Let’s Think Secondary Maths, Lesson 12 – Lines galore
Overview
Thinking Strands: Number and ratio (exploring direct proportion)
Lines galore focuses learners upon the ratio as constant multiplicative relations. This is begun
with a discussion of bikes and develops via cogs to the exploration of the ratio algebraically and
graphically.
Aims
 To develop understanding of ratio and proportion.
 To symbolise the relations between numbers in algebraic terms.
 To connect ratio to the gradient of lines.
Vocabulary
Equation, adding, multiplying, ratio, proportion, constant, graph, gradient.
In-school resource preparation
Materials – Supplied
 Resource Sheet A: Comparing bikes (Hook)
 Resource Sheet B: Similar ratios (Activity 1)
 Resource Sheet C: Blank graph axis (Activity 2)
Overview of Activities – please see video tutorial for full details of how to develop this lesson
with your class.
Hook: The mathematical focus for this lesson is introduced through a cartoon that taps into
learners intuitive understanding of the fact that bikes with gars are easier to ride. A real bike
could be used and explored at any point during this activity. Move into Activity 1 when the focus
upon ratio emerges from the resulting discussion.
Activity 1: Quickly show the mixed/random ratio pairs (including 1 mistake) asking the class
to describe what has just been shown. The discussion that follows develops in two ways:


A discussion of what the pairs of number mean including spotting the wrong result,
Asking pairs to organise the numbers into their own table.
This activity should develop from the ideas learners share to make sense of the data. The range of
ideas should be written down as they are shared for discussions of this type of activity lends itself
to a time of voting where learners chose the most helpful description of the ratio from the many
that have been shared. Allowing them to share why they prefer a certain method will provide
valuable insights into individual thinking concerning ratios. The video tutorial highlights the
many possible routes through this part of the lesson.
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Activity 2: The main aim for this part is for learners to discuss the key features of the graph:
its constant rate of increase and the steepness/gradient corresponding to the relationship within
the ratio. The actual direction taken by a group will depend on how much they understand about
linear relationships of this type. There are several possible steps here:
1.
2.
3.
4.
5.
Simply plotting the graph and describing how it links to the ratio, table of data and cogs,
Using the graph to predict new results (interpolation and extrapolation),
Using the symbols X and Y and writing the equation of the line,
Drawing new lines resulting from cogs with other ratios,
Exploring the nature of lines above and below y = x.
See the video tutorial for details of how this aspect of the lesson might be managed.
End of activity reflection: the bike image could productively used to encourage learners to
visulalise the link between different gear ratios and the resulting graph.
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