Year 7,8,9 Common Task: Functions
A first lesson
In the first lesson I introduce algebraic
notation as a way of labelling a rule that students are already
using. The motivation to look at graphing functions will come when
we have a disagreement about rules.
In silence, I write 3
4 on the board. Underneath, I write 5
8, and 9 .
I hold up the board pen. Several students put hands up. I nod to
one, motioning for them to take the pen. They come to the board,
take the pen and write 10. In a different colour, I draw
and take back the student's pen. The student returns to their seat.
More hands go up - I
nod to a different student who leaves their seat and writes 12 on
the board. Again, I draw
and invite a third student to come to the board. They write 16.
and point to
the space beneath the 9 (i.e. indicating that I want the student
to give the next starting number). They write 6, give me the pen
and return to their seat.
By inviting the student to give the next
start number, I hope to provoke an awareness that there is no pattern
in these numbers - i.e. we can put anything as an input.
I hold up the pen, and invite another student
to the board. This pattern is repeated, with different students
writing an answer and, if it gets a smiley face, writing a new starting
As more numbers are written and answered, I comment
"If anyone wants to write something that might help the others,
they can use the black pen on my desk and write on the right hand
side of the board". A student writes: x 2 – 2 with the
black pen, another writes: x 2 – 1. A third student writes:
add the number to itself and take two. As they are doing this the
game with the numbers continues.
Inviting students to write something that
might help others is a strategy for supporting students in the group
who may not know what is going on!
After some time, when it seems as though most
of the class could write answers, I take the pen from a student
and write N as the next starting ‘number’. I do not
comment on what they write and keep writing N as the starting number
until no on else wants to respond.
I am aiming here to get as many different
versions of the rule as possible, this can then provoke a need to
work at whether the rules are the same or different.
At this stage the board was as below:
Figure 5.2: Common board showing work on whether
the rules are equivalent
If there had not been differences in the rules that students wrote
I would have started another game with a different rule.
Having generated a difference in students' responses
to a task, it is now possible to work with those differences i.e.
it is possible to ask questions or invite students to ask questions.
- What do these mean? (Pointing to the rules.)
~ The first one says times your number by two and take away two.
- Can you show what you mean with one of the answers?
~ Like the four. Four times two is eight and eight take two is six.
~ On my one (x 2 - 1) I meant take away one first.
- What do you mean?
~ Four take away one is three, and three times two is six.
~ That's the same as the third rule.
~ Isn't that the same as times two and take away one?
This question (are the rules the same or are they
different?) may not come explicitly from the students, in which
case I might offer the question: "Which of these rules are
the same? Will these always be the same or always be different?"
We might write out the rules and try substituting
10 numbers into each of them. There is a chance to focus on the
conventions of writing algebra as the rules are re-written e.g.
'Mathematicians tend to write 2N instead of N x 2 but it means the
same thing', or, 'You need brackets around the N - 1 if you want
to show that needs to be done first'.
I might play several more function games, introducing
the idea of putting the right hand number first and writing the
arrow to the left.
Where this can go
At some stage in this series of lessons,
I will move the task onto drawing graphs of rules. This will happen
when, having played a function game, we again end up with differences
in the rule. This will happen when, having played a function game,
we again end up with differences in the rules the students write.
This can be particularly rich if one of the rules is quadratic (e.g.
I have played a game with the rule double take five, one student
writes the rule as 2N - 5, and another writes NN - 5).
- One way that mathematicians can tell if rules
are the same or not is to draw graphs of them. I'm going to show
you how to draw a graph of a rule. I want everyone to copy this
down exactly as I do it, because every time you draw a graph I want
you to do it in exactly the same way.
I write the rule on the board (in the illustrations
below, the rule is 2N - 5) and underneath write 5 ,
down to -5 .
Getting answers from the class, I fill in this table and next to
each line write the pair of numbers as co-ordinates. Everyone then
copies a set of axes, which I draw on the board. Students come to
the board to plot the co-ordinates from the table, ticking them
off as they go. Students copy the points onto their axes.
Some discussion is necessary with the negative
co-ordinates. I do not labour the point about rules for multiplying
negatives - I see part of this activity as setting up a situation
in which students can work at making their own sense of multiplying
negatives, whilst their attention is on something else (i.e. drawing
the graphs). Linear rules introduce an element of self - checking
- if the points do not lie in a straight line then something has
~ All the points are in a straight line.
- If you agree with that, you can join up the points using a ruler.
- Now, in your books, do a table for our other
rules (e.g. (N - 5) x 2 or whatever else they have come up with)
and plot the points on the same set of axes. Are the rules the same
or are they different?
The difference in the rules provides a motivation
to plot the graphs. If there is a quadratic rule, this will need
to be done altogether in order to discuss issues such as negative
two times negative one.
After plotting the first rule, the board might
be as below:
Figure 5.3: Graphing a rule
Once students have begun the process of drawing
graphs from rules, many possible challenges are possible. If encouraged,
the students themselves will come up with questions they want to
- What other rules will give the same line as
- If I give you any rule, can you predict what the graph will look
like without having to plot it?
- Can you be organised in the rules you try?
- Given two rules, are there any numbers that will give the same
- Which rules give straight lines, which rules give curved lines?
One way of structuring this work is to get students to draw their
graphs on paper (not writing the table of values but including the
rule, written large), which they pin up on a display board. At some
stage, one or two students can be given the task of organising the
graphs. When organised in some way, students can be invited to gather
around the display board.
- These graphs have all be organised by a student.
What does anyone else see or notice?
Categorising the graphs students have drawn and
discussing classifications is a strategy for provoking questions,
e.g. someone might notice all the curved graphs have NN or N2 in
them, question: will this always be true?
A good lesson start part way through is to have
a set of axes drawn on the board with some points drawn in a straight
line and ask students what they notice and eg can they give other
points on the line, can they give the equation, what about the equation
of a parallel line.
- If I have a rule N->aN + b, can you describe
what the graph will look like? What do a and b tell you?
- If I have a rule N->a N2 + bN + c, can you
describe what the graph will look like? What do a, b and c tell
Guidance for notes at the end of the
2n means ...
3n means ...
4n means ...
n2 means ...
n3 means ...
n4 means ...
2(n+1) means ...
[Include any other bits of notation that have arisen in the topic.]
[You might also end by asking students to write down what questions
remain for them at the end of this topic, which they will be able
to work on next year.]
[If there have been conjectures for the equivalent of y=mx+c then
write these down, as well as conjectures concerning which rules
give straight lines or not.]
Typical mathematical content
- Reading algebra
- Substituting into formulae
- Plotting co-ordinates
- Drawing axes
- Arithmetic with negatives
- Gradients and intercepts of straight line graphs
- Recognising when a graph will be straight or not