A lack of fluency with the multiplication tables can make playing this game difficult and tedious. As you move around the room, you may wish to visit some of the pairs and ask them the following questions:. Pulling the class together for a processing session is important and necessary after the children have had the opportunity to play several rounds of the game. Processing the game gives mathematical meaning to the activity. The children need to realize that although games can be great fun, as this one certainly is, good mathematical games also have purpose.

Crafting, asking, and answering good questions can further the mathematical understanding of just about any activity. Good questions can set the stage for meaningful classroom discussion and learning. Students are no longer passive receivers of information when they asked questions that deepen and challenge their mathematical understandings and convictions. Good questions. Questions such as those that follow can help to scaffold and articulate new understandings that have come about as a result of playing the Factor Game.

Processing questions in a whole-class format also gives you the opportunity to implement talk moves. You can help to establish respectful discourse by asking for agreement or disagreement. Revoicing can emphasize important mathematics, insights, or strategies. You can have follow-up lessons that draw upon the understandings constructed from the Factor Game.

My class explores perfect, abundant, and deficient numbers as well because of the connections they can make to number choices on the Factor Game game board. Exploring and applying divisibility rules also now have a place and purpose in the curriculum. Being mathematically proficient goes far beyond being able to compute accurately and proficiently. It involves understanding and applying various relationships, properties, and procedures associated with number concepts Math Matters, Chapin and Johnson The Factor Game and the lessons that it subsequently supports can do just that. The Factor Game Game Board for For this lesson, I planned to have the students work individually to solve a measurement problem involving fractions.

Two students snapped the cubes together.

They matched the train to the line segment and found that the train was about 2 inches longer than the line segment. Then I asked them to make a different estimate. I held the train up to the line segment and removed three cubes so that their lengths matched. Then I split the train into tens. There were thirty cubes in all.

### Create a List

There were no more questions. When you record your answer, be sure to explain why it makes sense. The room became quiet with the kind of quiet that test taking often produces. Some students started to write about their ideas; some did calculations on their papers; others gazed into the distance, apparently thinking.

I think this is a good way of doing this because all you have to do is multiply the numbers and you have your answer. Karine came up with an interesting beginning. She wrote: I know its less than 30 inches because the cubes are smaller than 1 inch. She was then stumped and had no place to turn.

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Mark made a good start but then took a false turn. He wrote: Two cubes make 1 inch. So 8 x 3 makes 24 inches. I multiplied and then I reduced to get my answer. 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.

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## Grades 2nd-3rd-4th

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. 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? If not, how much more will I need to buy?

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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.

## Series: Pete and Penny's Pizza Puzzles

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. 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.

### Lesson Outline

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!

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.

## Case of the Secret Sauce by Aaron Rosenberg

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. I get it now! This proves my answer is right. They just thought about it a little differently.