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**Welcome to Making Sense of Math: Thinking Rationally**

A number of different graphs are posted around the room. Put “yourself” on each graph. Look around the room. Introduce yourself to three people that you don’t know. Introduce yourself to the people at your table.

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**Making Sense of Math: Thinking Rationally**

Amy Lewis Math Specialist IU1 Center for STEM Education

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Goals for the course Use a variety of tools to deepen their understanding of rational numbers and explore proportional relationships to connect fractional meanings and representations. Participate collaboratively in solving problems in other base systems to strengthen reasoning skills. Connect new understandings of ratios and fractions to classroom practice.

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**Who are we? Look at the graphs about us. What do you notice?**

With a partner, write a single sentence that best describes the group.

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**Pre-Test Please take a few minutes to complete the pre-test.**

Although you should do the best that you can, please do not feel pressure to get all of the questions perfect. This is only a measure of growth from the start of the course until the end.

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Day 1: Use physical models to represent and manipulate fractions in order to visualize their meaning and better understand relationships between fractions. Consider types of fractional representations and their bearing on understanding. Examine methods for operating with fractions.

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Making Halves Each person: Find at least three different ways to show halves on your geoboard. Record each of your halves on geoboard paper. Share your work with others in your group explaining how you know your ways show halves. As a group, pick one example to present to the entire group. Setting up the problem – ask the teachers to take a rubber band and mark off the 5X5 region to define it as the whole. Do the “obvious halves” as a group---horizontal line, vertical line, diagonal line. Their task is to find other ways to show halves. Explicitly tell them that their halves should divide the whole into two (and only two) subregions. (That is, don’t want them to show fourths, eighths, sixteenths, etc.). Keep your eye out for this as you monitor their work – this comes up with nearly every group. During the debrief, you may want to discuss why you placed this restraint on the problem (keeping cognitive demand high). Sometimes, teachers have trouble getting started. Most often, someone in each group “gets it” and provides an example. Occasionally, that is not the case. In that situation, look for an individual or group that has an example of “atypical” halves and ask that person to show his or her example to the entire group. That is usually enough to get people started. Pass out the recording sheet

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**Share methods with class**

During debriefing, be sure to ask presenters how they know their geoboard shows halves. Be sure to have presented examples of halves that are NOT congruent shapes. If necessary, specifically ask if any group has an example of halves that are not congruent shapes.

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**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. Mathematical ideas: Fractional parts must have equal areas, but are not necessarily congruent pieces. Participants may also say things like: There are many ways to show halves Finding areas of irregular shapes Questions to ask to promote understanding: How do we define “halves”? Is it equal areas or congruent shapes that define “halves”? This activity is designed to dispel the myth that equal parts have to be congruent. Pictures in the elementary textbook almost always show congruent halves. This also asks learners to create the model.

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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

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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?

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**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.

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**Fraction Interpretations**

Part-Whole Parts of a region Parts of a set or group Measurement Quotient Ratio Rate Multiplicative Operator The interpretations highlighted in green are those that are going to be focused on during the DMU session.

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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.

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**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.

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**“Fraction” Sense Magnitude/Quantity Sense-Making Representation**

Making sense of symbols Ordering and comparing Benchmarking Equivalence Representation Physical Pictorial Words Symbols Sense-Making Estimation Operation sense Interpreting fractions in context 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.

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**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

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**Selling Advertising Space**

Teachers will need at least 11 strips for this activity. Please have plenty available and accessible for participants when they make mistakes.

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Ad Fractions You decide to sell advertising space at the bottom of each page of 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 ad. For your first sale, you want to have one ad take up the whole page. Your teacher will provide you with a strip of paper that represents one whole ad. 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.

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Ad Fractions Fold a new strip of paper in half. Without opening up the strip, fold the strip in half again. 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.

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Ad 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.

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**Ad Fractions thirds fifths sixths ninths tenths twelfths**

Make strips to show the fractions listed below. Describe the folds you used to make each strip. thirds fifths sixths ninths tenths twelfths

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**Ad Fractions Which strips helped you make other strips?**

Making Connections Which strips helped you make other strips? Explain the underlying mathematical relationships between these strips.

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Ad 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”).

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**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).

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Sharing Quesadillas As part of your school’s international foods festival, a classmate brings quesadillas that he made for the entire class. However, he only brought 21 quesadillas for the 28 students in your class. Because your class normally works in groups of four, your teacher suggests that you give the same number of quesadillas to each group of four students. How many quesadillas should each group receive? The content of this problem is very similar to the problem in the BTA text. Use the Teachers Implementation Guide if you need additional guiding questions or formative assessments.

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Sharing Quesadillas Each group must then decide how to share their quesadillas equally among the group members. How would you share the quesadillas equally among the group members? With a partner, find two different ways to solve the problem. Use a picture or diagram in at least one of your solutions. Have the participants solve this in two or more ways with at least one involving a picture or diagram representation. Look for different solutions to share with the class.

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Sharing Quesadillas Compare solutions with the other people at your table. Take turns sharing your solutions. Are all solutions the same? If not, do all solutions give the same answer? Do all solutions work? Choose a solution to share with the class. Explain the solution that you chose.

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Sharing Quesadillas Paula wants to have at least one piece that is one half of a quesadilla, so she starts by dividing all of the quesadillas in half. Dwayne says that because each group has three quesadillas, he will divide each quesadilla into thirds. Clifton wants to divide each quesadilla into eighths because he says that each person will get more pieces. Juanita decides that she will divide the quesadillas into sevenths because = 7. The pieces might be tiny, but they won’t make as big a mess. Participants should now investigate the four students’ theories. Note that Juanita’s theory is slightly different from what you find in the BTA textbook. Paula will divide the three quesadillas into halves so she will have 6 halves and give each person ½. She will have two halves left ( 4 x ½ = 4/2) over so she can divide them in half and give each person one of those pieces which are ¼ pieces. Each person will receive ½ + ¼ = ¾ Dwayne will divide his quesadillas into thirds so he will have 9 thirds. He can give everyone 2/3 (4 x 2/3 = 8/3) and he will have 1/3 left over. He now can split the 1/3 piece into 4 pieces which are 1/12th of the quesadilla and give everyone 1/12th. Each person will receive 2/3 + 1/12 = 9/12 = ¾ Clifton will divide the three quesadillas into 8ths. He will have 24/8’s. Each person will receive 6/8’s or ¾’s of a quesadilla. Juanita will divide her quesadillas into 7ths. She will have 21/7’s she will give each person 5/7 and have one piece or 7th left. She will need to give everyone a 4th of that 7th which is 1/28th. 5/7 + 1/28 = 21/28 = 3/4

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**Sharing Quesadillas Analyze each student’s method and determine:**

Does the method work? Why or why not? What would you have to do to make the method work? Write a number sentence that describes the amount of quesadilla each person gets using his or her method. Is this the same amount as in your group’s solution? The second question is very important and often overlooked by the participants. This question is where the understanding of common denominators will be built upon in upcoming lessons.

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Sharing Quesadillas Darnell claims that it doesn’t matter what number of pieces the quesadillas are initially cut into—any number will work. Investigate Darnell’s method. Is he correct? Why or why not? Use mathematics to explain whether or not he is correct. If the previous 4 examples are not sufficient to generate a general strategy, have participants try to cut the quesadillas into different fractions For example: If I cut the quesadillas into 10ths then I will have 30 pieces or 30/10 and I can give each each person 7/10ths and the two left over 10ths will be split evenly again into 4 pieces-4/20- and each person will receive one 20th. So, each person gets 7/10 + 1/20=15/20 = 3/4. It is sufficient for participants to describe a general strategy such as: Cut the quesadillas into any number of pieces. Divide the number of pieces evenly among the group members. Then, cut the leftovers into 4 pieces and divide them equally among the participants.

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Sharing Quesadillas Bobbie Jo wonders if there is an easy way to figure out the amount each person gets. She wants a way that would work even if the number of quesadillas and/or number of group members changed. Try several other combinations of quesadillas and group members. How can you easily figure out the amount each person gets? The goal of this task is for participants to a form a generalization about determining the amount each person would receive in any situation.. I.e., that the solution to the division problem: Number of quesadillas ÷ number of group members can be expressed as a fraction: number of quesadillas / number of group members. And, conversely, the fraction a/b represents a division problem, a ÷ b.

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**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? The slide that follows drives home the reason why this problem is worthwhile.

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Sharing Quesadillas Recap Fractions as Division is critical to a student’s understanding of fractions. Students find this meaning of fractions unusual. It is different from the meaning that has been carefully developed in the earlier grades—that of fractions as amounts or parts of wholes, not as operations. ¼ of 24, 24/4, and 24 ÷ 4 all mean exactly the same thing. They are all expressions for Van De Walle, 2004 Spend some time talking about the relationship between fractions and divisions and ask participants how this is explicitly developed in the classroom (if it is at all).

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**Hexagon Fractions Use a hexagon as a base.**

Cover the hexagon with other pattern block pieces. Take another hexagon and build a different design on top of it. Make as many designs as you can that cover the hexagon. How many different designs can you build? How do you know you found all of them? Make fraction number sentences to describe each of your designs, e.g., 1 = ½ + ½. Partner activity Solutions: 7 (if other pieces); 8 if allow hexagon to cover a hexagon 2 trapezoids: 1 = ½ + ½ 3 blue rhombus: 1 = 1/3 + 1/3 + 1/3 6 triangles: 1 = 1/6 + 1/6 + 1/6 + 1/6 +1/6 + 1/6 Trapezoid and 3 triangles: 1 = ½ + 1/6 + 1/6 + 1/6 Trapezoid, rhombus, and triangle: 1 + ½ + 1/3 + 1/6 2 blue rhombuses and 2 triangles: 1 = 1/3 + 1/3 + 1/6 + 1/6 Blue rhombus and 4 triangles: 1 = 1/3 + 1/6 + 1/6 + 1/6 + 1/6 May need to prompt participants to create combinations of different shapes. People typically cover the hexagon using only one shape first. Be sure to take time to discuss how participants knew they found all the ways.

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**Triple Hexagon Make a triple hexagon shape.**

Use that shape as the whole. (The ONE) Determine what fractional part each pattern block shape represents: Hexagon Trapezoid Rhombus Triangle Hopefully participants will make different shapes with their three hexagons. Ask what shapes were used as wholes. Point out the different shapes of wholes. If there are no differences, ask whether it is possible to make a different shape with three hexagons? Do the hexagons have to be touching to form a single region? Were the fraction names of the different pattern block pieces the same or different?

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**Large Hexagon Use the large hexagon shape as the whole. (The ONE)**

Determine what fractional part each pattern block shape represents: Hexagon Trapezoid Rhombus Triangle One of the challenges of this larger shape is that it cannot be covered completely by hexagons. Participants will have to determine the value of the hexagon block based on the values of the other pattern blocks. Be sure to ask how participants determined the fraction for the hexagon. Let participants realize this added constraint and grapple with the mathematics. Another important idea is that the sum of the all pieces is one. One way to get at this is to ask: What could you do to check that you labeled all your pieces correctly?

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How is This Possible? From her work with pattern blocks in third grade, Lynn always thought that the trapezoid was called ½. But when she made her triple hexagon, the trapezoid wasn’t called ½ anymore! What happened? How is this possible? The key idea is that the wholes are different. This is an example of where the same shape represents different fractional parts because the wholes are different. Names of fractional parts depend on the relationship between the parts and the whole. It is important to emphasize that fractions can be meaningless unless one thinks of them in reference to the whole. Half a minute is different from half an hour, etc. It is extremely important that students have experiences in which the whole varies. Explicitly ask about the relationship between the fraction and the size of the whole, e.g., In the first activity, the trapezoid was ½. What was it in triple hexagons? What was it in the large hexagon? Can you predict what it would be in a whole that was equivalent to 6 hexagons? What is the pattern? Why? The goal is to get participants to articulate the relationship between the scale factor relating the wholes and the fractional part, e.g., When the size of the whole is increased by a factor of 3, the fraction represented by the same sized piece is 1/3 the original fraction (e.g., the piece that was ½ becomes 1/6.)

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How is This Possible? Lynn was trying to figure out which was larger, 1/3 or 1/2. “My third grade teacher said that in fractions, larger is smaller and smaller is larger, so 1/2 is larger than 1/3.” But then she looked at the three pattern block problems she just did. “The hexagon is 1/3 and the trapezoid is 1/2. The hexagon is bigger than the trapezoid. So, 1/3 IS larger than 1/2. I knew larger couldn’t be smaller!” What happened? How is this possible? Again, the key idea is that the wholes are different and names of fractional parts depend on the relationship between the parts and the whole. This is an example of how students can develop misconceptions if they do not pay attention to the whole in naming and comparing fractions.

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Making Connections Why do the same pattern blocks have different values for the hexagon, triple hexagon, and large hexagon? What is the relationship between the size of the whole shapes and the fractional value of the pattern block pieces?

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**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?

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**From Parts to Wholes What is the whole if… the rhombus is 1/2?**

the trapezoid is 3/4? the hexagon is 2/3? the hexagon is 3/5? the rhombus is 2/9? Traditionally, we have only shown students the whole and asked them what fractional part a certain piece is. This fractional reasoning or “fraction sense” is more challenging because we don’t often ask students to reverse their thinking. The first three questions use unit fractions so their solutions should be straightforward—simply iterating the unit fraction. The last 4 should be more challenging. Probe for participants’ strategies. End by asking for a general solution, I.e., “How could you find the whole, given any fractional part?”

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Pattern Block Puzzles Use your pattern block pieces to build the following shapes. Sketch your shape on the recording paper: A triangle that is 1/3 green and 2/3 red. A triangle that is 2/3 red, 1/9 green, and 2/9 blue. A parallelogram that is 3/4 blue and 1/4 green. A parallelogram that is 2/3 blue and 1/3 green. A trapezoid that is 1/2 red and 1/2 blue. Build larger versions of your solutions with the same fractional parts. Sample solutions: For #1, a very common incorrect solution is a green triangle on a red trapezoid.

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**From Parts to Wholes What strategies did you use to solve the puzzles?**

Debriefing the Fraction Block Puzzles What strategies did you use to solve the puzzles? What happened when you tried to build a larger version of your puzzle? What patterns did you notice? Have participants share their solutions-are everyone’s the same? If they are different-How are they different? Have participants share their strategies for solving the puzzles in the larger group. Make sure everyone understands the thinking of others. Asking participants to build larger models with the same fractional parts of the original puzzle is also challenging. Insist on similar shapes, not any triangle, parallelogram or trapezoid will do.The larger versions must be mathematically similar--same shape, proportional dimensions--different size. Mathematical similarity may need some definition/clarification; e.g., that all rectangles are not similar shapes may be a new idea for some participants. Investigate the larger shapes and how they relate to the original. Are they twice as large? The sides should be twice the length but the actual shape will be 4 times as large in area. For example, a shape that is 3 times the side length would be 9 times as large in area.

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**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?

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**Looking Back What mathematics have we explored today?**

How have these activities shaped your understanding of place value? How would you describe the cognitive demand of the tasks we explored today?

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**Homework Complete the following problem.**

Look at how the students solved the problem. What place value concepts do they seem to struggle with? How would you address these struggles?

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**Making Sense of Math Wiki**

Join the wiki to ensure that you can post comments on the discussion board.

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Final Project To receive 1 CPE Credit for this course, participants must complete a Final Project. Each participant can choose a Final Project from the following three choices. The Final Project is due March 17th.

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**Final Project Unit of Instruction Rubric is posted on the wiki.**

How might you teach rational number concepts different based on your learning from this course? Write a unit of instruction to incorporate the strategies into your mathematics instruction. Describe an action plan for implementation. Rubric is posted on the wiki.

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**Final Project Student Work**

Collect 5 pieces of student work that demonstrate varying levels of fractional misunderstandings. Identify the mathematical misconceptions in the work. For each artifact, write and implement an action plan that describes how you are going to use the strategies used in this course to address the misunderstanding. Reflect on the successes and challenges faced when implementing each action plan. Rubric is posted on the wiki.

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**Final Project Self-Study**

Do you have an idea/topic for a project you’d like to explore which is not listed above? Please discuss your idea with the instructor in order to receive permission to pursue your own line of study.

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Questions? Amy Lewis (724)

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Copyright © Allyn and Bacon 2010

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