NAME THE TOOL

The pile is everything added up

Coach

Remember the rate? It says how fast coins drop in. Now ask a different question: how many coins are in the bank right now?

Count what was there at the start. Then add every coin that dropped in since. That total is the accumulation, the pile.

Quick check: The bank starts with 2 coins and gains 3 coins a day. After 1 day, is the pile 3 or 5?

THE SYMBOLS

Start + everything added

pile = start + (everything added)

When the rate is steady, everything added is rate × time:

pile = start + rate × time

A negative rate means spending. Adding a negative makes the pile smaller.

Quick check: True or false: the 3 in "3 coins a day" is the pile.

WORKED EXAMPLE

Start with 3 coins. Add 4 a day for 6 days. How many coins?

Strategy: Start first, then rate × time

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

1

Understand

Start 3. Rate 4 per day. Time 6 days. We want the pile.

Why? "How many coins" asks for the pile.

2

Plan

pile = start + rate × time.

Why? A steady rate adds the same each day.

3

Solve

4 × 6 = 24 added. 3 + 24 = 27.

Why? The start rides along underneath.

4

Look back

Count up by 4 from 3: 7, 11, 15, 19, 23, 27. ✓

Why? Two ways land on 27.

Tower way

Symbol way

3 + 4 × 6 = 3 + 24 = 27

Quick check: If you forgot the start, which wrong answer would you get?

SEE IT

The start rides along

Every day's tower stands on top of the starting block. Leave out the start and the whole tower drops.

Trap: Forgetting the start

Start 2, add 3 a day for 5 days. 3 × 5 = 15 is only what was added. The pile is 2 + 15 = 17.

Trap: Adding the rate and the time

2 + 3 + 5 = 10 mixes three different things. Rate × time first.

Trap: Counting the days wrong

From day 1 to day 5, only 4 days pass: 5 + (−1) = 4.

Quick check: Start 20 coins. Spend 3 a day for 4 days. Is the pile 8 or 32?

WHAT TO LOOK FOR

Clue words and easy mix-ups

starts with, already has, at firstThe start. It goes in the pile.
a day, per day, each dayThe rate. Multiply by the time.
spends, loses, leaksA negative rate.
how many in the bank, now holdsThe pile: start + everything added.

Quick check: "The bank already has 6 coins." Is 6 the start or the rate?

HOW THIS TRIPS PEOPLE

I fall for it first, so you don't

Coach

Start with 5 coins and add 2 a day for 4 days. 2 × 4 = 8. Done: 8 coins.

Coach

I dropped the 5 at the bottom. The bank held 5 before day 1 even began. 5 + 8 = 13.

Check habit: Write the start first, before you multiply anything.

Quick check: Start 5, add 2 a day for 4 days. Pile?

TRAP CHECK

Run this before every answer

Find the start. Write it down first.
Find the rate and its sign.
Find the time that passed.
pile = start + rate × time. Answer the pile, not the rate.