My second and third grade students explored a pattern. They were given the first three steps in the pattern. It looked like this:

After drawing each step, students were asked what they noticed. Here is what they said:

“I noticed they all look like L’s. Each one is getting bigger.”

“The first one is 3 high with 1 on the side. The second one is 4 high with 1 on the side. The third one is 4 high with one on the side.”

“Each one is one more high.”

“Step one has 4 blocks, step two has 5 blocks, step 3 has 6 blocks.”

Then students were allowed to explore what the 6th one in the pattern might look like. This came easy to many of the students who where able to follow the pattern to create the sixth step.

The next challenge was for students to predict what the 2oth step might be. Some students choose to continue to draw out each step in the pattern. As I was watching I noticed that the kids who were just drawing the pictures stuck to drawing the pictures. The kids who started to think about the relationship between how many blocks and the step number began to move toward a rule they could use to figure out future steps in the patter. The students who were going down this path, made their own version of a chart as a way for them to organize their information using numbers. Here is some of what they came up with:

By looking at the progression of student thinking, it really made a clear progression of thought. I could see how the kids who started to just think about the image, would easily be able to relate to the kids who started thinking about the number of blocks at each step. I could also see how the kids who started to think about the number of blocks at each step would benefit from seeing another group organize the information in a chart so they can see the pattern. A conjecture made by another group was that there are always three more blocks than the step number in the L-shape. Another group who noticed this pattern wanted to see if they could use this rule to predict how many in the 100th step. It was easy for him to say how many blocks total. Then I asked him what it would look like. How many tall would it be? He was really excited to explore.

I have been recently thinking a lot about conjectures and how to help kids think about conjectures and get exited about proving or disproving them. Since I introduced my 3rd graders to conjectures, the made a connection to inferences in reading. To hear one of my students raise her hand and say, ” I would like to make a conjecture.” is so great to hear. Another one of my students said, “I am making a math inference.”

We will be sharing out what they discovered and what they wondered during this activity. As I think about this pattern activity, I realize that this is an activity that could really be used at the beginning of the year to celebrate conjectures students are making and giving them time to prove or disprove with evidence. This also makes me think about why conjectures are important. From this activity I realized that once students proved a conjecture was true, their level of efficiency increased. For example, the students who used the rule they discovered would not have to draw out all the blocks to find out how many blocks were in step 100. My goal is to continue to help students be excited about their discoveries and really focus on their conjectures to help students increase in efficiency. A wonder I am left with after this is how to help students who didn’t discover a conjecture learn from and began to trust and apply a conjecture that was found.

Jamie

I am so glad you got to do this activity with your students. I notice you really honored each student’s approach to solving the problem. I also noticed you made the problem about more than just “the answer”. I love how you encouraged one student to estimate how big the 100th step would be. I wonder what the students would say if you asked them where the “three more” were in the image? I wonder if they could come up with any other rules that would also work.

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This is so cool that you did this with 3rd graders! I love that they were creating a table by putting the step number under the drawing and then the total number of blocks above. The student was able to see the relationship of the step number to the total number of blocks. I am wondering if your student who was excited about the 100th step was able to draw it.

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