Focus on Math

Helping children become mathematicians!

Candy Bars: a grade 2 or 3 question November 15, 2011

I recently gave both my grade two classes at Charlie Lake School a question involving groups of six. We had been working on a variety of problem-solving strategies including these:

  • using counters/objects
  • drawing pictures
  • using ten-frames
  • using a 100-dot array
  • using a 100 chart
  • using a blank number line
  • breaking numbers apart
  • using operations such as adding and subtracting
  • looking for patterns
  • using charts or tallies

I posed this question to the students:
Luke is buying candy bars to share with his classes. They come in packages of 6. How many packages will he need to buy if there are 25 students in his class?

The students glued in their question strips and we read the question together to make sure they understood what was being asked.

I will stop here and give my opinion on an issue. I know there are a lot of math text books out there with lots of writing in them, and teachers have told me that some students who are not good readers, but are better with numbers, have little success using books that require a great deal of reading. I like using a problem-based approach to mathematics, and find that, when I am focusing the lesson primarily on a single “rich” question, then I can read the question with the students, make sure students understand what is being asked, and set the students to work. Although I am a great proponent of literacy and want students to be accomplished in that area, I do not want reading to hold a student back in my math classroom. A rich question, again in my opinion, is one with multiple possible solutions OR a single solution with multiple strategies for finding the solution (or both!). I use many of the latter, and encourage students to find as many solutions as they can. In many cases, the more ways they can solve the problem, the greater their understanding of the concept.

So, back to this particular question. The students went to work solving the problem of candy bars bought in packages of 6. The photo shows some of the strategies shared in the one class. The discussion was quite interesting. The students realized that buying four packages of candy bars would get Luke 24 bars, but most did not want him to purchase another full package. They were suggesting it would be better if he then went to a convenience store and bought only one more bar, which, in real life, is a great idea!. I asked the students, what if Luke were in a hurry and had to buy only packages, and they all agreed that he would need to buy 5 packages to have enough. There was then a discussion around the extra bars: he could save them; he could sell them; he could give them to his family. Lots of good ideas!

It is important to have those kinds of discussion around division and any remainders that come up, because in real life things are much more likely not to divide evenly than to do so. The remainder must be considered carefully. If students were doing this in a “standard” way, they might be likely to say that 25 divided by 6 is “4 remainder 1” without ever considering what the remainder of 1 would stand for. In this case, it would be a student without a candy bar! The actual answer to this question is 5, not 4 remainder 1. This is a case of the answer being forced up to the next whole number.

Remember, have your student discuss the remainders!
Mathematically yours,
Carollee

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Missing sheep: a grade one problem November 7, 2011

I recently got to do a problem-solving lesson with some wee folks in a grade one class. (Thanks, Alison, for inviting me into your classroom!)
The teacher had been working a lot with 10-frames and various number relationships, but as of yet had not done math in the format of giving a question for the students to solve using strategies of their choice.

Before I gave the word problem, I introduced the “tool” that I had brought along for the students to use: mini blank 10 frame. I showed them the little 10-frames, already cut apart, and explained that they could be glued into their math journals where they would show their thinking.We spent some time discussing that students could either draw circles in the boxes of the 10-frame, or colour in the boxes, to represent a particular number.

I also introduced the word “strategy” (and its plural “strategies”) to the students. I stressed that once a solution was found using a particular strategy, their job was to look for another strategy that would solve the problem.

I gave the students the problem. Each students received a slip of paper with the question on it which they glued into their journals. I had a large copy of the question which I put on the board with a magnet (see photo). We read the question together and I made sure each student understood what it was they were to find. Here is the question:

A farmer had 12 sheep in a pen. Someone left the gate open and some sheep got out. When the farmer counted his sheep, he only had 5 in the pen. How many had escaped?

This is a subtraction question, as something is “lost” or removed, but it is an interesting question because the change number is missing. It sounds like it should be written in this form: 12 – ? = 5.  The missing-change format allows a problem to be solved by both addition (5 + ? = 12) and subtraction (12 – 5 = ?). I should mention here that I deliberately chose 5 as one of the numbers for the question, knowing that 5 is a very easy number to work with on a 10-frame.

I encouraged students to draw a picture to help them figure out how many sheep had gone missing, but I also suggested that the little 10-frames might help them. From there I “turned them lose” to do the hard thinking math sometimes requires. Alison and I circulated during the working time to prompt, question, suggest, and encourage. A number of students had difficulty getting started, and I continued to suggest the two “highlighted” strategies. Most students were able to do one or both of those strategies. A few students used other strategies that used more symbolic notation.

During this work time I was also making mental notes of which students I wanted to invite to share their strategies. I have found, especially with week folks who have the attention span of a gnat, that I need to make the best possible use of the limited sharing time. Pre-selecting students allows me to make sure a wider variety of strategies is shared to the whole group. Students were given about 12 minutes to work, and then I asked several students to share.

As students shared I wrote on the chalkboard. You will see the ten-frame strategy represented. One boy told how he used a 10 and a 2 to represent the 12 sheep. He circled five in a row to represent the sheep who did not leave the pen, then counted the ones not circled as being those that left.

Another student shared about drawing a picture. She drew a rectangle to represent the sheep pen, and then put 12 circles in the pen to represent the 12 sheep. She put X’s on the 5 sheep that stayed, they counted the ones not crossed out as the ones that had left.

One boy I talked to during the work time had nothing on his paper, but when I talked to him he proceeded to tell me how he manipulated numbers mentally to get the answer. I scribed for him as he told me this: “I know that 6 + 6 = 12, and that one of the numbers I have is 5. It is like taking one away from six, so I had to put that one with the other 6.” Thus he turned the double 6 + 6 into 5 + 7. It is important that students who do the work in their head learn to put their thinking down on paper. Mathematicians need to be able to represent their thinking~

One other boy shared that he knew if he did 12 – 5 (which represented the five sheep still there in the end) that the answer, 7, would show how many sheep had left.

Did you notice the word “represent” showing up in the sharing? That is another word I use regularly with students. I actually stress when students are drawing pictures to solve a problem, that we are NOT in ART class and that the pictures do not have to be fancy. We only need to represent the problem so we can think about it. If you use the word “represent” with students, students will use the word, too!

I was delighted with the thinking the class did during this first experience with problem solving! If we teach/model/share tools and strategies with students, they can become powerful problem solvers.

Mathematically yours,
Carollee