activity set 5.3 computing with fraction...

Activity Set 5.3 Computing with Fraction Bars 137

Activity Set 5.3 COMPUTING WITH FRACTION BARS

PURPOSE

VirtualManipulatives

To use Fraction Bars to perform the basic operations of addition, subtraction, multiplication, anddivision in a visual and intuitive manner.

MATERIALS- I~-~I I.~IIIII

!

Fraction Bars from the Manipulative Kit or from the Virtual Manipulatives.

INTRODUCTIONPiaget has charted the cognitive development of preadolescents,and his research indicates that even at the age of 12, most chil-dren deal only with symbols that are closely tied to their per-ceptions. For example, the symbolic representation -k + ~= %mas meaning for most elementary school children only if theycan relate it directly to concrete or pictorial representations.P

The following table documents the lack of conceptual understanding of fractions and frac-tion addition exhibited by 13-year-olds when they were asked to estimate the answer to :; + ~.7

Over 50 percent of the students responded with the incorrect answers 19 and 21. What is a rea-onable explanation of how students may have arrived at the incorrect answers 19 and 21?

www.mhhe.corn/bbn

01.2019

021

7

Responses Percent Responding, Age 13

242827

o I don't know 14

Students may learn fraction skills at a rote manipulation level, but when their memory failsor they encounter a nonstandard application, they have no conceptual basis to fall back on.

In this activity set, Fraction Bars will be used to develop visual images offraction operations.These images will provide a conceptual ba is for computations, estimations, and problemsolving.

Addition and Subtraction1. Addition and Subtraction: Determine the missing fractions and the sum or difference of

the fractions for each pair of bars on the page 139. Use Fraction Bars to determine the sumsby comparing the addend bars to a sum bar. Use Fraction Bars to determine the differencesby drawing two vertical lines on the bars to indicate the difference on the difference bar asillustrated in the examples on page 139.

6M. 1. Driscoll, "The Role of Manipulatives in Elementary School Mathematics," Research within Reach: ElementarySchool Mathematics (St. Louis, MO: Cernrel Inc., 1983): l.

7T. P. Carpenter, H. Kepner, M. K. Corbitt, M. M. Lindquist, and R. E. Reys, "Results and Implications of the SecondNABP Mathematics Assessment: Elementary School," Arithmetic Teacher 27 (April 1980): 10-12,44-47.


Sum Example:

312

+.1.37

12

Difference Example:

b.*a.

d.*c. 1"3+--=

f.*e. %+--=

_LI45-

IIIIIIIIIJ2. Select three pairs of Fraction Bars .at random. Complete the following equations for the sum

and difference of each pair of fractions. (Subtract the smaller from the larger if the fractionsare unequal.)

----+----

----+----

----+----

140 Chapter 5 Integers and Fractions

3. Obtaining Common Denominators: If each part of the t bar is split into 4 equal parts andeach part of the i bar is split into 3 equal parts, both bars will have 12 parts of the same. Th b h that 2 + I - 11 d 2 1 - 5size, ese new ars s ow "3"4 - 12 an "3- "4- U'

2'3

,~,=.~-=.-.- --=--- ~-

.- ~~ - --

~-"""""t' ~. -:~ r·.....,... -)

!:.', .~ -< I- _. --~ - - - --.--3

121"4

Use the fewest number of splits for the following pairs of Fraction Bars so that each pair ofbars has the same number of parts of the same size. Then write two fraction equalities undereach pair. For example, in part a, % = -t% and i = ?2'

b.*a., c.

':'":-~Tl-----,---'--- :-.-~~;

Jch..... ....:l___ ~. • _ _ ~

Use the bars from parts a, b, and c above to compute the following sums and differences.

*d. 2 + 1 =6 43 2_

e. "5 + 3"- f 1+1=. 9 6

*9· t - i =3 2_

1.6 "9-

4. Sketch Fraction Bars for each pair of fractions, and then split the paL1Sof the bars to carryout the operation. Use your diagrams to explain how you reached your conclusions.

*a. 1 + 22 6 b 1+1·34

3 3c·4-61 2d. "2-7


Multiplication

5. You can determine t X i by splitting the shaded part of a i bar into 3 equal parts. One ofthese maller parts is -fs of a bar, because there are 18 of these parts in a whole bar. (Youcan think of this as having i of an amount and taking t of it.)

16

To determine ± X %, plit each shaded part of the % bar into 3 equal parts. One of these splitparts is /5 of the whole bar, and 4 of the e split parts is J~ of the whole bar. (You can think ofthis a having ~ of an amount and taking 4- of each fifth.)

) j

1 12"X3"= .1X.1=2 5

.1X.£=2 8

a. Split each part of each bar into 2 equal parts. Use this result to complete the givenequations.

*b. Split each shaded part of each bar into 3 equal parts, and complete the equations.

~X.l=3 6

~X.i=3 5

i:'<""""""'" - •.•• ---, - ---- ~ - --~

-~ -- -~- - --

c. Split each shaded part of each bar into 4 equal parts, and complete the equations.

.1X.l=4 2

- - - - - -

- - - - -

6. Sketch a Fraction Bar for % and visually determine the product iX %. Write a description ofthe procedure you used.

142 Chapter 5 Integers and Fractions

Division

7. Using the measurement approach to division (see Activity Set 3.4), we can write ~ -i- ~ = 3,because the shaded portion of the ~ bar can be measured off (or fits into) exactly 3 times onthe shaded part of the ~ bar.

- -- -

- -

Similarly, % -i- t = 2~, because the shad_edportion of the t bar can be measured off (or fitsinto) 2~ times on the shaded part of the % bar.

Sketch Fraction Bars to determine the following quotients. Use your sketch to explain yourreasoning.

*d. 1..:...I =. 3 *f. 1..:...1=8 .4

Estimation

8. It is often helpful to draw a sketch or visualize a fraction model when you are trying to esti-mate with fractions. !'or example, to estimate the quotient It -i- t, you can compare FractionBars representing 1~ t? bars representing eighths. There are 8 eighth in 1 whole bar andabout 6 eighths in the ~ bar. So, i can be measured off about 14 times, and It -;-i = 14.

IJIIIIJJITIIIIIIJUse a visual approach to estimate the following fraction operations. Draw Fraction Barsketches and explain how you arrived at your estimation.

*b 1 1. 8 "6

1.+1c. 5 2 *d. I X 1

3 6


All Four Operations

9. *a. The following 10 fractions are from the set of Fraction Bars:

Using each fraction only once, place these fractions in the 10 blanks to form four equations.

----+---- 3x _

____ =2

b. Spread your bars face down and select any 10. Using the fraction from each bar onlyonce, complete as many of these four equations as possible.

Fractions selected: ~ ~ _

----+---- 3x _

____ =2

c. Solitaire: The activity in part b can be played like a solitaire game. See how many turnsit takes you to complete tbe four equations by selecting only 10 bars on each turn.

JUST FOR FUNThese three games provide opportunities to perform opera-tions on fractions.

selects bars one at a time, trying to get close to 1 withoutgoing over. (A player may wisb to take only 1 bar.) Eachplayer finishes hi or her turn by saying "I'm holding." Afterevery player in turn has finished, the players show their bars.The player who is closest to a sum of 1, but not over, winsthe round.

Examples: Player 1 has a sum greater than 1 and is over.Player 2 bas a greater sum than player 3 and wins the round.

FRACTION BAR BLACKJACK (Addition-2 to4 players)

Spread the bars face down. The object is to select one ormore bars so that the fraction or the sum of fractions is asclose to 1 as possible, but not greater than 1. Each player

Player 1- -T. ~- --- - -- -- -----;

-~-------~- --- - -- -.l+~2 4

Player 2Player 3

56'

activity set 5.3 computing with fraction...

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