Express counts of dots, lines, faces

4.OA.C.5 · take · grade 4

Archetype: Systematically Count Shapes in a Figure · step in a 5-type progression

The shapes below are made by connecting dots and line segments in a regular pattern. Find how many line segments the $100$ th shape has.

Position	1st	2nd	3rd
Number of dots	$4$	$7$	$10$
Number of line segments	$6$	$12$	$18$

Show solution

Understand

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.

Givens

1st shape: 6 line segments (4 dots)
2nd shape: 12 line segments (7 dots)
3rd shape: 18 line segments (10 dots)
Each step adds 6 more line segments than the previous one

Unknowns

The number of line segments in the 100th shape

Constraints

The number of segments grows by a constant 6 each step
The count at step 1 is 6

Plan

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

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.

Execute

#9 Solve an Easier Related Problem 4.OA.C.5

The segment counts are 6, 12, 18, which are 6x1, 6x2, 6x3. So the nth shape has 6 times n line segments.

6 = 6 \times 1,\ 12 = 6 \times 2,\ 18 = 6 \times 3

Starting at 6 and adding 6 each step gives the multiples of 6, so the position number times 6 is the count.

#5 Look for a Pattern 4.OA.C.5

Substitute n = 100 into 6 times n.

6 \times 100 = 600

The 100th shape is just 100 groups of 6 segments.

Answer: 600 line segments

Review

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.

Evaluate finite differences (tool 14): the common difference is 6, and since the count at n = 0 would be 0, the formula is 6n, giving 600 at n = 100.

Standards · min grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Finding the 6n rule for line segments and evaluating it at n = 100

💡 This only needs Grade 4 pattern sense: every step adds 6, so the 100th shape is 100 sixes!