## KenKen for Younger Learners

- Lesson

In this lesson, students practice addition (subtraction optional) and deductive reasoning skills to solve KenKen puzzles.

In case you are not familiar with the rules to solving a KenKen puzzle, they are listed below. These rules are available on the KenKen Puzzle Rules Overhead.

### KenKen Puzzle Rules

- Every square in the grid will contain one number.
- In a 3 x 3 puzzle, the only numbers that can be used are 1, 2, and 3.
- In a 4 x 4 puzzle, the only numbers that can be used are 1, 2, 3, and 4.
- In a 5 x 5 puzzle, the only numbers that can be used are 1, 2, 3, 4, and 5.
- And so on… The largest possible KenKen puzzle is a 9x9.

- You may not repeat the same number in a row or column.
- The areas that are outlined with a thick dark line are called cages. Each cage will contain one square with a number on the top left, which is called the target number. The numbers in the squares within a cage must create the target number using addition, subtraction, multiplication, or division. The symbol next to the target number tells you what operation to use.

For subtraction and division, the numbers can be in any order as long as the smaller number subtracted from (or divided into) the larger number equals the target. - Cages with just one square should be filled in with the target number in the top corner. These are called freebies. Freebies will not have a symbol next to the target number. Many KenKen puzzle solvers like to start with the freebies, if there are any.

A KenKen puzzle with a few labels is shown below. For emphasis, a variety of colors were used to denote the different cages outlined by the thick dark lines.

### Introducing the Puzzle

For the introduction, use a 3x3 addition only puzzle (Activity Sheet 1), as it focuses on the operation that is simplest for students as they begin to understand the nature of the puzzle.

Begin by passing out Activity Sheet 1 to your students and ask them what they notice about the puzzle and how they think it works. Sample guiding questions:

- "What do you think the areas surrounded by bold lines are?"

[The areas surrounded by bold lines are groups of squares, called cages, which will contain numbers that when either added, subtracted, multiplied, or divided will result in the target number.] - "What does the 5+ symbol mean?"

[The 5+ symbol means that the target number is 5 and the numbers within the squares of the cage must be added to obtain 5.] - "What numbers will we be using?"

[We will be using numbers from the range of 1-3 that add up to 5.]

Asking students to think about a novel situation, in this case a puzzle, encourages them to think about constraints and to speculate on the structure of a given activity.

After discussing some possible rules and meanings for the numbers and symbols in the puzzle (Activity Sheet 1), project the KenKen Rules Overhead so they can follow along.

Read and go over each KenKen puzzle rule while making sure to take any questions from your students as you move on to a new rule. You may also choose to print these for the students to reference during the class. Once the constraints have been established, ask your students where they would start this particular puzzle. One of the interesting aspects of KenKen puzzles is that there are many possible starting points and possible routes through any puzzle. Let students guide the process.

Some students may suggest starting with the freebie, which is a logical place to start. However, another student may suggest starting with another cage. This puzzle is the most basic possible puzzle, and it is actually reasonable to start with any cage. As your students proceed to more challenging puzzles, they will start to realize that certain cages are preferable to others, depending on the number of options to make that cage.

The following is one possible way to solve this KenKen puzzle, but the questions and strategies can be used for any scenario.

After filling in the freebie, if a student chooses the 4+ cage, ask them what numbers they believe should go in that cage. They may initially say 2+2, but this would be a violation of the rules. However, instead of just telling students why 2+2 cannot be used, you should guide them towards realizing why that is not a valid answer. Here are some sample guiding questions you may use at this point:

- "Does anyone have a different answer they would like to share?"
- "Does this comply with all the KenKen rules?"

Eventually, they will realize that 1+3 is the only viable option. The question then becomes, “Do we know where the 1 and the 3 go?” [No.] 1+3 and 3+1 are both equal to 4. This is an important juncture in students’ development as KenKen solvers. Beginning solvers will be inclined to guess and just place the 1 and 3 randomly, hoping that they have been placed correctly. If not, they will erase later and try something else.

### Introducing the Concept of Candidates

At this point, it is a good idea to introduce the concept of candidates. Candidates refer to possible numbers that could go into a cage. The example below shows how candidates could be used for the 4+ cage.

There are two good reasons why establishing the candidate method from the beginning is advisable.

- It encourages students to think about what they know for certain and what they are still uncertain about. This is a very important idea in solving KenKen puzzles. When students guess, particularly with larger puzzles, and are incorrect, the guess will start a chain of incorrect assumptions.
- The candidates actually enable the solver to determine another number at this point. Here are some sample guiding questions you can ask at this point:
- “Which other cages should we find candidates for?”
- “What are the candidates for this cage?”(point to cage where placement isn’t obvious)
- “Do the candidates from the 4+ cage tell us anything about other cages?”

From here, students may notice that the candidates from the 4+ cage tell us that the number in the lower right hand corner must be a 2. Since the 1 and the 3 are taken with the 4+ cage, the only remaining number in the column must be a 2. The number to the left of the 2 must be a 1 because 2+1=3.

At this point, we still do not know about the 1 and the 3 in the 4+ cage, so suggest going elsewhere. Students may elect the 5+ cage. Ask them,

- "What are the ways to decompose, or 'break up' 5?” [3+2 and 4+1.]
- "Which combination should we use? Why?" [3+2 because the only numbers that can be used in a 3x3 KenKen puzzle are 1, 2, and 3.]
- "Do we know which square contains the 3 and which one the 2?" [Yes.]

Look carefully….

The square to the left must be a 2, because placing a 3 would result in a repeated number in the first column. Here is what the puzzle would look like after placing the 2 and the 3:

### Wrapping up the Example

At this stage, you will undoubtedly have many students enthusiastically wanting to share what numbers they can now fill in. In fact, the puzzle now can be completed using deductive reasoning alone. In each row and column, there is only one number remaining that is possible. The finished puzzle will look like this:

At this point, a great way to finish the example is by having students verify that the puzzle was solved correctly. Ask students, “How do we check that our work is correct?”[Sample answer: We can check our work by verifying that none of the three KenKen puzzle solving rules were violated].

Once students mention that you can check the three KenKen puzzle solving rules, you may call on one student to name a KenKen Puzzle rule and then call on another student to check that this rule is correct. Once you have verified the puzzle’s correctness, you can “Ken-gratulate” your students!

After completing this puzzle as a class, feel free to allow students to independently solve KenKen puzzles using Activity Sheets 2, 3, and 4. Be sure to provide the following explanation for a subtraction cage:

A 2- cage means that there must be 2 numbers that have a difference of 2. The order of the numbers in a cage is not important, just that their difference is 2.

Activity Sheet 2—3x3 Addition and Subtraction Activity Sheet

Activity Sheet 3—4x4 Addition Only Activity Sheet

Activity Sheet 4—More 3x3 Addition Only Activity Sheet (3x3-addition only)

Alternatively, use the suggestions listed in the Assessments and Extensions to wrap up the class.

- KenKen Puzzle Rules Overhead
- Activity Sheet 1—3x3 Addition Only Activity Sheet
- Activity Sheet 2—3x3 Addition and Subtraction Activity Sheet (optional)
- Activity Sheet 3—4x4 Addition Only Activity Sheet (optional)
- Activity Sheet 4—More 3x3 Addition Only Activity Sheet (3x3-addition only) (optional)
- Activity Sheet 5—Incomplete KenKen Activity Sheet (optional)
- Activity Sheet 6—Correcting KenKens Activity Sheet (optional)

**Assessment Options**

- Use any of the Activity Sheet 4, completed individually by students, as a form of an assessment.

Activity Sheet 4 - Pass out Activity Sheet 6, and allow students to identify what is wrong with each puzzle. Each puzzle could violate one or more of the three rules of solving KenKen puzzles (only using 1-3, no repeating in a row or column, and the mathematical correctness in a cage).

Activity Sheet 6

**Extensions**

- Introduce students to a new and more challenging set of KenKen puzzles by using the 4x4 Addition only Activity Sheet.

Activity Sheet 3 - Pass out Activity Sheet 5, a partially filled out KenKen puzzle. The
things that would not be filled in are the target numbers, freebies, and
some of the numbers in the squares themselves. Allow students to fill
out the remainder of puzzle.

Activity Sheet 5 - Allow students to download the NCTM KenKen app from the Apple app store and solve KenKen puzzles in “play” mode.

NCTM KenKen - Allow students to create their own KenKen puzzle and verify that it is a valid puzzle by having them create an answer key for it.

**Questions for Students**

- “If you do not know the order of the candidates in a cage, what should you do?”

[You should look at the candidates for that cage and other cages and see if they allow you to figure out the placement of other numbers in the KenKen puzzle. Eventually, the information from other cages may allow you to figure out the order of the candidates of the cage you originally did not know.] - “What is the best way to start a KenKen puzzle and why?”

[There is no “best way,” but filling in the freebies first allows you to use given information that you could use to help fill out the other cages.] - “Once you are done solving a KenKen puzzle, how would you check that your answer is correct?”

[We can check that our answer is correct by adding the numbers in each cage and making sure they add up to the target number, verifying that none of the rows or columns have repeating numbers in them, and checking that only numbers in the allowed range were used.]

**Teacher Reflection**

- Which part of the KenKen puzzle rules did students have the most difficulty with? What changes would you make to the lesson to make it more understandable to them?
- What do you think was the most helpful aspect of using a game to learn? Why?
- How strongly would you consider integrating games into a lesson plan in the future? Why?

### KenKen

### Learning Objectives

Students will:

- Learn to decompose numbers in different ways in order to make different number combinations.
- Develop their numeracy and deductive reasoning skills in the context of a fun, engaging puzzle.

### NCTM Standards and Expectations

- Develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers.

- Understand the effects of adding and subtracting whole numbers.

- Develop and use strategies for whole-number computations, with a focus on addition and subtraction.

- Develop fluency with basic number combinations for addition and subtraction.

### Common Core State Standards – Mathematics

-Kindergarten, Algebraic Thinking

- CCSS.Math.Content.K.OA.A.3

Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

Grade 1, Algebraic Thinking

- CCSS.Math.Content.1.OA.B.3

Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP1

Make sense of problems and persevere in solving them.

- CCSS.Math.Practice.MP7

Look for and make use of structure.