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

Making Sense of Math:Thinking Rationally

Amy LewisMath Specialist

IU1 Center for STEM Education

Goals for the courseGoals 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.

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.

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.

Making HalvesMaking 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.

Share methods with classShare methods with class

Looking through Looking through Teacher LensesTeacher 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?

Collection Box: ¾Collection 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.

• 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?

Collection Box: ¾Collection Box: ¾ Gallery Walk

• Part-Whole– Parts of a region– Parts of a set or group

• Measurement• Quotient• Ratio

– Ratio– Rate

• Multiplicative Operator

Fraction InterpretationsFraction Interpretations

• Part-Whole– Parts of a region– Parts of a set or group

• Measurement• Quotient• Ratio

– Ratio– Rate

• Multiplicative Operator

Fraction InterpretationsFraction 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?

Collection Box: ¾Collection Box: ¾ Analyzing Fraction Interpretations

Key Fraction ConceptsKey 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

““Fraction” SenseFraction” 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

Fractional Parts of RegionsFractional 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

Selling Advertising SpaceSelling Advertising Space

Ad FractionsAd 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.

• 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.

Ad FractionsAd 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?


• thirds• fifths• sixths

• ninths• tenths• twelfths

Make strips to show the fractions listed below. Describe the folds you used to make each strip.


• Which strips helped you make other strips?

• Explain the underlying mathematical relationships between these strips.

Ad FractionsAd Fractions Making Connections

• 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”).






Sharing QuesadillasSharing 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 manyquesadillas should each group receive?

• 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.

Sharing QuesadillasSharing 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.


• 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 3 + 4 = 7. The pieces might be tiny, but they won’t make as big a mess.


• 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?


• 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.


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?






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 6.

Van De Walle, 2004

Sharing QuesadillasSharing QuesadillasRecap

Hexagon FractionsHexagon 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.

• 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

Triple HexagonTriple Hexagon

• Use the large hexagon shape as the whole. (The ONE)

• Determine what fractional part each pattern block shape represents:– Hexagon– Trapezoid– Rhombus– Triangle

Large HexagonLarge Hexagon

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?

How is This Possible?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?

How is This Possible?How is This Possible?

• 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?

Making ConnectionsMaking Connections





From Parts to WholesFrom Parts to WholesWhat is the whole if…• the rhombus is 1/2? • the rhombus is 1/3?• the rhombus is 1/4?• the trapezoid is 3/4?• the hexagon is 2/3?• the hexagon is 3/5?• the rhombus is 2/9?

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.

Pattern Block PuzzlesPattern 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?

From Parts to WholesFrom Parts to WholesDebriefing the Fraction Block Puzzles





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?

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?

Making Sense of Math Wiki

http://makingsenseofmath.iu1.wikispaces.net

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

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.

Final Project

• 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.

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.

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.

Questions?

Amy [email protected]

(724) 250-3330

Documents

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