Practice · Two colours, two separate rules · Sensim Math

Variant 1 easy answer: 92

Game stones are placed following the pattern shown below. For the $12$ th figure, what is the difference between the number of white stones and the number of black stones?

The figures show stones arranged in growing squares. The 1st figure is a $3\times3$ array: the outer border, $1$ stone thick, is black and the single center stone is white. The 2nd figure is a $4\times4$ array: the border is still $1$ stone thick and the inner $2\times2$ stones are white. The 3rd figure is a $5\times5$ array with an inner $3\times3$ white block. In this way, each step adds one more stone to the side of the white block, and the black border keeps the same thickness.

Show solution

1 · Understandwhat's really being asked

Each figure is a square of stones with a white block inside and a black border 1 stone thick. We need how far apart the two counts are in the 12th figure.

Givens

In figure n the white block is n by n.
The black border is 1 stone thick all round.
We want figure 12.

Unknowns

The difference between the white count and the black count.

Constraints

Every stone is either white or black, and none is counted twice.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

The two colours do not grow at the same speed, so one rule cannot cover both. Find a rule for each colour separately from the small figures, then run both at the figure we want.

3 · Execute4 carry out the plan

1Find the rule for white stones

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

The white stones form the inner square. Figure 1 has a 1x1 white block, figure 2 a 2x2, figure 3 a 3x3. So in figure n the white block is n by n.

\text{white}(n) = n \times n

White grows like a square number.

2Find the rule for black stones

#9 Solve an Easier Related Problem 5.OA.A.2

The black stones are the border of the whole square. In figure n the whole square is (n + 2) by (n + 2), and the white inside is n by n, so the black border is the difference.

\text{black}(n) = (n+2) \times (n+2) - n \times n = 4n + 4

Black grows in a straight line, not as a square.

3Use both rules for the 12th figure

#5 Look for a Pattern 5.OA.B.3

Apply each rule at n = 12. White is 12 x 12 = 144 stones. Black is 4 x 12 + 4 = 52 stones.

\text{white}(12) = 144,\quad \text{black}(12) = 4 \times 12 + 4 = 52

144 white against 52 black.

4Subtract to get the difference

#5 Look for a Pattern 5.OA.A.2

There are more white stones than black, so the difference is 144 - 52.

144 - 52 = 92

They differ by 92 stones.

Answer: 92

4 · Reviewdoes it hold up?

Both counts should add back to the whole square: 144 + 52 = 196, and the square is 14 x 14 = 196. They match, so nothing was missed or double-counted.

Another way: Sanity-check the rules on figure 1: white(1) = 1 and black(1) = 8, which is exactly the border of the 3 by 3 square in the first picture.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern that follows a given rule — Reading each colour's growth off the first three figures.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the black count as a whole square minus the white one.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Running two separate rules at the same figure number and comparing them.

💡Takeaway. When two things in a pattern grow at different speeds, give each one its own rule before you try to compare them.

Variant 2 easy answer: 89

Game stones are placed following the pattern shown below. For the $15$ th figure, what is the difference between the number of white stones and the number of black stones?

The figures show stones arranged in growing squares. The 1st figure is a $5\times5$ array: the outer border, $2$ stones thick, is black and the single center stone is white. The 2nd figure is a $6\times6$ array: the border is still $2$ stones thick and the inner $2\times2$ stones are white. The 3rd figure is a $7\times7$ array with an inner $3\times3$ white block. In this way, each step adds one more stone to the side of the white block, and the black border keeps the same thickness.

Show solution

1 · Understandwhat's really being asked

Each figure is a square of stones with a white block inside and a black border 2 stones thick. We need how far apart the two counts are in the 15th figure.

Givens

In figure n the white block is n by n.
The black border is 2 stones thick all round.
We want figure 15.

Unknowns

The difference between the white count and the black count.

Constraints

Every stone is either white or black, and none is counted twice.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

The two colours do not grow at the same speed, so one rule cannot cover both. Find a rule for each colour separately from the small figures, then run both at the figure we want.

3 · Execute4 carry out the plan

1Find the rule for white stones

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

The white stones form the inner square. Figure 1 has a 1x1 white block, figure 2 a 2x2, figure 3 a 3x3. So in figure n the white block is n by n.

\text{white}(n) = n \times n

White grows like a square number.

2Find the rule for black stones

#9 Solve an Easier Related Problem 5.OA.A.2

The black stones are the border of the whole square. In figure n the whole square is (n + 4) by (n + 4), and the white inside is n by n, so the black border is the difference.

\text{black}(n) = (n+4) \times (n+4) - n \times n = 8n + 16

Black grows in a straight line, not as a square.

3Use both rules for the 15th figure

#5 Look for a Pattern 5.OA.B.3

Apply each rule at n = 15. White is 15 x 15 = 225 stones. Black is 8 x 15 + 16 = 136 stones.

\text{white}(15) = 225,\quad \text{black}(15) = 8 \times 15 + 16 = 136

225 white against 136 black.

4Subtract to get the difference

#5 Look for a Pattern 5.OA.A.2

There are more white stones than black, so the difference is 225 - 136.

225 - 136 = 89

They differ by 89 stones.

Answer: 89

4 · Reviewdoes it hold up?

Both counts should add back to the whole square: 225 + 136 = 361, and the square is 19 x 19 = 361. They match, so nothing was missed or double-counted.

Another way: Sanity-check the rules on figure 1: white(1) = 1 and black(1) = 24, which is exactly the border of the 5 by 5 square in the first picture.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern that follows a given rule — Reading each colour's growth off the first three figures.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the black count as a whole square minus the white one.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Running two separate rules at the same figure number and comparing them.

💡Takeaway. When two things in a pattern grow at different speeds, give each one its own rule before you try to compare them.

Variant 3 easy answer: 248

Game stones are placed following the pattern shown below. For the $18$ th figure, what is the difference between the number of white stones and the number of black stones?

The figures show stones arranged in growing squares. The 1st figure is a $3\times3$ array: the outer border, $1$ stone thick, is black and the single center stone is white. The 2nd figure is a $4\times4$ array: the border is still $1$ stone thick and the inner $2\times2$ stones are white. The 3rd figure is a $5\times5$ array with an inner $3\times3$ white block. In this way, each step adds one more stone to the side of the white block, and the black border keeps the same thickness.

Show solution

1 · Understandwhat's really being asked

Each figure is a square of stones with a white block inside and a black border 1 stone thick. We need how far apart the two counts are in the 18th figure.

Givens

In figure n the white block is n by n.
The black border is 1 stone thick all round.
We want figure 18.

Unknowns

The difference between the white count and the black count.

Constraints

Every stone is either white or black, and none is counted twice.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

The two colours do not grow at the same speed, so one rule cannot cover both. Find a rule for each colour separately from the small figures, then run both at the figure we want.

3 · Execute4 carry out the plan

1Find the rule for white stones

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

The white stones form the inner square. Figure 1 has a 1x1 white block, figure 2 a 2x2, figure 3 a 3x3. So in figure n the white block is n by n.

\text{white}(n) = n \times n

White grows like a square number.

2Find the rule for black stones

#9 Solve an Easier Related Problem 5.OA.A.2

The black stones are the border of the whole square. In figure n the whole square is (n + 2) by (n + 2), and the white inside is n by n, so the black border is the difference.

\text{black}(n) = (n+2) \times (n+2) - n \times n = 4n + 4

Black grows in a straight line, not as a square.

3Use both rules for the 18th figure

#5 Look for a Pattern 5.OA.B.3

Apply each rule at n = 18. White is 18 x 18 = 324 stones. Black is 4 x 18 + 4 = 76 stones.

\text{white}(18) = 324,\quad \text{black}(18) = 4 \times 18 + 4 = 76

324 white against 76 black.

4Subtract to get the difference

#5 Look for a Pattern 5.OA.A.2

There are more white stones than black, so the difference is 324 - 76.

324 - 76 = 248

They differ by 248 stones.

Answer: 248

4 · Reviewdoes it hold up?

Both counts should add back to the whole square: 324 + 76 = 400, and the square is 20 x 20 = 400. They match, so nothing was missed or double-counted.

Another way: Sanity-check the rules on figure 1: white(1) = 1 and black(1) = 8, which is exactly the border of the 3 by 3 square in the first picture.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern that follows a given rule — Reading each colour's growth off the first three figures.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the black count as a whole square minus the white one.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Running two separate rules at the same figure number and comparing them.

💡Takeaway. When two things in a pattern grow at different speeds, give each one its own rule before you try to compare them.

Variant 4 easy answer: 316

Game stones are placed following the pattern shown below. For the $20$ th figure, what is the difference between the number of white stones and the number of black stones?

The figures show stones arranged in growing squares. The 1st figure is a $3\times3$ array: the outer border, $1$ stone thick, is black and the single center stone is white. The 2nd figure is a $4\times4$ array: the border is still $1$ stone thick and the inner $2\times2$ stones are white. The 3rd figure is a $5\times5$ array with an inner $3\times3$ white block. In this way, each step adds one more stone to the side of the white block, and the black border keeps the same thickness.

Show solution

1 · Understandwhat's really being asked

Each figure is a square of stones with a white block inside and a black border 1 stone thick. We need how far apart the two counts are in the 20th figure.

Givens

In figure n the white block is n by n.
The black border is 1 stone thick all round.
We want figure 20.

Unknowns

The difference between the white count and the black count.

Constraints

Every stone is either white or black, and none is counted twice.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

The two colours do not grow at the same speed, so one rule cannot cover both. Find a rule for each colour separately from the small figures, then run both at the figure we want.

3 · Execute4 carry out the plan

1Find the rule for white stones

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

The white stones form the inner square. Figure 1 has a 1x1 white block, figure 2 a 2x2, figure 3 a 3x3. So in figure n the white block is n by n.

\text{white}(n) = n \times n

White grows like a square number.

2Find the rule for black stones

#9 Solve an Easier Related Problem 5.OA.A.2

The black stones are the border of the whole square. In figure n the whole square is (n + 2) by (n + 2), and the white inside is n by n, so the black border is the difference.

\text{black}(n) = (n+2) \times (n+2) - n \times n = 4n + 4

Black grows in a straight line, not as a square.

3Use both rules for the 20th figure

#5 Look for a Pattern 5.OA.B.3

Apply each rule at n = 20. White is 20 x 20 = 400 stones. Black is 4 x 20 + 4 = 84 stones.

\text{white}(20) = 400,\quad \text{black}(20) = 4 \times 20 + 4 = 84

400 white against 84 black.

4Subtract to get the difference

#5 Look for a Pattern 5.OA.A.2

There are more white stones than black, so the difference is 400 - 84.

400 - 84 = 316

They differ by 316 stones.

Answer: 316

4 · Reviewdoes it hold up?

Both counts should add back to the whole square: 400 + 84 = 484, and the square is 22 x 22 = 484. They match, so nothing was missed or double-counted.

Another way: Sanity-check the rules on figure 1: white(1) = 1 and black(1) = 8, which is exactly the border of the 3 by 3 square in the first picture.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern that follows a given rule — Reading each colour's growth off the first three figures.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the black count as a whole square minus the white one.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Running two separate rules at the same figure number and comparing them.

💡Takeaway. When two things in a pattern grow at different speeds, give each one its own rule before you try to compare them.

Variant 5 medium answer: 368

Game stones are placed following the pattern shown below. For the $24$ th figure, what is the difference between the number of white stones and the number of black stones?

The figures show stones arranged in growing squares. The 1st figure is a $5\times5$ array: the outer border, $2$ stones thick, is black and the single center stone is white. The 2nd figure is a $6\times6$ array: the border is still $2$ stones thick and the inner $2\times2$ stones are white. The 3rd figure is a $7\times7$ array with an inner $3\times3$ white block. In this way, each step adds one more stone to the side of the white block, and the black border keeps the same thickness.

Show solution

1 · Understandwhat's really being asked

Each figure is a square of stones with a white block inside and a black border 2 stones thick. We need how far apart the two counts are in the 24th figure.

Givens

In figure n the white block is n by n.
The black border is 2 stones thick all round.
We want figure 24.

Unknowns

The difference between the white count and the black count.

Constraints

Every stone is either white or black, and none is counted twice.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

The two colours do not grow at the same speed, so one rule cannot cover both. Find a rule for each colour separately from the small figures, then run both at the figure we want.

3 · Execute4 carry out the plan

1Find the rule for white stones

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

The white stones form the inner square. Figure 1 has a 1x1 white block, figure 2 a 2x2, figure 3 a 3x3. So in figure n the white block is n by n.

\text{white}(n) = n \times n

White grows like a square number.

2Find the rule for black stones

#9 Solve an Easier Related Problem 5.OA.A.2

The black stones are the border of the whole square. In figure n the whole square is (n + 4) by (n + 4), and the white inside is n by n, so the black border is the difference.

\text{black}(n) = (n+4) \times (n+4) - n \times n = 8n + 16

Black grows in a straight line, not as a square.

3Use both rules for the 24th figure

#5 Look for a Pattern 5.OA.B.3

Apply each rule at n = 24. White is 24 x 24 = 576 stones. Black is 8 x 24 + 16 = 208 stones.

\text{white}(24) = 576,\quad \text{black}(24) = 8 \times 24 + 16 = 208

576 white against 208 black.

4Subtract to get the difference

#5 Look for a Pattern 5.OA.A.2

There are more white stones than black, so the difference is 576 - 208.

576 - 208 = 368

They differ by 368 stones.

Answer: 368

4 · Reviewdoes it hold up?

Both counts should add back to the whole square: 576 + 208 = 784, and the square is 28 x 28 = 784. They match, so nothing was missed or double-counted.

Another way: Sanity-check the rules on figure 1: white(1) = 1 and black(1) = 24, which is exactly the border of the 5 by 5 square in the first picture.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern that follows a given rule — Reading each colour's growth off the first three figures.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the black count as a whole square minus the white one.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Running two separate rules at the same figure number and comparing them.

💡Takeaway. When two things in a pattern grow at different speeds, give each one its own rule before you try to compare them.

Variant 6 medium answer: 409

Game stones are placed following the pattern shown below. For the $25$ th figure, what is the difference between the number of white stones and the number of black stones?

The figures show stones arranged in growing squares. The 1st figure is a $5\times5$ array: the outer border, $2$ stones thick, is black and the single center stone is white. The 2nd figure is a $6\times6$ array: the border is still $2$ stones thick and the inner $2\times2$ stones are white. The 3rd figure is a $7\times7$ array with an inner $3\times3$ white block. In this way, each step adds one more stone to the side of the white block, and the black border keeps the same thickness.

Show solution

1 · Understandwhat's really being asked

Each figure is a square of stones with a white block inside and a black border 2 stones thick. We need how far apart the two counts are in the 25th figure.

Givens

In figure n the white block is n by n.
The black border is 2 stones thick all round.
We want figure 25.

Unknowns

The difference between the white count and the black count.

Constraints

Every stone is either white or black, and none is counted twice.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

The two colours do not grow at the same speed, so one rule cannot cover both. Find a rule for each colour separately from the small figures, then run both at the figure we want.

3 · Execute4 carry out the plan

1Find the rule for white stones

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

The white stones form the inner square. Figure 1 has a 1x1 white block, figure 2 a 2x2, figure 3 a 3x3. So in figure n the white block is n by n.

\text{white}(n) = n \times n

White grows like a square number.

2Find the rule for black stones

#9 Solve an Easier Related Problem 5.OA.A.2

The black stones are the border of the whole square. In figure n the whole square is (n + 4) by (n + 4), and the white inside is n by n, so the black border is the difference.

\text{black}(n) = (n+4) \times (n+4) - n \times n = 8n + 16

Black grows in a straight line, not as a square.

3Use both rules for the 25th figure

#5 Look for a Pattern 5.OA.B.3

Apply each rule at n = 25. White is 25 x 25 = 625 stones. Black is 8 x 25 + 16 = 216 stones.

\text{white}(25) = 625,\quad \text{black}(25) = 8 \times 25 + 16 = 216

625 white against 216 black.

4Subtract to get the difference

#5 Look for a Pattern 5.OA.A.2

There are more white stones than black, so the difference is 625 - 216.

625 - 216 = 409

They differ by 409 stones.

Answer: 409

4 · Reviewdoes it hold up?

Both counts should add back to the whole square: 625 + 216 = 841, and the square is 29 x 29 = 841. They match, so nothing was missed or double-counted.

Another way: Sanity-check the rules on figure 1: white(1) = 1 and black(1) = 24, which is exactly the border of the 5 by 5 square in the first picture.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern that follows a given rule — Reading each colour's growth off the first three figures.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the black count as a whole square minus the white one.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Running two separate rules at the same figure number and comparing them.

💡Takeaway. When two things in a pattern grow at different speeds, give each one its own rule before you try to compare them.

Variant 7 medium answer: 776

Game stones are placed following the pattern shown below. For the $30$ th figure, what is the difference between the number of white stones and the number of black stones?

The figures show stones arranged in growing squares. The 1st figure is a $3\times3$ array: the outer border, $1$ stone thick, is black and the single center stone is white. The 2nd figure is a $4\times4$ array: the border is still $1$ stone thick and the inner $2\times2$ stones are white. The 3rd figure is a $5\times5$ array with an inner $3\times3$ white block. In this way, each step adds one more stone to the side of the white block, and the black border keeps the same thickness.

Show solution

1 · Understandwhat's really being asked

Each figure is a square of stones with a white block inside and a black border 1 stone thick. We need how far apart the two counts are in the 30th figure.

Givens

In figure n the white block is n by n.
The black border is 1 stone thick all round.
We want figure 30.

Unknowns

The difference between the white count and the black count.

Constraints

Every stone is either white or black, and none is counted twice.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

The two colours do not grow at the same speed, so one rule cannot cover both. Find a rule for each colour separately from the small figures, then run both at the figure we want.

3 · Execute4 carry out the plan

1Find the rule for white stones

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

The white stones form the inner square. Figure 1 has a 1x1 white block, figure 2 a 2x2, figure 3 a 3x3. So in figure n the white block is n by n.

\text{white}(n) = n \times n

White grows like a square number.

2Find the rule for black stones

#9 Solve an Easier Related Problem 5.OA.A.2

The black stones are the border of the whole square. In figure n the whole square is (n + 2) by (n + 2), and the white inside is n by n, so the black border is the difference.

\text{black}(n) = (n+2) \times (n+2) - n \times n = 4n + 4

Black grows in a straight line, not as a square.

3Use both rules for the 30th figure

#5 Look for a Pattern 5.OA.B.3

Apply each rule at n = 30. White is 30 x 30 = 900 stones. Black is 4 x 30 + 4 = 124 stones.

\text{white}(30) = 900,\quad \text{black}(30) = 4 \times 30 + 4 = 124

900 white against 124 black.

4Subtract to get the difference

#5 Look for a Pattern 5.OA.A.2

There are more white stones than black, so the difference is 900 - 124.

900 - 124 = 776

They differ by 776 stones.

Answer: 776

4 · Reviewdoes it hold up?

Both counts should add back to the whole square: 900 + 124 = 1024, and the square is 32 x 32 = 1024. They match, so nothing was missed or double-counted.

Another way: Sanity-check the rules on figure 1: white(1) = 1 and black(1) = 8, which is exactly the border of the 3 by 3 square in the first picture.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern that follows a given rule — Reading each colour's growth off the first three figures.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the black count as a whole square minus the white one.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Running two separate rules at the same figure number and comparing them.

💡Takeaway. When two things in a pattern grow at different speeds, give each one its own rule before you try to compare them.

Variant 8 medium answer: 929

Game stones are placed following the pattern shown below. For the $35$ th figure, what is the difference between the number of white stones and the number of black stones?

The figures show stones arranged in growing squares. The 1st figure is a $5\times5$ array: the outer border, $2$ stones thick, is black and the single center stone is white. The 2nd figure is a $6\times6$ array: the border is still $2$ stones thick and the inner $2\times2$ stones are white. The 3rd figure is a $7\times7$ array with an inner $3\times3$ white block. In this way, each step adds one more stone to the side of the white block, and the black border keeps the same thickness.

Show solution

1 · Understandwhat's really being asked

Each figure is a square of stones with a white block inside and a black border 2 stones thick. We need how far apart the two counts are in the 35th figure.

Givens

In figure n the white block is n by n.
The black border is 2 stones thick all round.
We want figure 35.

Unknowns

The difference between the white count and the black count.

Constraints

Every stone is either white or black, and none is counted twice.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

The two colours do not grow at the same speed, so one rule cannot cover both. Find a rule for each colour separately from the small figures, then run both at the figure we want.

3 · Execute4 carry out the plan

1Find the rule for white stones

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

The white stones form the inner square. Figure 1 has a 1x1 white block, figure 2 a 2x2, figure 3 a 3x3. So in figure n the white block is n by n.

\text{white}(n) = n \times n

White grows like a square number.

2Find the rule for black stones

#9 Solve an Easier Related Problem 5.OA.A.2

The black stones are the border of the whole square. In figure n the whole square is (n + 4) by (n + 4), and the white inside is n by n, so the black border is the difference.

\text{black}(n) = (n+4) \times (n+4) - n \times n = 8n + 16

Black grows in a straight line, not as a square.

3Use both rules for the 35th figure

#5 Look for a Pattern 5.OA.B.3

Apply each rule at n = 35. White is 35 x 35 = 1225 stones. Black is 8 x 35 + 16 = 296 stones.

\text{white}(35) = 1225,\quad \text{black}(35) = 8 \times 35 + 16 = 296

1225 white against 296 black.

4Subtract to get the difference

#5 Look for a Pattern 5.OA.A.2

There are more white stones than black, so the difference is 1225 - 296.

1225 - 296 = 929

They differ by 929 stones.

Answer: 929

4 · Reviewdoes it hold up?

Both counts should add back to the whole square: 1225 + 296 = 1521, and the square is 39 x 39 = 1521. They match, so nothing was missed or double-counted.

Another way: Sanity-check the rules on figure 1: white(1) = 1 and black(1) = 24, which is exactly the border of the 5 by 5 square in the first picture.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern that follows a given rule — Reading each colour's growth off the first three figures.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the black count as a whole square minus the white one.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Running two separate rules at the same figure number and comparing them.

💡Takeaway. When two things in a pattern grow at different speeds, give each one its own rule before you try to compare them.

Variant 9 hard answer: 1436

Game stones are placed following the pattern shown below. For the $40$ th figure, what is the difference between the number of white stones and the number of black stones?

The figures show stones arranged in growing squares. The 1st figure is a $3\times3$ array: the outer border, $1$ stone thick, is black and the single center stone is white. The 2nd figure is a $4\times4$ array: the border is still $1$ stone thick and the inner $2\times2$ stones are white. The 3rd figure is a $5\times5$ array with an inner $3\times3$ white block. In this way, each step adds one more stone to the side of the white block, and the black border keeps the same thickness.

Show solution

1 · Understandwhat's really being asked

Each figure is a square of stones with a white block inside and a black border 1 stone thick. We need how far apart the two counts are in the 40th figure.

Givens

In figure n the white block is n by n.
The black border is 1 stone thick all round.
We want figure 40.

Unknowns

The difference between the white count and the black count.

Constraints

Every stone is either white or black, and none is counted twice.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

The two colours do not grow at the same speed, so one rule cannot cover both. Find a rule for each colour separately from the small figures, then run both at the figure we want.

3 · Execute4 carry out the plan

1Find the rule for white stones

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

The white stones form the inner square. Figure 1 has a 1x1 white block, figure 2 a 2x2, figure 3 a 3x3. So in figure n the white block is n by n.

\text{white}(n) = n \times n

White grows like a square number.

2Find the rule for black stones

#9 Solve an Easier Related Problem 5.OA.A.2

The black stones are the border of the whole square. In figure n the whole square is (n + 2) by (n + 2), and the white inside is n by n, so the black border is the difference.

\text{black}(n) = (n+2) \times (n+2) - n \times n = 4n + 4

Black grows in a straight line, not as a square.

3Use both rules for the 40th figure

#5 Look for a Pattern 5.OA.B.3

Apply each rule at n = 40. White is 40 x 40 = 1600 stones. Black is 4 x 40 + 4 = 164 stones.

\text{white}(40) = 1600,\quad \text{black}(40) = 4 \times 40 + 4 = 164

1600 white against 164 black.

4Subtract to get the difference

#5 Look for a Pattern 5.OA.A.2

There are more white stones than black, so the difference is 1600 - 164.

1600 - 164 = 1436

They differ by 1436 stones.

Answer: 1436

4 · Reviewdoes it hold up?

Both counts should add back to the whole square: 1600 + 164 = 1764, and the square is 42 x 42 = 1764. They match, so nothing was missed or double-counted.

Another way: Sanity-check the rules on figure 1: white(1) = 1 and black(1) = 8, which is exactly the border of the 3 by 3 square in the first picture.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern that follows a given rule — Reading each colour's growth off the first three figures.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the black count as a whole square minus the white one.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Running two separate rules at the same figure number and comparing them.

💡Takeaway. When two things in a pattern grow at different speeds, give each one its own rule before you try to compare them.

Variant 10 hard answer: 1449

Game stones are placed following the pattern shown below. For the $45$ th figure, what is the difference between the number of white stones and the number of black stones?

The figures show stones arranged in growing squares. The 1st figure is a $7\times7$ array: the outer border, $3$ stones thick, is black and the single center stone is white. The 2nd figure is a $8\times8$ array: the border is still $3$ stones thick and the inner $2\times2$ stones are white. The 3rd figure is a $9\times9$ array with an inner $3\times3$ white block. In this way, each step adds one more stone to the side of the white block, and the black border keeps the same thickness.

Show solution

1 · Understandwhat's really being asked

Each figure is a square of stones with a white block inside and a black border 3 stones thick. We need how far apart the two counts are in the 45th figure.

Givens

In figure n the white block is n by n.
The black border is 3 stones thick all round.
We want figure 45.

Unknowns

The difference between the white count and the black count.

Constraints

Every stone is either white or black, and none is counted twice.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

The two colours do not grow at the same speed, so one rule cannot cover both. Find a rule for each colour separately from the small figures, then run both at the figure we want.

3 · Execute4 carry out the plan

1Find the rule for white stones

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

The white stones form the inner square. Figure 1 has a 1x1 white block, figure 2 a 2x2, figure 3 a 3x3. So in figure n the white block is n by n.

\text{white}(n) = n \times n

White grows like a square number.

2Find the rule for black stones

#9 Solve an Easier Related Problem 5.OA.A.2

The black stones are the border of the whole square. In figure n the whole square is (n + 6) by (n + 6), and the white inside is n by n, so the black border is the difference.

\text{black}(n) = (n+6) \times (n+6) - n \times n = 12n + 36

Black grows in a straight line, not as a square.

3Use both rules for the 45th figure

#5 Look for a Pattern 5.OA.B.3

Apply each rule at n = 45. White is 45 x 45 = 2025 stones. Black is 12 x 45 + 36 = 576 stones.

\text{white}(45) = 2025,\quad \text{black}(45) = 12 \times 45 + 36 = 576

2025 white against 576 black.

4Subtract to get the difference

#5 Look for a Pattern 5.OA.A.2

There are more white stones than black, so the difference is 2025 - 576.

2025 - 576 = 1449

They differ by 1449 stones.

Answer: 1449

4 · Reviewdoes it hold up?

Both counts should add back to the whole square: 2025 + 576 = 2601, and the square is 51 x 51 = 2601. They match, so nothing was missed or double-counted.

Another way: Sanity-check the rules on figure 1: white(1) = 1 and black(1) = 48, which is exactly the border of the 7 by 7 square in the first picture.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern that follows a given rule — Reading each colour's growth off the first three figures.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the black count as a whole square minus the white one.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Running two separate rules at the same figure number and comparing them.

💡Takeaway. When two things in a pattern grow at different speeds, give each one its own rule before you try to compare them.

Variant 11 hard answer: 2296

Game stones are placed following the pattern shown below. For the $50$ th figure, what is the difference between the number of white stones and the number of black stones?

The figures show stones arranged in growing squares. The 1st figure is a $3\times3$ array: the outer border, $1$ stone thick, is black and the single center stone is white. The 2nd figure is a $4\times4$ array: the border is still $1$ stone thick and the inner $2\times2$ stones are white. The 3rd figure is a $5\times5$ array with an inner $3\times3$ white block. In this way, each step adds one more stone to the side of the white block, and the black border keeps the same thickness.

Show solution

1 · Understandwhat's really being asked

Each figure is a square of stones with a white block inside and a black border 1 stone thick. We need how far apart the two counts are in the 50th figure.

Givens

In figure n the white block is n by n.
The black border is 1 stone thick all round.
We want figure 50.

Unknowns

The difference between the white count and the black count.

Constraints

Every stone is either white or black, and none is counted twice.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

The two colours do not grow at the same speed, so one rule cannot cover both. Find a rule for each colour separately from the small figures, then run both at the figure we want.

3 · Execute4 carry out the plan

1Find the rule for white stones

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

The white stones form the inner square. Figure 1 has a 1x1 white block, figure 2 a 2x2, figure 3 a 3x3. So in figure n the white block is n by n.

\text{white}(n) = n \times n

White grows like a square number.

2Find the rule for black stones

#9 Solve an Easier Related Problem 5.OA.A.2

The black stones are the border of the whole square. In figure n the whole square is (n + 2) by (n + 2), and the white inside is n by n, so the black border is the difference.

\text{black}(n) = (n+2) \times (n+2) - n \times n = 4n + 4

Black grows in a straight line, not as a square.

3Use both rules for the 50th figure

#5 Look for a Pattern 5.OA.B.3

Apply each rule at n = 50. White is 50 x 50 = 2500 stones. Black is 4 x 50 + 4 = 204 stones.

\text{white}(50) = 2500,\quad \text{black}(50) = 4 \times 50 + 4 = 204

2500 white against 204 black.

4Subtract to get the difference

#5 Look for a Pattern 5.OA.A.2

There are more white stones than black, so the difference is 2500 - 204.

2500 - 204 = 2296

They differ by 2296 stones.

Answer: 2296

4 · Reviewdoes it hold up?

Both counts should add back to the whole square: 2500 + 204 = 2704, and the square is 52 x 52 = 2704. They match, so nothing was missed or double-counted.

Another way: Sanity-check the rules on figure 1: white(1) = 1 and black(1) = 8, which is exactly the border of the 3 by 3 square in the first picture.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern that follows a given rule — Reading each colour's growth off the first three figures.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the black count as a whole square minus the white one.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Running two separate rules at the same figure number and comparing them.

💡Takeaway. When two things in a pattern grow at different speeds, give each one its own rule before you try to compare them.

Variant 12 hard answer: 3104

Game stones are placed following the pattern shown below. For the $60$ th figure, what is the difference between the number of white stones and the number of black stones?

The figures show stones arranged in growing squares. The 1st figure is a $5\times5$ array: the outer border, $2$ stones thick, is black and the single center stone is white. The 2nd figure is a $6\times6$ array: the border is still $2$ stones thick and the inner $2\times2$ stones are white. The 3rd figure is a $7\times7$ array with an inner $3\times3$ white block. In this way, each step adds one more stone to the side of the white block, and the black border keeps the same thickness.

Show solution

1 · Understandwhat's really being asked

Each figure is a square of stones with a white block inside and a black border 2 stones thick. We need how far apart the two counts are in the 60th figure.

Givens

In figure n the white block is n by n.
The black border is 2 stones thick all round.
We want figure 60.

Unknowns

The difference between the white count and the black count.

Constraints

Every stone is either white or black, and none is counted twice.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

The two colours do not grow at the same speed, so one rule cannot cover both. Find a rule for each colour separately from the small figures, then run both at the figure we want.

3 · Execute4 carry out the plan

1Find the rule for white stones

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

The white stones form the inner square. Figure 1 has a 1x1 white block, figure 2 a 2x2, figure 3 a 3x3. So in figure n the white block is n by n.

\text{white}(n) = n \times n

White grows like a square number.

2Find the rule for black stones

#9 Solve an Easier Related Problem 5.OA.A.2

The black stones are the border of the whole square. In figure n the whole square is (n + 4) by (n + 4), and the white inside is n by n, so the black border is the difference.

\text{black}(n) = (n+4) \times (n+4) - n \times n = 8n + 16

Black grows in a straight line, not as a square.

3Use both rules for the 60th figure

#5 Look for a Pattern 5.OA.B.3

Apply each rule at n = 60. White is 60 x 60 = 3600 stones. Black is 8 x 60 + 16 = 496 stones.

\text{white}(60) = 3600,\quad \text{black}(60) = 8 \times 60 + 16 = 496

3600 white against 496 black.

4Subtract to get the difference

#5 Look for a Pattern 5.OA.A.2

There are more white stones than black, so the difference is 3600 - 496.

3600 - 496 = 3104

They differ by 3104 stones.

Answer: 3104

4 · Reviewdoes it hold up?

Both counts should add back to the whole square: 3600 + 496 = 4096, and the square is 64 x 64 = 4096. They match, so nothing was missed or double-counted.

Another way: Sanity-check the rules on figure 1: white(1) = 1 and black(1) = 24, which is exactly the border of the 5 by 5 square in the first picture.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern that follows a given rule — Reading each colour's growth off the first three figures.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the black count as a whole square minus the white one.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Running two separate rules at the same figure number and comparing them.

💡Takeaway. When two things in a pattern grow at different speeds, give each one its own rule before you try to compare them.

Two colours, two separate rules · 12 practice problems

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