Problem
Build a table to enumerate cases
A coin is tossed 30 times: heads climbs 4 stairs, tails drops 1. Heads plus tails is 30. After all 30 tosses the position must be 20 stairs above the start. Find how many heads are needed.
Reasoning · Grade 3-2 Solving Problems with Tables
Your answer
Watch out An extra head looks like 4 more stairs, but a tail also disappears — the position climbs 5.
How to solve
Strategy Make a Systematic List — Build a table that pairs each possible number of heads with the matching number of tails (they must add to 30), compute the resulting position, and scan for the row that lands on +20. Guessing a starting value and adjusting, plus noticing how the position changes by a fixed amount each step, makes the table quick to fill.
1STEP 1
Set up the table columns
Tails is always 30 minus the heads.
tails = 30 - heads
Whatever flips are not heads must be tails, so once you pick the number of heads the number of tails is fixed.
2.OA.A.1Make A Systematic List2STEP 2
Write the rule for the final position
Position is heads × 4 minus tails.
position = 4 × heads - 1 × tails
Adding up all the ups and taking away all the downs gives the net climb.
3.OA.D.8Make A Systematic List3STEP 3
Try a guess to find the neighborhood
Eight heads gives 10, and each extra head adds 5.
4 × 8 - 22 = 10
Replacing a tails by a heads swaps a -1 for a +4, a jump of 5, so the position climbs steadily as heads increase.
3.NBT.A.2Guess And CheckTrading one tails for one heads swaps a step down for a step up, lifting the final position by exactly 5 every time.
Why?
Because every trade is worth the same 5, the finishing positions march upward by a fixed step as the number of heads grows.
🧱A constant step reaches any termWhen a list climbs by the same amount every time, any term is the starting value plus that step repeated, so you never have to write out the ones between.
Why?
Turning a down-step into an up-step first cancels the fall and then adds the climb, which is why one trade is worth both amounts together.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
4STEP 4
Fill in rows until the position hits 20
10, 15, 20 — so 10 heads reaches 20.
4 × 10 - 20 = 40 - 20 = 20
Because each added heads bumps the total by exactly 5, you can count up 10, 15, 20 and land right on the target.
3.OA.D.9Look For A PatternAnswer
10 times
4 × 10 − 20 = 20
With 10 heads and 20 tails: up 4 x 10 = 40 stairs and down 1 x 20 = 20 stairs, so net = 40 - 20 = 20 stairs up, matching the goal. The counts 10 and 20 add to 30 flips, so everything fits.
Takeaway
A neat little table that swaps tails for heads one at a time shows the climb jump by 5 each step until it lands right on 20!
- Set up the table columns
- Write the rule for the final position
- Try a guess to find the neighborhood
- Fill in rows until the position hits 20