# Focus on Math

## Helping children become mathematicians!

### Questions Regarding Snowmen: gr 2-4January 5, 2012

Yesterday five of my seven math classes at Charlie Lake School did questions that had to do with building snowmen. The two grade 2 classes did this version:

Some children at Charlie Lake School are having a contest to see who can make the best snowman. Each snowman is to be made with three big snowballs and then decorated. If there are 16 snowmen being built at the school, how many snowballs have to be rolled?

I was delighted with the thinking and figuring of the students. One grade 2 girl used two mini 10-frames to solve this. She began to skip count by threes, writing a number in each of the spaces until she had written 16 numbers.  Other students used the 100-dot array, while others added 3 16’s explaining that there would be 16 snowmen heads, 16 “middles” and 16 “bottoms”. Many students drew pictures of all 16 snowmen and use their pictures to count the number of snowballs. As I have stated before, I like giving word problems such as this that are open-ended regarding the strategies that students can use.

The other three classes (grade 2/3, grade 3, grade 3/4) did this version of the snowmen question:

Some grade 2, 3 and 4 classes at Charlie Lake School are building snowmen. Grade 2’s will use two snowballs, grade 3’s will use three snowballs, and grade 4’s will use 4 snowballs. If the grade 2’s are building 9 snowmen, the grade 3’s are building 16 snowmen, and the grade 4’s are building 15 snowmen, how many snowballs in all have to be rolled?

This question had more numbers in it than any other question I had given to date and thus was a bit more complex. In the grade 2/3 class we created a chart showing for each grade in the question what their snowmen would look like and how many were to be built. I suggested to those students that they work on the grade 2 and 3 snowmen and than go on to the grade 4 snowmen as they had time. In all three of the classes doing this version, students wanted to solve the problem by adding 9 + 16 + 15, which would give the total number of snowmen but not the total number of snowballs. In many cases students had to draw pictures to sort out all the numbers. As always, students who solved the problem before the working time was up were to find other strategies/methods for solving the problem.

In all it was a great day of solving snowmen problems. Too bad that the weather did not cooperate so the children could go outside and build some real ones!

Mathematically yours,

Carollee

### Candy Bars: a grade 2 or 3 questionNovember 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

### Missing sheep: a grade one problemNovember 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

### Communication: Recording a Problem Solving DiscussionJune 29, 2011

One of the things I learned this year was the advantage of recording student ideas as they shared strategies and solutions after solving an “rich” math problem [and I will restate my personal definition of a “rich” problem as one with a) many solutions; b) one solution but many strategies for finding it; or c) both many solutions and many possible strategies] .

For most of the school year as I would be working with students, I would record the students’ strategies and solutions on the chalkboard. My general rule of thumb is to have students tell me what they did, and I would do the recording. This was done very specifically so that the students had to practice verbalizing their thoughts — they usually found it easier to write things down with pictures and symbols that to tell me how to write things down.

I carefully recorded all that the students would tell me, but then, before my next class of students would come, I had to erase the board and get ready for another round. It occurred to me during the year that I was missing out on an important scaffolding step for students: if I were to record the work on large chart paper rather than the chalkboard, then the work could be hung for all to see could be referred to in later classes.

I should point out that in the particular case depicted in the photo, the T-chart on the right was done after the students had shared various solutions. It is important that, toward the end of the discussion, the teacher pose questions that can help the students move to a “bigger picture” — in this case I was moving toward an “n-rule” with the class.

So, that is how I began recording the discussion. I even got to the point where I printed off a large copy of the problem the students were working on. (NOTE: Thanks to Sharlene K’s brilliant idea, I now always write the student problems on the computer and copy and paste to fill a page. I usually only have to print off a few sheets, then use a paper cutter to cut them apart. The students begin each session by gluing the question strip into their exercise books — no one has to take the time to copy out a question, while it ensures that there is a copy of the question on the page.) Making a large copy of the problem and gluing it onto the recording sheet was easy since I was typing out the question anyway. I hung the chart paper on the side wall where it could be viewed. Each week, on Wednesdays, as I worked with the students, I taped the new work for a class on top of that class’ previous week’s work.

It was soon apparent that recording the discussions of solutions and strategies was a good idea. Students referred to solving previous problems (knowing the solution was still visible) making connections between one problem and another. I also referred to the previous problems, reminding them of strategies they could not clearly recall.

Even though I was doing this in an elementary setting, the principle of recording student solutions would work at ANY grade level. I HIGHLY recommend recording the class discussions on chart paper! I think you, too, will find it valuable for students and yourself.

Mathematically yours,
Carollee

PS:The problem on the page, used for a grade 3 class, is this:

Jacob, Charlie, Kara, and Heather shared a bag of Skittles.
They each ate the same amount. There were 2 left over, and
they gave those to Jacob’s little sister. How many Skittles could
have been in the bag?

### Problem Solving Question: Intermediate — Water JugsFebruary 15, 2011

Filed under: Intermediate Math Ideas & Problems — Focus on Math @ 4:00 pm
Tags:

I had a great time in Melissa’s grade 5/6 class yesterday doing a logic problem. Melissa had mentioned that she had done a few logic problems with her class and wanted me to facilitate one for her to observe.

I chose a classic “Water Jug” problem, one which, according to one story, was the inspiration for the 19th Century mathematician Simeon Denis Poisson’s pursuit of mathematics:

• Two friends have and eight-litre jug of water and wish to share it evenly. They also have two empty jugs, one which holds exactly five litres and another which holds exactly three litres. How can they measure four litres using these jugs?

The students went to work on the problem, but clearly some were having more difficulty than others (as is often the case). After a short while we stopped to have a quick discussion about which operations were likely to be used in solving the problem (addition and subtraction, and not multiplication or division). A second discussion a bit later focused on how students might organize their thinking to keep track of the “pouring”. This is a perfect example of when making a chart is a useful problem-solving strategy. Some students got stuck in a “cycle” of pouring, where they would have a line on their charts exactly like a previous line (or stage in the pouring) and they could not figure out how to get out of the cycle. Still, a number of students were able to find solutions in the time we had — and there was a real excitement in the air.

There are many variations of this problem — different sized jugs, different numbers of jugs, different target measure — something to fit nearly every level! If you haven’t yet tried logic problems with your class, I encourage you to give it a go!

I’ll not post the solution here (it is always good for YOU to practice your problem solving skills, too!), but would love to hear what your students do with the problem.

Mathematically yours,
Carollee