Problem
Express counts of dots, lines, faces
Shapes are built by nesting triangles in a regular pattern. The number of line segments goes 6, 12, 18 for the 1st, 2nd, 3rd shapes. Find how many line segments the 100th shape has.
Operations
Your answer
segments
Watch out It's easy to multiply 6 by 99 (counting only the jumps after the 1st shape) instead of by 100, the shape's own position number.
How to solve
Strategy Look for a Pattern — The segment counts 6, 12, 18 are the multiples of 6, so the nth shape has 6 times n segments. Confirm the rule on the small cases, then apply it to n = 100.
1STEP 1
Find the rule from the small cases
The segment counts are 6, 12, 18, which are 6x1, 6x2, 6x3. So the nth shape has 6 times n line segments.
6 = 6 × 1, 12 = 6 × 2, 18 = 6 × 3
Starting at 6 and adding 6 each step gives the multiples of 6, so the position number times 6 is the count.
4.OA.C.5Solve An Easier Related ProblemThe nth shape has 6 times n line segments.
Why?
The first shape already has 6 segments and every next shape adds another 6, so the nth shape is 6 counted n times over.
Why?
The nth shape's segments are n equal parts of 6 joined with no gaps or overlaps, so the total is simply those n sixes added together.
🧱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?
Adding 6 to itself n times is the same as n equal groups of 6, and that count equals 6 times n.
Why?
Making n equal groups of 6 and adding them up is exactly what n times 6 means.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
n groups of 6 counts the same dots as 6 groups of n, so n times 6 is the same number as 6 times n.
🧱Commutativity of multiplicationa rows of b and b rows of a count the same array of dots — rows and columns just swap.
2STEP 2
Apply the rule to the 100th shape
Substitute n = 100 into 6 times n.
6 × 100 = 600
The 100th shape is just 100 groups of 6 segments.
4.OA.C.5Look For A PatternAnswer
600 segments
6 × 100 = 600
The rule 6n gives 6, 12, 18 for n = 1, 2, 3, exactly the table values, so 6 x 100 = 600 for the 100th shape is consistent.
Takeaway
This only needs Grade 4 pattern sense: every step adds 6, so the 100th shape is 100 sixes!
- Find the rule from the small cases
- Apply the rule to the 100th shape
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested