Fractions

We have been exploring fractions lately. We started out cutting up an apple and talking about what fraction of the apple the pieces were. The students were able to recognize that two parts needed to be equal for them to be halves, and so forth. We also worked with blackberries to begin talking about fractions of a group of things. The additional value of these materials was that we could eat them when we finished.
We went on to make fraction kits, an activity from Marilyn Burn's About Teaching Mathematics: A K-8 Resource. In the activity, the students cut identically sized and multi-colored strips of paper up different ways. First they cut one in half and label each piece 1/2. The next strip is cut in 4 equal pieces and labeled appropriately. By the time we get to the fifth piece, the children have cut and written a lot. They are worried that they will need to cut this piece into 32 pieces, so they are relieved that this piece stays intact to represent 1 or 1/1.

After we made our fraction kits, we learned the game Cover Up, in which they roll a fraction cube to get pieces to cover the whole piece. On Friday we used the kits to explore equivalent fractions as well as how fractions add together. Later we will learn the game Uncover. It is a great hands on activity.

How do we measure liquids?

Friday I brought a container of water to math group and asked the students how we could measure a liquid. One student suggested that we could measure the temperature with a thermometer (calling up a S'math lesson we did earlier this school year.) Another said we could find out how much it weighed. Then someone pointed out it would be hard to find out because if we weighed it on a scale, we would also be weighing the container it was in. We discussed how to get around that problem. Then someone suggested measuring liquid with a measuring cup. I brought out some measuring cups and liters, and we looked at those.

Then I brought out a tall, thin glass and a short, fat glass. I asked which they thought would hold the most water. One student said he thought they would hold the same amount. The others were divided between thinking the tall glass and the wide glass would hold the most. When we carried out the experiment, it turned out they each held exactly 1 cup.

As we moved on the Math Choice time, some students chose to experiment some more with the water and the different containers.

Measurement Explorations

We have been working on finding items in the classroom that are about as long as the white and orange cuisinaire rods. Once students found 5 items for each, we learned that the white rod is one centimeter long and the orange rod is 10 centimeters or 1 decimeter long. Another day we did the same activity using inch cubes. After these explorations, students had to measure some things at home with rulers or yardsticks for homework. Where possible they were to measure in English and metric units. As we shared the results of their measurements last Friday, we developed a clear sense of the relationship between an inch and a centimeter.

This activity was drawn from Marilyn Burn's work.

Even and Odd and Crackers

Winter break is a wonderful time for I get to spend lots of time with family. One day at lunch my almost five year old granddaughter, Rachel, was staring intently at her cracker. Then she announced that it had an odd number of holes. "There are seven holes, and seven's an odd number." How do you know, she was asked. Well she had partnered up the holes and there was one left over. She said her teacher had taught her that. So hooray for Rachel's teacher who knew preschoolers are ready to learn about numbers and hooray for Rachel for taking the odd/even lesson and testing it out in real life. She continued her explorations of odd and even over our time together.

Equal Sign Explorations

Last spring when I attended the NCTM conference in D.C. I went to a workshop on Number of the Day for Algebra. The leaders of the workshop shared some research done with elementary math students. Students were given the open number sentence 8 + 4 = ☐ + 5. They were asked what number should go in the box. Not only did a majority of younger elementary students provide a wrong answer (usually answering either 12 or 17), but a majority of students through sixth grade did. Fascinated by the workshop, I sought out the book they referred to in their presentation, Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School.

So how can we help students develop a more accurate understanding of what the equal sign means? The book suggests in part a framework of using true and false number sentences and “open number sentences” (such as in the research above) to get children thinking and talking together about this topic. For children who “get” what the sign means and what that means in terms of the relationships between the numbers involved, it provides a chance to solidify their thinking as they work to explain it. For children who do not fully understand what the equal sign means, it provides a chance for them to hear their classmates explanations, which can help their developing understanding.

I have begun exploring true and false number sentences with my math group. Over time I will also introduce open number sentences in which they need to provide a missing number. It will be interesting to see how these activities and the math talk that they generate will affect the student’s thinking about numbers and equality.

Smathing Pumpkins

In Sky we do math in different ways at different times of day. It doesn't just happen in Math Group. It is a big part of what we call "S'math," science and math together. We draw a lot of our activities from AIMS Education Foundation (http://www.aimsedu.org/). AIMS stands for Activities Integrating Math & Science. We also draw on other sources, and we like to draw in any themes we are pursuing as well as student interests.

Last year we did an AIMS activity about apples. This year I thought I'd put a twist on it and do it with mini-pumpkins. After dividing into groups of 3 (at least one boy, at least one girl, at least one old-timer, at least one new-timer) we worked with the mini-pumpkins. First we looked at them with our "scientist's eye," observing closely and noticing details. Each student drew the group's pumpkin, using observation skills. Then they came up with estimates for how many teddy bear counters they thought it would take to balance the pumpkin in a scale. Once estimates were done, each group worked together to weigh the pumpkin and record how many teddy bear counters it took.

The last part of the activity was to think about the distance around the pumpkin, the circumference. Students estimated how many teddy bear counters would make a train as long as a string that went around the pumpkin in its fattest part. They also estimated how many unifix cubes would make a train that long. Then they worked together to cut a string that went around the pumpkin and to do the actual measurements and recording.

Math Talk

I often give my math homework in a "problem of the day" format. I draw a lot of the problems from Think About It! Primary Math Problems of the Day by Marcy Cook, published by Creative Publications. There are different kinds of problems. Some involve estimation. Last week students were asked to estimate how many of our math binders it would take to make a stack 1 meter high. On Fridays we review the homework as a group, sharing answers and strategies. It's a great time for the "math talk" that both helps individual students learn to put math thinking into words and gives students a chance to hear other students' ideas to broaden their perspectives.

The math binder estimation problem sparked a lot of discussion. First I asked, as I often do, what information can help us make a smart guess? Answers included how long a meter is and how thick a math binder is. Then we pulled out a meter stick and tackled the problem of how to find the actual amount. As we stacked the binders we could see that even if we got all the other math groups' binders, we wouldn't have enough binders to get to a meter. Several students showed us their ideas of what we could do. One pointed out that five binders reached the 10 centimeter mark. He explained that there are 100 centimeters in a meter, so 10 groups of 5 would be 50. Another pointed out that 10 binders were 17 cm. and she said we could add by 17's until we got to 100 (though we got to 102 rather than 100 even.) We used both ways and discovered we came up with 2 different answers. What was going on? Several students then recognized that the binders were thicker on one side, so how we stacked them effected the outcome. It was a lively math investigation.

The other problem from this week that sparked a lot of interest was the one that asked how many legs were on 4 octopi. Students were asked to write an equation to represent the problem and answer. Several students volunteered to write their equations on the board, and we talked about the different equations. Two people added up 4 eights, but one got 31 and the other got 32, so we all checked the math. Several had used multiplication equations, and we agreed that 4 X 8= 32 and 8 X 4= 32 were both correct and could both represent the problem. One person had 16 + 16= 32 and a classmate suggested an additional equation of 8 + 8= 16 would make the connection between that equation and the problem clearer.