← Express counts of dots, lines, faces · Systematically Count Shapes in a Figure

Express counts of dots, lines, faces · 10 practice problems

4.OA.C.5

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Generated variants — 10

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy

The shapes below are made by connecting dots and line segments in a regular pattern. Find how many line segments the $8$ th shape has.

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

Your answer

Show solution

1 · Understandwhat's really being asked

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 8th shape has.

Givens

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

Unknowns

The number of line segments in the 8th shape

Constraints

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

2 · Planchoose the strategy

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

The segment counts form a steady arithmetic pattern, so the nth shape's count follows a simple linear rule. Confirm the rule on the small cases, then apply it to the target.

3 · Execute2 carry out the plan

1Find the rule from the small cases

#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.

2Apply the rule to the 8th shape

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

Substitute n = 8 into 6 times n.

6 \times 8 = 48

The 8th shape just follows the same rule one more time.

Answer: 48 line segments

4 · Reviewdoes it hold up?

The rule 6n gives 6, 12, 18 for n = 1, 2, 3, exactly the table values, so 6 x 8 = 48 for the 8th shape is consistent.

Another way: 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 48 at n = 8.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Finding the linear rule for line segments and evaluating it at n = 8

💡Takeaway. This only needs Grade 4 pattern sense: every step adds 6, so the 8th shape is 8 6s!

Variant 2 easy

The shapes below are made by connecting dots and line segments in a regular pattern. Find how many line segments the $15$ th shape has.

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

Your answer

Show solution

1 · Understandwhat's really being asked

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 15th shape has.

Givens

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

Unknowns

The number of line segments in the 15th shape

Constraints

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

2 · Planchoose the strategy

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

The segment counts form a steady arithmetic pattern, so the nth shape's count follows a simple linear rule. Confirm the rule on the small cases, then apply it to the target.

3 · Execute2 carry out the plan

1Find the rule from the small cases

#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.

2Apply the rule to the 15th shape

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

Substitute n = 15 into 6 times n.

6 \times 15 = 90

The 15th shape just follows the same rule one more time.

Answer: 90 line segments

4 · Reviewdoes it hold up?

The rule 6n gives 6, 12, 18 for n = 1, 2, 3, exactly the table values, so 6 x 15 = 90 for the 15th shape is consistent.

Another way: 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 90 at n = 15.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Finding the linear rule for line segments and evaluating it at n = 15

💡Takeaway. This only needs Grade 4 pattern sense: every step adds 6, so the 15th shape is 15 6s!

Variant 3 medium

The shapes below are made by connecting dots and line segments in a regular pattern. Find how many line segments the $7$ th shape has.

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

Your answer

Show solution

1 · Understandwhat's really being asked

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 7th shape has.

Givens

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

Unknowns

The number of line segments in the 7th shape

Constraints

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

2 · Planchoose the strategy

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

The segment counts form a steady arithmetic pattern, so the nth shape's count follows a simple linear rule. Confirm the rule on the small cases, then apply it to the target.

3 · Execute2 carry out the plan

1Find the rule from the small cases

#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.

2Apply the rule to the 7th shape

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

Substitute n = 7 into 6 times n.

6 \times 7 = 42

The 7th shape just follows the same rule one more time.

Answer: 42 line segments

4 · Reviewdoes it hold up?

The rule 6n gives 6, 12, 18 for n = 1, 2, 3, exactly the table values, so 6 x 7 = 42 for the 7th shape is consistent.

Another way: 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 42 at n = 7.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Finding the linear rule for line segments and evaluating it at n = 7

💡Takeaway. This only needs Grade 4 pattern sense: every step adds 6, so the 7th shape is 7 6s!

Variant 4 medium

The shapes below are made by connecting dots and line segments in a regular pattern. Find how many line segments the $12$ th shape has.

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

Your answer

Show solution

1 · Understandwhat's really being asked

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 12th shape has.

Givens

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

Unknowns

The number of line segments in the 12th shape

Constraints

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

2 · Planchoose the strategy

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

The segment counts form a steady arithmetic pattern, so the nth shape's count follows a simple linear rule. Confirm the rule on the small cases, then apply it to the target.

3 · Execute2 carry out the plan

1Find the rule from the small cases

#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.

2Apply the rule to the 12th shape

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

Substitute n = 12 into 6 times n.

6 \times 12 = 72

The 12th shape just follows the same rule one more time.

Answer: 72 line segments

4 · Reviewdoes it hold up?

The rule 6n gives 6, 12, 18 for n = 1, 2, 3, exactly the table values, so 6 x 12 = 72 for the 12th shape is consistent.

Another way: 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 72 at n = 12.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Finding the linear rule for line segments and evaluating it at n = 12

💡Takeaway. This only needs Grade 4 pattern sense: every step adds 6, so the 12th shape is 12 6s!

Variant 5 easy

The shapes below are made by connecting dots and line segments in a regular pattern. Find how many line segments the $10$ th shape has.

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

Your answer

Show solution

1 · Understandwhat's really being asked

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 10th shape has.

Givens

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

Unknowns

The number of line segments in the 10th shape

Constraints

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

2 · Planchoose the strategy

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

The segment counts form a steady arithmetic pattern, so the nth shape's count follows a simple linear rule. Confirm the rule on the small cases, then apply it to the target.

3 · Execute2 carry out the plan

1Find the rule from the small cases

#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.

2Apply the rule to the 10th shape

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

Substitute n = 10 into 6 times n.

6 \times 10 = 60

The 10th shape just follows the same rule one more time.

Answer: 60 line segments

4 · Reviewdoes it hold up?

The rule 6n gives 6, 12, 18 for n = 1, 2, 3, exactly the table values, so 6 x 10 = 60 for the 10th shape is consistent.

Another way: 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 60 at n = 10.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Finding the linear rule for line segments and evaluating it at n = 10

💡Takeaway. This only needs Grade 4 pattern sense: every step adds 6, so the 10th shape is 10 6s!

Variant 6 medium

The shapes below are made by connecting dots and line segments in a regular pattern. Find how many line segments the $20$ th shape has.

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

Your answer

Show solution

1 · Understandwhat's really being asked

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 20th shape has.

Givens

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

Unknowns

The number of line segments in the 20th shape

Constraints

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

2 · Planchoose the strategy

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

The segment counts form a steady arithmetic pattern, so the nth shape's count follows a simple linear rule. Confirm the rule on the small cases, then apply it to the target.

3 · Execute2 carry out the plan

1Find the rule from the small cases

#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.

2Apply the rule to the 20th shape

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

Substitute n = 20 into 6 times n.

6 \times 20 = 120

The 20th shape just follows the same rule one more time.

Answer: 120 line segments

4 · Reviewdoes it hold up?

The rule 6n gives 6, 12, 18 for n = 1, 2, 3, exactly the table values, so 6 x 20 = 120 for the 20th shape is consistent.

Another way: 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 120 at n = 20.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Finding the linear rule for line segments and evaluating it at n = 20

💡Takeaway. This only needs Grade 4 pattern sense: every step adds 6, so the 20th shape is 20 6s!

Variant 7 medium

The shapes below are made by connecting dots and line segments in a regular pattern. Find how many line segments the $25$ th shape has.

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

Your answer

Show solution

1 · Understandwhat's really being asked

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 25th shape has.

Givens

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

Unknowns

The number of line segments in the 25th shape

Constraints

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

2 · Planchoose the strategy

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

The segment counts form a steady arithmetic pattern, so the nth shape's count follows a simple linear rule. Confirm the rule on the small cases, then apply it to the target.

3 · Execute2 carry out the plan

1Find the rule from the small cases

#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.

2Apply the rule to the 25th shape

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

Substitute n = 25 into 6 times n.

6 \times 25 = 150

The 25th shape just follows the same rule one more time.

Answer: 150 line segments

4 · Reviewdoes it hold up?

The rule 6n gives 6, 12, 18 for n = 1, 2, 3, exactly the table values, so 6 x 25 = 150 for the 25th shape is consistent.

Another way: 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 150 at n = 25.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Finding the linear rule for line segments and evaluating it at n = 25

💡Takeaway. This only needs Grade 4 pattern sense: every step adds 6, so the 25th shape is 25 6s!

Variant 8 hard

The shapes below are made by connecting dots and line segments in a regular pattern. Find how many line segments the $50$ th shape has.

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

Your answer

Show solution

1 · Understandwhat's really being asked

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 50th shape has.

Givens

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

Unknowns

The number of line segments in the 50th shape

Constraints

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

2 · Planchoose the strategy

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

The segment counts form a steady arithmetic pattern, so the nth shape's count follows a simple linear rule. Confirm the rule on the small cases, then apply it to the target.

3 · Execute2 carry out the plan

1Find the rule from the small cases

#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.

2Apply the rule to the 50th shape

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

Substitute n = 50 into 6 times n.

6 \times 50 = 300

The 50th shape just follows the same rule one more time.

Answer: 300 line segments

4 · Reviewdoes it hold up?

The rule 6n gives 6, 12, 18 for n = 1, 2, 3, exactly the table values, so 6 x 50 = 300 for the 50th shape is consistent.

Another way: 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 300 at n = 50.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Finding the linear rule for line segments and evaluating it at n = 50

💡Takeaway. This only needs Grade 4 pattern sense: every step adds 6, so the 50th shape is 50 6s!

Variant 9 hard

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	$6$	$12$	$18$

Your answer

Show solution

1 · Understandwhat's really being asked

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
3rd shape: 18 line segments
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

2 · Planchoose the strategy

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

The segment counts form a steady arithmetic pattern, so the nth shape's count follows a simple linear rule. Confirm the rule on the small cases, then apply it to the target.

3 · Execute2 carry out the plan

1Find the rule from the small cases

#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.

2Apply the rule to the 100th shape

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

Substitute n = 100 into 6 times n.

6 \times 100 = 600

The 100th shape just follows the same rule one more time.

Answer: 600 line segments

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin grade 4

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

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

Variant 10 hard

The shapes below are made by connecting dots and line segments in a regular pattern. Find how many line segments the $200$ th shape has.

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

Your answer

Show solution

1 · Understandwhat's really being asked

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 200th shape has.

Givens

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

Unknowns

The number of line segments in the 200th shape

Constraints

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

2 · Planchoose the strategy

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

The segment counts form a steady arithmetic pattern, so the nth shape's count follows a simple linear rule. Confirm the rule on the small cases, then apply it to the target.

3 · Execute2 carry out the plan

1Find the rule from the small cases

#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.

2Apply the rule to the 200th shape

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

Substitute n = 200 into 6 times n.

6 \times 200 = 1200

The 200th shape just follows the same rule one more time.

Answer: 1200 line segments

4 · Reviewdoes it hold up?

The rule 6n gives 6, 12, 18 for n = 1, 2, 3, exactly the table values, so 6 x 200 = 1200 for the 200th shape is consistent.

Another way: 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 1200 at n = 200.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Finding the linear rule for line segments and evaluating it at n = 200

💡Takeaway. This only needs Grade 4 pattern sense: every step adds 6, so the 200th shape is 200 6s!

Where next?

← Back to the problem: Express counts of dots, lines, faces