Problem
Separate fixed and changing parts of a pattern
Matchstick pentagons are placed in a row, each new one sharing one side with the one before it. I need to find how many matchsticks it takes to build 20 of them.
Operations
Your answer
How to solve
Strategy Look for a Pattern — The figure repeats the same step over and over: one starting pentagon, then +4 each time. That is exactly a pattern problem. I will separate the fixed part (the first 5) from the changing part (the 4 added per extra pentagon), check the rule on a few small cases I can draw, then apply it to 20.
1STEP 1
Build small cases and spot the rule
One pentagon needs 5 sticks, two need 9, three need 13, so each new one adds 4.
5, 9, 13, 17, … (+4 each step)
Trying the smallest cases first lets a young learner SEE the +4 rule instead of guessing it.
4.OA.C.5Solve An Easier Related Problem2STEP 2
Separate the fixed part from the changing part
The first pentagon's 5 sticks never change, and each of the 19 later pentagons adds 4.
total = 5 + 4 × (number of pentagons - 1)
Naming the part that stays and the part that grows turns a picture into one clean expression.
5.OA.A.2Look For A PatternThe row's matchsticks are the first pentagon's 5 plus 4 more for every pentagon after it.
Why?
Every pentagon after the first is built onto a side that is already standing, so it needs only 4 new sticks rather than 5.
Why?
The row's sticks are the first pentagon's five together with the new sticks each later pentagon brings, and the shared side is counted once, never twice.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Each of those later pentagons brings the same 4 sticks, so the added part is that many equal groups of 4.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
3STEP 3
Plug in 20 pentagons
Those 19 extra pentagons add 4 x 19 = 76 sticks on top of the fixed 5.
5 + 4 × 19 = 5 + 76 = 81
Once the rule is written, the answer is just one multiplication and one addition.
4.NBT.B.5Look For A PatternAnswer
81 matchsticks
If every pentagon needed its own 5 matchsticks, 20 pentagons would need 100. Sharing one side per join removes 19 matchsticks (one per shared side), and 100 - 19 = 81, which matches. The answer is a count of matchsticks, so a whole number near but below 100 makes sense.
Takeaway
Keep the part that never changes (the first 5) separate from the part that grows (4 per new shape), and a long pattern becomes one quick calculation.
- Build small cases and spot the rule
- Separate the fixed part from the changing part
- Plug in 20 pentagons
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested