Repeating Patterns Worksheet - Students identify and extend sequences of shapes, arrows, letters, and numbers.

To solve these repeating patterns, we need to find the "core" part of the pattern that repeats over and over again. Once we identify that core group, we can just copy it to fill in the blanks.

Step-by-Step Solution:

a. Shapes: Square, Circle, Circle, Square...
1. Look at the start: $\square \bigcirc \bigcirc \square$.
2. Look at what comes next: $\bigcirc \bigcirc \square$.
3. The repeating group is Square, Circle, Circle ($\square \bigcirc \bigcirc$).
4. The sequence ends with a Square ($\square$), which is the start of a new group.
5. To finish that group, we need the next two shapes: Circle, Circle.
6. Then we start a new group: Square.
7. So the next three shapes are: $\bigcirc, \bigcirc, \square$.

b. Arrows: Up, Down, Down, Up...
1. Look at the arrows: $\uparrow \downarrow \downarrow \uparrow \downarrow \downarrow \uparrow \downarrow \downarrow$.
2. The repeating group is Up, Down, Down ($\uparrow \downarrow \downarrow$).
3. The last arrow shown is a Down arrow ($\downarrow$), which is the end of a group.
4. So, we need to start a brand new group from the beginning.
5. The next three arrows are: Up, Down, Down ($\uparrow, \downarrow, \downarrow$).

c. Letters: A, B, A, C, A, D...
1. Look at the letters: $A, B, A, C, A, D, A, E, A, F, A$.
2. Notice that the letter A appears every other time. It acts like a separator.
3. The letters between the A's are increasing alphabetically: $B, C, D, E, F$.
4. The pattern is: $A$, [Next Letter], $A$, [Next Letter]...
5. The last letter shown is $A$. The previous letter was $F$.
6. The letter after $F$ in the alphabet is G.
7. So the next spot is G.
8. After $G$, the pattern puts an A.
9. The letter after $G$ is H.
10. So the next three letters are: $G, A, H$.

d. Shapes: Hexagon, Diamond, Diamond, Hexagon...
1. Look at the shapes: $\hexagon \diamond \diamond \hexagon \diamond \diamond \hexagon \diamond \diamond$.
2. The repeating group is Hexagon, Diamond, Diamond ($\hexagon \diamond \diamond$).
3. The last shape shown is a Diamond ($\diamond$), which is the end of a group.
4. We need to start a new group from the beginning.
5. The next three shapes are: Hexagon, Diamond, Diamond ($\hexagon, \diamond, \diamond$).

e. Numbers: 1, 2, 3, 3, 2, 1...
1. Look at the numbers: $1, 2, 3, 3, 2, 1, 1, 2, 3, 3, 2$.
2. Let's find the repeating chunk. It looks like it goes up and then down.
3. Group 1: $1, 2, 3, 3, 2, 1$.
4. Group 2 starts: $1, 2, 3, 3, 2...$
5. The repeating pattern is 1, 2, 3, 3, 2, 1.
6. The sequence ends with $2$. In our pattern ($1, 2, 3, 3, 2, 1$), the number after $2$ (the second one) is 1.
7. That finishes the group. Now we start the group again from the beginning: 1, 2.
8. So the next three numbers are: $1, 1, 2$.

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Final Answer:
a. $\bigcirc, \bigcirc, \square$
b. $\uparrow, \downarrow, \downarrow$
c. G, A, H
d. $\hexagon, \diamond, \diamond$
e. 1, 1, 2