Exploring Krypto

• Lesson
3-5,6-8
1

The rules of Krypto are amazingly simple — combine five numbers using the standard arithmetic operations to create a target number. Finding a solution to one of the more than 3 million possible combinations can be quite a challenge, but students love it. And you’ll love that the game helps to develop number sense, computational skill, and an understanding of the order of operations.

Briefly explain the rules of Krypto to students. The complete rules are included in the Krypto Rules & Strategies Activity Sheet. This sheet is provided for the teachers reference and should not be distributed to students at the start of the lesson. As students begin to play, they may ask questions such as

• Do we need to use all the cards? [Yes.]
• Do we need to use them in the same order that they appear? [No.]
• Do we have to use all four arithmetic operations? [No.]
• Can we use parentheses? [Yes.]
• Can we connect two digits to form a 2-digit number? [No, concatenating digits is not allowed.]

You should be familiar enough with the rules to be able to answer questions that arise when students start playing the game.

To play a sample hand with all students, display the following 5 numbers for all students to see:

Tell students that they need to use those 5 cards to create a target number of 10. Ask students to work on this individually for 45 seconds, which is usually enough time for most students to find at least one solution. Then, ask them to work with a partner for another 90 seconds. They should compare solutions, noting any differences. If neither of them found a solution, they should work together to find one. If they both found solutions, they should work together to find a third solution.

For the cards 3, 2, 6, 1, and 9 with target number 9, there are many possible solutions, several of which are shown below:

• 6 + 9 – 3 – 2 – 1 = 9
• 3 + 2 + 1 + 9 – 6 = 9
• 9 × (1 + 2 + 3) ÷ 6 = 9
• 9 + (3 × 2 × 1) ÷ 6 = 9
• (3 × 6) × (2 – 1) – 9 = 9
• (9 × 2 × 1) ÷ (6 ÷ 3) = 9
• (9 × 2 × 1) ÷ 6 × 3 = 9
• 9 – 6 × (2 – 1) × 3 = 9

Note that the first 2 solutions in the list above use only addition and subtraction. Solutions using addition and subtraction only are generally easier to find than those involving multiplication and division. A method for determining whether a solution using only addition and subtraction is possible is discussed in the Extensions section.

After sharing the above example with students, you may want to let them play the game for a while on their own, in pairs or in small groups. There are several options for doing this:

• Order a pack of Krypto cards from your local educational supply store. You may also choose to purchase a card set through a third party, which can be found using a search on the internet.
• Download the Krypto Spreadsheet, which can be used to generate a set of 5 numbers and a target number.
• Play the Illuminations' online version of Primary Krypto (which only uses the numbers 1 to 10). Unlike the paper and spreadsheet options above, this online version also provides a solution with every hand.

• Are there any cards that are particularly good? Why?
• Are there any cards that are troublesome?
• Can you ever use just some of the cards to get started? That is, can you use just 2 or 3 cards to make a good number? If so, how?

You’ll certainly need to ask other questions to get students thinking, but you want to elicit some of the strategies that are discussed on the Krypto Rules & Strategies handout.

Math Concepts from Krypto Strategies

Some of the strategies that students discover will provide opportunities to discuss math concepts in a fun context. In particular, two concepts are readily apparent in this game:

• Multiplicative Identity Property — This is the algebraic property that says any number multiplied by 1 is equal to itself. Symbolically, a × 1 = a. In Krypto, students realize that getting an intermediate result of 1 can often be helpful. In particular, if four of the cards can be used to make 1, and the fifth card equals the targer number, then multiplying the fifth card by 1 gives a solution.
• Additive Identity Property— This is the algebraic property that says any number added to 0 is equal to itself. Symbolically, a + 0 = a. An intermediate result of 0 can often be helpful for the same reason that 1 is helpful. If four of the cards can be used to make 0, and the fifth card equals the target number, then adding the fifth card to 0 gives a solution.

Order of Operations

Krypto can also be used to introduce the order of operations and to reinforce the idea of translating verbal statements into mathematical expressions and equations.

When identifying solutions, students often use "implicit parentheses." That is, they introduce grouping symbols when explaining their solutions, although they may have no idea that they are doing so. For example, one of the solutions given above was

9 × (1 + 2 + 3) ÷ 6 = 9

To explain this solution, students may grab the 1, 2, and 3 in their left hand and say,

1 plus 2 plus 3 makes 6…

They will then grab the 6 with their right hand and say,

…then 6 divided by 6 is 1…

And finally, they will grab the 9 and say,

… and 9 times 1 is 9.

Without even realizing it, students are using their hands as grouping symbols. Holding the 1, 2, and 3 in one hand, they are implicitly putting parentheses around these cards before dividing by 6. The cards shown in the figure above correspond to the symbolic expression (3 + 2 + 1)÷ 6, which is equal to 1, and students can then multiply the leftover 9 by 1. (Alternatively, if grouping symbols were not used, the expression would be 1 + 2 + 3 ÷ 6, which is equal to when the order of operations is applied.)

Because students do grouping naturally, it provides a nice introduction to grouping symbols and the order of operations. Use this as a teaching opportunity to show students how the solutions they describe verbally can be translated into a written mathematical expression. It’s also important for them to understand why it should be translated that way. If that same expression were given to a mathematician or engineer, or entered into a calculator without parentheses, the result would be calculated as follows:

9 × 1 + 2 + 3 ÷ 6
9 + 2 +
11 +
11

The difference between this answer and the previous answer is a result of applying the order of operations, in which multiplication and division are done before addition and subtraction.

Allow students to play several more hands of Krypto, but this time challenge them to translate their solutions into mathematical expressions that use grouping symbols and adhere to the order of operations. Circulate among students as they work. It may be necessary to work through one or two examples with those groups who are struggling.

The Order of Operations Activity Sheet will allow students to practice Krypto and writing the resulting expressions. Note that the examples on this worksheet progress from relatively easy (Question 1) to very difficult (Question 10).

Completing this activity sheet as a self-guided activity will allow students to practice or discern the order of operations on their own. When they solve a Krypto challenge and generate an expression, they can check their work by entering the same expression into a scientific calculator. If they made a mistake with the order of operations when generating their expression, the calculator will not give the correct result.

Closure

A follow-up discussion can be used to check answers on the activity sheet. In particular, students can discuss their solutions to the Krypto challenges, and the class can discuss the correct expression to represent each solution.

Solutions — Order of Operations Activity Sheet

1. (5 + 2) × (6 – 4 – 1) = 7
2. (3 × 7) ÷ (20 + 1) × 12 = 12
3. (9 + 3) ÷ (2 × 6) × 17 = 17
4. (9 – 4 – 2) × 11 – 22 = 11
5. (6 × 21 – 7 × 17) × 2 = 14
6. (16 – 8 – 4 – 2) × 5 = 10
7. (21 ÷ 3) × (3 × 4 ÷ 12) = 7
8. (11 + 15 + 24) ÷ 2 – 17 = 8
9. (4 × 7 × 11 + 15) ÷ 19 = 17
10. 11 × 18 × 3 – 23 × 25 = 19

Note that other solutions may be possible.

Assessments

1. The Order of Operations Activity Sheet can be used for assessment. Review the sheet with the class, or collect it to review each student's work.
2. It is probably unfair to evaluate students based on their ability to solve a Krypto challenge. However, evaluating their ability to turn a verbal description into a mathematical expression is reasonable. You may want to choose one Krypto challenge with multiple solutions and present it to the entire class. Students will likely find different solutions, but you can evaluate students based on their ability to form a correct expression. You could have students enter their expressions into a calculator to check themselves.

Extension

One of the strategies on the Rules & Strategies Activity Sheet explains how to find a solution that involves only addition and subtraction. That strategy is presented without an explanation. For students who are ready for this level of mathematics, the following proof can be presented for why this strategy works.

Let's say there are 5 numbers dealt in a hand of Krypto, and it's possible to use only addition and subtraction to solve the challenge. The answer might look like

__ – __ + __ + __ – __ = Target,

or

__ + __ – __ + __ + __ = Target,

or any of a variety of possibilities. But in all cases, the solution could be written so that all of the numbers to be added are in one group and all of the numbers to be subtracted are in another group, like this:

(__ + __ + __) – (__ + __) = Target

Note that there could be more or fewer than 3 terms in the first group, and there could be more or fewer than 2 terms in the second group. But regardless of the number of terms, this will be the general format, with some numbers to be added and some to be subtracted.

Now let's say that the sum of the numbers to be added is x and the sum of the numbers to be subtracted is y. For simplicity, let's represent the target number by T. This leads to the equation

xy = T

If we call the sum of all 5 cards S, then we also have this equation:

x + y = S

If these 2 equations are added together, the following result is obtained:

2x = S + T,  OR  x = ½(S + T)

Said another way, the subset of cards to be added must have a sum equal to half the sum of the 5 cards and the target number. For instance, if the 5 cards are 3, 6, 11, 14, and 18, and the target number is 16, then

S = 3 + 6 + 11 + 14 + 18 = 52

T = 16

S + T = (3 + 6 + 11 + 14 + 18) + 16 = 68

½(S + T) = ½(68) = 34

This means that if a subset of the 5 cards exists with a sum of 34, then you have found a solution. For this set of cards, such a subset does exist: 3 + 6 + 11 + 14 = 34. Therefore, a solution involving only addition and subtraction is

3 + 6 + 11 + 14 – 18 = 16

The beauty of this proof lies in the fact that a high-school level algebraic explanation can be used to describe a strategy for an elementary school level game.

Questions for Students

1. What are some strategies that can be used to find a solution in Krypto?

[If one of the 5 cards is equal to the target number, try to get 0 or 1 with the other four cards. Then, you can multiply by 1 or add 0 to find the solution. More generally, both 0 and 1 are good intermediate results to obtain, because of the additive identity property and multiplicative identity property.]

2. Why is the order of operations important?

[It ensures that the values of expressions are always computed in the same way. For instance, the value of 3 + 4 × 5 would be 35 if calculations were done left-to-right, but its value is 23 if the order of operations is used. There is less likelihood of confusion if everyone uses one consistent method for evaluating expressions.]

Teacher Reflection

• What modifications would you make if you were to teach this lesson again?
• How did you challenge the high achievers while still providing adequate support to struggling students?
• How did students demonstrate that they understood the mathematics of the lesson? That is, how did you ensure that they were learning something instead of just playing a game?

Learning Objectives

By the end of this lesson, students will:

• Investigate the game of Krypto and develop strategies for finding solutions efficiently.
• Explore the order of operations using Krypto challenges.

Common Core State Standards – Mathematics

• CCSS.Math.Content.3.OA.C.7
Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.