# Using Problem Based Learning in Maths… Again

Creating wonder in the classroom is even better when you’re discovering things with the students.

My terrible habit of preparing for lessons 5-10 minutes before they start is becoming more frequent, and scarily, more successful. Yesterday morning, in the ritual dance of having a shower, getting dressed, and making my coffee before leaving for work, I googled “how does the number plate lettering system work in South Australia” (I actually had to go into my browsing history and copy it from there). I remembered conversations that I used to have with my dad on long drives to cricket matches growing up. My dad, a mechanic who has continued the family business for over 50 years, has seen and fixed hundreds of cars. He would often talk about how you can estimate the year a certain car was made just by its registration plate. So, just before the lesson, I wrote up a quick Three Act Maths Lesson (a model of Problem Based Learning used by Dan Meyer).

Act One

I opened the lesson by showing this image to the class:

We then had some great discussions based on the following questions:

• What do you notice about the picture shown?
• What differences can you see between the registration plates?
• Have you seen any of these styles of plates before?
• Which ones wouldn’t you see so often?
• What other plates may you see apart from these?
• What appears on all the plates?
• What only appears on some?
• Can you have two registration plates with the same letters and numbers? (Some of “clever” students may say that there will always be two – one on each end of the car)

This open discussion, although not all about mathematical concepts or ideas, allows all students to engage with the lesson from the beginning. I think it is critical to engage all students at the start of the lesson, whatever level or context. Tasks therefore need a low floor and high ceiling (low level entry – high level exit).
I then asked the students to, individually, determine:

• Which State or Territory has the most registered vehicles, based on the format of the registration plate?

Most students decided that Tasmania had the least and Western Australia had the most, based on the number of digits/letters on their plates (Tasmania with 5 and WA with 7). Some students may determine that Northern Territory may have the most and, if they don’t, it is a great talking point, and a lead in to Act Two. I asked them all to ponder on this:

• Why would some people think that the Northern Territory has the most registered vehicles?

This allows students to compare this format and realise that a letter allows you to have 26 possible values, rather than the 10 possible values for a number.
After looking at the different States and Territories’ vehicle registration plates, I then asked the students to compare the two below:

Discussion stemmed from the following questions:

• What do you initially notice about the number plates?
• Which State has more registered vehicles?

Extension question (for students who find the answer quicker than others):

• Which is a later model car?

We then shared methods of solving the problem. This was elicited by using questions like:

• Who has an answer that seems reasonable/unreasonable?
• Who is confident that their method was the correct one?
• What values did people use in their calculations?
• How did you use these values?

Act Two

This is where the bulk of the learning happened for the students. Here I gave them some time to think about and try to solve the following problem:

In 1966, South Australia brought in the style of the top registration plate pictured above. The first plate was RAA-000 and the last registered vehicle with this style plate was in November 2008 with the registration XUB-299. The bottom registration plate style pictured above came in after November 2008.

How many vehicles were registered between 1966 and November 2008?

Extension question:

• What number vehicle was the bottom registration plate?

Students needed varied levels of support, with the extension question allowing extra time for all students to try and solve the problem, and more opportunity for the teacher to support them. Like in Act One, methods are shared, discussed, and different solutions explained. Other questions help students understand their answers in the context of the problem:

• What vehicles do our calculations not take into account?
• Can one vehicle have more than one registration plate?
• What is the significance of the new style of registration plate?

Act Three

Here, I usually try to either extend the students’ understanding or apply it to a different context, leaving them with a question or problem to ponder on and spend extra time investigating. This time, I showed the students the following picture with two questions:

• Which State has the most possible registration plates?
• Which State has the least possible registration plates?

The students that I used this lesson with was a Year 11 Maths class. I ran it with two different classes on the same day, same Year level, and the notes that the pre-service teachings who were observing my lessons would have been quite different. This task worked exceptionally well and I was pleasantly surprised with the level of engagement of all of the students in the class. By knowing my students, and how they learn and behave during lesson, I was able to adapt the types of questions asked, how and when they were asked, and how much support was needed from me as the teacher. Only having created the lesson ten minutes before the lesson meant that I didn’t have the answers, but I knew that the “open-ness” of the questions, discussions, and problems would allow all students to feel safe to investigate and come up with a solution of their own. The teacher’s role in facilitating discussion and guiding struggling students with key questions is essential, and will vary class to class and teacher to teacher.

The lesson can be found here: bit.do/numberplatemaths

For more lessons like this, I try to keep an up-to-date collection on a Google Sheet with links to download pages. The lessons are as I have used them, I would strongly recommend to take them and make them into your own to suit your teaching style and the learning needs of your students. This link can be found here: tiny.cc/jrowe