Observing Student Problem Solving
 Introduction | How Many Vehicles? -- Using Counters | Problem Reflection #1 | How Many Vehicles? -- Using Numbers and Cubes | Problem Reflection #2 | Classroom Practice | Observe a Classroom | Your Journal

Gabriella and Javier are working together and have chosen to use numbers to solve the problem. They have no difficulty getting started:

Gabriella: Since each car has four wheels, we can just keep adding 4 until we get to the 24 wheels.
Javier: Okay, if we have one car, that's four wheels.
Gabriella: Two cars would be four more wheels, so that would be eight wheels.
Javier: Let's just keep adding 4.

The students' paper looks like this:

Teacher: Explain what you did.
Javier: Each 4 stands for a car. We just kept adding on 4's until we got to 24. So, there are five cars.

Teacher: Let's check: 4 + 4 is . . .?
Students: Eight.

Teacher: And four more . . . ?
Students: Um, um . . .

Teacher: What can help you find out how many you have now?
Gabriella: We can look at the hundreds chart and count up from 8.

Teacher: Okay -- so where do you end up?
Gabriella: Twelve and four more is 16, and four more is 20, and four more is 24.

Teacher: Is there another way you can show the wheels on each car?
Javier: We could use the cubes.

Teacher: How many cubes will you need?
Students: Twenty-four.

Teacher: Right -- now show me with the cubes.

The students continue to work with the blocks. They call the teacher back when they have this configuration:

Teacher: I see what you have done. What is a stack of cubes for?
Javier: Each stack is a car, since there are four wheels on a car.

Teacher: I see. And how many cars are there?
Gabriella: Six.

Teacher: Now look back at the first way you wrote it. Do you have six cars there?
Javier: No -- we need another four.

Teacher: How do you know that?
Javier: Because we only have five, and there have to be six cars.

Teacher: Okay. Now I have a challenge for you. Can you find another way to do it so that you have different kinds of vehicles?
Students: Yeah.

Teacher: Good. Now tell me what kinds of vehicles you might have.
Gabriella: Buses and cars.
Javier: And bikes, and there was a motorcycle.

Teacher: You got it! Now go to work. Remember, you need to show me your thinking . . .

The teacher moves on to another group while the students get to work. Later, they call her back with this configuration of cubes:

Teacher: What have you got here?
Javier: Six bikes and two buses.

Teacher: How do you know that this will work?
Gabriella: Because there are 24 wheels all together.

Teacher: Do you think there might be more combinations?
Javier: Yeah -- let's find them.

The students continue working as the teacher moves on to another group of students.

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