What I had done was put the students in a testing situation, not a learning situation. Cathy Humphreys presented the same problem to seventh graders. She introduced it as I had. However, rather than have the students solve the problem and write individually, she had them work in groups of four. That way, the students could talk with one another and draw from their collective thinking. To promote further communication in the class, Cathy gave each group an overhead transparency and marker.

Group 6 wrote:. Then we multiply 30, because there is 30 cubes by 1, which equals to We drew ten sticks. See Figure 1. Group 6 figured the length of the train if the cubes were 1 inch long and then adjusted. Grade 7. See Figure 2.

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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. Group 5 gave two solutions, first figuring the length of six cubes and then figuring the length of two cubes. In order to introduce my students to problems that involve division with fractions, I use problem situations that draw on familiar contexts.

I keep the focus of their work on making sense of the situation and explaining their strategies and solutions. Will there be enough for each person in the class?

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If not, how much more will I need to buy? After students shared their answers and the methods they used, I gave them other problems to solve, using other amounts for the sizes of the large and small bags. This helped students connect the original situation to the correct mathematical representation. As before, the students were asked to explain the methods they used. The problems they wrote helped me assess their ability to connect an equation involving division with fractions to a real-world context.

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Tom was buying wood for his woodshop class. How much wood is left over? Each candy bar has five equal parts. Betty went to the local fabric store for fabric to make curtains.

## Kindergarten I

How much fabric is left over? The students shared their word problems, resulting in some very interesting discussions. After hearing the problem about Betty buying fabric for curtains, for example, I pointed out that if I went to buy fabric to make curtains, I would measure and know ahead of time how much fabric to buy and how many curtains I would be making.

Charles makes Pinewood Derby kits from 8-foot stock. How many 8-foot pieces of stock are required to fill an order for kits? After that, no one knew what to do next. I encouraged them to make a model. Then we measured and marked with masking tape 8 feet or 96 inches on the classroom floor. At this point, the students were off and running. Here is how one student expressed her thinking in writing. All of the considerations, from storing to rolling them, were an interesting challenge. In seven weeks, we collected , pennies, and we plan to continue at least until the end of the year to see how close we get to 1,, When students bring in the pennies, they toss them into a tub that is about the size of a file drawer.

That must be a million pennies. Then we figured out that we needed more than thirty tubs of pennies to make 1,, That shocked them — and me, too! I created an open-ended activity to do with my class:. If one million fifth graders each bought a Big Grab Bag of Hot Cheetohs, the Cheetohs would completely fill three of our very high ceiling classrooms that are about 10m-bym-by If one million fifth graders lined up fifteen feet apart and passed a football from one end of the line to the other, the ball could travel from Merced, California, to Antarctica!

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If one million fifth graders each ate a paper plate of lasagna and threw the plates away, the garbage would weigh as much as three blue whales and would fill a hole that is seventy-three cubic feet. 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:. 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.

## Teaching Place Value is Important

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.

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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. Materials A collection of coins dated before , placed in a clear plastic bag Overview of Lesson Marilyn is always on the lookout for ways to provide students experience with computing mentally. Her colleague Jane Crawford gave her the idea of presenting older students with the problem of figuring out the ages of coins.

To prepare for the lesson, Marilyn collected loose change for several days, choosing coins that were made before Marilyn planned to ask the students to figure in their heads rather than use paper and pencil. Her goal was for them to focus on making sense of the numbers and to discuss the different strategies they used for figuring.

Show the class the plastic bag of coins. List on the board how many of each coin are in the bag. For example, my bag contained the following:. Record their answers on the board as they report.