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Making Sense of Math: Thinking Rationally

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Making Sense of Math:Thinking Rationally

Amy Lewis

Math Specialist

IU1 Center for STEM Education

- 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.

- Use physical models to represent and manipulate fractions in order to make sense of the operations.
- Use and compare fractions and ratios.
- Examine student work to analyze uses of fractions and ratios.

- Examine the student responses.
- Which artifact demonstrates the strongest understanding of fractional representations? Why?
- Which artifact demonstrates the weakest understanding of fractional representations? Why?
- How would you address one of the misconceptions demonstrated by one of these responses?

- Think carefully about the following question. Write a complete answer. You may use drawings, words, and numbers to explain your answer. Be sure to show all of your work.
- José ate ½ of a pizza.
- Ella ate ½ of another pizza.
- José said that he ate more pizza than Ella, but Ella said they both ate the same amount. Use words and pictures to show that José could be right.

Minimal

Satisfactory

Partial

Extended

Satisfactory

- 1992 Mathematics Assessment
- 4th Grade
- ECR – Extended Constructed Response
- “Hard”

- Minimal – 18%
- Partial – 2%
- Satisfactory – 8%
- Extended – 16%
- Omitted – 57%

- Addition
- Subtraction
- Multiplication
- Division

Comparing

Candy

Students on the 7th grade field trip committee at STEM Middle School are planning a fund raising event. They can’t decide what candy to sell -- Blocko Choco or Choca Latta. Both candies cost the same so it is a matter of which candy will sell more. They use a lunchtime survey to determine the preference of their customers, the students in their school. When the results were tabulated each member of the committee designed an ad to present their results.

- 3 out of 5 students prefer Blocko Choco to Choca Latta.
- The students have spoken, students who preferred Blocko Choco outnumbered those who preferred Choca Latta by a ratio of 3 to 2.
- The students have spoken, students who preferred Blocko Choco outnumbered those who preferred Choca Latta by a ratio of 792 to 528.
- 264 more students preferred Blocko Choco to Choca Latta.
- 60% prefer Blocko Choco to Choca Latta.
- 3/5 of the students prefer Blocko Choco.
- 40% of the students prefer Choca Latta to Blocko Choco.
- The number of students who prefer Blocko Choco is 1.5 times the number who prefer Choca Latta.

- Examine the 8 statements.
- Discuss the questions with your partner or group.

- What do each of the 8 statements mean? How does each show a comparison? What is being compared and how is it being compared?
- Could all of the 8 statements be based on the same survey data?

- What information is lost in each form of comparison?
- Which is the most accurate way to compare the data? Why?

- Which would be the most effective way to advertise for Blocko Choco?
- Which would be the most effective way to advertise for Choca Latta?

- Which statements are misleading?
- If one statement was going to be used for a newspaper ad, as a consumer, which statement would influence you the most?

- How are ratios like fractions? How are they different?
- What is being compared with decimals and with percents?
- What does a scaling comparison tell you?

Many important mathematical applications involve comparing quantities. In instances where we need to know which quantity is greater or how much greater, we subtract to find a difference. Since addition and subtraction come first in a students’ experience with mathematics, this way of thinking becomes pervasive in any situation requiring comparison.

Beyond subtraction we can compare quantities using ratios, fractions, decimals, percents, unit rates, and scaling. Students must learn different ways to reason proportionally and to recognize when such reasoning is appropriate.

Making

Punch

Each year, your class presents its mathematics portfolio to parents and community members. This year, your homeroom is in charge of the refreshments for the reception that follows the presentations.

Making Punch

Four students in class give their recipes for lemon-lime punch. The class decides to analyze the recipes to determine which one will make the Fruitiest-tasting punch. The recipes are shown below.

Adam’s RecipeBobbi’s Recipe

4 cups lemon-lime concentrate 3 cups lemon-lime concentrate

8 cups club soda5 cups club soda

Carlos’ RecipeZeb’s Recipe2 cups lemon-lime concentrate1 cups lemon-lime concentrate3 cups club soda4 cups club soda

In your group, determine:

- Which recipe has the strongest taste of lemon-lime? Explain your reasoning.
- Which recipe has the weakest taste of lemon-lime? Explain your reasoning.
Record your solutions on poster paper and display it near your table.

Making Punch

Participate in a gallery walk and contrast the methods used by the groups.

Whole group discussion:

- What methods of comparison were used by the different groups?
- What are the similarities and differences across all the methods?
- Did everyone get the same answers?

In your group examine each team’s solution.

- What comparison model was used?
- What misconceptions if any are evident?
- Which solutions are mathematically equivalent?

Examining Student Work

Every participant will receive ½ cup of punch. For each recipe, how much concentrate and how much club soda are needed to make punch for 240 people? Explain your answer.

Summarize the work that you have done so far in the table below.

- What patterns do you notice in the table?

Making Connections

Each person in the group should take one of the recipes and make a table.

Next, each person in the group should make a graph of the cups of concentrate as a function of cups of punch.

- What type of relationship is shown by each graph?
- Write an equation for each graph.
- How can the equations or graphs be used to solve the original problem?
- In the graph, what does the steepness of the data represent?

Candy Jar

The Candy Jar (right) contains Jawbreakers (the circles) and Jolly Ranchers (the rectangles).

Use this Candy Jar to solve the following problems.

- Suppose you had a new treat tin with the same ratio of Jawbreakers to Jolly Ranchers as shown above, but it contained 10 Jolly Ranchers. How many Jawbreakers would you have?
- Suppose you had a treat tin with the same ratio of Jolly Ranchers to Jawbreakers as shown above, but it contained 720 candies. How many of each kind of candy would you have?
- Suppose that you are making treats to hand out to trick-or-treaters on Halloween. Each treat is a small bag that contains 5 Jolly Ranchers and 13 Jawbreakers. You have 50 Jolly Ranchers and 125 Jawbreakers. How many small bags could you make up?

- Record your solutions on poster paper.
- Reflect on your strategies:
- Did you use the same or different strategy for each problem? Why?
- How do your strategies for these problems compare to those you used for the Making Punch problem?

Kandies-R-Us sells a super-sized 500-piece Tub-O-Treats. It contains chocolate kisses in addition to Jawbreakers and Jolly Ranchers. If the ratio of Jolly Ranchers to Jawbreakers to kisses is 7:8:10, how many of each candy are in the Tub-O-Treats?

Fractions

and

Ratios

The treat tin containsJawbreakers (circles)and Jolly Ranchers (rectangles).

- Write as many different ratios as you can to describe the contents of the tin.
- Which, if any, of your ratios are fractions? Why?
- Which, if any, of your ratios are NOT fractions. Why?

- A ratio is a comparison of any two quantities.
- Part-to-Whole Ratios
- Part-to-Part Ratios
- Rates as Ratios
- A rate is a comparison of two different things or quantities, e.g., mile per hour.

- A fraction is a comparison of parts-to-wholes.

- There are nine women for every two men in the mathematics methods class.
- Two out of every five students in the methods class plan to be middle school teachers.
- National Brand orange juice costs 7 cents per ounce while Community Store orange juice only costs 6.5 cents per ounce.
- One fourth of the candies in the jar are jawbreakers.

Ratio, Fraction or Both?

- Katie’s average speed while from Pittsburgh to Philadelphia was 57 mph.
- Grandma’s favorite chocolate cake recipe calls for ¾ cup of milk.
- According to the Kool-Aid directions, for each quart of water you should add one scoop of Kool-Aid and 2 scoops of sugar.

In the world of ratios, can1/2 + 1/3 = 2/5 ?Create an example to support your conclusion.

Jason completed 2/5 of his mathematics homework and 3/7 of his science homework. How much of his mathematics and science homework did he complete?

- 29/35
- 5/12
- Some other number
- Cannot be determined

- 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?

http://makingsenseofmath.iu1.wikispaces.net

- Contains the rubrics for grading the Final Projects.

- 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.

- Unit of Instruction
- 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.

- 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.

- 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.

- 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.

Amy Lewis

alewis@washjeff.edu

(724) 250-3330