Unit 2 Number and Operations Fractions: Multiplying and Dividing Fractions


 Willis Byrd
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1 Unit Number and Operations Fractions: Multiplying and Dividing Fractions Introduction In this unit, students will divide whole numbers and interpret the answer as a fraction instead of with a remainder. They will multiply fractions (including improper fractions and mixed numbers), divide fractions by whole numbers, and divide whole numbers by unit fractions. Students will also solve realworld problems and use benchmark fractions and number sense of fractions to compare, estimate mentally, and assess the reasonableness of answers. The realworld problems involve addition and subtraction of fractions, multiplication of fractions and mixed numbers, division of unit fractions by nonzero whole numbers, and division of whole numbers by unit fractions. A note about division. Division with remainders is only used when dividing whole numbers by whole numbers. When they are dividing decimals by whole numbers or by decimals, do not ask students to decide for themselves what to do when the corresponding whole number division leaves a remainder. To deal with this situation, students need to know that they can continue the dividend by writing zeros after the decimal point. This technique is not taught in this course because it leads to the concept of repeating decimals, which is not required in Grade. Please note in particular that it is not true that = R, even though the corresponding whole number division is 7 = R. If anything, the remainder would be 0.. However, the convention is that remainders are only used when dividing whole numbers by whole numbers (although decimal answers can also be used, as in 7 =.). Preparation. Cut out the fraction pieces (wholes, halves, thirds, and fourths) from BLM Fraction Parts and Wholes (pp. M6 9). You will need them in Lessons NF and NF. Keep the pieces sorted according to their size to facilitate distribution to students. COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION Number and Operations Fractions M
2 NF0 Fractions and Division Pages 0 STANDARDS.NF.B. Goals Students will recognize a fraction as division of the numerator by the denominator. Vocabulary division fraction improper fraction mixed number unit fraction PRIOR KNOWLEDGE REQUIRED Understands division as equal sharing Can convert an improper fraction to a mixed number Can draw pictures representing proper fractions, improper fractions, and mixed numbers Sharing one object among friends. SAY: Two friends want to share one pie. Draw the circle in the margin on the board. Draw a vertical diameter, and while you are shading one half, SAY: Each friend gets half of the pie. Write the fraction / beside the picture, as shown below. Draw another circle on the board and ASK: If three friends want to share this pie, how many pieces would there be? () Divide the pie into three pieces and ask a volunteer to shade a piece and write the fraction / in stack form, as in the margin. SAY: This is an example of a unit fraction. When one thing is being shared equally, the result is a unit fraction. Remind students that we use division for equal sharing for example, if three friends share apples equally, each friend gets = apples. SAY: You can use division for equal sharing, and this applies whether the answer is a whole number or a fraction. For example, if two friends share a pie equally, each friend gets / of the pie, so = /. Exercises: Shade how much each person gets and then write the fraction. Four people share a pancake Six people share a pizza. Three people share a chocolate bar Sample answer: 6 Sharing more than one object. Write on the board: people share pies COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION M Teacher s Guide for AP Book.
3 Point out that you are drawing circles for pies. ASK: How many circles should I draw? () Draw the three circles on the board and then SAY: They decide to share each pie equally. ASK: How many pieces should I divide each circle into? () Divide the pies and then have a volunteer shade the amount that one person gets, as shown in the margin. Four people are sharing three whole pies, so one person will take the shaded pieces. Exercises: Describe the number of pieces and the number of pies. people share pies people share pies people share pies d) people share pies Answers: pieces in each pie and whole pies, pieces in each pie and whole pies, pieces in each pie and whole pies, d) pieces in each pie and whole pies Exercises: For the previous exercises, draw a picture to show how much one person gets. Sample answer: Each person gets / of a pie. Using division for equal sharing with a fraction for the answer. ASK: If two people share six pies, how much does each person get? ( pies) Write on the board: people share 6 pies people share pies 6 = = Ask a volunteer to fill in the blanks ( = /) and have students signal whether they agree (thumbs up) or disagree (thumbs down). Point out that the number of objects being divided goes first in the equation, and the number of people sharing goes second. The answer is how much each person gets. Point out also that now the answer is a fraction. COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION Have a volunteer write the division equation to show seven friends sharing two pies. ( 7 = /7) Explain to students that you can say seven friends share two pies or you can say two pies shared among seven friends. Exercises: Write the division equation. people share pies pies shared among people pies shared among 6 people d) 6 people share pies Answers: = /, = /, 6 = /6, d) 6 = /6 Number and Operations Fractions 0 M
4 Dividing whole numbers without a picture. Write on the board: 7 = 9 = = Challenge students to predict the answers to these questions without using a picture. Point out that the first number in the division is the top number of the fraction, and the second number in the division is the bottom number of the fraction. Exercises: Divide. Write your answer as a fraction d) 7 Bonus:,000 Answers: /0, /7, /9, d) 7/, Bonus: /,000 Writing the answer as a mixed number. Now tell students that three people are sharing five apple pies. ASK: How much does each person get? (/ pies) Draw on the board: Explain to students that they can reorganize the five shaded parts into one whole pie that is shaded and two shaded parts of another pie. Remind students that we can write the improper fraction / as the mixed number /. Ask a volunteer to shade / to find the mixed number, as shown below. = Exercises: Divide. Write the answer as an improper fraction and as a mixed number. Show your answer with a picture Bonus: Answers: 9/ = /, 7/ = /, 6/ = / or / = /, Bonus: / = 7/ (MP.) Exercises: Write the answer as an improper fraction and a mixed number. Five people share granola bars. How many granola bars does each person get? Three people share 6 pounds of rice. How much rice does each person get? Four people share nine pounds of flour. How much flour does each person get? Answers: / = /, 6/ = /, 9/ = / COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION M Teacher s Guide for AP Book.
5 Review multiplying fractions. Review multiplying a fraction by a whole number. 9 = ( 9) = (MP.) = 6 Exercises: Use multiplication to check the answer. = = 7 = 7 Selected solution: / = / = Extensions. When you divide granola bars among four people, you divide items among four groups. If you calculate, the answer you get represents the number of items in each group. total number of items number of groups = number of items in each group = Number of items We make groups because there are people Each person gets whole items Plus / of an item In general, when the divisor (in this case ) is the number of groups, the answer is the number of items in each group or the size of the group. However, when the divisor is the size of the group, the answer is the number of groups of that size. For example: if you cut m of rope into m pieces, how many pieces can you make? COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION whole pieces of meters in length meters left over ( meters is / of meters) As the diagram above shows, the whole number () in the answer tells you how many pieces of m each (or groups each made of four m pieces) can be made (here two groups). The fraction in the answer (/) does not mean that you have / of a meter left over, but instead means that you have / of the thing you were dividing by left over. The thing you are dividing by is m long, so you have / of m left over. (/ of m is m, which is exactly how much rope is left over.) Ask students to draw a picture to interpret the quotient as the number of groups. ASK: What does the fraction part of the answer mean? 7 = = Number and Operations Fractions 0 M
6 NF Multiplying Unit Fractions Pages STANDARDS.NF.B. Goals Students will develop and apply the formula for multiplying unit fractions by unit fractions. Vocabulary denominator numerator of (to mean multiply ) unit fraction PRIOR KNOWLEDGE REQUIRED Can multiply fractions by whole numbers Understands that of can mean multiply Review half of a number. Remind students that they can find half of a number by finding half of a set. For example, to find half of six, make a set with six objects and take half of it. ASK: How many are in half of the set of six? To illustrate, draw this diagram on the board: Ask a volunteer to circle half of the dots. SAY: So half of 6 is. Exercises: Find half. of of of 0 d) of Answers:,,, d) 6 Review that of can mean multiply. Remind students that of can mean multiply. SAY: means groups of things and / means / of a group of things. Exercises: Multiply. 0 Bonus: 00 Answers:,,, Bonus: 00 Half of a unit fraction. Explain to students that, just as we can talk about half of a whole number, we can also talk about half of a unit fraction. Demonstrate finding half of / by dividing an area model of the fraction / into a top half and a bottom half: of SAY: Let s extend the horizontal lines to find out what fraction of the whole rectangle is shaded. Draw the dotted line as shown in the margin and ASK: How many parts are there in total? (0) How many are shaded parts? () Write on the board: of = 0 COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION M6 Teacher s Guide for AP Book.
7 Exercises: Draw a dotted line to find out how much of the whole rectangle is shaded. Answers: of =, of = 6 Exercises: Find half of the fraction. 6 7 Answers: /, /, /, d) /, e) /, f) / 9 d) e) f) Finding a unit fraction of a unit fraction. Tell students you want to find / of /. Draw the picture in the margin on the board and SAY: Here is half of a rectangle. Divide the shaded part into three equal pieces as shown in the margin. SAY: I would like to find / of /, so I have to take one piece. Draw a bold box around the first shaded piece and remove the shading from the two other pieces (see example in margin). of Then extend the horizontal lines to find out what fraction of the whole rectangle is shaded, as shown in the margin. SAY: Here is one third of half the rectangle. ASK: What fraction of the rectangle is one third of half of it? (one sixth) Write on the board: of = 6 Exercises: Extend the horizontal lines in the picture. Then write what fraction of the rectangle is shaded. d) COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION Sample solution e) Answers:, 6,, d) Selected solution: Exercises: Draw a picture to find the fraction of the fraction. of of of Answers: /, /, /, d) /0, e) / d) of e) of Number and Operations Fractions  M7
8 Have volunteers show their pictures, and point out that the total number of pieces in the rectangle is the product of the denominators. That is because one of the denominators tells you the number of rows, and the other denominator tells you the number of columns. Also point out that the answer is a unit fraction that is, it is one piece and is the numerator. So the answer is always the unit fraction with a denominator equal to the product of the denominators. Exercises: Find the fraction of the fraction without using a picture. of of 6 of Answers: /, /, /0, Bonus: /,6 Bonus: of, Multiplying unit fractions. Remind students that of can mean multiply. SAY: To multiply / times /, you can find / of /. / of / is equal to /6, so / / is equal to /6 (see example below). ASK: When you think of a rectangle, how can you get the total number of pieces in the whole rectangle from the two fractions? (multiply the denominators) Write on the board: = 6 SAY: Multiply the denominators to get the answer s denominator. Exercises: Multiply without using a picture. of of of Bonus: 000, of Answers: /, /, /0, Bonus: /,000 Practice word problems. Have students solve several word problems. Check the solutions as a class. Kim drinks of a bottle of orange juice each week. What fraction of orange juice does she drink in a day? (/ /7 = /) To make a big muffin, a recipe calls for of a cup of blueberries. Farah is making a small muffin that is the size of a big muffin. What fraction of a cup of blueberries does Farah need? (/ / = /0 of a cup) COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION M Teacher s Guide for AP Book.
9 Extensions. Here is another way of dividing the fraction /7 in half. Instead of dividing a model into a top half and bottom half, divide into a left half and right half of 7 = of 7 = No matter how you find half of /7, the answer should always be the same. Draw two pictures to find the fraction of the fraction. of of. Find the missing number. = 6 = 0 = d) 0 =. Multiply three unit fractions. Answers: /, /0, /00. Find as many answers as you can = 6 0 Answers:, 6;, ;, ; 6, COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION Number and Operations Fractions  M9
10 NF Multiplying Fractions Pages 6 STANDARDS.NF.B. Goals Students will develop and apply the formula for multiplying fractions. Vocabulary denominator improper fraction lowest term numerator unit fraction PRIOR KNOWLEDGE REQUIRED Can multiply unit fractions Understands that of can mean multiply Multiplying fractions by unit fractions. Demonstrate finding half of / by dividing an area model of the fraction into a top half and a bottom half, as shown below: Exercises: Find half of the fraction. 9 7 of = 0 Answers: / = /9, /, /, d) /0 = /, e) /, f) / = /7 d) e) 6 f) 7 Explain to students that to find half of /, you divided the area model showing / into two parts. So, to find a quarter of /, you can divide the area model into four parts. Draw on the board: Exercises: Find the unit fraction of a fraction. of of of Answers: /, /0, / = /6, d) / = / of = 0 d) of Multiplying fractions in general. Tell students you want to find / of /. Draw the picture in the margin on the board. SAY: The shaded portion is twothirds of a rectangle. Then change the picture to show / of /, as shown below: of SAY: Now the shaded portion is four fifths of two thirds of the rectangle. ASK: What fraction of the rectangle is four fifths of two thirds? (/) COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION M0 Teacher s Guide for AP Book.
11 PROMPT: How many parts are within the thick outline? () How many parts are in the whole rectangle? () Write on the board: = of = Exercises: Draw pictures to find the fraction of the fraction. d) Answers: /6 or /, / or /6, /, d) 6/0 or /0, e) / e) ASK: How can you get the total number of pieces in the whole rectangle from the two fractions? (multiply the denominators) Point to the shaded, outlined portion of the rectangle and ASK: How can you get the number of pieces in this shaded rectangle from the two fractions? (multiply the numerators) Write on the board: = SAY: Multiply the numerators to get the answer s numerator and multiply the denominators to get the answer s denominator. Exercises: Multiply without using a picture. 7 d) 6 7 Answers: 9/0, /, /, d) /, e) /6 e) 7 Multiplying improper fractions. SAY: You can also multiply fractions greater than. Draw on the board: = Tell students that you want to know what / of / is. Draw this new diagram on the board: COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION of ASK: How many pieces are in one whole? (0) So then how many pieces are in one half? () Now how many pieces are in fourfifths of threehalves? () Write on the board: = 0 Exercises: Multiply. 7 6 Answers: /0, 0/7 or /, 6/ 7 Number and Operations Fractions  M
12 Practice word problems. Have students solve several word problems. Check the solutions as a class. Tina drinks of a bottle of water each day. How many bottles of water 6 does she drink in seven days? (/6 7 = /6 = /6 bottles) i) One lane of a swimming pool is the length of an Olympic swimming pool. How many lengths of an Olympic swimming pool does Orlando swim when he swims the lane five times? (/ = / = 7/ of an Olympic swimming pool) ii) An Olympic swimming pool for a long course is 0 m long. How many meters did Orlando swim? (0 m / = 70/ = 9 6/ = 9 / m) Extensions (MP.). Multiply. Reduce your answer to lowest terms. What do you notice? Why does this make sense? d) Answer: The answer in lowest terms is always /, because / is always being multiplied by a fraction equivalent to.. Jan bought cups of sugar. She used of the sugar to bake a cake. Each person eats of the cake. How much sugar does each person 6 eat? Is that more or less than cup of sugar? Answer: Each person eats /6 cups of sugar, which is less than / cup. (MP.). Multiply 9 and 9, and then compare your answer to 9 to check your answer. Answer / /9 = 6/6 = / = /6 < /9, because / < / = /9; / /9 = /7 > /9, because /7 > 6/7 = /9 NOTE: Multiplying /9 by a number greater than should produce an answer greater than /9 because you have more than one /9. If you are multiplying /9 by a number less than, the answer should be less than /9 because you have less than one /9. COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION M Teacher s Guide for AP Book.
13 NF Multiplying Mixed Numbers Pages 7 STANDARDS.NF.B. Goals Students will learn how to multiply mixed numbers. Vocabulary denominator improper fraction mixed number numerator remainder PRIOR KNOWLEDGE REQUIRED Can multiply fractions Can convert improper fractions to mixed numbers and vice versa Changing mixed numbers to improper fractions. Review how to turn a mixed number into an improper fraction using a concrete model and an example. Point to the diagram below, and SAY: I multiply because three whole pies are shaded, which each have four pieces in them. This gives shaded pieces. = SAY: I add one more piece because the remaining pie has one piece shaded in it. This gives pieces altogether. + = pieces COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION SAY: I keep the denominator the same because it tells the number of pieces in each whole pie, and that doesn t change. = equal parts shaded equal parts in one whole Exercises: Write the mixed number as an improper fraction. Explain how you found the answer. d) Answers: /7, /6, 9/9, d) 6/, e) /6 e) 6 6 Writing improper fractions as mixed numbers. SAY: If I have the improper fraction /, how do I know how many whole pies there are and how many pieces are left over? I want to divide into sets of two pieces, and I want to know how many full sets there are and if there are any extra pieces. ASK: What operation should I use? (division) What is the leftover part called? (the remainder) Number and Operations Fractions  M
14 Write on the board: = 7 Remainder, so = 7 = 7 R For fourths, draw the following picture on the board with three number statements: = + = = Remainder Multiplying mixed numbers by whole numbers. Tell students that they can now multiply mixed numbers by whole numbers by first changing any mixed number to an improper fraction. Exercises: Multiply. Leave the answer as an improper fraction. 7 Answers: 9/, 0/, 66/ Exercises: Write your answers in the previous question as mixed numbers. Answers: /, 6 /, / Multiplying mixed numbers. Write on the board: = = Have volunteers write the missing numerators and then multiply the improper fractions. (/ 9/ = / and / / = /0) Tell students that, when a question gives the numbers as mixed numbers, they should usually give the answer as a mixed number too. Have volunteers change the answers to mixed numbers. ( 9/ and 6 /0) Exercises: Multiply by changing the mixed number to an improper fraction. Write your answer as a mixed number. Bonus: Answers: 6/ = /, 77/0 = 7 7/0, / = /, Bonus: 9/0 = 6 9/0 Realworld problems. Tell students that you are making / of a recipe that calls for / cups of flour and you want to calculate how much flour to use. Have a volunteer show what expression you need to evaluate. (/ /) Have students evaluate the product and then ask a volunteer to tell you the answer. ( / or /) Now tell students that you have / cups of flour. ASK: Is that enough? (no) What if I used all my flour? Will the COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION M Teacher s Guide for AP Book.
15 cups cups cup cups cups (MP.) recipe turn out? PROMPT: How close to / is /? (it is quite close so the recipe will likely turn out) Draw the diagram in the margin on the board. SAY: / cups of flour is very nearly as much as / cups of flour. Exercises: Will the recipe turn out? I m making batches of gravy. Each batch needs cup of flour. I use cups of flours. I m making of a recipe for cupcakes. The recipe calls for cups of flour. I use cup of flour. I m making batches of cookies. Each batch needs flour. I use cups of flour. d) I m making batches of cookies. Each batch needs flour. I use cups of flour. cups of cups of Answers: yes, I need /6 = /6 cups, which is almost the correct amount; yes, I need /6 cups and cup is slightly more; no, I need 9/ = / cups, which is much less than cups; d) yes, I need /6 = /6 cups, which is very close to cups Extension (MP.) Anna needs these ingredients to make muffins. cups flour cup sugar teaspoon salt teaspoons cinnamon cup milk 7 tablespoons butter teaspoons baking powder egg tablespoons brown sugar COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION Anna has an eightmuffin pan so she would like to make a smaller batch of the recipe. What fraction of the ingredients should she use? Anna needs to use a whole egg. Her egg has a volume of about cup. How much extra liquid does this create in her muffin mix? Anna needs to reduce the milk by the amount of extra liquid she used for the egg to keep the total amount of liquid ingredients the same. How much milk should she use? Answers: /; / of an egg would be / cup, but she used / cup, so she used / cup extra liquid; / of cup = / cup less / cup extra liquid is / cups, or just over / cup milk Number and Operations Fractions  M
16 NF Multiplication and Fractions Pages 9 0 STANDARDS.NF.B. Vocabulary denominator distributive property fraction of a whole number improper fraction larger mixed number numerator product smaller Goals Students will understand that multiplying a given whole number by a fraction greater than one results in a product greater than the whole number, and that multiplying a given whole number by a fraction smaller than one results in a product smaller than the whole number. PRIOR KNOWLEDGE REQUIRED Can multiply fractions Can convert improper fractions to mixed numbers and vice versa Review multiplying whole numbers by fractions. Review situations where the word of means multiply. For example, with whole numbers, groups of means objects. Of can also mean multiply with fractions: / of 6 means / of a group of 6 objects, or / 6. Review finding a fraction of a whole number, and then have students use this method to multiply a fraction by a whole number. of 9 = (9 ) = Exercises: Find the product. 9 d) 7 Answers: 9,, 0, d) Understanding the meaning of the product. Remind students that, to find / or / of, they need to represent, identify four equal groups (so groups of two each), and then circle three of the four groups. Draw on the board: of = 6 ASK: If you divide into four groups and then circle three of the groups, is the circled part smaller or larger than? (smaller) SAY: It is smaller because three groups of something is smaller than the entire four groups. Point to the fraction / and SAY: In the fraction /, the numerator is less than the denominator, so / of is smaller than. Exercises: Find the product. Compare your answer to the whole number. Is the answer larger or smaller? 0 7 d) 9 Answers:,,, d) 6; all answers are smaller COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION M6 Teacher s Guide for AP Book.
17 Multiplying by improper fractions gives larger numbers. Write on the board: 6 Explain to students that to find / 6, they can find / of 6 first. Draw the picture on the board: of 6 = Point to the left group of two dots and SAY: Now take four of these groups. Draw the next picture on the board: 6 of 6 = ASK: Which one is greater: / 6 or 6? (/ 6) Explain that / 6 is greater because / is an improper fraction and improper fractions are always bigger than. Exercises: Find the product. Compare your answer to the whole number. 6 7 d) 0 7 Answers: 6, 0, 7, d) 0; all answers are greater. COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION Multiplying by mixed numbers using distributive property. Explain to students that multiplying a whole number by a mixed number results in a number greater than the whole number because any mixed number can convert to an improper fraction and from the previous section we know that multiplying by an improper fraction gives a larger number than the whole number. SAY: I would like to use another method for multiplying by mixed numbers. Write on the board: 6 Remind students that you can write the mixed number / as + /, and write on the board: 6 = + 6 = = Point to the last expression and SAY: Now you can see this is 6 plus something, so the answer will be greater than 6. Number and Operations Fractions  M7
18 Exercises: Rewrite the product in expanded form. 6 d) 7 Answers: + /, 6 + / 6, + /, d) + /7 Extensions. Writing fractions using their distance from. To find how far a fraction is from, subtract it from. For example, the distance of from is equal to  =. Using  = and fact families, another way of representing is  =. Write the fraction using its distance from. Answers: / = /, / = /, / = /, d) /7 = /7 d) 7. 0 = 0 = 0 0 = 0 0. The result is 0 minus something, and it s less than 0. Rewrite the product in expanded form. 6 d) 7 Sample answers / = ( /) = / = / / 6 = ( /) 6 = 6 / 6 = 6 / 6 COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION M Teacher s Guide for AP Book.
19 NF Scaling Pages STANDARDS.NF.B. Goals Students will use multiplication as resizing. Vocabulary part scale factor whole PRIOR KNOWLEDGE REQUIRED Can multiply fractions by whole numbers Can multiply fractions by fractions and mixed numbers MATERIALS rulers Finding the whole from the part. Draw a shaded square, as shown below. Ask students to extend the picture so the shaded part is half the size of the extended rectangle. Repeat this exercise so that the shaded square is / the size of the extended rectangle. Repeat again for /. ASK: How many equal parts are needed? (three for /, four for /) How many parts do you already have? () So how many more equal parts are needed? (two more for /, three more for /) Extra Practice: Extend the squares above so that / of them are shaded. Repeat this exercise for /7, /7, etc. Exercises: Use a ruler to solve the puzzle. Draw a line cm long. The line represents 6 the whole line looks like. of the whole. Show what Line: cm long. The line represents. Show the whole. COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION Line: cm long. The line represents. Show the whole. d) Line: cm long. The line represents. Show the whole. Bonus e) Line: cm long. The line represents. Show the whole. f) Line: cm long. The line represents. Show. g) Line: cm long. The line represents. Show. Number and Operations Fractions  M9
20 Finding a part from the whole. Ask students to use a ruler and draw three rectangles in their notebooks, each with length 6 cm and width cm. Ask students to shade / of the first rectangle, / of the second rectangle, and /6 of the third rectangle. Draw the answers on the board: 6 Point to the first rectangle and SAY: / of the whole rectangle is shaded. Then write on the board: part = whole ASK: How many shaded parts are needed to cover this whole? () Write on the board right under the previous equation: whole = part Exercises: Write similar equations for the second and the third rectangles. Answers: part = whole /, whole = part ; part = whole /6, whole = part 6 Scale factors. Point to the equation part = whole / on the board and remind students that, in this equation, the number / is called a scale factor. Point to the second equation whole = part and SAY: in this equation, the scale factor is equal to because, to find the whole from a part, you must multiply the part by a number greater than. Scale factors on the number line. Draw a number line from 0 to 0 on the board and bold the number, as shown below Remind students that double means two times, or twice, a number. ASK: What is the double of? () What is half of? () Circle and box, as shown below SAY: If I multiply by the scale factor, I get, and if I multiply by the scale factor /, the answer is. Exercises: Multiply the bold number by the scale factor. Circle the answer on the number line. Scale factor of: Answer: Scale factor of: Answer: COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION M0 Teacher s Guide for AP Book.
21 Scale factor of: Answer: d) Scale factor of: Answer: 6 e) Scale factor of: Answer: 9 Practice word problems. On a map, cm represents 00 m. How many meters in real life does cm represent? ( 00 = 00 m) If two towns are cm apart on the map, what is the actual distance between them? ( 00 =,000 m = km) Extensions. For the following numbers: and 6 and and d) and 0 i) What scale factor do you multiply the first number by to get the second number? ii) What scale factor do you multiply the second number by to get the first number? Answers i), ii) / i), ii) / i), ii) / d) i), ii) /. On a map, cm represents 00 m. If two towns are 7 km apart in real life, how far apart are they on the map? COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION Solution cm = 00 m, 00 m = cm, so 7 km = 7,000 m; 7, = cm Number and Operations Fractions  M
22 NF6 Word Problems with Fractions and Page Multiplication STANDARDS.NF.B.6 Goals Students will solve word problems involving multiplication of fractions. Vocabulary fraction mixed number PRIOR KNOWLEDGE REQUIRED Can add, subtract, and multiply fractions by fractions Review word problems. A review of multiplying fractions involving word problems can be found on AP Book. p.. You may wish to introduce words problems with these exercises. Exercises Michael ate pizza for dinner and had of the pizza left over. The next day, he ate of what was left. How much pizza did Michael eat on the second day? Answer: / / = /6 In Grade, of students have a brother. Of the students with a brother, also have a sister. What fraction of students in Grade have both a brother and a sister? Answer: / / = / Lina has a cake. She gives of what she has to her brother, Joe. What fraction of the whole pie does Joe get? How much is left for Lina? Answer: / / = /6 or / for Joe, / / = /6 /6 = /6 for Lina COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION M Teacher s Guide for AP Book.
23 NF7 Dividing Fractions by Whole Numbers Pages 6 STANDARDS.NF.B.7 Vocabulary denominator fraction improper fraction mixed number numerator unit fraction whole number Goals Students will use models to divide unit fractions and then fractions by whole numbers, and will develop the formula for this type of division through examples. Eventually, students will divide mixed numbers by whole numbers. PRIOR KNOWLEDGE REQUIRED Can divide whole numbers by whole numbers Can name fractions of models when parts shown are unequal Understands division as equal sharing Understands fractions of areas Review dividing whole numbers by whole numbers. Remind students that division can be used for sharing equally. Tell students to pretend that two people are sharing pizzas. Write on the board: 6 = = = Have volunteers use shading to show how much one person gets, and then answer the division. (, / =., /) Dividing unit fractions by whole numbers. Write on the board: = COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION Have a volunteer show how much each person gets by dividing the shaded part in two, as shown below left, and then ASK: Does this picture show a fraction? (yes) Ask students to point out the parts in the picture on the board. Then ask them if they are equal parts. (no, they are /6, /6, /, /) ASK: How can we change the picture to show equal parts? (divide the unshaded parts into equal parts too, see below, right) ASK: What fraction of the pizza is each piece? (/6) Finish the equation, / = /6, and SAY: When one third of a pizza is shared between two people, each person gets one sixth of the pizza. For the first picture shown on the next page, ask a volunteer to divide the shaded half into four equal parts. Draw stripes on one piece and SAY: This is how much each person would get if four people were sharing half a square piece of cake. ASK: What fraction of the whole cake is that? (/) Have a volunteer extend the horizontal lines to show this. Have a volunteer Number and Operations Fractions 7 M
24 write the division equation. (/ = /) PROMPTS: What fraction is being divided? (/) How many equal parts is it divided into? () What fraction of the whole is each equal part? (/) Exercises. Use the picture to divide. Answers: /6, /6. Instead of acutally drawing the horizontal line, imagine extending it. What fraction is each part? d) Bonus: Using pictures, divide in three ways. Answers: /, /9, /, d) /, Bonus: see margin. Write the division equation shown by the picture. Answers: / = /, / = /0 Using a rule to divide unit fractions by whole numbers. Write on the board: SAY: This picture shows / of the rectangle divided into three equal parts. ASK: How can we find out what fraction of the whole rectangle one of the three parts is? (extend the lines) Have a volunteer extend the lines. Then ASK: How many equal parts are there? () COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION M Teacher s Guide for AP Book.
25 Write on the board: = ASK: Without using the picture, how can you get from and? (multiply) Point out that you drew five columns to show / and three rows to find / divided by, so there are = parts altogether. That means each part is / of the rectangle. Exercises: Divide without drawing a picture. d) e) 7 Bonus f) 7 g) 9 6 h) i) j) 00 0 Answers: /0, /6, /, d) /, e) /, Bonus: f) /6, g) /, h) /6, i) /69, j) /6,000 Dividing any fraction by a whole number. Draw on the board: Ask a volunteer to divide the shaded parts into four equal rows. Mark one of the shaded rows with stripes (see first diagram below) and SAY: The amount in this one group shows / divided by. ASK: How can we change the picture so that it is easy to see what fraction of the whole this is? (extend the lines) Have a volunteer extend the lines, as shown in the second diagram. COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION SAY: Now all the parts are equal. ASK: How many parts did I mark with stripes? () How many parts are there altogether? (0) So threetwentieths are striped. Write on the board: = 0 Exercises: Divide without using a picture. d) e) 7 Bonus f) 7 g) 9 h) 00 i) 00 j) Answers: /6, /, /, d) /, e) /, Bonus: f) /6, g) /7, h) /,00, i) /,600, j) 7/,000 Number and Operations Fractions 7 M
26 Dividing improper fractions by a whole number. Draw on the board: SAY: If I want to divide / by, I can divide each half into, mark a / part in each /, and then consider all parts together. Revise the drawing on the board and describe what you are doing, as shown below: = = 0 Explain to students that there 0 parts in each whole and just three parts are striped, so the answer is /0. Exercises: Divide without using a picture. 7 9 d) e) Answers: /6, 7/, 9/, d) /0 or /, e) / Dividing mixed numbers by a whole number. Write on the board: ASK: How can I divide a mixed number by a whole number? If some students say that you can divide the whole part and the fractional part separately and then consider both parts together, suggest instead that they use a method with which they have a lot of experience: convert the mixed number to an improper fraction. SAY: Yes, that s right but it is easier to use a wellknown method. You know how to divide an improper fraction by a whole number, so you can change the mixed number to an improper fraction and then divide the improper fraction by a whole number. Write on the board and describe the steps in each line: Change the mixed number to an improper fraction. ( + ) = Divide the improper fraction by the whole number, which means multiplying the denominator by the whole number. ( ) = = SAY: In the last step, if the answer was an improper fraction, you can change the improper fraction to a mixed number. Write on the board: = 7 = R 7 COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION M6 Teacher s Guide for AP Book.
27 Exercises: Divide by changing the mixed number to an improper fraction. 7 d) e) 9 Answers: /6, 9/ or /, 9/ = /, d) /6 = /6, e) /6 = 7/6 (MP.) Practice word problems. Three people share of a cake equally. What fraction of the cake does each person get? (/) Eight people share pounds of chocolate equally. How much chocolate does each person get? (/6) Four people share of a meat pie. What fraction of the pie does each person get? (/6) Bonus: Five people share of a pie. Do they each have more or less than of the pie? (They each have /, which is less than / (/ = /; / = /; / is more than / because, while the fractions have the same numerator, the parts are bigger in /. A second method is to find a common denominator: / = /00 < /00 = /.) Extensions. Write the fact family for =. Answers / = /, / / =, / = /, / = /. Anna divided 6 9 by in this way: 6 9 =. What is her mistake? COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION (MP.) Answer: Anna divided both the numerator and the denominator by. The correct answer is 6/7 or /9.. Teach students a shortcut to divide when the numerator is a multiple of the whole number being divided by. For example, show them this equation: 6 sevenths = sevenths Then change the sevenths to other words to show what is the same within the equation. For example: 6 apples = apples Number and Operations Fractions 7 M7
28 Have students divide and write the fraction notation for the division equation. fifths = fifths ninths = ninths thirds = thirds d) fifths = fifths Answers: / = /, /9 = /9, / = /, d) / = /. You can divide improper fractions by whole numbers using the same rule that you used to divide proper fractions by whole numbers. Draw a picture to show why this works to divide. (MP.) Use the distributive property to divide mixed numbers by whole numbers. For example: = + = ( ) + = + 6 = = 6 = 6 (MP.) Investigate with several examples to check whether you get the same answers both ways: by changing to improper fractions AND by using the distributive property. = + = ( ) + = + = + = = COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION M Teacher s Guide for AP Book.
29 NF Dividing Whole Numbers by Unit Fractions Pages 7 STANDARDS.NF.B.7 Goals Students will divide whole numbers by unit fractions. Vocabulary division unit fraction whole number PRIOR KNOWLEDGE REQUIRED Understands division as fitting into Understands /n as one of n equal parts of a whole Can use number lines to represent whole numbers Understands fractions of lengths, areas, and number lines MATERIALS BLM Fraction Parts and Wholes (pp. M6 9) dice NOTE: In advance, prepare BLM Fraction Parts and Wholes by copying and cutting out the pieces so that each student will receive cutouts for whole, halves, thirds, and fourths. Review division as fitting into. Remind students that you can look at division as something fitting into something else. For example, to divide 6, you can ask how many inch long objects fit into the length of a 6inch long object: SAY: Three s fit into 6, so 6 =. Dividing by a unit fraction. Give students the prepared cutouts from BLM Fraction Parts and Wholes. ASK: How many /s fit into? () Students should show their answer by lining up pieces over the length of the whole. Write on the board: COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION = ASK: How many /s should fit into? () Have students check this with their cutouts. Ask a volunteer to write the division equation. ( / = ) Repeat for how many fourths fit into ( / = ). Exercises: Divide. 6 7 Bonus: 7 Answers: 6, 7, 0, d) 9, Bonus: 7 0 d) 9 Number and Operations Fractions  M9
30 Dividing a whole number by a unit fraction. Have students work in groups of four. Ask students to use their fraction pieces from BLM Fraction Parts and Wholes to determine, in their groups, how many /s fit into,,, and. Then show students how to write the division equations: = = = 6 = Ask the groups to repeat for thirds and fourths but, this time, have the groups write the division equations themselves. Take up the answers on the board. Point out that, no matter how many unit fractions fit into, twice as many will fit into as fit into, three times as many will fit into, and four times as many will fit into. ASK: How many sixths will fit into? (6) How many sixths will fit into? ( 6 = ) Write on the board: 6 = 6 so 6 = 6 = Exercises: Divide. 7 d) 6 e) 9 f) 0 7 g) 7 h) 9 Bonus i) 00 j) 000, k) 00 l) 00 7, 000 Answers: 0, 0,, d) 0, e), f) 70, g) 6, h) 7, Bonus: i) 00, j),000, k),00, l),00,000 0 Showing division on a number line. Draw the number line in the margin on the board. SAY: The size of a step is half a foot. How many steps fit into feet? (6) Write on the board: = 6 Tell students that drawing number lines is another way to show how many halves fit into. Ask a volunteer to extend the number line to find how many halves fit into. () Then draw a number line from 0 to, divided into fourths. Write on the board: = ASK: How big is each step? (/) Fill in the first blank with /. How many steps of / fit into? () Fill in the second blank. ( / = ) Exercises. Write the division statement to show how many steps fit into the number line. 0 0 COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION M0 Teacher s Guide for AP Book.
31 Bonus: 0 Answers: / =, / = 0, Bonus: / =. Draw a number line to determine. Answer: 0 = 6 Bonus: Draw pizzas divided into fourths to determine. Answer = ACTIVITY Students work in pairs. Player and Player decide together on a unit fraction to use: /, /, or /. Then Player rolls a die to get a number from to 6. Player walks the number of steps rolled on the die. Player then covers the same distance by taking steps that are / (or / or /) the size. Ask pairs to write the result as a division equation. For example, decide on /, roll, take steps, take / steps for the same distance: / = 0. Player and Player switch roles to play again. Extensions (MP.). Explain how you could use... a yard stick to show that = 6. two hundreds blocks to show that 00 = 00. your hands and fingers to show that = 0. COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION (MP.) Answers: Student answers will note how many inches are in feet (or yard), how many hundredths are in hundreds blocks, how many fingers are on hands. Six people share three oranges. Each orange is cut into eighths. How many pieces does each person get? Answer:. Discuss why it makes sense to think of dividing by whole numbers as sharing equally, but dividing by fractions as fitting into. Sample answer: It is hard to see how many pieces of size fit into /, so / would be hard to find by thinking of division as fitting into. In contrast, / would be hard to think of as sharing equally among many / people. Number and Operations Fractions  M
32 NF9 Word Problems with Fractions and Division NF0 Word Problems with Fractions, Pages 9 Multiplication, and Division STANDARDS.NF.B.6 Vocabulary fraction mixed number Goals Students will solve word problems involving multiplication and division of fractions. PRIOR KNOWLEDGE REQUIRED Can add, subtract, and multiply fractions by fractions Can divide fractions by whole numbers Can divide whole numbers by unit fractions MATERIALS / cup measure cup measure enough counters to fill a cup (MP.) Different contexts for dividing fractions. Show students a /cup measure, a cup measure, and enough counters to fill up the cup measure. Tell students that the small measure is labeled as / cup and the big measure as cup. ASK: How many small cupfuls should fill up the big cup? () Ask a volunteer to check that this is the case. Tell students that a recipe calls for cups of flour but you only have the /cup measure. ASK: How many cupfuls do you need? (6) Have a volunteer write the division equation. ( / = 6) Exercises. Tegan needs cups of sugar. She only has a How many cupfuls does she need? cup measure. Alex needs cups of water for a recipe. He only has a cup measure. How many cupfuls does he need? Mary has feet of ribbon. She uses of a foot for each gift. How many gifts can she put ribbon on? d) Rosa has two apples. She cuts them each into fourths. How many pieces does she have? e) Miki has six muffins. He cuts them into halves. i) How many pieces does he have? ii) Four people share the muffins. How many pieces does each person get? Answers: 0; ; ; d) ; e) i), ii) COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION M Teacher s Guide for AP Book.
33 . A satellite makes six revolutions of Earth in one day. How many revolutions would it make in days? Solution: 6 / = 6 9/ = 7. Julie baked five muffins that weigh each muffin weigh? pounds total. How much does Solution: / = 7/ = 7/0. Darya read 70 pages of a book, which is pages long is the entire book? of the book. How many Solution / of the book is 70 pages, so the whole book is 70 = 0 pages Review word problems. Reviews of fractions involving word problems can be found on AP Book. pp. 9. COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION Number and Operations Fractions 9, 0 M
34 NF Comparing Fractions Using Benchmarks Pages STANDARDS.NF.A. Goals Students will compare pairs of fractions by first comparing both to a common benchmark, such as or /. Vocabulary benchmark comparing fractions denominator double equivalent half numerator PRIOR KNOWLEDGE REQUIRED Understand the relationship between half and double Can use a number line to compare fractions Can multiply to find equivalent fractions MATERIALS small and large school supplies (for example, paper clip, pencil sharpener, glue stick) index cards with the fractions 6/7 and / written on them Review the many ways to write one half. Draw several pictures of one half on the board: Have a volunteer write the different names for one half. (/, /, /6, /) Then point out that the fraction / can have many different names. SAY: In a picture showing one half, there are always twice as many parts in the whole as there are in the shaded part. So, you can double the top number (numerator) to get the bottom number (denominator). Exercises: Write the missing denominator. = = 7 = Bonus: =, Answers: 0,,, Bonus:,6 SAY: If you know the bottom number of a fraction equivalent to /, you can divide by to get the top number. Exercises: Write the missing numerator. = 0 = 6 Answers:,,, d) 60 = 0 d) = 0 There are many ways to write as a fraction. SAY: Just like one half, you can also write the number in different ways. A whole pie is a whole pie, no matter how many pieces it is divided into. Draw on the board three fully shaded circles, as shown in the margin. Point to the first circle and ASK: How many parts are shaded? () How many parts are in the whole circle? () Write = / on the board. Repeat for the second and third circles, but COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION M Teacher s Guide for AP Book.
35 have volunteers write the fraction names for : = = = Point out that a fraction is equal to if the numerator is the same as the denominator because that means that all the parts are included. Exercises: Write the missing number to make the fraction equal to. 6 9 d) Bonus: 7 0 Answers: 7, 0, 6, d) 9, Bonus: Fractions greater than. Draw the picture in the margin on the board. SAY: I have a whole pie and one fourth of another pie. ASK: How many fourths of a pie do I have altogether? () Write the fraction / on the board. ASK: Is / more than one whole or less than one whole? (more) SAY: I only need four fourths to make a whole, but I have five fourths. Write on the board several fractions with the denominator : 7 6 For each fraction, have students signal whether the fraction is greater than (thumbs up) or less than (thumbs down). PROMPT: If you had the top number of fourths, would you have more than four fourths? Now write on the board the fraction /. ASK: How many fifths are in a whole? () How many fifths do you have? () Do you have more than a whole? (yes) Point out that, if the top number is more than the bottom number, then the fraction is more than. Write on the board: For each fraction, have students signal whether the fraction is greater than (thumbs up) or less than (thumbs down). Exercises COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION Which fractions are greater than? 6 I R T 7 6 L C Y 6 What country name did you spell? Answers: I, T, A, L, Y; ITALY A 9 0 Fractions greater than on number lines. Draw a number line on the board from 0/ to /. Ask a volunteer to circle the fraction equal to. Then ask another volunteer to circle all the fractions greater than. N E 0 Number and Operations Fractions  M
36 SAY: Fractions are like other numbers: they become greater as you move to the right on a number line. Point out how the denominators are all the same, but the numerators increase as you move to the right. SAY: This is another way of seeing that a fraction with the numerator bigger than the denominator is greater than. Exercises: Circle the fractions that are greater than Answers: /, /; /, /, 6/ Using as a benchmark. Secretly put an object smaller than an eraser in your left hand, such as a paper clip, and an object larger than an eraser, such as a pencil sharpener or a glue stick, in your right hand. Tell students that you have an object smaller than an eraser in your left hand and an object bigger than an eraser in your right hand. ASK: Which is bigger, the object in my left hand or the object in my right hand? Then show students the objects. 6 7 < SAY: Numbers are like objects. Hide an index card labeled 6/7 in your left hand and an index card labeled / in your right hand. Tell students that you have a fraction smaller than in your left hand and a fraction bigger than in your right hand. ASK: Which is bigger, the fraction in my left hand, or the fraction in my right hand? (your right hand) ASK: How do you know? (because the fraction in the right hand is bigger than ) Tape the cards to the board as shown in the margin and write < between them. List pairs of fractions and have students point to the greater fraction in each pair, for example: or 7 6 or 9 Exercises: Which fraction is greater? or 7 or 6 9 or 7 Bonus: 0 00 or , Answers: /, /7, 7/, d) 0/, Bonus: 0/00 d) 0 or Comparing a fraction to one half. Draw a circle with two out of five equal parts shaded on the board (see example in the margin). ASK: How many parts are shaded? () How many parts are not shaded? () ASK: Is more shaded or not shaded? (not shaded) Is / more than half or less than half? (less) Repeat for a circle with two out of three equal parts shaded (see example in the margin). This time, there are more parts shaded () than not shaded (), so / is more than half. Write on the board: < and < COPYRIGHT 0 JUMP MATH: NOT TO BE COPIED. CC EDITION M6 Teacher s Guide for AP Book.
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