Adding Integers on a Number Line

Transcription

1 NS7-5 Adding Integers on a Number Line Pages Standards: 7.NS.A.1b Goals: Students will add integers using a number line. Students will interpret addition of integers when the integers are written in brackets. Prior Knowledge Required: Can represent integers on a number line Can add whole numbers on a number line Vocabulary: integer, negative, positive Materials: pre-cut arrows (made from Bristol board, see below for details) BLM Number Lines from 6 to 6, 2 copies for each student (p. C-85) Preparation. Using Bristol board, cut out eight arrows, one of each length, in 6-inch intervals: 6 inches, 12 inches, 18 inches, and 24 inches. Label the arrows, front and back, with the positive integers from 1 to 4 showing arrows pointing right and the negative integers 1 to 4 showing arrows pointing left. Keep them for a later lesson (NS7-11: Adding and Subtracting Fractions with the Same Denominator) as well Using arrows to represent integers on number lines. Draw on the board a number line from 4 to 4 with the numbers 6 inches apart, and show students how to represent 3 by starting the 3 arrow at SAY: We can use the arrows starting at 0 to show the number. An arrow 3 units long will point at 3 if it points right. ASK: Where will the same arrow point if it points left? (at 3) Turn the arrow around to show this (your arrow should have 3 written on the back). Ask volunteers to show other integers using the arrows: 2, + 4, 1. C-20 Teacher s Guide for AP Book 7.1 Unit 2 The Number System

2 Using arrows to add integers on number lines. Write on the board: (+3) + ( 4) Using the same number line as above, demonstrate how to use arrows to add 3 and 4. SAY: You have to start at 0 and find where you end up after adding both numbers. So, after adding the 3 arrow, you start the 4 arrow where the 3 arrow finishes. Show this on the board: So (+3) + ( 4) = ASK: So what is positive three plus negative four? (negative one) Have volunteers use the arrows and the same number line to add other pairs of numbers: ( 2) + (+3) (+1) + ( 3) ( 2) + ( 1) (+3) + (+1) (+1, 2, 3, + 4) Now provide students with BLM Number Lines from 6 to 6 to do the exercises below (give each student two copies double-sided). Point out that when using a number line to add, students don t need to draw the arrows on the number line because the fact that the number line can continue isn t needed to solve the problem. Exercises: Add using a number line. a) ( 4) + (+1) b) (+3) + ( 6) c) ( 4) + ( 2) d) ( 3) + (+2) e) (+2) + ( 3) Bonus: ( 5) + (+3) + ( 2) Answers: a) 3, b) 3, c) 6, d) 1, e) 1, Bonus: 4 (MP.1) Tell students that in parts d) and e) they are adding the same two numbers, so they should get the same answers. Bonus: Use other pairs of numbers to check that adding the same numbers gets the same answer, no matter which order you add them in. SAY: You can also add integers by thinking of them as gains and losses. (MP.1) Exercise: Do the previous exercises without using a number line. Make sure you get all the same answers. Teacher s Guide for AP Book 7.1 Unit 2 The Number System C-21

3 (MP.7) A shortcut way to add on a number line. Tell students that there is a shortcut way to add on a number line. ASK: What can you do instead of drawing the first arrow? (rather than the first arrow starting at 0 and ending at the first number, you can just start at the first number and draw only the arrow for the second number) Exercises: Use a number line to add. Draw only one arrow. a) ( 3) + (+5) b) (+4) + ( 5) c) ( 5) + ( 1) Bonus: Draw only two arrows to add ( 3) + (+7) + ( 5). Answers: a) +2, b) 1, c) 6, Bonus: 1 A different notation is sometimes used for adding integers. Tell students that they can add integers by writing the integers without brackets, the same as when adding gains and losses. Write on the board: (+7) + ( 4) = SAY: Adding +7 is just like adding a gain of 7, and adding 4 is just like adding a loss of 4. It s easy to change the notation to not have brackets. Write on the board: +(+) = + +( ) = Exercises: Write the addition without brackets. a) ( 3) + (+4) b) (+5) + ( 6) c) (+5) + (+1) d) ( 3) + ( 4) Answers: a) 3 + 4, b) + 5 6, c) , d) 3 4 Students will need BLM Number Lines from 6 to 6 for the exercises below. (MP.1) Exercises: Add the integers in two ways by thinking of them as gains and losses, then by using a number line. Make sure you get the same answer both times. a) ( 5) + (+3) b) (+6) + ( 3) c) ( 4) + (+6) d) ( 2) + ( 3) Answers: a) 2, b) +3, c) +2, d) 5 Extensions 1. a) Extend the pattern using BLM Number Lines from 6 to 6. i) +4, +3, +2,,, ii) +5, +3, +1,,, iii) 6, 3, 0,, Answers: i) +1, 0, 1; ii) 1, 3, 5; iii) +3, +6 C-22 Teacher s Guide for AP Book 7.1 Unit 2 The Number System

4 b) Use the pattern to add. i) (+5) + (+3) +8 ii) (+3) + ( 7) (+5) + (+2) +7 (+2) + ( 7) (+5) + (+1) +6 (+1) + ( 7) (+5) + (0) +5 (0) + ( 7) (+5) + ( 1) + ( 7) ( 7) ( 7) Answers: i) (+5) + (+3) +8 ii) (+3) + ( 7) 4 (+5) + (+2) +7 1 (+2) + ( 7) 5 (+5) + (+1) +6 1 (+1) + ( 7) 6 (+5) + (0) +5 1 (0) + ( 7) 7 (+5) + ( 1) +4 1 _( 1)_ + ( 7) _( 2) _( 2)_ + ( 7) _( 3)_ +2 _( 3)_ + ( 7) Solve the puzzle by placing the same integer in each shape. a) + + = 6 b) + + = 30 Answers: a) 2, b) 10 (MP.1) 3. In the square, the integers in each row and column and the two diagonals (these include the center box) add up to +6. Find the missing integers Answers: Teacher s Guide for AP Book 7.1 Unit 2 The Number System C-23

5 NS7-6 Using Pictures to Subtract Integers Pages Standards: 7.NS.A.1c, 7.NS.A.1d Goals: Students will use pictures to subtract integers. Prior Knowledge Required: Can use pictures to represent integers Vocabulary: positive, negative, integer Materials: 10 + cards and 10 cards BLM Number Lines from 6 to 6 (p. C-85, see Extension 2) Preparation. Create 10 cards with a + sign in a circle, and 10 cards with a sign in a circle, that you can tape to the board and remove easily. Tell students that the positive sign in a circle can represent a gain of $1 or a positive charge. The negative sign in a circle can represent a loss of $1 or a negative charge. Using pictures to subtract positives from positives and negatives from negatives. Tape to the board five + cards. ASK: What integer does this represent? (+5) SAY: In the same way you can add integers by adding positives and negatives, you can subtract integers by taking some away. Now take away two of the cards and ASK: Now what integer does this represent? (+3) Write on the board: (+5) (+2) = +3 Then repeat with five cards, again taking away two cards. Have a volunteer write the equation on the board: ( 5) ( 2) = 3 Then draw on the board: For each picture, ASK: What integer does this picture represent? (the first picture represents +3 and the second picture represents 4) Then show this on the board: C-24 Teacher s Guide for AP Book 7.1 Unit 2 The Number System

6 ASK: When I take away +2 from +3, what do I get? (+1) Have a volunteer write the equation under the picture: (+3) (+2) = +1 When I take away 1 from 4, what am I left with? ( 3) Have a volunteer write the equation under the picture: ( 4) ( 1) = 3 SAY: When you subtract a number that has a positive or negative sign in front, you always have to put brackets around the number you are subtracting so that you don t have to write the subtracting symbol right next to the positive or negative sign. But you don t always have to put brackets around the first number. Write on the board: 4 ( 1) = 3 Exercises: Write a subtraction for the picture. a) b) Answers: a) 5 3 = 3 or +5 (+3) = +2, b) 6 ( 2) = 4 Representing an integer with both positives and negatives. Draw on the board: ASK: What number does this represent? ( 3) Then add a positive and a negative as shown below and ASK: Now what number is represented? (still 3) SAY: The positive and negative that I added cancel each other (circle them), so we are back where we started. Exercises: What number is represented? Hint: Circle amounts that cancel. What s left? a) b) c) Answers: a) +3, b) +2, c) 3 Teacher s Guide for AP Book 7.1 Unit 2 The Number System C-25

7 Subtracting a negative number from a positive number. Write on the board: +3 ( 1) SAY: I want to take away one negative, but there aren t any negatives to subtract. ASK: What can I do? Have students work in pairs to try to find a solution. As students work, provide guidance as needed. For example, you can PROMPT: How can I add a negative and keep the value represented the same? (add a positive, too) Show doing so on the board. Then SAY: Now I can take away 1. Cross out the minus sign. ASK: I started with +3 and took away 1. What s left? (+4) Now write on the board: +3 ( 1) = ( 3) = Ask a volunteer to draw a picture (or use the cards) of +2 that has 3 negatives. ASK: If I remove the three negatives, what s left? (+5) Write that answer on the board. Exercises: a) Draw a picture of +5 that has i) 1 negative ii) 2 negatives iii) 3 negatives iv) 4 negatives b) Use your pictures from part a) to subtract. i) +5 ( 1) ii) +5 ( 2) iii) +5 ( 3) iv) +5 ( 4) (MP.8) c) Predict +5 ( 5). Answers: a) i) ii) iii) iv) b) i) 6, ii) 7, iii) 8, iv) 9 c) 10 Have volunteers explain their prediction for part c). (all the numbers are becoming 1 bigger) C-26 Teacher s Guide for AP Book 7.1 Unit 2 The Number System

8 Exercises: Add enough positives and negatives to the picture of the first number so that you can subtract the second number. a) +3 ( 1) b) +1 ( 2) c) +2 ( 2) d) +1 ( 4) Answers: a) 4, b) 3, c) 4, d) 5 Subtracting a positive number from a negative number. Write on the board: 2 (+3) SAY: I have to add 3 positives to make enough to subtract. ASK: What else do I need to add if I want to keep the values the same? (3 negatives) Demonstrate doing so: 2 (+3) Have a volunteer do the subtraction by removing the 3 positives (if you used cards, the volunteer can literally remove them): 2 (+3) = 5 Exercises: Draw a picture of the first number so that you can subtract the second number. Then subtract. a) 2 (+1) b) 1 (+3) c) 4 (+2) d) 3 (+2) (MP.1, MP.3) Bonus: How do your answers to parts c) and d) compare? Why does this make sense? Answers: a) 3, b) 4, c) 6, d) 5, Bonus: the answer to part d) is one more that the answer to part c) because 3 is one more than 4 Subtracting when there is not enough to subtract more cases. Write on the board: 2 ( 3) Teacher s Guide for AP Book 7.1 Unit 2 The Number System C-27

9 ASK: Are there enough negatives to subtract? (no) I already have two negatives: how many more do I need? (1 more) If I add one negative, what else do I need to add? (one positive) Ask a volunteer to do that and another volunteer to do the subtraction: 2 ( 3) 2 ( 3) 2 ( 3) 2 ( 3) = +1 Repeat with more examples for volunteers to solve: + 2 (+4), 1 ( 3), + 2 (+5). ( 2, 2, 3) Exercises: Change the picture of the first number so you can subtract the second number. Then subtract. a) 1 (+2) b) +2 ( 4) c) 3 ( 4) d) +1 (+3) Bonus: How do the answers to parts b) and c) compare? Why does this make sense? Answers: a) 3, b) +6, c) +1, d) 2, Bonus: the answer to part c) is 5 less than the answer to part b) because 3 is 5 less than +2 SAY: Now you need to draw the picture of the first number yourself. Add enough positives and negatives to it so you can subtract the second number. Exercises: Subtract. a) +3 (+5) b) ( 1) (+2) c) 2 ( 6) d) (+3) ( 3) Answers: a) 2, b) 3, c) +4, d) +6 Extensions 1. a) Complete the picture to show 3. i) ii) iii) b) Use the pictures you made in part a) to subtract. i) 3 (+1) ii) 3 (+2) iii) 3 (+3) Answers: a) i) ii) iii) b) Removing the positives results in i) 4, ii) 5, iii) 6 NOTE: After students finish Extension 1, point out that another way of looking at what they did is to solve missing addend problems. i) +1 + = 3 ii) +2 + = 3 iii) +3 + = 3 For example, they had to add 4 negatives to +1 to get 3. So, for integers it s the same as for whole numbers subtracting is the same as finding the missing addend. C-28 Teacher s Guide for AP Book 7.1 Unit 2 The Number System

10 (MP.4) 2. Give students a copy of BLM Number Lines from 6 to 6. Demonstrate how they can use the BLM to determine the time in one city when given the time in another city in a different time zone. For example, tell students that it is 3:00 p.m. in the +4 time zone. Tell students that you want to determine the time in the 5 time zone. (p.m.) Point out that it is earlier to the left on the number line and later to the right on the number line. Together, fill in the number line to the left to find the time in the 5 time zone. (a.m.) (p.m.) So, it is 6 a.m. in the 5 time zone. Now display the time zones below. Time Zones Muscat, Oman +4 Helsinki, Finland +2 Rome, Italy +1 New York City, USA 5 Chicago, USA 6 a) A sporting event is being held in Rome at 2 p.m. Abdul lives in Muscat. At what time should he turn on his television to watch the event live? b) An acting awards ceremony is being held in New York City at 7:30 p.m. Alexa lives in Helsinki. At what time should she turn on her television to watch the event live? c) Jin lives in Chicago. His friend Clara lives in Helsinki. Jin wants to call Clara when he gets home from school at 4 p.m. Clara goes to bed at 10:30 p.m. Will she get the call before she goes to bed? Answers: a) 5 p.m.; b) 2:30 a.m.; c) no, it will be 12:00 midnight where Clara lives when Jin calls Teacher s Guide for AP Book 7.1 Unit 2 The Number System C-29

11 NS7-7 Subtracting Integers on a Number Line Pages Standards: 7.NS.A.1c Goals: Students will use a number line to subtract integers. Students will recognize that the distance between two integers is the absolute value of their difference. Prior Knowledge Required: Knows how to subtract by counting up Vocabulary: absolute value, integer, negative, positive Materials: BLM Subtraction on a Number Line (p. C-86) BLM Number Lines from 6 to 6 (p. C-85) Review using a number line to subtract two ways. Write on the board: ASK: How would you use the number line to subtract 5 3? Have a volunteer show their work. (most likely, students will start at 5 and move left 3 spaces to reach 2) Then let another volunteer show a different way. If no one suggests it, remind students that they can count up from 3 to 5. SAY: You can ask: How can you get from 3 to 5 on the number line? You need to move 2 units right, so the answer is +2. Show this on the board as follows: 5 3 = SAY: In this case, the answer isn t where you end up. This time, the answer is how you get from 3 to 5. Now write on the board: 3 5. Have two volunteers each show a different way to solve this on a number line. Make sure that both solutions come up. C-30 Teacher s Guide for AP Book 7.1 Unit 2 The Number System

12 Solution 1: Start at 3 and move left 5 spaces to reach Solution 2: How do you get from 5 to 3? Move two places left, so 3 5 = 2 If needed, you can PROMPT: To subtract 5 3, you asked how you can get from 3 to 5; what question would you ask to subtract 3 5? (How can you get from 5 to 3?) 3 5 = SAY: Both ways of subtracting 3 5 are correct and they both get the answer 2. In this lesson, we will use the second way to subtract integers. The answer isn t where you end up on the number line, but how you get from the second number to the first number. You have to start at the second number and ask yourself: How do I get to the first number? If you have to move right, the answer is positive, because you re subtracting a smaller number from a bigger number. Use the 5 3 example to illustrate. Then SAY: But if you have to move left, then you re subtracting a bigger number from a smaller number, so the answer is negative. Use the 3 5 example to illustrate. Give students BLM Subtraction on a Number Line. Exercises: Do Question 1 on the BLM. Answers: a) +4, b) 6, c) 1, d) +3 SAY: Where the arrow ends is where you start the subtraction. Exercises: Do Question 2 on the BLM. Answers: a) 1 ( 5) = +4, b) 2 ( 1) = +3, c) 5 ( 4) = 1, d) 1 (+5) = 6 Students will need a copy of BLM Number Lines from 6 to 6 to do the exercises below. SAY: Where you start the subtraction is where the arrow ends. Exercises: Draw the arrow that shows the answer. For parts e) to h), do the subtraction. a) +4 ( 1) = +5 b) 3 (+2) = 5 c) 6 (+3) = 9 d) 2 ( 5) = +3 e) 2 (+4) = f) 5 ( 2) = g) +1 ( 1) = h) +3 (+6) = Answers: a) from 1 to +4, b) from +2 to 3, c) from +3 to 6, d) from 5 to 2, e) from +4 to 2, 6; f) from 2 to 5, 3; g) from 1 to +1, +2, h) from +6 to +3, 3 Teacher s Guide for AP Book 7.1 Unit 2 The Number System C-31

13 Subtracting the same numbers in opposite orders gets opposite answers. Draw on the board: 5 2 = = (MP.1, MP.2) Have volunteers draw the arrows for both subtractions, then give the answers. ASK: How are the arrows the same? (they have the same length) How are the arrows different? (the directions are different) How are the answers the same? (the number part is the same) How are the answers different? (the sign is different) Point out that the length of the arrow, or how far apart the numbers are, tells you the number part of the answer and the direction tells you the sign. SAY: The distance between the integers, or how far apart they are, is always a positive number. But the subtraction is positive only if you subtract a smaller number from a bigger number. Exercises: Which subtraction tells you how far apart the numbers are? a) 3 5 or 5 3 b) +2 ( 1) or 1 (+2) c) 3 ( 5) or 5 ( 3) Answers: a) 5 3, b) +2 ( 1), c) 3 ( 5) Introduce absolute value notation. Remind students that the absolute value of a number is its distance from 0. Tell students that because absolute value is used a lot, mathematicians have created a notation for it so that you don t have to keep writing out the words. Write on the board: 3 = 3 Read the equation as: The absolute value of negative three is three. Point out that the absolute value is the number part without the sign. Exercises: Write the number. a) 9 = b) +10 = c) 12 = d) +3.4 = 1 e) - 3 = 2 f) 8.7 = Answers: a) 9, b) 10, c) 12, d) 3.4, e) 3 1/2, f) 8.7 C-32 Teacher s Guide for AP Book 7.1 Unit 2 The Number System

14 Subtraction and distance apart. Write on the board: 3 (+1) = ASK: Is the answer positive or negative? (negative) SAY: The number I m taking away (point to +1) is bigger than the number I m taking it from (point to 3), so the answer is negative. Write the negative sign to begin the answer. ASK: What is the number part? (4) How do you know? (the numbers are 4 units apart) SAY: The distance apart is always the number part of the subtraction so, if you know the answer to the subtraction, then the absolute value is the distance apart. Write on the board: 3 (+1) is the distance between the integers 3 and +1 Exercises: Subtract the second number from the first. Then take the absolute value to find the distance between the integers. a) 4 and +5 b) 3 and 6 c) 18 and 3 d) 100 and 124 Answers: a) 9 = 9, b) 3 = 3, c) 15 = 15, d) 24 = 24 (MP.4) Exercises: Find the distance apart. Show your answer with a subtraction equation. a) How much warmer is +6 F than 4 F? b) How much lower is 30 ft than +42 ft? c) Mark lives 8 blocks north (+) of the school and Jenny lives 3 blocks south ( ) of the school. How far apart do they live? d) Ravi lives 5 blocks south of the school and Sandy lives 12 blocks south of the school. How far apart do they live? Answers: a) +6 ( 4) = 10, so 10 F; b) 30 (+42) = 72, so 72 ft; c) +8 ( 3) = 11, so 11 blocks; d) 5 ( 12) = 7, so 7 blocks apart NOTE: There are two different ways to subtract on a number line. For example, to subtract 17 2, you can start at 17 and count 2 places left. The answer is where you end up. Or, you can find the distance between 2 and 17 by counting how many steps you have to take to get from 2 to 17. The answer is the number of steps. If you want to explore this further, students can do Extension 1. Extensions 1. Subtract 12 3 in two ways on a number line. Make sure you get the same answer both ways. Answer: 9 (MP.1) 2. Show students the connection between the fact that has the same answer as and the fact that addition is commutative: is really just another way of writing 10 + ( 5) + 3 which, by commutativity, is ( 5). Have students use the commutativity of addition to make the calculation easier. a) b) Answers: a) = = 14, b) = 40 6 = 34 Teacher s Guide for AP Book 7.1 Unit 2 The Number System C-33

15 3. a) Find the distance between each pair of integers. i) 3 and 7 are 3 ( 7) = = units apart. +3 and +7 are +3 (+7) = = units apart. ii) 3 and +2 are 3 (+2) = = units apart. +3 and 2 are +3 ( 2) = = units apart. iii) 4 and +6 are 4 (+6) = = units apart. +4 and 6 are +4 ( 6) = = units apart. (MP.8) b) If you know the distance between two integers, how can you find the distance between the two opposite integers? Answers: a) i) +4, 4, 4, 4; ii) 5, 5, +5, 5; iii) 10, 10, +10, 10; b) they are the same 4. Which integer, +7 or 10, is farther from 2? Answer: +7 (MP.1) 5. The distance between a positive integer and a negative integer is 5. What might the integers be? Bonus: List all possible solutions. Answers: +4 and 1, +3 and 2, +2 and 3, +1 and 4 (MP.1) 6. The distance between two negative numbers is 5/7. Give one pair of numbers that work. Sample answer: 1/7 and 6/7 7. The distance between 62 and a negative integer is 100. What is the negative integer? Answer: 38 C-34 Teacher s Guide for AP Book 7.1 Unit 2 The Number System

16 NS7-8 Patterns in Subtraction Pages Standards: 7.NS.A.1c, 7.NS.A.1d Goals: Students will understand subtraction of integers as adding the additive inverse. Prior Knowledge Required: Can subtract integers on a number line Can use a picture to subtract integers Can add integers Vocabulary: integer, negative, opposite, positive Materials: 10 + cards and 10 cards BLM Adding and Subtracting Integers on a Number Line (pp. C-87 88, see Extension 2) Subtracting a number from 0 gets the opposite of the number. Draw on the board: ( 3) SAY: These subtractions start at 0, so the arrow ends at 0 and starts at the other number. Have two volunteers draw the arrows and two more volunteers do the subtractions (0 2 = 2 and 0 ( 3) = +3). Then ask volunteers to predict: 0 (+5) 0 ( 7) 0 (+80) 0 ( 900) PROMPTS: Is +5 greater than 0 or less than 0? (greater) So if the number I m subtracting is bigger than 0, is the difference positive or negative? (negative) SAY: Subtracting a bigger number from a smaller number gets a negative answer. Repeat for the other subtractions (the answers are 5, +7, 80, and +900). Point out that subtracting from 0 changes the sign, but keeps the number part the same. Write on the board: 0 ( 3) = +3 0 (+3) = 3 Teacher s Guide for AP Book 7.1 Unit 2 The Number System C-35

17 Exercises: Subtract from 0. a) 0 4 b) 0 ( 4) c) 0 18 d) 0 ( 43) e) 0 (+71) Answers: a) 4, b) +4, c) 18, d) +43, e) 71 Have students show their answers to parts a) and b) on a number line (they can draw the number line themselves or you can provide one for them). Subtracting a number is like adding its opposite. Write on the board, using cards for the negatives: 5 (+2) Have a volunteer show how to change the picture so that you can subtract +2. Then SAY: We needed to add two positives and two negatives. Now we take away the two positives. Do so, then SAY: It s like all we did was add the two negatives. So subtracting two positives gets the same answer as adding two negatives. Exercises: Subtract +2 and add 2 to the same number. Make sure you get the same answer. a) +3 (+2) and +3 + ( 2) b) 3 (+2) and 3 + ( 2) c) +5 (+2) and +5 + ( 2) d) 0 (+2) and 0 + ( 2) Answers: a) +1 and +1, b) 5 and 5, c) +3 and +3, d) 2 and 2 Write on the board: +4 ( 1) Have a volunteer show how to change the picture so you can subtract 1. Then SAY: We needed to add one positive and one negative. Now we take away the one negative. Do so, then SAY: It s like all we did was add the one positive. So subtracting one negative gets the same answer as adding one positive. Exercises: Subtract 1 and add +1 to the same number. Make sure you get the same answer. a) +3 ( 1) and +3 + (+1) b) 3 ( 1) and 3 + (+1) c) +5 ( 1) and +5 + (+1) d) 2 ( 1) and 2 + (+1) Answers: a) +4 and +4, b) 2 and 2, c) +6 and +6, d) 1 and 1 SAY: You can subtract any number by adding its opposite number. Exercises: Fill in the blank so that both questions have the same answer. Then check your answer by evaluating both expressions. a) 5 ( 3) and 5 + b) +5 (+6) and +5 + c) 3 ( 2) and 3 + d) +3 (+4) and +3 + e) +4 (+1) and +4 + f) +2 ( 3) and +2 + C-36 Teacher s Guide for AP Book 7.1 Unit 2 The Number System

18 Answers: a) 5 ( 3) = 5 + (+3) = 2, b) +5 (+6) = +5 + ( 6) = 1, c) 3 ( 2) = 3 + (+2) = 1, d) +3 (+4) = +3 + ( 4) = 1, e) +4 (+1) = +4 + ( 1) = +3, f) +2 ( 3) = +2 + (+3) = +5 (MP.1) Demonstrate the connection between adding opposites and subtracting a number from itself. Write on the board: ( 3) + ( ) = 0 ( 3) ( ) = 0 ASK: What do I have to add to 3 to get 0? (+3) What do I have to subtract from 3 to get 0? ( 3) SAY: Any number added to its opposite gets 0, and any number subtracted from itself also gets 0, so adding its opposite does the same thing as subtracting itself. Exercises: Fill in the blanks. a) ( 5) + ( ) = 0 and ( 5) ( ) = 0 b) (+4) + ( ) = 0 and (+4) ( ) = 0 c) ( 9) + ( ) = 0 and ( 9) ( ) = 0 Answers: a) +5, 5, b) 4, +4, c) +9, 9 Rewriting bracket notation without brackets. SAY: You can write brackets around both numbers when writing addition or subtraction. Write on the board: (+7) ( 8) SAY: You can change it to notation without brackets by writing it as an addition of gains and losses. Write on the board: SAY: Subtracting 8 is the same as adding +8 or just adding 8. Removing a loss is like adding a gain. Then write on the board: ( 8) (+5) ( 8) ( 5) (+3) (+7) ( 4) (+3) Ask volunteers to write the subtractions as additions without brackets. ( 8 5, 8 + 5, + 3 7, 4 3) Exercises: Write the subtraction as an addition without brackets. Then evaluate. a) ( 3) ( 4) b) (+3) ( 2) c) ( 4) (+3) d) (+2) (+8) Answers: a) = +1, b) = +5, c) 4 3 = 7, d) = 6 SAY: Any addition or subtraction written in brackets can be written as an addition without brackets. The rules for changing notation are: +(+) = + +( ) = (+) = ( ) = + Teacher s Guide for AP Book 7.1 Unit 2 The Number System C-37

19 Exercises: Write the expression as an addition without brackets. Then evaluate. a) ( 2) + ( 7) b) ( 2) ( 7) c) ( 2) + (+7) d) ( 2) (+7) e) (+2) + ( 7) f) (+2) ( 7) g) (+2) + (+7) h) (+2) (+7) Answers: a) 2 7 = 9, b) = +5, c) = +5, d) 2 7 = 9, e) = 5, f) = +9, g) = +9, h) = 5 Subtracting integers in real-world contexts. Write on the board: What is the temperature now if a) the temperature was 40 F, then dropped 30 F? b) the temperature was 40 F, then dropped 50 F? c) the temperature was 40 F, then dropped 30 F? Have a volunteer answer the first question. (10 F) ASK: How did you know to subtract? (because dropped 30 F tells us it became less by 30 F) Have a volunteer answer the second question and write a subtraction equation for it. (40 50 = 10) Repeat for the third question ( = 70) Point out that sometimes knowing how to solve a problem in a familiar situation with positive numbers helps you to figure out how to solve the same type of problem in a more unfamiliar situation, such as when there are negative numbers. (MP.4) Exercises: a) The temperature in California is 75 F and in New York is 40 F colder. What is the temperature in New York? b) The temperature at the South Pole in January can reach as high as 9 F. In July, it can get 100 F lower. What is the temperature in July? c) Celsius is a temperature scale in which 0 C is the freezing point of water. When Sal adds some salt to the water, the mixture freezes at a temperature 15 C colder. What is the freezing temperature of the new mixture? d) On January 15, 1972, in Loma, Montana, the temperature rose from 54 F to 49 F, all in one day. How much of a temperature change was that? e) On January 22, 1943, the temperature in Spearfish, South Dakota, rose from 4 F to 45 F in only 2 minutes. How much of a temperature change was that? Answers: a) 35 F, b) 91 F, c) 15 C, d) 49 F ( 54 F) = 49 F + 54 F = 103 F, e) 45 F ( 4 F) = 45 F + 4 F = 49 F Extensions 1. (MP.7) a) Subtract the positive numbers. Then subtract the negative numbers by continuing the pattern. 6 3 = 6 2 = 6 1 = 6 0 = 6 ( 1) = 6 ( 2) = 6 ( 3) = b) Use a pattern to subtract (+5) ( 3). C-38 Teacher s Guide for AP Book 7.1 Unit 2 The Number System

20 NS7-9 Adding and Subtracting Multi-Digit Integers Pages Standards: 7.NS.A.1d Goals: Students will use place value to add and subtract multi-digit integers. Prior Knowledge Required: Can add and subtract integers Knows how to subtract an integer by adding the opposite integer Can add multi-digit numbers by lining up the place values Vocabulary: absolute value, expanded form, hundreds, ones, place value, tens, whole number Materials: overhead projector transparency of grid paper BLM Balance Model (p. C-89, see Extension 1) Using expanded form to add 3-digit numbers. Show how to add using the expanded form: 452 hundreds + tens + ones hundreds + tens + ones After regrouping: hundreds + tens + ones hundreds + tens + ones Point out that students need to regroup until all place values have only a single digit. Exercise: Add using expanded form. Bonus: Add 5, ,571 using expanded form. Answers: 1,106; Bonus: 8,417 Standard algorithm for addition with regrouping. Now demonstrate using the standard algorithm alongside the place value addition for the first example you did together ( ): You can project a transparency of grid paper onto the board to show the outlines. C-40 Teacher s Guide for AP Book 7.1 Unit 2 The Number System

21 Point out that after regrouping the tens, you add the 1 hundred that you carried over from the tens at the same time as the hundreds from the two numbers, so you get = 7 hundreds. NOTE: When adding and subtracting multi-digit numbers, students should be encouraged to use grid paper to line up the place values. Exercises: Use the standard notation to add. a) b) c) d) e) 1, ,217 f) 4, ,347 g) 94, ,647 h) 862, ,857 Bonus: Answers: a) 575; b) 815; c) 953; d) 1,486; e) 8,575; f) 13,005; g) 113,005; h) 1,061,452; Bonus: 1,995 Students who are struggling can use place value charts alongside the standard algorithm. Bonus: Find a short way to do part d) that doesn t require regrouping. Answer: the answer is 1 less than ,000 (1,487), so = 1,486 SAY: You have to make sure the place values are lined up, the ones with the ones, tens with tens. This can be tricky when the numbers have a different number of digits, but you just have to make sure the ones digits are aligned and the commas are aligned. Demonstrate the alignment in a) below: 32, ,736 Exercises: a) 32, ,736 b) 789, ,896 c) 999, ,322 d) 94, ,647 e) 652, f) 5, ,588 Bonus: 17, , ,117 Answers: a) 42,141; b) 833,000; c) 1,001,000; d) 103,005; e) 653,520; f) 303,561; Bonus: 590,309 Word problems practice. Exercises: a) Jayden ran 1,294 km one year and 1,856 km the next. How many kilometers did he run altogether? b) In an election with three candidates, the candidate who won received 1,052,817 votes. The other two received 972,435 votes and 71,095 votes. Did the candidate who won get more than the other two combined? Answers: a) 3,150 km; b) yes; the other two combined received only 1,043,530 votes Teacher s Guide for AP Book 7.1 Unit 2 The Number System C-41

22 Using the standard algorithm to subtract 3-digit numbers with regrouping. Write on the board: SAY: 7 2 is easy to subtract, but we can t subtract since we want a positive digit in the answer. (Students will see in Extension 3 how they can use = 30 to solve this question.) So let s take away 100 from 300 and add it to the SAY: Now it is easy to subtract each place value. Exercises: Subtract, then check by adding. a) b) c) d) Answers: a) 229, b) 163, c) 605, d) 182 Do the following example ( ) together as a class: Emphasize that you write the second regrouping above the first one, not over the first regrouping, so that you can see each step easily. Exercises: Subtract. Then check by adding. a) b) c) d) Bonus: Make up your own subtraction question that requires regrouping twice. Ask a partner to solve your question. Selected answers: a) 388, b) 268, c) 64, d) 253 C-42 Teacher s Guide for AP Book 7.1 Unit 2 The Number System

23 Borrowing from zero. Write on the board: ASK: Do I have enough ones to subtract? (no) What do I need to do? (regroup 1 ten as 10 ones) There are no tens to take from; what can I do? (regroup 1 hundred as 10 tens) Have a volunteer do so (or do it yourself if no one volunteered the strategy): SAY: Now we can regroup a ten as 10 ones. Have a volunteer do so, then have another volunteer show the subtraction: Exercises: Subtract using the standard algorithm. a) b) c) d) 62,187 41,354 e) 54,137 28,052 f) 9,319 6,450 g) 4,037 2,152 h) 90,319 6,405 i) 145,207 1,128 j) 3,695 1,697 k) 1, l) 10,000 4,356 Answers: a) 233; b) 278; c) 366; d) 20,833; e) 26,085; f) 2,869; g) 1,885; h) 83,914; i) 144,079; j) 1,998; k) 146; l) 5,644 Word problems practice. Exercises: 1. Construction of the Statue of Liberty began in France in When it was completed, the statue was shipped to the United States and rebuilt there in How long ago was it built in France? How long ago was it rebuilt in the United States? Answers: The answer depends on current year; e.g., = 133 years ago, and = 128 years ago 2. In 1810, the population of New York City was 96,373. In 2010, the population of New York City was 8,175,133. How much did the population grow in those 200 years? Answers: 8,078,760 Teacher s Guide for AP Book 7.1 Unit 2 The Number System C-43

24 Adding multi-digit integers with the same sign. Tell students that they can add integers with the same sign by adding their absolute values. Write on the board: = = 8 SAY: Adding two gains gets a greater gain and adding two losses gets a greater loss. Tell students that they can use grid paper to do their rough work. Exercises: Add the integers. a) b) 5 8 c) d) Bonus: ( 376) + ( 145) + ( 309) + ( 822) Answers: a) +11; b) 13; c) +1,585; d) 813; Bonus: 1,652 Remind students that they can subtract an integer by adding its opposite. Exercises: Subtract. a) 3 (+17) b) +31 ( 482) c) 142 (+7,483) d) +8,160 ( 752) Answers: a) 20; b) +513; c) 7,625; d) +8,912 Word problems practice. (MP.4) Exercises: Write an addition equation to show the answer to the word problem. a) The Boomerang Nebula, with a temperature of 458 F, is the coldest place in the universe known to scientists. The coldest possible temperature is only 2 degrees colder. What is the coldest possible temperature? b) The average temperature on Jupiter is 244 F. The average temperature on Neptune is 122 F colder. What is the average temperature on Neptune? Answers: a) 458 F 2 F = 460 F, b) 244 F 122 F = 366 F Adding multi-digit integers with different signs. Tell students that they can add integers with different signs by subtracting their absolute values. Write on the board: = = 2 SAY: Adding a gain and a loss reduces the gain or the loss. If the gain is bigger, you get an overall gain and if the loss is bigger, you get an overall loss. Write on the board: + 3,482 21,674 ASK: What s bigger, the gain or the loss? (the loss) SAY: Take away the smaller absolute value from the bigger absolute value. C-44 Teacher s Guide for AP Book 7.1 Unit 2 The Number System

25 Write the subtraction in grid form on the board and ask a volunteer to do the multi-digit subtraction: Write on the board: So, + 3,482 21,674 = 18,192 SAY: The difference tells you the number part and the number with the bigger absolute value tells you the sign. Exercises: Add the integers. a) b) c) 18, ,563 d) 9, ,512 Answers: a) 104; b) +761; c) 868; d) +9,036 SAY: Now you will need to decide whether to add or subtract the absolute values to add the integers. Exercises: Add the integers. a) b) c) + 81, ,517 d) 21, ,183 Answers: a) 1,255; b) 169; c) +84,933; d) +690,767 (MP.4) Exercises: Write an addition equation to show the answer to these word problems. a) The temperature on Mercury can get as high as 869 F. It can also get 1,167 F colder. What temperature is that? b) The average temperature on Neptune is about 340 F and on Venus, it is about 1,210 F warmer than that. What is the average temperature on Venus? Answers: a) F 1,167 F = 298 F; b) 340 F + 1,210 F = +870 F Exercises: Subtract by adding the opposite number. a) 174 ( 311) b) +854 (+5,142) c) 853 ( 215) d) 514 (+312) e) +704 (+908) f) +360 ( 412) Answers: a) +137; b) 4,288; c) 638; d) 826; e) 204; f) +772 Teacher s Guide for AP Book 7.1 Unit 2 The Number System C-45

26 Extensions (MP.4) 1. On BLM Balance Model, students will use a pan balance model to understand that an integer can be subtracted by adding its opposite. Answers: 1. b) down, c) down, d) up; 2. b) 1, c) 1, d) +1; 3. a) a negative integer, b) a positive integer, c) a negative integer; 4. a) negative, b) positive (MP.4) 2. Show students a paper person who starts at sea level (0 inches), goes down 1 inch for every brick added and goes up 1 inch for every helium balloon added. a) Where is the person if they are holding i) 2 bricks? ii) 3 helium balloons? iii) 8 helium balloons? iv) 5 bricks? v) 3 bricks and 2 helium balloons? vi) 4 bricks and 1 helium balloon? vii) 2 bricks and 5 helium balloons? b) What happens if you take away 3 bricks? c) What happens if you take away 4 helium balloons? d) How is taking away a helium balloon like adding a brick? Answers: a) i) 2, ii) +3, iii) +8, iv) 5, v) 1, vi) 3, vii) +3, b) moves up 3 inches, c) moves down 4 inches, d) they both make the person move down an inch (MP.1, MP.7) 3. Show students a way to subtract positive numbers using what they know about integers instead of regrouping. Example: = ( ) + (30 60) + (4 8) = = = 166 Students can try this technique with some of the questions they have already done and verify that it gets the same answer. C-46 Teacher s Guide for AP Book 7.1 Unit 2 The Number System

27 NAME Number Lines from -6 to 6 DATE COPYRIGHT 2015 JUMP MATH: TO BE COPIED. CC EDITION Blackline Master The Number System Teacher s Guide for AP Book 7.1 C-85

28 NAME Subtraction on a Number Line DATE 1. Write the answers to the subtraction. a) 3 - (-1) = b) -4 - (+2) = c) -3 - (-2) = d) -1 - (-4) = 2. Write the subtraction equation shown by the picture. a) Subtraction equation: b) Subtraction equation: c) d) Subtraction equation: Subtraction equation: COPYRIGHT 2015 JUMP MATH: TO BE COPIED. CC EDITION C-86 Blackline Master The Number System Teacher s Guide for AP Book 7.1

29 NAME DATE Adding and Subtracting Integers on a Number Line (1) Subtracting -2 does the opposite of adding -2. To add +3 + (-2), move 2 units left from +3 To subtract +3 - (-2), move 2 units right from Add the integer. Then subtract the same integer by doing the opposite. a) (-3) + (-2) = -5 (-3) - (-2) = b) (-2) + (+1) = (-2) - (+1) = c) (-4) + (-1) = (-4) - (-1) = (+) = + +(-) = - -(+) = - -(-) = + move right move left move left move right Example: -5 - (-2) = COPYRIGHT 2015 JUMP MATH: TO BE COPIED. CC EDITION -7 7 So -5 - (-2) = Write the subtraction as addition. Then use the nubmer line to find the answer (-3) = = Blackline Master The Number System Teacher s Guide for AP Book 7.1 C-87

30 NAME DATE Adding and Subtracting Integers on a Number Line (2) 3. Write the addition or subtraction without brackets. Then use the number line to find the answer. a) (-2) - (+3) = = b) (+5) - (+2) = = c) (-2) + (-3) = = d) (+7) - (+5) = = a) Would you move left or right on a number line? To add +5, move To add -5, move To subtract +5, move 5 units. 5 units. 5 units. To subtract -5, move 5 units. b) Look at your answers in part a). Subtracting +5 gives the same result as adding. Subtracting -5 gives the same result as adding. 5. Draw a number line to add or subtract. a) (-5) - (-1) b) (-3) + (+1) c) (+5) - (-4) d) (+3) + (-6) COPYRIGHT 2015 JUMP MATH: TO BE COPIED. CC EDITION C-88 Blackline Master The Number System Teacher s Guide for AP Book 7.1

31 NAME Balance Model DATE You can use integers to measure how the pan balance moves up and down. If the same number of marbles is in both pans, When you add a marble to the left side, the right side starts at the right side moves up Does the right side move up or down if you a) add a marble to the left side? up b) add a marble to the right side? c) remove a marble from the left side? d) remove a marble form the right side? 2. The right side starts at 0. Where does the right side end up if you a) add a marble to the left side? +1 b) add a marble to the right side? c) remove a marble from the left side? d) remove a marble form the right side? 3. Adding a marble to the left side is like adding a positive integer. Fill in the blanks. a) Adding a marble to the right side is like adding COPYRIGHT 2015 JUMP MATH: TO BE COPIED. CC EDITION b) Removing a marble from the left side is like adding c) Removing a marble from the right side is like adding 4. Write positive or negative. a) Subtracting a integer does the same thing as adding a positive integer. b) Subtracting a integer does the same thing as adding a negative integer. Blackline Master The Number System Teacher s Guide for AP Book 7.1 C-89