1. The document introduces the concept of "pop-ifying" and has students explore patterns in tables to develop rules for expressions using the symbol ⊙. 2. Students are split into expert groups to investigate expressions and come up with general rules for ⊙, then explain their rules to their initial groups. 3. The goal is for students to express rules for ⊙ in terms of arithmetic operations like addition and subtraction through exploring patterns.

1. POP ‘Rules’
Before we investigate some rules about pop-ifying, we need to know what it means to pop-ify!
Get into groups of 4. Explore the table below.
⊙ 1 2 3 4 5 6 7 8 9
1 3 4 5 6 7 8 9 10 11
2 5 6 7 8 9 10 11 12 13
3 7 8 9 10 11 12 13 14 15
4 9 10 11 12 13 14 15 16 17
5 11 12 13 14 15 16 17 18 19
6 13 14 15 16 17 18 19 20 21
7 15 16 17 18 19 20 21 22 23
8 17 18 19 20 21 22 23 24 25
9 19 20 21 22 23 24 25 26 27
Write an expression for a⊙b. Explain how you reasoned to this expression.
a⊙ b =
Evaluate: ! 10⊙5! 21⊙23! a⊙a
✦ Number yourselves off 1 to 4. Get into your expert groups.
✦ In your expert group, investigate the given expressions and come up with a general rule for each.
✦ When instructed to do so, return to your initial group and explain the rules you discovered.

2. ✦ Using the concept of pop-ifying, ✦ Using the concept of pop-ifying,
investigate each expression and develop investigate each expression and develop
a general rule expressed in terms of ⊙. a general rule expressed in terms of ⊙.
✦ Explain your reasoning. ✦ Explain your reasoning.
( a + c )  (b + d )
( a × k )  (b × k )
( a + k )  (b + k )
a b

k k ( a − c )  (b − d )
( a − k )  (b − k )
✦ Using the concept of pop-ifying, ✦ Using the concept of pop-ifying,
investigate each expression and develop investigate each expression and develop
a general rule expressed in terms of ⊙. a general rule expressed in terms of ⊙.
✦ Explain your reasoning. ✦ Explain your reasoning.
a0 even  even
0a even  odd
a 1 odd  even
1 a odd  odd