Group 6 figured the length of the train if the cubes were 1 inch long and then adjusted. Grade 7. See Figure 2. Group 3 used a combination of fractions and decimals. Group 1 wrote: The total inches are We think its And you get And then divide 90 by 4. They showed how they did the calculation. Group 7 had a different approach. Figure 3. Group 5 gave two solutions, first figuring the length of six cubes and then figuring the length of two cubes. Before I began this lesson, I checked with a local hamburger restaurant and learned that there are about forty french fries in a single serving.

So you could take a zero away from the forty to make four and a zero away from the one thousand and make it one hundred and then figure out how many fours in one hundred. I knew that by removing a zero from both the 40 and the 1,, Mia made a more manageable problem that was proportional to the original problem and, therefore, would produce the same answer.

But this is a difficult concept for students to grasp. I recorded on the board:.

## Problem Solving

Abdul raised his hand. There are five two hundreds in one thousand. So I think you could multiply five by five and that would make twenty-five servings. Mark did. I get it now! This proves my answer is right. They just thought about it a little differently. How many fries would be needed if everyone in our class ordered one bag of fries? I explained to the students what they were to do. You may use any of the ideas on the board or that you have heard before that you think would help you solve this problem.

You may also use your own ideas. Please be sure to show me your thinking clearly using words, pictures, and numbers. I circulated and gave help as needed. Later we had a discussion about the answer and the methods they used. Some children used the standard algorithm and I asked them to show me a second way they could solve the problem. Many made use of finding partial products in a nonstandard way. Carol used partial products to solve 52 x Josh also made use of finding partial products to solve the problem.

In this dice activity, the class works together to generate the numbers one through twelve in order. Students need to think flexibly and consider many possibilities in order to find the solutions to the computation challenges involved. Their response was positive. Our goal is to systematically get rid of each number I listed. We have to get rid of the numbers in order, beginning with one first and then moving to two, and all the way to twelve. Then you need to think about how you can use these numbers to make the number one.

You may use any operation—addition, subtraction, multiplication, or division—or a combination of operations. Here I go.

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Amy was surprised that students appeared to be struggling. Then she realized her directions might not have been clear enough. You can use one, two, or all three of the numbers rolled to make the number you need. Adriana suggested the need for parentheses around 5 — 3. Technically parentheses are not required in this situation; however, their use is not incorrect and adds clarity. Amy decided to include them. She then pointed out that since she had rolled a 1, they could use just that 1, and she recorded that on the board too.

### Stages of Problem Solving

Next up is two. Amy wrote equations for their ideas next to the 2 and then crossed off the 2. Now that the students understood the mechanics of the activity, she gave them time to talk in groups about ways to make the rest of the numbers. Soon the board filled up with a variety of solutions for the next few numbers. The students were more interested in finding multiple solutions for each number than moving on to the next number. To keep the game moving, Amy decided to limit responses to one equation for each number.

In a few minutes the board looked like this:. The class got stuck on eleven, so Amy told the students it was time to roll again. Before rolling, she asked a question to assess their comfort with the various operations.

## Mathematical economics - Wikipedia

What numbers would you like to get now? This quick question showed Amy that her students were comfortable breaking eleven apart in different ways but were not as comfortable thinking about operations other than addition. Perhaps with more time and practice, they would open up their thinking. She rolled the dice and got two 5s and a 1.

Then she turned to the class. Amy wrote the equation next to the 11 on the board. She pointed to the first 5 and the 1. Amy gave the students a minute to talk to a partner about how to use these rolls to make twelve. This time she got a 5, a 2, and a 1.

### Stages of Problem Solving

She wrote these numbers on the board below the previous roll. The students took a minute to consider the possibilities. Christine raised her hand. Amy wrote the equation on the board and had the class double-check to make sure everyone agreed. Amy crossed out the 12 on the board and congratulated the students for clearing the board.

She asked how many rolls it took for them to accomplish their task. Since she had written each roll on the board, the students had an easy reference.