Building the Foundation to Algebra

BUILDING THE FOUNDATION TO ALGEBRARational Numbers

Goals Examine the different interpretations of

fractions. Explore the meaning of "half". Represent fractions using regional parts

of a whole model. Develop conceptual understanding of

fractions as division.

Collection Box: ¾ 3

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?

Gallery Walk

Fraction Interpretations5

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

Measurement Quotient Ratio

Ratio Rate

Multiplicative Operator

Fraction Interpretations6

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

Measurement Quotient Ratio

Ratio Rate

Multiplicative Operator


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?

Key Fraction Concepts8

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” 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

9

• How would you rate your own “Fraction” Sense?• How would you rate your students’ “Fraction” Sense?

Fractional Parts of Regions10

“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

Making Halves

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.

Share methods with class

Making Fourths

Are These Eighths?

Looking through Teacher Lenses16

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?

Comic Strip Fractions

Comic Strip Fractions 18

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.”


Get a new strip. Fold it in half. How many equal parts do you

have? Label each of the parts with the

appropriate fraction. Fold a new strip of paper in

half. Without opening up the strip, fold

the strip in half again. Predict the number of frames and

check your prediction.

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.


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?

Comic Strip Fractions

thirds fifths sixths

ninths tenths twelfths

22

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.

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 Quesadillas

Sharing Quesadillas27

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?


Each group must 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 your solutions.


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 fifths The pieces may 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

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Building the Foundation to Algebra

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