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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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Fraction Chart Which is more, 3/4 or 4/5?

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Fraction Chart Which is more, 3/4 or 4/5? Which is more, 7/8 or 8/9?

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Fraction Chart The pattern of 1/2, 3/4, 4/5, 5/6, 6/7, 7/8, 8/9, 9/10.

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Fraction Chart How many fourths equal a half? Eighths? Sevenths?

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Fraction Chart How many fourths equal a half? Eighths? Sevenths?

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Fraction Chart What is half of a half?

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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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Fraction Chart

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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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Concentrating on One Game

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Concentrating on One Game What is needed with 3/8 to make 1? [5/8] 3 8

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Ruler Chart Especially useful for learning to read a ruler with inches.

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Ruler Chart Horizontal lines removed.

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Ruler Chart Fractions symbols removed.

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Ruler Chart Students are often surprised to see how a ruler is constructed.

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Ruler War Game A comparison game, using cards with ones, halves, fourths, and eighths.

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Ruler War Game Which is more, 1/8 or 1/4? 1 4 1 8

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Ruler War Game 3 4 5 8

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Ruler War Game 3 4 3 4 3 8 1 4

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Ruler War Game 3 4 3 4 3 8 1 4

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Fraction Chart Showing 9/8 is 1 plus 1/8.

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Mixed to Improper Fractions Each row of connected rectangles represents 1. Ask the student how she found the 11. 2 4 two 4s 3 + 3 11 = 11

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Mixed to Improper Fractions 2 4 two 4s 11 3 four 3s + 2 = 14 Each row of connected rectangles represents 1. + 3 = 11

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Mixed to Improper Fractions Each row of connected rectangles represents 1. 2 4 two 4s 11 3 four 5s + 3 = 23 four 3s + 2 = 14 + 3 = 11

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Improper to Mixed Fractions

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Improper to Mixed The correlation to division becomes obvious here.

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Improper to Mixed The correlation to division becomes obvious here.

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Fraction of Geometric Figures 1 2 2 3 1 4

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Fraction of Geometric Figures 1 2 2 3 1 4

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Fraction of Geometric Figures 1 2 2 3 1 4

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Fraction of Geometric Figures 1 2 2 3 1 4 A study showed that many students and adults though this was impossible.

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Making the Whole

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Making the Whole

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Making the Whole

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Fraction Chart What is one-half of 12? What is one-fourth of 12?

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Percents Percent means per hundred or out of 100.

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Percents Percent means per hundred or out of 100. 2 of 100 = = 50% 100 1 50

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Percents Percent means per hundred or out of 100. 4 of 100 = = 25% 100 1 25

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Dividing 100 100 50 50 25 25 25 25 10 10 10 10 10 10 10 10 10 10 33 1 3 1 3 33 1 3 1 3 33 1 3 1 3 12 1 2 1 2 12 1 2 1 2 12 1 2 1 2 12 1 2 1 2 12 1 2 1 2 12 1 2 1 2 12 1 2 1 2 12 1 2 1 2

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Simplifying Fractions 1 2

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Simplifying Fractions 3 6 = 1 2

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Simplifying Fractions 4 8 = 1 2

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Simplifying Fractions 8 12 = 2 3 4 Writing the common multiple in a circle, for example, the 4, helps students remember what they’re dividing by.

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Simplifying Fractions 9 12 = 3 4 3 Writing the common multiple in a circle, in this example, 3, helps students remember what they’re dividing by.

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Simplifying Fractions The fraction 4/8 can be reduced on the multiplication table as 1/2. 21 28 45 72

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Simplifying Fractions 12 16

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Skip Counting Patterns Twos Recognizing multiples necessary for simplifying fractions and doing algebra. 2 2 4 4 6 6 8 8 0 0

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Skip Counting Patterns Fours Notice the ones repeat in the second row. 4 4 8 8 2 2 6 6 0 0

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Skip Counting Patterns Sixes and Eights Second row repeats with the 6s and 8s. Also, the ones in the eights are counting by 2s backward, 8, 6, 4, 2, 0. 6 6 2 2 8 8 4 4 0 0 8 8 6 6 4 4 2 2 0 0

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© Joan A. Cotter, 2009 Skip Counting Patterns Sixes and Eights 6x4 8x7 6 x 4 is the fourth number (multiple).

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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 8 8 16 24 32 40 48 56 64 72 80 7 14 21 28 35 42 49 56 63 70

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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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Skip Counting Memory Game 8 16 24 32 40 48 56 64 72 80 8 8 7 14 7 14 21 28 35 42 49 56 63 70

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Skip Counting Memory Game 8 16 24 32 40 48 56 64 72 80 8 8 7 24 21 7 14 21 28 35 42 49 56 63 70 14

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Skip Counting Memory Game 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 We’ll never know who won. 8 8 7 14 24 21

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© Joan A. Cotter, 2009 Subtracting Fractions 4684 – 2372 4684 – 2879 2000 300 10 2 2312 2000 – 200 10 – 5 1805 4 thousand minus 2 thousand is 2 thousand. . . .

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© Joan A. Cotter, 2009 Subtracting Fractions 3 5 4 – 3 2 Using the previous subtraction method. 3 2 5 – 4 7 3 7 5 7 5 – 2 1 7

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© Joan A. Cotter, 2009 Multiplying Fractions 1 2 x = 1 2 The square represents 1.

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© Joan A. Cotter, 2009 Multiplying Fractions 1 2 x = 1 2 We are thinking 1/2 of 1/2. First find 1/2 of it vertically.

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© Joan A. Cotter, 2009 Multiplying Fractions 1 2 x = 1 2 1 4 Now find 1/2 of it horizontally. The solution is the double crosshatched area.

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© Joan A. Cotter, 2009 Multiplying Fractions 2 3 x = 3 4 Another example.

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© Joan A. Cotter, 2009 Multiplying Fractions 2 3 x = 3 4

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© Joan A. Cotter, 2009 Multiplying Fractions 2 3 x = 3 4

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© Joan A. Cotter, 2009 Multiplying Fractions 2 3 x = = 3 4 6 12 1 2

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© Joan A. Cotter, 2009 Multiplying Fractions 2 3 x = 3 4 The total number of of rectangles is 3 x 4.

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© Joan A. Cotter, 2009 Multiplying Fractions 2 3 x = 3 4 The number of double crosshatched rectangles is 2 x 3. The total number of rectangles is 3 x 4. That’s why to multiply fractions, we multiply the numerators and the denominators.

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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 1 ÷ 3 is simply the definition of a fraction. Notice the pattern. 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 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 To find ÷ = __ ÷ = 1 First find 2 3 3 4 3 4 4 3

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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