Fractions in Bridges in Mathematics

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1 Fractions in Bridges in Mathematics MODELS FOR ADDING & SUBTRACTING FRACTIONS Throughout Bridges and Number Corner, several models are used to visually represent fractions. As students progress from conceptualizing fractions as parts of a whole in grade through early operations with fractions in grade, they will encounter the same models used in increasingly powerful ways. In Bridges Grade, Unit 2, these models help students begin to develop intuition about common denominators. The models also provide them tools for adding and subtracting fractions that don t share the same denominators. Four particularly powerful models merit special consideration: money, clocks, the double number line, and ratio tables. This document covers these models in detail. What We Can Do with RatioTables Multiply numbers, Find equivalent ratios to determine cost of different amounts price pounds $6 $ ,600,68 $.20 $2 $20 $ Use equivalent ratios to determine the better buy 8 bars for $0 or 20 bars for $2 by scaling up 8 bars 40 bars 20 bars 40 bars $0 $0 $2 $46 by scaling down bars 8 $0 $ 4 price $.2 bars 20 0 price $2 $.0 $. $. To add and subtract fractions with unlike denominators, students have to understand the need for common denominators. For example, they must grasp that to solve /2 + /, you must rewrite the two fractions so they share the same denominator. In other words, you can add sixths to sixths, but adding halves to thirds is like trying to add apples to peaches. How to find common denominators: When both denominators are factors of 00, think about money. When both denominators are factors of 60, think about a clock. When clocks or money won't work, you can use a double number line or a ratio table to help rewrite fractions so they have common denominators. You can always multiply the denominators to get a common denominator. The Math Learning Center

2 THE MONEY MODEL Many students know the fraction of a dollar represented by each coin, as well as the value of the coin. They know, for example, that a quarter is one-fourth of a dollar, and also that it s worth 2 cents, or 2/00 of a dollar. They can use these understandings to add fractions that convert easily to money. For example, /4 + 2/0 is the same as a quarter plus two dimes, or $0.2 + $0.20, or 2/ /00 = 4/00. Money Value Pieces Money Value Pieces make it easy for students to see and understand the fractional relationships between each coin and a whole dollar. Calista They re like base ten pieces, but with money. Shawn You can see that the piece with the quarter on it really is onefourth of a dollar. Calista And the dime is like a strip in the base ten pieces. It takes 0 of them to make the dollar mat, so you can see they re each onetenth of a dollar. Because the dollar is shown as a grid of 00, the Money Value Pieces make it easier to understand decimals and decimal notation to hundredths half dollar = = = quarter = = = dime = = = nickel = = = 0.0 penny = 00 = 0.0 The Money Model for Fractions in Bridges Grade 4 Unit, Module, Session 2 Unit 6, Module, Session 4 Grade Unit 2, Module, Sessions, Unit 2, Module 2, Session 6 Unit 2, Module, Session 4 The Math Learning Center 2

3 THE CLOCK MODEL Like the money model, the clock model builds from a familiar idea to one less familiar. Many students know that an hour is 60 minutes. They also know that minutes is a quarter of an hour, 0 minutes is half an hour, and 4 minutes is three-quarters of an hour. They can use this information to add fractions that convert easily to time. For example, /4 + /2 is the same as minutes plus 0 minutes, or /60 + 0/60 = 4/60. The Clock Face The fact that an analog clock is a circle which can be easily divided into halves, thirds, fourths, sixths, twelfths, and sixtieths is a definite advantage. Although students may not know that 20 minutes is / of an hour, they can use the clock as a visual to make that discovery. The clock face is also useful in helping students rewrite fractions so they share common denominators, often in more than one way. For example, they can see that /2 + / can be thought of as /6 + 2/6 or 6/2 + 4/2 or even 0/ /60. Unit 2 Module Session 4 copy for display Clock Fractions Problem String page of minutes + 20 minutes = 0 minutes 2 + = = = 6 = = 2 = 6 = sets of 0 minutes = 0 6 sets of 0 minutes = 6 = 2 6 = 2 sets of 0 minutes = 20 6 sets of 0 minutes 60 Example = 0 minutes 60 minutes + minutes 60 minutes = minutes 60 minutes = sets of minutes 2 sets of minutes = 2 The Clock Model for Fractions in Bridges & Number Corner Grade April Calendar Grid April Calendar Collector Grade 4 Unit, Module, Session 2 Grade Unit 2, Module, Sessions 4 & Unit 2, Module 2, Session 6 Unit 2, Module, Session = = = 0 60 = 6 THE DOUBLE NUMBER LINE Building Intuition About Common Denominators The biggest drawback of clock and money models for fractions is that both only work well for a limited set of denominators factors of 60 or 00, respectively. If you re trying to add numbers like / and /, you need a more flexible model. The double number line offers this flexibility. When students choose a length to use for their double number line, they re basically finding the least common multiple (LCM) for two numbers so they can rewrite the fractions using a common denominator for example, / = / and / = 2/. Teachers and parents are likely to recognize the mathematics behind this model from their own learning in elementary school. The Math Learning Center

4 Students use the double number line to add fractions with unlike denominators. EXAMPLE: / + / Find a number that is divisible by both denominators. [ is divisible by both and.] Draw and label a number line with 0 at one end and the selected number at the other. 0 Find and mark / of the distance from one end of the number line to the other, both as a fraction and as a whole number. 0 Find and mark / of the length of the number line, both as a fraction and as a whole number. Scale up to find and mark / of the distance. Add the two lengths to determine the fraction of the total [26/] distance The Double Number Line in Bridges Grade Unit 2, Module 2, Sessions, 6 Unit 2, Module, Sessions 4 & Fourth graders use a single number line (with only one set of values, rather than one set of values on the top of the line and another set of values on the bottom) to represent fractions in Unit, Module 2, Session. That work helps develop foundations for the double number line in Grade. THE RATIO TABLE Fractions are special ratios the ratio of a part to the whole. Because of this, we can use ratio tables to rewrite fractions so they share the same denominator. For example, when a student sees a problem like /0 + /4, they can think of /0 in a ratio table (the ratio of to 0), and then think of /4 in a separate ratio table (the ratio of to 4). Then they can ask, What can I do to each ratio table to find an equivalent ratio, so I can add these two fractions? Amelie How can we use ratio tables to find equivalents for /0 and /4? Ramon Well, the only number I can think of right now that you can divide by 0 and also by 4 is 40. So let s use that. We just have to do the same thing on the top and on the bottom on both tables. Amelie Okay, so we multiply the top and bottom on /0 by 4 so we get 40 on the bottom. Then we have 2 on the top. Cole And we can multiply the top and the bottom on /4 by 0 so we get 40 on the bottom there, too. And then we have 0 on the top. Ramon So it s 0 plus 2 is 22 on the top that s 22/40. That s the same as /20. Oh, I guess we could have used 20 instead of 40! Well, it worked out anyway. Ratio Tables for Fractions in Bridges Grade Unit 2, Module 2, Sessions 4 & Unit 2, Module, Sessions, 2, 4 & 4 0 To add = The Math Learning Center 4

5 FRACTIONS IN BRIDGES IN MATHEMATICS The following tables list Bridges sessions and Number Corner workouts in which concepts and models for fractions are addressed. Fractions appear in other locations throughout the curriculum as well, but these sessions and workouts provide particularly indepth instruction. Grade LOCATION SESSION OR WORKOUT MODELS & NOTES Bridges Unit 4, Module Fair Shares, Unit Fractions Folding paper rectangles Fractions as Fair Shares 2 Comparing & Ordering Unit Fractions Ordering paper rectangles Pattern Block Fractions Pattern blocks 4 Fractions as Distances Class number line, handheld number line Bridges Unit 4, Module 4 Fractions on a Line Plot Line plots Bridges Unit 6, Module 4 2 Fractions on a Geoboard Geoboards Shapes & Fractions Geoboard Quilt Blocks Bridges Unit, Module Fractions on a Ruler Rulers as models for fractions Fractions as Parts of a Whole & Parts of a Set 2 Introducing Egg Carton Fractions Egg carton fractions Exploring Egg Carton Fractions 4 Equivalent Egg Carton Fractions Bridges Unit, Module 4 Fractions at Work Throughout this module, students apply what they ve learned about fractions to a new number line game, data collection and analysis, and division. Number Corner November Calendar Collector Unit Fraction Race Number lines, paper strips Number Corner April Calendar Grid More Equivalent Fractions Arrays, clocks, egg carton fractions, rulers Calendar Collector Collecting Fractions of an Hour Clock (fractions of an hour) Grade 4 LOCATION SESSION OR WORKOUT MODELS & NOTES Bridges Unit, Module Equivalent Fractions Bridges Unit, Module 2 Comparing, Composing & Decomposing Fractions & Mixed Numbers Bridges Unit, Module 4 Fractions & Decimals Bridges Unit 6, Module Line Plots, Fractions & Division A class number line is used throughout this module; students place fractions on the line. 2 Fair Shares Money, clocks, paper strips, number lines Fractions & Mixed Numbers Paper strips Egg Carton Fractions Egg carton fractions Exploring Fractions on the Geoboard Geoboards 4 Dozens of Eggs Egg carton fractions How Many Candy Bars? Number lines, ratio tables Decimal & Fraction Relationships 2 Fractions & Decimals Ordering Fractions & Decimals on a Number Line Students compare fractions and decimals using base ten pieces and grids. Students compare fractions and decimals using a class number line. Fraction Spin & Add Adding fractions with pattern blocks 4 Present Purchase Students work with decimals and fractions in the context of money in a problem string. The Math Learning Center

6 Grade LOCATION SESSION OR WORKOUT MODELS & NOTES Bridges Unit 2, Module Adding Using a Money Model Money & Subtracting Fractions Clock Fractions Clocks 4 Introducing the Clock Fractions Game Which Model Works Best? Clocks, money Bridges Unit 2, Module 2 River Trail Double number lines Introducing Common Denominators Double Number Lines 6 Fraction Checkpoint During a problem string, students use double number lines, clocks, or other models to solve addition and subtraction combinations involving fractions and mixed numbers. Bridges Unit 2, Module 2 Fractions Are Ratios Ratio tables Common Denominators 4 Fraction Strategies Poster Money, clocks, ratio tables, double number lines Common Denominators Double number lines Bridges Unit 2, Module 4 LCMs & GCFs Bridges Unit, Module Fraction Story Problems Students favorite models and strategies for Multiplying Whole Numbers by working with fractions will emerge as they solve 4 Sharing Strategies Fractions story problems and create a class poster listing their techniques. Bridges Unit, Module 2 Multiplying Fractions by Fractions Bridges Unit, Module More Fraction-by-Fraction Multiplication Geoboard perimeters The geoboard serves here as a gateway to the array/area model for multiplication with fractions. 2 Rectangles with Fractional Side Lengths Multiplying Fractions with the Area Model 4 Creating a Collection of Arrays Modeling Fraction Problems Bridges Unit, Module 4 Dividing Fractions & Whole Numbers In these sessions, students further explore the array/area model for multiplying fractions. Fraction Multiplication Story Problems Students further explore the array/area model for multiplying fractions. Objects, number lines, segmented or divided strips, geoboards, array/area model The Math Learning Center 6