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Updating Montessori Fractions October 24, 2009 2:00 - 3:30 p.m. Session 6 Garden Grove, CA by Joan A. Cotter, Ph.D. [email_address] Slides/handouts: ALabacus.com
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Fraction Chart At first ask child to remove and replace only first few rows. A linear model gives an overview and shows relationships.
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Fraction Stairs Notice what happens as the fractions get smaller and smaller. The curve is a hyperbola.
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Fraction Chart How many fourths in a whole? How many sixths? We use ordinal numbers, except for one-half, to name fractions.
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Fraction Chart What is more, 1/4 or 1/3? What is more, 1/9 or 1/10?
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Fraction Chart What is more, 1/4 or 1/3? What is more, 1/9 or 1/10?
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“ Fish Tank” Model 2 5 With this model, could you compare 2/5 and 1/4? Also, children will think fractions are two numbers, but they are one number just as 37 is one number.
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“ Words” Model This is fourths . This is thirds . The 1993 textbook using this model does not say the figures represent one. So fourths look like four. How do you compare fourths and thirds?
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“ Pie” Model Try to compare 4/5 and 5/6 with this model.
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“ Pie” Model Experts in visual literacy say that comparing quantities in pie charts is difficult because most people think linearly. It is easier to compare along a straight line than compare pie slices. askoxford.com Specialists also suggest refraining from using more than one pie chart for comparison. www.statcan.ca Even adults have difficulty with pie charts.
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“ Rounded Corner” Model The middle fractions are greater than the fractions at the ends! 1 3 1 3 1 3 1 4 1 4 1 4 1 4
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“ Color” Model 1 2 1 1 2 1 3 1 3 1 3 6 6 6 6 6 6 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 9 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 4 1 4 1 4 1 9 1 9 1 9 1 9 1 9 1 9 1 9 1 9 1 5 1 5 1 5 1 5 1 5 1 1 1 1 1 1 1 4 Notice how your eye tends to stay in the same row. Try, for example, to see how many eighths are in one half.
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Definition of a Fraction A fraction is part of a set or part of a whole. Textbook definition Another common meaning of fraction is fragment or a small part. 3 2 What about ?
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Definition of a Fraction An expression that indicates the quotient of two quantities. American Heritage Dictionary Do not use “over” as in 3 over 2. This definition is not appropriate for students, but it does emphasize the division aspect of fractions. “Over” does not indicate division.
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Definition of a Fraction or 3 ÷ 2. 1 1 1 1 1 1 2 1 2 1 2 1 2 3 2 means three s 1 2
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“ Missing 7ths & 9ths” Model Some curricula omit the sevenths and the ninths and add the twelfths. Look down the center. Fractions > 1/5 are always even numbers. 1 8 1 8 1 8 1 8 1 8 1 8 1 8 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 8
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“ Missing 7ths & 9ths” Model With the sevenths and the ninths omitted, the pattern is obscured. Math is the science of patterns.
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Concentrating on One Game A game to learn, for example, that 4 fourths and 8 eighths make a whole.
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Concentrating on One Game 5 3 When playing a memory game, the player must say what she is looking for before turning over the second card.
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Concentrating on One Game 3 5 2 5 Players use the fraction chart to find what they need. Don’t teach a rule.
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Skip Counting Patterns Nines Second row done backwards to see digits reversing. Also the digits in each number add to 9. 9 18 27 36 45 90 81 72 63 54
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Skip Counting Patterns 15 5 12 18 21 24 27 3 6 9 30 Threes 2 8 1 4 7 3 6 9 0 Threes have several patterns. First see 0, 1, 2, 3, . . . 9.
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Skip Counting Patterns 1 2 1 5 1 8 2 1 2 4 2 7 3 6 9 30 Threes The tens in each column are 0, 1, 2.
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Skip Counting Patterns 6 15 24 6 12 21 3 30 Threes The second row. [6] And the third row–the nines. Now add the digits in each number in the first row. [3] 18 27 9 18 27 9 12 21 3 30 15 24 6
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Skip Counting Patterns Sevens 28 35 42 49 56 63 7 14 21 70 Start in the upper right to see the 1, 2, 3 pattern. 8 9 7 0 5 6 4 2 3 1
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Skip Counting Memory Game The envelope contains 10 cards, each with one of the numbers listed. 7 14 21 28 35 42 49 56 63 70 A game for learning the multiples.
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Skip Counting Memory Game 7 14 21 28 35 42 49 56 63 70 Remove the cards from each envelope, shuffle slightly, and lay out face down. 8 16 24 32 40 48 56 64 72 80
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Skip Counting Memory Game 14 40 8 16 24 32 40 48 56 64 72 80 7 14 21 28 35 42 49 56 63 70 Players must collect the sets in order . Only one card is turned over per turn.
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Skip Counting Memory Game 8 16 24 32 40 48 56 64 72 80 8 8 56 7 7 14 21 28 35 42 49 56 63 70 Who needs 56? [both 7s and 8s] At least one card per game is a duplicate.
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Dividing Fractions One meaning is how many 2s in 6. 6 ÷ 2 = __ Relating to division.
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Dividing Fractions One meaning is how many 2s in 6. 6 ÷ 2 = __
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Dividing Fractions Since 12 is twice as much as 6, 12 ÷ 2 is twice as much as 6 ÷ 2. 12 ÷ 2 = __
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Dividing Fractions So 12 ÷ 2 = 2 x (6 ÷ 2) = 2 x 3 = 6. 12 ÷ 2 = __ Since 12 is twice as much as 6, 12 ÷ 2 is twice as much as 6 ÷ 2. 12 ÷ 2 = __ Notice that we are multiplying to solve the division problem.
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Dividing Fractions 9 ÷ 2 = __ Since 9 is one and one-half as much as 6, 9 ÷ 2 is one and one-half as much as 6 ÷ 2. So 9 ÷ 2 = 1 x (6 ÷ 2) = 1 x 3 = 4 . Let’s try 9 divided by 2. 1 2 1 2 1 2
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Dividing Fractions Since 9 is one and one-half as much as 6, 9 ÷ 2 is one and one-half as much as 6 ÷ 2. So 9 ÷ 2 = 1 x (6 ÷ 2) = 1 x 3 = 4 . 9 ÷ 2 = __ 1 2 1 2 1 2
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Dividing Fractions ÷ = 1 2 3 We’ll start by dividing 1 by various numbers. The last one can be thought of as how many 2/3s in 1 or half of 1 ÷ 1/3. 1 2 ÷ = 1 3 1
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Dividing Fractions ÷ = 1 2 1 2 ÷ = 1 3 1 3 Each white pair is a reciprocal, sometimes called a multiplicative inverse. When multiplied together, they equal 1. In the equation 6 ÷ 2 = 3 , 2 x 3 = 6. 1 3 1 4 ÷ = 1 3 ÷ = 1 4 3 4 ÷ = 1 4 3 ÷ = 2 3 1 3 2
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Dividing Fractions ÷ = 1 2 1 2 ÷ = 1 3 1 3 Sometimes textbooks put a 1 under a whole number to make it look like a fraction, but it’s really not necessary. 2 1 3 1 3 1 4 1 1 3 1 4 ÷ = 1 3 ÷ = 1 4 3 4 ÷ = 1 4 3 ÷ = 2 3 1 3 2
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Dividing Fractions ÷ = __ 5 To find Use the same principle for dividing by a fraction except use 1 as the reference, rather than 6. 2 3
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Dividing Fractions ÷ = 1 ÷ = __ 5 First find To find 2 3 3 2 2 3
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Dividing Fractions ÷ = 5 ÷ = __ 5 ÷ = 1 First find To find Then Does the answer make sense? About how many 2/3s are in 5? 2 3 = x = 3 2 1 2 5 7 2 3 2 3 3 2 5 2 3 ( 1 ) x ÷ 3 2
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Dividing Fractions ÷ = __ To find (Is the answer more or less than 1?) Another example: How many 3/4s are in 2/3. More or less than 1? 2 3 3 4
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Dividing Fractions ÷ = ÷ = 1 First find To find Then ÷ = __ The answer should be < 1 and it is. The extra step of dividing by 1 can later be omitted. 3 4 4 3 3 4 2 3 3 4 2 3 x ÷ ( 1 ) 3 4 2 3 = x = 4 3 8 9 2 3
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Dividing Fractions It’s ours to reason why We invert and multiply.
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Dividing Fractions It’s ours to reason why We invert and multiply. This presentation and handouts are available at ALabacus.com
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