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You said that fractions raised your anxiety level. In this fraction workshop I tried to draw pictures for all the basic fraction ideas so that people could see one way to look at what the symbols mean. Let me know if the pictures help you.Bruce
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Session 1. 1 and the definition of FractionIn session 1, the class will consider the concept of one. Using this concept, we
will define a fraction and look at various representations of common fractions.Session 2. Equivalent Fractions
In session 2, the class will consider the equivalence of fractions. We will see tworepresentations of the concept of equivalent fractions, one using tessellations of theplane into rectangles and the other using the multiplication table.Session 3. Inequality
In session 3, the class will consider inequality of fractions. We will take somefractions, such as 2/5 and 3/7, and investigate which is larger using the Math Enginetm.Session 4. Adding (Subtracting) Fractions with Like Denominators
In session 4, the class will see the hands-on representation of adding and sub-tracting two fractions with the same denominator.Session 5. Adding (Subtracting) Fractions with Unlike Denominators
In session 5, we will extend the idea of adding and subtracting two fractions tofractions with unlike denominators. We will also address the concept of finding theleast common denominator, showing how this is manifested with the manipulative.Session 6. Multiplying FractionsSession 7. Canceling Common Factors While Multiplying Fractions
In sessions 6 and 7, we will address the idea of multiplying two fractions usingthe math manipulative. We will look at multiplication, reciprocals, and the idea of can-celing like factors in the numerator and denominator when multiplying fractions.Session 8. Dividing Fractions
In session 8, the class will see how dividing fractions is essentially the same asdividing two whole numbers. We will see why invert and multiply works in dividingone fraction by another.Session 9. Ratio and Proportion
In session 9, the class will investigate the relationship between fractions andproportion. We will solve in a hands-on manner the proportion 2/5 =x/7. We will seehow this relates to other problems, such as percentage.Session 10. Wrap-up and Discussion.
In session 10, the class will review the ideas covered in the workshop. The in-structor will answer questions and lead a discussion about how the concepts learned inthe workshop can be used in the students’ classroom lessons and teaching.
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Hands-on Fractions Using Balls
The concepts of fractions can seem mysterious and frightening. What does itmean to say that 1/2 and 3/6 are the same fraction, i.e. they are equivalent? They do notlook the same. When we are told that to divide 2/3 by 4/5 we must invert 4/5 and thenmultiply it by 2/3, we might wonder why this is so. Where did this come from? Why isit true? Is there some way of seeing it? If fractions seem mysterious to us, what do theylook like to a 4th or 5th grader? In this workshop, we will try to use the Math Enginetm,a math manipulative, and some simple definitions to show why all these strange andmysterious rules are reasonable.
This workshop is a 10-hour study of fractions. I want you, the student, to try toput out of your mind all the rules and ideas you were taught about fractions We willdefine what a fraction is and then think about what it would mean to compare and dooperations on the fractions. We will try to demonstrate why all the rules, like thedivision rule invert and multiply, are true.
Each session will consist of an explanation, a lesson plan, and an exercise. Wewill discuss the definitions and then work out various problems and then do exercises. Iwould like you to ask questions about things that are not clear to you and dispute thingswith which you disagree. I want you to feel free to voice ideas and ask questionswithout wondering whether they are “stupid.” You could call this workshop,“Everything you wanted to know about fractions, but were afraid to ask.”
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Session 1 One and the Definition of Fraction
We will start by looking at where some of the words that we use with fractionsoriginated. I think it is always informative to see what a mathematical word means in aregular English non-mathematical setting. We can sometimes get a better understandingof the mathematical concept if we see where the word came from. A fraction is “a smallportion broken off; small part, amount, degree, etc.; a fragment; as, a fraction of time.”The word fraction comes from the Latin fractio, a breaking. It appears then that afraction is obtained by taking something, breaking it, and taking one of the parts.
In mathematics, we take this breaking and make it more precise. We takesomething which we call one or the unit and break it into a number of equal parts,which we call the denominator of the fraction. We then take a certain number of theseequal parts. The number of equal parts that we take is called the numerator. One of theimportant ideas that must be considered in the mathematical definition of denominatoris that the unit is broken into equal parts.
An interesting definition related to denominator is the one for the worddenominate. This comes from the past participle denominatus meaning to name. Theverb form in English means “to name; to give a name or epithet to; to call; to designate.The adjectival form in mathematics is interesting; “denoting a number which expressesa specific kind of unit; qualifying: opposed to abstract ; thus, seven pounds is adenominate number, while seven, without reference to concrete units, is abstract.”Thus, in 2/5, the 5 is denominate or the denominator because it tells us what we aretaking 2 of.
We can view the word numerator in a similar manner. Let’s look at the verbnumerate. It comes from the Latin word numeratus meaning to count, number. Tonumerate is either “to count, to enumerate” or “to read (a number or numbers expressedin figures).” The adjectival form, mainly British, is “able to deal with scientificconcepts, especially in a nathematical way; as, an increasingly numerate society.” Thenumerator of a fraction is “that term of a fraction which shows how many of thespecified parts of a unit are taken....” A second definition for a numerator is “a personor thing that counts.”
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ctio
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arFr
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Let’s look at a specific fraction, 3/4, and see what its different parts are.How do explain to a child what a fraction is? What is it that makes fractions soimportant? First note that there are three parts to the fraction, two numbers anda line. What are the different parts called and what do they represent?
A fraction is a mathematical representation of a physical action. We takean object, we break it into several equal parts, and then we take a certain number ofthose parts. Each part of the fraction indicates one of these actions. The fraction barrepresents taking an object, one thing, a unit and breaking it into equal parts.
Let’s take a rectangle that can hold 4 balls. This will be our unit.
We now break the rectangle into 4 pieces. Thefraction bar indicates that this fracturing has occurred.
The 4, called the denominator, in the fractiontells us that we have broken the rectangle into fourcompartments, each capable of holding 1 ball.
Finally, the 3 or numerator of the fraction tellsus how many of the balls we are taking out of the 4possible.
So now we have the building of the fraction, 3/4. We take a unit, we divide itinto four equal parts, and we take three of the parts.
3_4
3_4
3_4
3_4
Numerator-------Number------CountFraction Bar-----Fracture-----BreakDenominator-----Name--------Name
3_4
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The Math Engine can be used torepresent fractions using panes to show theunits. Black lines can be drawn on a pane toform a pattern on a rectangular grid. Redlines can be drawn in the patterns to breakthe shape into several equal sized parts. Wecan then choose a number of these parts toform a fraction. The diagram on the rightshows 6 different patterns. Each pattern hasbeen broken into equal sized parts. Finally,some of the parts have been colored in. Thefraction that is represented is marked nearthe appropriate pattern. See whether youagree with the fractions that are shown.
We do not have to have the entireMath Engine Screen on the page to showfractions. I will just show the shape on the Math Engine and the balls in the pattern toillustrate various fractions.
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Worksheet 1.1
1. What fractions are represented by the following pictures?
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Worksheet 1.2
1. Mark the diagrams to show the fractions?
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Lesson 1.1
Objective: To show the fraction 2/5Materials: Math Engine, 5 blue balls, 1 peg, 1 pane
1 Mark Math Engine or place a marked pane outlining thetop 5 ball positions in the left column.
2. Place a peg in the sixth hole down from the top of theleft hand column.
3. Place 5 blue balls above the peg and ask the studentshow many balls are in the Math Engine.
4. Point out that the balls fill the rectangle you marked.
5. Write 5 on the board to indicate that there are 5 balls inthe rectangle.
6. Put a peg under the two top balls and pull the lower peg,letting the bottom three balls drop out.
7. Ask the students how many balls are in the rectangle.
8. Point out that the rectangle can hold 5 balls but it onlyhas two balls in it.
9. Write the fraction 2/5 on the board.
10. Point to the 5. Say that the 5 shows how many balls can go into the rectangle.
11. Point to the 2. Say that the 2 shows how many balls are actually in the rectangle.
12. Tell the students that the fraction is called “two fifths.”
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Lesson 1.2
Objective: To show 1 in relation to a fractionMaterials: Math Engine, 12 blue balls, 6 red balls, 3 green balls, 1 black ball, 1 silverball, 1 yellow ball, 8 pegs, 2 panes
1. Place pane marked with 4 rectangles each enclosing sixballs, as shown to the right, on a Math Engine.
2. Put 2 pegs at the bottom of the first rectangle and put sixblue balls on top.
3. Point out to the students that we have one collection of 6balls, all of which are blue.
4. Put 2 pegs at the bottom of the second rectangle and putin 3 blue and 3 red balls, as shown to the left.
5. Point out to the students that we now have a collectionof 6 balls but not all are blue. There are two subcollections,one red and one blue, each having 3 balls.
6. There are 2 subcollections, one of which is blue so theblue balls are 1 out of two parts or 1/2 of the balls in thecollection.
7. Put 2 pegs at the bottom of the third rectangle and putblue, red, and green balls in as shown on the right.
8. Point out to the students that we now have a collectionof 6 balls but not all are blue. There are now threesubcollections, one green, one red, and one blue, eachhaving 2 balls.
9. There are 3 subcollections, one of which is blue, so theblue balls are 1 out of three parts or 1/3 of the balls in thecollection.
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121 1
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10. Put 2 pegs at the bottom of the fourth rectangle and putone blue, red, yellow, silver, black, and green ball in asshown on the left.
11. Point out to the students that we now have a collectionof 6 balls, not all blue. There are six subcollections, eachhaving 1 colored ball.
12. There are 6 subcollections, one of which is blue, so theblue balls are 1 out of six parts or 1/6 of the balls in thecollection.
13. Fill up all the rectangles with blue balls.
14. Place a pane with red lines which break up therectangle into smaller rectangles as shown to the right.
15. Point out to the students that we do not have to look atdifferent colors to see the subcollections. We can subdividethe rectangles into rectangles of the same size to see thatthe blue balls occupy the fractional part of the unitrectangles. 1
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Session 2. Equivalent Fractions
One of the most confusing yet powerful ideas with fractions is the concept ofequivalence. If I cut a pie into two equal pieces and take one of the pieces or I cut thepie into four equal pieces and take two of the pieces, I will get the same amount of pieno matter which one I choose. The single piece might be more manageable and lessmessy, so some people might prefer the choice in which the pie has been cut into thefewest number of pieces. This is the same mathematically as asking to pick the fractionin its lowest terms.
Let’s look at the definition of equality and equivalence. Equal comes from theLatin word aequalis meaning equal. Equivalent comes from the Latin words aequus,equal and valere to be strong. 1/2 and 2/4 are not the same but they have equalstrength as fractions. (Strengths is a nine letter English word with one vowel.)
Consider the different ways of looking at 3/4 we saw in the last session.
All of these are 3/4 because we have divided the figure into4 equal parts and colored 3 of them blue. What would thedifference be if we had divided them differently and coloredthe same number of balls. Look at the two fractions on theright. One is a rectangle divided into 4 equal parts and theother is the same rectangle divided into 8 equal parts. Wehave colored 3 of the parts in the former rectangle and 6 ofthe parts in the latter. Which is more? They are not equalbecause of the way we formed them, but they have equalpower in anything we would want to do with them. Theyare essentially the same. They are equivalent.
3_4
3_4
Which is more?
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1_2
2_4
3_6
6__12
Another question we can askis, “Given a fraction and arectangle representing thefraction, how many otherfractions which areequivalent to the first can weshow with the samerectangle?” We have anexample of this on the leftwith the 3 x 4 rectangle andthe fraction 1/2. There are 4fractions that we can showon the 3 x 4 rectangle which
are equivalent to 1/2. Note that the answer is the same as the number of even divisorsof 12. We can see that the number of balls that are green are the same in each instanceeven though we constructed the fraction in a different way.
We now turn our attention to a special collection of rectangles , a specialcollection of rectangular arrays of balls, and their relationship to the multiplicationtable. The special rectangles and rectangular arrays of ballsare those whose upper left hand corner is in the upper lefthand corner of the multiplication table. Take a rectanglewhose upper left hand corner is the upper left hand cornerof the multiplication table. The number of themultiplication table in the lower right hand corner of therectangle is the number of balls in the rectangle. If we lookat the example on the right, we see a blue 6 x 7 rectanglein the upper left hand corner of the multiplication table.The number of balls in the rectangle is the number in thelower right hand corner of the rectangle, namely 42.Similarly, if we have a rectangular array of balls in the upper left hand corner of themultiplication table, the number on the ball in the lower right hand corner of the array isthe number of balls in the array. Looking in the above example, we have a 3 x 7 arrayof green balls. The number on the ball in the lower right hand corner of the array ofgreen balls is 21, which is the number of green balls in the array. This gives us a fastway of figuring the fraction of green balls in the blue rectangle There are 21 green ballsout of the possible 42 balls in the rectangle. We have the fraction 21/42.
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Let’s take representingfractions on the multiplication tablewith balls and rectangles and show away of generating an endless supplyof equivalent fractions. We start withthe fraction 2/5. As we see on theright, 2/5 can be represented as 2balls out of a possible 5 balls in therectangle.
We take the rectangle andmake it 2 wide. We now have a 5x2rectangle which can hold 10 ballsand we take 4 of them. The rectanglerepresents 2/5 but it also represents4/10.
We can continue to addcolumns to the rectangle, generatingmore fractions equivalent to 2/5.Here we have 4 columns and see that8/20=2/5.
If we use a rectangle that isnine balls wide, we have 18/45=2/5.
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= 2x1___ 5x1
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2x4___ 5x4
2x9___ 5x9
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We can pull the numbersout of the preceding exampleand get a list of fractionsequivalent to 2/5. Weeliminate all the numbers in the multiplication table that arenot multiples of 2 or 5 and have two rows. We put one rowon top of the other row and put a fraction bar betweenthem and we have a list of fractions that are equivalent to2/5.
Another point we can notice is that thisidea works with columns as well as rows.Look at the partial multiplication table to theright. We can see the fractions that areequivalent to 2/5 by pulling out the second andfifth column of the multiplication table andstanding them side by side.
Of course, if we had a larger multiplication table, we could extend the list ofequivalent fractions as far as we wished.
The multiplication table can help us with anotherproblem encountered with equivalent fractions, namelyreducing fractions. If we can find the two numbers in thesame column (row), we can go to the first column (row)and look at those numbers. The fraction they form isequivalent to the original fraction. Take the fraction 35/63for example. By looking at the multiplication table to theleft, we see that 35 and 63 are both in the seventh column.By going to the first column, we find 5 and 9. This meansthat 35/63 = 5/9. This change of columns to get anequivalent fraction in lower terms works because we are
finding a common factor in the numerator and denominator. Note that this method willreduce a fraction but not necessarily to its lowest terms.
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Worksheet 2.1
1. Show as many fractions as you can which are equivalent to 1/3using a 4x3 rectangle.
1_3
_ _ _
2. What fractions does this show are equivalent?
_ _=
Explain:
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Worksheet 2.2
1. Use the multiplication table to show that 27/63=3/7. Explain allyour reasoning.
42 35 28 21 14 7
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Worksheet 2.3
1. Show that using the multiplication table to reduce a fractionmight give you a fraction that can be reduced again.
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Lesson 2.1
Objective: To show equivalent forms of 6/12Materials: Math Engine, 24 blue balls, 8 pegs, 1 pane
1. Mark a pane as shown on the right. There will befour 4x3 rectangles in the four corners with eachrectangle partitioned into different fractional parts byred lines.
2. Place 4 pegs on the bottom rectangles and fill thefirst two columns in each rectangle.
3. Place 4 pegs on the top rectangles and fill the firsttwo columns as before.
4. Each rectangle is the same size and holds the same number of blue balls butrepresents a different equivalent fraction.
5. Ask the students to choose the fraction which represents the largest fraction of thewhole. Let the students decide for themselves why dividing something into 12 parts andtaking 6 gives the same result as dividing the same object into 6 parts and taking 3, etc.
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Lesson 2.2
Objective: To show the relationship between rectangular arrays of balls and themultiplication table.Materials: 2 Math Engines, 21 green balls, 16 red balls, 9 pegs, 2 multiplication panes
1. Place multiplication panes on 2 Math Engines.
2. Put pegs in the first seven columns of the fourth row ofone of the Math Engines.
3. Fill the space above the pegs with the 21 green balls.
4. Point out to the students that the number of balls in the7 x 3 array of green balls is given by the number behindthe lower right ball in the array.
5. Put pegs in the first two columns of the ninth row of theother Math Engine.
6. Fill the space above the pegs with the 16 red balls.
7. Point out to the students that the number of balls in the2 x 8 array of red balls is given by the number behind thelower right ball in the array.
8. Ask the students if they could look at the multiplicationtable and find a rectangular array of balls that would have15 balls.
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Session 3. Inequality
How do we define inequality for fractions? When we are dealing with countingnumbers or integers the idea of inequality is fairly straightforward. We can look at 4and 6 or 4 and -3 and tell fairly easily if they are not equal. We also know for twonumbers a and b that a<b, a=b, or a>b. Can we figure out a way to extend this situationto fractions? Given two fractions a/b and c/d, do we have a/b<c/d, a/b=c/d, or a/b>c/d.In this session, we will look at a way to compare two fractions.
We want to compare two fractions. Let’s take two fractions and see if we canfigure a way to decide when one is bigger than the other. There are two cases that wehave to consider, the denominators are the same and the denominators are different.
Let’s dispense with the easy case first. Take anexample, say 2/5 and 3/5. We can look at the rectanglesrepresenting the two fractions. Since the denominatorsare the same, we are using the same rectangle for bothfractions. We can see that 3/5 has 1 more fifth than 2/5,so 3/5 is greater than 2/5, 3/5>2/5, and 2/5 is less than3/5, 2/5<3/5.
If we can figure out inequality when the denominatorsare equal, we have a way of approaching the idea ofinequality for two fractions with unlike denominators. If wecan take two fractions and find equivalent fractions to thetwo original fractions which have the same denominator,then we can compare the two fractions. We can ask whatfigure involves fifths and sevenths. Let’s look at a 5x7rectangle.
If we take a 5x7 rectangle, then we can representboth fifths and sevenths. Each row of the rectangle is 1/5 ofthe rectangle. Two rows of the rectangle are 2/5. If we lookat the columns, we are dealing with sevenths. Each columnof the 5x7 rectangle is 1/7. Hence, three columns of therectangle are 3/7. We now have 2/5 and 3/7 represented inthe same figure and we can compare them as before.
3_5
2_5
<
2_5
3_7
3_7
2_5
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15__35
<
2_5
2_5
14__35
= =
3_7=
3_7
=
We have shown in the representation above that 2/5 < 3/7. You may note that theonly math calculation knowledge that we needed was the ability to count to 35. We areable to speed the process if we do know the multiplication table.
We have worked using the multiplication table withequivalent fractions. Is there some way we can use it tofigure out inequalities? Let’s take two fractions and seewhether we can use the table to order them. 4/7 and 5/9seem like good candidates. We have marked 4/7 and 5/9off on the table to the right. Earlier in the session weformed a rectangle to compare two fractions, so let’scomplete a rectangle that encompasses the two fractions.
The smallest rectangle that contains both the fractionsis shown in the figure on the left. We have extended thetwo fractions to show the fractions that are equivalent tothem, namely 36/63 and 35/63, which have a commondenominator. We are now in a position to compare the twofractions. Since 36/63 and 35/63 have a commondenominator and 36 > 35, we know that 36/63 > 35/63.But 4/7 = 36/63 and 5/9 = 35/63.So, 4/7 = 36/63 > 35/63 = 5/9.
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== > 35__63
4_7
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1
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1/5
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Let’s look at fifths and sevenths again. We will usepart of the multiplication table to order all the proper fifthsand sevenths. If we arrange the multiples of 5 and 7 inascending order we have the following:
5 < 7 < 10 < 14 < 15 < 20 < 21 < 25 < 28 < 30.Hence,5/35 < 7/35 < 10/35 < 14/35 < 15/35 < 20/35 < 21/35 < 25/35 < 28/35 < 30/35.So, 1/7 < 1/5 < 2/7 < 2/5 < 3/7 < 4/7 < 3/5 < 5/7 < 4/5 < 6/7.
We finish this section with an ordering of proper fourths and sixths using themultiplication table. This ordering differs from the previous one because some of thefractions are equivalent.
4 < 6 < 8 < 12 = 12 < 16 < 18 < 20.Hence,4/24 < 6/24 < 8/24 < 12/24 = 12/24 < 16/24 < 18/24 < 20/24.So, 1/6 < 1/4 < 2/6 < 2/4 = 3/6 < 4/6 < 3/4 < 5/6.
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Worksheet 3.11. Show:
2. Write the fraction inequality that is shown by the diagram.Place a fraction in the and < or > in the .
<2_6
5_6<1_
32_3
>3_4
2_4
_ _ _ _
_
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Worksheet 3.21. Compare and explain:
2. Compare and explain:
< or = or >
3_4
2_3
< or = or >
3_4
7_9
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Worksheet 3.3
1. Use this part of the multiplication table to show whichfraction is smaller 2/5 or 3/8.
Explain your reasoning:
30 25 20 15 10 5
24 20 16 12 8 4
18 15 12 9 6 3
12 10 8 6 4 2
6 5 4 3 2 1
40 35
32 28
24 21
16 14
8 7
< or = or >
3_8
2_5
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Worksheet 3.4
1. Use the partial multiplication table to list the proper fourths andninths in ascending order:
Place a fraction in the and < or = in the .
Explain your reasoning.
24 20 16 12 8 4
18 15 12 9 6 3
12 10 8 6 4 2
6 5 4 3 2 1
36 32 28
27 24 21
18 16 14
9 8 7 1/4
19
29
39
49
59
69
79
89
99
2/4
3/4
4/4
_ _ _ _ _ _
_ _ _ _ _
_
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Lesson 3.1
Objective: To compare two fractionsMaterials: Math Engine, 14 blue balls, 15 green balls, pane
1. Put a pane that has two 5 x 7 rectangles onto the MathEngine.
2. Put pegs and green and blue balls so that there are greenballs in 3/7 of the first column in the first rectangle andblue balls in 2/5 of the first row in the second rectangle.
3. Ask the students which do they think is the largerfraction. Say that we can compare the fractions if they arefractions of the same unit rectangle.
4. Put pegs and balls so that 3/7 is represented by theequivalent fraction 15/35 and 2/5 by 14/35.
5. Ask the students which looks like the larger fractionnow.
6. Move the pegs and the balls in the side containing thesevenths. Moving the balls in the rectangle does not changethe fraction 3/7 or 15/35.
7. Ask the students which is the larger fraction. Since theunit rectangles are the same size and there are more greenballs by one, 3/7 > 2/5.
3 7__ 2
5__
= = 14 35__ 3
7__ 15
35__ 2
5__
>= = 14 35__ 3
7__ 15
35__ 2
5__
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Session 4. Adding (Subtracting) Fractionswith Like Denominators
In mathematics, when one does something new, it is helpful if the new action oridea can be linked to an idea or action one already knows. We are familiar with addingand subtracting whole numbers. Adding and subtracting fractions should be anextension of adding and subtracting whole numbers. Each whole number can be viewedas a fraction, for example, 2 = 6/3. When we add whole numbers by the rules foradding fractions, we should still get the same answer. 4/2 + 6/3 had better be 4.
When we deal with adding fractions with like denominators, we are essentiallyadding whole numbers. In this session, we will see that adding fractions is the same asadding whole numbers except that we change the picture just a bit.
We start by looking at what it means to add two numbers, say 1and 3. We can view this as 1 ball and 3 balls in the Math Engine andhave the picture look like the diagram on the right. Each ball is countedas one.
Let’s consider the problem 1/5 + 3/5 = 4/5. If we draw adiagram of this using balls we have the picture at the right. The firstcolumn has 1 ball out of a possible 5 in the unit. The middle columnhas 3 balls out of possible 5 in the unit. If we add them together weget the last column with 4 out of the possible 5. The diagram is thesame as the diagram for addition of whole numbers with respect to theballs. The only difference is the lines defining the units. In this casewe have a unit with 5 balls.
If we wanted to consider the problem 1 + 3 = 4 in terms offractions, we could redo the first diagram and make it clear that eachball counts as a whole unit and the unit is divided into one part. Thiswould change the way we look at the problem to 1/1 + 3/1 = 4/1. Thisis also a way to view whole numbers as fraction. 1 corresponds to1/1, 2 corresponds to 2/1, etc. We can see from the diagram on the leftthat if 1 + 3 = 4, then 1/1 +3/1 = 4/1.
1 + =3 4
15
+ =35
45
11
+ =31
41
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We almost forgot to look at the structure of the new words we have encountered,namely, add and subtract. The word add comes from the Latin ad to and dare to give.Hence, to add is to give to. Subtract also comes from the Latin sub under and trahere todraw, hence to draw under. It could also come from the Latin subtrahere to draw awayunderneath.
We now turn our attention to subtracting one fraction fromanother when they both have the same denominator. This follows thesame line of reasoning as the addition with like denominators. Wewill do one example to demonstrate the similarity.
Let’s do 7/9 - 5/9 and find the difference. We will show it assteps on a portion of the Math Engine as shown in the diagram on theright.. In the first column we show 7/9 with dark blue balls. In themiddle column, we lighten the color of the 5 balls we will subtract. Inthe last column, we show the result of removing the 5 balls, the resultof drawing away underneath the 5 light blue balls.
Since this is a fairly easy seesion to understand, we will look at one more ideaabout fractions. One could say that we are looking at fraction etiquette. The idea offraction is to break apart and take a piece. What if we take all the pieces or more thanall the pieces. This is not “proper.” We are thinking of fractions like 3/3 or 5/3. Howcan consider these. If we have 2 objects each broken into 3 pieces, we could take 4 ofthe six pieces. Now each piece is a third or 1/3, so the 4 pieces are 4/3. This couldcome up if we add 2/3 and 2/3. Let’s look at it from the point of view of the balls.
If the numerator is a larger number than the denominator, then this is considered“improper.” We then form a “mixed” number consisting of a whole number and a“proper” fraction. Mathematicians like to push the envelope and extend the conceptsbut they can still feel uncomfortable.
79
- =59
29
1_314
3=+ _2_3
2_3
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Worksheet 4.11.Do the indicated operation.
Show diagrams and explain.
2.Do the indicated operation.
Show diagrams and explain.
=+43_ 2_
3
=-94_ 3_
4
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Lesson 4.1
Objective: To add two fractions with a like denominatorMaterials: Math Engine, 4 blue balls, 3 green balls, 3 pegs, 1 pane
1. Put a pane marked with units containing 8 balls on theMath Engine.
2. Place pegs in the first and third columns to hold 4 and 3balls respectively.
3. Put 4 blue balls in the first column and 3 green balls inthe third column.
4. Place a peg in the second column at the bottom of therectangle containing 8 balls.
5. Put the blue balls and then the green balls in the secondcolumn.
6. Ask the students why we can just add the balls withoutany problems. (Every ball represents 1/8 of a unit.)
7. The answer we get is 7/8.
+48_ 3
8_
+48_ 3
8_ = 7
8_
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Session 5. Adding (Subtracting) Fractionswith Unlike Denominators
1/6 + 3/4 = ?. If every student could do this problem, understand why it works,and be able to explain it, how happy all teachers would be. This is the type of problemthat we will tackle in this session.
What do we know about adding fractions? We know that we can just add thenumerators if the denominators are the same. We also know how to find equivalentfractions by multiplying the numerators and denominators by the same number. We willcombine these two ideas using the Math Engine and add fractions. We will also seehow finding the lowest common denominator is represented using the balls.
We have seen, in the session on inequality, how we use rectangles to find figureson which we can represent 2 different denominators. Let’s look at 1/3 and 3/5. We willuse a 3 x 5 rectangle to deal with thirds and fifths. Each row is a third and each columnis a fifth.
We will now tackle subtraction of two fractions with unlike denominators. This isthe inverse of addition just as with the natural numbers. With addition we combine twoclumps of balls and count the result. With subtraction we start with a clump and takesome away. We use the same techniques as we do with addition. We can subtract if wehave common denominators so we use rectangles to find a common denominator.
=+1_3
3_5
1_3
3_5+
= +5__15
9__15
= __1415
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What have we done? We started with the simplest representation of the fractions3/5 and 1/3, 5 x 1 and a 1 x 3 rectangles. Since these two rectangles are different sizes,they are incompatible for adding or subtracting. We must find a rectangle which canrepresent 1/5 and 1/3 at the same time. If we look at the 5 x 3, we see that each columnis a fifth and each row is a third. We then change our point of view and consider eachball as 1/35 of the whole rectangle. We now have fractions which are equivalentfractions but which have a common denominator, 35. We can now add or subtract thetwo fractions to get an answer.
We turn our attention to the mysterious least common denominator. What in theworld is it and how can we show what it is? We have rules about finding the greatestcommon factor and dividing the product of the two denominators by it to obtain theleast common denominator. What does it all mean? What does it have to do withanything?
Let’s take the smallest two numbers that will give us a meaningful look at agreatest common divisor, namely 4 and 6. we will look at the sum 1/6 + 3/4. We knowby all the rules drummed into us that 4 x 6 = 24 is a common denominator, but 12 is theLEAST common denominator because 2 is the GREATEST common factor of 4 and 6and 24/2 = 12.
=- -= -
1_3
3_5
1_35__
15
3_59__
15
= __ 415
= __ 415
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We start with the two fractions, 1/6 and 3/4.Observe that there is no easy or obvious way to addthese fractions. We must change the way we look atthe fractions. We must find equivalent ways to lookat the fractions so that the fractions can be combined.
The way that we have done it in thepast is to form a rectangle whose sides arethe two denominators in length. In this case,we multiply 4 by 6 to get a 4 x 6 rectangle.If we look at the rows and columns of thisrectangle, we have the representation of 4thsand 6ths in one object.
Here is where we diverge from our previous technique and add one step to theprocess. We notice that 2 is a common divisor of 4 and 6. This means that we candivide the 4 x 6 rectangle into 2 pieces in two different ways. Each of the pieces has 12balls in it.
Even though the new rectangles are different shapes, they have the same numberof balls in them.
We are in a position to add the two fractions. We have denoted or named anobject from which we can easily take 1/6 and 3/4; we have found a commondenominator. Indeed, we have found the smallest object, the least commondenominator, for which this is possible. Let’s see what the addition looks like.
1_6
3_4
1_6
3_4
1_6
3_4
3_4
__ 912
1_6
__ 212
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We start with the two fractions and form a common rectangle containing both 4thsand 6ths.
We have not changed the size of the 4 x 6 rectangle before. We just used therectangle whose dimensions are the product of the denominators. Here, we havedivided by the greatest common factor of the two denominators and obtained thesmallest rectangle we could getting what will be the least common denominator.
We now take the equivalent fractions with a rectangle having 12, not 24, balls.The two rectangles are not the same shape but they contain the same number of balls.Each ball is one twelfth of its rectangle, so we can add the balls.
We can choose either of the two rectangles to represent the final sum.
= __1112
++ = 3_4
1_6
3_4
1_6
= + __ 912
__ 212
= + __ 3 4
__ 1 6
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Worksheet 5.1Add 1/4 and 2/5. Show all work. Explain each step.
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Worksheet 5.2Subtract 2/3 from 3/4. Show all work. Explain each step.
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Worksheet 5.3We want to add 1/6 and 4/9.
Picture 1/6 and 4/9 with rectangles showing equivalent fractions inthe least common denominator.
Add the 1/6 and 4/9 and represent the sum in a rectangle below.
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Lesson 5.1
Objective: To add two fractions with unlike denominatorsMaterials: Math Engine, 5 green balls, 9 blue balls, 8 pegs, 1 pane
1. Place pane with 5 x 3 rectangles tiling the Math Engine.
2. Place 1 green ball in the first column representing 1/3.
3. Place 3 blue balls in the first row representing 3/5.
4. Point out to the students that the green and blue ballsrepresent different fractions, the green, thirds and the blue,fifths.
5. Ask the students how we can change the fractions sothat we can add them.
6. Put pegs in the second row of the upper left rectangle toform 1/3 of the 15 ball unit rectangle.
7. Put pegs in the first three columns of the fourth row ofthe upper right rectangle to form 3/5 of the 15 ball unitrectangle.
8. Since each ball now represents one fifteenth, we canadd them all.
9. Place pegs along the bottom of the upper right unitrectangle and move the green balls to add the fractions.(The place the green balls came from is indicated.)
10. We have added the two fractions and find that theanswer is 14/15.
+13_ 3
5_
+13_ 3
5_ + 5
15 915__=
+ 515
915__ =14
15_
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Session 6. Multiplying Fractions
The rule for multiplying is the easiest of the rules for computing with fractions.To multiply two fractions, multiply their numerators and denominators. It is so simplethat one can overlook what it means and how it developed. We will look closely atmultiplication and see why it is appropriate to use the rule.
Multiply and multiplex come from similar roots. The Latin multus, many, andplicare, to fold, give us the image of many fold. In math then, to multiply would be tohave many fold copies of something.
With the Math Engine, it is easy to see whetheryou have multiple copies of the same number. As wesee in the first diagram on the right, there are 4 balls inthe first column. We can also see that there are 3columns of 4 balls in the second diagram, so we saythere are 4 balls taken 3 times or 3 times 4 balls. It isgenerally written 3 x 4. We can also think of 3 x 4 as taking 3 copies of the collectionof 4 balls or 3 of the 4 balls. So, 3 of something is the same as 3 times something. Notethat we form a rectangular array of balls when we line them up this way. Can we find asimilar array if we are multiplying fractions? We shall see that this is indeed the case.
There are at least two cases we must consider to make the transition frommultiplying whole numbers to multiplying fractions. The easier case is multiplyingsomething a whole number times. The more difficult case is multiplying something afractional amount of times. For example, we must decide what we mean by takingsomething 2/3 times.
The idea of taking a fraction a whole number oftimes is really no different from taking any number ofthings a whole number times. We are simply decidingto make a certain whole number copies of an object. Itis still the same rectangular ball area, the onlydifference is setting the unit. The diagram at the rightshows what happen when we choose the unit so thateach ball represents 1/5. If we think of 3 as 3/1 then the product would be 12/5, whichis the product, as we can see from the diagram.
4 3 x 4
4/5 3 x 4/5
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4 3 x 4
The second case is a little more delicate. What does it mean to take 2/3 of anobject or have an object 2/3 times.
Let’s look at taking 3 copies of 4 objects a littlecloser. If we take 3 copies of the four balls in the firstdiagram on the right, we find that we have 3 green balls,3 blue balls, 3 yellow balls, and 3 red balls in thesecond diagram. We have taken 3 copies of the fourballs by taking 3 copies of each ball in the collection.This gives us a way of looking at taking a collection ofobjects or units a fractional number of times. If we want to take 1/2 of a collection ofunits, we can do so if we take one half of each one. It is impossible to chop up the ballsbut this is not necessary. We are back to defining what we mean by the unit. Let’s lookat a way to take 4 units 1/2 times and 4 units 2/3 times. We start with 1/2.
If we want to take 1/2 of 4 units, then we want to take 1/2 of each unit. To take1/2 of a unit, we must have a good number of balls in a unit, namely a number that iseasily divisible by 2. 2 seems like the most appropriate number to take. In the firstrepresentation of 4 in the diagram above, we take a unit having 2 balls. We can easilytake 1/2 of each unit, as shown in the middle diagram above. Note that thisconfiguration is a rectangular array. In the third diagram above, we have moved theballs to make counting easier. We can see that 1/2 x 4 = 2 in this representation.
If we want to multiply 4 by 2/3 then each unit in the first diagram is divided into3 parts so that each ball is 1/3. We take 2 of them from each unit to get the middlediagram. Note that it is a rectangular array of balls. We have rearranged the balls in thelast diagram to make counting the units easier. We have 2/3 x 4 = 8/3 = 2 2/3. Alsonote that the multiplication rule holds if we consider the problem 2/3 x 4/1.
4 1/2 x 4 1/2 x 4 = 2
4 2/3 x 4 2/3 x 4 = 2 2/3
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We finish this session by multiplying a fraction by a fraction. Let’s take anexample. The general case would be similar. How about 4/5 x 6/7?
Let’s go over the entire process. We are trying to find 6/7, 4/5 times. This is thesame as 4/5 times 6/7 or 4/5 of 6/7. We first notice that we are dealing with fifths andsevenths. The first number we consider is 6/7. Since we are dealing with sevenths weset the unit on the far left to have 7 balls. 6/7 is marked out. We are also dealing withfifths. Each ball in the calculation will have to be split into five equal parts, so weextend the unit making each ball into five balls. The unit is now a 5 x 7 rectangle with35 balls in it. Each ball is now a thirty-fifth. We take 4 of the 5 balls that each seventhhas been broken into. We now have 4 x 6 balls in the product. Since each ball is 1/35,the product 4/5 x 6/7 = 24 / 35. Please note that using this definition of fractionmultiplication we have the product of the numerators is the numerator of the productand the product of the denominators is the denominator of the product. Also notice thatthe product is represented by a rectangular array.
4 / 5
6_7
6_7
4_5 x
6_7
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7_4
3_5 = __21
20x
18 15 12 9 6 3
12 10 8 6 4 2
21
14
6 5 4 3 2 1 7
6 5 4 3 2 1 7
3
2
1
42 35 28 21 14 7
36 30 24 18 12 6
30 25 20 15 10 5
24 20 16 12 8 4
60 50 40 30 20 10
54 45 36 27 18 9
48 40 32 24 16 8
70 63 56 49
60 54 48 42
50 45 40 35
40 36 32 28
30 27 24
20 18 16
100 90 80 70
90 81 72 63
80 72 64 56
10 9 8
10 9 8
7
6
5
4
10
9
8
We conclude this session by looking at the multiplication of two fractions, oneproper and one improper, on the Math Engine with the multiplication pane in place. Wecan see that the number in the lower right hand corner of the upper left unit is thedenominator of the product. The numerator is the number under the lower right ball inthe rectangular array representing the product. We note, once again, to multiply twofractions, multiply the numerators and multiply the denominators.
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Worksheet 6.1
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8 9 101 2 3 4 5 6 7
6
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5/9 x 7/2
Show the following:
3/8 x 9/2
Show what times 9/4 is 45/28.Show 1/9 x 1/5
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Lesson 6.1
Objective: To multiply one fraction by anotherMaterials: Math Engine, 2 green balls, 10 red balls, 6 blue balls, 2 panes
We will multiply 2/5 by 2/3, but first we have toreview what it means to multiply by a whole number.
1. Place 5 red balls in the first two columns of the MathEngine.
2. Tell the students that 2 x 5 is the same as 5 things taken2 times. That is, we take two copies of each ball and counthow many we have. Place 5 red balls in the second columnand point out we have 2x5 or 10 red balls.
3. Put on 1 x 5 rectangular grid and 2 green balls in thefirst column.
4. Ask the students how we can take 2/3 of those twoballs. Point out that multiplying by 2 means take 2 of eachball, so we should take 2/3 of each ball.
5. Have the students note that we can take a third of eachball if we change to an equivalent fraction where each ball
becomes three balls. We will change the color to blue and say that each green ball isworth 3 blue balls.
6. Put on a 3 x 5 rectangular grid and put 6 blue balls inthe upper left rectangle as shown to the right. We nowhave the blue balls as 2/5 of the 3 x 5 unit rectangle.
7. Take 2/3 of each row of the blue balls. The unitrectangle is 3 x 5 or 15 balls and the number of balls wehave is 2 x 2 or 4 balls, so the answer is 4/15.
2 x 5
25_
2 / 325_
23_ 2
5_ 2 x 2
3 x 5___x = = 4
15_
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Session 7. Canceling Common Factors WhileMultiplying Fractions
Canceling common factors when multiplying fractions can simplify calculationsand is an important concept to take into algebra. Symbolically, the rule for multiplyingfractions, commutativity of mulpltiplication, and rules of equivalent fractions are thejustification for canceling common factors when multiplying fractions. Is there aconcrete representation of canceling common factors on the Math Engine? Yes! We willexplore this representation in session 7.
Canceling common factors on the Math Engine when multiplying fractionsinvolves going back to the original definition of fraction and monkeying with the unit.Let’s take a closer look at the process of multiplying fractions and separate out some ofits important features.We will look first at 3/5 x 5/8 which is pictured on the left in thediagram below. We are going to concentrate on the units so they have been emphasizedand the ball color has been muted. A fraction has two basic numbers, the number ofelements the unit has been broken into (denominator) and the number of these units wehave taken (numerator). Let’s look more closely at the unit and play with it. In thediagrams below, we have taken the problem 3/5 x 5/8 and represented it on the MathEngine with the multiplication pane and a pane showing the unit rectangles. We rotatethe unit pane around the blue diagonal line and put it back on the Math Engine. Wehave NOT changed the fraction, but now it is in the form we have seen before forequivalent fractions. We have canceled out the common factor 5.
1
1
1
100
90
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35
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45
50
36
42
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54
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4 5 6 7
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8 9 104 5 6 7
6
7
8
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10
2
3
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5
3_5
5_8x = 3_
85_5
3_8x =
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We will next do a fraction multiplication wherethe factors are more deeply buried in the problem andnot quite as obvious as before. The multiplicationproblem 3/8 x 4/9 is pictured to the right. We will flipthe unit, look for different equivalences, and thenmultiply the reduced fractions.
We have rotated the unit pane around the diagonal inthe diagram to the left. This has interchanged thedenominators. Instead of 3/8 x 4/9, we have 3/9 x 4/8.This allows us to find the equivalences 3/9 = 1/3 and 4/8 = 1/2 by splitting up the unit into a different pattern.
We can regroup the balls into a pattern thatsimplifies the multiplication problem to 1/3 x 1/2 asshown in the diagram to the right. We have eliminatedtwo factors, 3 and 4, in the numerator and denominatormaking the problem a much easier one.
x 4_8
3_9
100
90
30
35
40
45
50
36
42
48
54
60
42
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63
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80
18
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81
90
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8 9
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2 3 4 5 6 7
2
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8 9 101 2 3 4 5 6 7
6
7
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6 2
2 1
31 2
1
2
x = 1_6
1_3
1_2
100
90
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8 9 101 2 3 4 5 6 7
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x3_8
4_9
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In the last part of this session we will look at the important mathematical conceptof the multiplicative inverse or reciprocal. Take a fraction. We want to find anotherfraction called the reciprocal so that, when we multiply the two, the product is 1. Let’spick a typical ordinary fraction like 5/7. What fraction do we have to multiply 5/7 by toget a product of 1? Considering what we just did with canceling we must have a factorof 7 in the numerator and a factor of 5 in the denominator. 7/5 looks like a logicalguess. We will see why this works.
The figure on the left shows the product 7/5 x 5/7. If we look at the unit, we findthat we have a 7 x 5 rectangle. If we look at the balls, we find that we have a 5 x 7rectangular array. The two are not the same but have the same number of spaces forballs. If we rotate the unit rectangular grid around the green diagonal, as we did before,the unit rectangle and rectangular grid align with each other so that the rectangulararray fills the unit rectangle exactly. This is the same as saying that the balls representthe number 1. Thus, 7/5 x 5/7 = 1. So 7/5 and 5/7 are reciprocals or multiplicativeinverses of each other.
x7 x = 17_7
5_5
_5
5_7
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Worksheet 7.11. Cancel common factors then multiply 5/6 x 4/7. Show a representation of the original problem and the solution. Explain what you are doing.
2. What is the reciprocal of 4/5? Show that it is the reciprocal. Explain what you are doing.
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Session 8. Dividing Fractions
Of all the operations, division of fractions is the most puzzling. Multiplying issimple, multiply numerators and denominators. Addition and subtraction arecomplicated and convoluted. But division is simple but strange. Invert and multiply.Why? We will see why in this session.
The Latin does not give us much help. Dividere is to divide. The concept is oldand important.
We must review what it means to divide wholenumbers since we will see that the operation carries over tofractions without modification. If we have 17 balls, asshown on the right, and want to divide them among 7persons, we would give one ball to each person. If therewere enough, we would give each person another ball....and so on. Each time we give a ball to each person, wesubtract 7 balls from the number we have to give. Dividingby 7 is asking how many groups of 7 balls are there in 16.
We can show this easily in the Math Engine as seenin the diagram to the left. Each column has 7 balls in it. Weare asking how many columns of 7 balls there are. We canthink of the answer as 2 with a remainder of 3. We can alsothink of the answer in terms of fractions. Since we areasking how many sevens there are in 17, we can view theproblem as one in which the unit has changed. Instead of aunit being one ball, we now have the unit consisting ofseven balls. Each ball is 1/7 of a unit. Dividing 17 by 7gives us an answer of 2 3/7. We will use this way of
looking at division when we divide one fraction by another.
We will study division by fractions in two steps. First we will divide wholenumbers by fractions. We will then consider dividing one fraction by another. We aregoing to start by taking the whole number 3 and dividing it successively by 1/2, 2/2,3/2, 4/2, 5/2, 6/2, 7/2, and 8/2. We will display the results all together so that we cansee the changes that result as we alter the denominator. Since we will be dividing 3 bymultiples of 1/2 we will display 3 in units having 2 balls, that is each ball is 1/2 a unit.We will then do the same thing but dividing by multiples of 1/3. With thirds we have tostart with a unit consisting of three balls.
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3
3/ = =14_2
6_4
2_4 3/ = =15_
26_5
1_5
3/ = =61_2
6_1 3/ = =32_
26_2
3/ = =23_2
6_3
3/ =8_2
6_83/ =7_
26_73/ = =16_
26_6
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3
3/ = =33_3
9_3
3/ = =16_3
9_6
3_6
3/ = =91_3
9_1
3/ = =24_3
9_4
1_4
3/ = =17_3
9_7
2_7
3/ = =42_3
9_2
1_2
3/ = =15_3
9_5
4_5
3/ = =18_3
9_8
1_8
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Let’s take a fraction and see what it would take todivide it by another fraction. Take 3/4 and divide by 5/6. Ifwe think about dividing 17 by 7, we had to ask how many7 unit collections we could take out of 17 units. The 7 and17 must be counted in the same units. We did the samething when we divided 3 by 4/3. We had 3 and 4/3 repre-sented in the same unit, namely a unit with 3 balls. Whatfigure allows us to deal simultaneously with 3/4 and 5/6?That’s right, a rectangle! In particular, a 4 x 6 or a 6 x4rectangle.
We first have to set up the 3/4 of a unit. We have donethis before. We take a 6 x 4 and fill 3 of its columns withballs. This represents the fraction 3/4.
We must now ask how many 5/6 units we can fillwith this 3/4 unit of balls. We have set up a grid of 4 x 5rectangles which has been superimposed on the 4 x 6 gridof rectangles that was our original unit. the 4 x 5 rectangleis 5/6 of the 4 x 6 rectangle. In the picture on the right, wehave the new 4 x 5 unit rectangular grid on top of the oldrectangular 4 x 6 unit grid and the 3/4 unit rectangular ballarray.
We move the balls so that they are all in one of thenew units. This allows us to calculate the answer to thedivision problem. The new unit can contain 4 x 5 or 20balls. We have 3 x 6 or 18 balls in the 3/4 so 18/20 is ananswer to dividing 3/4 by 5/6.
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We said at the beginning of the session that we would show that invert andmultiply follows from dividing a fraction by another fraction with the Math Engine.Let’s look more closely at the representation for division of filling rectangular areaswith rectangular arrays of balls. We will use the problem we just calculated, 3/4divided be 5/6. We have modified the diagram by not putting the whole grid of the newunit. We can see the old 4 x 6 rectangular unit grid in black. 3/4 represented by greenballs filling 3 out of a possible 4 columns in the old unit. The upper left rectangle in thenew rectangular grid indicating 5/6, marked in blue. To divide 3/4 by 5/6 we end updividing the number of green balls, which is 3 x 6, by the number of balls the new unitcan hold, which is 4 x 5. But this is the result we get if we invert 5/6 and multiply it by3/4.
3x6
4 x 5
=5_6 =3_
4/ 3 x 6 ____4 x 5
6_5
3_4 x
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18 12
15 10
12 8
9 6
6 4
3 2
24
20
16
12
8
4
6
5
4
3
2
1
We are going to add the multiplication table to theproblem 3/4 divided by 5/6 to make getting an answereasier. In the diagram to the right we have taken theoriginal unit 4 x 6 rectangle and placed the multiplicationtable on top. The green balls represent 3/4 and the bluerectangle represents 5/6. We can see that we can read theanswer, 18/20, right off the multiplication table.
We conclude this session by looking at the representations for various fractiondivision problems.
35 28 21 14 7
30
25
20
15
10
5
24 18 12 6
20 15 10 5
16 12 8 4
12 9 6 3
8 6 4 2
4 3 2 1
=2_7 =3_
5/ 3 x 7 ____5 x 2
7_2
3_5 x = 21__
10
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/ x== =6_5
7_4
7 x 5 ____4 x 6
5_6
7_4
35__24
36 30
30 25
24 20
18 15
12 10
6 5
42
35
28
21
14
7
24 18 12 6
20 15 10 5
16 12 8 4
12 9 6 3
8 6 4 2
4 3 2 1
=111__24
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Worksheet 8.11. Calculate: 2/3 divided by 1/2. Show allwork and explain.
1. Calculate: 1/3 divided by 3/4. Show allwork and explain.
3. Calculate: 4/3 divided by 4/3. Show allwork and explain.
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Lesson 8.1
Objective: To divide one fraction by anotherMaterials: Math engine, 18 green balls, 4 pegs, 2 panes
1. Put on pane with black 4x6 rectangular grid.
2. Place pegs in the first three holes on the second row ofholes in the Math Engine.
3. Put in 3 green balls, one above each of the pegs.
4. Tell the students that this represents the fraction 3/4,because we have 3 of the places filled in the first four.
5. Tell the students that we want to divide this fraction, 3/4, by the fraction 5/6, i.e.weare asking how many 5/6 are there in 3/4. We have to change the machine so that itrepresents fourths and sixths. If we consider the 4x6 rectangle, we see each row is asixth and each column is a fourth.
6. Add enough green balls to fill 3/4 of the top black4x6 unit rectangle, namely, 3 columns.
7. Place the pane with the red 4x5 unit rectangle on theMath Engine.
8. Tell the students that the new unit is 5/6 of the oldunit.
9. Rearrange the green balls so that they fill the redrectangles, starting at the top left.
10. Point out that there are 20 places for balls in each newunit rectangle and we have 18 balls, so the answer is 18/20.
3 / 4
3 / 4
56_
34_ 5
6_ 3 x 6
4 x 5___/ = = 18
20_
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Session 9. Ratio and Proportion
Ratio and proportion, Latin ratio et proportio, reckoning and analogy, areintimately related to fractions and incredibly important in real life. A ratio is a fractionand a proportion is an equivalence of fractions. When we say that 1/2 = 2/4, we arestating a proportion.
How do we encounter proportions in real life? We deal with ratio and proportionwhenever we encounter a rate.
Suppose we are traveling in a car on a trip andwe find that we have gone 525 miles in 10 hours and 24minutes. If we have another 326 miles to go and 7hours before we have to be at our destination, can weget there on time if we maintain our speed or do wehave to go faster?
A recipe that feeds 4 people calls for 1/4teaspoon of a spice. If we are having a dinnerparty for 7 couples, how much spice will weneed?
I can treat 2.5 acres of a crop with a quartof pesticide. How much pesticide will I need if I needto treat 18.5 acres?
These are just three examples. We could come up with scads more, anywhere arate would apply.
How do we solve a proportion? How does it appear on the Math Enginetm? Wewill start with a simple proportion.
= 14__?
___ 4 1/4
= ___ ? 18.5
___ 1 2.5
= ?_7
___52510.4
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1_2 = ?_
3Too difficult?
We might be tempted to start with theproportion, 1 is to 2 as what is to 3 because thenumbers are small and easy to work with. I think thiswould be a mistake, for we need some complexity tosee what is going on. We will come back to thisexample after we do an easier proportion problem.
We will start with an easier problem, namely, 2 is to 5 aswhat is to 7. How can we tackle this problem on the MathEnginetm; what does the problem look using balls?
We are dealing with fractions. The fractions are fifthsand sevenths. If we were working with equivalences oradding fractions, we would be looking for a structure thatwould handle fifths and sevenths. What about a 7 x 5rectangle (or a 5 x 7 rectangle)? We will use just part of theMath Engine, so we will just display the part that isinvolved and not picture the whole face.
The rows are fifths and the columns are sevenths. Inthe diagram to the right, the green balls are 2/5 of therectangle. If we move them around, they still are 2/5 of therectangle.
Let’s put the green balls in columns, as inthe diagram on the left. We have not added orsubtracted any balls, so the balls are 2/5 of therectangle. But now we can look at the balls assevenths and find we have 24/5 sevenths.
2_5 = ?_
7Easier?
2_5
2_5
24/5_7
2_5 = 24/5___
7
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Let’s go back and see how this relates to the solution of the proposition in theusual algebraic way.
What are we doing with the balls? We take a shape that can be used to representboth fifths and sevenths. We then move the shape to change 2/5 to the appropriatenumber of sevenths. If we look at the pictures and think of them in terms of whole
numbers we observe something interesting. The change in the pictures represents thealgebraic solution of the proportion. Take 2, multiply by 7, then divide by 5.
y_7
__ 2x75
=
=
2_5
y
= y24/5
2 2x72x7___5
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We are now in a position to solve the proportion, 1 is to 2 as what is to 3. Let’slook at a 3 x 2 rectangle.
What about the proposition, 1 is to 3 as what is to 2?
We can apply the idea of a proportion to percentage problems. First, let’s look atthe derivation of percent. In Latin, centum is 100. Percent is per centum or per 100. Sowe see that 15% is 15 percent is 15 per hundred is 15/100. 15% of 30 is 15/100 x 30. If15% x 30 = y, then this is the same as 15/100=y/30. We have a proportion which wehave learned how to solve.
1_2 = 11/2___
3
1_3 = 2/3___
2
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Worksheet 9.11. Take the proportion and draw the picture representing it. Whatother proportions can you form from this one. Explain.
3_4 = _
6
2. Solve the proportion and put the answer in the rectangle.Explain.
2_6 = _
93
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Worksheet 9.21. Take the fact that 4 x 9 = 6 x 6, as shownin the multiplication table on the right andgenerate as many proportions as you can.Picture and explain. 30 36 42
16
24
32
40
48
18
27
36
45
54
8 9
6 24 18 12
1
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35
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30 36 42
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8 9
6 24 18 12
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30 36 42
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8 9
6 24 18 12
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2 3 4 5 6 7
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30 36 42
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8 9
6 24 18 12
1
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2 3 4 5 6 7
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30 36 42
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8 9
6 24 18 12
1
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35
2 3 4 5 6 7
2
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9
8
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12
15 10
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Lesson 9.1
Objective: To work the proportion problem, 2/5 = x/7Materials: Math Engine, 14 blue balls, 8 pegs, 2 panes
1. Place pane on Math Engine with 1x5 unit rectangles.
2. Pace a peg and two green balls in the first column.
3. Tell the students that the balls represent the fraction2/5.
4. Ask them how many sevenths they think equal 2/5.
5. Tell them that we need to represent sevenths.
6. Replace the 1x5 unit rectangular grid with a 7x5 grid.
7. Put pegs and balls so that the green balls fill the toptwo rows in the 7x5 unit rectangle.
8. Point out to the students that the balls fill up 2/5 of theunit rectangle and will fill up 2/5 of the rectangle even ifwe move them.
9. Move the green balls so they fill columns in the 7x5 unit rectangle starting from theleft.
10. Point out to the students that there are 2 4/5 columnsthat are filled with green balls, so that 2 4/5 sevenths isthe same as 2/5.
25_
25_
25_
24/5
7__
2_5
____ 24/5
7 =
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Session 10. Wrap-up and Discussion
Discussion, Latin dis- apart and quatere to shake, consider and argue the prosand cons. What have we seen? We have the following:
1. A definition of fraction2. When fractions are really the same or equivalent3. When one fraction is less than another fraction4. How to add and subtract fractions5. How to multiply and divide fractions6. How to work with ratios and proportions
In this final seesion, I would like you to write a one-page paper exploring severalideas. I would like you to state anything that you saw in the first nine sessions whichwas new to you. We have all worked with fractions most of our lives. Was thereanything that you saw that put fractions in a new perspective? The second idea to writeabout is how you feel you could put what you have learned into use in your classroom.Is there anything that you have seen here that you would like to try with students,especially those who are not understanding the usual presentation? Finally, I would liketo have suggestions on how you would like to see the course changed. Which partswere useful, new and interesting? Which were ones you already knew and did not haveto go over? What hand-outs and blackline masters would you like to have and use inyour classroom?
You will write for half, (89/178), of an hour and then we will discuss what youhave written.
