26

27 Dividing Integers 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. CC.7.NS.2b Apply properties of operations as strategies to multiply and divide rational numbers. CC.7.NS.2c 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.) CC.7.NS.3 Lesson objectives Teachers' notes Use integer counters to find

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29 IN YOUR OWN WORDS Is the quotient of two integers positive, negative, or zero? How can you tell? Write general rules for dividing (a) two integers with the same sign and (b) two integers with different signs. 29

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31 The Coordinate Plane 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. CC.7.NS.1b Lesson objectives Teachers' notes 31

32 Coordinate Grids Coordinate grids are just 2 number lines put together

33 The x axis is horizontal the x axis Pull

34 The y axis is vertical the y axis Pull We can find any point on a coordinate plane using ordered pairs (also called coordinates). The point on the y axis always comes second. 5 4 ( 3,2) 3 2 The point on the x axis always comes first Pull

35 A coordinate plane is divided into 4 quadrants Quadrant II 2 Quadrant I Quadrant III 2 Click here to go to explore this further. 3 Pull Quadrant IV 4 5 Check out these websites for more practice plotting points on a coordinate plane. Locate the Aliens Billy Bug and His All New Quest for Grub 35

36 ( ( ( ( ( operations with Integers.notebook You Sank My Battleship! ( ( ( ( ( (x, y) Coordinates,,,,, Quadrant y axis y axis y axis y axis y axis x axis x axis x axis x axis x axis 1. Randomely choose x and y coordinates. 2. Slide a to the quadrant or axis where the point would lie. 3. Graph the point using a red dot y 5 36