2 Thinking algebraically
Thinking algebraically and using algebra in school involves recognising and analysing patterns and relationships, using symbols and developing generalisations. The ‘language of arithmetic’ focuses on finding answers, while the ‘language of algebra’ focuses on relationships. For example, ‘a + 0 = a’ is a symbolic representation for the generalisation that when zero is added to any number, it stays the same.
Algebra focuses on expressing a generalised relationship, whereas most mathematical lessons focus on finding the answer. So the first thing that needs to be understood is that algebra is different.
The activities in this unit will work on developing ideas about algebraic thinking:
- Activity 1 is about encouraging your students to play with numbers and making expressions that encourage them to think of the equals sign as meaning ‘is the same as’ rather than ‘find the answer’.
- Activity 2 begins to extend the students’ algebraic thinking, asking them to explore whether a statement is true or false and to make a conjecture about whether it is always true, sometimes true or always false.
- Activity 3 moves on to generalisation, encouraging students to consider whether their conjecture (or theory) works for all numbers. This means that they will be beginning to generalise about number properties in an algebraic way.
Before attempting to use the activities in this unit with your students, it would be a good idea to complete all, or at least part, of the activities yourself. It would be even better if you could try them out with a colleague as that will help you when you reflect on the experience. Trying them for yourself will mean you get insights into a learner’s experiences which can, in turn, influence your teaching and your experiences as a teacher.
Activity 1: The game of equality
This game is played by sets of two teams competing with each other. Decide how many sets of two teams you want to divide your class into.
For each set of two teams you will require the following:
- two number cards for each of the numbers 1 to 9
- operation cards for addition (+), subtraction (–), multiplication (×) and division (÷) – several of each
- a card for the sign of equality (=).
A number card can be made by writing the number on a large piece of paper. You can use a sketch or marker pen to write on the paper so that the ink is dark enough for all the students to see.
You will need some space for the students to move around. If the desks and benches cannot be moved sufficiently in your classroom then consider going outside. Regroup the class to create:
- two teams (A and B) with number cards.
- an Operation Team of four members who will have to hold the ‘operation’ cards
- one student with the ‘equals’ card (who will be called Professor Equals).
How the game is played
Team A makes a mathematical expression using any two of its members and the operation of addition or subtraction. For example:
9 + 8
7 – 4
Professor Equals then comes and stands at either end of the Team A expression.
Team B then makes another expression using any number of its members and any one of the remaining operations that has the ‘same value’ as the expression made by Team A. Members of Team B stand on the other side of Professor Equals.
For example, for the two expressions made by Team A above, Team B can make:
‘9 + 8 = 19 – 2’ or ‘9 + 8 = 21 – 4’, etc.
‘7 – 4 = 6 2’ or ‘7 – 4 = 9 – 6’, etc.
If Team B is successful in making an expression that equals the expression made by Team A, it earns as many points as the largest number it used in making its expression.
If Team B fails to make the expression that equals the expression made by Team A, then Team A gets as many points as the largest number it used in making its expression.
For the next move, Team B goes first. The two teams are allowed the same number of moves.
Case Study 1: Mrs Aparajeeta reflects on using the game of equality
This is the account of a teacher who tried Activity 1 with her elementary students.
This activity required a lot of management and did take some time before we could actually start. I got hold of some quite strong card and made the cards using this. When the activity was over I collected them all together and put them away in a safe place, determined to use them again.
One of my colleagues, Meena, saw me putting them away and asked about them. When I explained the activity [Figure 2], she said that she would like to use them as well, so they have been used twice now which makes the time spent more worthwhile. Next time I use cards – and I am sure that I will as the students learned so much – Meena and I will work together to make the cards.
Before the lesson we had to move the desks to make sure we had room to move. But it was well worth the time. In fact, all the students really enjoyed the activity and I think learned a lot about the equals sign.
To start with I explained the game using two teams of ten students but I had already decided that I would need to make four teams instead of two to actually play the game as there were too many students in my class and many would not have been involved otherwise. I also had four students appointed as evaluators to evaluate the expressions made by each group of students to see whether they were correct or not, and had two students to keep score. Team A played Team B, and Team C played Team D, at different ends of the classroom, then they swapped.
I noticed that Team B had some deep-thinking students. They decided to always use the biggest numbers that they could in order to get more marks and to put the other team out. This of course meant that they had to do some quite difficult arithmetic with their big numbers. It was their choice to challenge themselves but I was pleased to see the sums they set themselves and the trouble they took to make sure they were accurate and scored the points.
We decided to have a knock-out competition at the end of the lesson. Team A played Team B and the winner played Team C, and so on. I felt this worked very well as we stopped at each try and everyone evaluated the answers given, why they were correct or why they were wrong. This led to a great deal of discussion and again I was surprised at how much each student in the class challenged themselves to do the arithmetic in their heads quickly.
Reflecting on your teaching practice
When you do such an exercise with your class, reflect afterwards on what went well and what went less well. Consider the questions that led to the students being interested and able to progress, and those you needed to clarify. Such reflection always helps with finding a ‘script’ that helps you engage the students to find mathematics interesting and enjoyable. If they do not understand and cannot do something, they are less likely to become involved. Use this reflective exercise every time you undertake the activities, noting as Mrs Aparajeeta did, some quite small things that made a difference.
Pause for thought
Good questions to trigger such reflection are: