Operations with Fractions Learning goals: 1.Know the key teaching strategies for elementary mathematics 2.Understand the depth of the content regarding fractions in 4 th and 5 th grades 3.Consider additional teaching strategies for engagement and scaffolding higher order thinking
In which of the following are the three fractions arranged from least to greatest? NAEP 8 th Grade, 49% correct Why so few?
How did it go this past month?
Key Points 1.Use manipulatives and drawings. 2.Build knowledge of fraction operations on the underlying structures of word problems. 3.Help students reason with fractions. 4.Focus on the reasons behind operations, to develop the procedures. Problems that represent key content
Procedures 1.Adding or subtracting by finding equivalent fractions - How to find equivalent fractions - Why add numerators when the denominators are the same
Procedures 2.Multiplying fractions by multiplying the numerators and multiplying the denominators. - Where does this come from?
Marty made two types of cookies. He used 1/5 cup of flour for one recipe and 2/3 cup of flour for the other recipe. How much flour did he use in all? Is it greater than 1/2 cup or less than 1/2 cup? Is the amount greater than 1 cup or less than 1 cup? Explain your reasoning in writing.
When first learning Allow students to use manipulatives or drawings to figure this out. Use fraction circles or fraction bars to determine whether 1/5 + 2/3 is less than or greater than or 1. Eventually the image of the manipulatives or drawings will become second nature so students can see in their heads the fraction relationships.
Basic concepts A fraction is a part of a whole The numerator means the denominator means Unit fractions get smaller as their denominators get larger. Fractions are numbers on the number line (1/2 is half the way from 0 to 1) Fractions that are the same size are called equivalent fractions.
Reasoning questions Which is larger, 2/8 or 5/8? Why? Which is larger, 2/4 or 2/6? Why? How can you prove this? Which is larger, 2/3 or 3/4? 2/5 or 5/10? 3.NF.3 d. Compare two fractions with the same numerator or the same denominator by reasoning about their sizes. 4.NF.2 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.
Reasoning questions Where would you place 5/6 on the number line? Can you use other fraction pieces with different denominators to show 1/2? 1/4? 3/4? 4.NF.1 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.
C-R-A for equivalence ConcreteRepresentationalAbstract
Illuminations fraction game (Fraction Tracks) http://illuminations.nctm.org/ActivityDetail.aspx?ID=18 How can playing a game like Fraction Tracks help a student build understanding about the relative sizes of fractions? How can playing a game like Fraction Tracks help a student build understanding about the equivalence of fractions? What characteristics of the classroom environment would support students as they use a game like Fraction Tracks to help them deepen their understanding of fractions?
Smarter Balanced Assessment items Other virtual manipulatives on our Elementary Math Resources web pagesOther virtual manipulatives on our Elementary Math Resources web pages
Which approach? Is it A or B A: You can find equivalent fractions by multiplying the numerator and denominator by the same number. (Teacher explains procedure, shows worked out examples, students practice with new problems) 4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n a)/(n b) by using visual fraction models
Why does this work mathematically? Why dont they mention this?
Which approach? Is it A or B B: See pages 12-13 in Operations with Fractions packet
Fraction Addition and Subtraction
Learning Progression 4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. 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. 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. Write examples for each
4.NF.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. For example, express 3/10 as 30/100 and add 3/10 + 4/100 = 34/100. (Addition and subtraction with unlike denominators in general is not a requirement at this grade.)
Estimation and visualization are important. These abilities will help students monitor their work when finding exact answers.
Using reasoning about size For each of the following problems, explain if you think the answer is a reasonable estimate or not.
Estimation and visualization are important. These abilities will help students monitor their work when finding exact answers. Students need to experience acting out addition and subtraction concretely with an appropriate model before operating with symbols.
Learning Progression Step 1: Learn what it means to add fractions with the same denominator. Pictures, analogies, methods, etc. How does this generalize into adding fractions with different denominators? Step 2: one is a multiple of the other Step 3: both scale up to a common multiple
5.NF.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. 5.NF.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7 by observing that 3/7 < 1/2.
C-R-A for adding/subtracting ConcreteRepresentationalAbstract Use manipulatives to model this.
NLVM Fractions AddingFractions Adding
You try it ConcreteRepresentationalAbstract Story problem? 1.Adding to or putting together 2.Taking from or taking apart 3.Comparing 4.Part-whole by using visual fraction models or equations to represent the problem
Estimation and visualization are important. These abilities will help students monitor their work when finding exact answers. Students need to experience acting out addition and subtraction concretely with an appropriate model before operating with symbols. Making connections between concrete actions and symbols is an important part of understanding. Students should be encouraged to find their own way of recording with symbols.
Using circle fractions Page 494, last paragraph 1 st column, through end. Mark the text. For the problems in Figure 6, see the packet with rulers (like number lines).
For students who struggle Manipulatives and drawings Partner work Explicit teaching: Teacher verbalizes thought processes Works together with student Allows for practice with guided feedback Pair up and try this with Start with Can you show me how to make 2/5 from the fraction circle pieces?
Collaborative cards What do you think of this game as a teaching tool?
What about decimals? What does the common core say? How would you sequence this in a learning progression? What manipulatives and visual representations are helpful? How are decimals related to fractions? NLVM Place Value Number Line (3-5 Number and Operations)Place Value Number Line
4.NF.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. For example, express 3/10 as 30/100 and add 3/10 + 4/100 = 34/100. (Addition and subtraction with unlike denominators in general is not a requirement at this grade.) 4.NF.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. 4.NF.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or