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Because times 5 plus 6 [pointing to the y-intercept of 6] would just be there [sketching a parallel
line]. You can’t change the times table number because if you do it will be steeper and it won’t be
parallel.” Ellie began with a partial generalization that in order to get parallel lines for a function,
the value of the constant would have to be higher, but then amended this to include any other
constant than the one in the given function. Ellie used a point-wise approach when considering the
graph, for example, the link between changing the constant in the rule and the resulting change in
y-intercept on the graph. However, she also seemed to be able to consider the graph in a more
global manner, particularly when she sketched the line that would result from graphing the
function y=5x+6.
When asked for a function that would give a steeper line, Ellie again responded with a
generalized conjecture, this time not supported by any specific examples. Her answer
demonstrates an integrated understanding of the how both parameters of a composite function are
represented graphically. “A steeper number than this would be any higher number than 5 would be
the times table – the times part. If you have a different times table number then you would get a
steeper number – a steeper graph and if you use the same times table number and a different
addition number then you would get a parallel graph.”
To predict a function that would intersect y=5x+3, Ellie used two different strategies. The first
was based on her initial theory that a lower constant and a higher coefficient would result in a line
that intersects the given function (this is a viable conjecture given that Ellie only had experience
graphing in the first quadrant). “If you do, let’s say times 9 plus 0. So it would start at 0 [pointing
to the origin] then there [pointing to (1,9)] then 18 [pointing to (2,18)]. It crosses there
[sketching in the line for the function y=9x].” Once Ellie graphed the two functions she noticed
they intersected somewhere between 0 and 1 on the x-axis, and so estimated that they might be
intersecting at 0.6 or 0.7. “It’s not exactly in the middle, like 5 in the middle of 10, but it’s just over
the middle a little bit. It’s not exactly 1 or 0 so you have to go in decimals.” The fact that Ellie
estimated the intersecting point as a rational number indicates that, rather than viewing the graph
as a series of discrete points, Ellie saw the graphed function as continuous.
Ellie was then able to integrating her theory of increasing the coefficient and decreasing the
value of the constant with an ability to think of specific values in order to predict a function that
would intersect at the “second position”, (2,13). “If you do times 6 plus 1 then you’ll go from 1
[drawing a line from y-intercept to (1,7)] to 6 but then you have to go up 1, because it’s plus 1
(1,7), and then over to here [extends the line to (2,13)] that’s 12 but add 1 [points to the
intersection at (2,13)].”
For the last question, Ellie was able to reason about what it means to have two intersecting
lines, based on her experience of building geometric patterns. “If you built x5+3 and x6 +1 as
patterns they would have the same number of blocks at position 2 because it intersects, so, it has
the same number of blocks and… it’s just like a graph and blocks are two different ways to lay it
out.” Ellie recognized that a linear function underpins both a growing pattern and a graph.
Cassandra – Extending to Negative Integers
During her interview, Cassandra built on her existing knowledge of graphic representations of
the form y=mx+b and was able to integrate negative integers into her understanding. The students
had only ever worked with positive integers in their functions because it is difficult to represent a
Lamberg, T., & Wiest, L. R. (Eds.). (2007). Proceedings of the 29
th
annual meeting of the North
American Chapter of the International Group for the Psychology of Mathematics Education,
Stateline (Lake Tahoe), NV: University of Nevada, Reno.
Convention