# Making Sense of Problems: Part 2

This post is an extension of a previous post. For the background story to this post, it will be helpful to read THIS POST first.

The original Noticing and Wondering from the launch of the lesson:

Here are some expanded descriptions of the student work:

Chose numbers strategically to make it easier for themselves:

These two girls were great because they wrote out the paragraph first with the blanks left to fill in after they made a decision on their numbers. You can see the erased 5 in the second blank. When I asked them about it, they said 25 in a class seemed like too many but they couldn’t make the class too small to each get more than 1 bar.

These two were concerned with the number in each box. They said they knew 6 usually came in a box so they just did 4 boxes and then wrote the students in last to make it easy division.

These two were done SUPER fast so I gave them 5 more bars to try and decide what they wanted to do with them. They didn’t do any written work, but asked me how they divide something up into 5 pieces because then each student could get a piece. “We know halves and fourths, but that is not 5 pieces.” After playing around with “fiveths” I gave them the word fifths and they wrote down 1/5.

These two partners were so interesting because when I walked by the first time they had chosen their numbers together, but when I went back the second time, their computation was completely different. I absolutely loved that and asked them to explain their strategy to one another and asked how they were the same and how they were different. The difference was more about the look  of their work, but they agreed they were the same because it was still how many 30s were in 63.

Chose numbers randomly:

…and then they worked together on breaking the extra 3 into 10 pieces. Because they didn’t know how to name tenths, they went to something they obviously knew something about…percents! We ran out of time to ask how they knew that was 10%, but I have to make a point to go back and ask!

Dealt with the leftovers using fractions:

This one took a while for me to figure out. These two girls finished rather quickly, so I asked them if they could share the leftovers equally among the kids in the class. It looks like they multiplied the 22 by 2 to get how many pieces they would have if they split them in halves. They each person got an extra 1/2 and they were left with 18 halves. They multiplied by 2 to make them wholes again and ended with 9 bars left over. The sense-making in this one was so incredible to me.

Chose numbers strategically to make it harder for themselves:

When I asked these two girls why they chose the numbers they did and they said they wanted to make it hard! There are so many things I think continued to be fabulous after their initial number choice. The partial products for multiplication and then the repeated subtraction were amazing. I asked them why they were subtracting 26 every time. They said each time they subtracted 26, each student got 1, hence the growing list of 1,2,3,4,5… Absolutely awesome and something I would have never seen if I had given the original problem and the sense-making of what is happening whey you repeated subtract to divide just blows my  mind.

To me, every one of these examples, along with all of the papers in the class that day, demonstrated to me how we need to look critically at our math textbooks, think deeply about what we learn about students as they do that work, and adapt materials to allow students to make sense of problems and allow us to learn more about their understandings.

# 1st Graders Talking and Using Half

Yesterday, I had the chance to teach a 1st grade math class. The teacher told me they are about to start their fraction lessons so I thought it would be fun to do a quick check in on what they currently think about half and then do a numberless story problem to see if they incorporated anything about half in that work.

I launched with “Tell me everything you know about half.”

One student started by telling me it is like half a piece of pizza, so I asked what that looked like to her and she said it was the whole thing (big circle with hand) and then cut in half (hand straight down vertically). That springboarded into half of lots of things, cookies, strawberries…and many other things. Each time I asked if each half was the same in the different things and they said no, they were different sizes. So, I asked what was the same and someone said they were all cut in the middle. I got some cool sports references, I asked when halftime happened and they said in the middle of the game. Then one girl said “5 is half of 10.” Awesome. I asked how she knew that and she said, “because 5+5=10.” Hands shot up everywhere after that with other numbers and their halves.

Then I posed the following story on the board and read it to the class:

There was a pile of blocks on the table. Jimmy came into the room and took some of the blocks. He gave the rest of the blocks to his friend Kali.

We did a notice/wonder and they wondered the things I had hoped: How many blocks were there? How many did Jimmy take? How many did Kali get? It was really cool that one student noticed there were none left on the table because Kali got “the rest.” I didn’t expect that one!

I let the partners choose their own numbers and as they got their answers, I asked if they could write an equation for their work. My plan for the group share was to have groups share, some who split the blocks in half, others who did not. My backup plan, because I never know what could happen, was to compare addition and subtraction equations for the same story to see if they noticed a relationship. I ended up with enough half/not-half that I went with that!

Here were some of the groups who split the blocks in half. The number choices were really interesting. I would love to put all of them up there for the class to talk about…why did they choose numbers that end in zero? What do we know about those numbers?

Some did not choose a number they could easily split in half. The group on the right noticed the commutative property right away and drew lines to show the same numbers in their equations.

Some really wanted to write as many equations as they could that didn’t necessarily match what was happening in the story but was great mathematical reasoning in their work!

I cannot wait to see what this group does in their fraction unit, so many great thoughts and work about half!