Fractions: Fourth Grade What is a Fraction? •What is a fraction? •Think of a scenario to represent 1/3 and create a model. Explore Fractions Using Fraction Bars • http://www.mathsisfun.com/numbers/fraction-numberline.html • What fractions add up to one whole? • Create equivalent Fractions What patterns would you want your students to notice? • Number line • Area Model • Set Model • Develop understanding of fractions as numbers. • 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. • CCSS.Math.Content.3.NF.A.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram. • CCSS.Math.Content.3.NF.A.2.a Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. • CCSS.Math.Content.3.NF.A.2.b Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. Explain equivalence of fractions in special cases, and compare fractions by . reasoning about their size • CCSS.Math.Content.3.NF.A.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. • CCSS.Math.Content.3.NF.A.3.a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. • CCSS.Math.Content.3.NF.A.3.b Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model. • CCSS.Math.Content.3.NF.A.3.c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram. • CCSS.Math.Content.3.NF.A.3.d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model 4.NF.1 Equivalent Fractions • Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. • How would student explain equivalent fractions? • What do students need to understand in order to do this? • 2/4 ½ Equivalent Fractions • Students subdivide the equal parts of a fraction, resulting in a greater number of smaller parts. • Students discover that this subdividing has the effect of multiplying the numerator and denominator by n. 1 x 4 parts and 4 x 4 parts. 4.NF.2 Comparing Fractions • Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Comparing Fractions • Students create common numerators or common denominators by renaming one fraction or both. • When comparing fractions with the same numerator, they use prior knowledge about the relative sizes of fractional parts. 2 and 1 5 3 How would you rename these fractions to compare them? Comparing Fractions • Complete the following comparisons without using equivalent fractions. Make a note of how you did them… 1 and 8 8 9 7 and 5 8 6 4 and 3 9 4 What are some student misconceptions about comparing fractions? 4.NF.3 (a-b) Fractions as a Sum of Unit Fractions a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. Describe a common misconception for this problem: 2 + 1 5 5 Fractions as a Sum of Unit Fractions • All fractions can be seen as a sum or difference of two other fractions. • Think in terms of adding/subtracting copies of 1/b. “Just like 3 dogs + 9 dogs is 12 dogs, or 3 candies + 9 candies is 12 candies, or 3 children + 9 children is 12 children, 3 fifths + 9 fifths is 12 fifths” (Small, 2014, p. 51). • This line of thinking enables students to clearly understand why only the numerators are added or subtracted. Fractions as a Sum of Unit Fractions • Students rename mixed numbers as improper fractions and vice versa. • Students decompose fractions and mixed numbers in more than one way: mrs-c-classroom.blogspot.com Before teaching the “shortcut,” allow students plenty of time to reason with models. 4.NF.3 (c-d) Adding and Subtracting Mixed Numbers c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. Adding and Subtracting Mixed Numbers • Students rename mixed numbers as improper fractions, then add or subtract. But… • They notice that sometimes it is easier to add or subtract the whole number and fraction separately. How would you solve each problem? Why? Adding and Subtracting Mixed Numbers • Students can “count up” to find the difference between mixed numbers (see page 52 in Uncomplicating Fractions). OR • They might prefer to regroup part of the whole number in the greater mixed number. Explain how to regroup the first mixed number in this problem. 4.NF.B.4 Multiplying Fractions by a Whole Number • CCSS.Math.Content.4.NF.B.4a Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4). • CCSS.Math.Content.4.NF.B.4b Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.) • CCSS.Math.Content.4.NF.B.4c Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie? Interpreting the meaning of multiplication • It is important to let students model and solve these problems in their own way, using whatever models or drawings they choose as long as they can explain their reasoning. • Once students have spent adequate time exploring multiplication of fractions, they will begin to notice patterns. • Then, the standard multiplication algorithm will be simple to develop. Shift from contextual problems to straight computation. How can you solve the following problem? How many different ways can you solve it? Kristen ran on a path that was ¾ of a mile in length. She ran the path 5 times. What is the total distance that Kristen ran? Interpreting the meaning of multiplication • Adding 4/5 3 times ( 4/5 +4/5 +4/5) • 4 fifths + 4 fifths + 4 fifths = 12 fifths, or 12/5 • The result of three jumps of 4/5 on a number line, beginning at 0 • The number of fifths of a 2-D shape if 3 groups of 4 fifths are shaded. Understand decimal notation for fractions, and compare decimal fractions • CCSS.Math.Content.4.NF.C.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.2 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. • CCSS.Math.Content.4.NF.C.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. • CCSS.Math.Content.4.NF.C.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. Interpreting decimals • Representing tenths and hundredths and decimals as a sum. The 10-to-1 relationship continues indefinitely. What is a common misconceptions students have about decimal place value? Students must understand the equivalent relationship between tenths and hundredths. Representing Decimals on a Number line One of the best length models for decimal fractions is a meter stick. Experiences allow students to compare decimals and think about scale and place value. Comparing Decimals • Reason abstractly and quantitatively. Develop benchmarks; as with fractions: 0, ½, and 1. For example, is seventy-eight hundredths closer to 0 or ½, ½ or 1? How do you know? • Using decimal circle models. Multiple wheels may be used to conceptualize the amount. Or, cut the tenths and hundredths and the decimal can be built. • Why do many students think .4 < .19? Activity: The Unusual Baker • George is a retired mathematics teacher who makes cakes. He likes to cut the cakes differently each day of the week. On the order board, George lists the fraction of the piece, and next to that, he has the cost of each piece. This week he is selling whole cakes for $1 each. Determine the fraction and decimal for each piece. How much will each piece cost if the whole cake is $1.00? The Unusual Baker • CCSS 4.NF.A.1 Equivalent fractions • CCSS 4. NF.A.2 Compare two fractions • CCSS 4. NF.B.3a Understand addition and subtraction of fractions • CCSS4.NF.B.3b Decompose a fraction into a sum of fractions with the same denominator. • CCSS4. NF.C.6 Use decimal notation for fractions with denominators 10 or 100. • CCSS4.NF.C.7 Compare two decimals to hundredths by reasoning about their size.