Since beginning my math education journey, something I’ve prioritized is giving my students opportunities to work through sense making and problem solving mathematical puzzles.

One that I developed for my Algebra 2 class surprisingly had application across all ages of students that I teach, including my 6th graders whom I introduced to the puzzle on a whim. What a fun surprise it was to see them solve it in a way I wouldn’t have thought to teach!

Here is the question:

You look outside into the parking lot and see a total of 12 vehicles parked there. There is a combination of cars and motorcycles. You also happen to know that there are 42 tires in the parking lot. How many cars and how many motorcycles each are in the parking lot?

I developed this question with the intention of guiding my Algebra 2 students toward writing a system of two linear equations and solving by way of substitution or combination. I knew many of my 10th and 11th grade students would not immediately go this direction, and in fact, only a few guided by foreign exchange students tried it.

Despite not directly discovering the method which I eventually wanted them to learn, it warmed my heart to find that all of the groups of 2-3 teenagers standing around whiteboards in my classroom solved at least 4 different iterations of my original question, each with increasing levels of complexity, and in fact, they continued to ask for more for almost 45 minutes!

Some groups drew pictures. Some groups created tables of data. Others divided the total number of wheels by 4 and then guessed and checked from there. Still others as mentioned before wrote and solved a system of linear equations. One student led his group toward a series of steps he uses to solve a similar problem in a role playing video game he plays, and that I have yet to make sense of but worked for every question!

To increase complexity once again, I added the idea of there being tricycles in the parking lot, and groups of students found multiple possibilities of combination of vehicles from there.

Why Did I Love This Task?

This task follows the 5th Mathematics Teaching Practice Posing Purposeful Questions developed by the NCTM. The question allowed me to assess my students’ reasoning and sense making skills while also providing multiple entry points. While not all of my Algebra 2 students wrote linear equations, every single one fulfilled the purpose of the task repeatedly which was to solve a system of equations.

This task and its variations were easy to write. Here is how I wrote them:

I chose a random number of vehicles, for example, 15. Then I chose a random number of cars (I’ll choose 5 for a total of 20 tires) and fill in the rest with motorcycles (10 for a total of 20 tires). I will tell the students the following:

There are 15 vehicles, a combination of cars and motorcycles. There are 40 tires total. How many cars and how many motorcycles are in the parking lot?

The greater the number of vehicles you go with, the tougher the question tends to become for students. I think it would be best to start with small numbers to build confidence before moving on to great values.

What Did the 6th Graders Do?

Following the successful day with Algebra 2, I posed the same question to a few 6th grade students who completed their assignment early and were looking for something to do.

Despite some early hiccups, they persisted and found the most efficient way to solve the system was with a series of ratio tables. They had so much fun, they asked for more!