Sensim Math · Depth 한국어

2-2 · Finding Patterns

Find shape and color repeats separately

4.OA.C.5 · take · grade 4

Archetype: Repeating Cycle Patterns · step in a 6-type progression

▶ Practice — 8 problems

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 and the color cycles, each on its own steady period. The 8th item is blank. Find the shape and color that belong in that blank.

Givens
  • Position 1: red square, 2: yellow triangle, 3: red circle, 4: yellow square, 5: red triangle, 6: yellow circle, 7: red square.
  • Position 8 is blank.
  • Positions 9-12: red circle, yellow square, red triangle, yellow circle.
  • 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 8th item must be.

Execute

#5 Look for a Pattern 4.OA.C.5
Reading the shapes in order: square, triangle, circle, square, triangle, circle, square, ... they repeat every 3. Position 7 is a square, so position 8 is the next shape, a triangle.
square→triangle→circle→(repeat)\text{square}\to\text{triangle}\to\text{circle}\to(\text{repeat})
A 3-step shape cycle means position 8 follows position 7's square with a triangle.
#2 Make a Systematic List 4.OA.C.5
Reading the colors: red, yellow, red, yellow, ... they alternate every 2. Odd positions are red and even positions are yellow, so position 8 (even) is yellow.
red→yellow→(repeat), 8 is even\text{red}\to\text{yellow}\to(\text{repeat}),\ 8\text{ is even}
With a 2-step color cycle, every even spot lands on yellow.
#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.
The two independent cycles together fix exactly one shape-and-color for each spot.
Answer: A yellow triangle

Review

Checking neighbors: position 7 is red square, position 9 is red circle. A yellow triangle at position 8 keeps the shape order square-triangle-circle and the color order red-yellow-red, so it fits both cycles.

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 column 8.

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 position 8.
💡 Track the shape cycle and the color cycle separately, then put them together: spot 8 is a yellow triangle!