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Mathematics TEKS Refinement 2006 – K-5 Tarleton State University

Number, Operation, and Quantitative Reasoning Grade 5 Conversion Creations: From Mixed Numbers to Improper Fractions

Number, Operation, and Quantitative Reasoning Activity: Conversion Creations: From Mixed Numbers to Improper Fractions TEKS: (5.2) Number, operation, and quantitative reasoning. The student

uses fractions in problem-solving situations. The student is expected to: (B) generate a mixed number equivalent to a given improper

fraction or generate an improper fraction equivalent to a given mixed number;

(C) compare two fractional quantities in problem-solving situations using a variety of methods, including common denominators;

(5.14) Underlying processes and mathematical tools. The student

applies Grade 5 mathematics to solve problems connected to everyday experiences and activities in and outside of school

The student is expected to: (A) identify the mathematics in everyday situations; (D) use tools such as real objects, manipulatives, and technology to

solve problems. (5.15) Underlying processes and mathematical tools. The student

communicates about Grade 5 mathematics using informal language. The student is expected to: (A) explain and record observations using objects, words, pictures,

numbers, and technology; and (B) relate informal language to mathematical language and

symbols. (5.16) Underlying processes and mathematical tools. The student

uses logical reasoning. The student is expected to: (A) make generalizations from patterns or sets of examples and

nonexamples; and (B) justify why an answer is reasonable and explain the solution

process. Note: Portions of this lesson address TEKS at other grade levels as well;

however, the intent of the lesson fits most appropriately at the grade level indicated.

Overview: Students will have the opportunity to review the meaning of mixed and

improper fractions through the use of concrete and pictorial models. Then, they will use those models to create or invent procedures for generating improper fractions from mixed numbers and mixed numbers from improper fractions.



Pre-requisites:

Students will have had the opportunity to model fraction quantities greater than one using concrete objects and pictorial models (4.2B) and locate and name points on a number line using whole numbers and fractions (4.10). Likewise, students should have prior experience representing fractions with length, area, and set models.

Materials: Manipulatives that can be used to represent length, area, and set models

for fractions, such as the following: Pattern blocks Fraction bars Fraction circles Color tiles Geoboard and geobands Two-color counters Colored post-it dots Play money Adding machine tape Cuisenaire rods Measuring tapes Meter sticks Yard sticks Rulers Student number lines Measuring cups, spoons Liter cups

Poster or chart paper with crayons or markers, and/or blank overhead transparencies with vis-à-vis pens

Modeling Mixed Numbers Recording Sheet – Handouts/Transparencies 1a and 1b

Improper Fractions to Mixed Numbers Recording Sheet – Handouts/Transparencies 2a and 2b

Fraction Fakeout – Handouts/Transparencies 3a through 3k Mixing It Up Directions, Playing Cards, Recording Sheet –

Handouts/Transparencies 4a, 4b, 4c, and 4d Mixing It Up Example Problem – Handout/Transparency 5 Check for Understanding Mixed Numbers and Improper Fractions –

Handouts/Transparencies 6a, 6b, and 6c Check for Understanding Mixed Numbers and Improper Fractions Answer Key – Answer Keys 7a, 7b, and 7c Fraction Calculators Computer(s) with Internet Connection

Grouping: Individual, pairs, and whole group Time: 2-3 periods, teacher discretion



Lesson:

Procedures Notes 1. Modeling Mixed Numbers:

Place a variety of the listed manipulatives on each table. Be sure to include manipulatives which can be used to represent length, area, and set models for fractions and mixed numbers. Begin the lesson by asking students to use any of the materials on their table to independently construct or draw a model of one and three-fourths. Students should be prepared to share their models with the class.

This introductory activity can serve as a formative assessment as well as an instructional activity. If some students have difficulty getting started, prompt with questions such as the following: • What could you use to draw

and label a line that is

431 inches long?

• If I use this as my whole (pick one manipulative), how could

you model 431 ?

• How could you draw a picture

to show 431 cups of sugar?

• If 4 counters make one whole, how could you model 1 whole

and 43 more?

See Van de Walle (2006) for more information on the types of fraction models used in this lesson: 1) length (or measurement)

models; 2) area (or region) models; and 3) set models.

2. Call on several students to share their models with the class. Collect drawings of the various models on a chart, on the board, or on a transparency.

After the sharing session, you will have a collection of

representations for 431 . Look to

see if length, area, and set models have been shared. If any of these models are missing, ask questions to assist students in constructing them.

3. Have students examine the collection for similarities and differences.

If needed, prompt students by asking questions such as the



Procedures Notes following: • How are the models the same?• How are the models different? • Which models are represented

with some type of line? With some area or region? With sets of items?

• Where is the number 1 represented in each model?

• Where is the fraction 43

represented in each model? • Why are the 4th’s in this model

bigger than the 4th’s in this one?

Students need to see that each model includes one whole as well as an additional fractional part. The size of the fractional parts is determined by the size of the whole. The whole may or may not be subdivided into parts. Students may remark or describe how each

model also represents 47 . Do not

interject or elaborate on this comment (improper fraction) at this point.

4. Explain that 4

31 is an example of a mixed

number. Numbers like 431 are called mixed

numbers because they are a mixture of a whole number plus a fraction. Show and tell students how this mixed number is written

and read ( 431 , one and three-fourths).

Using several of the models on the posters,

illustrate and verbalize how 431 is 1 + 4

3 .

Ask students to relate where they have seen

Van de Walle (2004) emphasizes that fractional quantities should always be written with the horizontal division bar and not a slanting line. This reinforces the concept of denominator as divisor.



Procedures Notes or heard mixed numbers used in everyday life.

5. Ask students to work with a partner to choose or generate a mixed number and create at least two different models for that mixed number. They should create a poster by drawing their models on chart paper. They should not write the mixed number on their poster.

Students will have seen various models for a mixed number after completing procedures 1-4 above. Their new posters should reflect a variety of representations for their chosen mixed number. The more diverse the models, the better the understanding.

6. Give one or two pairs of students the opportunity to show their posters to the class. For each poster shared, ask the class the following: • What mixed number is represented by

the models? • How do we write and read this mixed

number? • Does this model represent a length

model? Area model? Set model?

When the correct mixed numbers have been identified by the class, have the student pairs record the mixed number (in symbolic and word form) on their posters.

7. Display the remaining posters around the room and number the posters so that they can be matched to the numbers on Modeling Mixed Numbers Recording Sheet – Handouts/Transparencies 1a and 1b. Ask students to record the mixed number (in symbolic and word form) for each poster on the recording sheet. Include the posters that have already been shared.

This recording sheet, as well as the posters, can serve as a formative assessment.

8. Converting Mixed Numbers to Improper Fractions: Choose and focus on any one of the models

Adapted from Van de Walle (2004). For students who are unable to



Procedures Notes (length, area, or set) on the posters. Example: A pair of students uses adding

machine tape to model 852 .

Ask students to find a single fraction that names the same amount. They can use the same manipulative, a different one, a drawing, or any other method as long as they can explain and model their results.

comprehend the task or cannot proceed after sufficient wait time, have them count out 8th’s (using an appropriate manipulative such as 8th’s of a fraction circle) until

they get to 88 . Have them write

this fraction and tell you how many

circles 88 represents. (1 circle).

Say: 88 is a single fraction which

names the same amount as 1. Now see if you can find a single fraction to name the same amount

as 2. ⎟⎠⎞

⎜⎝⎛

816

Finally, can you find a single fraction to name the same amount

as 852 ? ⎟

⎠⎞

⎜⎝⎛

821

9. Have students share their methods for

arriving at the new fraction ⎟⎠⎞

⎜⎝⎛

821 .

Ask: What does the denominator (8) tell us? (It tells us what fractional part we are counting – eighths.) What does the numerator (21) tell us? (It tells us how many eighths we have.)

One student’s method might be folding each of the 2 whole strips of adding machine tape into eighths, then, labeling and counting the total number of eighths,

1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16

17 18 19 20 21

or adding 88 + 8

8 + 85 = 8

21 .

10. Explain to students that this type of fraction

is called an improper fraction. The numerator is larger than the denominator. Sometimes they are described as “top heavy.”

A classroom teacher related the following analogy that she uses with her students when introducing improper fractions. She gives her students the analogy of trying to eat spaghetti with our fingers. It is “improper to



Procedures Notes do so,” but it still “gets the job done” because we are able to eat and fill our empty stomachs.

11. Repeat the task above several times with different mixed numbers from other posters. Have students share and verbalize their procedure for generating an improper fraction.

12. Have students contrast the use of improper fractions with that of mixed numbers. Ask students which form they see most often used in everyday life. Share and discuss real life situations in which improper fractions are utilized.

In our everyday lives, we express quantities more often as mixed numbers, even when the initial situation may utilize an improper fraction. Example:

Jean needs 21 yard of cloth for

each wall hanging she is making. If she is making 7 wall hangings, how many yards of cloth should

she purchase? ( 27 or 2

13 yards) In middle school, students will begin to compute with fractions. It is often easier to compute with improper fractions than with the corresponding mixed numbers.

13. Finally, ask students to work with a partner to determine a method for changing any mixed number to an improper fraction without the use of manipulatives. They should be able to develop and express a generalization or rule that makes sense to them. Student words might sound like the following:• Divide the wholes into the same type of

fractional parts as the fraction, and then count how many of those parts you have altogether.

• Find the number of fractional parts in each whole, and then add those to the

Van de Walle (2004) states, “there is absolutely no reason ever to provide a rule” for “converting mixed numbers to common fractions and the reverse.” In support of students developing their own procedures for conversion, see “Reflecting on Learning Fractions without Understanding,” (2003). NCTM On-Math e-resources http://my.nctm.org/eresources/view_article.asp?article_id=6430&pa



Procedures Notes numerator of the fraction to get the total number of parts.

• Multiply the whole number by the fractional part we are counting (the denominator), then combine that number of fractional parts with the parts in the fraction (numerator), and that tells how many parts there are in all.

ge=1 Or, see the IMAP version below: http://www.sci.sdsu.edu/CRMSE/IMAP/pubs/Reflections_on_Fractions.pdf

14. Have students share and demonstrate their rules for the class. Compare the various methods and draw attention to the similarities. Allow students to publish their rules in a class book. Let them illustrate their page and provide ideas for the cover. Keep the class book on the shelf for them to read and refer to throughout the year.

15. Converting Improper Fractions to Mixed Numbers: Prepare baggies with sets of identical fractional parts. Include a card on which the relationship of the part to the whole is identified. Students are to write the single fraction represented by the parts, and then change that single fraction into a mixed number that names the same amount. Rotate the baggies among the students. Number the baggies so that students can record their results for each one on a recording sheet. See Improper Fractions to Mixed Numbers Recording Sheet – Handouts/Transparencies 2a and 2b.

Prepare enough baggies for each student or each pair of students to have one. Examples might include: • 14 rhombi from a pattern block

set with a card labeled “thirds.” • 24 red Cuisenaire rods with a

card labeled “fifths.” • 33 base ten unit cubes with a

card labeled “tenths.” • 12 two-colored counters with a

card that shows a set of 3 equals “one-third.”

16. Discuss and verify or check student results.

17. Ask students to work with a partner to determine a method for changing any improper fraction to a mixed number without the use of manipulatives. Once again, they

Students will likely “work backwards” or reverse the procedures suggested in #13 above. For instance, they may



Procedures Notes should be able to develop and express a generalization or rule that makes sense to them.

describe a procedure whereby they group the parts into wholes and then record any leftovers as a fraction.

18. Have students share and demonstrate their rules for the class. Compare the various methods and draw attention to the similarities. Again, allow students to publish their rules in the class book.

19. Optional: Students can work in groups of 3-4 to play Fraction Fakeout. (See Fraction Fakeout – Handouts/Transparencies 3a through 3k.) Laminate and cut out as many decks of cards as needed.

This game provides students the opportunity to match mixed numbers and improper fractions with various pictorial representations. Fraction Fakeout is played like the card game, “Old Maid.” The goal is to make the greatest number of sets/ books (mixed or whole number, improper fraction, and pictorial representation).

20. Have participants play Mixing It Up. (See Mixing It Up Directions, Playing Cards, and Recording Sheet – Handouts/Transparencies 4a, 4b, 4c, and 4d.) Before the game is started, use Mixing It Up Example Problem (Handout/Transparency 5) to model the following problem: Terel needed to buy 32 ice cream sandwiches. He can only buy the ice cream sandwiches in packages of 6. Exactly how many boxes does Terel need to buy?

First demonstrate 326 using manipulatives.

Then demonstrate what this number would look like as a mixed number. Sample Demonstration:

This is a game to encourage practice of changing a mixed number to an improper fraction and vice versa. In addition to using the game to give students practice in converting between mixed numbers and improper fractions, you could use the scenarios to explore division and interpreting the remainder within a given context (5.3 C).



Procedures Notes Using pattern blocks designate that the yellow hexagon is the whole. Next take out 32 green triangular pieces. This represents 326 . Now group the triangular pieces into

wholes. You should now have 5 whole with 26 left over.

Let the participants work for 10 minutes playing Mix It Up.

21. Calculator Activities: Students can use a fraction calculator to practice conversions from mixed numbers to improper fractions and the reverse. Have students input a given mixed number and use the rule they have developed to mentally calculate the improper fraction. They can use the conversion key to check their results. Do the same with improper fractions.

Adapted from Van de Walle (2006). The TI-12 Explorer, TI-15, and Casio fx55 are all fraction calculators which can be utilized for this activity. You will need to program the calculators so they will not simplify fractions automatically.

22. Give students a targeted mixed number such

as 513 . Have students program their TI-15

calculator to count by 5th’s until they reach what they believe is the correct number of

5th’s ⎟⎠⎞

⎜⎝⎛

516 equal to the targeted number.

The TI-15 has the capability of storing fractions as “constants” in one or two operation keys. When you use these keys, the counter on the left shows how many times you have used the fraction, while the total appears on the right of the display.



Procedures Notes They can use the conversion key to check their answer. You may wish to provide fraction manipulatives for students who need concrete or visual support.

Example: Enter › + 1 5 ¥ ›. Count by pressing ›, ›, ›, etc. until you reach the desired total of 5th’s. Use the conversion key ¦ to change the improper fraction to a mixed number or vice versa. Be sure you are in manual mode and have the calculator set to fraction, not mixed number, mode.

23. Computer Activities: Students can complete the activities on each of the sites listed below and print out a report of their results. http://www.visualfractions.com/identify.htm Select the third bullet “Identify Mixed Numbers with Lines,” or “Identify Mixed Numbers with Circles.” http://www.visualfractions.com/rename.htm Select the first bullet “Mixed Numbers to Fractions with Lines,” or “Mixed Numbers to Fractions with Circles.”

The first link allows students to identify mixed numbers from line and circle models of fractional parts. The second link allows students to rename mixed numbers as improper fractions. NOTE: Students are able to access an explanation of the correct answer before submitting their own answer. In addition, the bottom part of the page (scroll down) offers a rule for converting an improper fraction to a mixed number. Therefore, it is recommended for use only as an extension beyond the lesson.

24. This applet allows students to explore circle, rectangular, and set models for fractions by adjusting the numerators and denominators. http://my.nctm.org/eresources/repository/2071/applet/FractionPie/index.html

Although this site includes topics beyond the scope of 5th grade TEKS (percents), it presents a variety of ways to examine patterns and relationships which would enhance 5th graders’ understanding of fractions and decimals. The following article elaborates on how the applet can be utilized in the classroom to promote understanding of these relationships among rational numbers.



Procedures Notes http://my.nctm.org/eresources/view_media.asp?article_id=2071

Homework: Ask students to find examples of mixed numbers and improper

fractions in everyday life. Students could create a collage or poster to present their findings.

Assessment: See Check for Understanding Mixed Numbers and Improper Fractions

– Handouts/Transparencies 6a, 6b, and 6c. Resources: Integrating Mathematics and Pedagogy (IMAP). (2002)

http://www.sci.sdsu.edu/CRMSE/IMAP/pubs/Reflections_on_Fractions.pdf National Council of Teachers of Mathematics. (ON-Math, Fall 2002)

http://my.nctm.org/eresources/view_media.asp?article_id=2071 National Council of Teachers of Mathematics. (ON-Math, Fall 2002)

http://my.nctm.org/eresources/repository/2071/applet/FractionPie/index.html National Council of Teachers of Mathematics. (ON-Math, Winter 2003)

http://my.nctm.org/eresources/view_article.asp?article_id=6430&page=1 Rand, R. E. (2006). Visual Fractions. Available online:

http://www.visualfractions.com/identify.htm http://www.visualfractions.com/rename.htm

Reys, R. E., Lindquist, M. M., Lambdin, D. V., Smith, N. L., Suydam,

M. N. (2004). Helping children learn mathematics. Hoboken: John Wiley and Sons, Inc.

Van de Walle, J. A. (2004). Elementary and middle school

mathematics: Teaching developmentally. Boston: Pearson Education, Inc.

Van de Walle, J. A. & Lovin, L. H. (2006). Teaching Student-Centered

Mathematics: Grades 3-5. Boston: Pearson Education, Inc.



Modeling Mixed Numbers Recording Sheet

Poster

Number Mixed

Number Mixed Number in Word

Form

Example

431

one and three-fourths

1

2

3

4

5

6

7

Handout/Transparency 1a



Modeling Mixed Numbers Recording Sheet

Poster

Number Mixed

Number Mixed Number in Word

Form

8

9

10

11

12

13

14

15

Handout/Transparency 1b



Improper Fractions to Mixed Numbers Recording Sheet

Bag Number

Common Fraction

Mixed Number

Example

821

852

1

2

3

4

5

6

7




Improper Fractions to Mixed Numbers Recording Sheet

Bag Number

Common Fraction

Mixed Number

8

9

10

11

12

13

14

15




Fraction Fakeout (Similar to the game Old Maid)

This game can be played by 3 or 4 players. The deck contains 72 cards (24 triples consisting of 1 mixed or whole number, its corresponding improper fraction, and a graphic representation) and two FAKE FRACTION cards. Each group of 3 or 4 students will play with half of the deck (36 cards made up of 12 triples) and one of the FAKE FRACTION cards. The object of the game is to make the most “books” with the three matching cards and not be caught with the FAKE FRACTION card. Rules: The dealer deals out all cards to the players even though some will have more cards than others. The players examine their cards and lay down any matching pairs or triples (books) they have. Players must verify each others’ matches. Each player’s pairs are placed face up until he is able to make a book. When a player completes a book, he turns it face down in front of him. The dealer begins the game. The dealer must offer his cards spread face down to the player on his left who selects a card without seeing it and adds it to his hand. If that player can make a pair or triple, he declares a match and lays the cards down appropriately. The player who just selected a card then offers his hand to the next player on his left. The process continues.




When a player completes his books and has no more cards, he is “safe.” The turn passes to the next player. Eventually, all of the cards will be matched except for the player holding the FAKE FRACTION card. The holder of that card loses and gets no points for the books he has made. The other players score 5 points for each book they have made. The game can be played multiple times and students can keep a running tally of their scores. Rotating decks among groups will give students the opportunity to work with different numbers and representations.




Handout/Transparency 3c



Handout/Transparency 3d



Handout/Transparency 3e



Handout/Transparency 3f



Handout/Transparency 3g



Handout/Transparency 3h



Handout/Transparency 3i



Handout/Transparency 3j



Handout/Transparency 3k



Mixing It Up Directions

Materials Needed: Mixing It Up Playing Cards

Counters Fraction bars Fraction squares Cuisenaire rods Pattern blocks Grid paper Recording sheet Pencil Procedures:

1. Divide students into groups of 4. 2. Have a set of Mixing It Up Playing Cards for each group. 3. Each person takes a turn drawing a card. The player has the option

of building with a manipulative of his or her choice, drawing pictures, or using other methods to represent the improper fraction or mixed number indicated.

4. If the number is an improper fraction, the participant must explain how it can be written as a mixed number and vice versa. All participants write the improper fraction and the mixed number on the recording sheet.

5. Play continues until all cards have been selected and discussed.




Mixing It Up Playing Cards

Donna is buying cupcakes for Molly’s birthday party. Seventeen children RSVP to the party. Donna wants to order just enough cupcakes so that each child will have one. How many dozen cupcakes does Donna need to order from the bakery? Represent this amount as a mixed number and as an improper fraction.

Sun-Yung is helping in her grandmother’s kitchen. There are seven extra pounds of sugar in a canister. Her grandmother asked her to divide the sugar evenly into separate bags so that she could deliver them to her neighbors, Mrs. Chang, Mrs. Chung, Mrs. Cibrario, and Mrs. Dickerman. How many pounds of sugar will each neighbor receive? Represent this amount as a mixed number and as an improper fraction.

Dorcas went shopping with her mother. She had to buy enough pizza for 12 people. If each pizza is cut into 8 pieces, she will need to buy 2 pizzas for everyone to get at least one piece. Exactly how many pieces of pizza will each person receive? Represent this amount as a mixed number and as an improper fraction.

Jarvis took 5 packages of chewing gum to the pitcher’s mound. There are six sticks of gum in each package. In the first inning, Jarvis chewed 3 sticks of gum. Exactly how many packages of gum does he have left? Represent this amount as a mixed number and as an improper fraction.




Valentina is cutting up bananas for pudding. She has 10 bananas to be divided evenly among 3 bowls of pudding. Exactly how many bananas will go in each bowl? Represent this amount as a mixed number and as an improper fraction.

Bethany went to the store to buy tortilla chips. There were 3 brands of chips.

The first brand weighed 7 and 12

ounces. Represent this amount as a mixed number and as an improper fraction.

Fan and Lai-Sang bought 2 and 12dozen cookies to share with their friends. Represent this amount as a mixed number and as an improper fraction.

Rahim worked for a local company selling cups of coffee. On Monday morning he sold 20 cups of coffee. If each pot holds 12 cups of coffee, exactly how many pots of coffee did Rahim sell? Represent this amount as a mixed number and as an improper fraction.




Mixing It Up Recording Sheet

Name Mixed

Number Improper Fraction

Name Mixed Number

Improper Fraction

Bethany

Jarvis

Donna

Rahim

Dorcas

Sun-Yung

Fan and Lai Sang

Valentina

Handout/Transparency 4d



Mixing It Up Example Problem

Terel needed to buy 32 ice cream sandwiches. He can only buy the ice cream sandwiches in packages of 6. Exactly how many boxes does Terel need to buy?

First demonstrate 326 using manipulatives.

Then, demonstrate what this number would look like as a mixed number.

Handout/Transparency 5



Check for Understanding Mixed Numbers and Improper Fractions

Change the following mixed numbers to improper fractions. 1.

873 =

2. 436 =

3. 521 =

Change the following improper fractions to mixed numbers. 4.

613 =

5. 722 =

6. 211 =

Use the symbols <, >, and = to compare the following numbers. 7.

38

313

8. 722

713

9. 211

216




10. How many fourths are in 5 wholes?

11. Mrs. Lara, the art teacher, has 434 yards of ribbon.

She needs to cut it so that each student will have 41

yard for the craft project. How many fourths of a yard will Mrs. Lara be able to cut?

12. Riley is making a cake for the cake walk. She needs

212 cups of flour, but she only has a

21 cup measuring

cup. How many half cups should she measure?




13. The new bakery cuts its square pans of brownies into ninths. How many pieces (ninths) can be cut from

942

pans of brownies? 14. The Draper family bought 2 medium pizzas. Scott ate

86 of a pizza. Steven ate

84 of a pizza, and Stephanie

ate 83 of a pizza. How many eighths did they eat?

15. What mixed number expresses the same amount of

pizza that the Drapers ate?




Check for Understanding Mixed Numbers and Improper Fractions

Answer Key Change the following mixed numbers to improper fractions.

1. 873 =

831

2. 436 =

427

3. 521 =

57

Change the following improper fractions to mixed numbers.

4. 6

13 = 612

5. 722 =

713

6. 211 =

215

Use the symbols <, >, and = to compare the following numbers.

7. 38 <

313

8. 722 =

713

9. 211 <

216

Answer Key 7a



10. How many fourths are in 5 wholes?

There are twenty fourths ⎟⎠⎞

⎜⎝⎛

420 in 5 wholes.

11. Mrs. Lara, the art teacher, has 434 yards of ribbon.

She needs to cut it so that each student will have 41

yard for the craft project. How many fourths of a yard will Mrs. Lara be able to cut?

Mrs. Lara will be able to cut nineteen fourths ⎟

⎠⎞

⎜⎝⎛

419 of a yard.

12. Riley is making a cake for the cake walk. She needs

212 cups of flour, but she only has a

21 cup

measuring cup. How many half cups should she measure?

Riley should measure five half-cups ⎟

⎠⎞

⎜⎝⎛

25 .

Answer Key 7b



13. The new bakery cuts its square pans of brownies into ninths. How many pieces (ninths) can be cut from

942 pans of brownies?

Twenty-two ninths ⎟

⎠⎞

⎜⎝⎛

922 can be cut from

942 pans of brownies.

14. The Draper family bought 2 medium pizzas. Scott ate

86 of a pizza. Steven ate

84 of a pizza, and

Stephanie ate 83 of a pizza. How many eighths did

they eat?

They ate thirteen eighths ⎟⎠⎞

⎜⎝⎛

813 of the pizza.

15. What mixed number expresses the same amount of pizza that the Drapers ate?

851

Answer Key 7c

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