# Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.

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1 Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material in NS7-72, NS7-73, and NS7-74. This BLM summarizes how to solve the different types of percent problems, with examples for each. Meeting Your Curriculum Students in Ontario study repeating decimals in Grade 9, so lessons NS7-58 and NS7-60 through NS7-63 are optional for them. For students working with the WNCP curriculum, lessons NS7-80 and NS7-81 are optional. Teacher s Guide for Workbook 7.2 L-1

2 NS7-55 Relating Fractions and Decimals (Review) Page 1 Curriculum Expectations Ontario: 7m5, 7m7, 7m11, 7m15 WNCP: 7N4, [CN, C] Goals Students will convert between terminating decimals and their equivalent fractions. PRIOR KNOWLEDGE REQUIRED Vocabulary decimal decimal fraction lowest terms equivalent fraction reduced to lowest terms numerator denominator Can write decimal fractions as decimals Can write decimals as decimal fractions Can write a fraction as a decimal fraction when possible Review reducing fractions using common factors. Remind students that you can make an equivalent fraction by multiplying both the numerator (top number) and denominator (bottom number) by the same number. But then dividing the numerator and denominator by the same number gives an equivalent fraction too. So, if the same number goes into both the numerator and the denominator, you can divide both terms by that number to find an equivalent fraction. Have students do this: a) (both are divisible by 5, so divide both by 5 to get 7 10 ) b) (both are divisible by 10, so divide both by 10 to get 3 4 ) c) 36 9 (both are divisible by 4, so divide both by 4 to get ) ASK: When a number divides into another number, what is it called? (a factor of the other number) Remind students that they are dividing by a common factor to reduce the fractions. ASK: If you want to get the smallest numbers you can in the numerator and denominator, what special number do you have to divide by? (the greatest common factor, GCF) When a fraction has the smallest possible numbers in the numerator and denominator, how do we describe that fraction? (we say it is in lowest terms) Summarize by saying that to reduce a fraction to lowest terms, you divide its numerator and denominator by their GCF. Review decimal fractions and decimals. Have students 23 convert decimal fractions to decimals (EXAMPLE: 1000 = 0.023). 3 convert decimals to decimal fractions (EXAMPLE: 5.03 = ). reduce given fractions to lowest terms (EXAMPLE: = 6 10 = 3 5 ). convert decimals to fractions in lowest terms (EXAMPLE: = 6 45 = 6 9 ) convert fractions in lowest terms, with terminating decimals, to decimal fractions (EXAMPLE: 3 4 = ). L-2 Teacher s Guide for Workbook 7.2

3 convert fractions with terminating decimals to their decimal equivalent (EXAMPLE: 3 5 = 6 10 = 0.6). Decimal hundredths that are equivalent to a fraction with denominator smaller than 100. Have students do these exercises. a) Write each decimal as a fraction with denominator 100. i) 0.35 ii) 0.56 iii) 0.13 iv) 0.75 v) 0.47 b) Which fractions from part a) can be reduced to a fraction with denominator smaller than 100? (i, ii, iv) How do you know? (when the numerator has a common factor with 100, then it can be reduced) connection Real world Process assessment 7m5, 7m7, [CN, C] Workbook Question 7 Contexts for fractions and decimals. Discuss situations where students have seen fractions or decimals being used. EXAMPLES: recipes, money, time (quarter hour). ASK: Would you write \$3/10 or \$0.30? Why do you think we use decimal notation for money? (because each dollar is divided into 100 equal parts, so place value makes sense) Would you use 1/4 hour or 0.25 hours? Why? (1/4, because an hour is divided into 60 minutes, not 100; also, we always think of an hour as divided into quarters 15-minute intervals) Repeat for 1/4 cup or 0.25 cup (1/4 because a cup isn t 10 or 100 or 1000 of anything) and 1/4 m or 0.25 m (0.25 because metres are divided into 100 cm so tenths and hundredths make sense). Number Sense 7-55 L-3

4 NS7-56 Fraction and Decimal Patterns Page 2 Curriculum Expectations Ontario: 7m1, 7m3, 7m5, 7m6, 7m11, 7m25 WNCP: 7N2, 7N4, 7N5, [CN, R, C] Goal Students will relate operations on fractions to operations on their equivalent decimals. PRIOR KNOWLEDGE REQUIRED Can add fractions with like or unlike denominators Can add and multiply (terminating) decimals Vocabulary unit fraction numerator denominator Review converting simple fractions to decimals. Have students convert each fraction to a decimal: 1/2, 1/4, 1/5, 1/8, 1/10, 1/20, and 1/25. ANSWERS: 1 2 = 5 10 = = = = 2 10 = = = = = = = = 0.04 Review adding fractions with numerator 1 and the same denominator. ASK: Three people each ate 1/8 of a pizza. How much of the pizza did they eat altogether? (3/8) Write on the board: 1/8 + 1/8 + 1/8 = 3/8. Have students add these fractions: 1 a) = 1 b) = 1 c) = 1 d) = Introduce multiplying a whole number by a fraction. Point out that students have been adding the same number over and over again. ASK: What is a shorter notation for this so that we don t have to keep writing the same number? (multiplication). Write on the board: = 4 5 L-4 Teacher s Guide for Workbook 7.2

6 NS7-57 Relating Fractions and Division Page 3 Curriculum Expectations Ontario: 7m1, 7m3, 7m5, 7m6, 7m7, 7m27 WNCP: 7N4, [R, T, CN, C] Goal Students will develop and apply the formula a/b = a b. PRIOR KNOWLEDGE REQUIRED Vocabulary fraction numerator denominator Doing a simpler problem first Can do long division of 3- and 4-digit numbers by 1-digit numbers Can divide decimal numbers by 1-digit whole numbers using long division Review long division with whole numbers and decimals. Remind students that when doing long division a b, they should be careful not to do b a. It doesn t matter which is bigger, a or b. What matters is which number is being divided by which number. ASK: How would you write the notation for long division to do 75 3? Is it 3) 75 or 75) 3? (it s 3) 75 ) How would you write the notation for 2 5? Which number is the 2 like the 3 or the 75? (the 75) How do you know? (because it comes first; it is the number being divided into) Note: Some students may answer the 3, because it is the smaller number. Emphasize that it doesn t matter which number is bigger the notation only tells you which number to divide into which. We are used to dividing a smaller number into a larger number but we can also divide a larger number into a smaller number. Since 2 replaces 75 and 5 replaces 3, we can use our familiarity with the simpler case to get the correct way of writing 2 5: 5) 2. Remind students that 2 is the same as 2.0 or 2.00 or Remind students that to divide a decimal by a whole number, simply line up the decimal point above the division sign and divide as though the decimal is a whole number. Have students use decimal long division to find:.... a) 5) 2. 0 b) 8) c) 25) d) 16) Relate fractions to division. Write on the board the fraction 12/3. ASK: What number does this represent? (4) How do you know? (because 12 3 = 4) Demonstrate why this works as follows. ASK: 12 is the numerator what does the numerator of a fraction tell us? (the number of parts we are considering) What does the denominator tell us? (the number of parts in one whole) ASK: The fraction is 12/3. How many parts are in one whole? (3) Then draw on the board: one whole L-6 Teacher s Guide for Workbook 7.2

8 Looking for a pattern Extending patterns in decimals. Have students divide to find 1/40, 2/40, 3/40, 4/40, 5/40. ANSWERS: (0.025, 0.05, 0.075, 0.1, 0.125) ASK: What kind of pattern do you see? PROMPT: Write the decimals to three decimal places. ASK: What is the rule for obtaining the next term in the pattern? (add 0.025) Write on the board: 6 40 = 7 40 = 8 40 = 9 40 = = Reflecting on other ways to solve a problem Process assessment 7m6, [CN] Workbook Question 3 7m1, [R] Workbook Question 4 Have students continue the pattern in the decimals. Then have students reduce to lowest terms the fractions that are not already in lowest terms (6/40 = 3/20, 8/40 = 1/5, 10/40 = 1/4) and create for all fractions an equivalent decimal fraction and then an equivalent decimal: 6 15 = = 0.15, = = 0.175, = = 0.2, = = 0.225; = = Do students get the same answer by turning the fraction into a decimal as they did by extending the pattern? (yes) L-8 Teacher s Guide for Workbook 7.2

9 NS7-58 Repeating Decimals Pages 4 5 Curriculum Expectations Ontario: 7m1, 7m2, 7m5, 7m6, 7m7, optional WNCP: 7N4, [R, CN, T, C] Goal Students will use division to represent fractions as repeating or terminating decimals. Students will compare and order decimals, including repeating and terminating decimals. Vocabulary quotient repeating decimal terminating decimal PRIOR KNOWLEDGE REQUIRED Can compare and order terminating decimals to thousandths Can use long division to write fractions with terminating decimal equivalents as decimals Materials 2-cm grid paper Compare the number of digits in the quotient to the number of subtractions in the long division. Remind students that to write 4/5 as a decimal, they can divide 4 5 and use long division. Go through this as a class to get 0.8. ASK: How did we know when to stop? (when the remainder is 0) Have students do 3 16 to write 3/16 as a decimal (0.1875). Provide grid paper to help students align the numbers correctly. ASK: How long did it take to get a remainder of 0 this time how many subtractions in the long division did you have to do? (4 subtractions) Making and investigating conjectures Have students do several long divisions, and then copy and complete this chart: Fraction Decimal Number of decimal places in the quotient Number of place values in the quotient Number of subtractions before getting 0 remainder 4 5 = = = = 3 8 = 98 5 = 7 10 = = Number Sense 7-58 L-9

11 ASK: Can you guess what the next digit is? After students guess, tell them it is 5, and write the next digit for them: π = Repeat having students guess the next digit and then writing what it actually is: If students have a scientific calculator, have them press the pi button. What number comes up? π = (Note: Different calculators may show a different number of digits, here and below. Just use whatever you see on your calculator display.) Now write on the board: 1 = ASK: Can you guess what the next digit is? (3) Repeat several times. (the next digit is always 3) Tell students that because there is a pattern, the next digit is easy to find. ASK: What type of pattern do you see in the digits? (a repeating pattern) Explain that because the digits form a repeating pattern, the next digit is easy to predict, and the decimal is called a repeating decimal. Some decimals, like π = , are not terminating and not repeating, but most decimals students will deal with this year will be either terminating or repeating. In fact, mathematicians have shown that any decimal that they can get from a fraction will be either terminating or repeating. Connecting Technology Finding repeating decimals. Have students find more decimals by long division and determine whether they are terminating or repeating: a) 5 b) 7 c) 5 d) 4 e) 5 f) ANSWERS: a) repeating b) repeating c) terminating d) repeating e) repeating f) terminating ASK: How are these repeating decimals different from 1/3 = ? Notice that the digit for parts a), b), and d) repeat, but not right away. In part e), there is not a single repeating digit but a repeating pattern (4, 5, then repeat), so this decimal is still called a repeating decimal. Calculators can be misleading. Have students calculate 3/7 using long division until they either get 0 as a remainder or they can prove that they won t because they find a pattern that goes on forever. They should get The repeating pattern is 4, 2, 8, 5, 7, 1, then repeat. Number Sense 7-58 L-11

13 large a number is (e.g., is larger than because, even though the digits are smaller, there are more digits before the decimal point). Process assessment 7m6, [CN] Workbook Question 8 Introduce how to compare repeating decimals. Show students how to compare decimals by writing their first few digits after the decimal point at least until they differ (you may need to add zeroes to terminating decimals). See the teaching box on Workbook page 5. Give students grid paper and have them do more problems like Questions 8 and 9 on Workbook page 5. Extra Practice for Workbook Question 8: a) b) c) Process assessment 7m1, [R] Workbook Question 10 d) e) f) g) h) i) Number Sense 7-58 L-13

14 NS7-59 Using Decimals to Compare Fractions Page 6 Curriculum Expectations Ontario: 7m1, 7m3, 7m6, 7m7, 7m11, 7m27 WNCP: 7N4, 7N7, [R, ME, C, CN] Vocabulary rounding fraction decimal equivalent Goal Students will use decimal equivalents to compare fractions and to determine fractions that are close to given decimals. PRIOR KNOWLEDGE REQUIRED Can compare fractions by finding a common denominator Can convert fractions to decimals (repeating or terminating) by using long division Can use subtraction to find the difference between two numbers Can subtract terminating decimals Can compare and order repeating decimals Can round decimal numbers NOTE: Many concepts in this lesson assume familiarity with repeating decimals. Ontario students who are not familiar with repeating decimals can still do most of this lesson. Because they will not know the notation for repeating decimals (the bar over digits that repeat), do not use it. To convert fractions to decimals, use a calculator and round the answers to 1, 2, or 3 decimal places, as required. Students can thus compare, order, and subtract terminating decimals to answer all but one question (5 b) on Workbook page 6. Using decimals to compare fractions. Now that students know how to convert fractions to decimals, and to compare decimals, have students compare fractions by first converting the fractions to equivalent decimals and then comparing the decimals. Allow students to use grid paper to help align the place values. See Workbook page 6 Questions 1, 3, and 7. Representing, Reflecting on other ways to solve a problem Process assessment 7m3, 7m7, [R, C] Workbook Question 4 Compare fractions and decimals two ways. Have students compare 4/9 and 0.435, first by writing both as fractions with a common denominator (4/9 = 4000/9000 and 435/1000 = 3895/9000, so 4/9 is larger) and then by writing both as decimals (4/9 = 0.4 = > because the largest place value where they differ is hundredths and 4 hundredths is more than 3 hundredths). ASK: Did you get the same answer both ways? (yes) Which way did you like better? Have students do more examples both ways, as in Workbook Question 4, but keep in mind that students have not yet learned to convert repeating decimals to fractions, so include only terminating decimals in decimal format for the comparison. Which one is closer? Have students convert these fractions to decimals: 3/11 and 3/13. ANSWERS: 0.27 and ASK: Which one is closer to 1/4 = 0.25? To guide students, write on the board: L-14 Teacher s Guide for Workbook 7.2

15 Changing into a known problem ASK: To know how far one number is from another, what do we have to do? (subtract them) PROMPT: What operation will we use? Then tell students that we don t yet know how to subtract repeating decimals. How can we do this problem without subtracting repeating decimals? PROMPT: What do we know how to subtract? (we know how to subtract terminating decimals) ASK: How can we change a repeating decimal into a terminating decimal and still keep the answer the same? (if we round the repeating decimals to a few decimal places after they start being different, and then subtract, we should still get the right answer as to which difference is smaller) Point out that when we compared repeating decimals, we didn t have to compare all the place values, only the early place values. It s the same here: we only need to compare the differences in the early place values, and rounding the repeating decimals will allow us to do that. Have students round each decimal to 5 decimal places and then find the differences: Reflecting on other ways to solve a problem, Looking for a pattern So the difference between 1/4 and 3/13 is smaller than the difference between 3/11 and 1/4. This means that 3/13 is closer to 1/4 than 3/11 is. Point out that 1/4 = 3/12, and write down this sequence of fractions: It is easy to see the decimal equivalents for the first few terms in this sequence they are all terminating decimals: Process assessment 7m6, [ME] Workbook Question 6 ASK: Do these fractions appear to be getting closer together or further apart? (closer together) Explain that the further we go in the sequence, the closer together the numbers are. ASK: Does this agree with the answer we got? (yes, we said that 3/18 is closer to 3/12 than 3/11 is) Discuss the differences between the two methods. Which method do students like better, and why? Have students do Workbook page 6 Questions 2 and 5. Number Sense 7-59 L-15

16 NS7-60 Is the Fraction a Terminating or Page 7 Repeating Decimal? Curriculum Expectations Ontario: 7m1, 7m2, 7m6, 7m7, optional WNCP: 7N4, [CN, R, C, T] Goal Students will determine whether a fraction s decimal equivalent will terminate or repeat by looking at the denominator of the fraction in reduced form. PRIOR KNOWLEDGE REQUIRED Vocabulary terminating decimal repeating decimal decimal fraction Can convert terminating decimals to decimal fractions Can convert decimal fractions to terminating decimals Introduce the lesson topic. Tell students that they will now investigate how to determine, by looking at a fraction, whether its decimal equivalent will terminate or repeat. (You will work through the Investigation on Workbook page 7 over the course of the lesson.) Review writing terminating decimals as decimal fractions. Have students write each of these terminating decimals as a decimal fraction: a) 0.3 b) 0.34 c) d) e) f) ANSWERS: a) 3 10 e) b) f) c) d) Have students do Parts A and B of the Investigation on Workbook page 7. Tell students that a terminating decimal can be very long, for example: Have students write that terminating decimal as a decimal fraction. ANSWER: / ASK: How did you know what to write in the denominator? (the number of zeroes is the number of decimal places in the decimal) Tell students that any terminating decimal can be written as a decimal fraction. For example, consider a decimal with two decimal places, such as Since there are two decimal places, we can read the number of hundredths: there are 312 hundredths, so 3.12 = 312/100. Similarly, has 3 decimal places, so we can read the number of thousandths: there are 45 thousandths, so = 45/1000. Tell students that terminating decimals can also look very much like the start of a repeating decimal. For example, is a terminating decimal. Have students write that decimal as a decimal fraction. 323, 232, 323, 232 ANSWER:. 1000,, 000, 000, 000 L-16 Teacher s Guide for Workbook 7.2

17 Connecting Review writing decimal fractions as terminating decimals. ASK: Can any decimal fraction be written as a terminating decimal? (yes) Ask students to articulate the reason, then explain that the smallest place value in the decimal will be determined by the denominator of the decimal fraction. For example, if the decimal fraction has denominator , then the smallest place value in the decimal will be ten thousandths, which means the decimal has only 4 place values. Have students write these decimal fractions as terminating decimals: a) 36 5 b) c) 2341 d) ANSWERS: a) 0.36 b) c) d) Representing, Technology Can you always use a calculator to decide whether a fraction has a terminating decimal equivalent? Have students calculate these quotients using a calculator: = 1 25 =? = 1 27 =? = 1 29 =? ASK: Which of the fractions looks like it has a decimal equivalent that terminates? (1/25 = 0.04 does) Can you tell for sure from the calculator display for all three fractions? Tell students that your calculator says that 1/27 = This could either be a very long terminating decimal or it could be the repeating decimal Also, tell students that your calculator says that 1/29 = This also could be a terminating decimal or it could be the repeating decimal SAY: Notice how close to the end of my calculator display we see the digits start to repeat. Some calculators may not even show enough digits for you to see that they repeat at all! In fact, 1/29 is a repeating decimal with a very long (28 digits!) block that repeats. Technology Decide whether a fraction has a terminating or repeating decimal equivalent. Have students do Workbook page 7 Investigation Part C. Tell students that if, after dividing a numerator by a denominator, the digits do not go to the end of the calculator display the most digits that the calculator can show then students can be sure that the equivalent decimal terminates, since the calculator would have shown more digits if there were any more to show. If the digits do go to the end of the calculator display, students won t be able to tell for sure whether the equivalent decimal repeats or not. Tell students to guess by looking at the digits whether there is a repeating pattern or not. After students complete the exercise, remind them that a fraction will have a terminating decimal equivalent if it has an equivalent fraction that is a decimal fraction. Then have students look at the fractions they said were repeating. Have them try to find an equivalent decimal fraction and to think about why they can t. Explain that to prove that it s not possible to do something, mathematicians often start by trying to do it! Sometimes only when they start trying to do something can they see why it won t work. After allowing students to work for several minutes, have students work in pairs Number Sense 7-60 L-17

18 Organizing data Looking for a pattern Investigating and making conjectures to articulate a reason why 7/15 has no equivalent decimal fraction. Then have students get into groups of four (two pairs) and agree on a reason. Allow each group of four to articulate their reason then summarize the results as follows: The fraction 7/15 is in lowest terms, so to make any equivalent fraction we have to multiply the numerator and denominator by the same number. Since we want to find an equivalent decimal fraction, we re looking for a fraction that has denominator a power of 10. Suggest trying each possibility in turn. Write on the board: 7? 15 = 7?? = 7?? ? = ? 15? = ? 15? = ? 15? = ASK: Does 15 divide evenly into any power of 10? Students can try this by long division and see that there will always be a remainder of 10. Then have students do Workbook page 7 Investigation Part D. After students finish, tell them that any power of 10 can always be written as a product of 2s and 5s. Now tell students that 15 = 3 5. ASK: Can 15 something be equal to a product of 2s and 5s? (no) Why not? (because of the 3) Tell students that 20 = 4 5, but 20 does divide evenly into 100, a power of 10. What makes the 3 different from the 4? (4 can be written as 2 2, so 20 is still a product of 2s and 5s, but 3 cannot be written as a product of 2s and 5s) Tell students that 3 is prime, and if a number divides evenly into a power of 10, the only prime numbers that divide into it are 2 and 5. ASK: Can 15 be written as a product of 2s and 5s? (no, because the prime number 3 divides into it) Now look again at the fractions from Investigation Part C. They are all in lowest terms. ASK: Which fractions have denominators that are products of only 2s and/or 5s? (5/8 because 8 = and 13/2000 because 2000 = students can find the answers by continually dividing by 2 and then by 5, or they can use prime factorizations if they are familiar with them). Are these the same as the fractions you said were terminating? (yes) Is 3/17 terminating? (no) How do you know? (because 17 is not a product of 2s and 5s) Now have students complete the Investigation on Workbook page 7. Extra Practice: 1. Write out all the fifteenths from 1/15 to 14/15, and then write them in lowest terms. Decide from the denominators which fractions will terminate as decimals, then check on a calculator. 2. Write these fractions in lowest terms and then decide if their decimal equivalents will terminate or not. a) 7 b) 13 9 c) d) 5 e) 6 4 f) Check your answers on a calculator. L-18 Teacher s Guide for Workbook 7.2

19 Does a calculator display the exact value for a fraction? Tell students that calculators never show a decimal as repeating you will never see the bar above repeating digits on a calculator, so calculators always make decimals look terminating. This means that when you calculate a fraction that has a repeating decimal equivalent on a calculator, it will not give you the exact answer, only an approximation. Have students do the following exercise: Process assessment 7m6, 7m7, [T, C] A calculator shows 50/97 = Is this an exact value or an approximation? How do you know? (it is an approximation because 97 is not a product of 2s and 5s, so its decimal equivalent is repeating, not terminating) Create fractions with a repeating or terminating decimal equivalent. Challenge students to create a fraction with a terminating decimal equivalent that Process assessment 7m7, [C] a) has a 1-digit denominator, b) has a 2-digit denominator, c) has a 3-digit denominator, d) has a 4-digit denominator. SAMPLE ANSWERS: a) b) c) d) Repeat for fractions with a repeating decimal equivalent. SAMPLE ANSWERS: 1 3, 1 33, 1 333, 1. Have students explain how 3333 they chose the denominators. Extension Why is the decimal representation of every fraction either repeating or terminating? Why can t it be like pi, which doesn t terminate or repeat? Think of what actually happens in the long division algorithm. Take for example 2/7 = ) Number Sense 7-60 L-19

20 Using logical reasoning At each step of the algorithm, we are dividing 7 into ten times whatever remainder we get. We start with remainder 2 because 2 7 = 0 Remainder 2, and divide 7 into 20 to get 2 Remainder 6. Our next step is to divide 7 into 60 and determine the remainder. As soon as we get a remainder that we ve already had, then the division algorithm becomes exactly the same as it was from the first time we saw that remainder. If we get 0 as a remainder, the algorithm stops and the decimal terminates, but if we never get 0, then we have to continue forever. But there are only 6 possible remainders when we divide by 7 (not including 0), so if we do the algorithm forever, and each time the remainder is either 1, 2, 3, 4, 5, or 6, we are eventually going to repeat a remainder. Once we do that, the decimal starts repeating. This reasoning works for any number. For example, to find 1/29 = 1 29, we are always dividing by 29, so our remainder will always be less than 29. It might take a while to find a repeat remainder, but if we go on forever, we eventually will. No matter what we divide by, if we keep dividing forever (because we never find a 0 remainder), we will eventually get the same remainder twice. The decimal will repeat from the point at which that remainder occurred the first time, because we are now doing the exact same divisions that we did to get all the digits in the decimal from the first time that remainder showed up! Note: Some numbers, like π, do not come from fractions, and hence cannot be calculated by using long division of one whole number by another. The decimal for π neither terminates nor repeats. L-20 Teacher s Guide for Workbook 7.2

23 Emphasize that they can subtract from left to right because there is no regrouping required. Have students practise. As above, students should write at least two cores in the answer, to be sure it repeats. EXAMPLES: a) b) c) d) Looking for a pattern Reflecting on other ways to solve a problem The pattern in ninths. Show students the beginning of the pattern: 1 9 = = = 0.3 Have students extend the pattern to express the rest of the ninths in decimal form. Then have students do Workbook page 8 Questions 2 and 3. When students finish, discuss the results: When we don t need to regroup, adding repeating decimals is easy, but when we need to regroup, we can t start at the left! So we have to add the decimals to more decimal places each time and hope we can see how the pattern continues. Sometimes it is easier to add the fractions. Adding and subtracting with regrouping when the decimal being subtracted is terminating. Show students how to subtract First, subtract up to hundredths, because that is where 0.27 ends: \ 4 \ Now we can subtract the rest: so = Have students practise: a) b) c) ANSWERS: a) b) c) Encourage students to check their answers on a calculator. Number Sense 7-61 L-23

25 Have students fill in a chart like the one below. Students can add their own decimals to the chart. (Point out that the repeating pattern in some of the sums will have a very long core, so students will have to write out a lot of digits to be sure that the decimal repeats.) First addend Length of repeating block Second addend Length of repeating block = Sum Length of repeating block The answer is that the length of the repeating block of the sum will be at most the lowest common multiple (LCM) of that for the two addends. The digits after the decimal point form a repeating pattern. If one pattern has core length 2 and another has core length 3, what happens when we add them? Consider the decimals and Start at the place value where both are repeating: Starting at the hundredths, the repeating patterns are 1, 2, repeat and 5, 4, 3, repeat. The first pattern will start over 2 places later, 4 places later, 6 places later, and so on (all the multiples of 2). The second pattern will start over 3 places later, 6 places later, and so on (all the multiples of 3). Simply find the smallest multiple of both to see when we can be sure they will both start over together. Then the sum will repeat from here on too (although it might start repeating earlier as well). Number Sense 7-61 L-25

26 NS7-62 Writing Repeating Decimals as Fractions Page 9 Curriculum Expectations Ontario: 7m1, 7m3, 7m5, 7m6, optional WNCP: 7N4, [R, CN, T] Goal Students will write repeating decimals that begin repeating immediately as fractions. PRIOR KNOWLEDGE REQUIRED Vocabulary terminating decimal repeating decimal Looking for a pattern Reflecting on the reasonableness of an answer Can use long division to find the decimal equivalent of a fraction Can multiply terminating decimals Write these fractions on the board: Have students use long division to write these fractions as decimals, as in Workbook page 9 Question 1 a). ASK: What do you notice about your answers? Is there a pattern? Can you predict what 5/11 will be as a decimal? PROMPT: Look at the first two digits after the decimal point: 09, 18, 27, 36. How should we continue this pattern? (add 9) So the next term is 45, which means we expect that 5/11 = ASK: Does this make sense? Let s compare both to 1/2 to check. Is 5/11 more or less than 1/2? (less) How much less a lot or a little? (just a little) How about 0.45 is it more or less than 1/2? PROMPT: What is 1/2 as a decimal? (0.5) Is 0.45 more or less than 0.5? (less) How much less a lot or a little? (just a little) They are both a little less than one half, so our prediction makes sense. Then have students extend the pattern to find 6/11, 7/11, 8/11, 9/11, 10/11, and 11/11, as in Workbook page 9 Question 1 b). Using products to write all the elevenths as decimals. Write down: 1 = 0.09 = Remind students that 5/11 = 1/11 + 1/11 + 1/11 + 1/11 + 1/11 = 5 1/11. So 5/11 = 5 1/11. But 1/11 = 0.09, so 5/11 = Have students calculate each product: Looking for a pattern Representing Then have students predict (ANSWER: 0.45) Express admiration: tell students that you knew they could multiply whole numbers by decimals with lots of decimal places, but you didn t know they could multiply whole numbers by decimals with infinitely many decimal places! Then verify that this result is the same as their prediction for 5/11. It is, because we predicted that the digits that repeat would be 9 5 = 45, and this is what we found. Have students finish writing all the elevenths, up to 11/11, as repeating decimals. L-26 Teacher s Guide for Workbook 7.2

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