Unit 7 Number Sense - JUMP Math for AP Book 8-1...Unit 7 Number Sense. ... − 3 − 2 − 6 c) + 2 + 5 − 2 d) − 4 + 6 + 2 e) ... progress to longer sequences. Suggest that students

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Teacher’s Guide for Workbook 8.1

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.

MaterialsSome students might benefit from using integer tiles when adding and subtracting integers. We do not rely on them in our lessons though.

Unit 7 Number Sense

H-2 Teacher’s Guide for Workbook 8.1

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NS8-64 Gains and LossesPages 179–180

CUrriCULUM ExPECTaTioNSOntario: 8m1, 8m2, 8m7, essential for 7m13, 7m14, 8m21, 8m22, 8m23WNCP: essential for 7N6, 8N7, [r, C]

VoCabULary gain loss net gain net loss cancel plus sign (+)minus sign (−)

GoalsStudents will add sequences of gains and losses.

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: + 5 − 3 = 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: − 3 + 5. 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 − 137 + 142

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: + 3 + 2 − 4. 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 + 3 + 2? (+5) Write: + 3 + 2 − 4 = + 5 − 4. aSK: What is + 5 − 4? (+1) Write:

+ 3 + 2 − 4 = + 5 − 4 = +1

Ask students to find the net gain or loss:

a) + 2 − 5 − 4 b) − 3 − 2 − 6 c) + 2 + 5 − 2 d) − 4 + 6 + 2 e) + 3 − 2 + 7

Solution for a) + 2 − 5 − 4 = − 3 − 4 = −7

Prior KNoWLEDGE rEQUirED

Has an intuitive understanding of gaining and losing money

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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) + 3 − 4 + 2 b) + 2 + 3 − 4 c) − 4 + 3 + 2 d) + 2 − 4 + 3

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)

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) − 3 + 7 + 3 b) + 3 − 3 + 7 c) + 7 + 3 − 3 d) + 3 + 7 − 3

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) − 4 + 7 + 4 b) + 5 − 5 + 7 c) + 7 + 6 − 6 d) + 2 + 7 − 2

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

ProCESS ExPECTaTioN

Changing into a known problem


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NS8-65 integersPage 181

CUrriCULUM ExPECTaTioNSOntario: essential for 7m13, 7m14, 8m1, 8m2, 8m3, 8m5, 8m6, 8m7, essential for 8m21, 8m22, 8m23WNCP: essential for 7N6, 8N7, [r, C, CN, V]

VoCabULary integer positive negative more than (>) less than (<) cancel opposite integer

GoalsStudents will represent, compare, and order integers.

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 0 + 5 = 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:

0 − 5 0 − 4 0 − 3 0 − 2 0 − 1 0 0 + 1 0 + 2 0 + 3 0 + 4 0 + 5

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:

0 − 5 0 − 4 0 − 3 0 − 2 0 − 1 0 1 2 3 4 5

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:


Understands the relationship between how many more (less) than and addition (subtraction)

CoNNECTioN

Real world

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Number Sense 8-65

5 4 3 2 1 0 1 2 3 4 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 8.

−8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8

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:

−8 −7 −6 −5 −4 −3 −2 −1 0 +1 +2 +3 +4 +5 +6 +7 +8

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: +5 −2 3 +4 −7 7 +2 (+3) (−4)

aSK: Which two numbers are the same? (3 and (+3))

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

ProCESS ExPECTaTioN

Modelling, Visualizing


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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

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

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 −5c) −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

ProCESS ExPECTaTioN

Using logical reasoning

ProCESS ExPECTaTioN

Looking for a pattern

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Number Sense 8-65

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 0 + 3 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

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.

ProCESS ExPECTaTioN

Connecting

CoNNECTioN

Real world

ProCESS aSSESSMENT

8m7, [C]

ProCESS ExPECTaTioN

Representing, Reflecting on the reasonableness of an answer

CoNNECTioN

Algebra


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review integers. Draw the number line from 0 − 5 to 0 + 5 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 0 + 5.

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 +3 − +4 or −5 + +3 + −2 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.

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. 184. Have students predict the result before doing the investigation.

NS8-66 adding integersNS8-67 adding integers on a Number LinePages 182–184

CUrriCULUM ExPECTaTioNSOntario: 7m13, 7m14, 8m1, 8m2, 8m5, 8m21, essential for 8m21, 8m22, 8m23WNCP: essential for 7N6, 8N7, [r, CN]

VoCabULary integer positive negative more than (>) less than (<) cancel opposite integer counter-example

GoalsStudents will add integers in different ways.


Understands the relationship between positive integers and how many more than 0 Understands the relationship between negative integers and how many less than 0

[R], 8m1, 8m2 Workbook p. 183, Question 9

ProCESS aSSESSMENT

Making and investigating conjectures

ProCESS ExPECTaTioN

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Number Sense 8-66, 8-67

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 person3 helium balloons +34 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


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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

NS8-68 Subtracting integersPages 185–186

CUrriCULUM ExPECTaTioNSOntario: 7m26, 8m1, 8m2, 8m5, 8m6, 8m7, essential for 8m21, 8m22, 8m23WNCP: 7N6, essential for 8N7, [r, CN, V, PS, C]

VoCabULary integer positive negative

GoalsStudents will subtract integers using a number line.


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)

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Number Sense 8-68

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).

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

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:

−5 −4 −3 −2 −1 0 1 2−5 −4 −3 −2 −1 0 1 2

(−3) + 2 = (−1) (−3) − 2 = (− 5)

−5 −4 −3 −2 −1 0 1 2

(−3) − (−2) =

−5 −4 −3 −2 −1 0 1 2

(−3) + (−2) = (−5)

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.

ProCESS aSSESSMENT

8m6, [V]

ProCESS aSSESSMENT

8m7, [C]


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Ask students to tell how many units right or left they will move from −4 to perform the following additions and subtractions:

(−4) + (−2) = (to add −2, move 2 units left)

(−4) − (−2) = (to subtract −2, move 2 units right, the opposite of how to add −2)

(−4) + 3 = (to add 3, move 3 units right)

(−4) − 3 = (to subtract 3, move 3 units left, the opposite of how to add 3)

(−4) + (−1) = (move 1 unit left)

(−4) − (−1) = (move 1 unit right)

Then have students do all the problems above in their notebooks.

aNSWErS: −6, −2, −1, −7, −5, −3

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) 5 + 3 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)

ProCESS ExPECTaTioN

Changing into a known problem

H-13

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Number Sense 8-68

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:

−3 − (−2) = −1 −1 −1 = −1

take away

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)

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:

−4683

−4386

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 83 − 46 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)

Problem Solving

ProCESS ExPECTaTioN

ProCESS ExPECTaTioN

Generalizing from examples

ProCESS ExPECTaTioN

Looking for a similar problem for ideas


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Now bring students’ attention back to the original problem: (−3) − (−5). Write: −3 = − 1 − 1 − 1. 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 = − 1 − 1 − 1 −1 −1 + 1 + 1. aSK: Can I take away five −1s now? (yes) Demonstrate doing so:

(−3) − (−5) = −1 −1 −1 −1 −1 +1 +1

= +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

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) = 54 − (−2) = 64 − (−3) = 7

aCTiViTiES 1–2

1. 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.

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.

4 − 4 = 4 − 3 = 4 − 2 = 4 − 1 = 4 − 0 =

ProCESS aSSESSMENT

Workbook p. 186, Question 6, [C, R], 8m2, 8m7

ProCESS aSSESSMENT

[CN, V], 8m5, 8m6

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Number Sense 8-68

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: −2 + 6 + (−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.

ExtensionUse 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


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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.

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: 5 + 2 = 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.

NS8-69 Subtraction Using a ThermometerPages 187–188

CUrriCULUM ExPECTaTioNSOntario: 7m26, 8m1, 8m2, 8m3, 8m4, 8m5, 8m6, 8m7, essential for 8m21, 8m22, 8m23WNCP: 7N6, essential for 8N7, [C, CN, r, T, PS, V, ME]

VoCabULary opposite integer

GoalsStudents 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.


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

ProCESS ExPECTaTioN

Modelling

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Number Sense 8-69

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 − 2 + 2 − 5 = 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 178 − 187= 234 − 342 = 3456 − 7890 =

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 2 + 5 = 5 + 2, or − 2 − 5 = − 5 − 2.

The relationship between (−a) − b and a + b. For example, since 3 + 4 = 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) − 67890 =

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

ProCESS aSSESSMENT

[R, T], 8m2 Workbook p. 187, Question 2


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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 +/− =

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 − 3 + 3 − 4 = 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.

another way to determine b − (−a). Now write on the board:

8 − 2 = 6 so 8 − 6 = 23 − 7 = −4 so

aSK: Using the reasoning of the first example, what should go in the blank? (3 − (−4) = 7)

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 4 + = 72 − (−3) = 5 2 + = 5

ProCESS ExPECTaTioN

Problem solving

ProCESS ExPECTaTioN

Making and investigating conjectures, Technology

ProCESS aSSESSMENT

8m1, 8m2, [PS, R]

ProCESS ExPECTaTioN

Reflecting on the reasonable-ness of an answer

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Number Sense 8-69

3 − (−3) = 6 3 + = 65 − (−3) = 8 5 + = 81 − (−3) = 4 1 + = 4

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, 3 + 2 = 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 + =

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 +/− =

Do they get the same answer as −3 + 5 = 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

ProCESS ExPECTaTioN


ProCESS ExPECTaTioN

Revisiting conjectures that were true in one context

ProCESS ExPECTaTioN

Mental math and estimation

ProCESS ExPECTaTioN

Technology

ProCESS ExPECTaTioN

Connecting, Modelling


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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)

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.

aCTiViTy

Extensions1. Students can use number lines to solve equations with integers:

x − (−3) = −6 y − (−2) = −4 z − 4 = −5a − (−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 −5.

−9 −8 −7 −6 −5 −4 −3 −2

−9 −8 −7 −6 −5 −4 −3 −2

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Number Sense 8-69

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

CoNNECTioN

Science


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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.

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.

NS8-70 More Gains and LossesNS8-71 Word ProblemsPages 189–190

GoalsStudents will use gains and losses to subtract integers and will solve problems involving addition and subtraction of integers.


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

+ ( + +

+ ( − −

− ( + −

− ( − +

CUrriCULUM ExPECTaTioNSOntario: 7m26, 8m6, 8m7, essential for 8m21, 8m22, 8m23WNCP: 7N6, essential for 8N7, [V, C]

VoCabULary opposite integers sum difference gain loss

ProCESS ExPECTaTioN

Visualizing

ProCESS aSSESSMENT

8m5, [CN] Workbook p. 189, Question 5 Workbook p. 190, Question 3

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Extensions1. 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.

−8 −7 −6 −5 −4 −3 −2 −1 0 +1 +2 +3 +4 +5 +6 +7 +8

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 +1 2

−2 and +2

−3 and +3

−4 and +4

−5 and +5

−6 and +6


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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, −31 342 and +31 342.

Extra bonus Calculate the distance between: −60 and +61, −349 and +350.

2. 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?

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 ZonesBecause 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) = +8They should change the time to be 8 hours later.

ii) They travel from Moscow to Athens. +2 − (+3) = −1They 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.

ProCESS ExPECTaTioN


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vi) They travel from Bangkok to Vancouver.

vii) They travel from Vancouver back to Toronto.

b) Count the total number of hours they moved their watches earlier and count the total number of hours they moved their watches later. Explain why these numbers should be the same.

c) When flying from Toronto to Moscow, the circus leaves Toronto at 7:00 a.m. (Toronto time) and arrives in Moscow at 11:00 a.m. (Moscow time) the next day.

i) How long did the trip take?

ii) The trip included an 8-hour stopover. How much time did the circus spend in the air?

4. Plus/Minus ratings in Hockey A hockey player’s plus/minus rating, or +/− rating, is determined as follows: If the player’s team scores while he is on the ice, +1 is added to his rating; if the other team scores while he is on the ice, −1 is added to his rating. Use the statistics for the fictional women’s team below to answer the following questions, or visit the website of the National Hockey League (www.nhl.com/ice/playerstats.htm) to find and substitute up-to-date statistics for your students’ favourite NHL players and teams.

Name Position Goals a PTS + / −

Susan Centre 31 47 78 7

Miki Defence 19 49 68 −1

Tania Defence 9 58 67 −1

Voula Left-wing 28 33 61 −12

Natalie Centre 17 43 60 −18

Haley Centre 18 27 45 −9

Kate Centre 11 34 45 0

Colleen Right-wing 19 19 38 −19

Deepa Left-wing 21 17 38 15

Cassie Centre 12 19 31 13

Yolanda Left-wing 17 11 28 −6

Rena Centre 15 12 27 5

Maggie Right-wing 5 11 16 −10

Joanne Defence 6 6 12 −11

Michelle Defence 0 8 8 −5

Gina Centre 1 7 8 −15

ProCESS ExPECTaTioN

Reflecting on the reasonable-ness of an answer


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a) What does a positive +/− rating mean? What does a negative +/− rating mean?

b) How many players had a positive +/− rating? How many players had a negative +/− rating?

c) If the players were listed in order from best +/− rating to worst +/− rating, write down the first five names and the last five names in the order they would occur.

d) Who had the best +/− rating and who had the worst +/− rating? How much did their ratings differ by?

e) You are the team’s coach. In practice, you play Miki, Natalie, Deepa, Maggie, and Joanne against Susan, Tania, Colleen, Yolanda, and Rena. Based on only the sum total of each “team’s” +/− rating, who do you expect to win: Miki’s team or Susan’s?

f) Make up the +/− rating all-star team by choosing the best three forwards (one centre, one right-wing, one left-wing) and the two players in defence.

g) There are no goalies in the list. In fact, goalies do not keep track of +/− ratings. Why is this?

h) For a game that the team won 5–2, a player had a +/− rating of +2. What is the maximum number of goals (by both teams) the player could have been on the ice for? (6; 4 from her own team and 2 from the other team) Another player had a rating of −1 during the same game. What is the maximum number of goals (by both teams) that player could have been on the ice for? (3; 2 from the opposing team and 1 from her own)

5. Tom and Sara played a card game where you win and lose points. Tom lost 3 points the first game, gained 4 points the second game, and lost 5 points the third game.

a) Write down Tom’s sequence of gains and losses. (− 3 + 4 − 5)

b) How many points did Tom gain or lose overall? (he lost 4)

c) Sara confesses that she cheated in the last game. They agree to cancel that game. How many points does Tom have now? Write an integer subtraction and solve it. (− 4 − (−5) = − 4 + 5 = + 1)

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Number Sense 8-72

review adding negative integers. Include problems where students add several negative integers at a time. ExaMPLE: (−2) + (−3) + (−5) + (−3)

Multiplication is a short form for repeated addition. Write on the board: 2 + 2 + 2 + 2 + 2 = 10. aSK: How can we write this in terms of multiplication instead of addition? Write × 2 = 10 on the board, so that students write the multiplication in the order shown (5 × 2 instead of 2 × 5). aSK: How would we write 5 × 7 in terms of addition—how many 7s are we adding? (7 + 7 + 7 + 7 + 7) Repeat with:

5 × 2/3 (2/3 + 2/3 + 2/3 + 2/3 + 2/3) and 5 × (−2) ((−2) + (−2) + (−2) + (−2) + (−2)).

Have students write the following multiplication statements as repeated addition and then use their knowledge of adding integers to answer these questions:

a) 5 × (−3) b) 3 × (−2) c) 4 × (−3) d) 2 × (−4) e) 3 × (−1) f) 2 × (−7)

aNSWErS: a) (−3) + (−3) + (−3) + (−3) + (−3) = − 3 − 3 − 3 − 3 − 3 = −15

b) −6 c) −12 d) −8 e) −3 f) −14

Compare a × (−b) to a × b. Write the following problems below the corresponding problems above—to encourage students to see their association and compare them—and have students solve them:

a) 5 × 3 b) 3 × 2 c) 4 × 3 d) 2 × 4 e) 3 × 1 f) 2 × 7

aSK: How are these questions the same as the ones above? How are they different? How are the answers the same? How are they different?

ExTra PraCTiCE for Workbook p. 191, Question 2: Have students multiply a positive integer with a negative integer by first multiplying the two positive integers.

a) 4 × 5 = so 4 × (−5) = b) 3 × 6 = so 3 × (−6) =

NS8-72 Multiplying integersPages 191–193

CUrriCULUM ExPECTaTioNSOntario: 8m1, 8m2, 8m3, 8m4, 8m7, 8m21, 8m22WNCP: 8N7, [r, C]

VoCabULary integers positive negative the distributive law the commutative law the associative law

GoalsStudents will develop and apply the formula for multiplying integers.


Can add and subtract integers Knows and can apply the distributive law Can use repeated addition to multiply whole numbers Can use the order of operations with brackets for these operations: +, −, ×, ÷

ProCESS ExPECTaTioN



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c) 7 × 5 = so 7 × (−5) = d) 8 × 2 = so 8 × (−2) = e) 9 × 7 = so 9 × (−7) = f) 7 × 8 = so 7 × (−8) =

review the order of operations with +, −, ×, ÷. Write on the board: 3 × 4 + 2 × 4. Then write different calculations that students you know used to calculate this expression:

Sara: 3 × 4 + 2 × 4 = 12 + 2 × 4 = 14 × 4 = 56

Tom: 3 × 4 + 2 × 4 = 12 + 2 × 4 = 12 + 8 = 20

Jacob: 3 × 4 + 2 × 4 = 3 × 6 × 4 = 18 × 4 = 72

Rosa: 3 × 4 + 2 × 4 = 4 + 4 + 4 + 4 + 4 = 5 × 4 = 20

aSK: Without saying whether Sara’s calculation is right or not, what was she thinking? (she calculated from left to right) What was Tom thinking? (he calculated the multiplication first and then the addition) What was Jacob thinking? (he calculated the addition first and then the multiplication) What was Rosa thinking? (she replaced the multiplications with addition and then she used addition only to calculate the answer) Who is right? (Tom and Rosa)

Explain that because multiplication is short for addition, whenever we see multiplication we can replace it with addition, the way Rosa did: 3 × 4 means 4 + 4 + 4 and 2 × 4 means 4 + 4. If we read from left to right, we can’t add 2 to 3 × 4, because the 2 tells us how many more 4s there are; it’s not telling us to add 2. This means that we have to calculate the multiplication before doing the addition. This is what Tom did.

review the distributive law for multiplication and addition. Have students use Rosa’s method to calculate these products:

a) 3 × 5 + 4 × 5 b) 3 × 2 + 6 × 2 c) 4 × 3 + 2 × 3

aNSWErS: a) 7 × 5 = 35 b) 9 × 2 = 18 c) 6 × 3 = 18

Go over part a) together. aSK: How many fives did you add together? (7)

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Number Sense 8-72

How did you know to add 7 fives together? (7 is 3 + 4) Explain that 3 fives and 4 fives is 7 fives. Write: 3 × 5 + 4 × 5 = 7 × 5. Point to part b) and aSK: How many twos are in 3 twos + 6 twos? (9) Write: 3 × 2 + 6 × 2 = 9 × 2. How did you get the 9? (3 + 6) Then write: 3 × 2 + 6 × 2 = (3 + 6) × 2. Have students write the expressions for c) and d) using brackets as well:

4 × 3 + 2 × 3 = (answer: (4 + 2) × 3)4 × 4 + 3 × 4 = (answer: (4 + 3) × 4)

Then have students do the same for these expressions:

4 × 2 + 5 × 2 = (answer: (4 + 5) × 2)5 × 4 + 4 × 4 = (answer: (5 + 4) × 4)

Now have students go in the other direction: start with the expression with brackets and write the expression without brackets.

(2 + 7) × 3 = (answer: 2 × 3 + 7 × 3) (3 + 3) × 5 = (answer: 3 × 5 + 3 × 5) (4 + 3) × 2 = (answer: 4 × 2 + 3 × 2)

Remind students of the commutative law of multiplication—order doesn’t matter. Show students how this tells us that 3 × (2 + 7) = 3 × 2 + 3 × 7. (We just changed the order of all the multiplication statements in (2 + 7) × 3 = 2 × 3 + 7 × 3.) Tell students that these are both examples of the distributive law for multiplication and addition:

a × (b + c) = a × b + a × c and (b + c) × a = b × a + c × a

review the distributive law for multiplication and subtraction. Write on the board: 7 × 2. Have a volunteer write the corresponding addition statement: 2 + 2 + 2 + 2 + 2 + 2 + 2. Then write: 7 × 2 − 4 × 2. Have a volunteer take away 4 twos. aSK: How many twos are left? (3) How did you get 3 from 7 and 4? (7 − 4) Write: 7 × 2 − 4 × 2 = (7 − 4) × 2. As for addition, have students work in both directions, from expressions without brackets to expressions with brackets and vice versa.

ExaMPLES: (5 − 3) × 7 = 8 × 4 − 5 × 4 =

Then mix up all types of questions. See Workbook p. 191, Question 4.

investigate the distributive law when the subtraction results in a negative integer. Write on the board these two expressions:

2 × (5 + 3) and 2 × (3 + 5)

aSK: How are these the same and how are they different? Repeat for:

2 × (5 − 3) and 2 × (3 − 5)

aSK: How is the second expression special? (the answer inside the brackets is negative)

Have students write the expression the distributive law would give for 2 × (3 − 5). (2 × 3 − 2 × 5) aSK: Do you think the distributive law will still


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hold? How can we check? Have half the class calculate 2 × (3 − 5) and the other half calculate 2 × 3 − 2 × 5. (The answers are 2 × (3 − 5) = 2 × (−2) = −4 and 6 − 10 = −4.) Have students pair up with a partner who did the opposite question. (It is a good idea to pair up stronger students with weaker students for this activity.) Are the answers the same? Yes, they are! If any pairs do not have the same answers, they should try to find each other’s mistake(s).

Have students write the following questions without brackets, do the calculations both ways, and check if the answers are the same:

a) 3 × (4 − 8) b) 2 × (3 − 6) c) 4 × (5 − 6) d) 3 × (2 − 5)

Multiplication of integers is defined so that the distributive law holds. Tell students that you want to multiply integers, and you want a rule for doing so, but you want multiplication of integers to satisfy the same basic properties that normal multiplication satisfies. aSK: What properties should we try to satisfy? (Students can answer with examples or with descriptions, such as: 3 × 4 = 4 × 3, multiplying the same numbers in different orders doesn’t change the answer, multiplication commutes; 3 × (2 + 5) = 3 × 2 + 3 × 5, multiplication distributes over addition, 2 threes and 5 threes is the same as (2 + 5) threes; 3 × (4 × 5) = (3 × 4) × 5.) Tell students that we want all of these to be satisfied. Ensure that examples of at least the commutative and distributive laws are brought up.

Show students the following problem: (−3) × 4. If the commutative law holds, what should the answer be? (−12) Why? (because 4 × (−3) = (−3) + (−3) + (−3) + (−3) = (−12)) Now tell students that the distributive law should hold too. Write on the board: (−3) × 4 = (2 − 5) × 4. Have students use the distributive law to calculate what (−3) × 4 should be:

(2 − 5) × 4 = 2 × 4 − 5 × 4 = 8 − 20 = −12

aSK: Do you get the same answer using the distributive law as you do using the commutative law? (yes) aSK: I used −3 = 2 − 5. What else could I have used for −3? (ExaMPLES: 3 − 6, 4 − 7, 1 − 4, 0 − 3)

Have students solve (−3) × 4 using different expressions for −3:

(3 − 6) × 4 (4 − 7) × 4 1 − 4) × 4 (0 − 3) × 4

aSK: Which way was easiest? Why? (using 0 × 3 was easiest because subtracting from 0 is easiest—you just have to subtract 3 × 4 = 12 from 0, so you get −12)

Have students do the questions below twice, once using the commutative law and once using the distributive law.

a) (−2) × 7 b) (−5) × 6 c) (−3) × 3 d) (−4) × 5

Emphasize that they should get the same answer both ways; if they don’t, they should compare answers with a neighbour and ask the neighbour to help them find their mistake. Encourage students to replace the negative integer with zero minus something (ExaMPLE: replace −2 with 0 − 2).

ProCESS ExPECTaTioN

Reflecting on what made the problem easy or hard, Selecting tools and strategies

ProCESS ExPECTaTioN


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Number Sense 8-72

Remind students that this was how we defined negative integers in the first place.

Have students look at their answers. aSK: How does (−2) × 7 compare to 2 × 7? How does (−5) × 6 compare to 5 × 6. Have students use the corresponding positive multiplication to do these questions:

a) (−4) × 10 b) (−8) × 8 c) (−7) × 6d) (−3) × 7 e) (−5) × 9

Multiplying two negative integers. Write on the board: (−3) × (−2). Tell students that you want to multiply these two numbers together. aSK: How is this different from the problems we’ve done so far? (both numbers are negative) What should we use to multiply these two numbers together, the distributive law or the commutative law? Allow students time to think, then aSK: What happens if we switch the numbers and multiply (−2) × (−3)—is that easier? (no, the numbers are still both negative) Explain that the commutative law doesn’t help us here; we have to use the distributive law and hope that we can find the answer.

Write on the board:

(−3) × (−2) = (−3) × (0 − 2) = (−3) × 0 − (−3) × 2 (emphasize that we know how to multiply a negative and a positive integer) = 0 − (−6) (emphasize that we know how to subtract negative integers) = 0 + 6 = 6

Have students use the distributive law to do the same question many ways, below. Do students always get the same answer?

a) (−3) × (1 − 3) b) (−3) × (2 − 4) c) (−3) × (3 − 5)d) (0 − 3) × (−2) e) (1 − 4) × (−2) f) (2 − 5) × (−2)

bonus (−3) × (98 − 100)

Emphasize that we can do these questions because we know how to multiply a negative integer with a positive integer and we know how to subtract integers.

Have students do the following questions by replacing one of the negative integers with zero minus the appropriate positive integer.

a) (−4) × (−5) b) (−3) × (−7) c) (−5) × (−6) d) (−2) × (−8)

After students finish, point to a), and aSK: How does (−4) × (−5) compare to 4 × 5? (they have the same answer) Repeat for b), c), and d). aSK: How does (−a) × (−b) compare to a × b? (they have the same answer)

Have students use multiplication of positive numbers to multiply two negative numbers. See Workbook p. 193, Question 13.

ProCESS ExPECTaTioN

Selecting tools and strategies

ProCESS ExPECTaTioN

Reflecting on other ways to solve a problem

ProCESS ExPECTaTioN



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Mix up all types of problems. Write on the board:

3 × 4 = (−3) × 4 = 3 × (−4) = (−3) × (−4) =

aSK: When is the answer positive? (when both numbers being multiplied are positive or both are negative) When is the answer negative? (when one number is positive and the other is negative) Have students use multiplication of positive integers and changing the sign when necessary to multiply negative integers. See Workbook p. 193, Question 14.

ExtensionHave students multiply sequences of integers by multiplying two at a time.

ExaMPLE: (−3) × 2 × (−5) × 10 = (−6) × (−5) × 10 = 30 × 10 = 300

Give students several problems like this and collect their answers in a chart with headings Multiplication, Number of Minus Signs, Sign (+ or −) in Answer. aSK: When is the answer positive? (when the number of minus signs is even) When is the answer negative? (when the number of minus signs is odd)

Have students multiply long sequences of negative integers by multiplying the positive integers and changing the sign when appropriate.

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review the relationship between multiplication and division. Have students write two division statements for each multiplication statement:

a) 3 × 5 = 15 b) 2 × 7 = 14

apply the relationship between multiplication and division to integers. See Workbook p. 194, Question 1.

review integer multiplication. Write on the board:

a) positive times positive = b) positive times negative = c) negative times positive = d) negative times negative =

Have students write the answers individually. Then provide several examples of multiplication problems that require students to decide the sign of the answer.

Find the missing sign in a product to find a quotient. Write on the board:

4 × (−5) = (−20)

Say: The answer is negative. What combinations produce a negative answer? (positive times negative or negative times positive) The second number is negative. What must the first number be? (positive) Repeat the line of questioning for:

a) 2 × (−3) = 6 b) 3 × 4 = (−12) c) (−5) × 6 = 30

Then have students do problems individually:

a) 2 × (−4) = −8 b) −3 × 5 = 15 c) (−7) × 2 = 14

Find the missing number in a product to find a quotient. Write on the board: × (−3) = 12. aSK: Without telling me whether the number

CUrriCULUM ExPECTaTioNSOntario: 8m1, 8m2, 8m3, 8m4, 8m5, 8m7, 8m21, 8m22WNCP: 8N7, [r, CN, C, PS]

VoCabULary integer positive negative sum difference product quotient

GoalsStudents will develop and apply the formula for dividing integers, and will perform operations on integers in problem-solving contexts.


Can multiply integers Knows the relationship between multiplication and division

MaTEriaLS

bLM operations with integers (p H-40)

NS8-73 Dividing integersNS8-74 Concepts in integersPages 194–196


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is positive or negative, what number should go in the blank? (4) (ProMPT: Pretend all numbers are positive—what times three equals twelve?) aSK: The answer is positive. What combinations produce a positive answer? (both numbers are positive or both numbers are negative) We know one of the numbers is negative three. Is the other number positive or negative? (negative) What is the other number? (−4) So what is 12 ÷ (−3)? (−4)

Repeat the line of questioning for × (−4) = (−20) and for × 3 = −18. Then have students do several problems individually (ExaMPLE: × 2 = (−10) so (−10) ÷ 2 = ). Provide lots of problems so that students will have data for the next exercise.

Look for a pattern to develop a rule for dividing integers. Have students list all the answers to problems in which they divided a positive number by a negative number. Was the answer positive or negative? Have students predict a rule for dividing a positive number by a negative number. Then have students create more problems to check their prediction. Have students solve several problems using their new rule: divide as though the two numbers are positive and make the answer negative.

When students are finished, point out that we now know how to divide a positive number by a negative number. What other types of problems do we need to know how to do before we can say that we know how to divide integers by integers? Tell students there are four different types.

(aNSWEr: Divide positive by positive, positive by negative, negative by positive, and negative by negative.) Summarize as follows:

(+) ÷ (+) = (+) ÷ (−) = (−) ÷ (+) = (−) ÷ (−) =

Fill in the first two blanks together, and then have students determine the other two by looking at the answers to problems they have done so far and creating more of their own.

Challenge students to write a rule, in their own words, for dividing integers. After students finish, have them pair up and try to write a better sentence than either did individually.

applying the rule. Finally, have students use their rules to solve various division problems similar to those on Workbook p. 194, Question 4 (2-digit divided by 1-digit). Then give students calculators and have them divide 2-digit integers by 2-digit integers.

order of operations with integers. Provide bLM operations with integers for practice evaluating expressions involving brackets and integers, using the correct order of operations.

Before having students do Question 3 on the BLM, write on the board: “Add 3 and 9. Then multiply by (−2).” aSK: What are some ways to write this as an expression? (Answers: (3 + 9) × (−2) or (9 + 3) × (−2) or

ProCESS ExPECTaTioN


ProCESS ExPECTaTioN

Technology

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(−2) × (3 + 9) or (−2) × (9 + 3)) Repeat for “Add 3 and 9. Then divide by (−2).” aSK: Do these both mean the same thing: (3 + 9) ÷ (−2) and (−2) ÷ (3 + 9)? (No, one is dividing 12 by (−2), the other is dividing (−2) by 12; you can’t change the order of numbers in division like you can in multiplication.)

integers in context. We can think of future times as positive and past times as negative. So if now is time 0, then three hours from now will be +3 hours and three hours ago was −3 hours.

Tell students that you are trying to find the freezing and boiling points of cooking oil (freezing point less than 0°) by heating and cooling it. The oil’s temperature is currently 0°. Tell students that when the temperature increases, it will become positive, and when the temperature decreases, it will become negative. Right now, you are heating it and its temperature is increasing at a rate of 2° every minute. aSK: What will the temperature be in 3 minutes? (+2) × 3 = +6°. What was the temperature 3 minutes ago? (+2) × (−3) = −6°. Write on the board:

(rate of increase) × (number of minutes from now) = (new temperature)

If the number of hours from now is negative, because it was in the past, then the new temperature will be negative too. This makes sense because the temperature had to increase to get to the current temperature of 0°.

Now aSK: When will the temperature be 20°? Look at the formula already on the board:

(rate of increase) × (number of minutes from now) = (new temperature)

aSK: Which pieces of information do we know? (the first and third) Write: (+2) × = (+20) Have a volunteer rewrite this as a division statement: (+20) ÷ (+2) = . The answer is +10, so in 10 minutes from now, the temperature will be 20°.

aSK: When was the temperature −20°? Repeat the reasoning above to conclude that the answer is (−20) ÷ (+2) = −10, so the temperature was −20°, 10 minutes ago.

Now tell students that you put the temperature back to 0° and are cooling it, so that its temperature is decreasing at a rate of 2° every minute. Repeat the same types of questions, but with rate of increase −2 instead of +2.

Puzzles with adding and multiplying integers. Tell students that you found two integers that add to −5 and multiply to +6. Together, list the following pairs of integers that add to −5 (pairs that are both negative have been left out on purpose):

0, −51, −62, −73, −84, −9

CoNNECTioN

Real World

ProCESS ExPECTaTioN

Organizing data


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Have students find the products of the pairs listed so far: 0, −6, −14, −24, −36. aSK: Do you think the next pair will have an answer that is closer to +6 or further from +6? Should we continue as we are doing? (no, we are getting further from the product we want) Suggest to students that instead of going down the list, we should go up the list because as we go up the list, we are getting closer to the answer we want. Have students tell you what goes next, above the pair 0, −5. (−1, −4) Find the product (+4) and continue until you get the product +6. It will come next, with (−2, −3).

another way to solve the same problem. Point out that we needed to find all pairs that both add to −5 and multiply to +6. aSK: Instead of finding pairs that add to −5, what else could we look for? (pairs that multiply to +6) Together as a class, list all the pairs of numbers that multiply to +6:

+1, +6+2, +3−1, −6−2, −3

Have students add the numbers in each pair (7, 5, −7, −5) So the pair that adds to −5 is −2, −3. Discuss which method students like better. Emphasize that there are likely to be a lot fewer possibilities to check if students first find all pairs that multiply to the desired result than all pairs that add to the desired result.

ExtensionbLM T-tables (pp H-38–H-39) applies integers to growing and shrinking patterns—the gap is positive for increasing patterns and negative for decreasing patterns.

ProCESS aSSESSMENT

8m1, 8m4, [R] Workbook p. 195, Question 2

8m1, [R] Workbook p. 196, Question 8

8m2, [R] Workbook p. 196, Question 5

8m5, [CN] Workbook p. 196, Question 7

8m1, [PS] Workbook p. 196, Question 9

8m7, [C] Workbook p. 196, Question 10

ProCESS ExPECTaTioN

Looking for a pattern, Selecting tools and strategies

ProCESS ExPECTaTioN

Reflecting on other ways to solve a problem

Unit 7 Number Sense - JUMP Math for AP Book 8-1...Unit 7 Number Sense. ... − 3 − 2 − 6 c) + 2 + 5 − 2 d) − 4 + 6 + 2 e) ... progress to longer sequences. Suggest that students

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