Problem
Rule for a growing matchstick figure
Matchsticks build bigger and bigger equilateral triangles. The 1st holds 1 small triangle, the 2nd 4, the 3rd 9. The pattern keeps going the same way. Find how many matchsticks the 9th shape needs.
Reasoning · Grade 4-1 Working with Sequences
Your answer
Watch out 81 triangles times 3 overcounts badly, because neighbours share their sides.
How to solve
Strategy Look for a Pattern — Drawing the 9th shape and counting 135 sticks one at a time would be slow and easy to get wrong. Instead I read the three small shapes I am given, list their stick counts, and look at how much is added each time. Once I can say exactly which new sticks appear when the shape grows by one row, the rule is trustworthy and I can jump straight to the 9th shape without drawing it.
1STEP 1
Count the sticks in the three shapes shown
The printed shapes use 3, 9 and 18 sticks.
3, 9, 18, …
Reading numbers straight off a picture is the safest first move — the three counts are the evidence everything else has to agree with.
4.OA.C.5Draw A Diagram2STEP 2
See how much is added each time
The additions 6 then 9 grow by 3.
9-3=6, 18-9=9
Looking at the gaps instead of the totals turns a hard-to-read list into an easy one, which is the standard way a Grade 4 student hunts for a shape rule.
4.OA.C.5Look For A Pattern3STEP 3
Find out why the added amount grows by 3
Reaching shape n adds 3 × n sticks.
sticks added at step n = 3 × n
Checking the rule against the picture of one small step is enough to trust it for every step, because every step is built the same way.
4.OA.C.5Solve An Easier Related Problem4STEP 4
Add up row by row to reach the 9th shape
Adding row by row, the 9th needs 135.
3, 9, 18, 30, 45, 63, 84, 108, 135
Writing out all nine terms is only eight additions, and each one is a two-digit sum a Grade 4 student can do in a line.
4.NBT.B.4Make A Systematic List5STEP 5
Get the same answer with one multiplication
One multiplication gives the same: 3 × 45 = 135.
3×(1+2+…+9)=3 × 45=135
Pulling the common factor 3 out of nine products turns a long addition into one easy multiplication, and 1 + 2 + ... + 9 = 45 is a sum a Grade 4 student can pair up (1+9, 2+8, 3+7, 4+6 make four 10s, plus the 5).
4.NBT.B.5Look For A PatternPulling the shared factor 3 out of the nine products turns a long addition into one easy multiplication.
Why?
Three of this plus three of that is three of the two together, so a shared factor can be lifted out of a whole sum at once.
🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.
Why?
The sum left inside the bracket runs 1 up to 9, which is quickly totalled by pairing the ends into equal amounts.
🧱Pairing from both ends gives a constant sumIn an evenly spaced list the smallest with the largest, the next with the next, always make the same total, so a long sum becomes a multiplication.
6STEP 6
Test the rule on the shapes I can see
The rule matches all three printed shapes.
3 × 1=3, 3 × 3=9, 3 × 6=18
A rule that reproduces every case you can actually see is a rule you are allowed to trust on the case you cannot see.
4.OA.C.5Guess And CheckAnswer
135 matchsticks
3 × (1 + 2 + … + 9) = 3 × 45 = 135
135 is a count of matchsticks, so it must be a whole number bigger than 18 — and it is. A rough size check also works: the 9th shape holds 9 x 9 = 81 small triangles, and 81 triangles would need 81 x 3 = 243 sticks if no stick were shared. Since most sticks are shared by two triangles, the true answer should be somewhere between half of 243 and 243, and 135 sits right in that window. Finally, 135 divides evenly by 3, as it must, because the sticks come in three equal groups of 45 — one group for each of the three directions a stick can point.
Takeaway
When a shape grows, count what each new row adds instead of recounting the whole picture — the 9th shape is just 3 x (1+2+...+9).
- Count the sticks in the three shapes shown
- See how much is added each time
- Find out why the added amount grows by 3
- Add up row by row to reach the 9th shape
- Get the same answer with one multiplication
- Test the rule on the shapes I can see