Washington Park School Math Content Workshop

Washington Park School Math Content Workshop
Noreen Fleming Amy Jones Lewis 5th Grade WPS Math Content Workshop 11/13/2018

Share representations with people at your table.
Collection Box: ¾ Individually, create at least 3 representations of ¾. Use pictures, diagrams, symbols, etc. Share representations with people at your table. Create a poster of your table’s representations. Facilitate this as a Think-Pair-Share. Ask groups to come up with as many meanings for ¾ as they can. Collect their names on poster paper and display the posters for the remainder of the week. The Thinking Mathematics article has a complete collection of interpretations/representations for rational numbers. This article is for the facilitator’s reference only (do not hand out copies to the participants). Some possible representations: Equivalent fractions measurement contexts-time-3/4 of an inch, foot, yd… 3:4 ratio pictures of ¾ of a whole pictures of ¾ of a set .75 75% 3  4 5th Grade WPS Math Content Workshop 11/13/2018

Rest of group members view other posters.
Collection Box: ¾ Gallery Walk One person from each group “mans” the group’s poster to answer questions. Rest of group members view other posters. Most common representations? Most unusual/surprising representations? 5th Grade WPS Math Content Workshop 11/13/2018

Fraction Interpretations
Part-Whole Parts of a region Parts of a set or group Measurement Quotient Ratio Rate Multiplicative Operator This list includes that 5 interpretations of rational numbers. These are defined in more detail in the Thinking Mathematics Handout. Use the examples from the article to better define and describe each interpretation. Again, the article is for the facilitator’s use and not to be distributed to the participants. 5th Grade WPS Math Content Workshop 11/13/2018

Collection Box: ¾ Analyzing Fraction Interpretations Identify the fraction interpretation illustrated by each of your collection box entries. Denote the interpretation with a colored pencil. Red: Part-whole region Blue: Part-whole set Green: Measurement Orange: Ratio Purple: Rate Brown: Operator Which interpretations were most common? Least common? Which do you typically address in your mathematics curriculum? Have the groups go back to their posters and identify the different fraction interpretations that they have represented, using colored pencils (or markers, if that’s all that is available). After participants have finished this, have them discuss the last two questions as a group and then share out to all of the participants. 5th Grade WPS Math Content Workshop 11/13/2018

Key Fraction Concepts Identifying the “Whole”, “One”, or “Unit”
Relationships Whole to Part Part to Whole Regions Sets Equal size pieces Congruent Area Equivalent Fractions Comparing Fractions This is a overview slide. Go through it with the participants, but don’t spend a lot of time on it now. It just foreshadows fraction concepts that participants will be exploring during the workshop. 5th Grade WPS Math Content Workshop 11/13/2018

“Fraction” Sense How would you rate your own “Fraction” Sense?
Magnitude/Quantity Making sense of symbols Ordering and comparing Benchmarking Equivalence Representation Physical Pictorial Words Symbols Sense-Making Estimation Operation sense Interpreting fractions in context How would you rate your own “Fraction” Sense? How would you rate your students’ “Fraction” Sense? Facilitator can ask participants: What does it mean to have “fraction” sense? What should a student be able to do to demonstrate that they have fraction sense? This list represents components of “fraction sense”. Again, don’t dwell on these ideas here. This slide just provides an anticipatory set for what participants will investigate over the next four days. Participants will also be developing their own “fraction” sense—I.e., number sense for fractions. 5th Grade WPS Math Content Workshop 11/13/2018

Fractional Parts of Regions
“I’ll take a large pizza with half-onion, two-thirds olives, nine-fifteenths mushrooms, five-eighths pepperoni, one-eighth anchovies, and extra cheese on five-ninths of the onion half.” Close to Home by John McPherson, 1993 5th Grade WPS Math Content Workshop 11/13/2018

Making Fractions Concrete
Comic Strip Fractions Teachers will need at least 11 strips for this activity. Please have plenty available and accessible for participants when they make mistakes. Making Fractions Concrete 5th Grade WPS Math Content Workshop 11/13/2018

Comic Strip Fractions You decide to create a comic strip for your school’s newspaper. To do this, you cut strips of paper that are a little narrower than the width of a newspaper page. The strip represents one whole comic. For your first comic, you want to have one frame. Label this strip “one whole.” There are several slight changes from the BTA version of Comic Strips. In this version, the first strip that teachers make is the “1 whole” strip. 5th Grade WPS Math Content Workshop 11/13/2018

Comic Strip Fractions Fold a new strip of paper in half.
Without opening up the strip, fold the strip in half again. Without unfolding the strip, predict the number of equal parts. Now unfold your strip. How many equal parts do you have? Label each of the parts with the appropriate fraction. Get a new strip. Fold it in half a total of three times. Predict the number of frames and check your prediction. Label each of the parts with the appropriate fraction. Repeat the folding in half process with a new strip of paper. Fold the strip in half a total of four times. Predict the number of frames and check your prediction. Label each of the parts with the appropriate fraction. 5th Grade WPS Math Content Workshop 11/13/2018

5th Grade WPS Math Content Workshop
Comic Strip Fractions Diane was puzzled about the way the folding activity contradicted what she was thinking. When Diane folded her “whole” strip into halves then halves again she got fourths just as she expected. But when she folded her strip a third time into halves, she expected to get 6ths because 3 times 2 is 6. When she folded it 4 times she expected 8ths because 2 times 4 is 8. She was surprised to find out that she was wrong! How would you explain to Diane the mathematical relationship between the number of folds and the number of pieces? The facilitator can differentiate this question for the participants through the types of answers that are accepted: A description of a recursive relationship (each strip is double the previous strip) – although this is correct, push teachers to relate the number of folds to the number of pieces. A written/verbal description of the relationship An algebraic function Many participants might feel like they can’t write the algebraic function. Use questions to guide them to the function without giving away the answer. 5th Grade WPS Math Content Workshop 11/13/2018

Comic Strip Fractions Make strips to show the fractions listed below. Describe the folds you used to make each strip. thirds fifths sixths ninths tenths twelfths 5th Grade WPS Math Content Workshop 11/13/2018

Which strips helped you make other strips?
Comic Strip Fractions Making Connections Which strips helped you make other strips? Explain the underlying mathematical relationships between these strips. 5th Grade WPS Math Content Workshop 11/13/2018

Comic Strip Fractions Arrange your strips in rows so that all of the left edges are lined up and the strips are ordered from the strip with the largest parts to the strip with the smallest parts. Write as many number sentences as you can that relate the sizes of your fraction pieces. We will be using these fraction strips throughout this workshop, so be sure to keep them in their envelope (Your “Fraction Kit”). 5th Grade WPS Math Content Workshop 11/13/2018

Looking through Teacher Lenses
How would you characterize the level of this task: High or low cognitive demand? What mathematical ideas are embedded in the task? What makes this worthwhile mathematics? If teachers say “low level” because they already know how to name fractional parts, ask them to consider the level of the tasks for their students. Additionally ask participants to consider the problem solving that was required to make the more complicated strips and the connections that can be made to multiplication (dividing the halves into thirds is the same and dividing the thirds into halves). 5th Grade WPS Math Content Workshop 11/13/2018

Candy Bar Cut-Ups Use blank fraction strips to represent candy bars.
How should you divide a candy bar so that: I get ½ of what you get? I get twice what you get? I get ⅓ of your share? I get 3 times as much? I get ¼ of your share? I get 4 times as much? I get ⅔ of what you get? This is a partner activity. Partners will need lots of blank strips, because they will have to attempt each one several times until they see the pattern. For each designated sharing, they should share one strip between them. The questions go together in pairs: If I get half as much as you, you get twice as much as me. The sharings in each pair involve dividing the candy bar the same way. The only difference is which partner gets the larger piece. This is one of the relationships the participants are supposed to learn in the activity, so don’t give it away. This can be a facilitator led activity---going through each pair of questions revealing one question at a time with the PowerPoint animation feature, then discussing each one. Or the activity sheet can be distributed and pairs can work independently, with group debriefing after everyone has attempted all the problems. BE SURE TO WATCH HOW PARTICIPANTS SOLVE THE FIRST PROBLEM. Many simply divide their strip in half—not realizing that that DOES NOT give one person ½ as much as the other. Helpful prompt: “Ask yourself: “Do I have half as much as you have?” “Do you have twice as much as me?” In the generalizing discussion, be sure to ask: How did you determine if you divided your candy bar correctly? When you did that, what was the whole? (e.g., For ½ as much, whole was larger piece; for twice as much, whole is the smaller piece.) What fractional parts did you divide the original candy bar into for each sharing problem? Generalize: To share so that you have 1/nth of mine or I have n times yours (answer: n+1 pieces); To share so that you have m/n’s of mine Can you generalize this? 5th Grade WPS Math Content Workshop 11/13/2018

Equivalent Fractions Why Do Our Rules Work?
5th Grade WPS Math Content Workshop 11/13/2018

Equivalent Fractions This morning, we investigated equivalent fraction concepts. Two fractions are equivalent if they are representations for the same amount or quantity—if they are the same number. We will now investigate the mathematics underlying the rules (algorithms) for generating equivalent fractions. We worked with equivalent fraction concepts on Days 2 and 3. Might ask if anyone can describe when they were used— Day 2: Comic Strip Fractions—making Fraction Kits, Fraction Wall, playing fraction games. Also, in Whole to Parts (pattern block fractions). Day 3: Chocolate fractions—writing as many different fraction sentences as you could to describe your candy “whole”, etc. These are examples of how you can develop the concept of equivalent fractions. Today, we’re going to look more formally at equivalent fractions. We’re going to examine the mathematics underlying the typical rules (algorithms) for creating equivalent fractions. It’s important that students understand the mathematics behind the rules, as well as know the rules. 5th Grade WPS Math Content Workshop 11/13/2018

Writing Equivalent Fractions
3/4 3/4 1/3 2/5 The purpose of this activity is to investigate the mathematical relationships underlying the rules for writing equivalent fractions. Explain and model the process for making equivalent fractions. Slide is animated to show each step. First, draw vertical lines and shade each square to show the indicated fraction. Then slice each square into an equal number of horizontal slices, using anywhere from two to eight slices. For each sliced square, write an equation showing the equivalent fraction. Create 4 equivalent fractions for ¾, 1/3 and 2/5. It is CRUCIAL that participants follow the directions in the correct order. Participants have the tendency to just create models of the fractions they want to create. By doing this, the participants don’t connect that when they divide the model into equal parts, they are dividing both the whole and the part of the fraction (i.e., the numerator and the denominator). We discuss this concept on the next page. 3/4 = 6/8 5th Grade WPS Math Content Workshop 11/13/2018

What patterns do you notice?
Equivalent Fractions What patterns do you notice? When you sliced the squares, how did the total number of parts change? How did the number of shaded parts change? The pattern to notice is that the number of shaded parts increased by the same factor as the number of total parts, e.g., when you cut ¾ with a vertical slice, you get double the number pieces, etc. 3 = 3 x 2 = 6 4 x Useful to write several equivalent fractions in this way. 5th Grade WPS Math Content Workshop 11/13/2018

Equivalent Fractions Analyzing Rules for Writing Equivalent Fractions Multiplication Rule To find an equivalent fraction, multiply both the numerator and denominator of the fraction by the same number. Division Rule To find an equivalent fraction, divide both the numerator and denominator of the fraction by the same number. How can you use sets of equivalent fractions to explain why these rules work? Make explicit the connection between the patterns and the rules. Also, ask whether the fraction is being increased by a factor of 2, 3, etc. when the numerator and denominator are multiplied by 2, 3, etc. This is the place to point out that the original fraction doesn’t change. It is being multiplied or divided by 1 in the form n/n. 5th Grade WPS Math Content Workshop 11/13/2018

Equivalent Fractions Aaron was puzzled.
Is this Possible? Aaron was puzzled. “When I start with a fraction, and multiply the numerator and denominator by the same number, I always get an equivalent fraction. But, if the rule is correct, how can 6/8 and 9/12 be equivalent? There is no number I can multiply the numerator and denominator of one fraction by to get the other one.” Is Aaron correct? Are there some situations where the rules do not apply? Participants will probably say both of these fractions are equivalent to ¾, so they are equal. But some may say there is no number to multiply one by to get the other. In other words, the rule doesn’t always apply Big idea is that “number” is not limited to whole numbers. There is a non-whole number that can be used, i.e.,. 8 x 1.5 (or 1 ½) = 12 So, x 8 = 8 x = 12 5th Grade WPS Math Content Workshop 11/13/2018

Comparing and Ordering Fractions
Benchmark Fractions 5th Grade WPS Math Content Workshop 11/13/2018

Order from Least to Greatest
Raise your hand when you have them in order. Many participants probably only know “rules” for comparing and ordering fractions, e.g., rewrite all fractions with common denominator. The purpose of this opening task is to illustrate that other methods can be used—and are easier than the “convert to a common denom.” rule. These fractions can be easily ordered if one notices that they all have a common numerator-- Tell the participants you are going to show them a list of fractions to put in order from least to greatest. They are to raise their hands when they have the answer. Show only the first set of fractions. Watch what participants do. Many probably will start trying to find a common denom. Give them only a few minutes. Most often, few if any people have their hands up. Stop the group and ask what is taking them so long. Then show second set. Should see hands immediately. Ask why is this so much easier than the first---someone will say you are comparing unit fractions, so can just look at denominators--- Then, direct attention back to the first set---look at the numerators, etc. Couldn’t we take the same approach? (If some participants solved the first set quickly, let them explain how they did it, etc. Illustrates how blindly following rules can make an easy problem difficult and that there are other ways to compare fractions. 5th Grade WPS Math Content Workshop 11/13/2018

Fraction Number Sense “The focus on fractional parts is an important beginning [for understanding fractions]. But number sense with fractions demands more—it requires that students have some intuitive feel for fractions. They should know ‘about’ how big a particular fraction is and be able to tell easily which of two fractions is larger.” Van de Walle, 2004, p. 251 5th Grade WPS Math Content Workshop 11/13/2018

Benchmarks: Fractions Close to 0, ½, and 1
1 Write 3 fractions that are: Close to 0. Close to ½ but not more than ½. Close to ½ and more than ½. Close to 1 but not more than 1. Close to 1 and more than 1. Benchmarks are reference points. The most important fraction benchmarks are 0, ½ and 1. Close to 0: After participants have written their 3 fractions, ask for an example. Be sure to ask how the participant knows the fraction is close to 0. Then ask if anyone has a fraction closer to 0—and how they know it is closer. Continue this a few more times. Use this procedure for each “close to” statement. For the Close to ½ statements: Prompt for a fraction with an odd number denominator if necessary. Discuss how they know it is close to ½, and whether it is more or less than ½. 5th Grade WPS Math Content Workshop 11/13/2018

Benchmarks: Fractions Close to 0, ½, and 1
1 Complete the statements: A fraction is close to 0 when . . . Close to ½ when . . . A fraction is close to 1 when . . . A fraction is close to 0 when . . . the numerator is very small compared to the denominator. Close to ½ when . . . the numerator is about half the size of the denominator. A fraction is close to 1 when . . . the numerator is very close in size to the denominator. Be sure to use numerator and denominator vocabulary. Include these statements on your classroom list of Important Concepts (a poster paper list) if you are keeping one. 5th Grade WPS Math Content Workshop 11/13/2018

Estimate the Sum + Purpose of this is to use benchmarks to estimate: Sum is close to, but less than 6, since you are adding 6 fractions that are close to, but less than 1. Probes: If they say “close to 6”, ask “How close?” Etc. If “between 5 and 6”, ask closer to 5 or 6? Etc. Be sure to ask “Can you tell if the sum is more or less than 5 ½ ? Answer: Sum is greater than 5 ½. If all addends were 11/12, sum would equal 5 ½ , because the sum would be 6 twelfths less than 6, and 6/12 = ½. Because five of the addends are greater than 11/12, sum must be greater than 5 ½ (because distance the sum is from 6 is less than ½.) 5th Grade WPS Math Content Workshop 11/13/2018

Fraction Number Sense “You have probably learned rules or algorithms for comparing two fractions…” “If children are taught these rules before they have had the opportunity to think about the relative size of various fractions, there is little chance that they will develop any familiarity with or number sense about fraction size.” Van de Walle, 2004, p. 252 5th Grade WPS Math Content Workshop 11/13/2018

Conceptual Comparisons
Fraction Number Sense Conceptual Comparisons Determine the greater fraction in each pair. Do not use drawings, models, or algorithms. Rely on concepts. 4/7 or 5/7 4/5 or 4/9 3/8 or 4/10 5/3 or 5/8 3/4 or 9/10 3/8 or 4/7 7/12 or 5/12 3/5 or 3/7 5/8 or 6/10 9/8 or 4/3 4/6 or 7/12 8/9 or 7/8 Can do this as partner activity with handout, or as whole group activity from PowerPoint. If the latter, structure so everyone participates, .e.g., use slates or paper and have everyone write down the greater fraction and hold it up when they are done. Or ask for show of hands—how many think x is greater? Y is greater? Etc. Always ask if anyone solved it another way to get all approaches for each one. Common denom already there—so have same size pieces, 5 > 4. Push for concept underlying common denom; don’t accept just rule. ½ as benchmark: 4/5 > ½, in fact is close to 1; 4/9 < ½. Ask how you know 4/9 < ½. Also, common numerators—have 4 of each piece, but 5ths are greater than 9ths. (Ask how you know 5th are greater than 9ths.) Use ½ as benchmark, distance from ½: both are 1 unit fraction less than ½, I.e., 3/8 is 1/8 less than ½; 4/10 is 1/10 less than ½. 1/10 is less than 1/8; so 4/10 is closer to ½ than 3/8. 4/10 is greater. Use 1 as benchmark: 5/3 > 1; 5/8 < 1. Be sure to ask how you know that. Or, use common numerators—5 of each ,etc. Use 1 as benchmark; distance from 1 (like #3) Or, 7.5/10 is same as ¾, so 9/10 is larger. Use ½ as benchmark; 3/8 < ½, 4/7 > ½. Be sure to ask how they know this. Already common denom, like #1. Or 7/12 > ½; 5/12 < ½. Use ½ as benchmark, 3/5 > ½, 3/7 < ½. Or, common numerators. Use ½ as benchmark and distance from ½. Same as #3, except since both are larger than ½, the fraction closest to ½ is the lesser one. 5/8 > 6/10. Use 1 as benchmark and distance from 1. Same as 9. 4/3 > 9/8. Use ½ as benchmark and distance from ½. Same as #9. Use 1 as benchmark and distance from 1. Same as #3. 8/9 > 7/8. 5th Grade WPS Math Content Workshop 11/13/2018

Strategies for Comparing and Ordering Fractions
Benchmarks Common Denominators Common Numerators 5th Grade WPS Math Content Workshop 11/13/2018

Frac-O For 2 players Object of the game: Be the first player to arrange five fraction cards from smallest to largest. Closest to zero Closest to one 5th Grade WPS Math Content Workshop 11/13/2018

Frac-O What strategies did you use? What mathematics did you use?
5th Grade WPS Math Content Workshop 11/13/2018

Exploring Numerators and Denominators
Suppose a, b, and c represent whole numbers different from 0. Also, suppose that a > b > c. What can you say about each of these fractions? a/b b/a b/c c/b a/c c/a This activity focuses on students’ understanding of fractions. Specifically, students are asked to make generalizations about fractions by comparing the numerator and the denominator. This is done right from the PowerPoint—there is no student sheet. Get participants started by having them consider specific values, for example, a=3, b=2, c=1. Solutions: a/b > 1 b/a < 1 b/c > 1 c/b < 1 a/c > 1 c/a < 1 5th Grade WPS Math Content Workshop 11/13/2018

Exploring Numerators and Denominators
Suppose a, b, and c represent whole numbers different from 0. Also, suppose that a > b > c. If possible, tell which is larger. Justify your thinking. a/c or b/c a/b or b/b a/b or a/c Solution: a/c > b/c a/b > b/b a/b < a/c 5th Grade WPS Math Content Workshop 11/13/2018

Ticket Out of the Door How has this experience with fractional representations changed the way you think about fractions? How will it change how you teach fractions? 5th Grade WPS Math Content Workshop 11/13/2018

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