4 Moving on to more formal generalisations
Moving from conjecturing about statements to generalising using symbols might seem to be a big step, but if your students have been working with games like the ones in Activities 1 and 2 they may have already started to use symbols.
For example, they may have said things like ‘If you take 2 from any number and then add five to it, the answer is always going to be three more’. Using x or n as a convenient way to show any number may well seem entirely natural in this context.
The next activity starts to encourage more formal generalisations.
Activity 3: Generalising
Create flash cards of two types:
- S-cards – These have specific arithmetic statements that may or may not be true.
- G-cards – These have generalised statements (conjectures) which correspond to the statements on the S-cards.
Resource 3 provides examples of S-cards and G-cards. You may alter these to suit the level of your class.
Divide the class into groups. Groups of six to ten students work well for this activity. Shuffle all the S-cards and all the G-cards separately. You may want to look at the key resource ‘’ when thinking about organising your class into groups. Divide each group into two halves. Distribute S-cards to one half of the group and G-cards to the other half.
Ask your students to create pairs of S- and G-cards and then explore if the conjecture made is always true, sometimes true or false. Another idea is to ask the students to work in groups of five or six, hand them six assorted S- and G-cards. If they have a specialised (S) card then they have to generalise it (make a G card). If they have a G-card they have to create an S-card for it and then discuss if it is always true, sometimes true or never true.
Resource 4 provides some examples for the content of each type of card.
Ask the students to make up their own S-cards and G-cards.
Video: Using groupwork
Case Study 3: Mrs Agarwal reflects on using Activity 3
I used the suggestions [in Resource 3] to make the S- and G-cards. I liked the activity because I thought it would encourage the students to compare the expression and then see how each expression could be referred to in mathematical language.
I made the groups such that each group had one student who already seemed to have a good understanding of algebra. I then asked them to see that all the students in their group participated in the discussion of whether their conjectured generalisations were completely false, sometimes true (and if so, when), or always true. The groups were also warned that the explanation for their group would have to be given by any student, so they all had to have a consensus.
This worked very well. The level of discussion that I overheard in the groups was quite exceptional for the class as they each tried to match up the specific with the generality and work out whether it was always true or not. I asked each group to present one pair of cards and give their reasons for what they had decided. This took some time but only because they were discussing so much with each other! We had to leave the second part of the activity until the next day.
Making their own S- and G-cards was also a good exercise, as I made them do these individually, and then they talked about what each had made in their groups. So they gained a lot of input from their classmates about whether they were right about their statements or whether they had some misunderstandings.
Pause for thought