6-8 Number and Operation

7/28/2019 6-8 Number and Operation

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Number and Operation, Grades 6-8 1

Number andOperation

TABLE OF CONTENTS1 Exploring Fractions ……………………………………………………. 2

2 Equivalent Fractions ………………………………………………...9

3 Adding and Subtracting Fractions ………………………………….14

4 Exploring Decimals …………………………………………………....18

5 Operations with Decimals ………………………………………….. 23

6 Exploring Percents …………………………………………………….29

7 Fraction, Decimal, and Percent Equivalents ……………………..34

8 Problem-Solving with Fractions, Decimals, and Percents …….…38

9 Proportional Reasoning and Rates ………………………………….43

10 Proportional Reasoning and Mixtures ………………………....…. 47

Student Pages ………………………………………………………………53



Number and Operation, Grades 6-82

Exploring Fractions

Mathematical Focus8 Fractional parts of wholes and sets8 Fractional parts of different-size sets8 Relationships among different fractional parts of same-size sets Students find and compare fractional parts of wholes and sets. Studentsrepresent sets of objects using counters and find fractional parts of the set

by dividing the counters into equal subsets. For example: to find 1/4 of

20, students divide the set of twenty counters into equal subsets of 5counters each. Students practice finding fractional parts of different sizesets: 1/3 of 30, 2/5 of 25, 3/4 of 24, etc. As they gather and recordinformation about fractional parts of different size sets, students begin tolook for patterns in their data. They explore ways of moving from theconcrete method of using counters for finding fractional parts of sets to amore abstract, symbolic approach. Two games (Fraction Four and Largest

Number) at the end of the activity provide additional practice with theconcepts.

Preparation and Materials8 Student Page 1: Fraction-of-a-Set Cards8 Student Page 2: Fraction-of-a-Set Chart8 Student Page 3: Game Boards, 1 copy per student 8 Counters, about 50

Part 1 of this activity provides a quick review of comparing and orderingfractions. If you feel that your students need additional work in this area,Activity 7: Recognizing and Comparing Fractions in the Grades 3-5

Number and Operation Unit provides more in-depth coverage of theseconcepts.

Cut out a set of Fraction-of-a-Set Cards (Student Page 1) ahead of time.




Fractional Parts of a Whole

1. Tell what you already know about fractions.

Review the following important points about fractions if students do notmention them:

The denominator, or bottom number, of a fraction tells the totalnumber of equal parts the whole or set is divided into. Thenumerator, or top number, tells the number of parts being

considered. The parts into which the whole is divided must be equal. To

divide something into fractional parts means the pieces of thesame denominations, e.g. thirds, are the same size.

2. Compare fractions of same-size wholes.

Write the following statement on a piece of paper: 2/3 > ¾

Ask students to read the statement aloud: Two thirds is greater than three fourths.

Ask: Is this statement true or false? Have students sketch a picture showing two large pizzas of equal size. Ask them to shade 2/3 of one pizza and 3/4 of theother pizza.

Write the following statement: ____ < 2/3 < 3/4 < ____

Ask students to think of fractions that could be put into each blank to make the statement true.

Have students add two additional pizzas to their previoussketch to illustrate the statement. Again, emphasize that the

pizzas are to be the same size.




Ask: How did you come up with a fraction that is less than2/3?

Ask: How did you come up with a fraction that is greater than¾?

Ask: What things do you know to be true about fractions?

3. Compare fractions of different-size wholes.

Present this scenario: Suppose someone tells you that you can have 1/4 of a pizza they have ordered. Will you get the same amount to eat if the

pizza ordered was a large pizza or a small pizza? Draw a sketch toillustrate your thinking.

Emphasize that the same fractional part of different wholes can bedifferent sizes. For example: 1/4 of a large pizza is more pizza than 1/4of a small pizza.

Fractional Parts of a Set1. Determine the size of a fraction of a set of objects.

Write the fraction ¼ and place 20 counters on the table. Present thefollowing scenario: Suppose a large package of crackers holds 20crackers. If someone tells you that you can have 1/4 of the crackers, howmany will you have? Use the counters to show 1/4 of the crackers.

Compare finding fractional parts of a set to division. Be sure studentsunderstand that the denominator of the fraction 1/4 tells how many equal

parts the whole set of crackers must be divided into: 4. Dividing 20 into 4equal subsets means 5 crackers in each subset: 1/4 of 20 = 5 crackers.Ask questions such as the following: (Have additional counters availablefor students to use.)




Suppose you were to get 2/4 of a box of 20 crackers, how manywould you get? Write the problem as a mathematical

statement. [2/4 of 20 = 10] Which is greater: 1/5 of this pack of crackers or 1/4 of

a pack crackers? Use counters to illustrate and compare 1/4 of 20 and 1/5 of 20. Emphasize that the sets of crackers being compared are the same size.

1/5 of 20 = 4 1/4 of 20 = 5

1/5 of 20 < 1/4 of 20

Which is greater: 3/4 of a pack of crackers or 2/5? Usecounters to show the different fractional parts. Explain your thinking.

Which is greater: 3/5 of a pack of crackers or 3/10? Usecounters to show the different fractional parts. Explain your thinking.

How many ways can a set of 20 crackers be evenly divided?Can it be divided into 2 equal sets? 3 equal sets? 4 equal

sets? Any other sets?

2. Look for fractions of a set of 24 pens.

Have students use counters to represent the pens. Ask students:

Find 7/8 of the set of pens. Find 3/5 of the set. Find 2/6 of the set. Which is greater: 1/3 of 24 or 2/6 of 24?

3. Compare fractions of different-size sets.

Tell students that a large package of crackers contains 20 crackers and asmall package contains 12 crackers. Ask:

Is 1/4 of the large package the same as 1/4 of the small package? Explain your thinking.

How many crackers are in 1/4 of a small package? Usecounters to show your answer.

Have students recall the two different sized amounts of pizza, large andsmall, that they created in Part 1. One quarter 1/4 of the large pizza wasnot the same as 1/4 of the small pizza. The same concept holds true for sets of objects—fractional parts of different size sets may be differentsizes.




Give students an opportunity to work through a few more examples:

Which is greater: 3/4 of a small package of crackers or 2/5 of a large package? Use counters or a sketch to show your answer.

3/4 of 12 = 9 2/5 of 20 = 8

3/4 of 12 > 2/5 of 20

Which is greater: 2/3 of a small package of crackers, or 1/2 of a large package?

4. Compare the sizes on Fraction-of-a-Set cards (Student Page 1).

Have each student draw two cards from a stack of Fraction-of-a-Set cards(Student Page 1). Ask students to determine which number is greater. For example: 1/4 of 20 or 2/4 of 12. Encourage students to use counters tosolve the problem and/or check their answers. Draw more pairs of cards,each time having students determine which number is greater.

5. Fill in the Fraction-of-a-Set chart (Student Page 2).

Explain that the chart can be used to record fractions of different size sets.Look at the “1/2 of” row. Students should be able to determine 1/2 of each of the different size sets without using counters. For example: 1/2 of 10 = 5, 1/2 of 12 = 6.

6. Explain why some boxes in the Fraction-of-a-Set chart (StudentPage 2) are shaded and others are not.

Ask: Why do you think the “1/2 of 15” box is not shaded? Is it possible to find 1/2 of 15? Or 1/3 of 10? If students are unable to provide anexplanation, do not tell them the answer. Return to the question after students have had an opportunity to fill in more of the chart.Give students an opportunity to draw cards from the Fraction-of-a-Setstack (Student Page 1), determine the correct fraction of the set, and recordtheir answer on the chart. It is not necessary that students go through thewhole stack or complete the chart.

7. Describe patterns on the Fraction-of-a-Set chart (Student Page2).

Have students look at their chart and select a row in which they have filledin 2 or more shaded squares. Challenge them to describe any patterns theysee in the numbers. For example, take the “1/3 of” row:

10 12 15 16 18 20 24 25 30 36 40




1/3 of 4 5 6 8 10 12

Students may notice that the shaded boxes are under numbers that aredivisible by 3. Students may also notice that to find 1/3 of a number, yousimply divide the number by 3. If this idea seems clear to students, ask:

Does the same hold true for finding 1/4 of a number (divide the number by4)? Or 1/5 of a number (divide the number by 5)? It is not essential thatstudents move to this level of abstraction at this point.Have students again consider the boxes that are not shaded. Ask: Is it

possible to find 1/3 of 10? The answer is yes, but the result will not be awhole number: 1/3 of 10 = 3 1/3 or 3.33 In this activity, students willonly work with fractional parts of sets that are whole numbers.

8. Compare rows that contain fractions with the samedenominators.

For example, the “1/3 of” row and the “2/3 of row”:

10 12 15 16 18 20 24 25 30 36 401/3 of 4 5 6 8 10 122/3 of 8 10 12 16 20 24

Ask questions such as: What patterns do you notice? If you know 1/3 of a set, for example 1/3 of 15, how could you

find 2/3 of the set without having to use counters? [1/3 + 1/3 =2/3, therefore 5 + 5 = 10]

Give students plenty of opportunities to explain their thinking andideas about fractional parts of sets. Encourage students to usecounters, sketches, addition, subtraction, multiplication, and divisionto describe their methods.

For students who are having difficulty with the concepts in thissection, continue to use concrete modeling with counters or other manipulatives. For students who are able to abstract the ideas,

provide time for filling in the remainder of the chart without the useof manipulatives.

Fraction Games

Fraction Four




Goal: To cover four squares in a row.Players: 2 or moreMaterials: A complete deck of Fraction-of-a-Set cards (Student Page 1),Game Boards (Student Page 3), counters or paper markersInstructions: Place the shuffled deck of cards face down and give each

player a copy of a game board. First player draws a Fraction-of-a-Setcard, for example “2/3 of 24.” The player determines that 2/3 of 24 is 16.If 16 is on the board he or she places a marker on it. If 16 is not on the

board, no marker is placed, and the next player takes a turn. The first player to get four markers in a row wins.

Largest Number

Goal: To win all the cards.Players: 2

Materials: A complete set of Fraction-of-a-Set cards (Student Page 1)Instructions: This game is similar to the well-known card game “War.”The dealer shuffles the cards and deals out the full deck to both players.Each player takes a card from the top of his or her pile and places it faceup. Each player determines the value of the card drawn (for example, 2/3of 24 would have a value of 16). If the cards have different values, the

player who played the highest card wins both. If the cards have equalvalues, each player puts three cards face down on top of the first card and

plays the fourth card face up. The player with the highest card wins all.When all the cards in one player’s deck are used, the cards taken are madeinto a deck and play resumes. Play continues until one player has all thecards.




Equivalent Fractions

Mathematical Focus8 Number sense for fractions8 Equivalent fractions

Students use fraction strips to investigate equivalent fractions. The strips provide a concrete way for students to visualize and represent equivalent

fractions. Once students are comfortable using strips to find equivalentfractions, they explore a method of finding equivalent fractions withoutthe strips. Students learn that multiplying or dividing a fraction by 1, or another name for 1, such as 4/4, produces an equivalent fraction. For example: 1/3 x 4/4 = 4/12. In this example, 4/12 is equivalent to 1/3.

Preparation and Materials8 Student Page 4: Fraction Strips, several copies8 Student Page 5: Fraction Match Cards, 2 copies

Cut out Fraction Match Cards (Student Page 5) ahead of time. You maywant to copy the Fraction Match Cards onto index cards to create asturdier, more easily manipulated set of playing cards.




Use Fraction Strips to Find EquivalentFractions

1. Compare fraction strips and describe things that you notice.

Have students cut out individual fraction strips from Student Page 4:Fraction Strips and compare them. Have an additional copy of StudentPage 4 available. Ask questions such as:

How many fourths does it take to make 1/2? By placing thefourths strip directly below the halves strip, students can easilysee that it take two fourths to make one half; therefore 1/2 is

equivalent to 2/4. Or students may take a copy of Student Page4 that has not been cut apart and shade in 1/2 and 2/4 on thestrip chart.

½ 1/21/4 ¼ 1/4 1/4

What is the name in sixths of a fraction equivalent to 1/2? Write:

1 = ?_ 2 6

Can you use fraction strips to find three more fractions that areequivalent to 1/2? [4/8, 5/10, 6/12].

1 = _ = _ = _ 2

What is the name in sixths of a fraction equivalent to 2/4?

2 = ?_ 4 6

Use fraction strips to find four fractions equivalent to 2/3. [4/6,6/9, 8/12, 10/15]

2 = _ = _ = _ = _ 3




1/3 1/31/6 1/6 1/6 1/6

1/9 1/9 1/9 1/9 1/9 1/9

How many fourths in 3 wholes?

2. Find more equivalent fractions.

Ask students to use their fraction strips or an uncut copy of Student Page4: Fraction Strips to find and record additional pairs of equivalentfractions. Suggest that students begin by finding equivalent fractions for the following fractions:

3/4 [possible equivalent fractions: 6/8, 9/12, 12/16]

3/9 [possible equivalent fractions: 1/3, 4/12]6/15 [possible equivalent fractions: 2/5, 4/10]

2/10 [possible equivalent fractions: 1/5, 3/15]

8/12 [possible equivalent fractions: 2/3, 6/9, 10/15]

Ask students to describe any observations or discoveries about theequivalent fractions.

Equivalent Fractions without strips

1. Find equivalent fractions by multiplying the numerator and thedenominator by the same number.

Introduce a method for finding equivalent fractions without using thestrips. On a piece of paper write:

1/3 = 5/15

Ask: Is this statement true? If students are unsure, have them use fractionstrips to confirm that the statement is true.

1/3 1/3 1/3

1/15 1/15 1/15 1/15 1/15 1/15 1/15 1/15 1/15 1/15 1/15 1/15 1/15 1/15 1/15

Ask: What number could you multiply the numerator of 1/3 by to get

5? [5]




What number could you multiply the denominator of 1/3 by to get 15? [5]

Explain that multiplying the numerator and the denominator of a fraction by the same number will give an equivalent fraction because it is the sameas multiplying the whole fraction by 1 (x/x = 1).

1 x 5 = 53 x 5 = 15

Multiplying both the numerator and the denominator of 1/3 by 5 is thesame as multiplying 1/3 by 5/5. So, since 5/5 is another name for 1, andmultiplying 1/3 by 1 gives you 1/3, multiplying 1/3 by 5/5 gives you anequivalent fraction in a different form: 5/15.

1/3 x 5/5 = 5/15

Ask students to try a few examples:

Start with the fraction 1/3. Multiply 1/3 by another name for 1,in this case, 2/2.

1/3 x 1 = 1/3 x 2/2 = 1 x 2 = 23 x 2 = 6

Is 2/6 equivalent to 1/3? Have the students use fraction strips to confirmtheir answer.

Explain to students that another name for 1 is 4/4. Multiplying thenumerator and the denominator of 1/3 by 4 to create another equivalentfraction is like multiplying 1/3 by 1.

1 x 4 = 43 x 4 = 12 1/3 = 4/12

Ask: Find fractions equivalent to 1/2. Multiply 1/2 by other names for 1, such as 3/3, 4/4, 5/5 or 6/6.

2. Find equivalent fractions by dividing the numerator and thedenominator by the same number.

Explain that dividing the numerator and the denominator of a fraction bythe same number (another name for one) also produces an equivalentfraction, provided the numerator and the denominator are both divisible bythat number.

For example, take the fraction 6/8. Ask: 6 and 8 are both divisible bywhat number ? [2]




6/8 ÷ 2/2 = 6 ÷ 2 = 38 ÷ 2 = 4

6/8 = 3/4 If students are not convinced, have them use fraction strips tocompare 6/8 and 3/4.

Have students use division to find one or more equivalent fractions for each of the fractions below:

12/15 [possible equivalent fraction: 4/5] 5/10 [possible equivalent fraction: 1/2] 8/12 [possible equivalent fraction: 2/3, 4/6]

Fraction Games

Fraction MatchGoal: To create pairs of equivalent fractions.Players: 2 or moreMaterials: A set of Fraction Match Cards from Student Page 5Instructions: Fraction Match is similar to the traditional game of “GoFish.” Each player is dealt eight Fraction Match Cards face down.Players look at the hand they have been dealt and try to create pairs of equivalent fractions. Equivalent pairs are placed face up on the table. All

players must agree on equivalent pairs. To begin, the first player may ask any other player for an equivalent fraction card that will make a pair with

a card in the first player’s hand. For instance: Do you have any cardsequivalent to 1/2? If the other player has any cards equivalent to 1/2, heor she gives one to the first player. If not, the first player may draw a newcard from the stack. If the new card can be matched with any card in hisor her hand to form an equivalent pair, the player may take another turn.If not, play proceeds to the next player. First player to match all of his or her cards wins the game.




Adding and

Subtracting Fractions

Mathematical Focus8 Addition of fractions8 Subtraction of fractions8 Like and unlike denominators

Students add and subtract fractions with like and unlike denominators.They find common denominators for pairs of fractions and use their knowledge of equivalent fractions to rewrite the fractions using thecommon denominator.

Preparation and Materials 8 Student Page 4: Fraction Strips8 Student Page 6: Close to 1 Score Card, several copies8 Make up a set of fraction numeral cards for the game, Close to 1. The

set should include 3 each of the following cards: 1/2, 1/3, 1/4, 1/6,1/8, 1/12, 3/8, and 5/12.

If your students have very little experience adding and subtractingfractions, you may want to begin with Activity 8: Adding and SubtractingFractions from the Grades 3-5 Number and Operation Unit




Adding and Subtracting Fractions

1. Add fractions by finding equivalent fractions with the samedenominators.

Write the following equation:

3/8 + 4/8 = ?

Have students draw a sketch to represent the problem. Since thedenominator of both addends is 8, simply add the numerators. Studentsshould readily see that 3/8 + 4/8 = 7/8.

Present another problem:1/5 + 3/10 = ?

In this case, the denominators are different. Ask: Can you find anequivalent fraction for one of the fractions that uses the same denominator as the other fraction?

Students may recall from their work in Activity 2 that multiplying 1/5 by2/2 will produce an equivalent fraction: 2/10.

2/10 + 3/10 = 5/10.

At this point, it is not necessary that students reduce their answers to lowestterms; however, if students wish to do so, show them how. In the exampleabove, 5/10 can be reduced to 1/2.

Consider another example:

1/2 + 1/3 = ?

In this case, neither fraction can be written in terms of the other. Ask :Can you find equivalent fractions for 1/2 and 1/3 that use the samedenominator or a common denominator? Encourage students to use thefraction strips from Student Page 4 to guide their thinking. Students maysee that 1/2 and 1/3 can both be written in terms of sixths.




6 and 1/3 = 2/6

3/6 + 2/6 = 5/6.

Work through the following examples with students. Each problemrequires students to rewrite one or both fractions using a commondenominator. Have fraction strips from Student Page 4 available for students to use in writing equivalent fractions. Once students have had achance to work through several problems, ask: Can you describe a rule

for finding a common denominator for two fractions?

1/4 - 2/12 = ?

3/15 + 2/5 = ?

1/4 + 3/8 = ?1/5 - 1/7 = ?

Fraction Games

Close to 1

Goal: To get 5 points, where a point is earned for having the sum closestto one.Players: 2Materials: A shuffled set of fraction numeral cards (see Preparation andMaterials); Student Page 6: Close to 1 Score Card provides a place for students to record the fractions they are dealt for each round of the game.Students circle the three fractions they are adding for that round andrecord the sum of the three fractions in the shaded box at the bottom of thecolumn.

Round 1 Round 2 Round 3 Round 4 Round 5 Round 6

¼3/8

1/12

1/619/24

Instructions: Each player is dealt four fraction cards. Players addtogether three of the fractions in their hand to get a sum that is as close to1 as possible (or equal to 1) without being greater than one. The player with the sum closest to 1 but not over 1 gets a point for that round. Cardsare returned to the stack and re-shuffled. Four new cards are dealt to each




player and play resumes. The first player to get 5 points wins. Sharestrategies for determining which three fractions will be added to produce asum that is closest to one.Variations: Play with the goal being to get as close to zero as possible.Players start with 1 and subtract three of the fractions from 1 to get as

close to zero as they can.Largest Sum

Goal: To win all the cards.Players: 2Materials: A shuffled set of fraction numeral cards (see Preparation andMaterials)Instructions: The dealer shuffles the cards and deals out the full deck to

both players. Each player takes two fraction cards from the top of his or her deck, places them face up and calculates the sum of the two fractions.If the sums are different values, the player with the highest sum wins bothsets of cards. If the sums are equal value, each player draws a third cardand adds it to the previous sum. The player with the highest total (sum of all three fractions) wins all the cards. If the sum is again equal, a fourthcard is added to the total, etc. When all the cards in one player’s deck areused, the cards taken are made into a deck and play resumes. Playcontinues until one player has all the cards.




Exploring Decimals

Mathematical Focus8 Decimal number sense8 Representation of decimal numbers8 Comparison and ordering of decimal numbers Students explore the relationship between fractions and decimals. Theyengage in a series of activities and games in which they represent,

compare, and order decimals. In Deci-O! (Decimal Order Game) studentsarrange a set of five or more decimal cards in order from largest tosmallest. In Decimal Windows, students place digits in window panes tocreate decimal numbers. The goal can be to make the largest value, thesmallest value, or a number close to a target value.

Preparation and Materials 8 Student Page 7: Number Line (tenths)8 Student Page 10: Decimal Cards8 Create two sets of digit cards, 0-9

Cut out Decimal Cards (Student Page 10) ahead of time. You may want tocopy the decimals onto index cards to make them easier to manipulate.




Decimals Between 0 and 1

1. Label a number line with tenths and compare values.

Point out that the number line, between 0 and 1, is divided into ten equalsections, or tenths. Ask students to show and label 2/10 on the number line and 7/10 on the number line.

Ask: Which is greater: 2/10 or 7/10? Students should readily see that7/10 is greater than 2/10.

Ask: Is 6/10 greater than ½ or less than ½? How do you know? Can you show me ½ on the number line? How many tenths is ½ equal to? Students may recall from their work with equivalent fractions in Activity 2that ½ and 5/10 are equivalent fractions. Therefore, 6/10 is greater than ½,or 5/10.

Try a few more problems:

Which is greater 10/10 or 1? If you could extend this number line, where would 15/10 be?

20/10? If students have trouble visualizing this, tape another piece of paper onto the student page and extend the number line.

2. Discuss how the number line would look if divided intohundredths.

Ask: How many parts would there be between each tenth ? [10]. About where would 50/100 be ? Ask students to explain their thinking. Studentsmay simply reduce 50/100 to 1/2, or they may count by 10’s on thenumber line until they reach 50/100 or 5/10.

Ask: Which is greater: 40/100 or 5/10? 3/100 or 30/10? Explain your

thinking.

3. Identify the hundreds place, the tens place, the ones place, thetenths, the hundredths, and the thousandths for a particularnumber.

Sketch the following chart (without labels) and have students identify thedifferent places:




2 3 6 0 1 5

236.015

4. Practice reading numbers aloud and writing the equivalentfractions.

Write a few numbers for students to say and find equivalent fractions:

3.5 three and five tenths 3 5/10 4.25 four and twenty-five hundredths 4 25/100 2.3 two and three tenths 2 3/10 .6 six tenths 6/10

.60 sixty hundredths 60/100 .06 six hundredths 6/100

For the last three decimal numbers, have students show their approximatelocation on the number line.

5. Compare the size of pairs of decimal numbers.

Show students the set of decimal cards you have prepared from StudentPage 8. Place the .5 card and the .25 card on a piece of paper. Havestudents read the decimal numbers : five tenths and twenty-fivehundredths. Ask: Which is greater? Students may want to locate 5/10and 25/100 on the number line to compare them. [.5 > .25]

Explain that in comparing decimals, it is often easier to change thedecimals so that they have the same number of places after the decimal

point. This is similar to rewriting fractions so that they have the samedenominator. Adding zeros after the last digit to the right of the decimal

point will not change the number. Have students again consider thedecimals .5 and .25.

Ask students to add a zero to .5: .50. Have students read the two numbersaloud: five tenths and fifty hundredths , and then confirm by locating thetwo numbers on the number line that they are indeed equivalent.

Place each pair of cards listed below on a piece of paper. Have studentsadd a zero when necessary to change the decimals so that they have thesame number of places after the decimal point (students may want torewrite the decimal numbers on a scratch sheet of paper). Students shouldcompare each pair of decimals and then write the appropriate symbol ( < ,> , = ) between them.




.5 .05

.15 . 2.3 .04

.80 . 8

The same principles apply to comparing thousandths: adding zeros after the last digit to the right of the decimal point will not change the number.Present the following example, asking students to order the decimals fromsmallest to largest:

.05 .5 .005

By adding zeros: .050 .500 .005, the seemingly difficult task of comparing the three decimals becomes much simpler: 5/1000 < 50/1000< 500/1000. The cards can be arranged in the following way:

.005 < .05 < .5

6. Compare the sizes of sets of 3 decimal cards.

Shuffle the decimal cards from Student Page 8. Have student draw threecards from the stack and to write appropriate symbols (<, >, and =)

between the cards to create a true statement. For example:

.002 < .4 < .75

For each statement the students create, have them read the statement aloudand explain their thinking in solving the ordering problem. Encouragestudents to work through several examples before introducing Deci-O!, thedecimal order game described in the next section.

Decimal Games

Deci-O!

Goal: To arrange a set of five (or more) Decimal Cards in order fromsmallest to largest.Players: 2Materials: A set of Decimal Cards from Student Page 8Instructions: Deal five (or more) cards to each player, face up, in avertical row. The cards must remain in the order they were placed. After dealing, players take turns drawing cards from the stack. They maychoose to replace one of the cards in the column with the card they drew,




so that the remaining cards are closer to being in order. If a player chooses not to exchange a card, it is discarded. The player who first re-orders his or her column of cards from lowest to highest is the winner.

Decimal Windows

Goal: To make the largest decimal number Players: 1 or moreMaterials: A set of digit cards (see Preparation and Materials); Each

player needs to draw a 1 x 4 window with a decimal point between thefirst and second panes.

Instructions: Shuffle the digit cards and place them face down. Turnover the first card. Each player writes the digit in a pane of their 1 x 4window. Once a digit has been written in a pane, it cannot be erased or moved. When everyone has written the first digit in a pane, turn over asecond card and have the players write that digit in a pane. Continue untilfour cards have been turned over. Compare decimal numbers to see whohas made the largest number. Ask students if the digits can be rearrangedto make an even larger number.Variations: Play again with the goal being to create the smallest four-digit number.

Make sure your students understand the meaning of “digit,” i.e. thenumbers 0 to 9 are digits. Example: The number 3.5 has two digits–thedigit 3 in the ones column and the digit 5 in the tenths column.




Operations with

Decimals

Mathematical Focus8 Decimals8 Addition, subtraction, multiplication, and division with decimals

This activity provides a collection of computational games in whichstudents must strategically add, subtract, multiply, and divide decimals.By engaging in the various games, students develop a conceptualunderstanding of the effects of different operations on decimal numbers.For example, they learn that multiplying a number by a decimal between 0and 1 will make the number smaller rather than larger.

Preparation and Materials 8 Student Page 6: Close to 1 Score Card, several copies8 Student Page 8: Decimal Cards8 Student Page 9: In the Range8 Student Page 10: Four-in-a-Row Cards, three sets8 Student Page 11: Four-in-a-Row Boards, several copies8 Calculator (optional)8 Counters

Cut out Decimal Cards (Student Page 8) ahead of time.




Decimal Games

1. Practice adding pairs of decimal numbers.

Give students pairs of the decimal cards created from Student Page 8 andask them to add the decimals. If students do not write the numbersvertically, encourage them to do so.

Placing each digit in the same place-value position as the digit aboveit will make addition or subtraction of decimals easier. Alsoencourage students to change the decimal numbers being added or subtracted so that they have the same number of places after thedecimal point. Adding zeros after the last digit to the right of thedecimal point will not change the number.

Example:

.95 + .707 can be re-written as follows:

.950+.7071.657

Close to 1

Goal: To get 5 points by having sums closest to one each round. Players: 2

Materials: A shuffled set of decimal cards (Student Page 8); a copy of Student Page 6: Close to 1 Score Card for each students (this provides a place for students to record the decimals they are dealt for each round of the game. Students circle the three decimals they are adding for that roundand record the sum of the three decimals in the shaded box at the bottomof the column.)





.02.7.4

.03

.75

Instructions: Each player is dealt four decimal cards. Players addtogether three of the decimals in their hand to get a sum that is as close to1 as possible (or equal to 1) without being greater than one. The player with the sum closest to 1 but not over 1 gets a point for that round. Cardsare returned to the stack and re-shuffled. Four new cards are dealt to each

player and play resumes. The first player to get 5 points wins. Sharestrategies for determining which three decimals will be added to produce asum that is closest to one.Variations: Play with the goal being to get as close to zero as possible.Players start with 1 and subtract three of the decimals from 1 to get asclose to zero as they can.

Decimal Jack

Goal: To have a set of cards that add up to an exact total of 2, or to have aset of cards that add up to the highest total less than 2.Players: 2 or moreMaterials: A complete deck of shuffled decimal cards from Student Page8.Instructions: Select one player to be the dealer. Deal two cards facedown and one card face up to each player. Players look at their own cardsand determine whether they want to be dealt an additional card or to pass(i.e., stick with the cards they have). Keep playing rounds until all playersstop requesting cards. Then turn over the face down cards. All playerswhose total is larger than 2 are out. All players who have an exact total of 2 win. If no player has an exact total of 2, the player(s) with the totalclosest to 2 wins.

In the Range

1. Think about the effect on numbers of multiplying them bydecimals.

Start with the number 10. Ask : If I multiply 10 by 5, will the product be greater than 10 or less than 10? [greater than 10] Suppose I multiply 10by .5. Will the product be greater than 10 or less than 10? [less than 10]

Explain your thinking.




2. Develop multiple ways of thinking about multiplication anddivision of decimals.

For example, students should recognize that multiplying a number by .5 isthe same as multiplying it by its fraction equivalent, ½, and is also thesame as adding .5 (or ½) ten times:

10 x 1/2 = 5

.5 + .5 + .5 + .5 + .5 + .5 + .5 + .5 + .5 + .5 = 5

3. Try a few more examples.

Ask: Will 10 x .2 be larger than 10 or smaller than 10? Can you describeone or more ways of thinking about this problem?

Remind students that .2 is the same as 2/10. Have students multiply 10 by

2/10: 10 x 2/10 = 20/10. Twenty tenths can be rewritten in decimal formas: 2.0. Or 20/10 can be reduced to 2. Another way to view the problemis to think of taking 2/10 of 10. (Taking a fraction of a set should befamiliar to students who worked on Activity 1.)

Ask students to consider the problem: 20 x .1 Have students read the problem aloud (twenty times one tenth) and ask: Can you describe one or more ways of thinking about this problem?

Twenty tenths1/10 twenty times.

1/10 of 20List the following problems and have students solve them.

5 x 100 =5 x 10 =5 x 1 =5 x .1 =5 x .01 =

Ask: Can you describe a pattern in the problems and their solutions?

What happens when you multiply 5 by a number greater than 1? What happens when you multiply 5 by a number between 0 and 1?

Have students look for a pattern in the division of numbers as well:

500 ÷ 100500 ÷ 10500 ÷ 1500 ÷ .1




500 ÷ .01

Students should understand that multiplying a whole number by a decimal between 0 and 1 will make the number smaller rather than larger.

Dividing a whole number by a decimal between 0 and 1 has the oppositeeffect: it makes the number larger.

Some students may have already learned to use a whole-number algorithm with decimal numbers in which they multiply the numbers and keep track of where to place the decimal point by counting the number of places after the decimal point in the numbers being multiplied. If students use this method, emphasize the importance of

checking the reasonableness of answers to validate the placement of the decimal point. For example, students may solve the problem: .2 x .3 by saying, “Two times three is six, move the decimal point to the left two places, the answer is .06.” Students should understand why .06 is a reasonable answer; because tenths times tenths give you hundredths.

4. Check the reasonableness of answers to multiplication problems.

Challenge students to determine whether or not the answer is reasonableand explain their thinking without actually doing the calculations.Students can use a calculator to check the answers.

.8 x 16 = 128 Is the answer reasonable? [not reasonable becausetaking 8/10 of a number will give you a smaller number rather thana larger number]

200 x .1 = 20.06 x 120 = 7200

In the Range

Goal: To multiply the starting number by another number so that the product of the two numbers falls within the target range.

Players: 2 or moreMaterials: Calculator (optional); a copy of Student Page 5: In the RangeInstructions: Students begin with a starting number and a target range.Begin with Game 1. The first player multiplies the starting number, in thiscase .4, by a number. The goal is for the product of the starting number and the multiplier to fall within the target range, .10-.20. The first player’smultiplier and product are recorded on the chart. If the first player’s

product is within the range, that round is over and player 1 gets a point. If




not, play moves to the second player. The second player takes the first player’s product and multiplies it by a number, the goal again being to geta product within the range. If the second player’s product is within therange, he or she gains a point, otherwise play moves to the next player (or

back to the first player). The first player to get a product within the range

wins a point. The first player to get three points wins the game.

Starting Number: .4Target Range: .10-.20

Multiplier Product3 1.2

10 12.1 1.2.1 .12

As you are playing the game, invite students to share the strategies theyare using. Ask questions such as: Do you need to make the number bigger or smaller? How much bigger? Twice as big? Less than that?

Do you need to make the number smaller? Half the size? How could youdo that? Variations: Encourage students to create additional games on the blank charts specifying a starting number and a target range.

Four in a Row

Goal: To get four products in a row on the game board.Players: 2 or moreMaterials: A set of A cards and a set of B cards made from Student Page10; a Four-in-a-Row Board (Student Page 11) for each player; counters or markersInstructions: Place the two stacks of cards (A cards and B cards) facedown beside each other. Each player draws one A card and one B card.

Numbers on the cards are multiplied together. If the product of the twodecimals is on a player’s game board he or she places a marker on thatnumber. If the product of the two decimals is not on the game board, nomarker is placed. Cards are placed in the discard pile and new cards aredrawn. The first player to get four in a row wins.Variations:




8 Four in a Row can be played with the same deck of cards, usingaddition or subtraction. Have students help make game boards byusing the cards to put sums and differences on two new boards.

8 Division Variation: Play Four in a Row with division. For thisversion of the game, have students create a new deck of cards and two

new game boards. Try out the new game and, after play, revise thecards or game boards if necessary.




Exploring Percents(The 100-Grid Model)

Mathematical Focus8 Percents8 Percents represented on a 100-Grid

Students use a 100-Grid to explore percentages. They use the 100-grid torepresent and/or determine percentages in various situations. By usingthis concrete representation of percents, students will gain a better understanding of this concept.

Preparation and Materials 8 Student Page 12: Switchboard (several copies)8 Student Page 13: 100-Grid Cards I8

Student Page 14: 100-Grid Cards II8 Student Page 15: 100-Grid8 Student Page 16: Parking Lot

Cut out 100-Grid Cards I and II (Student Pages 13 and 14) ahead of time.Make a copy of the 100-Grid (Student Page 15) on a transparency.




Switchboard Percentages1. Determine the percentage of people talking on the phone by

looking at a hotel switchboard.

Explain that a hotel operator looks at this grid of a switchboard (StudentPage 12), which has a square to represent each of the 100 rooms in thehotel. If the phone in one of the hotel rooms is being used, then the squareon the grid for that room lights up (is shaded), and if it is not being used, itstays blank. Shade in 30 of the squares on a copy of the grid and ask:

How many people are talking on the phones in the hotel right now?

How many phones are there in the hotel? What percentage of the phones are in use? If five more people get on their phones, what percentage will

be using their phones? What percentage will not be using their phones?

Give students a new copy of the switchboard and ask:

If at 6 p.m. three-quarters (3/4) of the phones are in use, what percentage of the phones are in use at that time? Shade in that percentage of the phones. [75]

If 2/3 of the people who are on the phone are talking to friendsin other towns, what percentage of all the rooms in the hotel have people who are talking to friends in other towns? [50]

If the system has a bug where it shuts down if more than 85%of the phones are in use, then how many more people can get on their phones right now without the phone system shutting down? [10]

Shade different portions of the switchboard and have students explainwhat percentage of the phones are in use and what percentage are not inuse.

Ask students to show different ways that the switchboard could look if acertain percentage of the phones were in use. For example, if you toldthem that 37% of the phones were in use, they could shade in theswitchboard in many different ways. Two examples follow:




Largest Grid Percentage

Goal: To win all the cards.Players: 2 or moreMaterials: A set of 100-Grid Cards (Student Pages 13 and 14)Instructions: Deal out the 100-Grid Cards so that all players have anequal number of cards. During each round, all players turn over the topcard in their piles, and the player who has the card with the highest

percentage of the grid shaded or the highest percentage number writtengets to keep all of the cards that were played. The winner places the cards

back at the bottom of his or her pile. If two or more of the cards that are played represent equal percentages, the players who played those cards laythree cards face down and play another card face up. The player who laysdown the card with the highest percentage gets to keep all of the cards thatare now on the table. Play continues until someone runs out of cards. Atthat point, the player with the most cards wins.

Percentage Estimates

1. Estimate then check what percentage of a square is shaded.

Draw a square the same size as the 100-Grid from Student Page 15: 100-Grid on a separate piece of paper. Shade in some portion of the squareand ask students to estimate what percentage of the square they think isshaded. After students have made an estimate, give them a transparencycopy of Student Page 15 to place over the shaded square. They can nowcount the squares that are shaded or not shaded in order to check theaccuracy of their estimates.

Shade in a square:




Place transparency over shaded square to check estimates:

Try the estimation of the percentage shaded problem a few times withdifferent shaded squares.

Parking Lot Problem

1. Help the owner of a store figure out whether to expand herparking lot, whether to leave it the same size, or whether tomake it smaller.Give students a copy of Student Page 16: Parking Lot and explain thefollowing situation to them:

The store owner wants to check how much of the parking lot is filled with cars at various points in the day so that she can decide whether it is close to full and should be expanded, close to empty and should be made smaller, or somewhere in-between. Look at the pictures that the store owner took of the parking lot at three different times of day, and make a chart for her that shows the

percentage of the lot that is full at each time of day. Also write your recommendation for what the owner should do with the parking lot (how many spots should be in the

parking lot) and why you would make that recommendation. Finally, have fill some more cars into each of the parking lot pictures to make one parking lot




that is 45% full, one that is 55% full, and one that is 80% full.

Give students some percentages of percentages problems to figure out,such as the following:

What percentage of a pie would you be eating if you had 35% of 80% of a pie?

If the adults used 50% of the 10 tickets to a movie, and the children used 50% of the 10 tickets to a movie,what percentage of the children’s tickets did one child

use? If during a half-hour television program, 30% of the

time is used for advertisements, and 50% of those advertisements are for toys, then what percentage of the total half-hour is used for toy advertisements?

Ask students to make up their own word problems that involve percentages.

Give students more estimation of percentage problems like the ones in

found in this activity, but spread the shaded part of the square out over thegrid more.




Fraction, Decimal,

and Percent Equivalents Mathematical Focus

8 Conversions from fractions to decimals to percents8 Representations of fractions, decimals, and percents

This activity gives students the chance to examine the relationship between fractions, decimals, and percents. Students begin by constructinga fraction, decimal, and percent converter, which shows how fractions,decimals, and percentages can be compared. Students use this converter as a reference, moving from its construction to an examination of fractions, decimals, and percents on a number line. Students then play agame to practice conversions between fractions, decimals, and percents.Throughout this activity, students build an understanding of fractions,decimals, and percents.

Preparation and Materials 8 Student Page 17: Conversion Table8 Student Page 18: Concentration Cards8 Pennies or counters, 100




Fraction, Decimal, and Percent Converter 1. Fill in the first and last rows of a Conversion Table with the

appropriate fractions, decimals, and percentages.

One way students could think about the conversion table (found onStudent Page 17) is as a measuring stick for a glass of water. If they placethe conversion table in a glass that is filled all the way to the top withwater, then it is 100% full. If there is no water in the glass, then it is 0%full. Ask students to write 100% at the top of the percentage column of the conversion table, 1.00 at the top of the decimal column, and 10/10=1at the top of the fraction column. Students should then write 0 at the

bottom of the percentage columns, 0.00 at the bottom of the decimalcolumn, and 0 at the bottom of the fraction column.

Percent Decimal Fraction100% 1.00 10/10 = 190% ? ?

? ? 8/10? 0.70 ?? ? 6/10? ? ?

? 0.40 ?30% ? ?20% ? ?

? 0.10 ?0% 0.00 0

2. Fill in the middle row of the Conversion Table.

Ask: What percentage of the glass would be full, what fraction of the glass would be full, and what decimal portion of the glass would be full if the water came to this height? [pointing at the middle]

3. Fill in the rest of the Conversion Table.

Continue to point at different points along the conversion table and ask students to fill in the columns for percent, decimal, and fractions at these

points, based on whatever fractions, decimals, and percents are alreadywritten.




4. Find equivalent fractions for the fractions you have written inthe Conversion Table.

Out of 100

1. Record the fraction of pennies that are located on each of threesheets of paper.

Divide 100 pennies or other objects into three groups (the groups do nothave to be equal in size) and place each group on a piece or paper (labeledA, B, and C). Students may refer to the conversion table they made earlier if they have trouble with this or the following questions.For example, in the picture below: A = 40/100 of the pennies; B = 50/100of the pennies; and C = 10/100 of the pennies.

2. Reduce the fractions to lowest terms, if possible.

In the example above, A can be reduced to 2/5, B can be reduced to 1/2,and C can be reduced to 1/10.

3. Convert the fractions for each piece of paper to a decimal, andwrite the number of pennies on each sheet of paper as apercentage of the total number.

In the example above, A is 0.4 or 40%, B is 0.5 or 50%, and C is 0.1 or 10%.

4. Find the new fraction, decimal portion, and percentage of

pennies on each sheet.Regroup the pennies on the three sheets of paper and ask students torecord the fractions, decimal portions, and percentages on each piece.

5. Divide the pennies amongst the sheets to represent givenfractions, decimals, or percents.

Tell students what fractions, decimal portion, or percentages of the pennies are on each sheet (e.g., 1/3, 1/6, and ½) and ask them to divide




group the pennies according to those portions. Students should write the portions in all forms—as decimals, fractions, and percentages.

Fraction/Decimal/Percentage Concentration

Goal: To have the most cards at the end of the game.Players: 2 or moreMaterials: Concentration cards (Student Page 18)Instructions: Place Concentration Cards face down and spread out on atable. Players alternate turns. During a turn, a player:

Flips over two cards Examines the cards to decide whether the two cards represent

the same numeric value of a unit (a pie, a pizza, a 100-grid, or anything)

Explains why he or she thinks the two cards have the samevalue or why one is bigger than the other

Keeps the cards if they are the same, turns them back over if they are different sizes

Play continues until all of the cards have been removed from the table.The winner is the player with the most cards at the end of the game.

Discuss with students how fractions, decimals, and percentages arecommonly used. Brainstorm together examples of situations where one

would most commonly use fractions, where one would most commonlyuse decimals, and where one would most commonly use percentages.Give students problems that mix fractions, decimals, and percentages, sothat they have to decide to either convert all numbers to one type of representation, or think conceptually about how to use the differentrepresentations together. An example problem is: It took a man 12.6minutes to run a race. For 1/3 of that time he was running up or downhills in the racecourse. Of the time that he spent running on hills 30% wasspent running down hills. How much time did he spend running downhills?Ask students to make up their own problems that use fractions, decimals,and percents.




Problem-Solving with

Fractions, Decimals, andPercents

Mathematical Focus8 Fractions, decimals, and percents8 Operations with decimals

This activity gives students the chance to try out their skills in workingwith fractions, decimals, and percents in a few problem-solving situations.Students start by planning out the use of floor space in a classroom to

practice the concept of using fractions, decimals, and percents to represent portions of space. The rest of the activity is devoted to practice usingfractions, decimals, and percents to operate on numbers through situationsof bargain hunting and calculating taxes and tips.

Preparation and Materials 8 Student Page 19: The Classroom8 Student Page 20: Bargain Hunt8 A restaurant menu

Cut out cards from Student Page 20: Bargain Hunt ahead of time.




Classroom Floor Plan on a 100-Grid

1. Make a chart to show what fraction, decimal portion, andpercentage of classroom space is taken up by various pieces of furniture in The Classroom (Student Page 19).

Students should make their charts with one column for the name of thetype of furniture, and three other columns where the portion of the spacein the room that will be taken up can be recorded as a fraction, a decimal,and a percentage. Make sure that students understand that all of the floor

space in the classroom could be thought of as 100% of the classroom, or 1.0 floors, or 1/1 floors. When they talk about portions of the floor spacein the classroom, they will be talking about some percentage of the space,or a decimal or fraction representation of the portion of 1 whole floor.

2. Shade given pieces of furniture into the floor plan of TheClassroom (Student Page 19) and record the fraction, decimalportion, and percentage of space used.

Have students shade the following pieces of furniture into the floor plan,recording the portion of space used.

Sink, 4 blocks of space Teacher’s table, 10 blocks of space

3. Figure out what percentage of the space, what fraction of thespace, and what decimal portion of the space is taken up so far.

4. Brainstorm other pieces of furniture that should be marked off in the classroom.

Each new area of the classroom that they designate for a piece of furniture

should be shaded in, labeled, and marked on the chart that shows the portion of space that is used. Periodically ask students to calculate howmuch space is being used up in total.

5. Think about questions referring to all the furniture in theclassroom.

What percentage of the space is taken up by furniture? What percentage of the space is left open?




What fraction of the space is taken up by furniture? Represent this as a decimal as well.

What fraction of the space is left open? Represent this as a decimal as well.

Fractions, Percentages, and Decimals withMoney

Shopping for Bargains Game

Goal: To have the most cards at the end of the game.Players: 2 or moreMaterials: Bargain Hunt Cards (Student Page 20)Instructions: Deal out all of the Bargain Hunt Cards so that each player has an equal number of cards. Play begins with each player turning over the top card in his or her pile. All players should discuss which cardshows the best bargain for buying a shirt, and whoever has the best

bargain gets to keep all of the cards on the table. That player should placeall of the cards that he or she just won at the bottom of his or her own pile.Play continues with each round consisting of everyone playing the nextcard in his or her pile and the player who played the best bargain keepingall of the turned over cards.If two players turn over bargains that are equivalent, then each of thosetwo players lays two cards face down and another card face up, and the

best of these two new bargains wins all the cards on the table. The gameends when one player runs out of cards or when some predetermined timelimit has been reached, and the winner is the player with the most cards.

1. Estimate what restaurant order will come to less than $10including tax and tip, then check your estimate.

Show students a copy of a menu (that they brought in, that you brought in,or that you created). Ask them to estimate what foods they could order without exceeding the limit. Then have them determine what their total

bill would come to with tax and tip.

Students may need a reminder about how to work with percents asthey calculate the tax and tip. Explain that they can convert the

percent to a fraction out of 100 or make it into a decimal by movinga decimal point two places to the left (for example 8% would be8/100 or .08).




2. Consider variations of the question about your order.

Ask questions such as: If the tax is 6%, what will the total bill come to? If you want to leave a 20% tip, what will the total bill

be? If you add in a glass of milk for $0.95, how will the

total bill change? What is the 6% tax on this glass of milk? What is the

total tax? If three people ordered this same meal, what would

the total bill be? If you have a coupon for 10% off the cost of the meal

(before tax and tip), then what will your total bill be?

3. Compare tips and taxes for two different meals.

Ask students to pick out two different meals they could order and toexplain how they would calculate the tax and tip for each. Also ask themto calculate the total bill with and without tip if both meals are orderedtogether.

4. Figure out which meals would result in particular tax amountsor tip amounts.

If the tax rate is 5% and the tax amount is $0.40 then what wasthe cost of the meal without tax or tip?

What would the cost of a meal without tax be if the tip was $1.50 and this represented 15% of the cost of the meal?

What would the tax on that same meal be if it is 5% of the cost of the meal?

What would the total cost of that meal be, including tax and tip?

5. Answer questions about tax, tip, and meal cost by making and

filling out a chart with columns for cost of meal, 5% tax amount,15% tip amount, and total cost of meal.

For each new row in the chart, start out by telling students what number would go in one of the columns (what the cost of the meal was, what thetax was, what the tip was, or what the total cost was) and then having themfigure out what amounts would go in all of the other columns.




Cost of Meal

5% Tax 15% Tip Total Cost

$10 $0.50 $1.50 $12

$8 ? ? ?? ? $0.90 ?? $0.45 ? ?? $0.36 ? ?? ? $1.80 ?? ? ? $13.20

$15.50 ? ? ?$5.70 ? ? ?

? $0.26 ? ?? ? ? $11.40? ? $1.23 ?

? ? ? $18

6. Figure out what meal could be ordered that would get you asclose as possible to spending specified amounts without goingover the specified amount.

What meal would you order to spend as close to $15 as possible if there is a 5% tax and a 15% tip?

What meal would you order to spend close to $10? What meal would get you close to $10 with a 6% tax

and 20% tip? What meal would get you close to $12 with an 8% tax

and 15% tip? How much money would you need to order the one

item on the menu that is most expensive with a 5% tax and 15% tip? With a 7% tax and 20% tip?

Ask students to make up their own word problems that involve fractions,decimals, and percents. Ask students to explain how fractions, decimals,and/or percents are involved in the problem, and the process that would beneeded to solve the problems. If there are multiple students, or if you havemade up some problems as well, then trade papers and solve each others’story problems.




Proportional Reasoningand Rates

Mathematical Focus8 Proportional Reasoning8 Rates

Students use different types of rate problems to develop their proportionalreasoning skills. During this activity, students will examine problemsabout food prices, apple-picking speeds, and telephone company rates.Students will have the opportunity to think about how differentcomponents of each problem, such as cost and number of items or timeand number of apples, will grow in proportion to each other. They willalso be given the opportunity to think backwards about how the differentcomponents would shrink in proportion to each other to maintain the samerates. Throughout this activity student’s thinking will be concentrated onthe proportionality of different factors that affect rates.

Preparation and Materials 8 Student Page 21: Catie and Carl’s Corner Store8 Student Page 22: Apple Rate Cards8 Student Page 23: Phone Rates8 Student Page 24: Phone Cards

Cut out Apple Rate Cards and Phone Cards (Student Pages 22 and 24)

ahead of time.




Rates1. Find the most and least expensive items in Catie and Carl’s

Corner Store (Student Page 21).

Students should be looking at the cost for one of each item.

2. Fill in the chart on Student Page 21 to show the cost of buyingdifferent numbers of each type of item.

Students should start by filling in the space in the chart that corresponds tothe prices given on the student page (if it is not already filled in), and then

work from there to fill in the rest of the chart.Comparing Rates

1. Add new products to the charts on Student Page 21.

Tell students that the store has Grand Slam candy bars which are 6 for $2.22 and Nuts Galore candy bars which are 2 for $0.70. Ask students tomake a chart that shows how much it would cost for each of these types of candy bars if any number were purchased between 1 and 6. Ask: Whichkind of candy bar is a better deal to buy?

Have students complete a similar chart for doughnuts. Doughnuts can be bought in Catie and Carl’s Corner Store at the price of $0.50 per doughnut, or you can buy 12 doughnuts for $5.50. Students should fill outthe chart for the two different doughnut prices for the numbers 1 through12 and then determine how much money is saved by buying 12 doughnutsat the price per dozen.

Explain that energy bars are on sale at Catie and Carl’s Corner Store at therate of 3 for $0.80. Ask students to make a chart that shows how much 6energy bars, 9 energy bars, or 7 energy bars would cost.

Apple-Picking Rates Game

Goal: To have the most pairs of cards on the table at the end of the game.Players: 2 or moreMaterials: Apple Rate Cards (Student Page 22)Instructions: Deal out 5 Apple Rate cards to each player. Place theremainder of the cards face down in a pile. Each card tells how manyapples a particular person picked in a certain amount of time. Each




player’s job in this game is going to involve translating the number of apples picked in a certain amount of time on each card into the rate of apples picked per hour for each card. If necessary, review with studentsthe process of finding this rate before playing.Player 1 picks one of the cards in his or her hand and asks one of the other

players for a card that shows an apple picker who picked apples at thesame rate as the apple picker on his or her card. For example, if player 1is holding a card that says “picked 45 apples in 20 minutes,” then player 1could ask player 2, “Do you have an apple picker who picks apples at therate of 135 apples per hour?”If player 2 has an apple picker who picks apples at this rate, then he or shegives that card to player 1 who places the pair down on the table, and

player 1 gets to take another turn. If player 2 does not have an apple picker who picks apples at this rate, then he or she says, “Pick one!”Player 1 draws a card from the pile to add to his or her hand, and it is thenext player’s turn. However, if the card that player 1 drew made a pair

with one of player 1’s cards, player 1 plays that pair down on the table andcontinues his or her turn.Play ends when one player runs out of cards. The player with the most

pairs on the table at this point wins the game. However, each player must be able to explain how each of the pairs really is a match, with equal rateson the two cards.

Telephone rates

1. Create a chart showing monthly fees and rates for a list of phonecompanies based on the information on Student Page 23.

Students’ charts should show the name of each phone company, anymonthly fees that the company charges, and the amount per minute whichthe phone company charges.

2. Determine what kind of calling habits would make each of theplans the most economical choice.

For instance, a person who makes many calls during the day would notwant a plan that charges expensive rates during the day and cheap rates atnight. Students should explain the situation where each plan would work

best and write a quick version of their explanations in a column that theyadd to their chart.

Telephone Plan Finder.

Goal: To have the most cards at the end of the game.Players: 2 or moreMaterials: Phone Rates (Student Page 23), Phone Cards (Student Page24)




Instructions: Place the pile of Phone Cards face down on the table.Players take turns drawing a card from the pile and then identifying under which plan from Student Page 23 someone could have had the charge thatis shown on the card. If a player chooses the correct plan and explainswhy that charge matches with that plan and what their thinking process

was, then that player gets to keep the card. If not, the card goes back tothe bottom of the pile. Play continues until all of the cards are gone fromthe pile, and the player with the most cards at this point wins.

Ask students to figure out their own rates by using a stopwatch or a clock with a second hand to time how long it takes to complete some task. Havestudents make a chart to show how much time it would take to completethe task different numbers of times. Ask students to create questions thatthey could ask other students about their rates for completing this task.

Repeat the activity with other tasks.Play “Pass the Rate”. Each player creates four rate cards by writing a rate(e.g., 2:1) or a story that represents the rate (e.g., There is a baseball player who hits 2 homeruns in every 1 game) on four index cards. All four ratecards that each player creates should represent the same rate (e.g., if thefirst card had the ratio 3:5, the other three cards could represent 6:10,15:25, and 3:5). Some ideas of what players could write on their cardsinclude the following:

There is a type of orange with 12 seeds per orange (12:1) There are 16 ounces in every cup (16:1) There are 3 classes in every 5 rooms at the museum (or 3:5)

Everyone’s cards should be mixed in together, then dealt out evenly to all players. Players sit in a circle, and every player passes one card from hisor her hand to the player on his or her right. Players continue to pass onecard at a time to the player on their right, at the same time collecting cardsfrom the player on their left. The object is to collect cards that representthe same ratio. A player does not need to collect cards that represent theratio that he or she created. The first player that can show that he or sheholds cards that all represent the same ratio wins.




Proportional Reasoning

and Mixtures Mathematical Focus

8 Proportional reasoning8 Mixtures and their components

Students will develop skills in proportional reasoning by comparingdifferent kinds of mixtures. This process will include a consideration of recipes and how the amounts of various ingredients would change for adifferent flavor combination or for a different number of servings, and astudy of mixtures of shapes.

Preparation and Materials 8 Student Page 25: Shape Cards8 A die

Cut out Shape Cards (Student Page 25) ahead of time




Fluffy Buttermilk Pancakes

1. Figure out the amount of each ingredient needed to makedifferent numbers of servings of pancakes.

Create a chart like the one below and show it to students. Point out thatthe chart shows the recipe for making 4 servings of pancakes.

Servings Flour Eggs Buttermilk Oil1234 2 cups 2 1 ½ cups 4 Tbs.5678

101220

Have students fill out the corresponding rows in the chart as you ask:

How much of each ingredient would be needed to double the recipe, so that it would serve 8 people?

How much of each ingredient would be needed if the recipe was cut in half?

How many people would this serve? How does the amount of each ingredient change as you

change the number of people that the recipe serves? Explain your thinking.

2. Answer questions about the pancake recipe as you fill out moreof the chart.

Ask students: How many people would the pancake recipe serve if it

included 3 eggs?




How much buttermilk would be needed to serve 3 people with pancakes?

How much of each of the other ingredients would you need if you were using 6 eggs, and how many people would this batch of pancakes serve?

To make batches that serve certain numbers of people,a fraction of an egg would be involved in the recipe.What numbers of servings that are listed on the chart fit that category?

To make 6 servings of pancakes, how many of each ingredient would you need to use?

Fill in the rest of the chart that is still empty to show how much of each ingredient you would need to serve for different numbers of people.

3. Examine the effects of changing the pancake recipe on theingredients needed for different numbers of servings.

Explain to students that another recipe for fluffy buttermilk pancakes usesmore buttermilk to change the consistency of the pancakes. For thisrecipe, 4 servings of pancakes would be made from 2 cups flour, 2 eggs, 2cups buttermilk, and 4 tablespoons of oil. Point out to students that therecipe is the same except that more buttermilk is used. Ask:

How is the ratio of ingredients different in this recipe than in the last?

What will happen with this recipe as the number of servings changes? Will it change in the same way that the other recipe did?

Make a new chart that shows how much of each of the ingredients you would need for this recipe with different numbers of servings.

4. Create a trail mix recipe by determining the amount of ingredients such as peanuts, raisins, chocolate chips, andpretzels you would like to include.

Students should start by figuring out how much of each of theseingredients (or other ingredients of their choosing) would be needed tomake enough trail mix to serve three people. When students have madetheir recipes, ask them to each make a chart that shows the proportion of the different ingredients to each other.

5. Determine how much of each of the trail mix ingredients wouldbe needed for different numbers of servings.




Have students make a chart that shows the amount of each of theingredients that would be needed to serve various numbers of people.Finally, have students come up with some questions that they could ask other students about the trail mix and what would happen if proportionswere to change or number of people served were to change.

Shape Groups

1. Draw a set of 10 shapes, where all the shapes are squares,triangles, and circles.

For example, a student could draw 3 triangles, 4 squares, and 3 circles onhis or her paper. Explain that each student may decide how manytriangles, how many squares, and how many circles to draw, as long as thetotal number of shapes is ten.

2. Answer questions about your group of shapes.

Ask: What is the proportion of squares to triangles to circles

in your drawing? What is the proportion of triangles to total shapes? What is the proportion of circles to total shapes? What is the proportion of squares to total shapes?

If students are having trouble figuring out what the proportionswould be, explain that they need to count the number of each type of shape, and compare how the numbers of each of the shapes (or of the total number of shapes) relate to each other. For example, if someone drew 3 squares, 5 triangles, and 2 circles, then the

proportion of squares to triangles to circles would be 3 squares to 5triangles to 2 circles, and the proportion of each type to the totalwould be 3:10, 5:10, and 2:10.

When students have the idea of how the numbers of shapes in their

original pictures relate to each other, ask them questions about how manyof each of the shapes there would be in bigger or smaller groupings. Ineach case, the shapes remain in the same proportion to one another.

If you had a group of 20 shapes, how many squares would you have? How many circles? How many triangles? Explain how you figured this out.




If you had a group of 5 shapes, how many of each type of shape would you have? Would you have to have any fractions of shapes? Explain your thinking.

Make a chart that shows how many of each type of shape you would have for 6 other total groupings of

shapes. Explain as you make the chart how you figure out how many of each type of shape there would be.

Shape Ratios

Goal: To be the first to have a set of shape cards in the ratio determinedat the beginning of the game.Players: 2 or moreMaterials: A die; a sheet of paper for each player; A set of Shape Cardsfrom Student Page 25Instructions: Roll a die three times. These three numbers represent the

proportional relationship between squares, triangles, and circles that will be the goal in this game. So if a 3, then a 5, then a 1 are rolled, everyoneshould write at the top of his or her paper: 3 squares to 5 triangles to 1circle. The goal of the game is to create a pile of shape cards that achievethis ratio. Place the pile of Shape Cards face down on the table. Ask students to take turns drawing a card from the top of the pile (you can

participate too). Each time a player takes a card from the pile, he or shedraws the shape on the card onto his or her sheet of paper. The studentthen explains what the proportional relationships between the numbers of each kind of shape is on his or her paper. The winner is the first player toreach a mixture of shapes on his or her paper that is in the same proportion

as the goal (the numbers written at the top of everyone’s paper). In order to win, that player must explain how his or her mixture is in the same

proportion as the goal. For example, if the player has 6 squares, 10triangles, and 2 circles, he or she must explain how this is in the same

proportion as the goal of 3:5:1 (if that was the goal).

Show students the following recipe for 12 servings of Three Alarm BeanSalad:

Ingredients: Kidney beans 2 cups Chickpeas 2 cups Wax beans 2 cups Chile Peppers 1 tablespoon for “one alarm salad”

2 tablespoons for “two alarm salad”3 tablespoons for “three alarm salad”

Ask students questions such as the following:




How many total cups of beans will be in the salad? What is the proportion of chile peppers to beans in a

“one alarm salad”? In a “two alarm salad”? In a “three alarm salad”?

By looking at the proportions, which salad will be the hottest, and how can you tell?

How much of each ingredient would you need if you wanted a “two alarm salad” to serve 5 people?

How much of each ingredient would you need if you wanted a “three alarm salad” to serve 30 people?

How much of each ingredient would you need if you wanted a “one alarm salad” to serve 25 people?

What would be the proportion of chile peppers to wax beans if you created a “six alarm salad”?

Place a clear plastic cup of water and another cup of blue water (water plus 3 drops of blue food coloring) on a piece of paper labeled “A”, and place 3 cups of water and 1 cup of blue water on a piece of paper labeled“B”. Ask students what would happen if you poured both of the A cupsinto one bowl, and all of the B cups into another bowl. Ask:

Which bowl would have darker blue water? Why? How do you write the ratio of blue water to clear water

in each bowl? [A is 1 blue to 1 clear (1:1). B is 1 blue to3 clear (or 1:3).]

For each set of combinations of blue water and clear water listed below,ask students to predict which bowl will contain a more strongly bluemixture and to explain their predictions. Students should keep a chartshowing the proportion of blue water to clear water in each mixture. Thechart can be used to show why one mixture is more blue.

2 blue and 1 clear on A versus 1 blue and 1 clear on B 3 blue and 1 clear on A versus 2 blue and 1 clear on B 3 blue and 1 clear on A versus 3 blue and 2 clear on B 1 blue and 1 clear on A versus 2 blue and 2 clear on B




Fraction-of-a-Set Cards

1/2 of 10 1/2 of 12 1/2 of 16 1/2 of 18 1/2 of 20 1/2 of 24

1/2 of 30 1/2 of 36 1/2 of 40 1/3 of 9 1/3 of 12 1/3 of 15




1/3 of 18 1/3 of 24 1/3 of 30 1/3 of 36 1/4 of 12 1/4 of 16

1/4 of 20 1/4 of 24 1/4 of 36 1/4 of 40 1/5 of 5 1/5 of 10

1/5 of 15 1/5 of 20 1/5 of 25 1/5 of 30 1/5 of 40 2/3 of 12

2/3 of 15 2/3 of 18 2/3 of 24 2/3 of 30 2/3 of 36 2/4 of 12

2/4 of 16 2/4 of 20 2/4 of 24 2/4 of 36 2/4 of 40 2/5 of 10

2/5 of 15 2/5 of 20 2/5 of 25 2/5 of 30 2/5 of 40 3/4 of 12

3/4 of 16 3/4 of 20 3/4 of 24 3/4 of 36 3/4 of 40 3/5 of 10

3/5 of 15 3/5 of 20 3/5 of 25 3/5 of 30 3/5 of 40 4/5 of 10

4/5 of 15 4/5 of 20 4/5 of 25 4/5 of 30 4/5 of 40

Fraction-of-a-Set Chart

10 12 15 16 18 20 24 25 30 36 40




1/2 of

1/3 of

1/4 of

1/5 of

2/3 of

2/4 of

2/5 of

3/4 of

3/5 of

4/5 of

Game BoardsGame Board 1




2 5 10 3

9 24 6 18

30 12 8 20

6 15 5 4

Game Board 2

8 5 12 6

18 6 2 15

16 20 3 10

10 9 4 32

Fraction Strips




½ ½

1/3 1/3 1/3

¼ ¼ ¼ ¼

1/5 1/5 1/5 1/5 1/5

1/6 1/6 1/6 1/6 1/6 1/6

1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8

1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9

1/10 1/10 1/10 1/10 1/10 1/10 1/10 1/10 1/10 1/10

1/12 1/12 1/12 1/12 1/12 1/12 1/12 1/12 1/12 1/12 1/12 1/12

1/15 1/15 1/15 1/15 1/15 1/15 1/15 1/15 1/15 1/15 1/15 1/15 1/15 1/15 1/15

Fraction Match Cards




1

2

2

4

3

6

4

8

5

10

6

12

1

3

2

6

3

9

4

12

5

15

1

4

2

8

3

12

1

5

2

10

3

15

1

6

2

12

2

3

4

6

6

9

8

12

10

15

3

4

68

912

25

410

615




Close to 1 Score Card






Number Line (tenths)

0 1




Decimal Cards

.1 .2 .3 .4 .5

.6 .7 .8 .9 .01

.02 .03 .04 .05 .06

.07 .08 .09 .15 .25

.45 .50 .65 .707 .025

.800 .008 .080 .005 .091

.001 .75 .076 .14 .95

.002 .024 .303 .333 .101

.21 .66 .404 .044 .590




In the RangeStarting Number: .4Target Range: .10-20

Multiplier Product

Starting Number: 1.5Target Range: .02-.03

Multiplier Product

Starting Number:Target Range:

Multiplier Product




Four-in-a-Row Cards A Cards

Cut out each card below and write the letter “A” on the back of each card.

1.1 2.3 4.5

6.7 8.8 9.1

B CardsCut out each card below and write the letter “B” on the back of each card.

.1 .2 .5

.01 .03 .05




Four-in-a-Row BoardsGame Board 1

1.82 .46 .355 .045

3.35 .88 1.76 .069

.264 2.25 .11 .088

.33 .455 .225 .55

Game Board 2

.91 .273 .135 4.4

.011 .45 .9 .055

.067 .115 .264 .22




1.34 .023 .091 1.76

Hotel SwitchboardRoom 1 Room 2 Room 3 Room 4 Room 5 Room 6 Room 7 Room 8 Room 9 Room

10

Room11 Room12 Room13 Room14 Room15 Room16 Room17 Room18 Room19 Room20

Room21

Room22

Room23

Room24

Room25

Room26

Room27

Room28

Room29

Room30

Room31

Room32

Room33

Room34

Room35

Room36

Room37

Room38

Room39

Room40

Room41

Room42

Room43

Room44

Room45

Room46

Room47

Room48

Room49

Room50

Room51

Room52

Room53

Room54

Room55

Room56

Room57

Room58

Room59

Room60

Room61

Room62

Room63

Room64

Room65

Room66

Room67

Room68

Room69

Room70

Room71

Room72

Room73

Room74

Room75

Room76

Room77

Room78

Room79

Room80




Room81

Room82

Room83

Room84

Room85

Room86

Room87

Room88

Room89

Room90

Room

91

Room

92

Room

93

Room

94

Room

95

Room

96

Room

97

Room

98

Room

99

Room

100

100-Grid Cards I

95% 20% 75%

67% 3% 58%




100-Grid Cards II

99% 45% 32%

15% 84% 50%




100 Grid




Parking Lot

Parking Lot at 9:00 a.m.

Parking Lot at 12:00 p.m.

Parking Lot at 5:00 p.m.







Conversion TablePercent

Decimal Fraction

90%

0.8

70/100 = 7/10

0.6

40%

30%

20/100 = 2/10 =1/5

0.1







Concentration Cards

89/89 4/5 10% 3/10

3/4 55% 1/2 20/80

20% 18/20 10/25 60%

75% 0.55 0.5 25%

1.0 80% 0.1 30%

0.2 0.9 40% 0.6




The Classroom




Bargain Hunt

2shirt s for

20

Just $50 for 5shirt

25%

off theorig inal

Was

$30,nowhalf o

20%

mar k-dow

$6

off the$70

Buy

oneat $48,

1/3

off the$36

2shirt s for

2shirt s for

2shirt s for

2shirt s for

2

shirt s for

2

shirt s or

2

shirt s or

2

shirt s for

2shirt s for

2shirt s for

2shirt s for

2shir ts




Catie and Carl’s Corner Store

Item CostLollipops $0.20 each

Doughnuts $0.50 eachTacos $1.25 eachSoda $3.60 for 6 cansChips $0.48 per bag

Oranges 5 for $2.00Pretzels 3 bags for $1.50

Number of Lollipops

1 2 3 4 5 6

Cost $0.20

Number of Doughnuts

1 3 6 12

Cost

Number of

Tacos

1 2 3 4 5 6

Cost

Number of Cans of Soda

1 2 3 4 5 6 7 8 9

Cost $3.60

Number of Bags of Chips

1 2 4 8 16 32

Cost

Number of Oranges 1 2 3 4 5 6 7 8

Cost

Number of Bags of Pretzels

1 3 6 9 10

Cost







Apple Rate Cards Picked

80applesin 2hours

Picked 8

applesin 12minutes

Picked

200applesin 4hours

Picked

150applesin 3hours

Picked 1appleevery 2

minutes

Picked 60apples

in 2hours

Picked 100apples in 5

hours

Picked 1appleevery 5

minutes

Picked 5applesin 5minutes

Picked 150applesin 2 and ½ hours


Picked 56 applesin 2 and 1/3hours

Picked 60applesin 1hour and 20minutes


Picked 78applesin 1hoursand 12minutes

Picked 130apples in 2

hours

Picked 180applesin 1.5hours


Picked 40applesin a half hour





Phone Rates

Phone Company A

$25 per monthunlimited free calling

Phone Company B

No monthly fee$0.07 per minute for all calls

Phone Company C

No monthly fee$0.16 per minute during

daytime hours (9am to 5 pm)$0.04 per minute during

nighttime hours (5 pm to 9am

Phone Company D

$5 per month$0.05 per minute for all calls

Phone Company E

No monthly fee$0.05 per minute Monday

through Friday$0.15 per minute Saturday

and Sunday

Phone Company F

$0.07 per minute for all callsor $26 for the month,

whichever is less




Phone Cards

No cost for a 3minute phonecall, but $25 for the month

$50 owed for two months nomatter howmany calls weremade

No calls madeduring themonth of January becauseon vacation, but$25 due

$0.49 for a 7minute callmade at 3 p.m.on a Friday

$0.35 for a 5minute callmade on aSaturday

Bill for themonth is for twenty 5 minutecalls and thetotal cost is$7.00

Two calls aremade onMonday, one at3 pm for 5minutes andone at 6 pm for

A 10 minutecall during theday cost $1.60

There is nomonthly fee,

but all of this person’s callswere charged at$0.16 (she

Only 1 call wasmade during themonth, for 30minutes. Thetotal bill was$6.50

Calls made anytime werecharge $0.05

per minute

The minimumcharge for themonth is $5.00

A call that costs

$0.50 on aFriday wouldcost 3 times asmuch the nextday

2 calls were

made onSaturday, andthe total cost for the 20 minutesof calling time

On Thursday at

6 p.m., a 40minute call cost$2.00. Whattwo plans doesthis charge

The maximumcharge for this

plan is $26 per

month

6 hours and 12minutes of talking costs the

same amount astalking for 12hours. What

10 minutes of calls for themonth costs

only $0.70.Which two plans satisfy




Pancakes

Servings Flour Eggs Buttermilk Oil

1

2

3

4 2 cups 2 1 ½ cups 4 Tbs.

5

6

7

8

10

12

20



Shape Cards

Documents

6-8 Number and Operation