Paging Doctor Math: Toothpick Task

This week my 6th grade worked on a modified version of a task dealing with a toothpick pyramid created by @ddmeyer.

http://threeacts.mrmeyer.com/toothpicks/

As you will notice, the primary standard for this task is from the high school set, so I changed some of the questions to fit my 6th graders while using the video. Here are the questions I asked my students to prove...

1) How many complete rows of toothpicks can you make?

2) Will there be any extra? How many?

3) How many more would you need to complete the next full row?

4) How many toothpicks would the container need to create 20 rows?

When focusing in on the standards, my questions align to the following:

6.RP.A.1

*The idea that row 1 has 3 toothpicks and row 2 has 6 toothpicks/row 1 has 1 triangle and row 2 has 2 triangles

6.RP.A.3

*Applying the above concept to an actual situation

6.EE.A.2

*Recognizing that each row is +3 for toothpicks or +1 for triangles

Even more important than the content standards are the Standards for Mathematical Practice. I have found that what any of my students have in knowledge of content coming to me from 5th grade to 6th grade, they are severely lacking in these process standards...and this includes your traditionally high achieving math students as well. I wrote a few NCTM Blogarithm post (part one and part two) on this topic and how hard you must work to ensure that these process standards are improving on a daily basis...and this task is just another opportunity.

Can your students make sense of the problem initially?

Can your students recognize patterns?

Can your students create an argument defending their solution?

Can your students prove a solution with some sort of mathematical model?

These are just a few of the questions I emphasis when doing a task like this. At this point in the year, my 6th graders are used to me listening to what they have to say, and then walking away to let them think a little more. At the beginning of the year they didn't understand why I would walk away when they were stuck or on the brink of a discovery. Now they smile when I walk away...because they are starting to get the point.

As far as student struggles go, one of the main culprits was thinking that the number in each row was the number of total toothpicks. I had a few poor students go to 83 rows because they thought that meant 249 total toothpicks. We had a good laugh when they realized they spent 20 minutes on that process for a minimal result. Those types of mistakes happen and can cause frustration, but all the mistakes can be turned into great teachable moments mathematically, which will ultimately help the student improve.

After looking this over, what are other questions I could have asked my 6th grade students?

Here are varied levels of student work for this task: