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different solutions and no one seems to care who gets points. But if you want to score, small groups can give one point to the first person to solve the hand correctly. I use a fun scoring method for whole-class teams – I divide the class in half and write the numbers for each hand on the board. The first person with a correct solution earns many points for the team’s score – the sum of the six numbers! If they call “Krypto” but cannot produce a correct solution, they have the sum of the six numbers SUBTRACTED from their team’s score. This minimizes impulsive, false claims. There are many possible benefits to playing this game in your classroom. May (1995) enthusi- astically describes and recommends the game of Krypto in an article on motivating activities for the math classroom. Way (2011) describes additional benefits of games to support mathematical cogni- tive objectives, including application of math skills in a context that is meaningful to students, building of positive attitudes towards math, increased skill levels, opportunities for students to participate at various levels of thinking, and opportunities to connect with families as students share the games at home. Lach and Sakshaug (2005) discuss their action research on games in a sixth grade class- room. Two of them, Muggins and 24, are similar to Krypto. After 12 weeks of playing math games the authors found students scored better in an assess- ment of algebraic reasoning. The game of Krypto does not present facts in the way flash cards do, but incorporates problem solving and pattern searching into fact practice. Beyond the obvious practice in mental arithmetic and developing number sense, it can be an environ- ment for applying properties of real numbers and the rules for order of operations. This game pro- motes the kind of number juggling used in factor- ing quadratic trinomials. Here are some examples:  Factoring quadratic trinomials . When we factor x 2 – 8x + 12 we are looking for numbers whose product is 12 and whose sum is -8. Once when I was teaching factoring to an 8 th grade algebra class, a student blurted out, “It’s easy. It’s just like Krypto!”

 Order of operations. Ask students to wr ite their solutions in correct notation, using rules for order of operations or “algebraic logic.” Once a student writes a solution, others can check. For example, if one student incorrectly writes 2 + 3 x 6 – (7 + 3) = 20 then others should note that parentheses are required around 2 + 3.  Commutative and Associative Properties . In comparing solutions, students will see that variations in grouping and order may produce the same result. For example, one student may write (3 + 2) x 1 + 4 + 5 = 14. Another may claim to have a different solution: 1 x (2 + 3) + 4 + 5 = 14. Students should recognize that two instances of commutative property are used – one for addition and one for multiplication. This can lead to a good discussion. Some stu- dents will say it is really the same solution and others will say they are different solutions.  Multiplicative Property of Zero. If you can make a zero from two of the numbers, you can eliminate other numbers that you don’t need.

Example: 6 4 3 6 22 Objective number is 7. Solution: (3 + 4) + (6 – 6) x 22 = 7

 Identity elements. If you can get the objec- tive number with one or two numbers, then try to get a zero or 1 with the remaining numbers. Multiply by one or add zero.

Example: 7 9 2 15 20 Objective number is 8.

Notice that 15 – 7 = 8. Can you get a 1 from the other three numbers? Yes, 20 ÷ 2 - 9. So the identity element for multiplication helps with a solution of (15 – 7)(20 ÷ 2 – 9) = 8 or 8 x 1 = 8.

Example: 2 4 5 3 7 Objective number is 5.

Notice that you are given a 5 in the middle of the set of five numbers. Can you get a

Virginia Mathematics Teacher vol. 44, no. 1

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