Category: algebra

Links Picnic #3

Links Picnic is an opportunity for us to share things from around the web that help and inspire maths teachers. Here are my picks. Please add yours in the comments section. You can leave a link to your own Links Picnic!

—Maths and Engineering fonts

—Answers to the question, “Why do I have to study algebra?”

–Seven posters answering the question, “When will I ever need maths?” I really like that they link to other school subjects, not careers to which students can’t relate.

–A collection of infographics, from deep to ditzy.

–The huge catalog of activity cards from SMILE is available from the UK’s National STEM Centre.

Equations of Straight Lines: Easy Mini Whiteboard Activity

My year 8 students did this task today as part of their lesson. The mini whiteboards (MWBs) have squares on one side, so they can use these for drawing axes.

I love using the MWBs because students feel less inhibited about making mistakes. Also, it gets them working faster; I seem to have a few students who want to draw perfect axes every time, and this can take up to five minutes!

This task is self-differentiating, since students choose the lines they want to draw. But one partner usually tries to outsmart the other, so most students get to try drawing (or guessing) the lines they find “harder”. Even though students can choose to stick to easy examples, they are encouraged not to by the structure of the task.

A pdf of this activity is available. Let me know if you try it!

Do you use mini whiteboards in class?

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?

Investigation: Things that Worked

I left my year 1o class feeling really positive about how it went and so I want to record a few ideas to remember later. It’s a very small group (eight) of lower ability students and at times I find them hard to motivate. I’m not sure if it was easier today because I was away on training during their last lesson. They looked genuinely pleased to see me–perhaps their cover lesson didn’t go very well? It turns out they had a non-specialist supply teacher. They worked through the sheet I left them–an investigation called Crossed Lines from New York Cop. But they found it hard as expected, and so I wanted to review it with them.

The first good part of the lesson was asking them to use the mini whiteboards (MWBs) to draw some of the patterns. They seem less afraid to make mistakes when using the MWBs and it also lets me see more quickly what they are thinking.

Another good thing was the way I felt quite upbeat (from coffee?!) and smiley. I kept encouraging them throughout the lesson to try new ideas. We recorded our ideas as a group on the board. One student seemed to me, in my caffeine-assisted state, to be more communicative than usual. And I was glad I mentioned to her (after class) how pleased I was with her contributions.

When I asked them to talk to their partner I heard more ideas being shared than usual. I tried hard to stay out of the conversations. One boy turned to me to share his idea with me, and I replied, “I am not your partner… Talk to your partner, please.”

I managed to explain the transition to algebra better than usual. We do these little investigations every fortnight; it’s the same sheet for all nine classes of year 10 students. (This means all the other students, in the higher ability classes, are also doing the same work. We try to stay realistic about doing as much as we can–which is usually about half the sheet.) Each time the investigation leads the students to find a rule for the patterns they see, whether number patterns or shape patterns. My students know this question is coming, so we are starting to anticipate it: “Write your rule in algebra.” Now that we are on the fourth one, they are more willing to look for and express the rule. They are not as scared by the n appearing! What went well today was they agreed that the algebra was easier and more elegant to write than the word formula. Success for Mrs A!

I have realised that I am the one responsible for making sure the correction notation is used. I am happy if they can describe the algebra they want to write and then I make sure it is correct. In time I think they will be responsible for the algebraic notation on their own, but while their algebra skills are still a bit weak I want them to see correct algebra from me.

What went well for you today?

Students Learning Techniques

Exam practice: a necessary evil; working through loads of past papers. Students need to be familiar with the types of questions and what they mean.

But before this, when students are learning techniques like adding fractions, do they need to do repetitious exercises? Do they need to practice twenty (or more) simple fraction addition questions? In my summer reading book, Developing Thinking in Algebra, the authors argue that students can gain fluency with a technique not by practicing it but by putting it into use. They state that another maths education writer argued that “in order to develop competence and fluency it is necessary to divert your attention away from what you are doing, rather than into it” (a summary of Caleb Gattegno in The Science of Education Part 1: Theoretical Considerations). This reminds me of what my department was trying to do with their year 7 fractions unit. Instead of the repetition of the adding fractions section in the textbooks, I used an nrich activity that investigates a fractions question based on adding fractions. Then the students are drawn into a task about whether all fractions can be written as the sum of unit fractions, and while doing their task they practice adding fractions more than enough.

“Learners do a lot of examples in pursuit of a greater goal,” Watson, Graham, and Johnson-Wilder say: and in doing so they gain the skills that will help them later (for example, to make connections with other areas of maths and also at exam time). They also they see when a technique is needed and how it can be used. And this is more valuable than simple, mindless practice.