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Learning to think and Reason in Mathematical Situations Glenda Lappan Michigan State University August 2007. In the middle years of schooling, students undergo cognitive changes that give us new opportunities in mathematics classrooms. Students Develop Mental Capacities For:. • Reasoning

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Learning to think and Reason in Mathematical SituationsGlenda LappanMichigan State UniversityAugust 2007
In the middle years of schooling, students undergo cognitive changes that give us new opportunities in mathematics classrooms.

Students Develop Mental Capacities For:

• Reasoning

• Abstraction

• Argument

• Controlling variables

• Examining consequences


Teaching Challenges

  • Meeting the needs of each student
  • • Connecting with students’ strengths and interests
  • • Creating a supportive classroom environment
  • • Harnessing students’ need to socialize
representation standard from principles and standards for school mathematics
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

Reading Representations

• 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)?

complete each statement using the table
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.
the sketches below show two members of the grump family who are geometrically similar
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
comparing cylinders
Comparing Cylinders
  • Start 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
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

Dividing Land

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.


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
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

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 to

the 1/2 mark on the number line above?


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?

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?


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


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.

Total 1.00




2 pts

0 pts

P(0) = .40

P(1) = .24

P(2) = .36

1 pt



One student raised his hand and said,

“But what about a three point foul?”


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:

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
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.

The number of studentswho go on a school trip is related to the price of the tripper student.

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


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.

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.