Learning to think and Reason in Mathematical Situations

Glenda LappanMichigan State University

August 2007

In the middle years of schooling, students undergo cognitive changes that give

us new opportunities in mathematics classrooms.

• Reasoning

• Abstraction

• Argument

• Controlling variables

• Examining consequences

Students Develop Mental Capacities For:

• Meeting the needs of each student

• Connecting with students’ strengths and interests

• Creating a supportive classroom environment

• Harnessing students’ need to socialize

Teaching Challenges

Representation Standard fromPrinciples and Standards for School

Mathematics

Instructional programs from prekindergarten through grade 12 should enable all students to –

• Create and use representations to organize, record, and communicate mathematical ideas

• Select, apply, and translate among mathematical representations to solve problems

• Use representations to model and interpret physical, social, and mathematical phenomena

• What do I “see”?

• What information can I extract?

• How can I use the information?

• How can I show my thinking and my solution(s)?

Reading Representations

Complete each statement using the table.

• The ratio of 7th graders who prefer comedies to 8th graders who prefer comedies is ____ to ____.

• The fraction of total students (7th and 8th) who prefer action movies is ____.

• The percent of 8th graders who prefer action movies is ____.

• Grade ____ has the greatest percent of students who prefer action movies.

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Reading Representations: Geometry Settings

Angles and Relationships

The sketches below show two members of the Grump family who are geometrically

similar.

• Write statements comparing the lengths of corresponding segments in the two Grump drawings. Use each concept at least once. – Ratio– Percent– Fraction– Scale factor

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Comparing CylindersStart with two sheets of paper. Tape the long sides of one sheet together to form a cylinder. Form a cylinder from the second sheet by taping the short sides together. Imagine that each cylinder has a top and a bottom.

Which cylinder has greater volume? Explain your reasoning.Which cylinder has greater surface area? Explain your reasoning.

How Do We Use Representations?

• To convey information• To catch the attention of a reader• To investigate a situation or

problem• To understand key ideas

When Tupelo Township was founded, the land was divided into sections that could be farmed. Each section is a square that is 1 mile long on each edge—that is, each section is 1 square mile of land.

There are 640 acres of land in a 1-square-mile section.

The diagram shows two side-by-side (adjacent) sections of land. Each section is divided among several owners. The diagram shows the part of a section that each person owns.

Dividing Land

What fraction of a section does each person own?

• If Fuentes and Theule combine their land, what fraction of a section would they own?

• Write a mathematical sentence to show your answer.

What Do Students Need To Know and be Able To Do With Representations in Mathematical

Situations?

• How to read information from a given representation

• How to represent information given in a situation or problem

• How to tinker with or see into a representation to understand a situation or problem

• How to convey one’s reasoning or ideas to others

Creating and Using Representations

a

Multiples of 5 Multiples of 4

Place the whole numbers from 1 to 40 in an appropriate place in the diagram.

Comparing fraction strips; What is equivalent?

Moving from fraction strips to number lines.

On the number line below, carefully label marks that show where 1/3 and 2/3 are located.

What is the distance from the 1/3 mark tothe 1/2 mark on the number line above?

Brownie Pans, Fraction Strips and Partitioning

to Support Multiplication

A pan of brownies costs $24 dollars. You can buy any fractional part of a pan of brownies. You pay that fraction of $24. For example, half a pan costs 1/2 of $24.

A. Mr. Sims asked to buy half a pan that was 2/3 full. What fraction of a whole pan did Mr. Sims buy and what did he pay?

B. Aunt Serena bought 3/4 of another pan that was half full. What fraction of a whole pan did she buy and how much did she pay?

Model of a Brownie Pan

A sixth-grade class raised 2/3 of their goal in 4 days.

What fraction of their goal did they raise each day on average?

Reading or Creating or Both!

How do the Areas and Perimeters Compare?

What is the size of the area in which the goat can graze?

40 ft

20ft

50 ft

20ft

40ft50ft rope

30 ft

10 ft

3/4 x π 50 + 1/4 π 30 + 1/4 π 10 = 2 2 2

6675.88

Each number sentence below models the formula for volume of a certain 3-dimensional figure. For each, name the figure being modeled, sketch and label the figure, and compute the volume.

a. 2 2/3 4 4/5 3 7/8

b. π (2.2)2 6.5

c. 1/3π (4.25)2 10

Using Area in a New Context

Nicky’s team is 1 point behind with 2 seconds left in the basketball finals.

Nicky is fouled, and gets a one-and-one foul shot.

Nicky’s free throw average is 60%.

Which of the following do you think is most likely to happen?

• Nicky will score 0 points.

• Nicky will score 1 point.

• Nicky will score 2 points.

Hit Miss

0 ptsH

itM

iss

2 pts

1 pt

P(0) = .40P(1) = .24P(2) = .36

Total 1.00

One student raised his hand and said,“But what about a three point foul?”

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A Series of Representations That Are Important in Algebra

• Atlantic City south to Cape May, New Jersey: five hours

• Ferry from Cape May

across the Delaware Bay to Lewes

• Bike to campsite

• Sarah recorded the following data about the distance traveled until they reached the ferry:

Time(hours)

Distance(miles)

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0

8

15

19

25

27

34

40

40

40

45

Day 2 Progress

Distance (miles)

90

80

70

60

50

40

30

20

10

0 0 1 2 3 4 5 6 7

Time (hours)

Distance from Lewes as the day progressed

Malcolm and Sarah’s Notes• We started at 8:30 a.m. and rode into a strong

wind until our midmorning break.• About midmorning, the wind shifted to our backs.• We stopped for lunch at a barbecue stand and

rested for about an hour. By this time, we had traveled about halfway to Norfolk.

• At around 2:00 p.m., we stopped for a brief swim in the ocean.

• At around 3:30 p.m., we had reached the north end of the Chesapeake Bay Bridge and Tunnel. We stopped for a few minutes to watch the ships passing by. Since bikes are prohibited on the bridge, the riders put their bikes in the van, and we drove across the bridge.

• We took 7.5 hours to complete today’s 80-mile trip.

What are the advantages and disadvantages of:

• A table?

• A graph?

• A written report?

For Each Graph Story

• Find the graph type that matches the story.

• Decide which variable is on each axis.• Explain what the graph tells about the

relationship.• Give the graph a title.• If another graph would better tell the

story, sketch the graph you have in mind.

aa

I II

III IV

V VI

A. The number of students who go on a school trip is related to the price of the trip per student.

B. When a skateboard rider goes down one side and back up the other of a half-pipe ramp, the skater’s speed changes as time passes.

aa

I II

III IV

V VI

C. When someone in your family takes a bath in a tub, the water level changes over the time between turning on the water and emptying the tub.

D. The number of customers at an amusement park with water slides, wave pools, and diving boards is related to the predicted high temperature for the day.

All kids are gifted; some just open their packages earlier than others.

- Michael Carr