## Can You KenKen?

- Lesson

The objective of this lesson is to use combinations to solve KenKen puzzles. An early solution strategy is for students to guess and check and use logic-based elimination. This lesson builds on those strategies by having students systematically list all possible combinations within each cage, the darkly outlined sections of the puzzle.

### Prerequisites

This lesson assumes that you and your students have played KenKen on an introductory level before. You may want to assess your students’ understanding of the basic rules and terms before beginning a puzzle (ex: “What is a cage?”). If necessary, review the rules of KenKen with your students:

**KenKen Rules**

- Every
square in the grid will contain one number.
- In a 3 x 3 puzzle, use the numbers 1–3.
- In a 4 x 4 puzzle, use the numbers 1–4.
- In a 5 x 5 puzzle, use the numbers 1–5.
- In a 6 x 6 puzzle, use the numbers 1-6.

- Do not repeat numbers in any row or column.
- A
*cage*is a heavily outlined set of squares. The numbers in the squares within a cage must combine (in any order) to produce the target number in the top corner. You must use the mathematical operation next to the target number. - Cages
with just one square should be filled in with the target number in the top
corner. These are called
*freebies*, and this is a good place to start solving the puzzle.

As a teacher, try to resist helping students too much while they are solving their puzzles. They will learn and gain more by struggling to solve it on their own rather than receiving help from the teacher.

### Preparation

To prepare for the lesson, prepare enough copies such that each student has access to each activity sheet:

*Answer keys can be found in the Materials section.*

### Main Lesson

Tell students, “Today we’re going to explore some strategies for solving KenKen puzzles. Remember, we want to avoid guessing, so we will build our logic skills to find other ways of solving the puzzle. Today, we will focus on listing combinations and use inductive reasoning to find an appropriate arrangement of the combinations.”

Distribute Puzzle 1 Activity
Sheet. Review the rules for completing a KenKen puzzle as well as the
vocabulary used (*cage, square, freebie*). This lesson should not be students’ first
activity with the puzzles so experience solving them is expected. The objective for this lesson is to be
explicit and thorough listing the possible combinations for each square,
including all possible arrangements within the cage. The goal of the Puzzle 1 activity is for
students to find ALL combinations and possible arrangements and develop
strategies for systematically eliminating those that do not work.

Next, distribute Puzzle 2 Activity Sheet. The objective of Puzzle 2 is to reflect on the difficulty of each cage and making conjectures about solving the puzzle, before attempting to solve it. By comparing all possible combinations to the possible arrangements of a cage, students make conjectures about the difficulty of each cage and justify their reasoning.

Puzzle #2 is designed to encourage students to reason about the solution process, make conjectures about the solution strategy, and defend their arguments,

Depending on time remaining, distribute Puzzles 3 & 4 Activity Sheet. Students should do these puzzles independently or in pairs. As students work on puzzles, circulate the room and assess whether they are using all possible combinations in a cage, or have discovered rules to limit the choices only to valid arrangements.

Use the Questions for Students portion as a guide. Also, try to avoid answering questions or helping students if they ask for help, unless it is just a rule clarifying question. This will allow the students to practice persevering.

### Wrapping It Up

To wrap up the class, discuss the following questions together as a class. A good alternative is to have students answer these questions as a journal entry.

- What are some strategies you have noticed for limiting the number of combinations that may fit in a square?
- When is it helpful to list all possible combinations for a square?
- What is the biggest challenge in solving a KenKen puzzle? Explain.
- How did listing the combinations help you?
- How did listing the possible arrangements help you?
- How did your knowledge of factors and addends help you?
- What other strategies could you use to solve KenKen puzzles?

**Assessment Options**

- Students solve KenKens. Start with a 3x3 and work up to KenKens of greater dimensions.
- Use the activity sheets as a form of assessment.
- Give students an incorrectly completed KenKen puzzle and ask them to find the errors and explain how they know.

**Extensions**

- Use the KenKen app for harder puzzles. The app also features bonus puzzles called “ NoOp “.

KenKen Downloadable App - Give student KenKens that have been filled in incorrectly. Have them find the errors and use math talk to explain why they think the error was made and how to fix it.
- Encourage students to create their own KenKen puzzle. Start with a 3x3. Keep a journal of what you discover regarding the process you use as you create your puzzle. Things to consider could include a puzzle with only one operation as well as puzzles including more than one operation.

**Questions for Students**

1. Why how did you decide that <#> belongs <point to a square in a puzzle>?

2. After a puzzle is complete. How could you check you work?

[Sample answer: Check to see that no numbers are repeated in every row and column.]

3. Have students keep track of the order in which they fill in the cages. When the puzzle is completed, compare order and reasoning with a partner. Defend your procedure as one based on knowledge rather than guessing. Statements could have a starter such as: “I know 3 has to be in this square because …”

**Teacher Reflection**

- How did students demonstrate their comprehension of using combinations in order to see the possibilities in each cage?
- How well were the students able to distinguish between a combination and an arrangement?
- Describe what you learned regarding students using this “combination” strategy to solve a KenKen.
- Describe the logic skills combined with arithmetic and factoring skills that you observed students using to generate combinations and fill in their puzzles.
- Where did students struggle the most?
- What additional guiding questions could you use to scaffold students’ struggle without giving too much information?
- What are some strategies that worked increase student perseverance and increase self-motivation?
- How does working within partner groups increase student accuracy?
- Does the explicit attention to combinations and arrangements encourage students to use logic-based reasoning instead of only guess and check methods?

### KenKen

### Learning Objectives

Students will:

- Define the different parts of KenKen puzzles.
- Solve 4x4 and 5x5 KenKens strategically by first listing all possible combinations within a cage.
- Reinforce their knowledge of their multiplication tables, factorization, and greatest common factors.
- Apply knowledge of factors to solving strategies.

### NCTM Standards and Expectations

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

- Develop fluency in adding, subtracting, multiplying, and dividing whole numbers.

- Use factors, multiples, prime factorization, and relatively prime numbers to solve problems.

### Common Core State Standards – Mathematics

Grade 4, Algebraic Thinking

- CCSS.Math.Content.4.OA.B.4

Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.

Grade 6, The Number System

- CCSS.Math.Content.6.NS.B.4

Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP1

Make sense of problems and persevere in solving them.

- CCSS.Math.Practice.MP2

Reason abstractly and quantitatively.

- CCSS.Math.Practice.MP3

Construct viable arguments and critique the reasoning of others.

- CCSS.Math.Practice.MP6

Attend to precision.

- CCSS.Math.Practice.MP8

Look for and express regularity in repeated reasoning.