Question. What is a fraction? Answer: A topic that scares many of our students


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1 Question What is a fraction? Answer: A topic that scares many of our students More seriously: Please write down your definition of a fraction. Then briefly discuss with a neighbor.
2 FRACTIONS are numbers too! Richard Bisk Ph.D. Professor of Mathematics Worcester State University
3 Imagine life without fractions? 6 pies 3 people = 2 pies per person 2 pies 3 people =? pies per person What would we do if we didn t have fractions? Skip the pie? No way!
4 Cut each pie into 3 pieces Each person now gets 2 of the 6 pieces But what is each of the pieces?
5
6 Unit Fractions Is this 1 unit shown 3 times? Or 1 unit broken into 3 pieces? 1/3 is the number that multiplied by 3 gives 1
7 It takes 3 one thirds to make 1. 2/3 means 2 of what it takes 3 of to make 1. A fraction is a point on a number line. 0 1/3 2/3 3/3 4/3 5/3 6/3 7/3 8/3
8 Common Core Grade 3 CCSS.Math.Content.3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
9 Common Core Grade 3 more concrete version CCSS.Math.Content.3.NF.A.1 Understand a fraction 1/3 as the quantity formed by 1 part when a whole is partitioned into 3 equal parts; understand a fraction 2/3 as the quantity formed by 2 parts of size 1/3.
10 MIF 2B
11 Common Core Grade 3 CCSS.Math.Content.3.NF.A.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
12
13 Expanding the number system A fraction is a number; not two numbers. They were developed to answer questions that can t be answered with whole numbers. They may look different, but they really behave just like whole numbers.
14 Fractions are Numbers And they have feelings too, just like whole numbers!
15 2/3 The denominator tells us our noun  what we have. The numerator tells us our adjective  how many we have. 2 dogs + 3 hats =??? 2 fifths + 3 fourths =????
16 Addition and Subtraction 4 sevenths + 2 sevenths = 6 sevenths 7 ninths  5 ninths = 2 ninths If we have the same fractional unit, we count, add and subtract fractions like whole numbers
17
18 Different Fractional Units Convert to equivalent fractions with like units. Students must understand why this must be done, before they learn the algorithm. Compare to place value conversions.
19
20 1/2 + 1/3
21 Multiplying Fractions 5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
22 Multiplying a Whole Number and a Fraction 6 x ½ or ½ x 6 We can think of this as 6 groups of ½. We often read ½ x 6 as ½ of 6. Compare to: 3 x 5 means 3 groups of 5.
23 Fraction as Area Consider 1/2 x 1/3 Start with a 1 inch square:
24 Divide it into 3 equal horizontal pieces and shade one them. This is 1/3 of the square.
25 Divide the square vertically into 2 equal pieces. Shade ½ of the 1/3.
26 The blue shaded area represent ½ of 1/3 of the square. Its area is 1/6 of the square. The blue shaded region is a rectangle of width ½ and height 1/3. Our area formula is length times width or ½ x 1/3 = 1/6
27
28
29 Your Turn: Draw a rectangle array model to represent 2/3 x 4/5
30 Division of Fractions Yours is not to reason why, just invert and multiply. (Old Proverb) If you believe in things that you don t understand, then you suffer (S.W.)
31 Division of Fractions Adult Explanation How did we define division of whole numbers? 6 2 =?? x 2 = 6
32 2 5 3? 7 3? 7 2 5
33 7 3 2??? 3 7 5
34 ?
35 ? 7
36 So WOW!
37 DIVISION OF FRACTIONS The Teaching Sequence 1. Review: Whole Number Whole Number 2. Teach: Fraction Whole Number 3. Teach: Whole Number Fraction 4. Teach: Fraction Fraction
38 Why this order? Which is easier to model? 2/ /3
39 Why this order? Step 2 is similar to Step 1. It can be done using either the partitive or measurement model. Steps 3 and 4 often use the measurement model
40 Example 1 Two children share 5 oranges equally. Draw two models for 5 2 Note relationship to multiplying by reciprocal.
41 Example 2 Four children shared 2/3 of a pie equally. What fraction of a pie did each get? To draw models for 2/3 4, we could use a. pie b. bar model c. number line d. rectangle. Note relationship to multiplying by reciprocal.
42
43
44
45 Example 3 Two pies are cut into pieces that are each ¼ of a pie. How many pieces are there? Draw models for 2 1/4 Note relationship to multiplying by reciprocal.
46
47 Example 4 Find 5/7 2/7 Draw a bar diagram. Note relationship to multiplying by reciprocal.
48
49 Conclusion Fractions are numbers too! Operations on fractions are extensions of the corresponding operations on whole numbers. Discrimination against fractions is a violation of regulation bisk3.14 and is punishable under numerous mathematical laws.
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