UK Video clip Fact Sheet: Investigating quadratic number sequences
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2009
UK
English
13-14 years
A higher achieving class in a state comprehensive school.
Investigating quadratic number sequences
Handheld technology (TI-Nspire handhelds)
Students working individually and in groups using handheld technology in a
mathematics classroom.
Male teacher experienced in mathematics teaching and in the use of
handheld technology in his teaching
Timotheus. (2009)
Clark-Wilson (2011)
Starting situation
This video clip is taken from a television documentary that told the story of a project in which a
group of teachers were developing and sharing their ideas about how they could use new
technologies to support the teaching and learning of topics in mathematics that they considered
‘hard to teach’.
Task description
The students begin by plotting a series of square numbers as a number sequence and establishing
that a curved graph is produced, which fits the quadratic function y = x2. The students are then given
a selection of quadratic number sequences of the form y = x2 ± a and are asked to try to model the
resulting sets of coordinate points using a quadratic function. The final part of the task requires the
students to record their sequence as a number sequence, graph, geometric pattern of squares and
algebraic function.
Clip description
The teacher introduced the task by writing a sequence of square numbers on the whiteboard and,
alongside each, a pattern of squares. He then demonstrated to the students how to enter this
sequence into a pre-designed TI-Nspire software file with automatically plotted each number on a
blank graphing page. The students were encouraged to try to model this sequence algebraically,
establishing the y = x2, gave the correct model. The students then worked independently on a chosen
sequence to try to establish an algebraic model and produce a display of their work. In the final
whole-class plenary, the students contributed some responses about what they had learned during
the lesson. The teacher used the software to emphasis a key feature of graphs of the type y = x2 ± a.
Transcript
Teacher: I was thinking really about algebra and sequences, and thinking about how I can link
together the many representations that link to sequences.
Narrator: Jay’s project involves using this handheld computer device to investigate quadratic
functions and number sequences.
Teacher: This function is x2…
Consultant: What made you pick this particular topic?
Narrator: Before he tackled the topic with his pupils, Jay and Alison discussed what usually makes it
so difficult.
Teacher: Quite often it is taught in different parts (of the scheme of work) and, although pupils learn
the square numbers, lower down the school, they often don’t look at the representation of quadratic
functions until a couple of years later. If it being done by pencil and paper, by the times the students
have actually drawn all of these out, three or four times, they might have lost the sense of what this
was all about. Whereas, what we are doing today, they will see the quadratic function plotted
instantly on the ICT and they will be able to make those connections a lot quicker.
Narrator: Jay then downloads the files the pupils will need from his laptop onto each of their
handheld devices.
Teacher: The number sequence we are going to look at first starts with a number 1, the next number
is the number 4, and the next number is the number 9, and then I’ve got the number 16…
Narrator: Jay starts the lesson by setting out a sequence of square numbers and the corresponding
patterns of squares. He then prepare to enter this sequence of square numbers into the computer
software.
Teacher: How many squares were there in the first one? Yes, Colin
Student: One
Teacher: In the second one there were…
Student: four
Teacher: four, thank you very much
Student: nine?
Teacher: Yes there were nine, very good. And in the fourth one there were, going over to Matthew?
Student: sixteen
Teacher: sixteen, thank you very much. So that one (coordinate point), represents one, that
represents four, that represents nine, that represents sixteen. I’d like you to type them into yours as
well, to make sure that they are appearing on your screens as well, That’s fantastic. And you can put
the fifth one in as well if you want to.
Narrator: Jay’s year 8 pupils have used these handheld devices before to look at straight line graphs,
but this is the first time they have encountered quadratic functions. Now they are trying to find the
graph of an equation that fits the points they have plotted.
Student: We could try the squared key and put a number in… two
Student: I know, I know, type x squared.
Student: I pressed x and then I pressed the x squared button, then I pressed enter I get a curved
graph
Teacher: Now let’s think about this for a minute. If those are your x numbers, x squared becomes
your y numbers and then you get a graph like that, okay?
Narrator: Jay then hands out some other quadratic number sequences.
Teacher: ..and you’re trying to find what equation will go through the points you get when you plot
these values.
Teacher: In your pair, choose one of the sequences and type them in and see if you can find the
equation.
Student: What shall we do?
Student: three, eight, fifteen, twenty-four
Student: We though at the start we do the same as the last… you know
Student: the last one which was x squared, the equations was x equals.. x squared
Student: But when we put x squared and pressed enter, it’s a bit off, so we thought now what can we
do, so we..
Student: so we realized that every number down here is one less than the square number in the last
one so we thought if we put minus one at the end it might give us the answer.
Student: minus one here.. and press enter and it’s on the line. We got it!
Teacher: What are you noticing?
Student: It’s just add ten
Teacher: Oh it’s add ten. So what was the equation we used last time?
Student: x squared
Teacher: x squared, and this is… you just said it was…
Student: add ten
Teacher: So what do you think you might try?
Student: x squared add ten
Teacher: Try it. This could be right. Oh wow! Well done.
Student: So we do zero, three, eight, fifteen and then we do twenty-four. And then you press control
and tab to go onto this part, that’s the equation side. So the equation would be x squared minus one.
Yes, that’s it, x squared minus one.
Teacher: What I would like you to do now, is just to pick one of the sequences that you have
explored, I would like you to draw for me the number pattern that goes with it, I’d like you to write
down what the numbers are, I’d like you to sketch the graph that you got, and write down the
equation that you used. And all of these four things here, one, two, three, four things, I’d like you to
write them down, they all represent the same thing.
Student: five six seven eight nine ten…
Student: five six seven
Teacher: Okay, now, what have you found out today?
Student: I found out that, you know graphs, I thought it would just be line, bar graphs, like
pictograms and telegrams, but this one is different because you are using square numbers. I didn’t
know that you could like transfer square numbers onto a graph. it is curved. graphs, could have
Teacher: Okay, and what was special about the graph that you got?
Student: It is curved. All of them are like lines and stuff but this one is curved.
Teacher: That was something new for you, fantastic. Did anyone notice anything interesting about
how the graphs varied, so if you had x squared minus one or plus five, or plus ten?
Student: If you did x squared it would be low, but if you did plus five it lifts up and goes higher.
Teacher: How high would it go then?
Student: Erm, five times more higher?
Teacher: It would go through five, okay... and if you did x squared plus twenty, what would happen
then?
Student: It would go… like lift up to twenty, go through twenty.
Teacher: and if you had x squared minus one?
Student: It would go through minus one.
Teacher: Shall we just have a look at that on the screen? Because I am not sure that everybody
spotted that earlier. What will happen if that was x squared plus one instead of x squared plus ten?
Student: Where the curve is, it will go lower.
Teacher: It would go lower, and what number would it go though on the y axis?
Student: It would go through one.
Teacher: It would go through one, let’s have a look. So going over to my computer again and
changing that… changing it to a one… oh look and you can just see that is one, there’s zero, there’s
five and that’s one. So I think you have worked very well today so we’ll stop there, thank you.
Additional information
An account of this lesson is written up in Clark-Wilson, A., & Oldknow, A. (2009). Teachers using ICT
to help with 'hard to teach' topics. Mathematics in School, 38(4), pp.3-6.