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

Transcription

1 Activity Set 5.3 Computing with Fraction Bars 137 Activity Set 5.3 COMPUTING WITH FRACTION BARS PURPOSE Virtual Manipulatives - I ~-~I I.~IIIII To use Fraction Bars to perform the basic operations of addition, subtraction, multiplication, division in a visual and intuitive manner. MATERIALS Fraction Bars from the Manipulative INTRODUCTION Kit or from the Virtual Manipulatives. Piaget has charted the cognitive development of preadolescents, and his research indicates that even at the age of 12, most children deal only with symbols that are closely tied to their perceptions. For example, the symbolic representation -k + ~ = % mas meaning for most elementary school children only if they can relate it directly to concrete or pictorial representations.p The following table documents the lack of conceptual understanding of fractions and fraction 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 reaonable explanation of how students may have arrived at the incorrect answers 19 and 21? and Responses Percent Responding, Age o I don't know 14 Students may learn fraction skills at a rote manipulation level, but when their memory fails or 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 problem solving. Addition and Subtraction 1. 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 sums by comparing the addend bars to a sum bar. Use Fraction Bars to determine the differences by drawing two vertical lines on the bars to indicate the difference on the difference bar as illustrated in the examples on page M. 1. Driscoll, "The Role of Manipulatives in Elementary School Mathematics," Research within Reach: Elementary School 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 Second NABP Mathematics Assessment: Elementary School," Arithmetic Teacher 27 (April 1980): 10-12,44-47.

2 Activity Set 5.3 Computing with Fraction Bars 139 Sum Example: Difference Example: *a. b. *c. 1 "3+--= d. *e. %+--= _LI f IIIIIIIIIJ 2. 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 fractions are unequal.)

3 140 Chapter 5 Integers and Fractions 3. Obtaining Common Denominators: If each part of the t bar is split into 4 equal parts and each 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 size, ese new ars s ow "3"4-12 an "3- "4- U' 2 '3 1 "4,~, =.~-= =--- ~-.- ~~ - -- ~-"""""t' ~. -:~ r...,... -)!:.',.~ -< I - _. --~ Use the fewest number of splits for the following pairs of Fraction Bars so that each pair of bars has the same number of parts of the same size. Then write two fraction equalities under each pair. For example, in part a, % = -t% and i =?2',*a. b. c. ':'":-~Tl-----,---'--- :-.-~~; Jch......:l ~. ~ Use the bars from parts a, b, and c above to compute the following sums and differences. *d = _ 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 carry out the operation. Use your diagrams to explain how you reached your conclusions. *a b c d. "2-7

4 Activity Set 5.3 Computing with Fraction Bars 141 Multiplication 5. You can determine t X i by splitting the shaded part of a i bar into 3 equal parts. One of these maller parts is -fs of a bar, because there are 18 of these parts in a whole bar. (You can think of this as having i of an amount and taking t of it.) 1 6 To determine ± X %, plit each shaded part of the % bar into 3 equal parts. One of these split parts is /5 of the whole bar, and 4 of the e split parts is J~ of the whole bar. (You can think of this a having ~ of an amount and taking 4- of each fifth.) ) j a. Split each part of each bar into 2 equal parts. Use this result to complete the given equations "X3"=.1 X.1= X. = 2 8 *b. Split each shaded part of each bar into 3 equal parts, and complete the equations. i:'<""""""'" , ~ - --~ -~ -- -~ ~X.l= 3 6 ~X.i= 3 5 c. Split each shaded part of each bar into 4 equal parts, and complete the equations X.l= Sketch a Fraction Bar for % and visually determine the product ix %. Write a description of the procedure you used.

5 142 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 on the shaded part of the ~ bar Similarly, % -i- t = 2~, because the shad_edportion of the t bar can be measured off (or fits into) 2~ times on the shaded part of the % bar. Sketch Fraction Bars to determine the following quotients. Use your sketch to explain your reasoning. *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 estimate with fractions.!'or example, to estimate the quotient It -i- t, you can compare Fraction Bars representing 1~ t? bars representing eighths. There are 8 eighth in 1 whole bar and about 6 eighths in the ~ bar. So, i can be measured off about 14 times, and It -;-i = 14. IJIIIIJJITIIIIIIJ Use a visual approach to estimate the following fraction operations. Draw Fraction Bar sketches and explain how you arrived at your estimation. *b " c. 5 2 *d. I X 1 3 6

6 Activity Set 5.3 Computing with Fraction Bars 143 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 only once, 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 turns it takes you to complete tbe four equations by selecting only 10 bars on each turn. JUST FOR FUN to perform opera- These three games provide opportunities tions on fractions. FRACTION BAR BLACKJACK (Addition-2 to 4 players) Spread the bars face down. The object is to select one or more bars so that the fraction or the sum of fractions is as close to 1 as possible, but not greater than 1. Each player selects bars one at a time, trying to get close to 1 without going over. (A player may wisb to take only 1 bar.) Each player finishes hi or her turn by saying "I'm holding." After every player in turn has finished, the players show their bars. The player who is closest to a sum of 1, but not over, wins the 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. Player 1 --T. ~ ; -~ ~ l+~ 2 4 Player 2 Player 3 5 6'