Unit 2 Number and Operations—Fractions: Multiplying and Dividing Fractions
IntroductionIn 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 real-world problems and use benchmark fractions and number sense of fractions to compare, estimate mentally, and assess the reasonableness of answers. The real-world problems involve addition and subtraction of fractions, multiplication of fractions and mixed numbers, division of unit fractions by non-zero 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 5. Please note in particular that it is not true that 0.7 ÷ 0.2 = 3 R 1, even though the corresponding whole number division is 7 ÷ 2 = 3 R 1. If anything, the “remainder” would be 0.1. 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 ÷ 2 = 3.5).
Preparation. Cut out the fraction pieces (wholes, halves, thirds, and fourths) from BLM Fraction Parts and Wholes (pp. M-46–49). You will need them in Lessons NF5-28 and NF5-32. Keep the pieces sorted according to their size to facilitate distribution to students.
VOcABULARy division fraction improper fraction mixed number unit fraction
GoalsStudents will recognize a fraction as division of the numerator by the denominator.
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 1/2 beside the picture, as shown below.
12
Draw another circle on the board and ASK: If three friends want to share this pie, how many pieces would there be? (3) Divide the pie into three pieces and ask a volunteer to shade a piece and write the fraction 1/3 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 12 apples equally, each friend gets 12 ÷ 3 = 4 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 1/2 of the pie, so 1 ÷ 2 = 1/2.
Exercises: Shade how much each person gets and then write the fraction.
a) Four people share a pancake b) Six people share a pizza.
Point out that you are drawing circles for pies. ASK: How many circles should I draw? (3) 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? (4) 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.
a) 4 people share 2 pies b) 3 people share 5 pies
c) 5 people share 3 pies d) 2 people share 3 pies
Answers: a) 4 pieces in each pie and 2 whole pies, b) 3 pieces in each pie and 5 whole pies, c) 5 pieces in each pie and 3 whole pies, d) 2 pieces in each pie and 3 whole pies
Exercises: For the previous exercises, draw a picture to show how much one person gets.
Sample answer: c) Each person gets 3/5 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? (3 pies) Write on the board:
2 people share 6 pies 5 people share 4 pies
6 ÷ 2 = 3 ÷ =
Ask a volunteer to fill in the blanks (4 ÷ 5 = 4/5) 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.
Have a volunteer write the division equation to show seven friends sharing two pies. (2 ÷ 7 = 2/7) 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.
a) 5 people share 3 pies b) 4 pies shared among 5people
c) 3 pies shared among 6 people d) 6 people share 4 pies
Answers: a) 3 ÷ 5 = 3/5, b) 4 ÷ 5 = 4/5, c) 3 ÷ 6 = 3/6, d) 4 ÷ 6 = 4/6
Dividing whole numbers without a picture. Write on the board:
1 ÷ 7 = 4 ÷ 9 = 3 ÷ 8 =
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.
a) 1 ÷ 10 b) 4 ÷ 7 c) 8 ÷ 9 d) 7 ÷ 8
Bonus: 13 ÷ 1,000
Answers: a) 1/10, b) 4/7, c) 8/9, d) 7/8, Bonus: 13/1,000
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? (5/3 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 5/3 as the mixed number 1 2/3. Ask a volunteer to shade 5/3 to find the mixed number, as shown below.
53
= 123
Exercises: Divide. Write the answer as an improper fraction and as a mixed number. Show your answer with a picture.
a) 9 ÷ 4 b) 7 ÷ 2 c) 6 ÷ 4 Bonus: 15 ÷ 8
Answers: a) 9/4 = 2 1/4, b) 7/2 = 3 1/2, c) 6/4 = 1 2/4 or 3/2 = 1 1/2, Bonus: 15/8 = 1 7/8
Exercises: Write the answer as an improper fraction and a mixed number.
a) Five people share 12 granola bars. How many granola bars does each person get?
b) Three people share 16 pounds of rice. How much rice does each person get?
c) Four people share nine pounds of flour. How much flour does each person get?
Answers: a) 12/5 = 2 2/5, b) 16/3 = 5 1/3, c) 9/4 = 2 1/4
Review multiplying fractions. Review multiplying a fraction by a whole number.
23
× 9 = (2 × 9) ÷ 3
= 18 ÷ 3
= 6
Exercises: Use multiplication to check the answer.
a) 3 ÷ 4 = 34
b) 2 ÷ 5 = 25
c) 7 ÷ 3 = 73
Selected solution: a) 4 × 3/4 = 12/4 = 3
Extensions1. When you divide 11 granola bars among four people, you divide
11 items among four groups. If you calculate 11 ÷ 4, 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
11 ÷ 4 = 2 34
Number of items
We make 4 groups because there are 4 people
Each person gets 2 whole items
Plus 3/4 of an item
In general, when the divisor (in this case 4) 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 11 m of rope into 4 m pieces, how many pieces can you make?
2 whole pieces of 4 meters in length
3 meters left over (3 meters is 3/4 of 4 meters)
4 4
As the diagram above shows, the whole number (2) in the answer tells you how many pieces of 4 m each (or “groups” each made of four 1 m pieces) can be made (here two groups). The fraction in the answer (3/4) does not mean that you have 3/4 of a meter left over, but instead means that you have 3/4 of the thing you were dividing by left over. The thing you are dividing by is 4 m long, so you have 3/4 of 4 m left over. (3/4 of 4 m is 3 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?
VOcABULARy denominator numerator of (to mean “multiply”) unit fraction
GoalsStudents will develop and apply the formula for multiplying unit fractions by unit fractions.
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 3.
Exercises: Find half.
a) 12
of 4 b) 12
of 8 c) 12
of 10 d) 12
of 12
Answers: a) 2, b) 4, c) 5, d) 6
Review that “of” can mean multiply. Remind students that “of” can mean multiply. SAY: 3 × 4 means “3 groups of 4 things” and 1/2 × 4 means “1/2 of a group of 4 things.”
Exercises: Multiply.
a) 12
× 4 b) 12
× 8 c) 12
× 10 Bonus: 12
× 200
Answers: a) 2, b) 4, c) 5, Bonus: 100
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 1/5 by dividing an area model of the fraction 1/5 into a top half and a bottom half:
15
12 of
15
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? (10) How many are shaded parts? (1) Write on the board:
Exercises: Draw a dotted line to find out how much of the whole rectangle is shaded.
a)
b)
Answers: a) 12
of 12
= 14
, b) 12
of 13
= 16
Exercises: Find half of the fraction.
a) 16
b) 17
c) 19
d) 14
e) 1
11 f ) 1
2Answers: a) 1/12, b) 1/14, c) 1/18, d) 1/8, e) 1/22, f ) 1/4
Finding a unit fraction of a unit fraction. Tell students you want to find 1/3 of 1/2. 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 1/3 of 1/2, 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).
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:
13
of 12
= 16
Exercises: Extend the horizontal lines in the picture. Then write what fraction of the rectangle is shaded.
a) b) c) d)
Answers: a) 18
, b) 16
, c) 1
12, d)
112
Selected solution: c)
112
Exercises: Draw a picture to find the fraction of the fraction.
a) 12
of 14
b) 13
of 14
c) 14
of 12
d) 15
of 12
e) 15
of 13
Answers: a) 1/8, b) 1/12, c) 1/8, d) 1/10, e) 1/15
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 1 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.
a) 13
of 18
b) 12
of 16
c) 14
of 15
Bonus: 12
of 1
2 134,
Answers: a) 1/24, b) 1/12, c) 1/20, Bonus: 1/4,268
Multiplying unit fractions. Remind students that “of” can mean multiply. SAY: To multiply 1/2 times 1/3, you can find 1/2 of 1/3. 1/2 of 1/3 is equal to 1/6, so 1/2 × 1/3 is equal to 1/6 (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:
12
× 13
= 16 2 × 3
SAY: Multiply the denominators to get the answer’s denominator.
Exercises: Multiply without using a picture.
a) 13
of 14
b) 13
of 15
c) 18
of 15
Bonus: 1
1000, of
123
Answers: a) 1/12, b) 1/15, c) 1/40, Bonus: 1/23,000
Practice word problems. Have students solve several word problems. Check the solutions as a class.
a) Kim drinks 12
of a bottle of orange juice each week. What fraction of
orange juice does she drink in a day? (1/2 × 1/7 = 1/14)
b) To make a big muffin, a recipe calls for 15
of a cup of blueberries.
Farah is making a small muffin that is 12
the size of a big muffin.
What fraction of a cup of blueberries does Farah need? (1/5 × 1/2 = 1/10 of a cup)
VOcABULARy denominator improper fraction lowest term numerator unit fraction
GoalsStudents will develop and apply the formula for multiplying fractions.
PRIOR KNOWLEDGE REQUIRED
Can multiply unit fractions Understands that “of” can mean multiply
Multiplying fractions by unit fractions. Demonstrate finding half of 3/5 by dividing an area model of the fraction into a top half and a bottom half, as shown below:
35
12
of 35
= 3
10
Exercises: Find half of the fraction.
a) 29 b)
57
c) 311
d) 25
e) 56
f ) 47
Answers: a) 2/18 = 1/9, b) 5/14, c) 3/22, d) 2/10 = 1/5, e) 5/12, f ) 4/14 = 2/7
Explain to students that to find half of 3/5, you divided the area model showing 3/5 into two parts. So, to find a quarter of 3/5, you can divide the area model into four parts. Draw on the board:
35
14
of 35
= 3
20
Exercises: Find the unit fraction of a fraction.
a) 13
of 25
b) 15
of 34
c) 14
of 23
d) 13
of 35
Answers: a) 2/15, b) 3/20, c) 2/12 = 1/6, d) 3/15 = 1/5
Multiplying fractions in general. Tell students you want to find 4/5 of 2/3. Draw the picture in the margin on the board.
SAY: The shaded portion is two-thirds of a rectangle. Then change the picture to show 4/5 of 2/3, as shown below:
45
of 23
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? (8/15)
PROMPT: How many parts are within the thick outline? (8) How many parts are in the whole rectangle? (15) Write on the board:
45
× 23
= 45
of 23
= 8
15
Exercises: Draw pictures to find the fraction of the fraction.
a) 12
× 23
b) 23
× 14
c) 34
× 12
d) 25
× 34
e) 25
× 23
Answers: a) 2/6 or 1/3, b) 2/12 or 1/6, c) 3/8, d) 6/20 or 3/10, e) 4/15
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:
45
× 23
= 8
15 5 × 34 × 2
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.
a) 35
× 34
b) 37
× 45
c) 45
× 23
d) 35
× 67
e) 38
× 78
Answers: a) 9/20, b) 12/35, c) 8/15, d) 18/35, e) 21/64
Multiplying improper fractions. SAY: You can also multiply fractions greater than 1. Draw on the board:
32
=
Tell students that you want to know what 4/5 of 3/2 is. Draw this new diagram on the board:
45
of 32
ASK: How many pieces are in one whole? (10) So then how many pieces are in one half? (5) Now how many pieces are in four-fifths of three-halves? (12) Write on the board:
Practice word problems. Have students solve several word problems. Check the solutions as a class.
a) Tina drinks 56
of a bottle of water each day. How many bottles of water
does she drink in seven days? (5/6 × 7 = 35/6 = 5 5/6 bottles)
b) i ) One lane of a swimming pool is 38
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? (3/8 × 5 = 15/8 = 1 7/8 of an Olympic swimming pool)
ii ) An Olympic swimming pool for a long course is 50 m long. How many meters did Orlando swim? (50 m × 15/8 = 750/8 = 93 6/8 = 93 3/4 m)
Extensions1. Multiply. Reduce your answer to lowest terms. What do you notice?
Why does this make sense?
a) 25
× 22
b) 25
×33
c) 25
× 44
d) 25
× 55
Answer: The answer in lowest terms is always 2/5, because 2/5 is always being multiplied by a fraction equivalent to 1.
2. Jan bought 52
cups of sugar. She used 34
of the sugar to bake a cake.
Each person eats 16
of the cake. How much sugar does each person
eat? Is that more or less than 13
cup of sugar?
Answer: Each person eats 5/16 cups of sugar, which is less than 1/3 cup.
NOTE: Multiplying 2/9 by a number greater than 1 should produce an answer greater than 2/9 because you have more than one 2/9. If you are multiplying 2/9 by a number less than 1, the answer should be less than 2/9 because you have less than one 2/9.
VOcABULARy denominator improper fraction mixed number numerator remainder
GoalsStudents will learn how to multiply mixed numbers.
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 3 × 4 because three whole pies are shaded, which each have four pieces in them. This gives 12 shaded pieces.
3 × 4 = 12
14
3
SAY: I add one more piece because the remaining pie has one piece shaded in it. This gives 13 pieces altogether.
12 + 1 = 13 pieces
14
3
SAY: I keep the denominator the same because it tells the number of pieces in each whole pie, and that doesn’t change.
14
3 = 134
13 equal parts shaded
4 equal parts in one whole
Exercises: Write the mixed number as an improper fraction. Explain how you found the answer.
a) 317
b) 516
c) 439
d) 758
e) 656
Answers: a) 22/7, b) 31/6, c) 39/9, d) 61/8, e) 41/6
Writing improper fractions as mixed numbers. SAY: If I have the improper fraction 15/2, how do I know how many whole pies there are and how many pieces are left over? I want to divide 15 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)
For fourths, draw the following picture on the board with three number statements:
154
= 334
4 × 3 + 3 = 15
15 ÷ 4 = 3 Remainder 3
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.
a) 312
× 7 b) 123
× 4 c) 3 × 425
Answers: a) 49/2, b) 20/3, c) 66/5
Exercises: Write your answers in the previous question as mixed numbers.
Answers: a) 24 1/2, b) 6 2/3, c) 13 1/5
Multiplying mixed numbers. Write on the board:
123
× 214
= 3
× 4
215
× 234
= 5
× 4
Have volunteers write the missing numerators and then multiply the improper fractions. (5/3 × 9/4 = 45/12 and 11/5 × 11/4 = 121/20) 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. (3 9/12 and 6 1/20)
Exercises: Multiply by changing the mixed number to an improper fraction. Write your answer as a mixed number.
a) 135
× 213
b) 312
× 215
c) 234
× 112
Bonus: 4 1325
× 112
Answers: a) 56/15 = 3 11/15, b) 77/10 = 7 7/10, c) 33/8 = 4 1/8, Bonus: 339/50 = 6 39/50
Real-world problems. Tell students that you are making 3/4 of a recipe that calls for 3 1/2 cups of flour and you want to calculate how much flour to use. Have a volunteer show what expression you need to evaluate. (3/4 × 3 1/2) Have students evaluate the product and then ask a volunteer to tell you the answer. (2 1/8 or 2 5/8) Now tell students that you have 2 1/2 cups of flour. ASK: Is that enough? (no) What if I used all my flour? Will the
recipe turn out? PROMPT: How close to 2 5/8 is 2 1/2? (it is quite close so the recipe will likely turn out) Draw the diagram in the margin on the board.
SAY: 2 1/2 cups of flour is very nearly as much as 2 5/8 cups of flour.
Exercises: Will the recipe turn out?
a) I’m making 512
batches of gravy. Each batch needs 38
cup of flour.
I use 2 cups of flours.
b) I’m making 38
of a recipe for cupcakes. The recipe calls for 212
cups of
flour. I use 1 cup of flour.
c) I’m making 112
batches of cookies. Each batch needs 112
cups of
flour. I use 3 cups of flour.
d) I’m making 212
batches of cookies. Each batch needs 123
cups of
flour. I use 4 cups of flour.
Answers: a) yes, I need 33/16 = 2 1/16 cups, which is almost the correct amount; b) yes, I need 15/16 cups and 1 cup is slightly more; c) no, I need 9/4 = 2 1/4 cups, which is much less than 3 cups; d) yes, I need 25/6 = 4 1/6 cups, which is very close to 4 cups
Extension
Anna needs these ingredients to make 12 muffins.
134
cups flour 12
cup sugar
14
teaspoon salt 113
teaspoons cinnamon
1 cup milk 7 tablespoons butter
2 teaspoons baking powder 1 egg
2 tablespoons brown sugar
a) Anna has an eight-muffin pan so she would like to make a smaller batch of the recipe. What fraction of the ingredients should she use?
b) Anna needs to use a whole egg. Her egg has a volume of about 38
cup. How much extra liquid does this create in her muffin mix?
c) 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: a) 2/3; b) 2/3 of an egg would be 2/8 cup, but she used 3/8 cup, so she used 1/8 cup extra liquid; c) 2/3 of 1 cup = 2/3 cup less 1/8 cup extra liquid is 13/24 cups, or just over 1/2 cup milk
VOcABULARy denominator distributive property fraction of a whole number improper fraction larger mixed number numerator product smaller
GoalsStudents 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, 2 groups of 3 means 2 × 3 objects. “Of” can also mean multiply with fractions: 1/2 of 6 means 1/2 of a group of 6 objects, or 1/2 × 6.
Review finding a fraction of a whole number, and then have students use this method to multiply a fraction by a whole number.
23
of 9 = 2 ×(9 ÷ 3) = 2 × 3
Exercises: Find the product.
a) 35
× 15 b) 29
× 18 c) 25
× 25 d) 47
× 21
Answers: a) 9, b) 4, c) 10, d) 12
Understanding the meaning of the product. Remind students that, to find 3/4 × 8 or 3/4 of 8, they need to represent 8, identify four equal groups (so groups of two each), and then circle three of the four groups. Draw on the board:
34
of 8 = 6
ASK: If you divide 8 into four groups and then circle three of the groups, is the circled part smaller or larger than 8? (smaller) SAY: It is smaller because three groups of something is smaller than the entire four groups. Point to the fraction 3/4 and SAY: In the fraction 3/4, the numerator 3 is less than the denominator 4, so 3/4 of 8 is smaller than 8.
Exercises: Find the product. Compare your answer to the whole number. Is the answer larger or smaller?
a) 13
× 12 b) 25
× 20 c) 37
× 28 d) 89
× 18
Answers: a) 4, b) 8, c) 12, d) 16; all answers are smaller
Multiplying by improper fractions gives larger numbers. Write on the board:
43
× 6
Explain to students that to find 4/3 × 6, they can find 1/3 of 6 first. Draw the picture on the board:
13
of 6 = 2
Point to the left group of two dots and SAY: Now take four of these groups. Draw the next picture on the board:
43
of 6 = 8
6
ASK: Which one is greater: 4/3 × 6 or 6? (4/3 × 6) Explain that 4/3 × 6 is greater because 4/3 is an improper fraction and improper fractions are always bigger than 1.
Exercises: Find the product. Compare your answer to the whole number.
a) 32
× 4 b) 53
× 6 c) 74
× 4 d) 107
× 14
Answers: a) 6, b) 10, c) 7, d) 20; all answers are greater.
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:
113
× 6
Remind students that you can write the mixed number 1 1/3 as 1 + 1/3, and write on the board:
113
× 6 = 1 + 13
× 6 = 1 × 6 + 13
× 6 = 6 + 13
× 6
Point to the last expression 6 + 13
× 6 and SAY: Now you can see this is
6 plus something, so the answer will be greater than 6.
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 1/3 the size of the extended rectangle. Repeat again for 1/4. ASK: How many equal parts are needed? (three for 1/3, four for 1/4) How many parts do you already have? (1) So how many more equal parts are needed? (two more for 1/3, three more for 1/4)
Extra Practice: Extend the squares above so that 2/5 of them are shaded. Repeat this exercise for 1/7, 3/7, etc.
Exercises: Use a ruler to solve the puzzle.
a) Draw a line 1 cm long. The line represents 16
of the whole. Show what the whole line looks like.
b) Line: 1 cm long. The line represents 13
. Show the whole.
c) Line: 2 cm long. The line represents 14
. Show the whole.
d) Line: 3 cm long. The line represents 12
. Show the whole.
Bonus e) Line: 1
12
cm long. The line represents 14
. Show the whole.
f ) Line: 3 cm long. The line represents 14
. Show 12
.
g) Line: 2 cm long. The line represents 18
. Show 14
.
GoalsStudents will use multiplication as resizing.
PRIOR KNOWLEDGE REQUIRED
Can multiply fractions by whole numbers Can multiply fractions by fractions and mixed numbers
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 1 cm. Ask students to shade 1/2 of the first rectangle, 1/3 of the second rectangle, and 1/6 of the third rectangle. Draw the answers on the board:
12
13
16
Point to the first rectangle and SAY: 1/2 of the whole rectangle is shaded. Then write on the board:
part = whole × 12
ASK: How many shaded parts are needed to cover this whole? (2) Write on the board right under the previous equation:
whole = part × 2
Exercises: Write similar equations for the second and the third rectangles.
Answers: part = whole × 1/3, whole = part × 3; part = whole × 1/6, whole = part × 6
Scale factors. Point to the equation “part = whole × 1/2” on the board and remind students that, in this equation, the number 1/2 is called a scale factor. Point to the second equation “whole = part × 2” and SAY: in this equation, the scale factor is equal to 2 because, to find the whole from a part, you must multiply the part by a number greater than 1.
Scale factors on the number line. Draw a number line from 0 to 10 on the board and bold the number 4, as shown below.
0 1 2 3 4 5 6 7 8 9 10 11 12
Remind students that “double” means two times, or twice, a number. ASK: What is the double of 4? (8) What is half of 4? (2) Circle 2 and box 8, as shown below.
0 1 2 3 4 5 6 7 8 9 10 11 12
SAY: If I multiply 4 by the scale factor 2, I get 8, and if I multiply 4 by the scale factor 1/2, the answer is 2.
Exercises: Multiply the bold number by the scale factor. Circle the answer on the number line.
GoalsStudents will solve word problems involving multiplication of fractions.
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 5.2 p. 23. You may wish to introduce words problems with these exercises.
Exercises
a) Michael ate pizza for dinner and had 13
of the pizza left over. The next
day, he ate 12
of what was left. How much pizza did Michael eat on the
second day?
Answer: 1/3 × 1/2 = 1/6
b) In Grade 5, 13
of students have a brother. Of the students with a
brother, 25
also have a sister. What fraction of students in Grade 5
have both a brother and a sister?
Answer: 1/3 × 2/5 = 2/15
c) Lina has 12
a cake. She gives 23
of what she has to her brother, Joe.
What fraction of the whole pie does Joe get? How much is left for Lina?
Answer: 1/2 × 2/3 = 2/6 or 1/3 for Joe, 1/2 – 1/3 = 3/6 – 2/6 = 1/6 for Lina
NF5-27 Dividing Fractions by Whole Numbers Pages 24–26
STANDARDS 5.NF.B.7
VOcABULARy denominator fraction improper fraction mixed number numerator unit fraction whole number
GoalsStudents 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 ÷ 2 = 3 ÷ 2 = 1 ÷ 2 =
Have volunteers use shading to show how much one person gets, and then answer the division. (3, 1 1/2 = 1.5, 1/2)
Dividing unit fractions by whole numbers. Write on the board:
13
÷ 2 =
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 1/6, 1/6, 1/3, 1/3) 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? (1/6) Finish the equation, 1/3 ÷ 2 = 1/6, 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? (1/8) Have a volunteer extend the horizontal lines to show this. Have a volunteer
write the division equation. (1/2 ÷ 4 = 1/8) PROMPTS: What fraction is being divided? (1/2) How many equal parts is it divided into? (4) What fraction of the whole is each equal part? (1/8)
|||||||| ||||||||
Exercises
1. Use the picture to divide.
a) ||||||||
12
÷ 3
b) |||||
13
÷ 2
Answers: a) 1/6, b) 1/6
2. Instead of acutally drawing the horizontal line, imagine extending it. What fraction is each part?
a)
14
÷ 2
b)
13
÷ 3
c)
12
÷ 2
d)
14
÷ 3
Bonus: Using pictures, divide 14
÷ 2 in three ways.
Answers: a) 1/8, b) 1/9, c) 1/4, d) 1/12, Bonus: see margin
3. Write the division equation shown by the picture.
a) ||||| b) ||||||||
Answers: a) 1/3 ÷ 4 = 1/12, b) 1/2 ÷ 5 = 1/10
Using a rule to divide unit fractions by whole numbers. Write on the board:
15
÷ 3||||||
SAY: This picture shows 1/5 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? (15)
ASK: Without using the picture, how can you get 15 from 5 and 3? (multiply) Point out that you drew five columns to show 1/5 and three rows to find 1/5 divided by 3, so there are 5 × 3 = 15 parts altogether. That means each part is 1/15 of the rectangle.
Exercises: Divide without drawing a picture.
a) 14
÷ 5 b) 12
÷ 8 c) 12
÷ 11 d) 1
11 ÷ 5 e)
15
÷ 7
Bonus
f ) 17
÷ 8 g) 19
÷ 6 h) 18
÷ 8 i ) 1
23 ÷ 3 j )
1200
÷ 30
Answers: a) 1/20, b) 1/16, c) 1/22, d) 1/55, e) 1/35, Bonus: f ) 1/56, g) 1/54, h) 1/64, i ) 1/69, j ) 1/6,000
Dividing any fraction by a whole number. Draw on the board:
35
÷ 4
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 3/5 divided by 4. 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.
|||||| |||||| |||||| |||||| |||||| ||||||
SAY: Now all the parts are equal. ASK: How many parts did I mark with stripes? (3) How many parts are there altogether? (20) So three-twentieths are striped. Write on the board:
35
÷ 4 = 3
20 5 × 4
Exercises: Divide without using a picture.
a) 38
÷ 2 b) 25
÷ 3 c) 34
÷ 2 d) 45
÷ 5 e) 57
÷ 2
Bonus
f ) 58
÷ 7 g) 58
÷ 9 h) 58
÷ 300 i ) 543800
÷ 2 j ) 1740
÷ 50
Answers: a) 3/16, b) 2/15, c) 3/8, d) 4/25, e) 5/14, Bonus: f ) 5/56, g) 5/72, h) 5/2,400, i ) 543/1,600, j ) 17/2,000
Dividing improper fractions by a whole number. Draw on the board:
32
÷ 5
SAY: If I want to divide 3/2 by 5, I can divide each half into 5, mark a 1/5 part in each 1/3, and then consider all parts together. Revise the drawing on the board and describe what you are doing, as shown below:
32
÷ 5 =
|||||||| |||||||| ||||||||
=3
10 2 × 5
Explain to students that there 10 parts in each whole and just three parts are striped, so the answer is 3/10.
Exercises: Divide without using a picture.
a) 53
÷ 2 b) 74
÷ 3 c) 92
÷ 2 d) 52
÷ 5 e) 81
÷ 3
Answers: a) 5/6, b) 7/12, c) 9/4, d) 5/10 or 1/2, e) 8/3
Dividing mixed numbers by a whole number. Write on the board:
112
÷ 5
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 well-known 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:
425
÷ 3
Change the mixed number to an improper fraction. (4 × 5 + 2)
= 225
÷ 3
Divide the improper fraction by the whole number, which means multiplying the denominator by the whole number. (5 × 3)
= 22
5 3×
= 2215
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:
Exercises: Divide by changing the mixed number to an improper fraction.
a) 123
÷ 2 b) 214
÷ 3 c) 547
÷ 2 d) 512
÷ 3 e) 589
÷ 4
Answers: a) 5/6, b) 9/12 or 3/4, c) 39/14 = 2 11/14, d) 11/6 = 1 5/6, e) 53/36 = 1 17/36
Practice word problems.
a) Three people share 58
of a cake equally. What fraction of the cake does
each person get? (5/24)
b) Eight people share 12
pounds of chocolate equally. How much
chocolate does each person get? (1/16)
c) Four people share 34
of a meat pie. What fraction of the pie does each
person get? (3/16)
Bonus: Five people share 35
of a pie. Do they each have more or less than 18
of the pie? (They each have 3/25, which is less than 1/8 (3/5 ÷ 5 = 3/25;
1/8 = 3/24; 3/24 is more than 3/25 because, while the fractions have the same numerator, the parts are bigger in 3/24. A second method is to find a common denominator: 3/25 = 24/200 < 25/200 = 1/8.)
Answers: a) 4/5 ÷ 2 = 2/5, b) 8/9 ÷ 2 = 4/9, c) 8/3 ÷ 4 = 2/3, d) 12/5 ÷ 4 = 3/5
4. a) 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 53
÷ 2.
b) Use the distributive property to divide mixed numbers by whole numbers. For example:
512
÷ 3 = 5 + 12
÷ 3
= (5 ÷ 3) + 12
÷ 3
= 53
+ 16
= 106
+ 16
= 116
= 116
c) 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.
NF5-28 Dividing Whole Numbers by Unit Fractions Pages 27–28
STANDARDS 5.NF.B.7
VOcABULARy division unit fraction whole number
GoalsStudents will divide whole numbers by unit fractions.
PRIOR KNOWLEDGE REQUIRED
Understands division as fitting into Understands 1/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. M-46–49) dice
NOTE: In advance, prepare BLM Fraction Parts and Wholes by copying and cutting out the pieces so that each student will receive cut-outs for 1 whole, 2 halves, 3 thirds, and 4 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 ÷ 2, you can ask how many 2-inch long objects fit into the length of a 6-inch long object:
2 2 2
SAY: Three 2s fit into 6, so 6 ÷ 2 = 3.
Dividing 1 by a unit fraction. Give students the prepared cut-outs from BLM Fraction Parts and Wholes. ASK: How many 1/2s fit into 1? (2) Students should show their answer by lining up pieces over the length of the 1 whole. Write on the board:
1 ÷ 12
= 2
ASK: How many 1/3s should fit into 1? (3) Have students check this with their cut-outs. Ask a volunteer to write the division equation. (1 ÷ 1/3 = 3) Repeat for how many fourths fit into 1 (1 ÷ 1/4 = 4).
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 1/2s fit into 1, 2, 3, and 4. Then show students how to write the division equations:
1 ÷ 12
= 2 2 ÷ 12
= 4 3 ÷ 12
= 6 4 ÷ 12
= 8
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 1, twice as many will fit into 2 as fit into 1, three times as many will fit into 3, and four times as many will fit into 4. ASK: How many sixths will fit into 1? (6) How many sixths will fit into 3? (3 × 6 = 18) Write on the board:
1 ÷ 16
= 6 so 3 ÷ 16
= 3 × 6 = 18
Exercises: Divide.
a) 5 ÷ 14
b) 2 ÷ 15
c) 3 ÷ 17
d) 5 ÷ 16
e) 9 ÷ 12
f ) 10 ÷ 17
g) 8 ÷ 17
h) 9 ÷ 18
Bonus
i ) 100 ÷ 13
j ) 5 ÷ 1
1000, k) 13 ÷ 1
100 l ) 400 ÷
17 000,
Answers: a) 20, b) 10, c) 21, d) 30, e) 18, f ) 70, g) 56, h) 72, Bonus: i ) 300, j ) 5,000, k) 1,300, l ) 2,800,000
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 3 feet? (6) Write on the board:
3 ÷ 12
= 6
Tell students that drawing number lines is another way to show how many halves fit into 3. Ask a volunteer to extend the number line to find how many halves fit into 4. (8) Then draw a number line from 0 to 2, divided into fourths. Write on the board:
2 ÷ =
ASK: How big is each step? (1/4) Fill in the first blank with 1/4. How many steps of 1/4 fit into 2? (8) Fill in the second blank. (2 ÷ 1/4 = 8)
Exercises
1. Write the division statement to show how many steps fit into the number line.
Answers: a) 3 ÷ 1/4 = 12, b) 5 ÷ 1/2 = 10, Bonus: 5 ÷ 1/3 = 15
2. Draw a number line to determine 2 ÷ 13
.
Answer: 0 1 2
2 ÷ 13
= 6
Bonus: Draw pizzas divided into fourths to determine 3 ÷ 14
.
Answer
3 ÷ 14
= 12
AcTIVITy
Students work in pairs. Player 1 and Player 2 decide together on a unit fraction to use: 1/2, 1/3, or 1/4. Then Player 1 rolls a die to get a number from 1 to 6. Player 1 walks the number of steps rolled on the die. Player 2 then covers the same distance by taking steps that are 1/2 (or 1/3 or 1/4) the size. Ask pairs to write the result as a division equation. For example, decide on 1/2, roll 5, take 5 steps, take 1/2 steps for the same distance: 5 ÷ 1/2 = 10. Player 1 and Player 2 switch roles to play again.
Extensions1. Explain how you could use ...
a) a yard stick to show that 3 ÷ 1
12 = 36.
b) two hundreds blocks to show that 2 ÷ 1
100 = 200.
c) your hands and fingers to show that 2 ÷ 15
= 10.
Answers: Student answers will note a) how many inches are in 3 feet (or 1 yard), b) how many hundredths are in 2 hundreds blocks, c) how many fingers are on 2 hands
2. Six people share three oranges. Each orange is cut into eighths. How many pieces does each person get?
Answer: 4
3. 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 3 fit into 1/2, so 1/2 ÷ 3 would be hard to find by thinking of division as fitting into. In contrast, 3 ÷ 1/2 would be hard to think of as sharing equally among many 1/2 people.
GoalsStudents 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
1/3 cup measure 1 cup measure enough counters to fill a cup
Different contexts for dividing fractions. Show students a 1/3-cup measure, a 1-cup measure, and enough counters to fill up the 1-cup measure. Tell students that the small measure is labeled as 1/3 cup and the big measure as 1 cup. ASK: How many small cupfuls should fill up the big cup? (3) Ask a volunteer to check that this is the case. Tell students that a recipe calls for 2 cups of flour but you only have the 1/3-cup measure. ASK: How many cupfuls do you need? (6) Have a volunteer write the division equation. (2 ÷ 1/3 = 6)
Exercises
1. a) Tegan needs 5 cups of sugar. She only has a 12
cup measure. How many cupfuls does she need?
b) Alex needs 3 cups of water for a recipe. He only has a 14
cup measure. How many cupfuls does he need?
c) Mary has 5 feet of ribbon. She uses 13
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: a) 10; b) 12; c) 15; d) 8; e) i ) 12, ii ) 3
(MP.4)
NF5-29 Word Problems with Fractions and Division NF5-30 Word Problems with Fractions, Pages 29–31
GoalsStudents will compare pairs of fractions by first comparing both to a common benchmark, such as 1 or 1/2.
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 5/4 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. (1/2, 2/4, 3/6, 4/8) Then point out that the fraction 1/2 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.
a) 12
= 5
b) 12
= 7
c) 12
= 12
Bonus: 12
= 4,132
Answers: a) 10, b) 14, c) 24, Bonus: 8,264
SAY: If you know the bottom number of a fraction equivalent to 1/2, you can divide by 2 to get the top number.
Exercises: Write the missing numerator.
a) 12
= 10
b) 12
= 36
c) 12
= 50
d) 12
= 120
Answers: a) 5, b) 18, c) 25, d) 60
There are many ways to write 1 as a fraction. SAY: Just like one half, you can also write the number 1 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? (2) How many parts are in the whole circle? (2) Write 1 = 2/2 on the board. Repeat for the second and third circles, but
Point out that a fraction is equal to 1 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 1.
a) 7
b) 10
c) 6
d) 9
Bonus: 182
Answers: a) 7, b) 10, c) 6, d) 9, Bonus: 182
Fractions greater than 1. 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? (5) Write the fraction 5/4 on the board. ASK: Is 5/4 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 4:
34
74
14
64
For each fraction, have students signal whether the fraction is greater than 1 (thumbs up) or less than 1 (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 8/5. ASK: How many fifths are in a whole? (5) How many fifths do you have? (8) 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 1. Write on the board:
98
37
103
310
For each fraction, have students signal whether the fraction is greater than 1 (thumbs up) or less than 1 (thumbs down).
Exercises
a) Which fractions are greater than 1?
I 65
R 38
T 154
A 98
E 1011
L 75
c 68
y 86
N 8
10
b) What country name did you spell?
Answers: a) I, T, A, L, Y; b) ITALY
Fractions greater than 1 on number lines. Draw a number line on the board from 0/4 to 8/4. Ask a volunteer to circle the fraction equal to 1. Then ask another volunteer to circle all the fractions greater than 1.
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 1.
Exercises: Circle the fractions that are greater than 1.
a)
02
12
22
32
42
b)
03
13
23
33
43
53
63
Answers: a) 3/2, 4/2; b) 4/3, 5/3, 6/3
Using 1 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.
SAY: Numbers are like objects. Hide an index card labeled 6/7 in your left hand and an index card labeled 5/4 in your right hand. Tell students that you have a fraction smaller than 1 in your left hand and a fraction bigger than 1 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 1) 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:
35
or 76
85
or 911
Exercises: Which fraction is greater?
a) 23
or 54
b) 87
or 1114
c) 69
or 75
d) 108
or 1215
Bonus: 101100
or 980
1000,
Answers: a) 5/4, b) 8/7, c) 7/5, d) 10/8, Bonus: 101/100
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? (2) How many parts are not shaded? (3) ASK: Is more shaded or not shaded? (not shaded) Is 2/5 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 (2) than not shaded (1), so 2/3 is more than half. Write on the board:
Exercises: What fraction is shaded? Is it more or less than half?
a) b)
Answers: a) 3/8 is less than half, b) 6/10 is more than half
comparing a fraction to one half using doubling. Draw the diagram in the margin on the board and write the fraction 5/11 in stacked form below it. ASK: How many parts are there altogether? (11) How can you tell from the fraction how many parts are shaded? (the numerator tells us, so 5 parts are shaded) How many parts are not shaded? (6) Write on the board:
not shadedshaded altogether
5 + 6 = 11
5 + 5 = 10
Point out that double the shaded parts (5 + 5) is less than the whole number of parts (10 < 11), so the fraction is less than 1/2. Write on the board:
38
59
49
511
815
1225
Point to 3/8 and ASK: What is double 3? (6) Is 6 more or less than the total number of pieces? (less) Point out that double the number of shaded pieces is less than a whole, so 3/8 < 1/2. Repeat for 5/9: double 5 is 10, and 10 is more than 9, so 5/9 is more than 1/2. Then, for the remaining fractions, have students signal whether the fraction is greater (thumbs up) or less (thumbs down) than one half.
NOTE: Some students might find it easier to subtract the numerator from the denominator to find the number of parts not shaded, and compare the result to the numerator (the number of parts shaded). For example: 11/20 is greater than 1/2 because more parts are shaded (11) than not shaded (20 – 11 = 9). This is also an acceptable method to compare fractions to 1/2.
Exercises: Is the fraction more than 12
or less than 12
?
a) 58
b) 1120
c) 1735
d) 2245
Answers: a) more, b) more, c) less, d) less
Bonus: Which fractions are more than half? Write down the letters. What do they spell?
Point to each fraction and ASK: Is it more or less than half? (3/8 is less, 2/3 is more) Now draw the inequality in the margin on the board. Have a volunteer write the fractions in the correct boxes. SAY: 3/8 is less than 2/3 because 3/8 is less than 1/2, and 1/2 is less than 2/3.
Exercises: Compare the fractions. Write < or >. Hint: First compare them both to 1/2.
a) 59
6
15 b)
23
4
10 c)
1018
1230
d) 49
58
Answers: a) >, b) >, c) >,d) <, Bonus: <
Practice word problems.
a) Daniel walked 511
of a mile. Is that more than or less than half a mile? (less)
b) On a soccer team, 8
15 of the players are girls. Are there more boys or
girls on the team? (more girls)
c) Lina gave away 35
of a cake and kept the rest. Did she keep more or
less than half? (less)
d) Katie ate 4/9 of a chocolate bar and Jacob ate 611
of a chocolate bar. Who had more? (Jacob)
Extensions1. Write any number to make the fraction greater than 1.
a) 7
b) 12
c) 6
Bonus: 7
Answers: a) 8 or more, b) 13 or more, c) 7 or more, Bonus: 1 to 6
2. Write any number to make the fraction greater than 1/2, but less than 1.
a) 7
b) 12
c) 6
Bonus: 7
Answers: a) 4, 5, or 6; b) 7, 8, 9, 10, or 11; c) 4 or 5; Bonus: 8 to 13
3. John adds 25
+ 53
= 78
. What mistake did he make? How can you tell
by estimating that the answer is incorrect? Hint: Look for any fractions in the equation that are more than 1.
Answer: He added the numerators and denominators to add the fractions. You can’t do that when adding fractions; you have to change the fractions to have the same denominator before you add. He was probably thinking about the method for multiplying fractions. To estimate the answer, note that 5/3 is more than 1. When you add something to a number more than 1, you should get a number more than 1. 7/8 is less than 1, so the answer is incorrect.
GoalsStudents will estimate addition, subtraction, multiplication, and division by estimating fractions.
PRIOR KNOWLEDGE REQUIRED
Can use a number line to compare fractions Can multiply to find equivalent fractions
MATERIALS
BLM Fraction Parts and Wholes (pp. M-46–49) overhead transparency sheets markers for overhead transparency sheets grid paper scissors
Fractions on number lines. Draw or project this number line on the board:
010
110
210
310
410
510
610
710
810
910
1010
0 1
NOTE: Invite students, in small groups if necessary, to gather around as you go through the first part of the lesson.
Have a volunteer show where 1/2 is on the number line. Prepare another number line the same length divided into two equal parts on another transparency. Superimpose it over the number line above so that students see that 1/2 is exactly at the 5/10 mark. ASK: Which fraction is equal to 1/2? (5/10) Is 2/10 between 0 and 1/2 or between 1/2 and 1? (between 0 and 1/2) Is 7/10 between 0 and 1/2 or between 1/2 and 1? (between 1/2 and 1) What about 6/10? (between 1/2 and 1) 4/10? (between 0 and 1/2) 3/10? (between 0 and 1/2) 9/10? (between 1/2 and 1)
Go back to the fraction 2/10 and ASK: We know that 2/10 is between 0 and 1/2, but is it closer to 0 or to 1/2? (closer to 0) Draw above and below the number line as follows:
010
110
210
310
410
510
610
710
810
910
1010
0 12
1
Consider 6/10. ASK: Is 6/10 between 0 and 1/2 or between 1/2 and 1? (between 1/2 and 1) Which number is 6/10 closest to, 1/2 or 1? (1/2)
Have a volunteer show the distance to each number with arrows. Which arrow is shorter? (the arrow from 6/10 to 1/2)
Bonus: Which number is 4/10 closest to, 0, 1/2, or 1? (1/2) Which is 8/10 closest to? (1) Go through all of the remaining fractions between 0 and 1.
Make number lines. Have students draw a line on grid paper 10 squares long. Then have them cut out the line—leaving space above and below for writing—and fold it in half so that the two endpoints meet. Ask students to mark on their line the points 0, 1/2, and 1. Now have students refold the line in half, and then fold it in half a second time.
Have them unfold the line and look at the folds. ASK: What fraction is exactly halfway between 0 and 1/2? (1/4) How do you know? (because the sheet is folded into four equal parts so the first fold from 0 must be 1/4 of the distance from 0 to 1) Point out that the second half of the number line is also cut into two quarters. SAY: So every half is two quarters.
ASK: What fraction is halfway between 1/2 and 1? How do you know? (3/4 because the sheet is folded into four equal parts so the third fold from 0 must be 3/4 of the distance from 0 to 1)
Have students mark the fractions 1/4 and 3/4 on their number lines. Then have students write the fractions from 1/10 to 9/10 in the correct places on their number lines, using the squares on the grid paper to help them.
Tell students to look at the number lines they’ve created and to fill in the blanks in the following questions by writing “less than” or “greater than” in their notebooks.
a) 4/10 is 1/4 b) 4/10 is 1/2 c) 8/10 is 3/4
d) 2/10 is 1/4 e) 3/10 is 1/2 f ) 7/10 is 3/4
Answers: a) greater than, b) less than, c) greater than, d) less than, e) less than, f ) less than
Have students rewrite each statement using the “greater than” and “less than” symbols: > and <.
Estimating by rounding to 0, 1/2, or 1. Write on the board:
110
+ 3
SAY: I would like to estimate the sum. ASK: Is 1/10 closer to 0 or 1/2? (0) Then write on the board:
110
+ 3 ≈ 0 + 3 = 3
Exercise: Estimate by rounding one of the fractions to 0, 12
, or 1.
a) 1
10 +
57
b) 67
× 1317
c) 8
15 ÷
12
Answers: a) 0 + 5/7 = 5/7, b) 1 × 13/17 = 13/17, c) 1/2 ÷ 1/2 =1, Bonus: 28/27 – 0 = 28/27 or 1 – 1/10 = 9/10
Estimating sums of fractions using half. Now tell students that, even though it’s hard to find out exactly what amount the two fractions add to, you can still find out something about the sum; you can compare the sum to the number 1.
Cut out the pieces from BLM Fraction Parts and Wholes. Give each student one whole, two halves, three thirds, and four fourths. Have students verify that the cards are labeled correctly by checking that two halves fit exactly into one whole, as do three thirds and four fourths. Then have students line up different numbers of pieces from the BLM to check if they add to more or less than 1. For example, to check whether 2/3 + 1/2 is more or less than 1, students can take two 1/3 pieces and one 1/2 piece, lay them end to end, and compare the total length to one whole:
13
13
12
One whole
Because the pieces in total are longer than one whole, the fractions add to more than 1. Point out that, even without knowing the actual value of the answer, we can still determine whether the answer is more or less than 1. (In this case, it’s more.)
Exercises: Does the fraction add to more than 1 or less than 1? Explain your answer using 1/2 as a benchmark.
a) 12
+ 25
b) 12
+ 34
c) 34
+ 35
d) 23
+ 34
Answers: a) less than 1 because one fraction is 1/2 and the other is less than 1/2; b) more because one fraction is 1/2 and the other is more than 1/2; c) more because both fractions are more than 1/2; d) more because both fractions are more than 1/2
Mixed numbers on number lines. Draw on the board:
1 1 14
1 24
1 34
2 2 14
2 24
2 34
3 3 14
3 24
ASK: We know that 1 3/4 is between 1 and 2, but is it closer to 1 or to 2? Draw on the board:
1 1 14
1 24
1 34
2 2 14
2 24
2 34
3 3 14
3 24
ASK: What is the fraction part of 1 3/4? (3/4) ASK: Is it greater than 1/2 or smaller than 1/2? (greater) SAY: If the fraction part is greater than half, the nearest whole number is to the right.
ASK: Is 2 1/4 between 2 and 2 1/2 or between 2 1/2 and 3? (2 and 2 1/2) Which number is it closest to, 2 or 2 1/2? (2) ASK: What is the fraction part of 2 1/4? (1/4) ASK: Is it greater than 1/2 or smaller than 1/2? (smaller) SAY: If the fraction part is less than half, the nearest whole number is to the left.
Exercises: Round the mixed number to the nearest whole number.
a) 215
b) 435
c) 316
d) 759
Answers: a) 2, b) 5, c) 3, d) 8
Estimating subtraction, addition, multiplication, and division on mixed numbers. Write on the board:
759
- 215
SAY: By using the previous exercise about rounding up or rounding down, we can look at the numbers and say: 7 5/9 is almost equal to 8 and 2 1/5 is almost equal to 2, so the difference is almost equal to 6. Write on the board:
759
- 215
≈ 8 – 2 = 6
Exercises: Estimate by rounding both fractions to the nearest whole number (including 0).
a) 21
10 + 3
57
b) 318
× 34
c) 62
15 ÷ 1
45
Bonus: 37981
+ 1423
Answers: a) 2 + 4 = 6, b) 3 × 1 = 3, c) 6 ÷ 2 = 3, Bonus: 4 - 1 = 3
Extensions1. Which answers are less than 1? Explain how you know.
a) 125
- 34
b) 11121
- 1735
c) 149
100 -
3670
Answers: a) less than one because 2/5 is less than half and 3/4 is greater than half; b) greater than one because 11/21 is greater than half and 17/35 is less than half; c) less than one because 49/100 is less than half and 36/70 is greater than half
2. Which is greater, 45
or 56
? Hint: Think about the size of the missing
part of each fraction. The missing part, or missing fraction, is what you would get by taking the fraction away from the whole.
Answer: 5/6 is greater because it has a smaller missing fraction. The missing fraction of 4/5 is 1/5 and the missing fraction of 5/6 is 1/6, so 5/6 is closer to one than 4/5 is.
Interpreting mixed number answers when the answer has to be a whole number. Tell students that some problems that involve division will have a mixed number answer, but the answer needs to be a whole number. Write on the board:
Nomi can carry 16 lbs.
How many books weighing 112
pounds each can she carry?
Have volunteers tell you what to divide (16 ÷ 1 1/2), change the mixed number to an improper fraction (1 1/2 = 3/2), do the division (16 ÷ 3/2 = 16 × 2/3 = 32/3), and write the answer as a mixed number. (10 2/3) Point out that she can’t carry 2/3 of a textbook, so she has to only carry 10 books. Emphasize that the answer students got by dividing is a mixed number, but the answer to the problem has to be a whole number.
Exercises
NOTE: You can change the exercises as you wish to include potatoes, corn, cake, or any similar example that can be shared equally.
a) Ron can carry 1623
lb. How many textbooks weighing 223
lb each can
he carry at once?
b) Diba can carry 1512
lb. How many textbooks weighing 123
lb each can
she carry at once?
c) Lina has 35
lb of dry pasta. Each person needs 3
16 lb. How many
people can she feed?
d) Bilal has 78
lb of dry rice. Each person needs 3
16 lb. How many
people can he feed?
Answers: a) 50/3 ÷ 8/3 = 6 1/4, so he can carry six textbooks at once; b) 31/2 ÷ 5/3 = 9 3/10, so nine textbooks; c) 3/5 ÷ 3/16 = 3 1/5, so three people; d) 7/8 ÷ 3/16 = 4 2/3, so four people
NF5-33 cumulative Review Pages 37–38
STANDARDS 5.NF.A.2, 5.NF.B.6, 5.NF.B.7.c
VOcABULARy fraction mixed number remainder whole number
GoalsStudents will solve word problems involving multiplication and division of fractions.
PRIOR KNOWLEDGE REQUIRED
Can add, multiply, and divide fractions by fractions
Answers i ) The orders are reversed. ii ) Dividing the same number by a larger number gives a smaller answer.
d) Divide and then compare the fraction to 29
÷ 1 = 29
to check your answer.
i ) 29
÷ 34
29
÷ 43
ii ) 29
÷ 52
29
÷ 25
iii ) 29
÷ 74
29
÷ 47
Answers i ) 2/9 ÷ 3/4 = 8/27; 2/9 ÷ 4/3 = 6/36 = 1/6, so 2/9 ÷ 3/4 is larger than 2/9 and 2/9 ÷ 4/3 is smaller than 2/9
ii ) 2/9 ÷ 5/2 = 4/45; 2/9 ÷ 2/5 = 10/18 = 5/9, so 2/9 ÷ 5/2 is smaller than 2/9 and 2/9 ÷ 2/5 is larger than 2/9
iii ) 2/9 ÷ 7/4 = 8/63; 2/9 ÷ 4/7 = 14/36 = 7/18, so 2/9 ÷ 7/4 is smaller than 2/9 and 2/9 ÷ 4/7 is larger than 2/9
2. How do the answers in each pair compare? Which answers are greater than 1? Why does this make sense?
a) 23
÷ 35
and 35
÷ 23
b) 14
÷ 13
and 13
÷ 14
c) 92
÷ 52
and 52
÷ 92
d) 32
÷ 73
and 73
÷ 32
Answer: When the dividend is greater than the divisor, the answer is greater than 1. This makes sense because division is asking how many of the divisor fit into the dividend. The answer is more than 1 precisely when fitting a smaller object into a larger object.
3. Have students distinguish between problems where a mixed number answer should be rounded up or rounded down to a whole number.
For example, if Katie needs to carry 3412
pounds of groceries, but she
can only carry 15 pounds on each trip, how many trips does she need to make?
Answer: 34 1/2 ÷ 15 = 2 9/30; the mixed number answer must be rounded up because two trips would only let Katie carry 30 pounds; she needs to make three trips.
4. Calculate.
a) 45
÷ 13
b) 45
÷ 26
c) Compare your answers to parts a) and b). What do you notice? Why is this the case?
Answer: The answers are the same, 12/5 = 24/10, which makes sense because you were dividing 4/5 by equivalent fractions in the first place.
