NAME THE TOOL

How fast the pile grows

Coach

Remember Δ, the change? The bank gained 12 coins. That is the change. But did it take 1 day or 6 days? We cannot tell how fast it filled yet.

A rate shares the change out over the time. 12 coins over 3 days is 4 coins each day. The rate is how fast the pile grows.

Quick check: The bank gains 12 coins in 3 days, the same amount each day. How many coins each day?

THE SYMBOLS

Rate = change ÷ time

rate = Δcoins ÷ Δdays

Read the word per as "divided by." "Coins per day" means coins ÷ days. The coins go on top.

When the pile shrinks, Δcoins is negative, so the rate is negative too.

Quick check: The bank goes from 20 coins down to 8 in 4 days. Is the rate 3 or −3 coins per day?

WORKED EXAMPLE

Day 2: 5 coins. Day 5: 17 coins. What is the rate?

Strategy: Change first, then share it over the time

Name the rate or the pileTry a small caseDraw itAsk whyAnswer the exact thing asked

1

Understand

We want coins per day, not coins.

Why? The question says "rate."

2

Plan

Find Δcoins and Δdays, then divide.

Why? A rate is change ÷ time.

3

Solve

Δcoins = 17 + (−5) = 12. Δdays = 5 + (−2) = 3. 12 ÷ 3 = 4 coins per day.

Why? Divide by the time that passed, 3, not by day 5.

4

Look back

4 coins a day for 3 days is 12. 5 + 12 = 17. ✓

Why? Multiplying back lands on the new pile.

Share way

12 coins shared into 3 day-boxes: 4 in each box.

Symbol way

(17 + (−5)) ÷ (5 + (−2)) = 12 ÷ 3 = 4

Quick check: True or false: the rate here is 17 ÷ 5, because it is day 5 with 17 coins.

SEE IT

Same change, different speeds

Both banks gain 12 coins. The steep one does it in 3 days, the other in 6. Same change, different rates.

Trap: Giving the change, not the rate

"12 coins in 3 days" has the rate 4 coins per day. 12 is the change.

Trap: Flipping the fraction

3 ÷ 12 = 1/4 puts days on top. Coins per day puts coins on top.

Trap: Dividing by the last day

From day 2 to day 5 is 3 days, not 5.

Quick check: From day 1 to day 4, how many days passed?

WHAT TO LOOK FOR

Rate words and pile words

per, each, a day, an hour, how fastRate. Divide the change by the time.
how many, holds, in all, totalPile. A number of coins, no "per."
gained, lost, ΔChange. Divide it by the time to get a rate.
from day 2 to day 5Δdays = 5 + (−2) = 3.

Quick check: "The bank holds 20 coins and gains 4 coins per day." Which number is the rate?

HOW THIS TRIPS PEOPLE

I fall for it first, so you don't

Coach

A bank goes from 4 coins on day 1 to 10 coins on day 3. The change is 6. I divide by 3, because it is day 3: rate 2.

Coach

Day 3 is a label, not the time that passed. Only 3 + (−1) = 2 days passed. 6 ÷ 2 = 3 coins per day.

Check habit: Find Δtime as new day + (−old day) before you divide.

Quick check: Day 1: 4 coins. Day 3: 10 coins. Rate?

TRAP CHECK

Run this before every answer

Is the question asking for a rate (per) or a pile?
Find Δamount: new + (−old).
Find Δtime: new + (−old), not the last day.
Divide, amount on top. If the pile shrank, the rate is negative.