← Count all shapes hidden in a grid · Systematically Count Shapes in a Figure

Count all shapes hidden in a grid · 10 practice problems

1.G.A.23.OA.D.9

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Generated variants — 10

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

Variant 1 easy

How many squares, large and small, can you find in the figure below?

The figure is made of equal-sized unit squares joined together. the top row has $3$ unit squares, the bottom row has $4$ unit squares, all aligned at the left edge (so the rows form a stepped shape). Count every square that can be traced in the figure -- the $1$ -cell squares, the $4$ -cell ( $2\times2$ ) squares, the $9$ -cell ( $3\times3$ ) square, and so on.

Your answer

Show solution

1 · Understandwhat's really being asked

A stepped figure is built from 7 equal unit squares with row widths [3, 4], all left-aligned. I must count every square of every size that can be traced along the grid lines.

Givens

The figure is made of 7 equal unit squares.
The row widths from top to bottom are [3, 4].
All rows are aligned at the left edge, giving a stepped outline.

Unknowns

The total number of squares of all sizes in the figure.

Constraints

A counted square must have all of its cells present in the figure.
Squares may overlap and be of different sizes.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

To avoid missing or double-counting, organize the count by square size: first all 1x1, then 2x2, and so on. Splitting by size turns one tricky count into a few easy subproblems.

3 · Execute3 carry out the plan

1Count the 1x1 squares

#2 Make a Systematic List 3.OA.D.9

Every unit cell is a 1x1 square. The figure is made of 7 unit cells, so there are 7 squares of size 1x1.

3 + 4 = 7

Adding the cells in each row is Grade 3 addition you can do by counting the picture.

2Count the 2x2 squares

#7 Identify Subproblems 1.G.A.2

A 2x2 square needs a full 2-by-2 block of present cells. Scanning every position, 2 such block(s) fit inside the stepped shape.

2

Composing 4 small squares into one bigger square is what 'compose two-dimensional shapes' means.

3Add up all sizes

#2 Make a Systematic List 3.OA.D.9

Total squares = the per-size counts added together. No larger square fits because the figure is not tall or wide enough.

7 + 2 = 9

Collecting the size-by-size subtotals into one sum is Grade 3 addition.

Answer: 9

4 · Reviewdoes it hold up?

There must be more small squares than big ones, and the counts (7 of size 1x1, 2 of size 2x2) shrink as the size grows; the total 9 is at least the 7 unit cells, as expected.

Another way: Count from the full bounding grid and subtract the squares lost to the missing corner cells, getting the same total.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Adding the per-size subtotals and reasoning systematically.
1.G.A.2 Compose two-dimensional shapes or three-dimensional shapes — Recognizing that 4 or 9 unit squares compose a larger square.

💡Takeaway. Count squares one size at a time, smallest to biggest, and you will never miss one!

Variant 2 medium

How many squares, large and small, can you find in the figure below?

The figure is made of equal-sized unit squares joined together. the top row has $2$ unit squares, the middle row has $4$ unit squares, the bottom row has $4$ unit squares, all aligned at the left edge (so the rows form a stepped shape). Count every square that can be traced in the figure -- the $1$ -cell squares, the $4$ -cell ( $2\times2$ ) squares, the $9$ -cell ( $3\times3$ ) square, and so on.

Your answer

Show solution

1 · Understandwhat's really being asked

A stepped figure is built from 10 equal unit squares with row widths [2, 4, 4], all left-aligned. I must count every square of every size that can be traced along the grid lines.

Givens

The figure is made of 10 equal unit squares.
The row widths from top to bottom are [2, 4, 4].
All rows are aligned at the left edge, giving a stepped outline.

Unknowns

The total number of squares of all sizes in the figure.

Constraints

A counted square must have all of its cells present in the figure.
Squares may overlap and be of different sizes.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

To avoid missing or double-counting, organize the count by square size: first all 1x1, then 2x2, and so on. Splitting by size turns one tricky count into a few easy subproblems.

3 · Execute3 carry out the plan

1Count the 1x1 squares

#2 Make a Systematic List 3.OA.D.9

Every unit cell is a 1x1 square. The figure is made of 10 unit cells, so there are 10 squares of size 1x1.

2 + 4 + 4 = 10

Adding the cells in each row is Grade 3 addition you can do by counting the picture.

2Count the 2x2 squares

#7 Identify Subproblems 1.G.A.2

A 2x2 square needs a full 2-by-2 block of present cells. Scanning every position, 4 such block(s) fit inside the stepped shape.

4

Composing 4 small squares into one bigger square is what 'compose two-dimensional shapes' means.

3Add up all sizes

#2 Make a Systematic List 3.OA.D.9

Total squares = the per-size counts added together. No larger square fits because the figure is not tall or wide enough.

10 + 4 = 14

Collecting the size-by-size subtotals into one sum is Grade 3 addition.

Answer: 14

4 · Reviewdoes it hold up?

There must be more small squares than big ones, and the counts (10 of size 1x1, 4 of size 2x2) shrink as the size grows; the total 14 is at least the 10 unit cells, as expected.

Another way: Count from the full bounding grid and subtract the squares lost to the missing corner cells, getting the same total.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Adding the per-size subtotals and reasoning systematically.
1.G.A.2 Compose two-dimensional shapes or three-dimensional shapes — Recognizing that 4 or 9 unit squares compose a larger square.

💡Takeaway. Count squares one size at a time, smallest to biggest, and you will never miss one!

Variant 3 easy

How many squares, large and small, can you find in the figure below?

The figure is made of equal-sized unit squares joined together. the top row has $2$ unit squares, the middle row has $3$ unit squares, the bottom row has $4$ unit squares, all aligned at the left edge (so the rows form a stepped shape). Count every square that can be traced in the figure -- the $1$ -cell squares, the $4$ -cell ( $2\times2$ ) squares, the $9$ -cell ( $3\times3$ ) square, and so on.

Your answer

Show solution

1 · Understandwhat's really being asked

A stepped figure is built from 9 equal unit squares with row widths [2, 3, 4], all left-aligned. I must count every square of every size that can be traced along the grid lines.

Givens

The figure is made of 9 equal unit squares.
The row widths from top to bottom are [2, 3, 4].
All rows are aligned at the left edge, giving a stepped outline.

Unknowns

The total number of squares of all sizes in the figure.

Constraints

A counted square must have all of its cells present in the figure.
Squares may overlap and be of different sizes.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

To avoid missing or double-counting, organize the count by square size: first all 1x1, then 2x2, and so on. Splitting by size turns one tricky count into a few easy subproblems.

3 · Execute3 carry out the plan

1Count the 1x1 squares

#2 Make a Systematic List 3.OA.D.9

Every unit cell is a 1x1 square. The figure is made of 9 unit cells, so there are 9 squares of size 1x1.

2 + 3 + 4 = 9

Adding the cells in each row is Grade 3 addition you can do by counting the picture.

2Count the 2x2 squares

#7 Identify Subproblems 1.G.A.2

A 2x2 square needs a full 2-by-2 block of present cells. Scanning every position, 3 such block(s) fit inside the stepped shape.

3

Composing 4 small squares into one bigger square is what 'compose two-dimensional shapes' means.

3Add up all sizes

#2 Make a Systematic List 3.OA.D.9

Total squares = the per-size counts added together. No larger square fits because the figure is not tall or wide enough.

9 + 3 = 12

Collecting the size-by-size subtotals into one sum is Grade 3 addition.

Answer: 12

4 · Reviewdoes it hold up?

There must be more small squares than big ones, and the counts (9 of size 1x1, 3 of size 2x2) shrink as the size grows; the total 12 is at least the 9 unit cells, as expected.

Another way: Count from the full bounding grid and subtract the squares lost to the missing corner cells, getting the same total.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Adding the per-size subtotals and reasoning systematically.
1.G.A.2 Compose two-dimensional shapes or three-dimensional shapes — Recognizing that 4 or 9 unit squares compose a larger square.

💡Takeaway. Count squares one size at a time, smallest to biggest, and you will never miss one!

Variant 4 easy

How many squares, large and small, can you find in the figure below?

The figure is made of equal-sized unit squares joined together. the top row has $2$ unit squares, the middle row has $3$ unit squares, the bottom row has $3$ unit squares, all aligned at the left edge (so the rows form a stepped shape). Count every square that can be traced in the figure -- the $1$ -cell squares, the $4$ -cell ( $2\times2$ ) squares, the $9$ -cell ( $3\times3$ ) square, and so on.

Your answer

Show solution

1 · Understandwhat's really being asked

A stepped figure is built from 8 equal unit squares with row widths [2, 3, 3], all left-aligned. I must count every square of every size that can be traced along the grid lines.

Givens

The figure is made of 8 equal unit squares.
The row widths from top to bottom are [2, 3, 3].
All rows are aligned at the left edge, giving a stepped outline.

Unknowns

The total number of squares of all sizes in the figure.

Constraints

A counted square must have all of its cells present in the figure.
Squares may overlap and be of different sizes.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

To avoid missing or double-counting, organize the count by square size: first all 1x1, then 2x2, and so on. Splitting by size turns one tricky count into a few easy subproblems.

3 · Execute3 carry out the plan

1Count the 1x1 squares

#2 Make a Systematic List 3.OA.D.9

Every unit cell is a 1x1 square. The figure is made of 8 unit cells, so there are 8 squares of size 1x1.

2 + 3 + 3 = 8

Adding the cells in each row is Grade 3 addition you can do by counting the picture.

2Count the 2x2 squares

#7 Identify Subproblems 1.G.A.2

A 2x2 square needs a full 2-by-2 block of present cells. Scanning every position, 3 such block(s) fit inside the stepped shape.

3

Composing 4 small squares into one bigger square is what 'compose two-dimensional shapes' means.

3Add up all sizes

#2 Make a Systematic List 3.OA.D.9

Total squares = the per-size counts added together. No larger square fits because the figure is not tall or wide enough.

8 + 3 = 11

Collecting the size-by-size subtotals into one sum is Grade 3 addition.

Answer: 11

4 · Reviewdoes it hold up?

There must be more small squares than big ones, and the counts (8 of size 1x1, 3 of size 2x2) shrink as the size grows; the total 11 is at least the 8 unit cells, as expected.

Another way: Count from the full bounding grid and subtract the squares lost to the missing corner cells, getting the same total.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Adding the per-size subtotals and reasoning systematically.
1.G.A.2 Compose two-dimensional shapes or three-dimensional shapes — Recognizing that 4 or 9 unit squares compose a larger square.

💡Takeaway. Count squares one size at a time, smallest to biggest, and you will never miss one!

Variant 5 medium

How many squares, large and small, can you find in the figure below?

The figure is made of equal-sized unit squares joined together. the top row has $3$ unit squares, the middle row has $3$ unit squares, the bottom row has $3$ unit squares, all aligned at the left edge (so the rows form a stepped shape). Count every square that can be traced in the figure -- the $1$ -cell squares, the $4$ -cell ( $2\times2$ ) squares, the $9$ -cell ( $3\times3$ ) square, and so on.

Your answer

Show solution

1 · Understandwhat's really being asked

A stepped figure is built from 9 equal unit squares with row widths [3, 3, 3], all left-aligned. I must count every square of every size that can be traced along the grid lines.

Givens

The figure is made of 9 equal unit squares.
The row widths from top to bottom are [3, 3, 3].
All rows are aligned at the left edge, giving a stepped outline.

Unknowns

The total number of squares of all sizes in the figure.

Constraints

A counted square must have all of its cells present in the figure.
Squares may overlap and be of different sizes.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

To avoid missing or double-counting, organize the count by square size: first all 1x1, then 2x2, and so on. Splitting by size turns one tricky count into a few easy subproblems.

3 · Execute4 carry out the plan

1Count the 1x1 squares

#2 Make a Systematic List 3.OA.D.9

Every unit cell is a 1x1 square. The figure is made of 9 unit cells, so there are 9 squares of size 1x1.

3 + 3 + 3 = 9

Adding the cells in each row is Grade 3 addition you can do by counting the picture.

2Count the 2x2 squares

#7 Identify Subproblems 1.G.A.2

A 2x2 square needs a full 2-by-2 block of present cells. Scanning every position, 4 such block(s) fit inside the stepped shape.

4

Composing 4 small squares into one bigger square is what 'compose two-dimensional shapes' means.

3Count the 3x3 squares

#7 Identify Subproblems 1.G.A.2

A 3x3 square needs a full 3-by-3 block of present cells. Scanning every position, 1 such block(s) fit inside the stepped shape.

1

Composing 9 small squares into one bigger square is what 'compose two-dimensional shapes' means.

4Add up all sizes

#2 Make a Systematic List 3.OA.D.9

Total squares = the per-size counts added together. No larger square fits because the figure is not tall or wide enough.

9 + 4 + 1 = 14

Collecting the size-by-size subtotals into one sum is Grade 3 addition.

Answer: 14

4 · Reviewdoes it hold up?

There must be more small squares than big ones, and the counts (9 of size 1x1, 4 of size 2x2, 1 of size 3x3) shrink as the size grows; the total 14 is at least the 9 unit cells, as expected.

Another way: Count from the full bounding grid and subtract the squares lost to the missing corner cells, getting the same total.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Adding the per-size subtotals and reasoning systematically.
1.G.A.2 Compose two-dimensional shapes or three-dimensional shapes — Recognizing that 4 or 9 unit squares compose a larger square.

💡Takeaway. Count squares one size at a time, smallest to biggest, and you will never miss one!

Variant 6 hard

How many squares, large and small, can you find in the figure below?

The figure is made of equal-sized unit squares joined together. the top row has $4$ unit squares, the middle row has $4$ unit squares, the bottom row has $3$ unit squares, all aligned at the left edge (so the rows form a stepped shape). Count every square that can be traced in the figure -- the $1$ -cell squares, the $4$ -cell ( $2\times2$ ) squares, the $9$ -cell ( $3\times3$ ) square, and so on.

Your answer

Show solution

1 · Understandwhat's really being asked

A stepped figure is built from 11 equal unit squares with row widths [4, 4, 3], all left-aligned. I must count every square of every size that can be traced along the grid lines.

Givens

The figure is made of 11 equal unit squares.
The row widths from top to bottom are [4, 4, 3].
All rows are aligned at the left edge, giving a stepped outline.

Unknowns

The total number of squares of all sizes in the figure.

Constraints

A counted square must have all of its cells present in the figure.
Squares may overlap and be of different sizes.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

To avoid missing or double-counting, organize the count by square size: first all 1x1, then 2x2, and so on. Splitting by size turns one tricky count into a few easy subproblems.

3 · Execute4 carry out the plan

1Count the 1x1 squares

#2 Make a Systematic List 3.OA.D.9

Every unit cell is a 1x1 square. The figure is made of 11 unit cells, so there are 11 squares of size 1x1.

4 + 4 + 3 = 11

Adding the cells in each row is Grade 3 addition you can do by counting the picture.

2Count the 2x2 squares

#7 Identify Subproblems 1.G.A.2

A 2x2 square needs a full 2-by-2 block of present cells. Scanning every position, 5 such block(s) fit inside the stepped shape.

5

Composing 4 small squares into one bigger square is what 'compose two-dimensional shapes' means.

3Count the 3x3 squares

#7 Identify Subproblems 1.G.A.2

A 3x3 square needs a full 3-by-3 block of present cells. Scanning every position, 1 such block(s) fit inside the stepped shape.

1

Composing 9 small squares into one bigger square is what 'compose two-dimensional shapes' means.

4Add up all sizes

#2 Make a Systematic List 3.OA.D.9

Total squares = the per-size counts added together. No larger square fits because the figure is not tall or wide enough.

11 + 5 + 1 = 17

Collecting the size-by-size subtotals into one sum is Grade 3 addition.

Answer: 17

4 · Reviewdoes it hold up?

There must be more small squares than big ones, and the counts (11 of size 1x1, 5 of size 2x2, 1 of size 3x3) shrink as the size grows; the total 17 is at least the 11 unit cells, as expected.

Another way: Count from the full bounding grid and subtract the squares lost to the missing corner cells, getting the same total.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Adding the per-size subtotals and reasoning systematically.
1.G.A.2 Compose two-dimensional shapes or three-dimensional shapes — Recognizing that 4 or 9 unit squares compose a larger square.

💡Takeaway. Count squares one size at a time, smallest to biggest, and you will never miss one!

Variant 7 hard

How many squares, large and small, can you find in the figure below?

The figure is made of equal-sized unit squares joined together. the top row has $3$ unit squares, the middle row has $3$ unit squares, the bottom row has $4$ unit squares, all aligned at the left edge (so the rows form a stepped shape). Count every square that can be traced in the figure -- the $1$ -cell squares, the $4$ -cell ( $2\times2$ ) squares, the $9$ -cell ( $3\times3$ ) square, and so on.

Your answer

Show solution

1 · Understandwhat's really being asked

A stepped figure is built from 10 equal unit squares with row widths [3, 3, 4], all left-aligned. I must count every square of every size that can be traced along the grid lines.

Givens

The figure is made of 10 equal unit squares.
The row widths from top to bottom are [3, 3, 4].
All rows are aligned at the left edge, giving a stepped outline.

Unknowns

The total number of squares of all sizes in the figure.

Constraints

A counted square must have all of its cells present in the figure.
Squares may overlap and be of different sizes.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

To avoid missing or double-counting, organize the count by square size: first all 1x1, then 2x2, and so on. Splitting by size turns one tricky count into a few easy subproblems.

3 · Execute4 carry out the plan

1Count the 1x1 squares

#2 Make a Systematic List 3.OA.D.9

Every unit cell is a 1x1 square. The figure is made of 10 unit cells, so there are 10 squares of size 1x1.

3 + 3 + 4 = 10

Adding the cells in each row is Grade 3 addition you can do by counting the picture.

2Count the 2x2 squares

#7 Identify Subproblems 1.G.A.2

A 2x2 square needs a full 2-by-2 block of present cells. Scanning every position, 4 such block(s) fit inside the stepped shape.

4

Composing 4 small squares into one bigger square is what 'compose two-dimensional shapes' means.

3Count the 3x3 squares

#7 Identify Subproblems 1.G.A.2

A 3x3 square needs a full 3-by-3 block of present cells. Scanning every position, 1 such block(s) fit inside the stepped shape.

1

Composing 9 small squares into one bigger square is what 'compose two-dimensional shapes' means.

4Add up all sizes

#2 Make a Systematic List 3.OA.D.9

Total squares = the per-size counts added together. No larger square fits because the figure is not tall or wide enough.

10 + 4 + 1 = 15

Collecting the size-by-size subtotals into one sum is Grade 3 addition.

Answer: 15

4 · Reviewdoes it hold up?

There must be more small squares than big ones, and the counts (10 of size 1x1, 4 of size 2x2, 1 of size 3x3) shrink as the size grows; the total 15 is at least the 10 unit cells, as expected.

Another way: Count from the full bounding grid and subtract the squares lost to the missing corner cells, getting the same total.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Adding the per-size subtotals and reasoning systematically.
1.G.A.2 Compose two-dimensional shapes or three-dimensional shapes — Recognizing that 4 or 9 unit squares compose a larger square.

💡Takeaway. Count squares one size at a time, smallest to biggest, and you will never miss one!

Variant 8 medium

How many squares, large and small, can you find in the figure below?

The figure is made of equal-sized unit squares joined together. the top row has $4$ unit squares, the middle row has $4$ unit squares, the bottom row has $4$ unit squares, all aligned at the left edge (so the rows form a stepped shape). Count every square that can be traced in the figure -- the $1$ -cell squares, the $4$ -cell ( $2\times2$ ) squares, the $9$ -cell ( $3\times3$ ) square, and so on.

Your answer

Show solution

1 · Understandwhat's really being asked

A stepped figure is built from 12 equal unit squares with row widths [4, 4, 4], all left-aligned. I must count every square of every size that can be traced along the grid lines.

Givens

The figure is made of 12 equal unit squares.
The row widths from top to bottom are [4, 4, 4].
All rows are aligned at the left edge, giving a stepped outline.

Unknowns

The total number of squares of all sizes in the figure.

Constraints

A counted square must have all of its cells present in the figure.
Squares may overlap and be of different sizes.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

To avoid missing or double-counting, organize the count by square size: first all 1x1, then 2x2, and so on. Splitting by size turns one tricky count into a few easy subproblems.

3 · Execute4 carry out the plan

1Count the 1x1 squares

#2 Make a Systematic List 3.OA.D.9

Every unit cell is a 1x1 square. The figure is made of 12 unit cells, so there are 12 squares of size 1x1.

4 + 4 + 4 = 12

Adding the cells in each row is Grade 3 addition you can do by counting the picture.

2Count the 2x2 squares

#7 Identify Subproblems 1.G.A.2

A 2x2 square needs a full 2-by-2 block of present cells. Scanning every position, 6 such block(s) fit inside the stepped shape.

6

Composing 4 small squares into one bigger square is what 'compose two-dimensional shapes' means.

3Count the 3x3 squares

#7 Identify Subproblems 1.G.A.2

A 3x3 square needs a full 3-by-3 block of present cells. Scanning every position, 2 such block(s) fit inside the stepped shape.

2

Composing 9 small squares into one bigger square is what 'compose two-dimensional shapes' means.

4Add up all sizes

#2 Make a Systematic List 3.OA.D.9

Total squares = the per-size counts added together. No larger square fits because the figure is not tall or wide enough.

12 + 6 + 2 = 20

Collecting the size-by-size subtotals into one sum is Grade 3 addition.

Answer: 20

4 · Reviewdoes it hold up?

There must be more small squares than big ones, and the counts (12 of size 1x1, 6 of size 2x2, 2 of size 3x3) shrink as the size grows; the total 20 is at least the 12 unit cells, as expected.

Another way: Count from the full bounding grid and subtract the squares lost to the missing corner cells, getting the same total.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Adding the per-size subtotals and reasoning systematically.
1.G.A.2 Compose two-dimensional shapes or three-dimensional shapes — Recognizing that 4 or 9 unit squares compose a larger square.

💡Takeaway. Count squares one size at a time, smallest to biggest, and you will never miss one!

Variant 9 medium

How many squares, large and small, can you find in the figure below?

The figure is made of equal-sized unit squares joined together. the top row has $3$ unit squares, the middle row has $4$ unit squares, the bottom row has $4$ unit squares, all aligned at the left edge (so the rows form a stepped shape). Count every square that can be traced in the figure -- the $1$ -cell squares, the $4$ -cell ( $2\times2$ ) squares, the $9$ -cell ( $3\times3$ ) square, and so on.

Your answer

Show solution

1 · Understandwhat's really being asked

A stepped figure is built from 11 equal unit squares with row widths [3, 4, 4], all left-aligned. I must count every square of every size that can be traced along the grid lines.

Givens

The figure is made of 11 equal unit squares.
The row widths from top to bottom are [3, 4, 4].
All rows are aligned at the left edge, giving a stepped outline.

Unknowns

The total number of squares of all sizes in the figure.

Constraints

A counted square must have all of its cells present in the figure.
Squares may overlap and be of different sizes.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

To avoid missing or double-counting, organize the count by square size: first all 1x1, then 2x2, and so on. Splitting by size turns one tricky count into a few easy subproblems.

3 · Execute4 carry out the plan

1Count the 1x1 squares

#2 Make a Systematic List 3.OA.D.9

Every unit cell is a 1x1 square. The figure is made of 11 unit cells, so there are 11 squares of size 1x1.

3 + 4 + 4 = 11

Adding the cells in each row is Grade 3 addition you can do by counting the picture.

2Count the 2x2 squares

#7 Identify Subproblems 1.G.A.2

A 2x2 square needs a full 2-by-2 block of present cells. Scanning every position, 5 such block(s) fit inside the stepped shape.

5

Composing 4 small squares into one bigger square is what 'compose two-dimensional shapes' means.

3Count the 3x3 squares

#7 Identify Subproblems 1.G.A.2

A 3x3 square needs a full 3-by-3 block of present cells. Scanning every position, 1 such block(s) fit inside the stepped shape.

1

Composing 9 small squares into one bigger square is what 'compose two-dimensional shapes' means.

4Add up all sizes

#2 Make a Systematic List 3.OA.D.9

Total squares = the per-size counts added together. No larger square fits because the figure is not tall or wide enough.

11 + 5 + 1 = 17

Collecting the size-by-size subtotals into one sum is Grade 3 addition.

Answer: 17

4 · Reviewdoes it hold up?

There must be more small squares than big ones, and the counts (11 of size 1x1, 5 of size 2x2, 1 of size 3x3) shrink as the size grows; the total 17 is at least the 11 unit cells, as expected.

Another way: Count from the full bounding grid and subtract the squares lost to the missing corner cells, getting the same total.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Adding the per-size subtotals and reasoning systematically.
1.G.A.2 Compose two-dimensional shapes or three-dimensional shapes — Recognizing that 4 or 9 unit squares compose a larger square.

💡Takeaway. Count squares one size at a time, smallest to biggest, and you will never miss one!

Variant 10 hard

How many squares, large and small, can you find in the figure below?

The figure is made of equal-sized unit squares joined together. the top row has $4$ unit squares, the second row has $4$ unit squares, the third row has $4$ unit squares, the fourth row has $4$ unit squares, all aligned at the left edge (so the rows form a stepped shape). Count every square that can be traced in the figure -- the $1$ -cell squares, the $4$ -cell ( $2\times2$ ) squares, the $9$ -cell ( $3\times3$ ) square, and so on.

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1 · Understandwhat's really being asked

A stepped figure is built from 16 equal unit squares with row widths [4, 4, 4, 4], all left-aligned. I must count every square of every size that can be traced along the grid lines.

Givens

The figure is made of 16 equal unit squares.
The row widths from top to bottom are [4, 4, 4, 4].
All rows are aligned at the left edge, giving a stepped outline.

Unknowns

The total number of squares of all sizes in the figure.

Constraints

A counted square must have all of its cells present in the figure.
Squares may overlap and be of different sizes.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

To avoid missing or double-counting, organize the count by square size: first all 1x1, then 2x2, and so on. Splitting by size turns one tricky count into a few easy subproblems.

3 · Execute5 carry out the plan

1Count the 1x1 squares

#2 Make a Systematic List 3.OA.D.9

Every unit cell is a 1x1 square. The figure is made of 16 unit cells, so there are 16 squares of size 1x1.

4 + 4 + 4 + 4 = 16

Adding the cells in each row is Grade 3 addition you can do by counting the picture.

2Count the 2x2 squares

#7 Identify Subproblems 1.G.A.2

A 2x2 square needs a full 2-by-2 block of present cells. Scanning every position, 9 such block(s) fit inside the stepped shape.

9

Composing 4 small squares into one bigger square is what 'compose two-dimensional shapes' means.

3Count the 3x3 squares

#7 Identify Subproblems 1.G.A.2

A 3x3 square needs a full 3-by-3 block of present cells. Scanning every position, 4 such block(s) fit inside the stepped shape.

4

Composing 9 small squares into one bigger square is what 'compose two-dimensional shapes' means.

4Count the 4x4 squares

#7 Identify Subproblems 1.G.A.2

A 4x4 square needs a full 4-by-4 block of present cells. Scanning every position, 1 such block(s) fit inside the stepped shape.

1

Composing 16 small squares into one bigger square is what 'compose two-dimensional shapes' means.

5Add up all sizes

#2 Make a Systematic List 3.OA.D.9

Total squares = the per-size counts added together. No larger square fits because the figure is not tall or wide enough.

16 + 9 + 4 + 1 = 30

Collecting the size-by-size subtotals into one sum is Grade 3 addition.

Answer: 30

4 · Reviewdoes it hold up?

There must be more small squares than big ones, and the counts (16 of size 1x1, 9 of size 2x2, 4 of size 3x3, 1 of size 4x4) shrink as the size grows; the total 30 is at least the 16 unit cells, as expected.

Another way: Count from the full bounding grid and subtract the squares lost to the missing corner cells, getting the same total.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Adding the per-size subtotals and reasoning systematically.
1.G.A.2 Compose two-dimensional shapes or three-dimensional shapes — Recognizing that 4 or 9 unit squares compose a larger square.

💡Takeaway. Count squares one size at a time, smallest to biggest, and you will never miss one!

Where next?

← Back to the problem: Count all shapes hidden in a grid