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Common Core Learning StandardsGRADE 5 Mathematics

NUMBER & OPERATIONS – FRACTIONS

Common Core Learning Standards Concepts Embedded Skills Vocabulary

Use equivalent fractions as a strategy to add and subtract fractions.

Adding and subtracting

fractions

Add fractions with unlike denominators (including mixed numbers).

Simplify common

denominators unlike

denominators fraction equivalent reduce mixed number improper

fraction numerator

Subtract fractions with unlike denominators (including mixed numbers).

5.NF.1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

Simplify fraction solutions.

Rewrite two fractions with unlike denominators to have common denominators in order to add or subtract fractions.

SAMPLE TASKSI. These rectangles have been divided into fourths. Use the rectangles below to divide into equivalent parts.

Example: Fourths Change to eighths Change to twelfths

Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.

II. These rectangles have been divided into thirds. Use the rectangles above to create equivalent parts

Thirds Sixths Ninths Twelfths

III. These rectangles have been divided into fifths. Use the rectangles above to create equivalent parts

Example: Fifths Tenths Twentieths

IV. After the painter had painted 2/3 of this painting, he found a mistake and had to scrape 1/6 of it off. How much was still painted? Show your work by dividing the rectangle into equal parts.


V. The painter painted 3/10 of a wall blue and 2/5 of the wall orange. How much of the wall is painted? Use the rectangle below to show your work.



Use equivalent fractions as a strategy to add and subtract fractions.

fractions and real world situations

Solve word problems involving addition and subtraction of fractions of unlike denominators referring to the same whole.

unlike denominators

benchmark fractions

estimation fraction equivalent reduce mixed number improper

fraction numerator

Justify the reasonableness of a solution using estimation and benchmark fractions.5.NF.2.

Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

SAMPLE TASKSI. You are making trim for a quilt. You need 5 and ½ feet of purple lace and 3 and ¾ feet of pink lace. How much lace do you need in all to complete this project?


II. Rob has one fruit roll up. The fruit roll up is one foot long. He wants to share it with his friends. He wants to keep 4/6 of it for himself, and share 1/9 of one to Jimmy and the rest with Tom. How much will Tom get? Reduce to lowest terms.

III. Sam was drawing on the white board. After he finished 3/4 of the picture, he decided he did not like his picture. He erased 2/8 of it. How much is left of the drawing? Reduce your answer.

IV. The fifth grade teachers bought three party pizzas to celebrate good behavior. Each party pizza has twenty-four slices. They order three kinds of party pizzas; one cheese, one pepperoni, and one sausage. They eat 20/24 of the pepperoni pizza, 7/8 of the cheese pizza and ¾ of the sausage pizza. A. What fraction of each pizza is left? ______of the cheese pizza ______ of the pepperoni pizza ______ of the sausage pizza

B. How many slices of each pizza are left? _______ slices of cheese pizza are left _______ slices of pepperoni pizza are left _______ slices of sausage pizza are left



Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

fractions Define a fraction as division of the numerator by its denominator.

Numerator Denominator Division Quotient

Solve word problems involving the division of two whole numbers where the solution is a fraction or mixed number.

5.NF.3. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

Explain between what two whole numbers the fraction solution lies.

SAMPLE TASKSI. Out of 40 pieces of fruit, 8 grapes were picked. Write a fraction to represent this information and reduce to simplest form.

_____________________________________________

Write this fraction as a decimal.

_____________________________________________




multiplication of fractions and whole numbers

Draw a fraction model to illustrate a product of a fraction by a whole number and a fraction by a fraction.

part of area tiling unit fractionRelate multiplying by a fraction as taking "part of" a

whole number.5.NF.4a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

SAMPLE TASKS1.( See answer key drawing)Using visual fraction models, multiply ½ x ¾.

2. Lena has some eggs. She uses 3/5 of the eggs to make waffles and scrambled eggs. She uses 2/3 of the eggs she took to make the waffles. What fraction of the total number of eggs does Lena use to make waffles?

3.Tasha finished a job in ¾ of an hour. Megan finished it in 4/5 of the time Tasha took. How long did Megan take to finish the job?


4.(Draw a fraction model to help you) Jenny spends 1/6 of her paycheck and saves 2/5 of the remaining amount. What fraction of her paycheck is saved?




Area of rectangles

with fractional

side lengths

Compute the area of a rectangle with fractional side lengths.

Tiling unit square equivalenceTile a unit square into unit fraction side lengths.

5.NF.4b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

Prove through tiling the equivalence of multiplication and area.

SAMPLE TASKS1. (See drawing in the answer key)

Find the area of a farmer’s field with a width a quarter of a mile and length of 2/3 of a mile. Calculate the area of the farmer’s field. Draw a picture using tiling to show your solution.

5/6 m

3/4 m

Find the area of this large picture frame with a length of 5/6 of a meter and a width of ¾ of a meter. Use tiling as a visual fraction model to answer the question.




size of products

Describe the size of a product in terms of how many times larger one factor is to another without multiplying.

Product Factor improper

fraction mixed number

5.NF.5a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.

Explain and show why multiplying by a fraction equal to 1 result in an equivalent fraction.

SAMPLE TASKS

I. Carl ate 34 of

12 of a pizza.

A. Explain without finding the exact product. Did Carl eat more or less than 34 of the pizza? Did he eat more or less than

12 of

the pizza?

______________________________________________________________________________________________________________

______________________________________________________________________________________________________________

______________________________________________________________________________________________________________


II. Molly drew the model below.

Using what you know about multiplying fractions, explain Molly’s model.

______________________________________________________________________________________________________________

______________________________________________________________________________________________________________

______________________________________________________________________________________________________________


=X



size of products

Explain and show why multiplying by a fraction less than one will result in a product less than the greater number.

Product Factor equivalent

fractionExplain and show why multiplying by an improper/mixed number will result in a product greater than the given number.5.NF.5b.

Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.

Rewrite the number 1 as an equivalent fraction i.e. 2/2, 3/3, 4/4, etc.

SAMPLE TASKSI. Without multiplying, identify which of the expressions has the greater product.

54 x 30 or 14 x 30

Explain how you know.

______________________________________________________________________________________________________________

______________________________________________________________________________________________________________

______________________________________________________________________________________________________________


II. Is the product of a fraction that is greater than one and a whole number:

less than the whole number OR greater than the whole number(circle one)

Explain your thinking. ______________________________________________________________________________________________________________

______________________________________________________________________________________________________________

______________________________________________________________________________________________________________




multiplication of fractions and mixed numbers

Solve word problems involving multiplication of fractions and mixed numbers.

Fractions mixed number visual modelsRepresent the product of fractions in simplest form.

5.NF.6. Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

Write equations to represent word problems involving multiplication of fractions.

Draw/show multiplication of fractions through visual models.

SAMPLE TASKS1.A baker bought 2 ½ pints of blueberries. He used ¾ of the blueberries to make tarts. How many pints of blueberries did the baker use?

2.There is 1 ½ of material on a bolt. You need ¾ of that material to make a pillow. How much material do I needs?


3.¾ of an Olympic size swimming pool is dedicated to swimming; the other ¼ is used for diving only. 1/3 of the swimming area is used only for swimmers under 10. What fraction of the pool is used for swimmers under 10?




division of unit fractions

and whole numbers

Define a unit fraction as fraction with a numerator of 1.

unit fraction whole number divide estimation quotients lowest terms

Divide a unit fraction by a whole number.

5.NF.7a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.

Draw/show division of a unit fraction by a whole number as dividing the unit fraction into smaller parts.Create a story in which division of a unit fraction by a whole number is used.

Explain the effects of dividing a unit fraction by a whole number.Justify the reasonableness of answer in the context of a problem. Simplify/reduce quotients to lowest terms.

SAMPLE TASKS1.Mr. Smith’s garden covers 1/5 of an acre of land. He divides the land into 6 equal sections, what fraction of an acre is each section of the garden?

2.Denise divided ½ pound of butter into 4 equal parts. How much butter was in each pan?




division of unit fractions

and whole numbers

Define a unit fraction as a fraction with a numerator of 1.

unit fraction whole number divide estimation quotients lowest terms numerator denominator

Divide a whole number by a unit fraction.

5.NF.7b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

Create a story in which division of a whole number by a unit fraction is used.Explain the effects of dividing a whole number by a unit fraction.

Define the reciprocal of a unit fraction for the purpose of division.Simplify/reduce quotients to lowest terms.

Justify the reasonableness of answer in the context of a problem.

SAMPLE TASKSI. Mario ordered 5 pizzas. How many slices did Mario order if each slice is ⅛ of a pizza?

A. 5B. 8C. 13D. 40


II. What is the quotient of 12 divided by ⅓?A. 36B. 24C. 12D. 3

III. Jen had 4 boxes of raisins. She needed ⅕ of a box for a batch of Kickin’ Raisin Cookies. How many batches of cookies can Jen make with the 4 boxes of raisins?

IV. Wilma is making a gallon of lemonade, which requires 4 cups of lemonade mix. She only has a ¼ measuring cup.

A. How many ¼ cup scoops of lemonade mix does Wilma need?

B. Use words and/or numbers to explain your answer to part A.




division of unit fractions and

whole numbers

Divide a whole number by a unit fraction (vice versa) in the context of word problems.

unit fraction whole number divide estimation quotients lowest terms numerator denominator

Solve a story/word problem in which division of a whole number by a unit fraction (vice versa) is used.

5.NF.7c.Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Explain the effects of dividing a whole number by a unit fraction (vice versa) in the context of a word problem.Justify the reasonableness of answer in terms of the context of the problem.

Simplify/reduce quotients to lowest terms.

SAMPLE TASKS

I. When you divide a unit fraction by a whole number, which statement is true? Explain why?

A. The quotient will be greater than the whole number.B. The quotient will be greater than the unit fraction.C. The quotient will be less than the unit fraction.D. The quotient will be equal to the unit fraction.

______________________________________________________________________________________________________________________


II. Tina, Lilly, and Jackie equally shared a

23 gallon carton of Ben & Jerry’s Ice Cream. How much ice cream did each girls eat?

A.

29

B.

13

C.

23

D.

63

III. Sarah has 9 pounds of cereal mix. She will divide it into small bags of

610 pound each. How many bags can she make? Show your work.

IV. A large punchbowl held 24 cups of fruit punch. If each ladle contained

35 of a cup of fruit punch, how many servings would the

punchbowl yield?

a. 15b. 24c. 40


d. 120

V. For the Superbowl party, Selma invites 23 people. If she makes pasta salad and gives everyone ½ cup, how many cups of pasta salad will she need to make?


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