Gap Closing. Measurement Algebra. Intermediate / Senior Student Book

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1 Gap Closing Number Sense Measurement Algebra Intermediate / Senior Student Book

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3 Gap Closing Fractions Intermediate / Senior Student Book

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5 Fractions Diagnostic...3 Comparing Fractions...6 Adding Fractions...13 Subtracting Fractions...21 Multiplying Fractions...28 Dividing Fractions...34 Relating Situations to Fraction Operations...40 Template Fraction Tower...46

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7 Diagnostic 1. List three fractions equivalent (equal) to each fraction. 2 a) 3 b) 8 10 c) Use a greater than (>) or less than (<) sign to make these statements true. a) b) c) d) e) f) Order these values from least to greatest: Draw a picture to show why each statement is true: 2 a) = 4 5 b) = Add each pair of numbers. a) b) c) d) e) f) Write a story problem that you could solve by adding 2 3 and September 2011 Marian Small, 2011 Fractions (IS)

8 Diagnostic 7. Draw a picture to show why each statement is true: 7 a) = 5 8 b) = Subtract: 7 a) b) c) d) e) f) Write a story problem that you could solve by subtracting from Draw a picture to show why each statement is true: 2 a) = 2 5 b) = Multiply each pair of numbers. a) b) c) d) Describe a situation where you might multiply September 2011 Marian Small, 2011 Fractions (IS)

9 Diagnostic 13. Draw a picture to show why each statement is true: 8 a) = 4 8 b) = Divide: 6 a) b) c) d) e) f) A painter uses cans of paint to paint 1 of a room. How much of a room could 4 he paint with 1 can of paint? 16. Write an equation involving fractions and an operation sign that you would complete to solve the problem. a) Mia read 5 of her book. How much of her book does she have left to read? 8 b) Mia read 5 8 of her book. She read 1 of that amount on Monday. What fraction 3 of the whole book did she read on Monday? c) Mia read 5 8 of a book. If she read 1 of the book each hour, how many hours 5 was she reading? 5 September 2011 Marian Small, 2011 Fractions (IS)

10 Comparing Fractions Learning Goal selecting a strategy to compare fractions based on their numerators and denominators. Open Question Choose two pairs of numbers from 3, 4, 6, 8, 9, 10, 12, 16, 20 to use as numerators and denominators of two fractions. For eample, you could use 4 6 and Make sure that some of your fractions are improper and some are proper. Make sure that some of your fractions use the same numerator and some do not. Tell which of your fractions is greater and how you know. Make more fractions following the rules above and compare at least si pairs of them. Tell how you know which fraction is greater each time. 6 September 2011 Marian Small, 2011 Fractions (IS)

11 Comparing Fractions Think Sheet Pairs of fractions are either equal or one fraction is greater than the other fraction. We can decide which statement is true by using a model, by renaming one or both fractions, or by using benchmarks. Using a Model To compare 3 5 and 5, we can use a picture that shows the two fractions lined 7 up, so we can see which etends farther. 3_ 5 5_ 7 3_ < 5_ 5 7 3_ 5 5_ 7 We can use parts of sets. For eample, use a number of counters, such as 35, that is easy to divide into both fifths and sevenths. 1 5 of 35 counters is 7 counters, so 3 of 35 counters is 21 counters of 35 counters is 5 counters, so 5 of 35 counters is 25 counters is more than 21, so 5 7 > September 2011 Marian Small, 2011 Fractions (IS)

12 Comparing Fractions Renaming Fractions To rename a fraction, we can think about how to epress the fraction as an equivalent fraction or equivalent decimal. Equivalent Fractions Two fractions are equivalent, or equal, if they take up the same part of a whole or wholes. For eample, 3 5 = _ 5 6_ 10 Each section of the 3 model is split into two sections in the 6 model. So, there 5 10 are twice as many sections in the second model, and twice as many are shaded. The numerator and denominator have both doubled. We can multiply the numerator and denominator by any amount (ecept 0) and the same thing happens as in the eample above. There are more sections shaded and more sections unshaded, but the amount shaded does not change. For eample, if we multiply by 3: 3 5 = = or If we multiply by 10: 3 5 = = Common Denominators and Common Numerators Renaming fractions to get common denominators or common numerators is helpful when comparing fractions. If we compare fractions with the same denominator, the one with the greater numerator is greater. If we compare fractions with the same numerator, the one with the lesser denominator is greater. To determine if 5 8 or 2 is more, we might use common denominators: = = = 4 6 = 6 9 = 8 12 = = = = Since 8 = and 2 3 = 16, 24 is a common denominator. 24 Since 15 < 16, then 5 8 < = = 4 6 = 6 9 = 8 12 = Since 8 = and 2 3 = 10, 10 is a common numerator. 15 Since fifteenths are bigger than siteenths, > 10 16, and 2 3 > September 2011 Marian Small, 2011 Fractions (IS)

13 Comparing Fractions Decimals Another way to rename a fraction is to represent it as a decimal. For eample, 2 = 2 3. Using a calculator, we see that 2 3 = The reason a = a b is because we can imagine a objects, each being shared b by b people. Each person gets one section of each object, so altogether each gets a sections of size 1 b ; that is a b. For eample, if four people share three bars each one gets one fourth from each of the 3 whole bars and that is 3 4. We can use decimal renaming to compare fractions, such as 2 3 to Since 3 = and 4 5 = 0.8, we see that 2 3 < 4 5. Using Benchmarks We can use benchmarks to compare fractions. Sometimes we can tell that one fraction is more than another by comparing them to 1. For eample, is more than 1, and 3 is less than 1, so > 3 4. Sometimes we can tell that one fraction is more than another since one is greater than one half and one is not. We can mentally rename one half to compare the other fractions to it. For the fraction 5 6, since one half is 3 6, 5 is more than one half. 6 For the fraction 3, since one half is , 3 is less than one half So, 6 > Sometimes we can tell one fraction is more than another by comparing them to whole numbers other than 1. For the fraction 11 4, since 2 is 8 4, 11 is more than 2. 4 For the fraction 5 3, since 2 is 6 3, 5 is less than So, 4 is more than September 2011 Marian Small, 2011 Fractions (IS)

14 Comparing Fractions 1. What fraction comparison is being shown? a) b) c) d) 2. Draw a picture to show why this statement is true. 3 a) 8 = b) 4 = Tell or show why 3 8 is not equivalent to Circle the greater fraction. Eplain why. 5 a) 9 or b) 9 or September 2011 Marian Small, 2011 Fractions (IS)

15 Comparing Fractions 5. How might you rename one or both fractions as other fractions to make it easier to compare them? Tell how it helps. 4 a) 5 and b) 5 and c) 8 and d) 3 and a) Why does it make sense that 4 = 4 9? 9 b) What are the decimal equivalents of 3 7 and 4 9? c) How could you use decimal equivalents to compare 3 7 and 4 9? 11 September 2011 Marian Small, 2011 Fractions (IS)

16 Comparing Fractions 7. How can each of these fraction pairs be compared without renaming them as other fractions or as decimals? 8 a) 3 and 2 5 b) 7 3 and c) 2 3 and d) 1 10 and e) 8 and 9 10 f) 8 9 and Describe a real-life situation when you might make each comparison: 3 a) 5 to 1 2 b) to A fraction with a denominator of 5 is between one fraction with a denominator of 3 and one fraction with a denominator of 4. Fill in all 9 blanks to show three ways this could be true. 3 < 5 < 4 3 < 5 < 4 3 < 5 < a) List all the fractions with a denominator of 3 between 2 and 3. b) List all the fractions with a denominator of 4 between 2 and 3. c) Is it possible to list all the fractions between 2 and 3? Eplain. 12 September 2011 Marian Small, 2011 Fractions (IS)

17 Adding Fractions Learning Goal selecting an appropriate unit and an appropriate strategy to add two fractions. Open Question Choose two different, non equivalent, fractions to add to meet these conditions. Their sum is a little more than 1. Their denominators are different. At least one denominator is odd. Tell how you predicted the sum would be a little more than 1. Calculate the sum and eplain your process. Verify that the sum is just a little more than 1. Repeat the steps at least three more times with other fractions. 13 September 2011 Marian Small, 2011 Fractions (IS)

18 Adding Fractions Think Sheet Adding fractions means combining them. Same Denominators To combine fractions with the same denominator we can count. For eample, is 3 fourths + 5 fourths. If we count the fourths, we get 8 4 fourths, so = 8 4. Another name for 8 is 2. 4 Notice we add the numerators, not the denominators, because we are combining fourths. If we combined the two denominators, we would get eighths not fourths. Using fraction pieces, we can see that = 8 4. If we combine the last 1 part of 4 5 the 4 with the 3 4, we see that 2 wholes are shaded. That shows that 8 is _ 4 5_ 4 Different Denominators To combine fractions with different denominators requires more thinking. For eample, look at this picture of The total length is not as much as 2 3 but it is more than _ 3 1_ 4 1_ 3 2_ 4 14 September 2011 Marian Small, 2011 Fractions (IS)

19 Adding Fractions If the fractions had the same denominator, we could count sections. We can create equivalent fractions for 1 3 and 1 that use the same denominator. 4 That denominator has to be a multiple of both 3 and 4, so 12 is a possibility because 3 4 = = 4 12 and 1 4 = 3, so we add 4 twelfths + 3 twelfths to get 7 twelfths. In this 12 picture, there are 7 sections shaded and each is a twelfth section. 4_ 12 1_ 3 1_ 4 3_ 12 1 So, = Use a Grid Another way to combine fractions is to use a grid. For eample, to add 2 3 and 1, we could create a 3-by-5 grid to show thirds and 5 2 fifths 3 is two rows and 1 is one column. 5 2_ 3 Use the letter to fill two rows to show 2. Use the letter o to fill a column to 3 1 show 5. Move counters to empty sections of the grid so each section holds only one counter. 13 Since 15 of the grid is covered, = Notice that 2 3 = and 1 5 = 3, so this makes sense. 15 1_ 5 o o o o o o 15 September 2011 Marian Small, 2011 Fractions (IS)

20 Adding Fractions Sometimes the sum of two fractions is greater than 1. To model this with fraction strips, we might need more than one whole strip. For eample, to show we can use 20 as the common denominator. 4 3_ 12_ = _ 15 _ = Move up to the first strip to fill the whole. Then, 7 is left in the second strip = or We can model the sum of two fractions that is greater than 1 with a grid. When you move the counters so there is only one counter in each section, the grid is filled, with 7 counters etra. 3 5 = 12 (marked with o s) = 15 (marked with s) 20 o o o o o o o o o o o o o o o o o o o o o o o o = September 2011 Marian Small, 2011 Fractions (IS)

21 Adding Fractions Improper Fractions To add improper fractions, we can use models or we can use equivalent fractions with the same denominators and count. For eample, = = 37 6 = We can check by estimating is close to = 6. 3 Mied Numbers To add mied numbers, such as , we could add the whole number parts 4 and fraction parts separately = 5 13 = a) How does this model show ? o o o b) What is the sum? 2. What addition is the model showing? a) b) 17 September 2011 Marian Small, 2011 Fractions (IS)

22 Adding Fractions c) d) o o o o o o o o o o o o o o o o o o e) o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 3. Estimate to decide if the sum will be more or less than 1. Circle MORE than 1 or LESS than 1. a) MORE than 1 LESS than 1 b) MORE than 1 LESS than 1 c) MORE than 1 LESS than 1 d) MORE than 1 LESS than 1 4. Add each pair of fractions or mied numbers. Draw models for parts c) and e). 2 a) b) c) d) e) f) September 2011 Marian Small, 2011 Fractions (IS)

23 Adding Fractions 5. The sum of two fractions is a) What might their denominators have been? Eplain. b) List another possible pair of denominators. 6. Choose values for the blanks to make each true. a) = 4 b) = 2 7. Lisa used cups of flour to make cookies and another 1 3 cups of flour to 4 bake a cake. a) How do you know that she used more than 3 cups of flour for her baking? b) How much flour did she use? 8. Write a story problem that would require you to add to solve it. Solve your 4 problem. 19 September 2011 Marian Small, 2011 Fractions (IS)

24 Adding Fractions 9. a) Use the digits 3, 5, 7 and 9 in the boes to create the least sum possible. + b) Use the digits again to create the greatest sum possible Kyle noticed that if you add two fractions, you get the sum s denominator by multiplying the denominators and you get the sum s numerator by multiplying each numerator by the other fraction s denominator and adding. For eample, for , the denominator is 5 8 and the numerator is a) Do you agree? b) Eplain why or why not. 11. You have to eplain why it does not work to add numerators and denominators to add two fractions. What eplanation would you use? 20 September 2011 Marian Small, 2011 Fractions (IS)

25 Subtracting Fractions Learning Goal selecting an appropriate unit and an appropriate strategy to subtract two fractions. Open Question Choose two fractions to subtract to meet these conditions. The difference is close to 1, but not eactly 1. Their denominators are different. One denominator is odd. Tell how you predicted the difference would be close to 1. Calculate the difference and eplain your process. Verify that the difference is close to 1. Repeat the steps at least three more times with other fractions. 21 September 2011 Marian Small, 2011 Fractions (IS)

26 Subtracting Fractions Think Sheet Subtracting can involve take away, comparing, or looking for a number to add. To subtract fractions with the same denominator we count. For eample, is 5 fourths 3 fourths. Since the denominators are fourths, if 4 we take 3 from 5, there are 2 left, so = 2 4 or 1 2. We subtracted the numerators because the fourths were there just to tell the size of the pieces. Nothing was done to the 4s. We can see that = 2 4 ( 1 2 ) using fraction pieces. To subtract fractions with different denominators requires a number of steps. For eample, look at the picture of The difference is the part of the 2 5 that etends beyond the 1 4. It is hard to tell how eactly long the dark piece is by looking at the picture. So we can create equivalent fractions for 2 5 and 1 that use the same 4 denominator. That denominator has to be a multiple of both 5 and 4, so 20 (5 4) is a possibility. 2 5 = 8 20 and 1 4 = 5, so we subtract 20 5 twentieths from 8 twentieths, leaving 3 twentieths. In the diagram there are 3 etra sections in shaded on top and each is a twentieth section = _ 5_ = _ 8_ = September 2011 Marian Small, 2011 Fractions (IS)

27 Subtracting Fractions To subtract fractions, we can use a grid. For eample, to subtract 1 4 from 5 6, we could create a 4-by-6 grid to show both fourths and siths. 5 is five of si 6 columns and 1 4 is one of four rows. Using counters to model 5, we rearrange 6 them so that 1 of the counters (one full row) can be removed. Since 14 sections 4 remain full, the difference is _ 20 _ = _ 24 6_ 24 = 14_ 24 If one of the fractions is greater than 1, more than one strip or grid may need to be used. For eample, to show : 5_ 3 Renaming the fractions as fifteenths can help. 5 3 = = = _ 3 Using two 3-by-5 grids to solve the same question, cover 5 by covering 5 rows. 3 Remove counters from 4 columns to remove 4 5 of a grid. 13 of a grid are left 15 covered. 23 September 2011 Marian Small, 2011 Fractions (IS)

28 Subtracting Fractions To subtract mied numbers, such as , we could add up. 4 For eample, if we add 1 4 to 1 3 4, we get to 2. If we add another we get to 4 1 3, 1 so = = _ 4 2 1_ _ 4 1_ 3 We could subtract the whole number parts and the fraction parts. Since 1 3 > 3, it 4 might make sense to rename as first = = = a) How does this model show ? b) How much is ? 24 September 2011 Marian Small, 2011 Fractions (IS)

29 Subtracting Fractions 2. What subtraction is the model showing? a) b) c) d) e) 3. Estimate to decide if the difference will be closer to 1 or 1. Circle your choice. 2 5 a) closer to 1 closer to 1 2 b) closer to 1 2 closer to 1 c) closer to 1 2 closer to 1 d) closer to 1 2 closer to 1 25 September 2011 Marian Small, 2011 Fractions (IS)

30 Subtracting Fractions 4. Subtract each pair of fractions or mied numbers. Draw models for parts c) and f). 9 a) b) c) d) e) f) The difference of two fractions is a) What do you think their denominators might have been? Eplain. b) List another possible pair of denominators. 6. Choose values for the blanks to make each equation true. a) 3 5 = 6 b) 12 1 = Sakura started with 3 cups of flour. She used 1 2 cups for one recipe. 3 a) About how much flour did she have left? b) How much flour did she have left? 8. Write a story problem that would require you to subtract from 3 1 to solve it. 3 Solve the problem. 26 September 2011 Marian Small, 2011 Fractions (IS)

31 Subtracting Fractions 9. a) Use the digits 2, 3, 5, 6, 7 and 9 in the boes to create a difference that is less than 1. b) Use the digits again to create a difference greater than Kyla noticed that if you subtract two fractions, you get the denominator of the answer by multiplying the denominators and you get the numerator of the answer by multiplying each numerator by the other fraction s denominator and subtracting. For eample, for 8 5 4, the denominator is 5 3 and the numerator is a) Do you agree? b) Eplain why or why not. 11. You have to eplain why it does not work to subtract numerators and denominators to subtract two fractions. What eplanation would you use? 27 September 2011 Marian Small, 2011 Fractions (IS)

32 Multiplying Fractions Learning Goal representing multiplication of fractions as repeated addition or determining area. Open Question Choose two fractions or mied numbers to multiply to meet these conditions. The product is a little more than 1. If a mied number is used, the whole number part is not 1. Tell how you predicted the product would be a little more than 1. Calculate the product and eplain your process. Verify that the product is close to 1. Repeat the steps at least three more times with other fractions. 28 September 2011 Marian Small, 2011 Fractions (IS)

33 Multiplying Fractions Think Sheet We have learned that 4 3 means 4 groups of 3. So it makes sense that means 4 groups of two-thirds: = = 8 or 2 thirds + 2 thirds + 2 thirds + 2 thirds = 8 thirds 3 We multiply 4 2 to get the numerator 8. The denominator has to be 3 since we are combining thirds. To multiply two proper fractions, we can think of the area of a rectangle with the fractions as the lengths and widths just as with whole numbers. For eample, 4 3 is the number of square units in a rectangle with length 4 and width 3. That is because there are 4 equal groups of 3. To multiply 2 3 4, we think of the area of a rectangle that 5 2 is 3 wide and 4 5 long = sections out of the total number of 3 5 sections are inside the rectangle. 4_ That means the area is and so = _ 3 The product of the numerators tells the number of sections in the shaded rectangle and the product of the denominators tells how many sections make a whole. We can multiply 5_ improper fractions 4 the same way as proper fractions. 1_ For eample, is the area of a rectangle 2 3_ 2 1 whole that is units wide and 1 1 units long. There 2 are 15 sections in the shaded rectangle and each is 1 8 of 1 whole, so the area is = _ 2 The shaded part of the diagram looks like it might be close to 2 wholes so the answer of 15 makes sense. Each shaded 8 square is 1 whole. 29 September 2011 Marian Small, 2011 Fractions (IS)

34 Multiplying Fractions To multiply two mied numbers, we can either rewrite each mied number as an improper fraction and multiply, or we can build a rectangle. For eample, 1 for _ 3 Legend: whole 6 3 1_ 2 3_ 3 2_ 2 1_ = = is the same as = 49 6 = How would you calculate 5 4? Why does your strategy make sense? 9 2. a) How does this model show ? b) What is the product? 30 September 2011 Marian Small, 2011 Fractions (IS)

35 Multiplying Fractions 3. What multiplication is being modelled? a) b) c) 1 1 d) Estimate to decide if the product will be closer to 1 or 1. Circle your choice. 2 a) closer to 1 2 closer to 1 b) closer to 1 2 closer to 1 c) closer to 1 2 closer to 1 d) closer to 1 2 closer to 1 31 September 2011 Marian Small, 2011 Fractions (IS)

36 Multiplying Fractions 5. Multiply each pair of fractions or mied numbers. Draw models for parts b) and e). You may use grid paper. 3 a) b) c) d) e) f) The product of two fractions is a) What fractions might they have been? Eplain. b) List another possible pair of fractions. 7. Choose values for the blanks to make this equation true. 3 1 = A recipe to serve 9 people requires 4 2 cups of flour. 3 a) About how much flour is needed to make the recipe for 6 people? b) How much flour is actually needed? 32 September 2011 Marian Small, 2011 Fractions (IS)

37 Multiplying Fractions 9. Write a story problem that would require you to multiply to solve it. 4 Then solve the problem. 10. a) Use the digits 1, 2, 3, 4, 5, and 6 in the boes to create a fairly high product. b) Use the digits again to create a fairly low product. 11. Is it possible to multiply two fractions and get a whole number product? Eplain. 12. You have to eplain why multiplying a number by 2 results in less than you started 3 with. What eplanation would you use? 33 September 2011 Marian Small, 2011 Fractions (IS)

38 Dividing Fractions Learning Goal representing division of fractions as counting groups, sharing or determining a unit rate. Open Question Choose two fractions to divide to meet these conditions. The quotient (the answer you get when you divide) is about, but not eactly, 3 2. The denominators are different. Tell how you predicted the quotient would be about 3 2. Calculate the quotient and eplain your process. Verify that the quotient is about 3 2. Repeat the steps at least three more times with other fractions. 34 September 2011 Marian Small, 2011 Fractions (IS)

39 Dividing Fractions Think Sheet Just as with whole numbers, dividing fractions can describe the result of sharing, can tell how many of one size group fits in another, or can describe rates. Sharing When we divide by a whole number, we can think about sharing. For 4 eample, 5 2 means that 2 people share 4 of the whole. 5 We divide the 4 sections into 2 and remember that we are thinking about fifths. Each person gets 2 5. Sometimes the result is not a whole number of sections. For eample, means that 2 people share Each gets 3. We can either multiply numerator and denominator by 2 to get the equivalent 5 fraction 6 or we can use an equivalent fraction model for 5 3. The thirds were split in half since there needed to be an even number of sections for 2 people to share them. Each person gets 5 6. If it had been 5 4, 3 the thirds could be split into fourths so that it would be possible to divide the total amount into four equal sections. Counting Groups Sometimes it makes sense to count how many groups. For eample, there are 5 groups of 1 6 in _ 6 2_ 6 3_ 6 4_ 6 5_ 6 1 5_ 1_ 6 6 = 5 35 September 2011 Marian Small, 2011 Fractions (IS)

40 Dividing Fractions 5 To model 6 2 6, we need to see how many groups of 2 6 are in 5. We figure out 6 how many 2s are in 5 or _ 6 2_ 6 3_ 6 4_ 6 5_ 6 1 5_ 2_ 1_ 6 6 = 2 2 If two fractions have the same denominator, all of the sections are the same size. To get the quotient, we figure out how many times one numerator fits into the other. For eample, for 7 8 3, we calculate 7 3 since that tells how many 8 groups of 3 of something are in 7 of that thing. For 3 8 6, we calculate 3 6, 8 1 which is 2, since that tells how much of the 6 8 fits in 3 8. If two fractions have different denominators, we can use a model to see how many times one fraction fits in another, or we can use equivalent fractions with the same denominator and divide numerators. For eample, for , the picture shows that the 1 3 section fits into the 1 2 section times. We could write as = 3 2 or 1 1. It is the same as representing the model 2 using equivalent fractions. Using a Unit Rate Another way to think about division is thinking about unit rates. For eample, if we can drive 80 km in two hours, we think of 80 2 as the distance we can travel in one hour. If a girl can complete 1 3 of a project in two days, we divide 1 2 to figure out 3 1 how much she can complete in one day. Similarly, if she can complete 3 of a project in 1 2 day, we divide to figure out how much she can complete 2 in 1 day. Since you know that we could also multiply 1 2; it makes sense 3 1 that = September 2011 Marian Small, 2011 Fractions (IS)

41 Dividing Fractions Suppose we know that 2 3 of a can of paint can cover 1 of a space. To figure 4 out how much of the space one full can of paint can cover, we calculate If 3 of a can covers 1 4 of a space, then 1 3 of a can covers 1 8 of the space ( ) and then 3 3, which is 1, can covers 3 8 of the space ( ). That means = 3 8. Since we know that = = 3, this makes sense. 8 What we did was divide the 1 by 2 and multiply by So = = We reversed the divisor and multiplied by the reciprocal, the fraction we get by switching the numerator and denominator. Another way to think about this is: = 3 since three are 3 groups of 1 in 1 whole = 3 2 since there are only half as many groups of 2 in 1 whole as there 3 would be groups of = since there are only one fourth as many groups of 2 3 in 1 4 as groups of 2 in a whole. 3 Improper fractions can be divided in the same way as proper fractions. Mied numbers are usually written as improper fractions to divide them. 1. How would you calculate each? Eplain why your strategy makes sense. 6 a) 8 4 b) a) How does this model show ? b) Estimate the quotient. 37 September 2011 Marian Small, 2011 Fractions (IS)

42 Dividing Fractions 3. What division is being modelled? a) 0 1_ 3 2_ 3 1 4_ 3 5_ 3 2 b) 0 1_ 4 2_ 4 3_ 4 1 5_ 6_ 7_ _ 4 4. Estimate each quotient as closer to 1 or 1 or 2. Circle your choice. 2 3 a) closer to 1 closer to 1 closer to 2 2 b) closer to 1 2 closer to 1 closer to 2 c) closer to 1 2 closer to 1 closer to 2 d) closer to 1 2 closer to 1 closer to 2 5. Divide each pair of fractions or mied numbers. Draw models for parts a) and c). You might use grid paper. a) b) c) d) e) f) You have 3 1 cups of flour to divide into equal batches for four recipes. 3 a) About how much flour is available for each batch? b) How much flour is that per batch? 38 September 2011 Marian Small, 2011 Fractions (IS)

43 Dividing Fractions 7. You can tile 2 5 of a floor area in 3 of a day. 4 a) How much of the floor can you tile in one full day? b) What computation can you do to describe this situation? 8. Write a story problem that would require you to divide 3 5 by 1 to solve it. Solve 8 the problem. 9. a) Choose values for the blanks to make this true. 3 3 = 4 b) List another possible set of values. 3 3 = a) Use the digits 2, 3, 5, 6, and 9 in the boes to create a small quotient. b) Use the digits again to create a large quotient. 39 September 2011 Marian Small, 2011 Fractions (IS)

44 Dividing Fractions 11. Kevin noticed that is 9 8, but = 8 9. a) Eplain why it makes sense that the first quotient is greater than 1, but the second one is less than 1. b) Eplain why it also makes sense that the quotients are reciprocals. 12. You have to eplain why dividing by 5 6 is the same as multiplying by 6. What 5 eplanation would you use? 40 September 2011 Marian Small, 2011 Fractions (IS)

45 Relating Situations to Fraction Operations Learning Goal connecting fraction calculations with real-life situations. Open Question Create two problems that could be solved using each of the equations below. Make the problems as different as you can. You do not have to solve them = = = = Eplain your thinking for each. 41 September 2011 Marian Small, 2011 Fractions (IS)

46 Relating Situations to Fraction Operations Think Sheet It is important to be able to tell what fraction operations make sense to use for solving fraction problems. Addition Addition situations always involve combining. For eample: I had 1 3 of my essay done. I did another 1 of the essay. How much of it is finished 2 now? Answer: = 5 of the essay 6 Subtraction Subtraction situations could involve take away, comparing, or deciding what to add. An eample of take away is: I had 3 4 cup of juice. I poured out 1 cup. How 3 much of a cup of juice is left? Answer: = 5 12 cup An eample of comparing is: I finished 5 8 of my project. Angela finished 1 of her 4 project. How much more did I finish? Answer: = 3 8 An eample of deciding what to add is: I have finished 5 of my project. How 8 much more is left? Answer: = 3 8 Multiplication Multiplication situations could involve counting parts of sets, determining areas of rectangles, or applying rates. An eample of counting parts of sets is: Jeff filled 8 cups, each 1 of the way. 3 If he had filled the cups to the top instead, how many cups would he fill? Answer: = 8 3. Another eample is: A class raised 2 of the money they needed to go on a trip. 3 The boys raised 2 of the class s contribution. What fraction of the whole amount 5 of money needed did the boys raise? 42 September 2011 Marian Small, 2011 Fractions (IS)

47 Relating Situations to Fraction Operations An eample of determining areas of rectangles is: One rectangle is 1 as long 3 2 and 3 as wide as a 4 8 rectangle. What is its area? Answer: = An eample of applying rates is: Jane can paint a wall in 2 of an hour. How 3 long would it take her to paint walls? Answer: = 5 3 hours Division Division situations could involve sharing, determining how many groups, or determining unit rates An eample of sharing is: 2 of a room must be painted. Four friends are going 3 to share the job. What fraction of the room will each paint if they work at the same rate? Answer: 2 4 = 2 of the room 3 12 An eample of determining how many groups is: You want to measure 5 3 of a cup of flour, but only have a 1 cup measure. How many times must you fill the cup measure? Answer: = times An eample of determining unit rate is: It takes you 1 2 of an hour to clean 2 5 of the house, how much of the house could you clean in 1 hour? Answer: = 4 5 of the house in 1 hour. More Than One Option Sometimes the same situation can be approached using more than one operation. For eample, if you can solve a problem by dividing, you can also solve it by multiplying. The last problem above could be solved by thinking: = 5. Every division can also be solved as a multiplication. Every subtraction can also be solved as an addition. 43 September 2011 Marian Small, 2011 Fractions (IS)

48 Relating Situations to Fraction Operations 1. Tell what operation or operations you could use to solve each problem. Write the equation to represent the problem. a) Cynthia has containers of juice. Each smaller container holds 2 as much 5 as a large container. How many small containers can she fill? 3 b) About 4 of the athletes in a school play basketball. About 1 of those players are 4 in Grade 9. What fraction of the students in the school are Grade 9 basketball players? c) Stacey read 1 5 of her book yesterday and 1 of it today. How much of the book 3 has she read? d) It takes Lea s mom about hours to drive to work in the morning and hours to drive home every afternoon. If it is Wednesday at noon and she went to work Monday, Tuesday and Wednesday (and is still there), about how many hours has she spent driving to and from work? e) The gas tank in Kyle s car was 7 full when they started a trip. Later in the day, 8 the tank registered 1 full. How much of the tank of gas had been used? 4 44 September 2011 Marian Small, 2011 Fractions (IS)

49 Relating Situations to Fraction Operations f) You can travel 2 3 of the way to your grandmother s home in 1 1 hours. How 2 much of the way can you travel in 1 hour? 2. a) What makes this a multiplication problem? Fred has times as much money as his sister. Aaron has 3 as much money as 4 Fred. How many times as much money as Fred s sister does Aaron have? b) What makes this a division problem? A turkey is in the oven for hours. You decide to check it every 1 of an hour. 3 How many times will you check the turkey? c) What makes this a subtraction problem? Ed has rolls of tape. He uses about 1 3 rolls to wrap gifts. How many rolls of 4 tape does he have left? 3. Write an equation you could use to solve each problem in Question 2. a) b) c) 45 September 2011 Marian Small, 2011 Fractions (IS)

50 Relating Situations to Fraction Operations 4. Why might you solve this problem using either multiplication or division? You have worked out hours this week. Each workout was 2 of an hour. How many times 3 did you workout? 5. How would you keep part of the information in Question 2b, but change some of it so that: a) it becomes a multiplication question b) it becomes a subtraction question 6. Finish this problem so that it is a division problem: Aleander has 3 1 dozen eggs What hints can you suggest to decide if a problem requires subtraction to solve it? 46 September 2011 Marian Small, 2011 Fractions (IS)

51 47 September 2011 Marian Small, 2011 Fractions (IS) Fraction Tower 1_ 20 1_ 20 1_ 20 1_ 20 1_ 20 1_ 20 1_ 20 1_ 20 1_ 20 1_ 20 1_ 20 1_ 20 1_ 20 1_ 20 1_ 20 1_ 20 1_ 20 1_ 20 1_ 20 1_ 20 1_ 10 1_ 10 1_ 10 1_ 10 1_ 10 1_ 9 1_ 9 1_ 9 1_ 9 1_ 9 1_ 9 1_ 9 1_ 9 1_ 9 1_ 8 1_ 8 1_ 8 1_ 8 1_ 8 1_ 8 1_ 8 1_ 8 1_ 6 1_ 6 1_ 6 1_ 6 1_ 6 1_ 6 1_ 5 1_ 5 1_ 5 1_ 5 1_ 5 1_ 4 1_ 4 1_ 4 1_ 4 1_ 3 1_ 3 1_ 3 1_ 2 1_ 2 1 1_ 18 1_ 18 1_ 18 1_ 18 1_ 18 1_ 18 1_ 18 1_ 18 1_ 18 1_ 18 1_ 18 1_ 18 1_ 18 1_ 18 1_ 18 1_ 18 1_ 18 1_ 18 1_ 12 1_ 12 1_ 12 1_ 12 1_ 12 1_ 12 1_ 12 1_ 12 1_ 12 1_ 12 1_ 12 1_ 12 1_ 15 1_ 15 1_ 15 1_ 15 1_ 15 1_ 15 1_ 15 1_ 15 1_ 15 1_ 15 1_ 15 1_ 15 1_ 15 1_ 15 1_ 15 1_ 10 1_ 10 1_ 10 1_ 10 1_ 10

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53 Gap Closing Decimals Intermediate / Senior Student Book

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55 Topic 2 Decimals Diagnostic...3 Multiplying Decimals...6 Dividing Decimals Order of Operations...16

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57 Diagnostic DO NOT USE A CALCULATOR FOR THIS DIAGNOSTIC. 1. Estimate each product, without calculating. a) b) c) Calculate each product. a) b) c) d) e) September 2011 Marian Small, 2011 Decimals (I / S)

58 Diagnostic 3. Suppose you know that = Eplain why has to be Estimate each quotient without calculating. a) b) c) Calculate each quotient: a) b) c) September 2011 Marian Small, 2011 Decimals (I / S)

59 Diagnostic 6. How can you predict why will be one tenth of without calculating either quotient? 7. In which order would you perform the calculations that are part of this question? ( ) a) Circle the correct equation = 15 or = b) Eplain why it is correct. 9. Circle the greater epression. Eplain your thinking [ ] or ( ) September 2011 Marian Small, 2011 Decimals (I / S)

60 Multiplying Decimals Learning Goal reasoning about the relationship between the products of decimals and related whole numbers. Open Question Wild Salmon $15.78 per kg GROUND BEEF $8.82 per kg All Natural! Grain Fed Hormone Free Antibiotic Free $14.29 per kg CHICKEN BREASTS BONELESS SKINLESS Choose two of the items. List five possible amounts of each you could buy that meet the rules below. The numbers of kilograms for each item must be in the form. kg or. kg. Altogether you must spend between $50 and $100. Tell the total price for each and tell why your answer makes sense. 6 September 2011 Marian Small, 2011 Decimals (I / S)

61 Multiplying Decimals Think Sheet We multiply a decimal by a whole number by adding the decimal the correct number of times. For eample, = since 4 a number means four groups of that number. Sometimes, we multiply a decimal by another decimal the same way. For eample, we think of as 1.5 groups of 4.2. That is half of 4.2. Since half of 4.2 is 2.1, we add = 6.3. But it is easier to think about multiplying decimals in terms of area. For eample, is the area of a rectangle that is 1.2 units wide and 1.5 units long Each section in the rectangle is 0.01 of a whole, since there are 100 squares in one whole. Notice that there are 12 rows of 15 sections. That makes 180 sections of = 1.80 since 1.80 is 180 hundredths Multiplying is the same as multiplying and knowing that the units are hundredths (2 decimal places) or realizing the answer is close to 2 1 = is one tenth of 1.2. So if we multiply instead, we could think of it as one tenth of Since = 1.80, must have as a product. If we know how to multiply fractions, we could think of as , which is also (125 14) thousandths. 7 September 2011 Marian Small, 2011 Decimals (I / S)

62 Multiplying Decimals 1. Suppose you know that = 322. Use that information to help you determine each of these products without using a calculator. a) b) c) d) Use estimation to eplain your answers to Questions 1c) and 1d). 3. Predict the number of decimal places that will be in each product without using a calculator. a) b) b) d) a) Sirloin steak costs $13.39 for 1 kilogram. How much do you pay for 2.1 kilograms? b) How could you figure out the price for 0.21 kg without using your calculator? 8 September 2011 Marian Small, 2011 Decimals (I / S)

63 Multiplying Decimals 5. What is the area of the parallelogram? 9.2 cm 4.5 cm 6. Suppose Shira is 1.12 times as tall as her sister, Lyla. If Lyla is 135 cm tall, how tall is Shira? 7. A car drives an average of 62.5 km per hour for 2.1 hours. How far did it go during that time? 8. Kellan used a calculator and said that = Eplain what is wrong with Kellan s thinking. 9 September 2011 Marian Small, 2011 Decimals (I / S)

64 Multiplying Decimals 9. Copy each number or symbol on a separate card. (Try this question without a calculator.) Rearrange the cards to create a multiplication a) How could you arrange the cards to create a product with three decimal places? b) How could you arrange the cards to create a product of about 200? c) How could you arrange the cards to create a product of about 2000? 10. Rhys says that when you multiply two numbers with digits after the decimal point, the number of digits after the decimal point in the product is the total of the number of digits after the decimal point in the two numbers you multiply. Do you agree? Eplain. 10 September 2011 Marian Small, 2011 Decimals (I / S)

65 Dividing Decimals Learning Goal reasoning about the relationship between the quotients of decimals and related whole numbers. Open Question A very large pot might hold L of soup. Imagine a pot that holds just a little more. Decide how much that amount will be, but make sure it is of the form. L. Choose a portion size for a bowl of soup for one person. It might be anywhere between 0.2 L and 0.4 L, but make it of the form 0. L. Decide how many portions (or parts of portions) the larger pot could hold. Justify how you know your answer makes sense. Repeat using two more pot sizes of the form. L and portion sizes of the form 0. L. 11 September 2011 Marian Small, 2011 Decimals (I / S)

66 Dividing Decimals Think Sheet When we divide 4 tens by 2 tens (40 by 20), the answer is the same as the answer to 4 2. That is because, if we divide numbers with the same units, we do not need to consider the units. The number of 2 ones in 4 ones is the same as the number of 2 tens in 4 tens or 2 hundreds in 4 hundreds can be thought of as 42 tenths 2 tenths; that is the same as 42 2 = 21. We can check this another way. Since 0.2 is the same as 1 5 and there are five 1 5 s in a whole, there should be sets of 0.2 in 4.2 and = We can record the computation like this: = = 2 42 Similarly, is 42 hundredths 2 hundredths; that is also 42 2 = 21. Sometimes, we might want to divide numbers where the number of decimal places is different; that means that the units are different. For eample, is 124 tenths 2 hundredths. 124 tenths = 1240 hundredths, so = 1240 hundredths 2 hundredths = = 620. You can check by realizing that there are 50 sets of 0.02 in one whole and that means there are (620) sets of 0.02 in = = September 2011 Marian Small, 2011 Decimals (I / S)

67 Dividing Decimals Sometimes the quotient is a decimal, not a whole number. For eample, consider : is 530 hundredths 25 hundredths = To divide 530 by 25, we can break the 530 into parts to divide by 25. For eample since 530 = , we can epress as: = = tenths So tenths = 21.2 When we divide, we rename the leftover piece in terms of tenths or hundredths or thousandths. 1. Suppose you know that = 51. Use that information to help you determine each of these quotients. a) b) c) d) Eplain why your answers to Questions 1c) and 1d) make sense. 3. Estimate each quotient. a) b) c) d) September 2011 Marian Small, 2011 Decimals (I / S)

68 Dividing Decimals 4. You bought 1.6 kg of meat. You paid $ How much did 1 kg cost? 5. What is the base length of the parallelogram? 4.5 cm cm 2 6. A car travelled km in 1.8 hours. What was the average hourly speed? 7. Is each statement possible when you divide tenths by tenths? Eplain each response. a) The quotient could be a whole number. b) The quotient could be a tenth. c) The quotient could be a hundredth. 8. Kevin used a calculator and said that = Eplain what is wrong with Kevin s thinking. 14 September 2011 Marian Small, 2011 Decimals (I / S)

69 Dividing Decimals 9. You are dividing 32.4 by 0.. You fill in the blank with a single digit. a) What is the least possible quotient? How do you know? b) What is the greatest possible quotient? How do you know? 10. For which division can you give an eact decimal answer? Eplain or Rhys says that you can only divide decimals if they have the same number of decimal places. Eplain whether you agree or disagree and why. 15 September 2011 Marian Small, 2011 Decimals (I / S)

70 Order of Operations Learning Goal recognizing that the same order of operations rules that apply to whole number calculations must apply to decimal calculations. Open Question Choose values of the form. or. between 1 and 10, one for each of the boes above. Choose operations to connect the boes. Add brackets if you wish. Your choice of operations should get you as close to 1.5 as possible when you use the order of operations rules. Repeat three more times using different sets of decimal values and operations. 16 September 2011 Marian Small, 2011 Decimals (I / S)

71 Order of Operations Think Sheet When an epression involves more than one operation, different people might interpret it different ways. There need to be rules so that everyone gets the same answer. For eample, consider : If we subtract 1.5 from 3.2 first and then multiply the result by 2, the answer would be 3.4, but if we multiply first and then subtract 3 from 3.2, the answer would be 0.2. The rules for order of operations are: Step 1: If there is a calculation within brackets (or parentheses), do what is inside the brackets first. For eample, for 1.5 [3 2.1], do the subtraction calculation inside the brackets first. If there are brackets inside brackets, work on the inside brackets first. Note: Sometimes brackets are round ( ) and sometimes they are square [ ]; the shape does not matter. Step 2: Perform all division and multiplication calculations net, in order from left to right. Note: It does not matter whether the division or multiplication comes first. For eample, for , first do and and then add. Step 3: Perform all addition and subtraction calculations net, in order from left to right. Note: It does not matter whether the addition or a subtraction comes first. For eample, for , first subtract and then add. If we are evaluating [ ], we: multiply 2.5 by 8, so the whole epression becomes (5.5) (1.1) + [20 4] subtract 4 from 20, so the whole epression becomes (5.5) (1.1) + 16 divide 5.5 by 1.1, so the whole epression becomes add 5 and 16, so the whole epression becomes 21 Some people call the rules for Order of Operations BEDMAS: B stands for brackets. E stands for eponents. (If there are squares or cubes, etc., do them before multiplying and dividing.) DM stands for dividing and multiplying. AS stands for adding and subtracting. 17 September 2011 Marian Small, 2011 Decimals (I / S)

72 Order of Operations 1. Tell which calculation you would perform first. a) ( ) 5 b) c) ( ) d) ( ) 2. Calculate each epression using the order of operations. a) b) c) d) 6.4 ( ) (4 3.1) e) [ ( )] f) ( ) September 2011 Marian Small, 2011 Decimals (I / S)

73 Order of Operations 3. Why are the answers to ( ) and ( ) different even though the calculations are the same? 4. Show that if you start at 0 and perform these three operations in different orders, you get different results: Divide by 1.5 Multiply by 3 Add Place brackets in the epression below to get 3.5 as a result a) Create an epression involving at least three operations that would give the same result if you calculated in order from left to right as if you used the proper order of operation rules. Make sure to use decimals. b) Eplain why the rules did not matter. 7. a) Create an epression involving decimal operations that would require knowing the order of operations rules to get a result of 4.5. b) What about the order of operations rules would you have needed to know? 19 September 2011 Marian Small, 2011 Decimals (I / S)

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75 Gap Closing Integers Intermediate / Senior Student Book

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77 Topic 3 Integers Diagnostic...3 Representing and Comparing Integers...5 Adding and Subtracting Integers...10 Multiplying and Dividing Integers...15 Order of Operations...21

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79 Diagnostic 1. Draw a number line from 10 to +10. Mark the locations of these integers: 2, 8, 0, Describe three things that the number 2 might represent. 3. Order these integers from least to greatest: 6, 2, 3, 8, 20, +15, 9, 9. Eplain how you know which number is the least. 4. Eplain why 2 < 1, even though +2 > +1. [Recall that < means less than and > means greater than. ] 5. Add each pair of integers. a) (3) + (8) b) (20) + (+16) c) (+9) + (13) d) (+13) + (3) 6. Use a model to show why your answer to Question 5c) makes sense. Eplain the model. 7. Subtract each pair of integers. a) 4 (2) b) 8 (+16) c) (9) (2) d) (11) (18) 3 September 2011 Marian Small, 2011 Integers (I/S)

80 Diagnostic 8. Use a model to show why your answer to Question 7d) makes sense. 9. Multiply each pair of integers. a) (3) 8 b) 9 (2) c) (5) (10) d) (9) (7) 10. Use a model to show why your answer to Question 9b) makes sense. Eplain the model. 11. Divide each pair of integers. a) (4) (2) b) (8) 4 c) 16 (4) d) (+20) (5) 12. Use a model to show why your answer to Question 11b) makes sense. Eplain the model. 13. Circle the correct equation. Eplain why it is right. (2) + 8 (4) = 24 or (2) + 8 (4) = Which of these epressions is greater? How much greater? (3) + 6 [4 (2)] or (3) (2) 4 September 2011 Marian Small, 2011 Integers (I/S)

81 Representing and Comparing Integers Learning Goal selecting a strategy to represent and compare integers depending on the values of those integers Open Question Integers include three groups of numbers: Positive integers: the numbers you say when you count (1, 2, 3, 4, 5, 6, ), although we often put a + sign in front of them when we are talking about them as integers (e.g., +1, +2, +3, ) Zero: 0 Negative integers: the opposites of the counting numbers (1, 2, 3, 4, ). Each negative integer is as far from 0 as its opposite on a number line, but on the left side of zero Choose eight integers, using these rules: Five of them are negative. When you place the integers on the number line, no two of the negatives are the same distance apart. Show them on a number line and check that you have used the rules. Order the eight integers from least to greatest. Choose two of the negative integers. Eplain how you positioned them on the number line. Show another way you might represent or describe those two negative integers. 5 September 2011 Marian Small, 2011 Integers (I/S)

82 Representing and Comparing Integers Think Sheet Representing Integers Integers include three groups of numbers: Positive integers: the numbers we say when you count (1, 2, 3, 4, 5, 6, ), although we often put a + sign in front of them when we are talking about them as integers Zero: 0 Negative integers: the opposites of the counting numbers (1, 2, 3, 4, ) The most familiar use of negative integers is for temperatures. For eample, 3 means 3 degrees below 0. Sometimes people use negative integers to describe debts; for eample, if you owe $5, you could say that you have 5 dollars. Negative numbers are sometimes used in golf scores and hockey statistics. The negative integers are to the left of 0 on the number line. They are opposites of the positive integers; opposite means equally far from 0. So, for eample (5) is eactly the same distance from 0 to the left as (+5) is to the right Both positive and negative integers can be far from 0. For eample, +200 is quite far from 0, but so is 200. They can also both be near to 0. For eample, +1 and 1 are both very close to 0. We can use an up-down number line, more like a thermometer, instead of a horizontal number line, to model the integers. Here the positive numbers are above the negative ones September 2011 Marian Small, 2011 Integers (I/S)

83 Representing and Comparing Integers We can also use counters to represent integers. We can use one colour to represent positive integers and a different colour to represent negative integers. For eample, +5 could be modelled as 5 light counters and 5 could be modelled as 5 dark ones Comparing Integers One integer is greater than another, if it is farther to the right on a horizontal number line or higher on a vertical number line. For eample, +8 > +2 and 2 > 6. [Recall that > means greater than and < means less than. } It is difficult to compare integers with counters; number lines make the most sense to use for comparison. 1. a) Draw a number line from 10 to +10. Mark these integers with dark dots on your number line: +2, 6, 8, +7. b) Mark each of their opposites with an X. 2. What integers are represented? (Remember that positive is light.) a) b) c) Draw the opposites of each of the integers in parts a) and b) and tell what the new integers are. 7 September 2011 Marian Small, 2011 Integers (I/S)

84 Representing and Comparing Integers 3. Name two integers to meet each requirement: a) 5 units from 2 on a number line b) 12 units from +4 on a number line c) 5 units from 8 on a number line 4. Two opposite integers are 16 apart on a number line. What could they be? 5. Describe something that one of the integers you named in Question 3a might represent. 6. Do you think that 0 has an opposite? If so, what is it? 7. Replace the with a greater than (>) or less than (<) sign to make these epressions true. a) 2 +2 b) c) Put these integers in order from least to greatest: 2, +8, +2, 6, 10, 1 9. List two integers to fit each description. a) between 4 and 3 b) between 4 and 10 c) between 12 and +1 d) a little greater than 4 e) a little less than 9 8 September 2011 Marian Small, 2011 Integers (I/S)

85 Representing and Comparing Integers 10. Fill in the blanks so that these temperatures are in order from coldest to warmest: a), 3,,, +1 b) 12,, 10,,, 5 c), 5, +2, 11. List four possible values to make the statement true or eplain why it is not possible. a) an integer greater than 2 and greater than 8 b) an integer greater than 12 and less than 2 c) an integer greater than 2 and less than Why is any negative integer less than any positive one? 9 September 2011 Marian Small, 2011 Integers (I/S)

86 Adding and Subtracting Integers Learning Goal selecting a strategy to add or subtract integers depending on the values of those integers Open Question The sum of two integers is between 20 and 4. If you subtract these two integers, the difference is a negative number close to 0. List four possible pairs of integers. Eplain how you know they are correct. Make up two other rules describing a sum and difference of two integers; either the sum or the difference or both must be negative. Choose four possible pairs of integers to satisfy those rules. Show that they satisfy the rules you made up. 10 September 2011 Marian Small, 2011 Integers (I/S)

87 Adding and Subtracting Integers Think Sheet Adding When we add two integers, we combine them, just as when we add whole numbers. We must also remember the Zero Principle: = 0. For eample: If a boy had $1 (+1) in his wallet and a debt of $1 (1), it is as if he had $0 (or no money). Also moving forward one step on a number line from 0 in the direction of +1 and then moving one step in the direction of 1 puts you back at When you have counters, the zero principle allows you to model 0 with any equal number of dark and light counters since any pair of light and dark counters is 0. If we add two numbers on a number line, we start at the first number and move the appropriate number of steps in the direction from 0 of the second number. The sum is the final location. For eample, 2 + (+8) means: Start at 2. Move 8 steps to the right (direction of +8 from 0). The landing spot is (2) + (+8) = (+6) (2) + (7) means: Start at 2 and move 7 steps to the left (direction of 7 from 0). The landing spot is (2) + (7) = (9) 11 September 2011 Marian Small, 2011 Integers (I/S)

88 Adding and Subtracting Integers If we add integers using counters, we represent both numbers and combine them. We can ignore any pairs of counters that make 0, since 0 does not affect a sum. For eample: +2 + (8) is shown at the right: +2 + (8) = 6 Subtracting On a number line, it is useful to think of subtracting by thinking of what to add to one number to get the total. For eample, 3 (7) asks: What do I have to add to 7 to get to 3? The result is the distance and direction travelled (7) = Using counters, it is useful to think of subtraction as take away. To model 3 (7), we show three dark counters, but we want to take away 7 dark counters. 3 Since there are not seven dark counters to take away, we add zeroes to the three dark counters until we have enough to take away. The result is what is left after the seven dark counters are removed. 3 (7) = +4 If numbers are far from 0, we imagine the counters or number line. For eample, on a number line (45) (+8) asks what to add to 8 to go to 45. We start at 8. We go 8 back to get to 0 and another 45 back to get to 45, so the result is 53. Or if we have 45 dark counters and want to remove eight light ones, we need to add 0 as eight pairs of dark and light counters. We remove the eight light ones and are left with 53 dark counters: (45) + (8) = 53. Some people think of a b as a + (b) and think of a + (b) as a b. 12 September 2011 Marian Small, 2011 Integers (I/S)

89 Adding and Subtracting Integers 1. Model and solve each addition. a) (7) + (+7) b) (+3) + (4) c) (3) + (4) d) (3) + (+4) e) (8) + (+7) f) (20) + (+19) 2. You add a number to (3) and the sum is negative. List four possible numbers you might have added and four you could not have added. 3. You add two integers and the sum is 4. a) List two possible negative integers. b) List two possible integers that are not both negative. 4. Katie added all the integers from 20 to +20 in her head. Eplain how could she have done that. 5. Is the statement sometimes true, always true, or never true? Eplain. a) The sum of two positive integers is positive. b) The sum of two negative integers is negative. c) The sum of a positive and negative integer is positive. 13 September 2011 Marian Small, 2011 Integers (I/S)

90 Adding and Subtracting Integers 6. Solve each subtraction. Model at least three of them. a) (7) (+7) b) (+7) (7) c) 8 (4) d) 4 8 e) 4 (8) f) 34 (19) g) 19 (+34) h) 19 (34) 7. How can you use a number line to show why 5 (4) is the opposite of (4) 5? 8. You subtract two integers and the difference is 8. What could those integers be? 9. Use a model to eplain why any number (7) is the same as that number Complete the statement to make it true. a) If you subtract a positive integer from a negative one, the result. b) If you subtract a negative integer from a positive one, the result. c) If you subtract a negative integer from a negative one, the result is negative if. d) If you subtract a positive integer from a positive one, the result is negative if. 14 September 2011 Marian Small, 2011 Integers (I/S)

91 Multiplying and Dividing Integers Learning Goal justifying the value and sign of a product or quotient of two integers Open Question The product of two integers is between 100 and 20. If you divide the same two integers, the quotient is an integer close to 0. List four possible pairs of integers. Make sure some of the divisors are positive, some are negative, and the quotients are all different. Show that your pairs satisfy the rules. Make up two other rules describing a product and quotient of two integers (some of the results must be negative). Choose four possible pairs of integers to satisfy your rules. 15 September 2011 Marian Small, 2011 Integers (I/S)

92 Multiplying and Dividing Integers Think Sheet Multiplying We know how to multiply two positive integers. For eample, (+3) (+2) means three groups of +2. That is the same as three jumps of 2 starting at 0 on the number line or three sets of two light counters In a similar way, (+3) (2) is three jumps of (2) starting at 0 on the number line or it could be modelled as three sets of two dark counters (+3) (2) = 6 Since it does not matter in which order you multiply numbers, (3) (+2) is the same as (+2) (3). That is 2 jumps of (3) starting from 0 or two groups of three dark counters (3) (+2) = (6) Notice that the products for (+3) (2) or (+2) (3) are the same and the opposite of the product for (+3) (+2). There is no simple way to model (3) (2), but it does make sense that the product should be the opposite of (+3) (2) and the result is +6. See Questions 5 and 6 for other ways to understand why (a) (b) = +ab. Notice that (3) (2) = (+3) (+2) and (3) (+2) = (+3) (2). 16 September 2011 Marian Small, 2011 Integers (I/S)

93 Multiplying and Dividing Integers Dividing We already know how to divide two positive integers. For eample, (+12) (+4) = (+3), since division is the opposite of multiplication and (+3) (+4) = (+12). In a similar way, (12) (+4) = (3) since (+4) (3) = 12. We model this calculation by thinking of dividing 12 dark counters into four equal groups and noticing that there are three dark counters (3) in each group. (12) (+4) = 3 (12) (4) = (+3) since (4) (+3) = (12). We model this calculation by thinking: How many groups of four dark counters are there in 12 dark counters? Since the answer is three groups, (12) (4) = +3. It is difficult to model (+12) (4), but it does make sense that the quotient is 3 since (4) (+3) = 12. See Question 11 for another way to understand why (+a) (b) = (a b). Notice that (12) (3) = (12) (3) and (12) (+3) = (+12) (3). 1. Solve each multiplication. Model parts a, b, and c. a) (6) (+2) b) (+3) (4) c) (3) (+4) d) (7) (+7) e) (8) (+7) f) (10) (+19) 17 September 2011 Marian Small, 2011 Integers (I/S)

94 Multiplying and Dividing Integers 2. You multiply a number by (3) and the product is negative. List four possible numbers you could have multiplied by 3 and four you could not have multiplied. 3. You multiply two integers and the product is (36). a) List four possible pairs of integers. b) Eplain why 4 (9) is not a solution. 4. Is the statement true or false? Eplain. a) The product of two positive integers is always positive. b) The product of two negative integers is always negative. 5. a) Complete this pattern. What do you notice? 3 (2) = 2 (2) = 1 (2) = 0 (2) = (1) (2) = (2) (2) = b) What pattern could you create to show why (3) (6) = (+18)? 18 September 2011 Marian Small, 2011 Integers (I/S)

95 Multiplying and Dividing Integers 6. Karan says that since 3 4 is the opposite of 3 4, then 3 (2) should be the opposite of 3 (2). a) Do you agree with Karan? b) How would that help Karan figure out (3) (2)? 7. Model and solve at least three of these. a) (49) 7 b) 49 (7) c) 36 (6) d) (81) 9 e) (22) (2) f) (40) (8) 8. Why does it make sense that 30 (6) is negative? 9. Two other integers have the same quotient as 40 (5). List three possible pairs of integers. 19 September 2011 Marian Small, 2011 Integers (I/S)

96 Multiplying and Dividing Integers 10. You divide two integers and the quotient is 12. List four possible pairs of integers. 11. a) Complete the pattern. What do you notice? (12) (4) = (8) (4) = (4) (4) = 0 (4) = 4 (4) = 8 (4) = b) What pattern could you create to show why (+9) (3) = 3? 12. Complete the statement to make it true. a) If you divide a positive integer by a negative one, the result b) If you divide a positive integer by a positive one, the result c) If you divide a negative integer by a negative one, the result d) If you divide a negative integer by a positive one, the result. 20 September 2011 Marian Small, 2011 Integers (I/S)

97 Order of Operations Learning Goal recognizing that the same order of operations rules that apply to whole number calculations must apply to integer calculations Open Question Choose values anywhere from 10 to + 10 for numbers to insert in the boes. Some of them should be negative. Then choose at least three different operations to connect the boes and add brackets if you wish. Your choice should result in a 5 when you use the order of operations rules. Repeat three more times using at least some different integers. 21 September 2011 Marian Small, 2011 Integers (I/S)

98 Order of Operations Think Sheet If an epression involves more than one operation, we could interpret it different ways. There need to be rules so that everyone gets the same answer. For eample, think about: (2) (8) (+2). If we subtract 8 from 2 first and then multiply the result by +2, the answer would be +12. But if we multiply (8) (+2) first to get 16 and then subtract 16 from 2, the answer would be +14. The rules for Order of Operations are: Step 1: Brackets first. For eample, for (2) [+3 (4)], subtract 4 from 3 first. Notes: i) The brackets are not the parentheses we use for positive and negative integers, but ones that include calculations. ii) Sometimes brackets are round ( ) and sometimes they are square [ ]; the shape does not matter. If there are brackets inside brackets, work on the inside brackets first. Step 2: Division and multiplication calculations net, in order from left to right. It does not matter whether division or multiplication comes first. For eample, for (12) (3) + (12) (2), first do (12) (3) and then (12) (2) and then add. Step 3: Addition and subtraction calculations net, in order from left to right. It does not matter whether addition or a subtraction comes first. For eample, for (12) (+3) + (4), first subtract and then add. If we were evaluating (15) (+3) [(2) (8) (4)], we would think: (15) (+3) [16 (4)] Some people call the Order of Operation rules BEDMAS: B stands for brackets. E stands for eponents. (If there are squares or cubes, etc., do them before multiplying and dividing.) DM stands for dividing and multiplying. AS stands for adding and subtracting. 22 September 2011 Marian Small, 2011 Integers (I/S)

99 Order of Operations 1. Calculate using the correct order of operations. a) (+12) + (4) (+3) (2) (6) b) (3) (4) + (12) (3) (5) c) 9 (8) (3) d) 50 (2 + (3)) (4 (2)) e) [6 + (3) (4 + 8)] 4 (2) f) 6 + (3) (4 + 8) 4 (2) 2. Why are the answers to Questions 1e) and 1f) different, even though the numbers and operations involved are the same? 3. Show that, if you start at 0 and perform the following three operations in different orders, you get different results: Divide by 4 Multiply by 2 Add 6 23 September 2011 Marian Small, 2011 Integers (I/S)

100 Order of Operations 4. a) Place brackets in the epression below to get a result of (2) b) Place brackets in the same epression to get a result of (2) a) Create an epression that would give the same result, if you calculated in order from left to right, as if you used the proper order of operation rules. Make sure to use integers and include both a division and a subtraction. b) Eplain why the rules did not matter in this case. 6. Create two of your own epressions, involving integer operations that would require knowing the order of operations rules, to get a result of September 2011 Marian Small, 2011 Integers (I/S)

101 Gap Closing Proportional Reasoning Intermediate / Senior Student Book

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103 Topic 4 Proportional Reasoning Diagnostic...3 Describing and Representing Ratios, Rates, and Percents...6 Equivalent Forms of Rates, Ratios, and Percents...12 Solving Ratio and Rate Problems...18 Solving Percent Problems...26 Template 10-Part Spinner...31

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105 Diagnostic 1. There are 8 boys and 3 girls on the Tech Team. a) Write the ratio of he number of girls to number of boys in the form :. b) Write the ratio of the number of boys to the number on the whole team. c) Another Tech Team of 11 students has a higher ratio of number of girls to number of boys. What could the ratio be? 2. The ratio of Vada s height to Melissa s height is 5 : 4. a) Who is taller? b) Is she twice as tall? How do you know? 3. Valene s running rate is 0.18 km/min. Eplain what that means. 4. a) What percent of the grid is shaded? b) What percent is not shaded? 3 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

106 Diagnostic 5. Indicate whether each statement does or does not make sense by circling your choice. a) 8% of something is a lot of it. MAKES SENSE DOES NOT MAKE SENSE b) 80% of something is a lot more than half of it. MAKES SENSE DOES NOT MAKE SENSE c) 35% of the people in a high school building on a school day are adults, not students. MAKES SENSE DOES NOT MAKE SENSE 6. Eplain your answer to Question 5c. 7. Complete the missing amounts so that the ratios are equivalent. a) 2 : 7 = : 14 b) 5 : 10 = : 8 c) 12 : = 3 : 5 8. Suppose your heart beats 144 times in 2 minutes. How many times would you epect it to beat in 5 minutes? 9. What fraction is equivalent to each percent? a) 40% b) 112% c) 3.5% 10. Three bars of soap cost $2.61. At this rate, how much would each number of bars below cost? a) 6 bars b) 8 bars 4 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

107 Diagnostic 11. A car goes 78 km in 45 minutes. At that speed, how far would it go in an hour? 12. A 2.6 L container of juice costs $3.00. How much are you paying for 1 L? 13. Suppose the ratio of the number of boys to the number of girls in a class is 7 : 3. What percent of the class is girls? 14. A T-shirt is priced at $ The store is offering a discount of 30%. How much will the shirt cost (before taes)? 15. Tell if each statement is TRUE or FALSE by circling the correct word. a) 40% of 120 is about 30. TRUE FALSE b) 20% of 83 is about 16. TRUE FALSE c) 11% of 198 is about 20. TRUE FALSE 16. Eplain your answer to Question 15a. 17. Lea spent $25 of the money she saved. She still has 60% of her money left. How much does she have left? 5 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

108 Describing and Representing Ratios, Rates, and Percents Learning Goal representing comparisons based on multiplying as either ratios, rates and percents Open Question Ratios, rates, and percents all describe comparisons. For eample: Ratio A recipe uses 3 parts flour for every 1 part sugar. Rate A painter uses one can of paint to cover 2 walls in 5 hours. Percent 52 is 52% of 100. Search the Internet and find three or four eamples for each type of comparison (ratio, rate, and percent) that are related to one of these topics: environmental issues sports the arts Each time, indicate what two things are being compared. Write down the url. 6 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

109 Describing and Representing Ratios, Rates, and Percents (Cont.) Think Sheet These epressions describe comparisons using a ratio, a rate and a percent: Three girls for every four boys is a ratio that compares the proportion of girls to boys. We can write that ratio as 3 : 4. 3 and 4 are called terms in the ratio: 3 is the first term and 4 is the second term. For eample, if a class has 12 girls and 16 boys, you could arrange them to show that there are 3 girls for every 4 boys. G G G B B B B [3 girls for 4 boys] G G G B B B B [3 girls for 4 boys] G G G B B B B [3 girls for 4 boys] G G G B B B B [3 girls for 4 boys] 12 girls 16 boys Notice that the ratio of girls to all the students is not 3 : 4; it is 3 : 7 since there are 7 students in total for every 3 girls. We could also say 3 of the 7 class is girls. 3 boes for $4 is a rate that compares an amount of goods to a dollar amount. For eample, if 3 boes of one brand of a product costs $4 and 3 boes of another brand of that product costs $5, we could choose the best buy by comparing the prices of 1 bo of each brand or by comparing how much of each brand $1 buys. A rate compares two things measured in different units. This time the units were boes and money. Sometimes rates describe speed (kilometres per hour or metres per second) or a map scale (1 centimetre on the map for every 12 kilometres of real distance.) 7 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

110 Describing and Representing Ratios, Rates, and Percents (Cont.) Saying that 20% of the game is over uses a percent to compare a part of the game to the whole game. A percent is a ratio where the second term is 100. We could also write 20% as the ratio 20 : 100. Since percents are ratios out of 100, a 100-grid is a good way to represent a percent. The model shows 20% (20 squares out of 100). Part of the game that is over Part of the game that remains Whole game To compare the part of the game that is over to the part that remains, you could use the ratio 20 : 80 and use the same picture. 1. Draw a picture to show each ratio. a) 2 circles : 3 squares b) 4 circles : 5 shapes 2. Which circle is more red? Eplain why. A: 2 parts red out of 6 equal parts. B: 2 parts red out of 7 equal parts C: 3 parts red out of 8 equal parts 8 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

111 Describing and Representing Ratios, Rates, and Percents (Cont.) 3. You can mi 1 cup of water with different numbers of cups of orange juice to get different tastes. Which of these ratios of water to orange juice will taste the most watery? Eplain why. 1 : 3 1 : 4 1 : : Yasir built a scale model of a bird. He decided to use a ratio of: 3 : 2 model lengths : real lengths a) Was the model bigger or smaller than the real bird? Eplain. b) If a claw on the bird was really 2 centimetres, how long was it on the model? c) What does the ratio 2 : 3 tell in this situation? 5. The ratio of the length to the width of a rectangle is 12 : 4. a) Is the rectangle almost square or not? Eplain. b) What is the ratio of the length to the perimeter? c) Why might the length and width be either 12 centimetres and 4 centimetres or 12 metres and 4 metres, but not 12 metres and 4 centimetres? 9 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

112 Describing and Representing Ratios, Rates, and Percents (Cont.) 6. One measure of fitness is based on comparing your body fat mass to your total mass. A low ratio suggests that you are more fit. Which ratio of body fat mass to total mass is better: 3 : 10 or 3.4 : 10? Eplain. 7. Why does your pulse describe a rate? 8. The word per is often used to describe rates. For eample, you might talk about kilometres per hour. It can also be shown as a / (e.g., km/hr). List at least three other rates you might describe using the word per. 9. What percent might you be representing on a 100-grid if you shade: a) every other square? Columns b) columns 1, 2, 4, 5, 7, and 8? c) every 4 th square? d) most, but not all, of the squares? e) just a few squares here and there? 10 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

113 Describing and Representing Ratios, Rates, and Percents (Cont.) 10. On a particular day, 5% of all of the people in a school building are adults. The rest are students. a) What percent are not adults? b) What is the ratio of students to adults? 11. Which of these statements make sense? Eplain your reasoning for each one. a) You eercise vigorously 30% of the day. b) If you flip a coin, it will land on heads 50% of the time. c) A penny is worth 1% of a loonie. d) If you buy a sweater and save 5%, you must have saved less than your friend who bought a sweater at 10% off. 11 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

114 Equivalent Forms of Rates, Ratios, and Percents Learning Goal representing comparisons based on multiplying in a variety of different ways Open Question Any fraction can be written in equivalent forms. For eample, 2 3 = 6 9, 9 15 = 3 5, and 4 = Ratios, rates, and percents can also be written in equivalent forms Spin the spinner 9 times to fill in the digits. Ratio Rate Percent : km/ h % Show that each can be written in an equivalent form that somewhere includes the number 10. Repeat twice more. Ratio Rate Percent : km/ h % Ratio Rate Percent : km/ h % 12 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

115 Equivalent Forms of Rates, Ratios, and Percents Think Sheet There are many ways to describe the same ratio or rate. The equivalent forms can be other ratios or rates, or they can be fractions or percents. Ratio For eample, if the gym has 4 soccer balls for every 2 basketballs, the ratio of soccer balls to basketballs is 4 : 2. That means for every 2 soccer balls, there must be 1 basketball. 4 : 2 = 2 : 1 But that same ratio could also be described as 8 : 4, since for every 8 soccer balls, there would be 4 basketballs. 4 : 2 = 8 : 4 Notice that each time, the first term is double the second. Since that relationship is the same, the ratios are equivalent. Just as with fractions, if we multiply the two terms by the same amount, we will have an equivalent ratio : 1 = 4 : 2 = 12 : 6 The equation that says that two ratios are equal is called a proportion. 13 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

116 Equivalent Forms of Rates, Ratios, and Percents Rate The same is true for rates. A rate of $3 for 4 boes is the same as a rate of $6 for 8 boes, $9 for 12 boes, $1.50 for 2 boes or 75 for 1 bo. The equivalent rate for one item is called a unit rate. Percent Percents can also be described in equivalent forms. For eample, 25% means 25 out of 100, or = 1 4 or any fraction equivalent to 1, for eample, 50 out of 200 or out of 400. When a ratio or percent is written in a form where the two parts have no common factor, the ratio is in lowest terms. For eample, 50 : 100 is not in lowest terms but the equivalent ratio 1 : 2 is in lowest terms. 1. Which of these ratios are equivalent ratios? a) 2 : 5 and 2 : 3 b) 2 : 5 and 4 : 10 c) 4 :10 and 6 : September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

117 Equivalent Forms of Rates, Ratios, and Percents 2. Draw a picture that eplains why the ratio 4 : 5 is equivalent to the ratio 8 : 10. Tell how the picture shows this. 3. Some of the columns in a place value chart are shown: Show that the ratios of any column heading to the column heading two columns to its right are equivalent. 4. i) These ratios are equivalent. Fill in the missing terms. a) 4 : 10 = 2 : b) 6 : 8 = 9: c) 8 : = 20 : 10 d) 52 : 13 = : 300 e) 3.5 : 10.5 = : 6 f) 5 : 8 = 1 : ii) Eplain your strategy for part c) and e). 5. One way to compare ratios is to use equivalent ratios. Suppose one dessert uses four cups of strawberries for every three cups of blueberries. Another uses two cups of strawberries for every one cup of blueberries. Which equivalent ratios might you use to decide which is more strawberry? Eplain your reasoning. 15 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

118 Equivalent Forms of Rates, Ratios, and Percents 6. The ratio of the number of boys to the number of girls in one class is 7 : 3. The ratio of the number of boys to the number of girls in another class is 3 : 2. Which class has a greater fraction of boys? How do you know? 7. One car drives 32 kilometres every 15 minutes. Another drives 120 kilometres per hour. Use equivalent ratios to decide which is going faster. 8. Five bars of soap cost $3.89. What is an equivalent description of that rate? 9. The heart rates of different animals are shown below. Dog Lion Elephant Chicken 200 beats in 2 minutes 40 beats in 1 minute 140 beats in 4 minutes 120 beats in 30 seconds a) Write each rate as a unit rate (number of beats in 1 minute). b) Which animal s heart beats fastest 16 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

119 Equivalent Forms of Rates, Ratios, and Percents 10. What fraction with a numerator of 1 or 2 would be good to estimate each percent? Eplain your thinking. a) 30% b) 15% c) 70% d) 11% 11. Canada s population is growing by 1.3% a year. a) Write 1.3% as an equivalent fraction. b) Is a growth of 25 people for every 2000 people a higher or lower rate of growth? Eplain. 12. Jamila earned 12 marks out of 15 on her project. What would her percentage mark be? 13. Fifty students tried out for a music competition. Twelve were Grade 9 students; ten were Grade 10 students; thirteen were Grade 11 students; and 15 were Grade 12 students. What percent of the students who auditioned were at each grade level? 14. A certain number is 20% of Number A and is also 40% of Number B. a) Which number is bigger Number A or Number B? b) What is the relationship between those numbers? 17 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

120 Solving Ratio and Rate Problems Learning Goal using an equivalent form of a ratio or rate to solve a problem Open Question Keena s brother told her these things were true: a) If you drive 30 kilometres in 13 minutes, then you would drive 32 kilometres in 15 minutes if you kept the same speed. b) If there are adults and children in a large group and the ratio of the number of adults to the number of children is 5 : 3, then the total number of people has to be a multiple of 8. c) If you buy $1 Canadian with $1.08 U.S., then you can buy $1 U.S. with 92 Canadian. d) If the length and width of two different rectangles are in a 5 : 2 ratio, the ratio of their diagonals, perimeters, and areas are also in a 5 : 2 ratio. With which do you agree? Eplain why. With which do you disagree? Eplain why. Make up a similar statement that is true. Prove that it is true. Make up a similar statement that sounds true, but really is not true. Prove that it is not true. 18 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

121 Solving Ratio and Rate Problems Think Sheet Sometimes a situation is described using a ratio or rate, but we need an equivalent form to be able to solve a problem. For eample: Ratio problem The Canadian flag s length-to-width dimensions are 2 : 1. We want to know how wide to make a flag that is 51 centimetres long. You want a ratio equivalent to 2 : 1 where the first term is 51.? We are trying to figure out if 2 : 1 = 51 :. One way to solve the problem is to figure out what we multiplied 2 by to get 51. Then we multiply 1 by the same amount. Since 51 2 = 25.5, we multiplied 2 by 25.5 to get 51. Then the width must be centimetres = 25.5 cm. Another way to solve the problem is to notice that the width is always half the length, so just take half of 51. This is also 25.5 centimetres. Rate problem We know that a family drove 130 kilometres in 1.6 hours. We want to know how far they would travel in 2 hours. We want an equivalent rate where the second value is 2 hours instead of 1.6 hours.? =? 2 We could figure out what to multiply 1.6 by to get 2 and then multiply 130 by the same amount = = km We could also divide 130 by 1.6 (which is 81.25) to figure out the number of kilometres they drove in one hour, the unit rate. In two hours, they would go twice as far = km 19 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

122 µ Solving Ratio and Rate Problems We can build a ratio table. A ratio table is a table where equivalent ratios or rates fill the columns. To get from one column to another, we multiply or divide both terms by the same amount. We can also add or subtract the pairs of terms in two columns to create another equivalent ratio. For eample, if both ratios are equivalent to 3 : 4, they both describe groups of 3 items matching groups of 4 items; if you combine them, there are still groups of 3 items matching groups of 4 items For the problem about the family driving, we start with a rate with terms 130 and 1.6. We manipulate the values, following the rules above, to try to get a 2 in the bottom row since you want the distance for two hours. There is always more than one way to build a ratio table. One eample is: 8 10 Distance Time A car uses 10 L of gasoline to go 100 kilometres. How much fuel will it use to go these distances? a) 300 km b) 450 km c) 75 km 20 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

123 Solving Ratio and Rate Problems 2. Tara s heart beats 32 times in 24 seconds. What is her heart rate in beats per minute? [Remember: 60 seconds = 1 minute] 3. A competitive runner goes 10 km in about 30 min. a) What is the unit rate in kilometres per minute? b) What is the unit rate in minutes per kilometre? 4. Some paint is made by miing 2 cans of white paint for every 5 cans of red paint. To make the same tint, how many cans of white paint would you need if you used 13 cans of red paint? Eplain how you solved the problem. 5. A certain recipe uses 908 g of meat for 8 servings. Figure out the amount of meat you would need for 4, 12, and 15 servings: a) 4 servings b) 12 servings c) 15 servings 21 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

124 Solving Ratio and Rate Problems 6. 6 cans of Brand A soup cost $ cans of Brand B soup cost $5.44. Describe three different ways to decide which brand is the better buy. 7. A computer can download a file that is 11.6 MB in one second. a) How long would it take to download a 5 MB file at that same speed? b) The same computer uploaded an 11.6 MB file in 103 seconds. How long would it take to upload a 5 MB file? 8. Canada s annual birth rate was reported as births per 1000 people. a) If the population of Canada is about 33 million, about how many children are born in a year? b) If the ratio of male births to female births is 1.06 : 1, about how many of those new babies were boys? 22 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

125 Solving Ratio and Rate Problems 9. A very cautious driver drives a certain distance in 35 minutes driving at a speed of 60 km/h. How much time would he save if he drove that distance at the legal limit of 70 km/h? Eplain your thinking. 10. The scale ratio on a map is reported as 1 : If two places are 3 cm apart on the map, how far apart are the real places? Describe your answer using two different metric units. 11. The ratio of the perimeter of a certain regular heagon (si sides equal) to the perimeter of a certain square is 6 : 2. What do you know about the relationship between the side lengths of the two shapes? 12. a) Draw a rectangle with a width to diagonal ratio of 1 : 3. Tell how you did it. b) Estimate the length to width ratio. 23 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

126 Solving Percent Problems Learning Goal using an equivalent form of a percent to solve a problem Open Question Keena s brother told her that these things were true: a) If you buy something on sale at 10% off and then you get another 15% off, you could have taken 25% off the original price. b) If you buy something on sale at 10% off and then you pay 13% ta, you could have just added 3% to the original price. c) If the discount at a store is 35%, you can calculate the sale price by using 65% of the original price. d) If a smaller number is 80% of a larger one, then the larger one is 120% of the smaller one. With which do you agree? Eplain why. With which do you disagree? Eplain why. Make up a similar statement that is true. Prove that it is true. Make up a similar statement that sounds true, but really is not. Prove that it is not true. 24 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

127 Solving Percent Problems Think Sheet Percents are ratios where the second term is 100. Sometimes, to solve a problem, we need an equivalent ratio with a different second term. When We Know the Whole For eample: Suppose a school has 720 students. At least 60% are required to participate in a fund-raiser before a sponsoring company will help with a school event. We want to know eactly how many students are needed. So instead of the ratio 60 : 100 (60%), we want an equivalent ratio in the form : 720. Solve the problem, using a fraction, decimal or ratio table: Using a fraction or a percent 60% = 3 5 So 60% of 720 = 3 5 of 720 = = = 2160 = We can use what we know about simple percents of a number (benchmarks) to calculate a more complicated percent. For eample, since 10% is 1, then 10% of 720 = % would be 6 times as much = 432. Using a decimal 60% = 60 = = 432 Using a Proportion Sometimes we solve by writing an equivalent ratio directly. If 60 : 100 = : 720, we divide 720 by 100 to see what we multiplied 100 by to get 72. Then we multiply 60 by that same amount = = September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

128 µ Solving Percent Problems Using a ratio table We can build a ratio table. A ratio table is a table where equivalent ratios or rates fill the columns. To get from one column to another, we can multiply or divide both terms by the same amount. We can also add or subtract the pairs of terms in two columns to create another equivalent ratio. For eample, if both ratios are equivalent to 3 : 4, they both describe groups of 3 items matching groups of 4 items; if we combine them, there are still groups of 3 items matching groups of 4 items. We start with the ratio 60 : 100, which is 60%, and build equivalent ratios. Step 1 It is easier to get to 720 from 20 than is from Step 2 Since 720 is , it would be helpful to get 700 in the bottom row. It is not difficult from Step 3 There is now 700 and 20 in the same row that could be combined to get µ /5 of sity + 7 sities = 7 1/5 sities /5 of a hundred + 7 hundreds = 7 1/5 hundreds + Note: We could also have multiplied column 2 by 36 (since = 720) instead of getting the third column that is there. Then = 432 and we would have had the same result. 26 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

129 Solving Percent Problems These same strategies can be used even if the percent is more than 100% or a decimal percent: for eample, if we wanted 132% of 420, we could multiply by the fraction 132 or decimal We could also start a ratio table with a column 100 reading 132 and 100 or we could figure out how = 420. When We Know a Part The same strategies we have been using can be adapted to solve problems in which we know a part and not the whole. For eample, suppose we know that 15% of the students in a school volunteer at a local soup kitchen. We know that the number of student volunteers is 39 students. We want to know how many students are in the school. Suppose t is the total number of students in the school. Using a decimal We know that 0.15 t = 39, so we divide 39 by 0.15 to figure out the missing value t. 39 = Using a proportion 15 We can write the proportion 100 = 39. Figure out what we multiplied 15 by to get 39 by dividing 39 by 15. Multiply 100 by the same amount = = 260 Using benchmarks If we know that 15% is 39, then we know that 5% is 1 of 39, which is 13. That 3 means 10% is 26 (2 5%) and 100% is 260 (10 10%). Using a ratio table We create equivalent ratios with a first term of 39: September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

130 Solving Percent Problems 1. Which of these make sense? Why? a) 30% of 58 is about 12. b) 74% of 82 is about 60. c) 110% of 93 is about Each of these calculations is a step in the solution of a different percent question. Using a percent, what might the question have been? a) b) 88 4 c) People often recommend giving a 15% tip for good service. If a meal costs $45.29, what tip would you leave if you wanted to leave about 15%? Eplain how to estimate without using a calculator. 28 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

131 Solving Percent Problems 4. The Harmonized Sales Ta (HST) is 13%. Suppose an item costs $89 and you have to pay HST. a) How might you estimate the HST on the item before calculating it? b) What is the HST on that item? c) How much is the total cost including the ta? 5. Anika calculated the price for an item including HST by multiplying by Why does that make sense? 6. Jayda bought a sweater on sale. The discount was 30%. Before the ta was added, the sale price of the sweater was $ Figure out the original price and describe the strategy you used. 7. Ethan had $550 in the bank. He took out $50 to spend on a present. a) What percent of his money is still in the bank? b) How much could he have taken out if he wanted to make sure that 70% was still in the bank? 29 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

132 Solving Percent Problems 8. The side length of one square is 112.5% of the side length of another. What percent of the area of the small square is the area of the larger one? Eplain why your answer seems reasonable. 9. A new employee in a company earns only 63% the salary of a more eperienced employee. If the eperienced employee earns $57 000, what would be the salary of the new employee? 10. A town of 4827 people is epected to grow by 3.2% net year. a) What is the epected population for the net year? b) What percent of the old population is the new one? c) What percent of that new population is the old one? 11. What is the same and what is different about how you solve these two problems? A: I spent $40 and it was 24% of what I had. How much did I have? B: I spent 24% of the $40 I had. What did I spend? 30 September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

133 10-Part Spinner September 2011 Marian Small, 2011 Proportional Reasoning (I/S)

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135 Gap Closing Powers and Roots Intermediate / Senior Student Book

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137 Powers and Roots Diagnostic...3 Perfect Squares and Square Roots...6 Powers Pythagorean Theorem...16

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139 Diagnostic 1. A perfect square is the result of multiplying a whole number by itself. For eample, 64 is a perfect square since it s 8 8. List three perfect squares between 1000 and The square root of a number is what you multiply by itself to get the number. For eample, 8 is the square root of 64 since 8 8 -= 64. Estimate each square root. Do NOT use a calculator. a) 250 b) 88 c) What picture would you draw to show why 16 is 4? 4. Why does it make sense that is 100 times as much as 25? 5. Which does 5 3 mean? A: B: C: D: What is the value of each power? a) 3 4 b) 4 3 c) 10 3 d) (2) 5 e) (0.2) 2 3 September 2011 Marian Small, 2011 Powers and Roots (IS)

140 Diagnostic 7. Without referring to their actual values, why does it make sense that 3 5 might be more than 5 3? 8. Use the thicker line and dotted liine divisions to help you use a power to represent the number of dark squares that could be fit into the largest square. Eplain your thinking. 9. How can you use information about the squares in the picture to tell you that this triangle is a right triangle? Do NOT use a protractor. 4 September 2011 Marian Small, 2011 Powers and Roots (IS)

141 Diagnostic 10. The two shortest sides of a right triangle are given. Determine the length of the longest side without measuring. a) 4 cm and 5 cm b) 5 cm and 12 cm 11. The longest side of a right triangle is 10 cm. One leg length is given. Determine the other leg length without measuring. a) leg is 3 cm b) leg is 5 cm 12. Determine the height of the triangle without measuring. 8 cm 8 cm 8 cm 5 September 2011 Marian Small, 2011 Powers and Roots (IS)

142 Perfect Squares and Square Roots Learning Goal relating numerical and geometric descriptions of squares and square roots. Open Question If we draw a square and the area has the value m, then the side length is called the square root of m. We write it m. If m is an integer, then m is called a perfect square. m m Choose eight numbers, all between 100 and 300 using the rules that follow. (Think of each number as the area of a square.) Three numbers are perfect squares and five are not perfect squares. At least one of the numbers is the double of another number. At least one number has many factors and at least one number does not have many factors. 6 September 2011 Marian Small, 2011 Powers and Roots (IS)

143 Perfect Squares and Square Roots Choose four numbers between 0 and 1 using the rules that follow. (Think of each number as the area of a square.) Make sure that two values are close to 0. Make sure that two values are close to 1. At least one should be a fraction and at least one should be a decimal. One should be the square of a fraction or decimal. For both groups of values: For each number that is the square of another whole number, fraction, or decimal, tell the value of the square root. For each other number, estimate the value of the square root and eplain your estimate. 7 September 2011 Marian Small, 2011 Powers and Roots (IS)

144 Perfect Squares and Square Roots Think Sheet A perfect square is the product of two identical whole numbers. For eample, 16 is a perfect square since 4 4 = 16. We can say 4 squared is 16. Other perfect squares are listed below: We can model perfect squares as square shapes. For eample, 3 3 looks like this: Notice that perfect squares get farther and farther apart. For eample, 1 and 4 are only three apart, but 4 and 9 are five apart and 9 and 16 are seven apart. The square root of a number is the number that we multiply by itself to result in that original number. The square roots of perfect squares are whole numbers. For eample, the square root of 16 is 4. Square roots do not have to be whole numbers. We write the square root of using the symbol. The square root of 8 is written as 8. It is between 2 and 3 since 2 2 = 4 and 3 3 = 9 and 8 is between 4 and 9. One way to model a square root is to think of it as the side length of a square with a given area. For eample, to show 12, think of a square with area 12. The square root of 12 is the side length. A = 12 cm 2 12 cm 8 September 2011 Marian Small, 2011 Powers and Roots (IS)

145 Perfect Squares and Square Roots We can use the button on a calculator to get the value of a square root. To estimate the square root of a number, we could start by relating the square to known perfect squares. For eample, since 125 is between 121 (11 11) and 144 (12 12), 125 is between 11 and 12. It is probably closer to 11 since 125 is closer to 121 than to 144. It helps to know some perfect squares as shown in the table. If we use a calculator, we learn that 125 is about We use the facts = 100 and = to help estimate square roots of larger numbers. For eample, since 15 is close to 4, then 1500 is close to 40 and is close to 400. Number Square Root a) List all of the perfect squares between 200 and 300. b) Eplain how you know you have all of them. 2. Eplain how you know that 250 cannot be a perfect square. 9 September 2011 Marian Small, 2011 Powers and Roots (IS)

146 Perfect Squares and Square Roots 3. a) Which of these powers of 10 are perfect squares? Eplain b) Why are they not all perfect squares? 4. a) You want to multiply by a number to make a perfect square. List three possible amounts you could multiply by and prove that each is a perfect square. b) When you factor those perfect squares down to primes, what do you notice? c) What is the least number you could multiply by to make a perfect square? Eplain. 5. What is the side length of each of three square gardens? (The areas are given.) How do you know? a) 64 m 2 b) 144 m 2 c) 200 m 2 6. The square root of a number is closer to 7 than to 8. What might the number be? How do you know? 7. Estimate each square root without using a calculator. Eplain your strategy. a) 30 b) 300 c) September 2011 Marian Small, 2011 Powers and Roots (IS)

147 Perfect Squares and Square Roots 8. a) What did you notice about the answers to Questions 7a) and 7c)? b) Why does that make sense? 9. How could factoring as help you figure out its square root? 10. How many digits could the whole number part of the square root of these whole numbers have? Eplain your thinking. a) a 3-digit number b) a 4-digit number 11. Why does it make sense that 1 4 is more than 1 4? 12. A number is related to its square root as indicated. What is the number? a) The number is 5 times its square root. b) The number is 1 of its square root. 3 c) The number is 90 more than its square root. d) The number is 4 less than its square root September 2011 Marian Small, 2011 Powers and Roots (IS)

148 Powers Learning Goal recognizing the efficiency of representing repeated multiplication using powers. Open Question The way to shorten a repeated multiplication is to use a power. For eample, 2 4 means ; 2 is multiplied by itself 4 times. It is a power since it is the product of a number multiplied by itself. The 2 is the base. The 4 is the eponent and 2 4 is the power. base 2 4 eponent Use the digits 1, 2, 3, 4, and 5 as bases and eponents. Also use the decimal 0.5 and the integer 2 as a base. List all the powers you can and calculate their values. Tell what you notice about the powers. 12 September 2011 Marian Small, 2011 Powers and Roots (IS)

149 Powers Think Sheet Multiplication is a short way to record repeated addition. For eample, it is quicker to write 4 5 than The way to shorten a repeated multiplication is to use a power. For eample, 2 4 means ; 2 is multiplied by itself 4 times. The 2 is the base. The 4 is the eponent and 2 4 is the power. base 2 4 eponent There are special names if the eponent is 2 or 3. For eample, 3 2 is read three squared. If the eponent is 3, we use the word cubed; e.g., 5 3 is read five cubed. Otherwise, we use ordinal words, e.g., we read 6 5 as si to the fifth (power). We relate a square (or a number to the second power) to the area of a square. That helps us understand why we use the unit cm 2 or m 2 for area. 9 (3 2 ) cm 2 is the area of this square. 3 cm We relate a cube (or a number to the third power) to the volume of a cube. That helps eplain why we use the unit cm 3 or m 3 for volume. 64 (4 3 ) cm 3 is the volume of this cube. 4 cm Powers of whole numbers grow very quickly. For eample, 3 4 = 81, but 3 5 = 243 and 3 6 = September 2011 Marian Small, 2011 Powers and Roots (IS)

150 Powers Powers of fractions or decimals less than 1 shrink as the eponent increases. For eample, 1 ( 2 ) 4 1 = 16, but 1 ( 2 ) 5 1 = 32. 1_ 16 1_ 32 Powers of negative numbers can be positive. If the eponent is even, the power is positive. If it is odd, the power is negative. For eample, (3) 3 = 27, but (3) 4 = Write each power as a multiplication. a) 3 4 b) 3 6 c) 4 6 d) 2 10 e) (3) 4 2. Draw a picture that shows the meaning of each power. a) 5 3 b) September 2011 Marian Small, 2011 Powers and Roots (IS)

151 Powers 3. A large bo holds 4 small ones. Each small bo holds 4 smaller boes. Each of the smaller boes holds 4 tiny boes. Use a power to tell how many tiny boes would fit in the large bo. Eplain your thinking. 4. Tell a story (as in Question 3) that might describe (2) is a positive number. What could be? 6. a) Order from least to greatest: 1 ( 2 ) 3 1 ( 2 ) 4 1 ( 2 ) 2 ( 1 2 ) 5 b) Order from least to greatest: c) What do you notice if you compare the answers to parts a) and b)? 15 September 2011 Marian Small, 2011 Powers and Roots (IS)

152 Powers 7. A certain power has a base that is one less than its eponent. The value of the power is about What could the power be? 8. Choose values to make these statements true: 4 a) = 9 2 b) 6 8 = 36 c) 25 3 = 6 9. Which value is greater each time? How do you know? a) 2 3 or 2 5 b) 10 3 or 9 3 c) 10 3 or To replace the boes below, you can repeat numbers or use different numbers. Tell what the numbers might be if: a) b) 3 < 2 3 is at least 100 more than 2 c) 3 is less than 1 5 of Eplain why 20 5 is 400 times as much as How would you write each of these as a single power? a) b) c) Why is it useful to write a number as a power? 16 September 2011 Marian Small, 2011 Powers and Roots (IS)

153 Pythagorean Theorem Learning Goal relating numerical and geometric descriptions of the Pythagorean theorem and applying the theorem to solve problems. Open Question Draw four right triangles and three non-right triangles. For each triangle, draw a square on each side of the triangle so that the side of the triangle is the full base of the square. Compare the total area of the two smallest squares with the area of the largest square for each triangle. What do you notice? How could that be useful if you knew two side lengths of a right triangle and wanted to know the third side length? 17 September 2011 Marian Small, 2011 Powers and Roots (IS)

154 Pythagorean Theorem Think Sheet Right triangles are special. When we know two of the side lengths and which two lengths they are, we automatically know the third one. [This is not true for just any triangle.] It is true in a right triangle because the total area of the two squares we can build on the smaller sides (the legs) of a right triangle is the same as the area of the square we can build on the longest side, the hypotenuse (the side across from the right angle). Since the area of the square is the square of the side length each time, we write a 2 + b 2 = c 2 if a, b and c are the side lengths of the triangles, and c is the hypotenuse. b b = b 2 b c a c c = c 2 This is called the Pythagorean theorem. If we know a triangle is a right triangle, we can use a a = a 2 the equation above to figure out the third side. For eample, if the hypotenuse is 12 units and one leg is 4 units, then a 2 = = 128; that means a = 128, or about 11.3 units. Since this relationship is only true for right triangles, it is a way to test whether a triangle is a right triangle without drawing it. For eample, if a triangle has side lengths: 9, 12 and 15; it is a right triangle, since = 225 = If a triangle has side lengths: 5, 7, and 9, it is not a right triangle since = 74 and not September 2011 Marian Small, 2011 Powers and Roots (IS)

155 Pythagorean Theorem 1. Draw a picture to show the squares on each side length of this right triangle. Tell how the Pythagorean theorem is shown. 2. For each right triangle, calculate the missing side length. a) b) 5 cm 4 cm 12 cm 8 cm c) 3 cm 12 cm d) 7 cm 25 cm 3. Decide whether these are the side lengths of a right triangle. Eplain your thinking for part c) a) 5 cm, 8 cm, 11 cm b) 24 cm, 32 cm, 40 cm c) 10 cm, 24 cm, 26 cm 19 September 2011 Marian Small, 2011 Powers and Roots (IS)

156 Pythagorean Theorem 4. A ramp rises 2 metres from a point 12 metres away. How long is the ramp? 2 m 12 m 5. Determine the height of this triangle without measuring. 10 cm 10 cm 6 cm 6. A hill is 80 metres high. What is the distance between the two points for viewing the hill? 80 m 1600 m 1500 m 7. Three rectangles with different side lengths all have a perimeter of 100 centimetres. Sketch and label the side lengths of the rectangles and figure out their diagonal lengths. 20 September 2011 Marian Small, 2011 Powers and Roots (IS)

157 Pythagorean Theorem 8. Calculate the lengths of the diagonals of each square. Divide by the side length. What do you notice? a) b) c) 9. List both leg lengths of three different right triangles, each with a hypotenuse of 10 cm. 10. Is there only one right triangle with one side length of 3 units and another of 5 units? Eplain. 11. A certain triangle is almost, but not quite, a right triangle. What could the side lengths be? How do you know? 21 September 2011 Marian Small, 2011 Powers and Roots (IS)

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159 Gap Closing Algebraic Epressions and Equations Intermediate / Senior Student Book

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161 Topic 6 Algebraic Epressions and Equations Diagnostic...3 Translating into Algebraic Epressions and Equations...6 Equivalent Epressions Evaluating Algebraic Epressions...16 Relating Pattern Rules to Epressions and Equations...20

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163 Diagnostic 1. Describe what each epression or equation tells you to do with a number represented by the letter j. The first one is modelled for you. a) 2j Double the number j represents b) 8 j c) 4j + 8 d) 20 2j = Use an algebraic epressionto say the same thing. a) triple a number and then add 2 b) multiply a number by 4 and then subtract the product from 30 c) three more than twice a number is 85 d) one number is four less than twice another number 3. Use an algebraic epression to describe each of the following: a) the perimeter of the square s w b) the perimeter of the rectangle l c) the total value of the money f bills t toonies 4. Eplain why 4a + (3a) = a. 5. Eplain why 5a 1 is not 4a. 3 September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

164 Diagnostic 6. Write the simplest form for each epression. a) 2a a + 8 b) 2a + (7) + 3a 8 c) 9t + (5) + (8s) Evaluate the following epressions for the given values. a) 4k 3, if k = 8 b) 20 3k, if k = 2 c) 6 + m + 2m 2, if m = 3 d) 3a 2, if a = 4 8. Two algebraic epressions involving the variable t have the value 20 when t = 2. What might they be? 9. Without substituting values, tell why each has to be true. a) 3m 20 > 2m 20, if m is positive b) 40 3t > 40 2t, if t is negative 10. Write a pattern rule for the number of tiles in each pattern using the variable f, where f is the figure number. a) Figure 1 Figure 2 Figure 3 Figure 4 4 September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

165 Diagnostic b) Figure 1 Figure 2 Figure 3 Figure Use an algebraic epression or equation to write the general term of each patterm. Use the variable n. a) 3, 6, 9, 12, b) 7, 12, 17, 22, 27, c) 50, 48, 46, 44, 12. How does the equation = 120 help you figure out where the number 120 appears in the pattern: 8, 12, 16, 20,? 5 September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

166 Translating into Algebraic Epressions and Equations Learning Goal representing numerical rules and relationships using algebraic epressions Open Question Algebraic Epressions t 100 m w 3n p 2y + 1 Algebraic Equations 100 2n = 48 6h = d = 80 5f + 2t = 200 P = 3s 2l + 2w = 600 Choose at least two of the algebraic epressions and two of the algebraic equations. For each one, describe at least three different real-world situations that the algebraic epression or algebraic equation might describe. 6 September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

167 Translating into Algebraic Epressions and Equations Think Sheet An algebraic epression is a combination of numbers, variables, and operations. Some eamples are: 4 2t 3n n t + (t 1) For the algebraic epression 4-2t: term each part of the epression 4 and -2t are terms coefficient what you multiply the variable by -2 is the coefficient constant a value that does not change 4 is the constant When a number (coefficient) sits right net to a variable, you assume those values are multiplied. For eample, 2n means two n s; that is 2 multiplied by n. Sometimes the coefficient 1 is not written. For eample, in the epression 3, the coefficient of is 1. Algebraic epressions allow you to describe what to do with a number quickly. The chart shows how different verbal epressions are said (or written) algebraically. Verbal The result after 2 less than a 1 more than triple The sum of two subtracting a epression number a number numbers in a row number from 10 Algebraic epression n 2 3n + 1 t + (t + 1) 10 s Notice that 10 s is what you write for subtracting a number from 10. But you would write s 10 to subtract 10 from a number. 7 September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

168 Translating into Algebraic Epressions and Equations An algebraic equation is a statement where two epressions, at least one of which is an algebraic epression, have the same value. There are 3 kinds of equations: Type of equation Eample What it means 3n + 1 = 4 There is a value of n for which 3n + 1 is 4. Ones where you figure out an unknown value Ones which are always true Ones describing relationships or 3n + 1 = 2n + 5 2t + t = 3t y = or A = lw There is a value of n for which 3n +1 and 2n +5 have the same value. The equation shows two ways of saying the same thing. There is a relationship between the two variables and y and you can calculate y if you know. There is a relationship between the length, width and area of a rectangle. 1. Match each algebraic epression, on the right, with an equivalent phrase in words. There will be one item in each column without a match. a) five more than a number 5 n b) a number is multiplied by five and four is added 4 + t + 5 c) a number is multiplied by four and five is added n 5 d) four more than a number and then five more 4n + 5 e) a number is subtracted from five n + 5 f) five less than a number 4 5n 2. Describe each algebraic epression using words. a) + 2 b) 3 + 4t 5 c) 20 2t d) 14 + t 8 September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

169 Translating into Algebraic Epressions and Equations 3. Write an algebraic epression to describe each situation. a) the total cost of p items, if each costs $5 b) the sum of a multiple of 3 and the number one greater than that multiple of 3 c) the total cost of m muffins that cost $1.50 each and p muffins that cost $1.29 each d) one person s share, if 5 people pay d dollars for something and share the cost equally e) your total savings after w weeks, if you had $100 saved and are saving another $10 a week 4. Write an equation to describe each situation. a) Three more than a number is equal to eighteen. b) Four more than double a number is equal to three times the same number. c) One number is always double another. d) If a number is subtracted from 10, the result is four times the original number. 5. a) Pick any day in the first half of the calendar month. Put a d in the bo. Eplain why the number directly below the d is d + 7. September S M T W T F S b) Pick a date in the last two rows of the calendar. Put a d in the bo. Where is d - 10? 9 September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

170 Translating into Algebraic Epressions and Equations 6. a b h a b a) Write an algebraic epression to describe the perimeter of the parallelogram. b) Write an equation that says that the perimeter is 84 cm. c) Write an algebraic epression to describe how much longer b is than a. d) Write an equation that says that the longer side length (b) is 8 cm longer than the shorter side length (a). e) Write an algebraic epression to describe the area of the parallelogram. f) Write an equation that says that the area is 42cm Choose one of the epressions: Describe a real-life situation where an equation involving your epression might be used. 10 September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

171 Equivalent Epressions Learning Goal reasoning that any algebraic epression can be represented in a variety of ways Open Question Gemma wants to find an equivalent epression to 3n (2n) + (2). She used the zero principle [that (+1) + (1) = 0 and (+n) + (n) = 0] to figure that out. + n n 3n + n n 2n + n = = 0 n + 2 Similarly, the epression 3m 2 + 4t 5 3t is equivalent to the epression 3m + t 7. Create at least three different algebraic epressions and their equivalent epressions to meet each of these conditions: One epression has 6 terms (6 separate parts) and the equivalent one has 4 terms. One epression has 6 terms and the equivalent one has 2 terms. One epression involves two variables, but the equivalent epression only involves one variable. You might need to create your own model for the second variable. Eplain why each of your pairs of epressions is equivalent. 11 September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

172 Equivalent Epressions Think Sheet Just like 6 2 is a quicker way to write , there are sometimes quicker ways to write algebraic epressions. The two ways to write the epressions are equivalent; they mean the same thing. For eample, n is equivalent to n + 3. p + p is equivalent to 2p. You can use algebra tiles to help you write simpler equivalent epressions. You could model variables and constants with these tiles. n -n 1-1 For eample : Epression n n + 2 2n 3 3n + 1 Model Then it is easy to see why 2n n... is equivalent to 5n Even though the letter n is used here, the tile could be used to represent any variable September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

173 Equivalent Epressions Sometimes there are positive and negative tiles and you can use the zero principle to simplify. The zero principle says that (+n) + (n) = 0 or (+1) + (1) = 0 + = 0 = + When you add or subtract zeroes, the amount does not change, so you can get rid of etra zeroes to simplify epressions. In an epression like 3n (2n) + (2): You can match up copies of (+n) and (n), and (+1) and (1) to eliminate zeroes. = 0 n + 2 So an equivalent epression for 3n (2n) + (2) = n + 2. Even without the models, we might have rearranged 3n (2n) + (2) as 3n + (2n) (2). Altogether there would be 1n (since 3 + (2) = 1) and +2 (since 4 + (2) = +2. You can combine identical variables, like terms, (e.g., (5t) and (+4t) or (+2n) and (+3n) or numbers), but you can t mi them together. If there are two different variables, think of them separately. 2 4y 13 September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

174 Equivalent Epressions 1. Use a model to show why 5q = 3q + 2q. 2. Model each. Then write an equivalent epression. a) 2n n + 8 b) 3n + (5) + (4n) + (3) c) 2n 8 + (6n) + 2 d) Do your equivalent epression use more or fewer tiles than the original epressions? Why does that make sense? 3. Write an equivalent epression for each, using two more terms. NOTE: If you use models, use different sizes for s and y. a) s + 8 b) 2s 4y + 3 c) Choose a number on the hundreds chart and replace the value with the variable s. a) Why does the epression s + 10 describe the number directly below s? September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

175 Equivalent Epressions b) Write the two numbers below s in terms of s. c) What algebraic epression would result from adding s to the two numbers directly below it? d) Write an equivalent epression for your answer to part c). e) Write two equivalent epressions for adding s to the two numbers on each side of that square. l 5. Write two equivalent epressions for the perimeter of the rectangle. w 6. a) Why does it make sense that 5 n is equivalent to n? 5 b) Write several other epressions equivalent to n and tell why each one is equivalent. 15 September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

176 Equivalent Epressions 7. Use equivalent representations to describe this sum: A number is added to three more than it and two less than it. 8. Write an algebraic epression with 5 terms that is equivalent to an algebraic epression with 3 terms. 16 September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

177 Evaluating Algebraic Epressions Learning Goal reasoning about how values of an algebraic epression will change when different values are substituted Open Question An algebraic epression involving the variable m has the value 2 when m = +4. One eample is m 6 since 4 6 = 2. What else could the algebraic epression be? List as many possibilities as you can think of, including some where the variable m appears more than once in the epression. You may want to use different operations in the different epressions. 17 September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

178 Evaluating Algebraic Epressions Think Sheet When an epression involves a variable, you can substitute values for the variable to evaluate the epression. To evaluate 2m when m = 4, substitiute 4 for m, and calculate 2 4 = 8. If the same variable appears more than once in an equation, or epression, you must use the same value in each of those places. For eample, to evaluate 3p 2 + p when p = 1, calculate 3 (1) 2 + (1). Equivalent epressions always have the same value when the same substitution is made. For eample, 3n + 2 = n n + 1. If n = 5, it is true that = Non-equivalent epressions might have the same value when the same substitution is made or might not. For eample, 25 n = 4n, but only when n = 5 and not for other values of n. If an epression involves more than one variable, these variables can be substituted with either different values or the same values. For eample: Substitute p = 6 and s = 3 into Substitute p = 4 and s = 4 into 5p 7s 5p 7s = = = = = 9 = (8) When evaluating epressions, the normal order of operations (BEDMAS) rules apply. For eample: Substitute m = 6 into 3 + 2m = = Evaluate each epression. a) 3 4m, when m = 3 b) m, when m = 1 c) j + 2j 2, when j = 4 18 September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

179 Evaluating Algebraic Epressions d) j + (2j) 2, when j = 4 e) 15 3p, when p = 2 3n + 2 f), when n = 0 10 n 2. Substitute the values of m = 0, then 1, then 2, then 3, then 4 into each epression. m = 0 m = 1 m = 2 m = 3 m = 4 a) 3m 4 b) 3m + 6 c) 10m 8 d) 5m + 9 e) m 2 f) 3m a) Evaluate the epression 3n for several different whole number values of n. b) What is true each time? c) How could you have predicted that? 4. Predict which value will be greater. Then test your prediction. a) 4 + 3m if m = 8 OR if m = 8 b) 30 8t if t = 1 OR if t = 10 c) 4t t 2 if t = 3 OR if t = 3 19 September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

180 Evaluating Algebraic Epressions 5. How could you predict that each is true even before you substitute? a) If m = 10, then 4m 2 has to be positive. b) If m = 10, then 4m 2 has to be even. c) 6m 200 is negative for small values of m d) 200 6m is negative for large values of m 6. For each part, create an epression involving p to meet the condition. a) It is even when p is 4. b) It is a multiple of 10 when p = 5. c) It is greater than 100 when p = September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

181 Relating Pattern Rules to Epressions and Equations Learning Goal representing certain algebraic epressions and equations using linear patterns Open Question The pattern 2, 8, 14, 20,. can be modelled as shown below. Notice that the numbers go up by 6 and so there is an etra row of 6 each time. Figure 1 Figure 2 Figure 3 Figure 4 The pattern rule for all the squares shown (including the faint ones) is 6f since if f is the figure number, there are f rows of 6. The pattern rule for the dark squares is 6f 4 since there are 4 squares not counted each time. Create at least four other algebraic epressions that could be pattern rules. Each pattern should include the number 30 somewhere. Show the first four terms of the pattern with pictures and then show the first four terms using numbers. Try to arrange your pictures to make the pattern rule easy to see. Write an equation that would help you figure out where the number 30 is in each pattern. 21 September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

182 Relating Pattern Rules to Epressions and Equations Think Sheet You can think of an algebraic epression as a pattern rule. For eample, p + 2 is the rule for this table of values or the associated picture. Position Value Figure 1 Figure 2 Figure 3 Figure 4 Notice that in the picture, the number of squares is always 2 more than the Figure number. The number of white tiles is the Figure number. The number of dark tiles is the 2. To model the algebraic epression 3p + 2, you can use this table of values which relates p to 3p + 2 or you can use the shape pattern shown. Position Value Figure 1 Figure 2 Figure 3 Figure 4 Notice that the way the pattern was coloured made it easier to see why the coefficient of p was 3. There were 3 times as many squares as the figure number. It was easy to see the + 2 of 3p + 2 using the 2 shaded tiles. Sometimes you can focus on the columns and other times you can focus on the rows to help figure out the rule. The pattern below has the rule 2f + 1 since there are twice as many white squares as the figure number and 1 etra shaded square. Figure 1 Figure 2 Figure 3 Figure 4 22 September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

183 Relating Pattern Rules to Epressions and Equations If you colour the figures differently, you might see an equivalent epression 2(f + 1) 1. The 1 is shown by fading out one square. The 2(f + 1) is shown by having two columns, each with f + 1 squares in it. Figure 1 Figure 2 Figure 3 Figure 4 You can think of an equation as a way of asking which figure in a pattern has a certain number of squares. For eample, 2f + 1 = 31 is solved by figuring out which Figure in the last pattern shown has eactly 31 squares. 1. Tell why the pattern rule describes each pattern. a) Figure 1 Figure 2 Figure 3 Figure 4 Pattern rule 3n + 1 b) Figure 1 Figure 2 Figure 3 Figure 4 Pattern rule: 2n September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

184 Relating Pattern Rules to Epressions and Equations c) Create two equivalent rules for this pattern. Figure 1 Figure 2 Figure 3 Figure 4 2. The pattern below can be described by the equivalent rules 2f + 1 or f + (f + 1). Figure 1 Figure 2 Figure 3 Figure 4 a) How does this way of colouring help you see the pattern rule 2f + 1? Figure 1 Figure 2 Figure 3 Figure 4 b) How does this way of colouring help you see the pattern rule f + (f + 1)? Figure 1 Figure 2 Figure 3 Figure 4 24 September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

185 Relating Pattern Rules to Epressions and Equations 3. Draw the first four figures of the pattern below. Shade or arrange squares in your figures to help make the rule as obvious as you can. a) Pattern: 4f + 2 b) Pattern: 2f 1 4. Describe a pattern rule for each pattern using the variable p. The rule should tell what the value in the pattern is based on its position p in the pattern. a) 2, 4, 6, 8, b) 4, 9, 14, 19, 24, 29, c) 28, 32, 36, 40, 44, d) 202, 200, 198, 196, e) 61, 58, 55, 52, 5. How are the pattern rules for these patterns alike? How are they different? Pattern 1: Pattern 2: 6, 11, 16, 21, 26, 31, 200, 195, 190, 185, 180, 25 September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

186 Relating Pattern Rules to Epressions and Equations 6. A pattern rule includes the term 8n. a) List two possible patterns. b) What would have to be true about any pattern you listed? 7. What equation would you use to find out the position in the pattern of the number 100? a) 2, 4, 6, 8, b) 4, 7,10, c) 12, 16, 20, 24, d) 121, 120, 119,118, e) 300, 290, 280, 8. a) Describe the first five terms of the pattern you might be thinking of when you write the equation 3n + 8 = 35. b) Check if the equation actually does make sense for your pattern. 9. Create your own algebraic epression. a) List the first few terms of the pattern for which it is a pattern rule. b) Create an equation that would make sense for your pattern rule. 26 September 2011 Marian Small, 2011 Algebraic Epressions and Equations (I / S)

187 Gap Closing Solving Equations Intermediate/ Senior Student Book