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← 2-2 · Find shape and color repeats separately · Repeating Cycle Patterns

Find shape and color repeats separately · 8 practice problems

4.OA.C.5

Generated variants — 8

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

Variant 1 answer: A orange triangle

Find the rule, then draw and color the shape that belongs in the blank.

The shapes are lined up in a single row. The shapes and the colors each repeat in their own steady pattern. Reading from the left, the shapes and colors are:

  1. green triangle
  2. orange circle
  3. green square
  4. blank
  5. green circle
  6. orange square
  7. green triangle
  8. orange circle
  9. green square
  10. orange triangle
  11. green circle
  12. orange square

Draw the shape that belongs in the blank (the fourth one) and color it the correct color.

green orange ?
Show solution

Understand

A row of shapes repeats two patterns at once: the shape cycles (period 3) and the color cycles (period 2), each on its own steady period. Position 4 is blank. Find the shape and color that belong in that blank.

Givens
  • The first 6 positions read: green triangle, orange circle, green square, green circle, orange square.
  • Position 4 is blank.
  • The shapes repeat in their own pattern and the colors repeat in their own pattern.
Unknowns
  • The shape that belongs in position 4.
  • The color that belongs in position 4.
Constraints
  • Shapes cycle triangle, circle, square (period 3).
  • Colors cycle green, orange (period 2).

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List

Shape and color each follow their own repeating cycle, so I find each cycle separately and list the positions to read off what the blank item must be.

Execute

#5 Look for a Pattern 4.OA.C.5
The shapes repeat every 3: triangle, circle, square. Position 4 is at cycle slot 1, so its shape is triangle.
triangle→circle→square→(repeat)\text{triangle}\to\text{circle}\to\text{square}\to(\text{repeat})
A 3-step shape cycle fixes the shape at every position.
#2 Make a Systematic List 4.OA.C.5
The colors repeat every 2: green, orange. Position 4 is at color slot 2, so its color is orange.
green→orange→(repeat)\text{green}\to\text{orange}\to(\text{repeat})
With a 2-step color cycle, each position lands on a fixed color.
#5 Look for a Pattern 4.OA.C.5
Position 4 is a triangle (shape cycle) and orange (color cycle), so the blank is a orange triangle.
orange triangle\text{orange triangle}
The two independent cycles together fix exactly one shape-and-color for each spot.
Answer: A orange triangle

Review

Checking the neighbor before it (position 3) is a square; a orange triangle at position 4 keeps both the shape order and the color order going.

Make a two-row table listing position numbers with the shape cycle in one row and the color cycle in the other, then read straight down the blank column.

Standards · min grade 4

  • 4.OA.C.5 Generate a number or shape pattern following a given rule — Continuing the shape cycle and the color cycle to fill the blank position.
💡 Track the shape cycle and the color cycle separately, then put them together: the blank is a orange triangle!
Variant 2 answer: A blue square

Find the rule, then draw and color the shape that belongs in the blank.

The shapes are lined up in a single row. The shapes and the colors each repeat in their own steady pattern. Reading from the left, the shapes and colors are:

  1. red circle
  2. yellow square
  3. blue circle
  4. red square
  5. yellow circle
  6. blank
  7. red circle
  8. yellow square
  9. blue circle
  10. red square

Draw the shape that belongs in the blank (the sixth one) and color it the correct color.

red yellow blue ?
Show solution

Understand

A row of shapes repeats two patterns at once: the shape cycles (period 2) and the color cycles (period 3), each on its own steady period. Position 6 is blank. Find the shape and color that belong in that blank.

Givens
  • The first 8 positions read: red circle, yellow square, blue circle, red square, yellow circle, red circle, yellow square.
  • Position 6 is blank.
  • The shapes repeat in their own pattern and the colors repeat in their own pattern.
Unknowns
  • The shape that belongs in position 6.
  • The color that belongs in position 6.
Constraints
  • Shapes cycle circle, square (period 2).
  • Colors cycle red, yellow, blue (period 3).

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List

Shape and color each follow their own repeating cycle, so I find each cycle separately and list the positions to read off what the blank item must be.

Execute

#5 Look for a Pattern 4.OA.C.5
The shapes repeat every 2: circle, square. Position 6 is at cycle slot 2, so its shape is square.
circle→square→(repeat)\text{circle}\to\text{square}\to(\text{repeat})
A 2-step shape cycle fixes the shape at every position.
#2 Make a Systematic List 4.OA.C.5
The colors repeat every 3: red, yellow, blue. Position 6 is at color slot 3, so its color is blue.
red→yellow→blue→(repeat)\text{red}\to\text{yellow}\to\text{blue}\to(\text{repeat})
With a 3-step color cycle, each position lands on a fixed color.
#5 Look for a Pattern 4.OA.C.5
Position 6 is a square (shape cycle) and blue (color cycle), so the blank is a blue square.
blue square\text{blue square}
The two independent cycles together fix exactly one shape-and-color for each spot.
Answer: A blue square

Review

Checking the neighbor before it (position 5) is a circle; a blue square at position 6 keeps both the shape order and the color order going.

Make a two-row table listing position numbers with the shape cycle in one row and the color cycle in the other, then read straight down the blank column.

Standards · min grade 4

  • 4.OA.C.5 Generate a number or shape pattern following a given rule — Continuing the shape cycle and the color cycle to fill the blank position.
💡 Track the shape cycle and the color cycle separately, then put them together: the blank is a blue square!
Variant 3 answer: A red triangle

Find the rule, then draw and color the shape that belongs in the blank.

The shapes are lined up in a single row. The shapes and the colors each repeat in their own steady pattern. Reading from the left, the shapes and colors are:

  1. red square
  2. blue triangle
  3. red circle
  4. blue square
  5. blank
  6. blue circle
  7. red square
  8. blue triangle
  9. red circle

Draw the shape that belongs in the blank (the fifth one) and color it the correct color.

red blue ?
Show solution

Understand

A row of shapes repeats two patterns at once: the shape cycles (period 3) and the color cycles (period 2), each on its own steady period. Position 5 is blank. Find the shape and color that belong in that blank.

Givens
  • The first 7 positions read: red square, blue triangle, red circle, blue square, blue circle, red square.
  • Position 5 is blank.
  • The shapes repeat in their own pattern and the colors repeat in their own pattern.
Unknowns
  • The shape that belongs in position 5.
  • The color that belongs in position 5.
Constraints
  • Shapes cycle square, triangle, circle (period 3).
  • Colors cycle red, blue (period 2).

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List

Shape and color each follow their own repeating cycle, so I find each cycle separately and list the positions to read off what the blank item must be.

Execute

#5 Look for a Pattern 4.OA.C.5
The shapes repeat every 3: square, triangle, circle. Position 5 is at cycle slot 2, so its shape is triangle.
square→triangle→circle→(repeat)\text{square}\to\text{triangle}\to\text{circle}\to(\text{repeat})
A 3-step shape cycle fixes the shape at every position.
#2 Make a Systematic List 4.OA.C.5
The colors repeat every 2: red, blue. Position 5 is at color slot 1, so its color is red.
red→blue→(repeat)\text{red}\to\text{blue}\to(\text{repeat})
With a 2-step color cycle, each position lands on a fixed color.
#5 Look for a Pattern 4.OA.C.5
Position 5 is a triangle (shape cycle) and red (color cycle), so the blank is a red triangle.
red triangle\text{red triangle}
The two independent cycles together fix exactly one shape-and-color for each spot.
Answer: A red triangle

Review

Checking the neighbor before it (position 4) is a square; a red triangle at position 5 keeps both the shape order and the color order going.

Make a two-row table listing position numbers with the shape cycle in one row and the color cycle in the other, then read straight down the blank column.

Standards · min grade 4

  • 4.OA.C.5 Generate a number or shape pattern following a given rule — Continuing the shape cycle and the color cycle to fill the blank position.
💡 Track the shape cycle and the color cycle separately, then put them together: the blank is a red triangle!
Variant 4 answer: A yellow triangle

Find the rule, then draw and color the shape that belongs in the blank.

The shapes are lined up in a single row. The shapes and the colors each repeat in their own steady pattern. Reading from the left, the shapes and colors are:

  1. red square
  2. yellow triangle
  3. red circle
  4. yellow square
  5. red triangle
  6. yellow circle
  7. red square
  8. blank
  9. red circle
  10. yellow square
  11. red triangle
  12. yellow circle

Draw the shape that belongs in the blank (the eighth one) and color it the correct color.

red yellow ?
Show solution

Understand

A row of shapes repeats two patterns at once: the shape cycles (period 3) and the color cycles (period 2), each on its own steady period. Position 8 is blank. Find the shape and color that belong in that blank.

Givens
  • The first 10 positions read: red square, yellow triangle, red circle, yellow square, red triangle, yellow circle, red square, red circle, yellow square.
  • Position 8 is blank.
  • The shapes repeat in their own pattern and the colors repeat in their own pattern.
Unknowns
  • The shape that belongs in position 8.
  • The color that belongs in position 8.
Constraints
  • Shapes cycle square, triangle, circle (period 3).
  • Colors cycle red, yellow (period 2).

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List

Shape and color each follow their own repeating cycle, so I find each cycle separately and list the positions to read off what the blank item must be.

Execute

#5 Look for a Pattern 4.OA.C.5
The shapes repeat every 3: square, triangle, circle. Position 8 is at cycle slot 2, so its shape is triangle.
square→triangle→circle→(repeat)\text{square}\to\text{triangle}\to\text{circle}\to(\text{repeat})
A 3-step shape cycle fixes the shape at every position.
#2 Make a Systematic List 4.OA.C.5
The colors repeat every 2: red, yellow. Position 8 is at color slot 2, so its color is yellow.
red→yellow→(repeat)\text{red}\to\text{yellow}\to(\text{repeat})
With a 2-step color cycle, each position lands on a fixed color.
#5 Look for a Pattern 4.OA.C.5
Position 8 is a triangle (shape cycle) and yellow (color cycle), so the blank is a yellow triangle.
yellow triangle\text{yellow triangle}
The two independent cycles together fix exactly one shape-and-color for each spot.
Answer: A yellow triangle

Review

Checking the neighbor before it (position 7) is a square; a yellow triangle at position 8 keeps both the shape order and the color order going.

Make a two-row table listing position numbers with the shape cycle in one row and the color cycle in the other, then read straight down the blank column.

Standards · min grade 4

  • 4.OA.C.5 Generate a number or shape pattern following a given rule — Continuing the shape cycle and the color cycle to fill the blank position.
💡 Track the shape cycle and the color cycle separately, then put them together: the blank is a yellow triangle!
Variant 5 answer: A red square

Find the rule, then draw and color the shape that belongs in the blank.

The shapes are lined up in a single row. The shapes and the colors each repeat in their own steady pattern. Reading from the left, the shapes and colors are:

  1. red square
  2. yellow triangle
  3. blue circle
  4. red square
  5. yellow triangle
  6. blue circle
  7. blank
  8. yellow triangle
  9. blue circle
  10. red square
  11. yellow triangle
  12. blue circle

Draw the shape that belongs in the blank (the seventh one) and color it the correct color.

red yellow blue ?
Show solution

Understand

A row of shapes repeats two patterns at once: the shape cycles (period 3) and the color cycles (period 3), each on its own steady period. Position 7 is blank. Find the shape and color that belong in that blank.

Givens
  • The first 9 positions read: red square, yellow triangle, blue circle, red square, yellow triangle, blue circle, yellow triangle, blue circle.
  • Position 7 is blank.
  • The shapes repeat in their own pattern and the colors repeat in their own pattern.
Unknowns
  • The shape that belongs in position 7.
  • The color that belongs in position 7.
Constraints
  • Shapes cycle square, triangle, circle (period 3).
  • Colors cycle red, yellow, blue (period 3).

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List

Shape and color each follow their own repeating cycle, so I find each cycle separately and list the positions to read off what the blank item must be.

Execute

#5 Look for a Pattern 4.OA.C.5
The shapes repeat every 3: square, triangle, circle. Position 7 is at cycle slot 1, so its shape is square.
square→triangle→circle→(repeat)\text{square}\to\text{triangle}\to\text{circle}\to(\text{repeat})
A 3-step shape cycle fixes the shape at every position.
#2 Make a Systematic List 4.OA.C.5
The colors repeat every 3: red, yellow, blue. Position 7 is at color slot 1, so its color is red.
red→yellow→blue→(repeat)\text{red}\to\text{yellow}\to\text{blue}\to(\text{repeat})
With a 3-step color cycle, each position lands on a fixed color.
#5 Look for a Pattern 4.OA.C.5
Position 7 is a square (shape cycle) and red (color cycle), so the blank is a red square.
red square\text{red square}
The two independent cycles together fix exactly one shape-and-color for each spot.
Answer: A red square

Review

Checking the neighbor before it (position 6) is a circle; a red square at position 7 keeps both the shape order and the color order going.

Make a two-row table listing position numbers with the shape cycle in one row and the color cycle in the other, then read straight down the blank column.

Standards · min grade 4

  • 4.OA.C.5 Generate a number or shape pattern following a given rule — Continuing the shape cycle and the color cycle to fill the blank position.
💡 Track the shape cycle and the color cycle separately, then put them together: the blank is a red square!
Variant 6 answer: A yellow square

Find the rule, then draw and color the shape that belongs in the blank.

The shapes are lined up in a single row. The shapes and the colors each repeat in their own steady pattern. Reading from the left, the shapes and colors are:

  1. blue square
  2. yellow triangle
  3. blue circle
  4. yellow square
  5. blue triangle
  6. yellow circle
  7. blue square
  8. yellow triangle
  9. blue circle
  10. blank
  11. blue triangle
  12. yellow circle

Draw the shape that belongs in the blank (the tenth one) and color it the correct color.

blue yellow ?
Show solution

Understand

A row of shapes repeats two patterns at once: the shape cycles (period 3) and the color cycles (period 2), each on its own steady period. Position 10 is blank. Find the shape and color that belong in that blank.

Givens
  • The first 12 positions read: blue square, yellow triangle, blue circle, yellow square, blue triangle, yellow circle, blue square, yellow triangle, blue circle, blue triangle, yellow circle.
  • Position 10 is blank.
  • The shapes repeat in their own pattern and the colors repeat in their own pattern.
Unknowns
  • The shape that belongs in position 10.
  • The color that belongs in position 10.
Constraints
  • Shapes cycle square, triangle, circle (period 3).
  • Colors cycle blue, yellow (period 2).

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List

Shape and color each follow their own repeating cycle, so I find each cycle separately and list the positions to read off what the blank item must be.

Execute

#5 Look for a Pattern 4.OA.C.5
The shapes repeat every 3: square, triangle, circle. Position 10 is at cycle slot 1, so its shape is square.
square→triangle→circle→(repeat)\text{square}\to\text{triangle}\to\text{circle}\to(\text{repeat})
A 3-step shape cycle fixes the shape at every position.
#2 Make a Systematic List 4.OA.C.5
The colors repeat every 2: blue, yellow. Position 10 is at color slot 2, so its color is yellow.
blue→yellow→(repeat)\text{blue}\to\text{yellow}\to(\text{repeat})
With a 2-step color cycle, each position lands on a fixed color.
#5 Look for a Pattern 4.OA.C.5
Position 10 is a square (shape cycle) and yellow (color cycle), so the blank is a yellow square.
yellow square\text{yellow square}
The two independent cycles together fix exactly one shape-and-color for each spot.
Answer: A yellow square

Review

Checking the neighbor before it (position 9) is a circle; a yellow square at position 10 keeps both the shape order and the color order going.

Make a two-row table listing position numbers with the shape cycle in one row and the color cycle in the other, then read straight down the blank column.

Standards · min grade 4

  • 4.OA.C.5 Generate a number or shape pattern following a given rule — Continuing the shape cycle and the color cycle to fill the blank position.
💡 Track the shape cycle and the color cycle separately, then put them together: the blank is a yellow square!
Variant 7 answer: A red square

Find the rule, then draw and color the shape that belongs in the blank.

The shapes are lined up in a single row. The shapes and the colors each repeat in their own steady pattern. Reading from the left, the shapes and colors are:

  1. red square
  2. yellow circle
  3. blank
  4. yellow circle
  5. red square
  6. yellow circle
  7. red square
  8. yellow circle

Draw the shape that belongs in the blank (the third one) and color it the correct color.

red yellow ?
Show solution

Understand

A row of shapes repeats two patterns at once: the shape cycles (period 2) and the color cycles (period 2), each on its own steady period. Position 3 is blank. Find the shape and color that belong in that blank.

Givens
  • The first 5 positions read: red square, yellow circle, yellow circle, red square.
  • Position 3 is blank.
  • The shapes repeat in their own pattern and the colors repeat in their own pattern.
Unknowns
  • The shape that belongs in position 3.
  • The color that belongs in position 3.
Constraints
  • Shapes cycle square, circle (period 2).
  • Colors cycle red, yellow (period 2).

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List

Shape and color each follow their own repeating cycle, so I find each cycle separately and list the positions to read off what the blank item must be.

Execute

#5 Look for a Pattern 4.OA.C.5
The shapes repeat every 2: square, circle. Position 3 is at cycle slot 1, so its shape is square.
square→circle→(repeat)\text{square}\to\text{circle}\to(\text{repeat})
A 2-step shape cycle fixes the shape at every position.
#2 Make a Systematic List 4.OA.C.5
The colors repeat every 2: red, yellow. Position 3 is at color slot 1, so its color is red.
red→yellow→(repeat)\text{red}\to\text{yellow}\to(\text{repeat})
With a 2-step color cycle, each position lands on a fixed color.
#5 Look for a Pattern 4.OA.C.5
Position 3 is a square (shape cycle) and red (color cycle), so the blank is a red square.
red square\text{red square}
The two independent cycles together fix exactly one shape-and-color for each spot.
Answer: A red square

Review

Checking the neighbor before it (position 2) is a circle; a red square at position 3 keeps both the shape order and the color order going.

Make a two-row table listing position numbers with the shape cycle in one row and the color cycle in the other, then read straight down the blank column.

Standards · min grade 4

  • 4.OA.C.5 Generate a number or shape pattern following a given rule — Continuing the shape cycle and the color cycle to fill the blank position.
💡 Track the shape cycle and the color cycle separately, then put them together: the blank is a red square!
Variant 8 answer: A purple circle

Find the rule, then draw and color the shape that belongs in the blank.

The shapes are lined up in a single row. The shapes and the colors each repeat in their own steady pattern. Reading from the left, the shapes and colors are:

  1. purple triangle
  2. orange square
  3. purple circle
  4. orange triangle
  5. purple square
  6. orange circle
  7. purple triangle
  8. orange square
  9. blank
  10. orange triangle
  11. purple square

Draw the shape that belongs in the blank (the ninth one) and color it the correct color.

purple orange ?
Show solution

Understand

A row of shapes repeats two patterns at once: the shape cycles (period 3) and the color cycles (period 2), each on its own steady period. Position 9 is blank. Find the shape and color that belong in that blank.

Givens
  • The first 11 positions read: purple triangle, orange square, purple circle, orange triangle, purple square, orange circle, purple triangle, orange square, orange triangle, purple square.
  • Position 9 is blank.
  • The shapes repeat in their own pattern and the colors repeat in their own pattern.
Unknowns
  • The shape that belongs in position 9.
  • The color that belongs in position 9.
Constraints
  • Shapes cycle triangle, square, circle (period 3).
  • Colors cycle purple, orange (period 2).

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List

Shape and color each follow their own repeating cycle, so I find each cycle separately and list the positions to read off what the blank item must be.

Execute

#5 Look for a Pattern 4.OA.C.5
The shapes repeat every 3: triangle, square, circle. Position 9 is at cycle slot 3, so its shape is circle.
triangle→square→circle→(repeat)\text{triangle}\to\text{square}\to\text{circle}\to(\text{repeat})
A 3-step shape cycle fixes the shape at every position.
#2 Make a Systematic List 4.OA.C.5
The colors repeat every 2: purple, orange. Position 9 is at color slot 1, so its color is purple.
purple→orange→(repeat)\text{purple}\to\text{orange}\to(\text{repeat})
With a 2-step color cycle, each position lands on a fixed color.
#5 Look for a Pattern 4.OA.C.5
Position 9 is a circle (shape cycle) and purple (color cycle), so the blank is a purple circle.
purple circle\text{purple circle}
The two independent cycles together fix exactly one shape-and-color for each spot.
Answer: A purple circle

Review

Checking the neighbor before it (position 8) is a square; a purple circle at position 9 keeps both the shape order and the color order going.

Make a two-row table listing position numbers with the shape cycle in one row and the color cycle in the other, then read straight down the blank column.

Standards · min grade 4

  • 4.OA.C.5 Generate a number or shape pattern following a given rule — Continuing the shape cycle and the color cycle to fill the blank position.
💡 Track the shape cycle and the color cycle separately, then put them together: the blank is a purple circle!