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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
• Controlling variables
• Examining consequences
Instructional programs from prekindergarten through grade 12 should enable all students to –
• 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)?
Which cylinder has greater volume? Explain your reasoning.
Which cylinder has greater surface area? Explain your reasoning.
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?
Place the whole numbers from 1 to 40 in an appropriate place in the diagram.
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.
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.
P(0) = .40
P(1) = .24
P(2) = .36
“But what about a three point foul?”
the day progressed
• 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.
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.
- Michael Carr