Unit 7 Number Sense H-1. Teacher s Guide for Workbook 8.1 COPYRIGHT 2010 JUMP MATH: NOT TO BE COPIED

Transcription

1 Unit 7 Number Sense In this unit students will use integers in context, compare and order integers, add and subtract integers, and make connections between different methods of addition and subtraction. Students will also multiply and divide integers and solve problems involving integers. Because operating on integers can be counterintuitive for many students, your students will need a lot of practice adding, subtracting, multiplying, and dividing integers more than can be provided in these lessons. To promote automaticity, we recommend doing problems for a few minutes each day, two or three days a week, until the end of the school year. Students need to perform integer operations automatically in order to be able to solve problems involving integers. At first, display the chart from Workbook p. 189 and display a chart showing the sign rules for multiplication and division (e.g., (+) ( ) = ( ) ), but eventually scaffold students away from requiring the charts. Materials Some students might benefit from using integer tiles when adding and subtracting integers. We do not rely on them in our lessons though. Teacher s Guide for Workbook 8.1 H-1

2 NS8-64 Gains and Losses Pages Curriculum Expectations Ontario: 8m1, 8m2, 8m7, essential for 7m13, 7m14, 8m21, 8m22, 8m23 WNCP: essential for 7N6, 8N7, [R, C] Goals Students will add sequences of gains and losses. PRIOR KNOWLEDGE REQUIRED Has an intuitive understanding of gaining and losing money Vocabulary gain loss net gain net loss cancel plus sign (+) minus sign ( ) Add integers informally using the familiar context of gains and losses. Introduce the plus (+) and minus ( ) signs as symbols for gains and losses. Then have students, given a gain and a loss, decide and indicate whether there is a net gain or net loss by using the appropriate sign. Students can signal their answer by forming a plus sign or a minus sign with their arms. Write a sequence of gains and losses using + and signs. See Workbook p. 179, Question 2. Translate a sequence of + s and s to gains and losses. EXAMPLE: = a gain of $5 and a loss of $3. Was more gained or lost? Write down a gain followed by a loss or a loss followed by a gain. EXAMPLE: a loss of $3 then a gain of $5. ASK: Was more gained or lost? Repeat with a sequence of + s and s interpreted as gains and losses. EXAMPLE: Emphasize that if the larger number has a + sign, then more was gained; if the larger number has a sign, then more was lost. Repeat with various examples. Bonus How much was gained or lost overall? See Workbook p. 179, Question 4. Have students do several examples, some of which result in no gain or loss. Then ASK: When was there no gain or loss? (when the amount of the gain was the same as the amount of the loss) Sequences of more than two gains and losses. Write on the board: ASK: How is this question different from the questions we have done so far in this lesson? (there are three gains and losses instead of two) ASK: What is ? (+5) Write: = ASK: What is + 5 4? (+1) Write: = = +1 Ask students to find the net gain or loss: a) b) c) d) e) Solution for a) = 3 4 = 7 H-2 Teacher s Guide for Workbook 8.1

3 Answers: b) 11 c) + 5 d) + 4 e) + 8 Changing the order of gains and losses results in the same net gain or loss. Ask students to solve these problems: a) b) c) d) ASK: What do you notice about your answers? (they are all the same) Why is that the case? (because we added the same gains and losses each time, just in a different order) To emphasize this point, ASK: If one week, I gain $3, then gain $2, then lose $4, and another week, I gain $2, lose $4, then gain $3, which week was a better week? (neither, they are the same) Changing into a known problem Longer sequences of gains and losses. Show students how to group all the gains (+ s) together and all the losses ( s) together. Start with a sequence of only three gains and losses, so that students need to group only two gains or two losses. See Workbook p. 179, Question 5. Then progress to longer sequences. Suggest that students start by circling all the gains, so that they can more easily see what they have to group together. If students miss some losses, have them use a different colour to circle the losses this will help ensure that they didn t miss any losses. By grouping all the gains and then all the losses, emphasize that we are changing the problem into a problem with just one gain and one loss and we already know how to do that kind of problem! Emphasize that changing one problem into another that we already know how to do is a strategy that mathematicians use every day. Cancelling gains and losses. Have students add these gains and losses: a) b) c) d) ASK: What do you notice about all these answers? (they are all the same) Why is this the case? (because all the same numbers are used each time) Now have students add these gains and losses: a) b) c) d) ASK: What do you notice about all these answers? (they are all the same) Why is this the case? (because adding the same gain and loss results in no change) Tell students that when you add the same gain and loss, you can cancel them, because together they add 0 to the result. Have students practise cancelling the same gain and loss, as on Workbook p. 180, Question 7; start with a sequence of three gains and losses and then progress to longer sequences of gains and losses. Include examples where more than one pair of gains and losses cancel each other. Number Sense 8-64 H-3

4 NS8-65 Integers Page 181 Curriculum Expectations Ontario: essential for 7m13, 7m14, 8m1, 8m2, 8m3, 8m5, 8m6, 8m7, essential for 8m21, 8m22, 8m23 WNCP: essential for 7N6, 8N7, [R, C, CN, V] Goals Students will represent, compare, and order integers. PRIOR KNOWLEDGE REQUIRED Understands the relationship between how many more (less) than and addition (subtraction) Vocabulary integer positive negative more than (>) less than (<) cancel opposite integer connection Real world NOTE: In this lesson, students will see that the minus sign can be used in two different ways: the in 0 5 means subtract, but the in front of a number (e.g., 5) indicates that the number is less than 0. Just as 5 represents 0 5, we can use +5 to represent = 5, but the signs are being used in a different way. Be aware that this new use of both the plus and minus signs may cause confusion at first, and students will need practise and time to become comfortable using the signs as both symbols of operations and indicators of position. Using a credit card to have less than $0. Tell students that you don t have any money, but you want to buy something. It costs $5. Tell students that you know that you will be getting money soon. ASK: How can I pay for it now? (use a credit card) Explain to students that even though you started off with nothing, you have even less now! Integers. Remind students that to get the number that is 5 less than 8, we can subtract: 8 5. ASK: How can you get the number that is 5 less than 0? (subtract: 0 5) Repeat for 8 less than 0 (0 8), and then 3 more than 0 (0 + 3). Draw this number line on the board: ASK: Does anyone know a shorter way to write these numbers (point to the numbers larger than 0)? When students give the answer, rewrite the number line as follows: ASK: What number is 3 less than 2? Demonstrate starting at 2 and moving 3 places left, to the marking labelled 0 1. Repeat with these questions: What number is 5 less than 3? (0 2) What number is 6 less than 1? (0 5) What number is 4 more than 0 2? (2) Tell students that you find it cumbersome to always write zero minus something for numbers less than zero. We don t have to write zero plus something for numbers larger than 0! Is there a shorter way to write the numbers that are zero minus something? Draw the following number line on the board and ask students if this is a good solution: H-4 Teacher s Guide for Workbook 8.1

5 ASK: What is wrong with this solution? (we can t tell the difference between 5 more than 0 and 5 less than 0!) Tell students that mathematicians have come up with a shorter way to write numbers less than 0 that allows us to tell the difference between 5 more than 0 and 5 less than 0. ASK: Does anyone know how they do it? Explain that, instead of 0 5, we can just write 5. The tells us that the number is less than 0, and the 5 tells us by how much. Draw a number line on the board, with integers from 8 to Tell students that the numbers less than zero are called negative numbers; for example, the number that is 5 less than 0 is called negative 5. ASK: What do you think numbers larger than zero are called? (positive numbers) Tell students that sometimes we want to emphasize the difference between positive and negative numbers. We can write +5 for the number that is 5 more than 0, to emphasize that the number is positive. Now draw this number line on the board: Explain that negative numbers always have the negative sign ( ) in front of them, but positive numbers sometimes have the positive sign (+) and sometimes don t. Distinguish between positive and negative numbers. Draw a chart on the board with the headings positive and negative. Write various numbers on the board and have students signal which column to put them in (the left or right column), either by pointing to the column or by raising their left or right hand. EXAMPLES: (+3) ( 4) ASK: Which two numbers are the same? (3 and (+3)) Modelling, Visualizing Then write 0 on the board and discuss where it should go. Some students might suggest putting it in both places, others might suggest putting it nowhere. Both are good answers, but mathematicians have decided to call 0 neither positive nor negative. Comparing numbers on a number line. ASK: Tom had no money but spent $2 on his credit card. How can we write how much money he has? ( $2) Demonstrate moving 2 places left from 0 on the number line. Then ASK: Ahmed has $6 but spent $11 on his credit card. How much money does Ahmed have? ( $5) Demonstrate moving 11 places left from +6 on the number line. ASK: Whose situation is better, Tom s or Ahmed s? (Tom s) How can you tell? ( 2 is to the right of 5) Explain that because Tom s amount is more to the right than Ahmed s amount, Ahmed would Number Sense 8-65 H-5

6 have to gain money to have as much as Tom. ASK: How much would Ahmed have to gain to have as much as Tom? ($3, because Tom s amount is 3 places to the right of Ahmed s) It might help some students to think about temperatures. Which temperature is lower, 2 C or 5 C? ( 5 C ) How much lower? (3 C the temperature would have to increase 3 C to get from 5 C to 2 C) Compare a negative number to a positive number. Referring to the number line from 8 to +8, ask students to circle the smaller integer in each pair by determining which number comes first (is to the left of the other): a) 3 or +2 b) 1 or +4 c) 1 or 3 d) 8 or 8 e) 5 or +2 Using logical reasoning ASK: If one number is positive and the other number is negative, how can you tell which number is smaller? (the negative number is always smaller) How do you know? (the negative number comes before 0 and 0 comes before the positive number, so the negative number comes before the positive number) Compare two positive numbers and two negative numbers. Emphasize that students already know how to compare two positive numbers. We know that 3 is less than 4 because 3 comes before 4. ASK: But how do you compare two negative numbers? Which comes first, 3 or 4? ( 4) Which is greater, 3 or 4? ( 3) Have students compare two positive numbers and then the opposite negative numbers (but do not use this term yet). EXAMPLE: 2 is than 5 and 2 is than 5 Looking for a pattern After doing several such examples, ASK: How can you tell by comparing the positive numbers how to compare the negative numbers? (the order is the opposite; since 5 is more than 2, 5 is less than 2) Encourage students to think about which one is further from 0; for positive numbers, the number that is further is larger, but for negative numbers, the number that is further is smaller. Then have students do these without looking at a number line: a) 3 is than 4 b) 7 is than 5 c) 6 is than 9 d) 2 is than 1 Bonus 134 is than 135 Opposite integers. Tell students that the same numbers with opposite signs are called opposite integers, so +2 and 2 (or 2 and 2) are opposite integers. Have students write the opposite of each integer: a) +3 b) 5 c) 4 d) 10 e) +7 f) 3 ANSWERS: a) 3 b) +5 or 5 c) 4 d) +10 or 10 e) 7 f) 3 H-6 Teacher s Guide for Workbook 8.1

7 Representing, Reflecting on the reasonableness of an answer Process assessment 8m7, [C] Connecting ASK: Which two answers are the same? (a and f) Why? (because +3 and 3 are the same number, just written two different ways) Emphasize that the number tells how far from 0 a number is and the sign (+ or ) tells in which direction. So opposite integers are each the same distance from 0, just in opposite directions. Tell students that you know that because 3 is less than 4, 3 is more than 4. Have each student individually write a statement that generalizes this, using the term opposite integers. Ensure that all students have a sentence, and then have students work in pairs to write a sentence they both agree is better than their individual sentences. For example: If one number is less than another, its opposite integer is more than the other s opposite integer. The special case of 0. Remind students that +3 means and that 3 means 0 3. ASK: What do you think +0 means? (0 + 0 = 0) What do you think 0 means? (0 0 = 0) Tell students that 0 is considered to be the opposite of 0 because +0 and 0 are both 0, and hence are both the same distance from 0. The signs for greater than (>) and less than (<). Remind students how to use these signs. Three possible mnemonics: 1. The side with two ends instead of one end always points to the greater number. 2. Think of the sign as an open mouth wanting to eat a greater number of [insert food students like]. 3. Think of the sign as an equal sign that moves: the two ends of the bars that face the smaller number are close together (smaller distance) while the two ends of the bars that face the larger number are farther apart (larger distance). Have students do several problems similar to Workbook p. 181, Question 4. Have students find and label several numbers on a number line and then use their answers to list the numbers in order from smallest to largest. EXAMPLE: A 3 B 8 C +7 D 2 E 5 connection Real world connection Algebra Temperatures as integers. Ask students where they have seen integers in real life. EXAMPLES: gains and losses (money), golf scores, +/ ratings in hockey, temperatures. Mention temperatures if students don t. Have students decide, for various pairs of temperatures, which one is warmer. Then have students put several temperatures in order from warmest to coldest. Work through Workbook p. 181, Question 9 together as a class. Number Sense 8-65 H-7

8 NS8-66 Adding Integers NS8-67 Adding Integers on a Number Line Pages Curriculum Expectations Ontario: 7m13, 7m14, 8m1, 8m2, 8m5, 8m21, essential for 8m21, 8m22, 8m23 WNCP: essential for 7N6, 8N7, [R, CN] Goals Students will add integers in different ways. PRIOR KNOWLEDGE REQUIRED Understands the relationship between positive integers and how many more than 0 Understands the relationship between negative integers and how many less than 0 Vocabulary integer positive negative more than (>) less than (<) cancel opposite integer counter-example Review integers. Draw the number line from 0 5 to from the previous lesson and have students draw and label the number line the more conventional way in their notebooks. ASK: What notation do we use for 0 5? Repeat for Use integers to represent gains and losses. See Workbook p. 182, Question 1. Using gains and losses to add integers. Progress as on Workbook p. 182, Questions 2 4. Explain that we use brackets when adding integers because the signs for adding and subtracting look exactly like the positive and negative signs in front of the integers, and it would be awkward to write and read expressions like or without brackets. Opposite integers add to 0. First review the definition of an opposite integer same number but opposite sign. Tell students that when you gain the same amount as you lose, then you end up with nothing, so opposite integers add to 0. Cancelling opposites to add integers. See Workbook p. 182, Question 5, and Workbook p. 183, Question 6. Using sums of +1s and 1s to add integers. See Workbook p. 183, Question 7. If any individual students would benefit from using concrete materials, you could give them integer tiles to work with, if you have them. process assessment [R], 8m1, 8m2 Workbook p. 183, Question 9 process expectation Making and investigating conjectures Adding integers mentally. See Workbook p. 183, Question 8. Remind students that if the gain is bigger, the result is positive, and if the loss is bigger, the result is negative. Using a number line to add integers. See Workbook p. 184, Question 1. The investigation on Workbook p Have students predict the result before doing the investigation. H-8 Teacher s Guide for Workbook 8.1

9 ACTIVITY In this activity, students use a vertical number line and the context of height above or below sea level to add integers. (You will be able to apply the same model to subtracting integers by using take away in NS8-68.) Draw a large vertical number line on the board, going from 6 to +6. Cut out a small paper person from white paper, eight small balloons from blue paper to represent helium balloons, and eight small sandbags from brown paper to represents sandbags. Ensure that each paper person, balloon, and sandbag can be easily taped to the board, by attaching tape to their backs ahead of time. Start the paper person at 0 on the number line and explain that the person is at sea level. The person moves up 1 m every time he is given a helium balloon and down 1 m every time he is given a sandbag. Start by giving the person 3 helium balloons and ASK: Where will the person be now? Have a volunteer move the person accordingly (to +3 on the number line). Then give the person 4 sandbags. SAY: The person now has 3 helium balloons and 4 sandbags. Where will they be? How can we write that with integers? (+3) + ( 4) = ( 1) Continue in this way, adding helium balloons (positive integers) and sandbags (negative integers) until you have none left to add. Record the addition of integers as you go. EXAMPLE: Add... Location of person 3 helium balloons +3 4 sandbags 1 because (+3) + ( 4) = ( 1) 2 helium balloons +1 because ( 1) + (2) = (+1) 3 sandbags 2 because (+1) + ( 3) = ( 2) 1 sandbag 3 because ( 2) + ( 1) = ( 3) 3 helium balloons 0 because ( 3) + (+3) = 0 Note that the person now has 8 helium balloons and 8 sandbags. Since 8 helium balloons represents +8 and 8 sandbags represents 8, we can write this as (+8) + ( 8) = 0. Process assessment [CN, R], 8m1, 8m5 Workbook p. 184, Question 2 Number Sense 8-66, 8-67 H-9

10 NS8-68 Subtracting Integers Pages Curriculum Expectations Ontario: 7m26, 8m1, 8m2, 8m5, 8m6, 8m7, essential for 8m21, 8m22, 8m23 WNCP: 7N6, essential for 8N7, [R, CN, V, PS, C] Vocabulary integer positive negative Goals Students will subtract integers using a number line. PRIOR KNOWLEDGE REQUIRED Can add and subtract positive integers using a number line Can add negative integers using a number line Can add integers using gains and losses Understands variables Understands that addition and subtraction are opposite operations Can write integers as a sum of +1s and 1s Materials BLM Subtraction as Take Away (p H-37) What can we do with integers? Tell students that, so far, we know how to compare integers to determine which is larger, order a list of integers, and add integers. Have students predict what else we can do with integers. Encourage students to say subtract, multiply, and divide, although other answers might arise as well (e.g., find the mean of a set of integers). Tell students that we will learn how to subtract integers in this section and later to multiply and divide them. What does it mean to subtract integers? Discuss different meanings for subtracting positive numbers. (take away, how many more than, distance between) Tell students that we will see different contexts for subtracting integers. For example, comparing temperatures: If it is +32 C in Florida and 10 C in Alaska, we can ask ourselves how much warmer it is in Florida than in Alaska. The difference between the temperatures is +32 C ( 10 C ). Subtraction is the opposite of addition. Remind students that some actions undo others. ASK: If I walk 3 blocks east, how can I get back to where I started? (walk 3 blocks west) Emphasize that it doesn t matter where you start if you walk 3 blocks east and then walk 3 blocks west, you always get back where you started. ASK: I add 3 to a number, how can I get back to the number I started with? (subtract 3 from the result) SAY: I start with a mystery number and add 3 to it. The answer is 10. What is the mystery number? (7) How did you get that? (subtracted 3 from 10 because subtracting 3 undoes adding 3) Have students work in pairs: Student 1 thinks of a number. Student 2 tells Student 1 what to add. Student 1 gives the answer and Student 2 has to find the original mystery number. Partners then switch roles. Review adding and subtracting a positive number on a number line. Draw a number line on the board. Ask your students how they move on H-10 Teacher s Guide for Workbook 8.1

11 a number line when they start at 7 and then add and subtract 2. Repeat, starting at 5. ASK: How do you add 2 to a number on a number line? (move 2 units right) How do you undo moving 2 units right? (move 2 units left) Emphasize that, no matter where you start, if you move 2 units right and then move 2 units left, you always get back to where you started. Then connect this to subtracting 2. Explain that to subtract 2, you have to undo adding 2, which is like undoing a move of 2 units right by moving 2 units left. Subtracting positive integers from negative integers. Write on the board 5 2 and ( 4) 2, and draw a number line from 10 to 10. First have a volunteer show how to do 5 2 on the number line (move left 2 units starting at 5) and then have another volunteer show how to do ( 4) 2 on the number line (move left 2 units starting at 4). Process assessment 8m6, [V] Have students practise subtracting more positive integers from negative integers by going left on a number line. EXAMPLES: a) ( 5) 2 b) ( 6) 1 c) ( 2) 3 d) ( 3) 4 e) ( 1) 5 Answers: a) 7 b) 7 c) 5 d) 7 e) 6 Process assessment 8m7, [C] Have students describe how they found their answers. For example, to find ( 5) 2, start at 5 on a number line and move 2 places left. To place these problems in context, suggest that the expression in part a) is like having a debt of $5 and then borrowing two more dollars, or like owing $5 on your credit card and then spending $2 more on your credit card. Subtracting a negative number means going right on a number line. Review with the students how adding a positive integer means moving right on a number line, and adding a negative integer means moving left, in the opposite direction. Subtracting a positive integer also means moving left because you are undoing adding a positive integer. Summarize the situation with the following diagram: ( 3) + 2 = ( 1) ( 3) 2 = ( 5) ( 3) + ( 2) = ( 5) ( 3) ( 2) = Ask students to guess which direction they need to move in when they subtract a negative integer. Ask them to explain their guesses. Emphasize that we go in the opposite direction to the direction we would go in when adding: to add a negative integer, move left, so to subtract the same negative integer, move right the number of places you would move left to add it. This is because subtracting undoes adding. Number Sense 8-68 H-11

12 Ask students to tell how many units right or left they will move from 4 to perform the following additions and subtractions: ( 4) + ( 2) = ( 4) ( 2) = ( 4) + 3 = ( 4) 3 = ( 4) + ( 1) = ( 4) ( 1) = (to add 2, move 2 units left) (to subtract 2, move 2 units right, the opposite of how to add 2) (to add 3, move 3 units right) (to subtract 3, move 3 units left, the opposite of how to add 3) (move 1 unit left) (move 1 unit right) Then have students do all the problems above in their notebooks. Answers: 6, 2, 1, 7, 5, 3 Changing into a known problem Writing subtraction questions as addition questions. Write these questions: a) ( 5) + 2 = b) ( 5) 2 = c) ( 5) + ( 2) = d) ( 5) ( 2) = Have students solve them and then describe how to do each question on a number line: Start at 5 and... a) move 2 units right b) move 2 units left c) move 2 units left d) move 2 units right ASK: Which questions have the same answer? Why? (( 5) + 2 has the same answer as ( 5) ( 2) because you are doing the same thing moving 2 units right from 5 to get both answers; also, ( 5) 2 has the same answer as ( 5) + ( 2) because you are doing the same thing moving 2 units left from 5 to get both answers.) Repeat the exercises above with these questions: a) b) 5 3 c) 5 + ( 3) d) 5 ( 3) Tell students that when faced with a subtraction question, you can always change it into an addition question. Write 7 ( 4). ASK: How do you subtract 4 on a number line? (move 4 units right) Why? (because you are undoing adding 4: you have to move left 4 units to add 4, so you have to move right 4 units to subtract 4) Moving 4 units right is the same as adding what number? (+4) Have a volunteer finish the equation: 7 ( 4) = 7 + (+4 or just 4) Have students use the same reasoning to answer these questions: a) ( 8) 3 = 8 + (answer: 3) b) ( 6) ( 2) = ( 6) + (answer: 2 or +2) c) 5 ( 3) = 5 + (answer: +3 or 3) d) 9 7 = 9 + (answer: 7) H-12 Teacher s Guide for Workbook 8.1

13 Generalizing from examples Writing subtraction questions with variables as addition statements. Have students write these questions as addition statements a) 5 ( 3) b) 4 ( 3) c) 3 ( 3) d) 2 ( 3) e) 1 ( 3) f) ( 5) ( 3) g) ( 3) ( 3) h) x ( 3) Emphasize that subtracting 3 is always like adding 3, no matter what you are subtracting 3 from. Then ask students to turn the following subtractions into additions: x ( 3) = y ( 2) = z 4 = a ( 5) = p (+1) = Bonus Find x ( 5) when x = 8. Subtraction as taking away. Write on the board: ( 3) ( 2). Remind students that subtraction also means taking away, so if you write 3 = 1 1 1, and 2 = 1 1, then you can take away two of the 1s from 3: take away 3 ( 2) = = 1 Have students solve these problems (some students might benefit from using integer tiles to literally remove the 1s): a) ( 8) ( 5) b) ( 4) ( 3) c) ( 6) ( 1) d) ( 7) ( 4) process expectation Problem Solving Then write on the board: ( 3) ( 5). Have students try to solve this question. ASK: What problem do you run into? (there are not enough 1s to take away: we need to take away five of them, but we only have three) Challenge students to find a way to overcome this problem. Suggest a few problem-solving strategies (EXAMPLES: look for a similar problem for ideas, generalize from known examples, look for a pattern). Allow students time to think. Then ASK: When have you ever had to subtract and found there s not enough of something to subtract from? Write on the board: Looking for a similar problem for ideas ASK: Do you remember in what grade you first learned how to do this question? (point to the first one) What made that problem harder than this one (point to the second one) ANSWER: In this one, there are enough ones to subtract from; in the first problem, there are not. Challenge students to think about how they solved 83 46, and to see if they can use a similar strategy to solve ( 3) ( 5). Again, allow students time to think. Remind students that to solve they had to add ones without changing the original number. ASK: How did you do that? (took a ten from the 8 and added ten ones to the 6) Number Sense 8-68 H-13

14 Now bring students attention back to the original problem: ( 3) ( 5). Write: 3 = Tell students that we want to take away five 1s but there are only three of them! ASK: How can we add more 1s but still keep the original number the same? PROMPT: If I add a 1, what do I have to add as well to keep the original number the same? (+1) How many 1s do I need to add? (2) Tell students that you will add two 1s and two +1s: 3 = ASK: Can I take away five 1s now? (yes) Demonstrate doing so: ( 3) ( 5) = = +2 Have students solve the following problems using this method: a) ( 2) ( 6) b) (+1) ( 2) c) ( 3) ( 7) d) ( 4) ( 9) e) (+3) ( 2) Extra Practice: BLM Subtraction as Take Away 4 4 = 4 3 = 4 2 = 4 1 = 4 0 = Use patterns to subtract. For example, to subtract 4 ( 3), have students first solve the sequence of problems shown in the margin. Then have students continue the pattern in both the question and the answer: The number being subtracted is getting smaller (4, 3, 2, 1, 0, 1, 2, 3, etc.) and the answer is getting bigger (0, 1, 2, 3, 4, 5, 6, 7, etc.). Continue until you reach 4 ( 3): 4 ( 1) = 5 4 ( 2) = 6 4 ( 3) = 7 Process assessment Workbook p. 186, Question 6, [C, R], 8m2, 8m7 ACTIVITIES Refer to the Activity in NS8-66/NS8-67, which uses a vertical number line to add integers. Using the same materials, review adding helium balloons and sandbags to the person and writing addition sentences to represent the person s new location. Process assessment [CN, V], 8m5, 8m6 Then put the person at sea level (0 m) holding all the helium balloons and sandbags, and take away a sandbag. ASK: When I take away a sandbag, what happens to the person? Why? (taking away a sandbag is the opposite of adding a sandbag, so if the person moves down 1 m when you add a sandbag, the person will move up 1 m when you remove a sandbag) Continue in this way, removing helium balloons (subtracting positive integers) and sandbags (subtracting negative integers) until there are no more left to take away. Have students write the integer subtraction that each take-away represents. EXAMPLE: Start at sea level (0 m above or below sea level) with 8 helium balloons and 8 sandbags. H-14 Teacher s Guide for Workbook 8.1

15 Take away... Location of person 1 sandbag +1, and we write this as 0 ( 1) = +1 3 helium balloons 2, and we write this as (+1) (+3) = ( 2) 2 helium balloons 4, and we write this as ( 2) (+2) = ( 4) 5 sandbags +1, and we write this as ( 4) ( 5) = (+1) 3 helium balloons 2, and we write this as (+1) (+3) = ( 2) 2 sandbags 0, and we write this as ( 2) ( 2) = 0 2. Integer card game for 2 players. Players will need a deck of cards (remove the face cards, but leave the jokers in). The black cards are positive, the red cards are negative, and jokers are zeros. Deal four cards to each player and lay the rest in a pile face down. The cards of each player are visible to both players. Players take turns drawing one card from the pile and laying the card face up. Players then look for cards at least one from the hand of each player that add to the card in the centre to result in 0. Example: If the card in the centre is a red (negative) 2, Player 1 can add a black (positive) 6, and Player 2 can add a red (negative) 4. Total: ( 4) = 0. All three cards are discarded. If players cannot produce a total of 0 with the card that is face up, the player who drew the card adds it to his or her hand and the other player draws a card from the pile. Players must have at least 4 cards in their hands at all times; if a player has less than 4 cards left, he or she takes more from the pile. The object of the game is to get rid of all the cards in the pile and in the players hands. When the cards in the pile are all played, players can add the cards in their hands and if the result is 0, they win (as a team). If it is not, they lose. (They will win if they have added the integers correctly.) Encourage students to notice that they simply need all the red cards to add to the same total as all the black cards in order to get an integer sum of zero. Extension Use opposite integers to subtract. Instead of adding 1s and +1s, add a large enough negative integer and its opposite so that you can perform the subtraction. EXAMPLE: ( 3) ( 5) = ( 3) + ( 2) + (+2) ( 5) since adding opposite integers adds nothing = ( 5) + (+2) ( 5) = (+2) since adding and subtracting the same integers adds nothing Number Sense 8-68 H-15

16 NS8-69 Subtraction Using a Thermometer Pages Curriculum Expectations Ontario: 7m26, 8m1, 8m2, 8m3, 8m4, 8m5, 8m6, 8m7, essential for 8m21, 8m22, 8m23 WNCP: 7N6, essential for 8N7, [C, CN, R, T, PS, V, ME] Vocabulary opposite integer Goals Students will subtract positive numbers from positive or negative numbers by using the model of temperature dropping on a thermometer. Students will then use patterns to predict how to subtract negative numbers from positive or negative numbers. PRIOR KNOWLEDGE REQUIRED Can add integers Knows that integers that add to 0 have the same number but a different sign Is familiar with negative temperatures Materials calculators The plan for the lesson. Explain to students that in the last lesson, they subtracted integers by using a number line. This time, they will subtract integers by using a thermometer model. Then, they will compare the two methods to see if they give the same answer. Modelling A thermometer as a vertical number line. Review adding and subtracting on a number line. Then turn the number line from a horizontal to a vertical position. Add by moving up and subtract by moving down. ASK: Where have you seen a vertical number line before? (EXAMPLES: a thermometer, the vertical axis on a graph, the scale on a measuring cup) If no one suggests thermometer, remind them. Then tell students that the temperature was 5 C and increased 2 C. ASK: What is the temperature now? (7 C) What operation did you use? (addition) Write on the board: = 7. Then tell students that the temperature was 5 C and dropped 2 C. ASK: What is the temperature now? (3 C) What operation did you use? (subtraction) Write on the board: 5 2 = 3. ASK: If 5 2 shows the temperature dropping from 5 C by 2 C, what would 2 5 show? (the temperature dropping from 2 C by 5 C) Explain that although we can t take away 5 objects when we have only 2 objects, temperature can drop by 5 C when it is only 2 C. Draw a thermometer on the board. Show starting at 5 C and dropping 2 C. Ask a volunteer to show starting at 2 C and dropping 5 C. ASK: What is the temperature now? ( 3 C) Write on the board: 2 5 = 3. Have students practise subtracting positive integers from smaller positive integers using the thermometer model. Students could copy a vertical thermometer from 5 C to 5 C into their notebooks to help them. H-16 Teacher s Guide for Workbook 8.1

17 EXAMPLES: 3 4 = 2 6 = 3 5 = 4 7 = The relationship between a b and b a. Have students discover this relationship by completing Workbook p. 187, Questions 1 and 2; students will subtract only positive numbers from positive numbers. Then explain the relationship as follows: = 5 5 = 0, so 5 2 and 2 5 add to 0. This means that they are opposite integers: they have the same numerical part, but opposite signs. Bonus = = = Process assessment [R, T], 8m2 Workbook p. 187, Question 2 The relationship between ( a) b and ( b) a. See Workbook p. 187, Questions 3 and 4. Have students subtract only positive numbers from negative numbers. Discuss how this subtraction fits with what they know about dropping temperatures. Whether the temperature was 2 C and dropped 5 C or whether it was 5 C and dropped 2 C, the temperature is now 7 C. This is just the fact that = 5 + 2, or 2 5 = 5 2. The relationship between ( a) b and a + b. For example, since = 7, ( 3) 4 = 7 (just change the answer from positive to negative). Show the symmetry of this on a vertical number line. First, start at 3 and move up 4 places, then start at 3 and move down 4 places. You are still 7 away from 0, but in opposite directions, so instead of reaching +7, you get to 7. Using temperatures, if the temperature starts at 3 C and increases by 4 C, it becomes 7 C; if the temperature starts at 3 C and decreases by 4 C, it becomes 7 C. Bonus ( 372) 327= ( 1234) 3421 = ( 34567) = Determine b ( a). Remind students that if they know 5 2, they can find 2 5 just by changing the sign. Tell students that you know that ( 3) 2 = 5 because if the temperature is 3 C and it drops by 2 C, then the temperature becomes 5 C (show this on a vertical number line). Write 2 ( 3) and ASK: Can I use the thermometer model to solve this problem? What does it mean for the temperature to drop 3 C? (Some students might think of this as an increase of 3 C. If so, tell them this could be one interpretation, and it will give the same answer, but because dropping a negative number of degrees might not make sense to everyone, you should explain it a different way.) ASK: How can I get 2 ( 3) from knowing ( 3) 2? PROMPT: When we found 2 5, how did knowing 5 2 help? Remind students that when they switch the two numbers being subtracted, they just change the sign of the answer: the answer to 5 2 is +3, so the answer to 2 5 is 3. Have students predict 2 ( 3). ASK: Since ( 3) 2 = 5, what should 2 ( 3) be? Write the following on the board: 8 1 = 7 so 1 8 = 7 ( 3) 2 = 5 so 2 ( 3) = ASK: Using the reasoning of the first example, what should go in the blank? (5) Why? (interchanging the numbers in the subtraction changes the sign Number Sense 8-69 H-17

18 Problem solving of the answer, so the answer should be +5) Then have students verify this using their calculators. First, ensure that students know how to add and subtract negative numbers on a calculator. The addition and subtraction buttons should be familiar, but the +/ button that allows students to input a negative number may not be. To find 2 ( 3), students should press: 2 3 +/ = Making and investigating conjectures, Technology Give students more problems to predict and check. EXAMPLES: ( 2) 4 = 6, so 4 ( 2) = ( 4) 2 = 6, so 2 ( 4) = ( 1) 3 = 4, so 3 ( 1) = Then have students make up their own examples to check. Finally, explain the result as follows: What is 4 ( 2) + ( 2) 4? Cup your hands around each subtraction statement in the expression: (4 ( 2)) + (( 2) 4). Then cover the 4s with your hands and explain that you are subtracting and then adding the same number ( 2), so they cancel: 4 ( 2) + ( 2) 4 = 4 4 = 0 (Analogy: = 4 4 = 0) Since 4 ( 2) and ( 2) 4 add to 0, they are opposite integers. This means that they have the same number, but opposite signs. Process assessment 8m1, 8m2, [PS, R] Another way to determine b ( a). Now write on the board: 8 2 = 6 so 8 6 = = 4 so ASK: Using the reasoning of the first example, what should go in the blank? (3 ( 4) = 7) Reflecting on the reasonableness of an answer Then have students redo examples of this form using this new method. That is, to find 3 ( 4) =, solve 3 = ( 4), i.e., how much do you have to drop from 3 C to get down to 4 C? Do students get the same answer? Compare subtracting a negative integer to adding a positive integer. First, have students subtract 3 from various positive numbers by first subtracting the positive numbers from 3: ( 3) 4 = so 4 ( 3) = (answer: 7) ( 3) 2 = so 2 ( 3) = (answer: 5) ( 3) 3 = so 3 ( 3) = (answer: 6) ( 3) 5 = so 5 ( 3) = (answer: 8) ( 3) 1 = so 1 ( 3) = (answer: 4) Now erase or cover the first part of each line above, so that students see only the second equation, and add new equations next to them as follows: 4 ( 3) = = 7 2 ( 3) = = 5 H-18 Teacher s Guide for Workbook 8.1

19 3 ( 3) = = 6 5 ( 3) = = 8 1 ( 3) = = 4 Generalizing from examples Revisiting conjectures that were true in one context Mental math and estimation Have students fill in the blank with the missing integer (the answer is always 3). ASK: What do you have to add to get the same answer as subtracting 3? (add 3) SAY: In these examples we subtracted 3 and saw that the answer is the same as adding +3. Do you think subtracting other negative numbers will be the same as adding the opposite positive numbers? How could we check? Have students do the ( a) b mentally to deduce b ( a) and fill in the missing number. EXAMPLE: 3 ( 2) = (think 2 3 = 5, so 3 ( 2) = 5) Have students suggest examples of their own and perform the subtraction. Then ask students to check whether adding the opposite integers in the same examples would produce the same answer (in the example above, = 5 gives the same answer). Finally, have students subtract negative integers by adding the opposite positive integer. EXAMPLES: 4 ( 2) = 4 + = 5 ( 3) = 5 + = 2 ( 5) = 2 + = Technology Subtracting a negative integer from a negative integer. Explain that so far in this lesson, we have only subtracted positive numbers from positive or negative numbers, and negative numbers from positive numbers. ASK: What kind of problem haven t we done? (subtract negative numbers from negative numbers) Write: ( 3) ( 5) = ASK: What would you predict we can add to 3 to get the same answer? Then write on the board: ( 3) ( 5) = ( 3) +. Have students use their prediction to fill in the blank and then check on a calculator. Students should press: 3 +/ 5 +/ = Connecting, Modelling Do they get the same answer as = 2? Moving up a thermometer to subtract a negative integer. Explain to students that they can subtract a positive number by imagining moving down a thermometer. They can subtract a negative number by adding the opposite positive number. To do this, they can imagine moving up a thermometer. See Workbook p. 188, Question 9. Connect the thermometer method to the number line method. Discuss how the thermometer is like a horizontal number line. Moving up the thermometer is like moving right on a number line. ASK: What is moving Number Sense 8-69 H-19

20 down the thermometer like? (moving left on the number line) ASK: How did you subtract a negative number on a number line? (moved right) What is moving right like? (moving up on a thermometer) Is that how you subtracted a negative number on a thermometer? (yes) ACTIVITY Repeat the Activity from NS8-68, which uses helium balloons and sandbags as a model, but with a temperature model. Draw a thermometer and a bathtub full of water on the board, and have ready several small red and blue cardboard circles. The red circles are hot stones that increase the temperature of the water by 1 C; the blue circles are cold stones that decrease the temperature by 1 C. Say the temperature in the tub is 20 C to start and mark that on the thermometer. Adding stones (and removing them!) changes the temperature of the water. Ask students to adjust the thermometer accordingly. EXAMPLE: add 2 red stones, temperature rises to 22 C; add 3 blue stones, temperature drops to 19 C; remove one of the blue stones, temperature rises to 20 C. Now tell students that corn oil has a freezing point of 20 C and that you filled the bathtub with corn oil instead of water. Repeat the exercises above using negative temperatures, starting at 10 C. Be sure to never reach temperatures below 20 C. If any students previously described dropping temperature by a negative amount as increasing temperature, this is a good place to point out why the students are correct: taking away something that makes the water colder will make the water warmer. Extensions 1. Students can use number lines to solve equations with integers: x ( 3) = 6 y ( 2) = 4 z 4 = 5 a ( 5) = 3 p (+1) = 6 EXAMPLE: x ( 3) = 5. To subtract 3, I have to go right 3 units. I end at 5. So to find the number I started with (x), I have to start at 5 and go left 3 units. This means x = 8. Indeed, if I move 3 places right from 8, I end up at H-20 Teacher s Guide for Workbook 8.1

21 connection Science 2. The boiling point of hydrogen is 253 C and the boiling point of oxygen is 183 C. Which gas has a lower boiling point, hydrogen or oxygen? How much lower? (Hydrogen has a lower boiling point by 70 C.) 3. Place the integers 1, 2, and 3 in the shapes so that the equation is true. a) + = b) = c) + = ( 2) d) + = ( 4) Answers: a) 1, 2, 3 or 2, 1, 3 b) 3, 1, 2 or 3, 2, 1 c) 3, 1, 2 or 1, 3, 2 d) 2, 3, 1 or 3, 2, 1 4. For each equation, put the same integer in all boxes. a) ( 4) + = ( 6) b) + = 2 Answers: a) 1 b) 2 Number Sense 8-69 H-21

22 NS8-70 More Gains and Losses NS8-71 Word Problems Pages Curriculum Expectations Ontario: 7m26, 8m6, 8m7, essential for 8m21, 8m22, 8m23 WNCP: 7N6, essential for 8N7, [V, C] Vocabulary opposite integers sum difference gain loss + ( ( ( + ( + Goals Students will use gains and losses to subtract integers and will solve problems involving addition and subtraction of integers. PRIOR KNOWLEDGE REQUIRED Understands subtracting positive numbers Understands that integers with the same number and different sign add to 0 Can compare and order integers Can write sums of integers as sequences of gains and losses Understands that variables represent numbers Subtraction using gains and losses. Tell students to think of subtracting a positive number as taking away a gain, and subtracting a negative number as taking away a loss. Taking away a gain of $5 is like losing $5, but taking away a loss of $5 is like gaining $5 (for example, if you owe me $5 but I say you don t have to give it to me, you re now $5 ahead of where you were yesterday). Draw the chart in the margin (also on Workbook p. 189) on the board. Have students use the chart to rewrite sums of integers as sequences of gains and losses. Using gains and losses to add and subtract integers. See Workbook p. 189, Question 2. Visualizing Process assessment 8m5, [CN] Workbook p. 189, Question 5 Workbook p. 190, Question 3 ACTIVITY If you have the equipment, videotape a student walking backwards. Play it, and then rewind it. What do you see when the tape is rewinding? (The student appears to be walking forwards.) Students can think of playing a tape as adding, rewinding the tape as subtracting, walking forwards as a positive integer, and walking backwards as a negative integer. Playing the tape (adding) is the opposite of rewinding (subtracting). This might help some students remember the rule that subtracting a negative integer does the same thing as adding a positive integer. H-22 Teacher s Guide for Workbook 8.1

23 Extensions 1. Subtraction Using Distance Apart. First review subtraction for positive numbers as distance apart: 7 2 is the distance between 2 and 7 on a number line. Then point out that if you subtract a smaller number from a larger number, the answer will be positive; if you subtract a larger number from a smaller number, the answer will be negative. To explain this, you could use the analogy of a credit card: If you have $2 but spend $7 on a credit card, you have less than nothing, so taking away 7 from 2 leaves a negative answer. If you have $7 but spend $2, you still have money left to spend, so taking away 2 from 7 leaves a positive answer. Then teach students to subtract in three steps. EXAMPLE: ( 8) (+5) Step 1: Find the distance between the two integers on a number line. That will be the number in the answer. Demonstrate this on a number line The distance between ( 8) and (+5) is 13. Step 2: Determine whether the answer will be positive or negative are you taking away a smaller or larger number? Because ( 8) is less than (+5), we are subtracting a larger number from a smaller number, so the answer will be negative. Step 3: Combine the sign of the answer from Step 2 with the number in the answer from Step 1. The answer to ( 8) (+5) is 13. Extra Practice: a) ( 5) ( 3) b) 8 ( 2) c) ( 3) ( 7) d) ( 9) (+4) e) ( 2) 1 Bonus Have students copy and complete this T-chart in their notebooks: Pair of Numbers Distance Apart 1 and and +2 3 and +3 4 and +4 5 and +5 6 and +6 Number Sense 8-70, 8-71 H-23

24 Students should extend the pattern and fill in the last two rows themselves. When students are finished, have them decide how far apart these pairs are: 50 and +50, 124 and +124, and Extra Bonus Calculate the distance between: 60 and +61, 349 and Patterns in Distance Apart a) How far apart are these pairs: 4 and +4 3 and +5 2 and +6 1 and +7 0 and +8 What do you notice about your answers? Why? Making and investigating conjectures b) How far apart are +3 and 4? How about 3 and +4? Repeat for +2 and 7, then 2 and +7. How far apart are +3 and +8? How about 3 and 8? Repeat for +4 and +6, then 4 and 6. What pattern do you notice? Make a conjecture and check your conjecture for more examples. 3. Time Zones Because of the Earth s rotation, different parts of the Earth have daylight at different times; when it is night in some places it is day in others. For example, when it is 3 p.m. in one place, it might be 1 a.m. in another place. Integers are used to show whether the time of day in a particular place is earlier or later than the time in London, England. Athens is +2, which means it is 2 hours later there than in London. Halifax is 4, which means that in Halifax, it is 4 hours earlier than in London. A circus is traveling the world. Here is a list of the time zones for the cities they travelled to: London, England 0 Toronto, Canada 5 Athens, Greece +2 Vancouver, Canada 8 Moscow, Russia +3 Cape Town, South Africa +1 Halifax, Canada 4 Bangkok, Thailand +6 a) Each time the circus enters a different time zone, performers need to change the time on their watches. Use a subtraction statement to determine how they should change the time after each move. i) They travel from Toronto to Moscow. +3 ( 5) = +8 They should change the time to be 8 hours later. ii) They travel from Moscow to Athens. +2 (+3) = 1 They should change the time to be 1 hour earlier. iii) They travel from Athens to Halifax. iv) They travel from Halifax to Cape Town. v) They travel from Cape Town to Bangkok. H-24 Teacher s Guide for Workbook 8.1