Ideas for the K-6 Classroom:

The Game of Krypto to Support Number Sense

Dana T. Johnson

The purpose of this article is to share a teaching activity that is useful and powerful for instruction, motivation of students, and enrichment in math classrooms for grades 3 – 8. It is also a great game to teach to families.

The game of Krypto is a card game that consists of 56 cards. In the deck there are three each of the numbers 1 – 6, four each of 7 – 10, two each of 11 – 17, and one each of 18 – 25. Play begins by dealing a set of six numbers. The first five are combined in any order along with any of the operations +, –, x, or ÷ to obtain a result equal to the sixth number (called the objective or target number).

For example, suppose the numbers dealt are 20, 15, 17, 3, 9, and 4. Here is a solution: 20 ÷ [(17-15) + (9 ÷ 3)] = 4. This game is similar to the game called “24,” which uses four numbers that are printed on a card to get the objective number 24.

In Krypto, the purpose of the cards is to generate numbers for use in the game. If you do not have cards in your classroom and you are playing as a whole-class activity, you may simply ask six students to choose a number between 1 and 25. Write the numbers on the board and have everyone work to find a solution. When a student finds a solution, s/he calls “Krypto” and explains it to the class.  Students may also play in small groups with a deck of cards. The game is also an excellent solitaire game. The National Council of Teachers of Mathematics has an online version of the game that can be found through internet search terms “NCTM and Krypto.”

Mental arithmetic is the preferred method of solving the hands. Some students may be allowed to work with paper and pencil. For young students or those who need a more concrete approach, you may have them write the six numbers on a long strip of scrap paper. Then they tear the numbers apart, thus creating their own set of mini-cards for the hand. Some students are more successful when they can physically rearrange the numbers.

When you work with young students, you may want to use only the numbers 1-10. For primary grade students, you may allow them to solve the hand using fewer than all five numbers. For example, suppose the numbers dealt are 4, 3, 6, 8, 1 with an objective number of 7. Students may find solutions such as:

• 2-number solutions: 4 + 3 = 7; 6 + 1 = 7; 8 - 1 = 7
• 3-number solution: 4 + 6 – 3 = 7
• 4-number solution: 6 ÷ 3 x 4 – 1 = 7
• 5- number solution: 6 ÷ (8 ÷ 4) + 3 + 1 = 7

This strategy allows students to differentiate the activity for their own level of comfort.

Over the last few decades I have played this game with students in grades 3 to 12. No one has ever cared much about scoring. The satisfaction seems to be in finding a solution or seeing someone else find one. Sometimes several students share 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) enthusiastically 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 cognitive 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 classroom. 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 assessment 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 environment for applying properties of real numbers and the rules for order of operations. This game promotes the kind of number juggling used in factoring quadratic trinomials.

Here are some examples:

• Factoring quadratic trinomials. When we factor x² – 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 write 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 students 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 objective 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 zero from the other four numbers? One way is 2 [(4 + 3) – 7] = 0. The

solution becomes 5 + 2 [(4 + 3) – 7] = 5 or 5 + 0 = 5.

Is it always possible to find a solution using all five numbers to get the objective? It is rare, but sometimes there is no solution. Here are the only two examples I have encountered in the 45 years I have been playing the game.

3, 23, 13, 16, 1      Objective = 20

9, 9, 7, 16, 4          Objective = 25

If you search the internet for a “Krypto solver” you will find some sites that will check your hand for you. If it is solvable, they do it for you. If not, they will tell you it is unsolvable.  Whenever we hit a tough hand in a classroom that no one seems to be able to solve, I write it on the corner of the board. By the next day, it always seems to be solved by someone!

As a challenge for middle school students who seem adept at this game, I give five numbers but no objective. They write the five numbers at the top of a piece of lined notebook paper. They write the numbers 1-25 down the margin on the left. This gives them 25 possible objectives. An example is given below. This activity can be done by individuals or groups. It is an excellent setting to motivate the use of order of operations notation.

	Use these five numbers to get each of the objectives listed on the left. Use correct Order of Operations symbols! Two of them are done for you. 3     6     8     25     22
1 =
2 =	(25 – 22) ÷ 3 x (8 – 6)
3 =
4 =	22 ÷ (25 – 14) + 8 – 6
5 =
6 =
7 =
8 =
9 =
10 =
...
23 =
24 =
25 =

The game of Krypto can also be used as a classroom management activity. I use it to fill the last minute or two before dismissal. I tell students to cross their arms when they are ready to leave. I quickly write the numbers on the board and ask them to solve the hand using only mental arithmetic. They stay focused and quiet while thinking. If it is solved before the bell, I generate a new hand. This strategy helps make every minute count in math class!

References

Lach, T. and Sakshaug, L. (2005). Let’s do math: Wanna play? Mathematics Teaching in the Middle School, 11(4), 172-176.

May, L. (1995). Motivating activities, Teaching PreK-8, 26(1), 26-27.

Way, J. (2011). Learning mathematics through games. Series 1: Why games? Retrieved from http://nrich.maths.org/2489.

 Dana T. Johnson

 Retired Faculty

 College of William and Mary

 dtjohn@wm.edu