Depth · one concept from first principles · step 2 of 4
Regular pentagons are being built out of matchsticks as shown below. How many matchsticks are needed to make regular pentagons?
The matchstick pentagons are joined side by side in a single row. The first pentagon uses matchsticks, and each additional pentagon shares one side with the neighboring pentagon, so more matchsticks are added each time.
One pentagon needs 5 sticks, two need 9, three need 13, so each new one adds 4.
Trying the smallest cases first lets a young learner SEE the +4 rule instead of guessing it.
4.OA.C.5Solve An Easier Related ProblemThe first pentagon's 5 sticks never change, and each of the 19 later pentagons adds 4.
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.
Every pentagon after the first is built onto a side that is already standing, so it needs only 4 new sticks rather than 5.
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.
Each of those later pentagons brings the same 4 sticks, so the added part is that many equal groups of 4.
Those 19 extra pentagons add 4 x 19 = 76 sticks on top of the fixed 5.
Once the rule is written, the answer is just one multiplication and one addition.
4.NBT.B.5Look For A PatternKeep 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.
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The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Geometric Series Share, Lattice / Grid Pattern from Small Cases. Why the shapes repeat →