Richland Parish School Board



Grade 6

Mathematics

Unit 4: Operating with Fractions

Time Frame: Approximately three weeks

Unit Description

This unit focuses on the refinement of the understandings of addition, subtraction, multiplication and division of fractions. Students use concrete materials to model these operations with fractions. Activities provide opportunities to develop an understanding of equivalent fractions and common denominators. Visual models can help students understand division of fractions before development of the algorithm.

Student Understandings

Students use various strategies for addition, subtraction, multiplication and division of fractions to solve real-world problems.

Guiding Questions

1. Can students use a variety of strategies to add and subtract fractions, writing answers in lowest terms when specified?

2. Can students apply addition, subtraction, multiplication and division of fractions to real-life situations?

3. Can students create models to explain how to multiply and divide fractions?

MATH: Grade 6 Grade-Level Expectations and Common Core State Standards (CCSS)

|Grade-Level Expectations |

|GLE # |GLE Text and Benchmarks |

|Number and Number Relations |

|3. |Find the greatest common factor (GCF) and least common multiple (LCM) for whole numbers in the context of problem-solving |

| |(N-1-M) |

|4. |Recognize and compute equivalent representations of fractions and decimals (i.e., halves, thirds, fourths, fifths, |

| |eighths, tenths, hundredths) (N-1-M) (N-3-M) |

|CCSS for Mathematical Content |

|CCSS# |CCSS Text |

|The Number System |

|6.NS.1 |Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., |

| |by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ |

| |(3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to |

| |explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will |

| |each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? |

| |How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? |

|6.NS.3 |Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. |

| |*For the Transitional Unit, this CCSS will include adding, subtracting and multiplying fractions. |

|ELA CCSS |

|Reading Standards for Literacy in Science and Technical Subjects 6–12 |

|RST.6-8.4 |Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific |

| |scientific or technical context relevant to grades 6–8 texts and topics. |

|RST.6-8.7 |Integrate quantitative or technical information expressed in words in a text with a version of that information expressed |

| |visually (e.g., in a flowchart, diagram, model, graph, or table). |

|Writing Standards for Literacy in History/Social Studies, Science, and Technical Subjects 6–12 |

|WHST.6-8.4 |Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and|

| |audience. |

Sample Activities

Activity 1: Landscaping (GLE: 3)

Materials List: Landscaping BLM, Grid Paper BLM, colored pencils, paper, pencil

Present the following problem.

Mr. Jones wants to plant a vegetable garden in his back yard. He has 12 tomato plants and 18 eggplant plants. Each row will have the same number of plants with no mixing of plant types. Sketch all possible arrangements.

What are the factors of 12? 1, 2, 3, 4, 6, 12

What are the factors of 18? 1, 2, 3, 6, 9, 18

What factors do they have in common? 1, 2, 3, and 6

Mr. Jones can plant rows of 1, 2, 3, or 6 plants.

Rows of 1 t = tomato e = eggplant

|t |t |t |t |t |

|t |t |e |e |e |

|t |t |e |e |e |

|t |t |e |e |e |

|t |t |e |e |e |

|t |t |e |e |e |

Have students discuss which layout they think Mr. Jones should use and why? Discuss the relationship between 1 row of 30 and 30 rows of 1. Are they the same? Why or why not? Yes, both arrangements would have 30 plants planted in one row.

Give students the Landscaping BLM and Grid Paper BLM. Have them work in groups to solve the problems. Students can use colored pencils to show the different possible combinations of plants and to represent the various fractional parts. Ask students to make up similar problems using flowers, trees, or other plants. Review GCF and LCM and how they apply to solving these problems.

Activity 2: Mental Giants (GLEs: 4; CCSS: 6.NS.3, RST.6-8.4)

Materials List: Fraction Table BLM, Fraction Operations BLM, fraction circles or other fraction manipulatives, pencil

Adding and subtracting fractions with unlike denominators should be a natural progression from work done in Unit 3 and Activity 1. Distribute the Fraction Table BLM and review with students the process of finding equivalent fractions. To assist students with equivalent fractions, consider having them cut the table into pieces or having them use fraction circles or any other fraction manipulative. Discuss the connections and applications that equivalent fractions have to finding GCF and LCM of numbers.

Distribute the Fraction Operations BLM and start with fractions and mixed numbers with common denominators by having the students complete problems 1 – 3 in small groups. Discuss the answers as a class. Then give them simple, related fractions to add or subtract, such as [pic] or[pic]. Have students use the Fraction Table BLM or manipulatives to help find equivalent fractions.

Have students complete the Fraction Operations BLM and discuss the processes they are using in their groups.

In their math learning logs (view literacy strategy descriptions), have the students explain the processes they used to add and subtract the fractions. This sometimes helps students to understand that adding and subtracting fractions requires a common “name” or denominator. As students become more proficient, have them add the fractions mentally. Include mixed numbers. Let students exchange log entries with a partner to check for logic and accuracy.

Activity 3: Adding and Subtracting Unlike Denominators (GLEs: 3, 4; CCSS: 6.NS.3; RST.6-8.7; WHST.6-8.4)

Materials List: Clock Face BLM, Ad Space BLM, pencil, paper

To facilitate understanding of the process of adding and/or subtracting fractions with unlike denominators, give students opportunities to discover the algorithm. Revisit the application of GCF and LCM in class discussions. Give students problems that reflect real-life situations which students can easily solve by drawing models. They should transfer the understanding gained by the models to relevant algorithms. Distribute the Clock Face BLM and Ad Space BLM for students to use with the following examples:

• Mary walked for [pic] of an hour and ran for [pic] of an hour. What fraction of an hour did she exercise? Solve the problem by using the Clock Face BLM to model, write the related problem, and explain your answer.

[pic] of an hour is 10 minutes. This could also be thought of as [pic] of an hour. [pic] of an hour is 15 minutes or [pic] of an hour. To add [pic] and [pic], rewrite the fractions as [pic] + [pic]. The related problem is [pic]

Mary exercised for[pic] of an hour. To check their work, have students add 10 minutes and 15 minutes. Ask them what fraction of an hour 25 minutes is. ([pic]hr)

• Lucy is baking cookies. If it takes [pic] of an hour to preheat her oven and [pic] of an hour to bake the cookies, what fraction of an hour did she use the oven? Solve the problem by using the Clock Face BLM to model, write the related problem, and explain your answer.

[pic] of an hour is 12 minutes. This could also be thought of as [pic] of an hour. [pic] of an hour is 18 minutes. The related problem is [pic]+ [pic]= [pic] Lucy used the oven for [pic]hour. To check their work, have students add 12 minutes and 18 minutes. Ask them what fraction of an hour 30 minutes is. ([pic]hr)

• Jerome rides the bus to school each morning. It takes him [pic] of an hour to walk to his bus stop and then he often has to wait [pic] of an hour for the bus. If the bus ride is [pic] of an hour, what fraction of an hour does it take for Jerome to get to school? Solve the problem by using the Clock Face BLM to model, write the related problem, and explain your answer.

[pic] of an hour is 5 minutes. [pic] of an hour is 10 minutes. [pic] of an hour is 20 minutes. The related problem is [pic]

It takes Jerome [pic] of an hour to get to school. To check their work, have students add 5 minutes, 10 minutes and 20 minutes. Ask them what fraction of an hour 35 minutes is. ([pic]hr)

• John always buys advertisement space in a local magazine. In the past, he has bought an ad the size of A. This year he would like to buy one the size of C + D + E. What fractional part of the page would he buy? How much more ad space did he buy this year than last year? Explain your work and write the problems you solved.

[pic]

Teacher Note: A & B are equal in size; C & F are equal in size; D & E together are the same size as C and F; and the height of A and C are equal. Students should determine that [pic][pic] therefore, the first problem is[pic]. The second is[pic]. To solve the problem, students will have to discover the need to subdivide the entire page into twelve equal parts. John buys 1/3 of the ad space and this is 1/12 more of the ad space than he bought last year.

Have students demonstrate their understanding of adding and subtracting fractions by completing a RAFT writing (view literacy strategy descriptions) assignment. This form of writing gives students the freedom to project themselves into unique roles and look at content from unique perspectives. From these roles and perspectives, RAFT writing is used to explain processes, describe a point of view, envision a potential job or assignment, or solve a problem. It’s the kind of writing that when crafted appropriately should be creative and informative.

Ask students to work in pairs to write the following RAFT:

R – Role

like denominators in an addition or subtraction problem

A – Audience

unlike denominators in an addition or subtraction problem

F – Form

advice column, song, or poem

T – Topic

how to become like denominators in an addition or subtraction problem

For example: The following song is sung to the tune of Row, Row, Row Your Boat

Rewrite your fractions with common denominators.

Then add your numerators.

Write the sum over your new denominator,

simplify and your done.

In their RAFTed advice column, song, or poem, students must include how to solve an addition or subtraction problem with unlike denominators. When finished, allow time for students to share their RAFTs with other pairs or the whole class. Students should listen for accuracy and logic.

Below are websites to provide additional practice adding and subtracting fractions.







Activity 4: Numbers in the News (CCSS: 6.NS.3)

Materials List: newspapers or Internet access, paper, pencil

The newspaper and the Internet are good resources for applications of fractions. It may even be possible to find problems that relate to content students are covering in other subjects. Following are a few examples:

• Have students research various stocks and follow the changes in price each day for a period of one week (or longer). Have them calculate the new closing price each day.

• Have students gather recipes for 3 different types of cupcakes. Have students find the total amount of each ingredient needed to make all 3 recipes.

Activity 5: Real-World Fractions (CCSS: 6.NS.3; WHST.6-8.4)

Materials List: paper, pencil

Divide students into small groups. Have them brainstorm (view literacy strategy descriptions) a variety of situations where fractions would be required. Examples for fractions: dividing/cutting food such as pizza, apples, oranges; weighing food; cooking; measuring lengths; reading a clock; understanding signs in a store advertising a sale. After discussing the examples above, have the students create a text chain (view literacy strategy descriptions) using real-life fraction situations. Put students in groups of four. On a sheet of paper, ask the first student to write the opening sentence for the math text chain. For example, a student might begin the chain with this sentence.

Mary is baking a cake for a friend’s birthday party.

The student then passes the paper to the student sitting to the right, and that student writes the next sentence in the story, which might read:

A recipe calls for ¼ cup of granulated sugar and ⅔ cup of brown sugar.

The paper is passed again to the right to the next student who writes the third sentence of the story.

What is the total amount of sugar needed?

The paper is now passed to the fourth student who must solve the problem and write out the answer: The other three group members review the answer for accuracy.

Answer: 11/12 cup of sugar

This activity allows students to use their writing, reading, and speaking skills while learning and reviewing the concept of adding and subtracting fractions. When text chains are completed, be sure students are checking them for accuracy and logic. Groups can exchange text chains to further practice reviewing and checking for accuracy.

Activity 6: Multiplying Fractions (CCSS: 6.NS.3, RST.6-8.7)

Materials List: Multiplying Fractions BLM, paper, pencil

Real-life situations and visual models can help students understand multiplication of fractions before development of the algorithm. By guiding students to the understanding instead of telling them what they should know, the students retain the information and develop a deeper understanding of the concept.

Example 1:

A recipe for Sugar Cookies calls for [pic]cup of sugar. You are making [pic]of the recipe. How much sugar should you use?

You will need [pic]cup of sugar to make the cookies because [pic] × [pic] = [pic].

Example 2:

Derrick brought a pan of brownies to school for his birthday. After the party, he still had [pic]of the pan of brownies left. He wants to give [pic]of what is left of the pan of brownies to the class next door.

[pic]= [pic] [pic] of the pan will be given to the class next door.

Have students use the drawing to explain why[pic].

Guide students to notice that to multiply fractions without the models, they can simply multiply the numerators, multiply the denominators and then simplify the fraction.

Distribute the Multiplying Fractions BLM. Have students work in pairs to complete the problems. Discuss the solutions as a class.

Activity 7: Dividing Fractions (CCSS: 6.NS.1, RST.6-8.7)

Materials List: Dividing Fractions BLM, paper, pencil

Real-life situations and visual models can help students to understand division of fractions before development of the algorithm. By guiding students to the understanding instead of telling them what they should know, the students retain the information and develop a deeper understanding of the concept.

Example 1:

Jack has to mow [pic]of his grandmother’s yard. He can mow[pic] of her yard in an hour. If Jack mows her yard at a constant rate without stopping, how long will it take him?

The problem is asking how many sixths are in ½. [pic] There are 3 sixths in ½ and each one-sixth takes him an hour to mow. It will take him 3 hours to mow ½ of the yard. [pic].

Example 2:

Ms. Jones had [pic] a yard of fabric to make book covers. Each book cover is made from [pic] a yard of fabric. How many book covers can Ms. Jones make?

The problem is asking how many eighths are in 3/4. [pic] There are 6 eighths in ¾, so she can make 6 book covers. [pic]

Example 3:

Represent [pic]in a real life problem and draw a model to show your solution.

Real-life problem:

The chocolate chip cookie recipe calls for [pic]cup of sugar. You have [pic]cup of sugar. How much of the cookie recipe can you make? You want to find the number of [pic] in [pic]

Model of solution:

First model represents the [pic]cup of

sugar you have.

Next model represents [pic] cup and also

shows [pic] cups horizontally.

Move the [pic] cup to fit in only the

area shown by [pic] of the model.

3 out of the 4 sections in the [pic]portion are shaded. A [pic] cup is [pic] of a [pic] cup portion, so you can only make [pic] of the recipe.

Distribute the Dividing Fractions BLM. Have students work in pairs to complete the problems. Discuss solutions as a class.

Activity 8: What’s Cooking? (CCSS: 6.NS.1)

Materials List: Recipe BLM, paper, pencil

Display the following recipe:

Sugar Cookie Recipe

• ⅓ cup butter • ⅛ teaspoon salt

• ⅓ cup butter shortening • 2 large eggs

• ¾ cup granulated sugar • 1 teaspoon vanilla extract

• 1 teaspoon baking powder • 2 cups all-purpose flour

Ask students what mathematical operation they are doing when they halve the recipe. (dividing) What are they dividing by when they halve the recipe? (dividing by 2) Have students draw a model to represent ⅓ cup of butter.

| | | |

Have students divide their model into 2 parts. Tell them that they are only using half of the amount they had.

| | | |

| | | |

Ask, “What fraction of the model is shaded?” (1/6)

Tell students that they will need 1/6 of a cup of butter if they halve the recipe.

Have students halve the remaining ingredients. Discuss the answers.

Ask students what mathematical operation they are doing when they double the recipe. (multiplying) What are they multiplying by? (multiplying by 2) Have students draw the model again to represent ⅓ cup of butter.

| | | |

Tell them if they double the amount of butter needed, they could add ⅓ + ⅓ = ⅔.

| | | |

| | | |

Tell students that another method that can be used is to write the whole number as a fraction. The whole number is the numerator and one is the denominator. For example, the number 2 would be written as [pic] because 2 ÷ 1 = 2. Multiply numerator by numerator and denominator by denominator and simplify if needed. Show the following examples.

[pic]. You need [pic]cup of butter if you double the recipe.

What would you do if you wanted to triple the recipe? Multiply by 3, [pic]or add, [pic]

What if you double the sugar? [pic]. You end up with an improper fraction. Why do you think it is improper? (The numerator is larger than the denominator. The fraction is greater than 1.) How many times does 4 divide into 6? (1 time) This is the whole number. How many are left? (2) This is the numerator. The denominator stays the same. What if you only had a ¼ cup measuring cup, how many scoops would you use? Six scoops of sugar. In some cases, an improper fraction is more useful than a mixed number.

[pic] [pic] can be simplified since both 2 and 4 can be divided by 2. [pic] [pic]

Have students double the remaining ingredients. Discuss the answers.

|Ingredient |Original Amount |Half recipe |Doubled recipe |

|Butter |⅓ |1/6 |⅔ |

|Shortening |⅓ |1/6 |⅔ |

|Sugar |¾ |⅜ |1½ |

|Baking Powder |1 |½ |2 |

|Salt |⅛ |1/16 |¼ |

|Eggs |2 |1 |4 |

|Vanilla |1 |½ |2 |

|Flour |2 |1 |4 |

Distribute the Recipe BLM. Have students work in pairs to solve the problems. Discuss the solutions as a class.

Sample Assessments

General Assessments

• Have students create a portfolio containing samples of experiments and activities.

• Have students give real-life examples of using fractions.

• Have students add or subtract several fractions with unlike denominators.

• Create extensions to an activity by increasing the difficulty or by asking “what if” questions whenever possible.

• Have students create and demonstrate original math problems by acting them out or by using manipulatives to provide solutions on the board or overhead.

• Facilitate during small group discussion to determine misconceptions, understandings, use of correct terminology, and reasoning abilities. Appropriate questions to ask to encourage reflection as students develop algorithms for addition/subtraction with unlike denominators are

o How did you get your answer?

o Does your answer seem reasonable? Why or why not?

o Can you describe your method to us all? Can you explain why it works?

o What if you had started with… rather than…?

o What if you could only use…?

o What have you learned or found out today?

o Did you use or learn any new words today? What do they mean? How do you spell them?

Ask probing questions to help students build confidence in their reasoning such as …

o Why is that true?

o How did you reach that conclusion?

o Does that make sense?

o Can you make a model to show that?

• The student will create journal writings using such topics as:

o The most important thing I learned in math this week was…

o The easiest thing about today’s lesson was…

Activity-Specific Assessments

• Activity 2: Have the student thoroughly answer the following: In adding two fractions, is it better to use a common denominator or the LCD to write the equivalent fractions? Why?

• Activity 3: Have the student determine which fractions must be renamed before adding or subtracting and explain his/her reasoning.

a. [pic] b. [pic] c. [pic] d. [pic]

• Activity 7: Have students explain how a model can help to divide fractions.

• Activity 8: Have students find a recipe that has at least five ingredients with at least 3 fractions in the ingredients and copy the recipe. Have students double the recipe and cut the recipe in half. Have them show their work. Have students explain in complete sentences how they would adjust the recipe to serve 20 people.

-----------------------

A

B

C

D

E

F

[pic]cup of sugar

[pic]cup of sugar

[pic]of the recipe

[pic]cup of sugar needed for the recipe

[pic] a cup

of sugar

[pic] pan of brownies

[pic] pan of brownies

[pic] brownies will be given to the class next door.

3 hours

yard divided into [pic]

yard divided into [pic]

6 book covers

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

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