Practice · Find shape and color repeats separately · Sensim Math

Variant 1 easy

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:

red square
yellow circle
blank
yellow circle
red square
yellow circle
red square
yellow circle

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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).

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Find the shape cycle

#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.

\text{square}\to\text{circle}\to(\text{repeat})

A 2-step shape cycle fixes the shape at every position.

2Find the color cycle

#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.

\text{red}\to\text{yellow}\to(\text{repeat})

With a 2-step color cycle, each position lands on a fixed color.

3Combine shape and 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.

\text{red square}

The two independent cycles together fix exactly one shape-and-color for each spot.

Answer: A red square

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. Track the shape cycle and the color cycle separately, then put them together: the blank is a red square!

Variant 2 easy

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:

red square
blue triangle
red circle
blue square
blank
blue circle
red square
blue triangle
red circle

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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).

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Find the shape cycle

#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.

\text{square}\to\text{triangle}\to\text{circle}\to(\text{repeat})

A 3-step shape cycle fixes the shape at every position.

2Find the color cycle

#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.

\text{red}\to\text{blue}\to(\text{repeat})

With a 2-step color cycle, each position lands on a fixed color.

3Combine shape and 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.

\text{red triangle}

The two independent cycles together fix exactly one shape-and-color for each spot.

Answer: A red triangle

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. Track the shape cycle and the color cycle separately, then put them together: the blank is a red triangle!

Variant 3 medium

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:

red circle
yellow square
blue circle
red square
yellow circle
blank
red circle
yellow square
blue circle
red square

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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).

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Find the shape cycle

#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.

\text{circle}\to\text{square}\to(\text{repeat})

A 2-step shape cycle fixes the shape at every position.

2Find the color cycle

#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.

\text{red}\to\text{yellow}\to\text{blue}\to(\text{repeat})

With a 3-step color cycle, each position lands on a fixed color.

3Combine shape and 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.

\text{blue square}

The two independent cycles together fix exactly one shape-and-color for each spot.

Answer: A blue square

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. Track the shape cycle and the color cycle separately, then put them together: the blank is a blue square!

Variant 4 medium

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:

purple triangle
orange square
purple circle
orange triangle
purple square
orange circle
purple triangle
orange square
blank
orange triangle
purple square

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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).

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Find the shape cycle

#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.

\text{triangle}\to\text{square}\to\text{circle}\to(\text{repeat})

A 3-step shape cycle fixes the shape at every position.

2Find the color cycle

#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.

\text{purple}\to\text{orange}\to(\text{repeat})

With a 2-step color cycle, each position lands on a fixed color.

3Combine shape and 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.

\text{purple circle}

The two independent cycles together fix exactly one shape-and-color for each spot.

Answer: A purple circle

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. Track the shape cycle and the color cycle separately, then put them together: the blank is a purple circle!

Variant 5 medium

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:

green triangle
orange circle
green square
blank
green circle
orange square
green triangle
orange circle
green square
orange triangle
green circle
orange square

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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).

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Find the shape cycle

#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.

\text{triangle}\to\text{circle}\to\text{square}\to(\text{repeat})

A 3-step shape cycle fixes the shape at every position.

2Find the color cycle

#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.

\text{green}\to\text{orange}\to(\text{repeat})

With a 2-step color cycle, each position lands on a fixed color.

3Combine shape and 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.

\text{orange triangle}

The two independent cycles together fix exactly one shape-and-color for each spot.

Answer: A orange triangle

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. Track the shape cycle and the color cycle separately, then put them together: the blank is a orange triangle!

Variant 6 medium

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:

red square
yellow triangle
red circle
yellow square
red triangle
yellow circle
red square
blank
red circle
yellow square
red triangle
yellow circle

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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).

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Find the shape cycle

#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.

\text{square}\to\text{triangle}\to\text{circle}\to(\text{repeat})

A 3-step shape cycle fixes the shape at every position.

2Find the color cycle

#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.

\text{red}\to\text{yellow}\to(\text{repeat})

With a 2-step color cycle, each position lands on a fixed color.

3Combine shape and 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.

\text{yellow triangle}

The two independent cycles together fix exactly one shape-and-color for each spot.

Answer: A yellow triangle

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. Track the shape cycle and the color cycle separately, then put them together: the blank is a yellow triangle!

Variant 7 hard

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:

red square
yellow triangle
blue circle
red square
yellow triangle
blue circle
blank
yellow triangle
blue circle
red square
yellow triangle
blue circle

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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).

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Find the shape cycle

#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.

\text{square}\to\text{triangle}\to\text{circle}\to(\text{repeat})

A 3-step shape cycle fixes the shape at every position.

2Find the color cycle

#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.

\text{red}\to\text{yellow}\to\text{blue}\to(\text{repeat})

With a 3-step color cycle, each position lands on a fixed color.

3Combine shape and 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.

\text{red square}

The two independent cycles together fix exactly one shape-and-color for each spot.

Answer: A red square

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. Track the shape cycle and the color cycle separately, then put them together: the blank is a red square!

Variant 8 hard

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:

blue square
yellow triangle
blue circle
yellow square
blue triangle
yellow circle
blue square
yellow triangle
blue circle
blank
blue triangle
yellow circle

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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).

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Find the shape cycle

#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.

\text{square}\to\text{triangle}\to\text{circle}\to(\text{repeat})

A 3-step shape cycle fixes the shape at every position.

2Find the color cycle

#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.

\text{blue}\to\text{yellow}\to(\text{repeat})

With a 2-step color cycle, each position lands on a fixed color.

3Combine shape and 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.

\text{yellow square}

The two independent cycles together fix exactly one shape-and-color for each spot.

Answer: A yellow square

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. Track the shape cycle and the color cycle separately, then put them together: the blank is a yellow square!

Find shape and color repeats separately · 8 practice problems

Generated variants — 8

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