In Saturday Workshop 1, we discussed three representations of proportionality as related to constant rate
of change:
If x is proportional to y, then
1. Scaling: If you scale one quantity by a number, you scale the other quantity by the same number.
(xia→ yia) .
• An example of this is when we were given that the bike traveled 11 feet every 0.5 second
traveling at a constant speed. We doubled the change in time and therefore doubled the
change in distance to find the unit rate of 22 feet per 1 second.
2. Constant Ratio: the ratio of the two quantities is constant. That is xy = m where we call m the
constant of proportionality.
• This is the form of proportionality we often use when determining the value of the
constant rate of change.
3. Constant Multiple: one quantity is a constant multiple of the other. That is, y = m i x .
• This is the form of proportionality we utilized to create our linear functions in the water
tank problem: Δh = 4.6 Δt .
( )
For example, suppose you have a picture frame with dimensions 4in (base) x 6in (height) and want to
enlarge the picture so that it is 9 inches in height. Students may approach this problem in different ways:
1. Scaling: They can find the scale factor of 9/6 = 1.5 and argue that since the height is 1.5 times as
large as the original height, the base also must be 1.5 times as large as the original base, or
1.5(4) = 6 inches. This may also mathematically look like 96 = 4b because setting those two
fractions or ratios equal is stating that the scale factor must be the same for both quantities.
2. Constant Ratio: Students may argue that the ratio of the two quantities must be constant. This
may lead them to set up the ratio 64 = 9b . Students may set up this equality without thinking about
the meaning of it, but the formulation is assuming a constant ratio between the quantities.
3. Constant Multiple: Students may comment that the base is 2/3 times as large as the height. Since
the new height is 9 inches, the base must still be 2/3 times as large, or (2/3)(9) = 6 in. This also
looks like b = 23 i h .
Though each formulation is related to one another, they are different ways of approaching, understanding,
and solving problems. They are often seen by students as separate ideas, much like students struggle
connecting different representations of the same relationships – tables, graphs, formulas, and words.
For homework, I would like you to give students a problem where they are forced to use proportionality
to solve. It doesn’t have to be a large problem, but is one that allows students to show the type of
reasoning they used to solve the problem. On Canvas, give the problem you used, and describe examples
of students solving it in each of the 3 ways listed above. Describe why you believe the student was
reasoning in that particular way (list your evidence). Answers should be submitted on Canvas by the
November Saturday workshop. Please also bring a copy of the student work to our session of the Saturday
Workshop to share.