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EMSE 3123 Math and Science in Education Not Whole Numbers I: Fractions Presented by Frank H. Osborne, Ph. D. © 2015 1 Teaching the Meaning of Fractions • So far, we have studied the teaching of whole number concepts. Frequently we need to express those that are not whole numbers. • These are called rational numbers. • One way to express rational numbers is via fractions. • Using manipulatives, we need first to have something we can consider as a whole, or ‘1’. • This can be divided in numerous ways. 2 Teaching the Meaning of Fractions • Example: Take a piece of clay and find its mass. – Now, break off a piece amounting to ¾ of the whole clay. – How would you demonstrate that you have in fact broken off ¾? • To do this, you need to break the clay into 4 equal parts and then take 3 of these. We express this in the following form. 3 Teaching the Meaning of Fractions We write number of parts (numerator) over the number of equal parts (denominator). 4 Teaching the Meaning of Fractions A way to show this pictorially is to take a square and identify it as the “whole”. The square can then be divided into any number of equal parts. We use shading to indicate how much our fraction has. 5 Teaching the Meaning of Fractions We use shading to indicate how much our fraction has. 6 Teaching the Meaning of Fractions If more than one of the parts of is shaded, we still use our definition to express the number as 7 Teaching the Meaning of Fractions Formal instruction for associating a fraction with a part of the whole should begin in third grade asking the students to express each not-whole-number as a fraction. 8 Teaching the Meaning of Fractions Formal instruction for associating a fraction with a part of the whole should begin in third grade asking the students to express each not-whole-number as a fraction. 9 Teaching the Meaning of Fractions Children should also have experiences working with objects, where a certain number of objects represents a whole, and each object is a fractional part of the whole. For example, we have 8 pears. 10 Teaching the Meaning of Fractions Each pear is ?/8 of the whole? Five pears are ?/8 of the whole? ?/8 of the pears are colored? 11 Teaching the Meaning of Fractions Each pear is ?/8 of the whole? 1/8 Five pears are ?/8 of the whole? 5/8 ?/8 of the pears are colored? 4/8 = 1/2 12 Teaching the Meaning of Fractions Circular or rectangular regions can also be used as models to illustrate fractions. Divide them into equal parts. Some examples of rectangular regions are shown in the activity on the next slides. 13 Teaching the Meaning of Fractions Write down different size relationships among fraction from the strips below. Use terminology of greater than, less than, equals. 14 Teaching the Meaning of Fractions Just a few: 1/9 is less than 1/6 1/6 is greater than 1/9 1/4 is greater than 1/9. 1/8 is less than 1/3. 15 Teaching the Meaning of Fractions • Children need to realize through use of manipulatives that fractions with the same numerator and denominator are all equal to 1. We can show that 9/9=6/6=3/3=8/8=4/4=1. 16 Teaching the Meaning of Fractions • With the aid of manipulatives, children should be able to order simple fractions, first with the same numerator: 1/9, 1/8, 1/6, 1/4, 1/3, • and then different numerators. 17 Teaching the Meaning of Fractions • After mastering size relationships for fractions less than 1, they can be introduced to fractions greater than 1. • For example, circular regions, like these below, can be used to introduce fractions greater than 1. 18 Teaching the Meaning of Fractions Rectangles, like those shown below, can demonstrate that a fraction can be expressed as a mixed number. 19 Developing Concepts for Fractions With Denominators of 10 and 100 • When working with fractions it is very important to introduce fractions in which the denominator is 10 or 100. • This work prepares students for decimals which are the next topic. • Children must learn that there is another way to write tenths. This permits early introduction of decimals as another way of expressing fractions with 100 or 100 as the denominator. 20 Developing Concepts for Fractions With Denominators of 10 and 100 c. Use grids to answer each question: 1/10 = ?/100 2/10 = ?/100 10/10 = ?/100 21 Developing Concepts for Fractions With Denominators of 10 and 100 c. Use grids to answer each question: 1/10 = 10/100 2/10 = ?/100 10/10 = ?/100 22 Developing Concepts for Fractions With Denominators of 10 and 100 c. Use grids to answer each question: 1/10 = 10/100 2/10 = 20/100 10/10 = ?/100 23 Developing Concepts for Fractions With Denominators of 10 and 100 c. Use grids to answer each question: 1/10 = 10/100 2/10 = 20/100 10/10 = 100/100 24 Developing Concepts for Fractions With Denominators of 10 and 100 d. How would you demonstrate on the grids which is bigger, 45/100 or 5/10? 25 Developing Concepts for Fractions With Denominators of 10 and 100 d. How would you demonstrate on the grids which is bigger, 45/100 or 5/10? 26 Developing Concepts for Fractions With Denominators of 10 and 100 e. Demonstrate the answer to 50/100 + 12/100 = ? 27 Developing Concepts for Fractions With Denominators of 10 and 100 e. Demonstrate the answer to 50/100 + 12/100 = 62/100 28 Developing Concepts for Fractions With Denominators of 10 and 100 Once children understand the differences between 1/10’s and 1/100’s, simple decimal notation can be introduced as a shorter way of writing fractions that have denominators of tenths or hundredths, e.g., 1/10 = .1 1/100 = .01 29 Developing Concepts for Fractions With Denominators of 10 and 100 Children should then be able to express any number of tenths or hundredths in terms of decimals e.g., 2/10 = .2 2/100 = .02 3/10 = .3 3/100 = .03 . . . . 9/10 = .9 9/100 = .09 30 Addition and Subtraction of Fractions Addition and subtraction of fractions using manipulatives is a natural extension of the activities we have already done. Just specify a piece to represent the whole, then represent each fraction in the addition or subtraction problem with fraction pieces. Bring the pieces together for addition, or take away for subtraction, and express the answer in terms of the whole. We will start with ½ + ¾ as a demonstration. 31 Addition and Subtraction of Fractions Demonstration of ½ + ¾ . 32 Addition and Subtraction of Fractions When studying addition and subtraction it is helpful to have all of the fraction pieces for a particular set of problems lined up. 33 Addition and Subtraction of Fractions We can do many fraction manipulations with the set of fraction pieces. For example, to add ½ and 1/3, we bring the corresponding pieces together. How much is this? 34 Addition and Subtraction of Fractions To find the answer we compare it to the whole by lining it up with the complete set. We see that the answer is less than the whole but more than one-half of the whole. The set also gives us the exact answer. 35 Addition and Subtraction of Fractions For the exact answer we use the other fraction pieces. We see that our answer is 10 of the twelfths (10/12) or 5 of the sixths (5/6). 36 Addition and Subtraction of Fractions We could also express each of the fractions in the problem in terms a smaller piece. Onehalf is the same lengths as 3 of the sixths while 1/3 is the same as 2 of the sixths. The longest rod that will fit into both pieces is called the common denominator. This is a concrete demonstration of how it is found. 37 Addition and Subtraction of Fractions Subtraction proceeds in a similar way. For example, in order to subtract 1/6 from ½ (1/2 – 1/6) 38 Addition and Subtraction of Fractions To see how much the answer is we line it up with the rest of the pieces. We find that it represents 2 of the sixths, or 1 of the thirds. 39 Addition and Subtraction of Fractions Or we could have also used our common denominator method seeing that ½ is the same as 3 of the sixths. Take 1 sixth away and you have two sixths left which is 1/3. 40 Teaching Fractions with Cuisenaire Rods Size Relationships • We can assign any rod to represent a whole or the number one. • The other rods will have a corresponding value in terms of the whole. • Assume that the dark green (6) rod equals one. What are the number values of all the other rods? 41 Teaching Fractions with Cuisenaire Rods Size Relationships Line the other rods up against the dark green one to see the answer. White = Red = Green = Purple = Yellow = Dark Green = Black = Brown = Blue = Orange = 42 Teaching Fractions with Cuisenaire Rods Size Relationships Line the other rods up against the dark green one to see the answer. White = 1/6 Dark Green = 1 Red = 1/3 Black = 7/6 Green = 1/2 Brown = 4/3 Purple = 2/3 Blue = 3/2 Yellow = 5/6 Orange = 5/3 43 Teaching Fractions with Cuisenaire Rods Size Relationships Using the rods to answer, which is larger? a. 2/3 or ½? b. 2/3 or 5/6? c. 3/2 or 4/3? 44 Teaching Fractions with Cuisenaire Rods Size Relationships Using the rods to answer, which is larger? a. 2/3 or ½? b. 2/3 or 5/6? c. 3/2 or 4/3? 45 Teaching Fractions with Cuisenaire Rods Size Relationships Now we assume that brown (8) represents the number one. What are the values of all the rods? White = Red = Green = Purple = Yellow = Dark Green = Black = Brown = Blue = Orange = 46 Teaching Fractions with Cuisenaire Rods Size Relationships Now we assume that brown (8) represents the number one. What are the values of all the rods? White = 1/8 Dark Green = 3/4 Red = 1/4 Black = 7/8 Green = 3/8 Brown = 1 Purple = 1/2 Blue = 9/8 Yellow = 5/8 Orange = 5/4 47 Teaching Fractions with Cuisenaire Rods Size Relationships Using the rods to answer, which is larger? a. ¾ or 5/8? b. ½ or 3/8? c. ¾ or 7/8? 48 Teaching Fractions with Cuisenaire Rods Size Relationships Using the rods to answer, which is larger? a. ¾ or 5/8? b. ½ or 3/8? c. ¾ or 7/8? 49 Teaching Fractions with Cuisenaire Rods Size Relationships How many eighths make up 1 1/4? 50 Teaching Fractions with Cuisenaire Rods Size Relationships How many eighths make up 1 1/4? 51 Teaching Fractions with Cuisenaire Rods Addition and Subtraction of Fractions Let the Orange (10) rod represent the number one. What will represent a. ½ b. 1/5 c. 1/10 52 Teaching Fractions with Cuisenaire Rods Addition and Subtraction of Fractions Let the Orange (10) rod represent the number one. What will represent a. ½ b. 1/5 c. 1/10 Note that ½ = 5/10 and 1/5 = 2/10. We need to keep the common denominator in mind when we add or subtract fractions. 53 Teaching Fractions with Cuisenaire Rods Addition and Subtraction of Fractions Use your rods to verify that ½ + 1/5 = 7/10. 54 Teaching Fractions with Cuisenaire Rods Addition and Subtraction of Fractions Use your rods to verify that ½ + 1/5 = 7/10. In terms of the common denominator, we have 5/10 + 2/10 = 7/10. 55 Teaching Fractions with Cuisenaire Rods Addition and Subtraction of Fractions Use your rods to answer the following a. ½ + 2/5 b. 3/5 + 1/10 c. 1/5 + 7/10 56 Teaching Fractions with Cuisenaire Rods Addition and Subtraction of Fractions Use your rods to answer the following a. ½ + 2/5 b. 3/5 + 1/10 c. 1/5 + 7/10 Answers are: a. 5/10 + 2/10 = 9/10 b. 6/10 + 1/10 = 7/10 c. 2/10 + 7/10 = 9/10 57 Teaching Fractions with Cuisenaire Rods Addition and Subtraction of Fractions Finding a common denominator is most important. You cannot add fractions unless the denominators are the same size. Example: We wish to add 1/3 + 1/2. We see that the denominators are different. So we need to find a common denominator. 58 Teaching Fractions with Cuisenaire Rods Addition and Subtraction of Fractions We take threes and twos and lay them end to end until they line up evenly. It takes 2 threes and 3 twos to make two even rows. Therefore, the common denominator is 6. Now we can deal with the numerators. 59 Teaching Fractions with Cuisenaire Rods Addition and Subtraction of Fractions We used 2 threes, so we need to have 2 ones. We used 3 twos, so we need 3 ones. When we line them up, we get a total numerator value of 5. Answer: 1/3 + 1/2 = 2/6 + 3/6 = 5/6 60 Teaching Fractions with Cuisenaire Rods Addition and Subtraction of Fractions As the children practice this procedure they can move up to more complicated examples. However, this is done the exact same way. Example: We wish to add 2/7 + 3/5. We see that the denominators are different. So we need to find a common denominator. 61 Teaching Fractions with Cuisenaire Rods Addition and Subtraction of Fractions We take sevens and fives and lay them end to end until they line up evenly. It takes 5 sevens and 7 fives to make two even rows. Therefore, the common denominator is 35. Now we can deal with the numerators. 62 Teaching Fractions with Cuisenaire Rods Addition and Subtraction of Fractions We used 5 sevens, so we need to have 5 twos. We used 7 fives, so we need 7 threes. When we line them up, we get a total numerator value of 31. Answer: 2/7 + 3/5 = 10/35 + 21/35 = 31/35 63 Teaching Fractions with Cuisenaire Rods Addition and Subtraction of Fractions The same concept applies to subtraction. You can only subtract fractions with equal size denominators. Example: We wish to subtract 2/3 – 1/5. We see that the denominators are different. So we need to find a common denominator. 64 Teaching Fractions with Cuisenaire Rods Addition and Subtraction of Fractions We take threes and fives and lay them end to end until they line up evenly. It takes 5 threes and 3 fives to make two even rows. Therefore, the common denominator is 15. Now we can subtract the numerators. 65 Teaching Fractions with Cuisenaire Rods Addition and Subtraction of Fractions We used 5 threes so we need 5 twos. We used 3 fives so we need 3 ones which will be subtracted from the twos. Answer: 2/3 – 1/5 = 10/15 – 3/15 = 7/15 Remember that addition and subtraction of fractions require a common denominator. 66 Multiplication of Fractions • Just as we did for whole number multiplication, we want to teach the meaning of multiplying by a whole number or another fraction before teaching algorithms. • This is done using concrete objects. • We can use the repetitive addition model, or the area model. 67 Multiplication of Fractions The addition model for multiplying fractions. Using repetitive addition, multiplying a whole number by a fraction such as 3 x ½ is the same as 3 x ½ = ½ + ½ + ½ = 3/2 = 1 ½ For another example, 2x3½=3½+3½=3+3+½+½=7 These can easily be demonstrated using fraction pieces or Cuisenaire rods. 68 Multiplication of Fractions How would we teach the meaning of multiplying a fraction by a whole number such as ½ x 3? ½x3=3x½=1½ A fraction multiplied by a whole number is the same as taking the fractional part of the whole number ½ x 3 = ½ of 3 = 1 ½ 69 Multiplication of Fractions So, 3 x ½ is the same as taking half of the number 3, which is 1 ½. This means that if we take three wholes and divide into two equal parts, each part is 1 ½. 70 Multiplication of Fractions As another example, multiplying the fraction 1/3 by 3 1/3 x 3 = 1/3 of 3 = 1 This means that if we have three wholes, and divide them in to three equal parts, each is 1. Multiplications such as 2/3 x 3 follow logically. 71 Multiplication of Fractions In words, the problem 2/3 x 3 means that we find the fraction piece which is 1/3 of 3 and take two of these pieces. As each third is 1, then 2/3 is equal to 2. 1/3 of 3 = 1 2/3 of 3 = 2 (two 1/3 pieces) 3/3 of 3 = 3 (three 1/3 pieces) 72 Multiplication of Fractions • Essentially, in multiplying a fraction by a whole number, a certain number of objects represents your whole, and you are taking a fractional part of these objects as we demonstrated above. • After students understand the meaning of multiplying a whole number by a fraction they can be introduced to the algorithm which is cross-cancellation and multiplication. 73 Multiplication of Fractions Cross-cancellation and multiplication. Example: ¾ x 12 74 Multiplication of Fractions Multiplication of two fractions can follow the same approach. ½ x 1/3 = ½ of 1/3 Which means to take a 1/3 fraction piece and divide it into two equal parts (or, take ½ of this 1/3 piece). Each part represents what part of the whole? 75 Multiplication of Fractions ½ x 1/3 = ½ of 1/3 Each part represents what part of the whole? We see that if we take ½ of the 1/3 piece we get a piece that corresponds to 1/6 of the whole. 76 Multiplication of Fractions In Lab, we will work with exercises based on this pattern. 77 Multiplication of Fractions The area model for multiplying fractions. We used the area approach in the multiplication of whole numbers. We can use the same method for multiplying fractions. ½ x 1/3 = As shown above, the answer is 1/6. 78 Multiplication of Fractions As another example, we multiply 2/3 x 3/5 = As shown, the answer is 6/15. 79 Multiplication of Fractions • Division of fractions has the same interpretation as division of whole numbers. X ÷ Y means, “How many Ys fit into X?” In the case of a whole number divided by a fraction, such as 2 ÷ ½ means, “How many 1/2’s fit into 2?” 80 Division of Fractions 2 ÷ ½ means, “How many 1/2’s fit into 2?” One way to demonstrate the answer is by repetitive subtraction: 2–½-½-½-½=0 We see that four ½’s fit into 2. This can be illustrated by fraction pieces or Cuisenaire rods. We will use the purple (4) to represent the whole. 81 Division of Fractions 2 ÷ ½ means, “How many 1/2’s fit into 2?” Or, “How many ½’s fit into 2 wholes?” 82 Division of Fractions Dividing two fractions together can be done in a similar manner. ½ ÷ ¼ means how many ¼’s fit into ½. Using successive subtraction ½-¼-¼=0 Using Cuisenaire rods it is done this way. We can see that two ¼’s fit into ½. 83 Division of Fractions For a set of exercises we will use orange 10 and red 2 together to make a whole. What is the value of each color? White = Dark Green = Red = Black = Green = Brown = Purple = Blue = Yellow = Orange = 84 Division of Fractions For a set of exercises we will use orange 10 and red 2 together to make a whole. What is the value of each color? White = 1/12 Dark Green = 1/2 Red = 1/6 Black = 7/12 Green = 1/4 Brown = 2/3 Purple = 1/3 Blue = 3/4 Yellow = 5/12 Orange = 5/6 We can use the rods to divide 1÷1/6. How many reds (1/6) make up one whole? 85 Division of Fractions Simple activities can begin as early as 2nd or 3rd grade. Have the students learn the meaning of mathematical operations using manipulatives before teaching the algorithms. 86 The End 87