An Introduction to Proportion Slideshow 24, Mathematics Mr Richard Sasaki

Objectives Introduce the meaning of proportion Understand the meaning of “direct proportion” and correct notation for this

Proportion What is proportion? Proportion literally means that as one thing changes, another thing does. It is a very loose meaning without further definition. So if two things (let’s say 𝑥 and 𝑦) relate to each other, we say “𝑦 is proportional to 𝑥”. 𝑥 is also proportional to 𝑦.

Proportion For two things to vary, they cannot be fixed. So proportion deals with variables (unknowns with changeable values). Example Explore the total number of legs on a group of sheep. 1 2 3 4 # of Sheep Total # of legs 4 8 12 16 We can easily see that as the number of sheep goes up by 1, the number of legs goes up by . 4

Proportion Proportion simply means that two things relate to each other but in mathematics, proportion is somewhat more specific. For two variables, 𝑥 and 𝑦 to be proportional, one of the following must be true: When 𝑥=0, 𝑦= When 𝑥=0, 𝑦 tends to ∞ Note: We shouldn’t write a value is equal to infinity. Really 𝑦=∞ means 𝑦 tends to ∞. We can write this as lim 𝑥→0+ 1 𝑥 =∞ but this is obviously not taught in Grade 7.

Proportion State for each equation below whether 𝑥 and 𝑦 are proportional to one another or not. 𝑦= 3 𝑥 2𝑦=𝑥 Yes! Yes! 𝑦= 𝑥 3 Yes! Yes! 𝑦= 𝑥 No… 𝑦=3(𝑥+2) No… 𝑦=𝑥+2 Yes! 𝑦= 𝑥 2 Yes! 𝑦=𝑥(𝑥+2)

Proportion State whether the two variables in each case are proportional or not (with good reason). Yes! A rectangles width and its area. No… A man’s height and his wife’s height. No… A young boy’s age and his height. Yes..? A car’s engine size and its top speed. (Well yes they are, but we would need to consider , , , car weight car size engine build aerodynamics and many other things.)

1 2 3 4 5 6 4 8 12 16 20 24 1 2 3 4 5 6 3 6 9 12 15 18 Edges are shared so the rate would differ. (0, 4, 7, …)

Different children grow at different rates Different children grow at different rates. Plus babies are born different sizes. The two variables are not proportional. A 0 day old baby isn’t 0 𝑐𝑚. Numbers on dice, edges on hexagons, etc… When 𝑥=0, 𝑦≠0 or ∞. ∴, they are not proportional. Yes, as when 𝑥=0, 𝑦 tends to ∞.

Direct Proportion Most of what we have seen so far is direct proportion. Direct proportion has a very precise relationship between two variables. So what is direct proportion? Direct proportion is when two things are related and only a coefficient exists between them. An example could be… Number of wheels = × Number of bicycles 2

Direct Proportion So if two variables can be written in the form 𝑦=𝑘𝑥 where 𝑦 and 𝑥 are variables and 𝑘 is some constant value, they are directly proportional. If two variables 𝑦 and 𝑥 are directly proportional, we can write… ∝ 𝑦 𝑥 (𝑦 is directly proportional to 𝑥.) Note: We see 𝑦=𝑘𝑥 written as 𝑦=𝑎𝑥 too where 𝑎 is a constant. This appears in graphs and appears more in Japanese maths.

150÷2×6=450 Yen The pencils all cost the same amount. 13÷100×10=1.3 seconds His speed was constant. 30÷150×200=40 seconds The cat eats the same speed at all times. Part 𝑏. There is some acceleration when running. No. Here, 𝑦 2 ∝𝑥 so the rate would change. We can’t write it in the form 𝑦=𝑘𝑥. Yes, as 3𝑦= 4𝑥 3 ⇒𝑦= 4 9 𝑥 𝑘= 4 9 As 𝑥 increases by 1, 𝑦 increases by 4 so 𝑦=4𝑥.