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Compare Proportional Relationships in Different Forms

Students compare two proportional relationships presented in different forms—one as a graph, one as an equation, or one as a table—by extracting and comparing unit rates. They decide which relationship has a greater rate of change without converting both to the same form.

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Compare Proportional Relationships in Different Forms

Grade 8 · Proportional Relationships and Slope · question 1 of 3

0 of 20

20 rungs. A star at the top.

Which of these makes blank = 8 ÷ 4 true: 4 ÷ 4, 2 ÷ 1, 12 ÷ 4 or 16 ÷ 4?

How it is taught

0:00 of 4:41

A proportional relationship has a constant k.

k is the rate that y divided by x stays.

Read it: Comparing rates

36 ÷ 4 = 9 and 20 ÷ 2 = 10. On one line, 10 is further right, so 20 ÷ 2 is the greater rate, though 36 is more than 20.

  1. Work each rate out as one number.
  2. Place the numbers on one line.
  3. Further right is greater; the same place is the same rate.

A rate is how much for one, and dividing finds it.

On one line the numbers grow to the right, so their places put them in order.

45 ÷ 5 = 9 is the same rate as 36 ÷ 4: the same amount for one.

Compare rates by their values, never by their first numbers.

Calling 36 ÷ 4 greater because 36 is more than 20.

Work them out: 9 and 10, so 20 ÷ 2 is greater.

Which is greater: 24 ÷ 4 or 14 ÷ 2? — 14 ÷ 2. 24 ÷ 4 = 6 and 14 ÷ 2 = 7: 7 is greater, though 24 is more than 14.

Maya's graph vs. Liam's equation y = 7x

Maya and Liam both run at steady paces. Maya's run is shown as a graph and Liam's as an equation. Who's faster?

Here's Maya's run. The straight line through the origin tells us her speed is constant.

Pick an easy point on the line. After one hour, Maya has run six miles.

So Maya's unit rate is six miles per hour.

Now look at Liam. His distance is given by the equation y equals seven x.

The number in front of x — the slope — is the unit rate.

Who runs faster?Maya (graph) vs. Liam (equation)Maya126012hoursmiles(1, 6)6 miles / hourLiamy = 7xslope = unit rate7 miles / hour7 > 6FASTERLiamby 1 mph
Grade 8 · Comparing unit rates across forms

Liam runs seven miles per hour.

Now we compare the rates: seven is greater than six.

Liam is faster — by one mile per hour. We compared two proportional rates without converting forms!

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics
Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal
1 star — one skill mastered, every rung of its ladder climbed
Climbing now
Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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