Completing the Square: Starter Question

Here’s an idea that worked for me last week. My A-level class learned how to complete the square earlier in the year and I wanted to know if they still remembered how to do it. Instead of asking them to just do a couple of completing the square questions, I put this up on the board.

complete the square

This appeals to me as a starter question because it’s more broadly accessible than a couple of examples and also allows for more in-depth extension work.

1. For the student who sees this and thinks, “Oh no! I can’t remember how to complete the square,” there’s a quadratic expression there for them to have a go at. While I’m circulating I can give tips to these students.

2. For the student who thinks, “Oh yes! I know how to complete the square,” there is a quadratic expression there for them to try and also permission to make up any quadratic expression at all to try. While I’m circulating, I’ll encourage them to proceed to generalisation.

3. For the student who thinks, “Oh yes! It is possible no matter what the quadratic expression is,” there is the permission there to say why and give evidence. While I’m circulating I can ask questions and push them to generalise with justification or evidence.

a mathematics lesson that worked


When I was writing this question I was trying to create something that would push forward my students’ thinking no matter what they could remember about completing the square. I think this worked for me and my students and I hope I can think of more good questions like this one!

What maths question worked for you this week?


Using Exit Slips: an #eduread post

My grade 11 class (Mathematics SL year 1) are getting ready for an exam in a week and a half. I was reading this week’s #eduread article about exit slips while they were doing a quiz. I got to the end of the article a few minutes before they finished and I was pondering the last two sentences of the article:

Exit slips are easy to use and take little time away from instruction. Many teachers use them routinely—even daily—and attest to their positive influence on student achievement.

It’s been a while since I used exit slips so I thought, well, there is no time like the present! And I wrote these three questions on the board to use immediately with my grade 11s.


Screen Shot 2014-04-25 at 11.23.44 AM


I passed out some small pieces of scrap paper, and voila!, exit slips.

The article mentions four main uses for exit slips. First, to get formative assessment data. My first two questions are of this type. Students give feedback on what they have learned. I now know that my students feel somewhat prepared; the median and modal level was 3. I need to plan more review about trigonometric functions and applications of differentiation.

Secondly, exit slips can be used to have students reflect on their learning strategies or effort. An example question would be “How hard did you work today?” I am planning to use this question soon–it could be illuminating.

Thirdly, the slips can be used to get feedback about my teaching. In the past I have often asked how my pace was during that lesson. My third question today is also of this type. Some students asked for more exam-style questions, several others want me to do tricky stuff on the board.

Last, exit slips can be a place for open communication with the teacher. In the past, I have frequently asked, “What is your foremost question or concern?” This prompt allows students to say whatever it is they want to about mathematics, our class, or anything else. The responses have ranged from useful to hilarious.

This post is for a group of mathematics teachers who read an article and chat about it each week using the hashtag #eduread. You are welcome to join in; our chat about exit slips is on Wednesday night at 8pm in North America/Thursday morning at 9am in Singapore (and the time where you are).

What questions would you ask on exit slips?

A Pythagoras’ Theorem Open Question

An open question like this one:

  • is easy to pose,
  • is easy to understand,
  • can lead to lots of practice (if that’s your goal for the students),
  • can be kept simple or extended to suit lots of students,
  • can lead to an interesting discussion,
  • can lead to a generalisation.

I used this with a year 11 class that are revising for their Foundation IGCSE exam. I asked students to come up to the board and sketch some triangles that fit the criteria. Some triangles had 10 cm as the length of one of the shorter sides. Others had 10 cm as the length of the hypotenuse. Some students had chosen to draw isosceles triangles; other students drew scalene triangles.

This task can be made more simple by asking students to draw accurate triangles and measure the sides. Then they can extend this to checking using Pythagoras’ theorem.

On the other hand, this task can be extended by asking students to generalise what they have found. If 10 cm is length of the hypotenuse, what can be said about the other two sides? If the triangle is isosceles, there is only one answer. But if it is scalene, perhaps students will call the length of the second side x. Then they can come up with a formula for the third side in terms of x. And how does this change if 10 cm is not the length of the hypotenuse but of one of the other sides?