# Number Theory

### Objective

This game is similar to 20 questions and can be adjusted to students' level of understanding of numbers and number theory. It provides students an opportunity to think about the many ways numbers can be described (odd/even, whole number, fraction less than or greater than one, prime number, multiple of n, etc.). Students will apply a variety of mathematical concepts and skills to solve problems and use mathematical reasoning to determine whether a number fits a generalization. It requires students to use logical reasoning to identify a number—they narrow down choices based on the answers to questions they ask.

The level of this lesson can be adjusted by limiting or expanding the range of the numbers (1 to 50, 1 to 1000, etc.), the number types (whole numbers, fractions, integers, etc.), and the number of questions asked. For primary students, you may want to make a copy of a hundreds chart.

### Procedure

Draw 20 tally marks on the chalkboard. Tell students you are thinking of a whole number between 1 and 100 and that they may ask up to 20 questions to identify the number. The questions must be phrased so that the answer is either yes or no. Point out that they can use the hundreds chart to keep track of the numbers they eliminate. Proceed with the game, erasing or crossing out one tally mark for each question asked.

After the number has been identified, discuss the different types of questions asked. Elicit that while a no response to a question such as, "Is the number 48?" eliminates only the number 48, a no response to a question such as, "Is the number even?" eliminates fifty numbers—all the even numbers from 1 to 100. Ask students to brainstorm other questions they could ask. Play the game again. Compare the number of questions it took for students to identify the number in the first game with the number it took in the second game.

Brainstorm questions that would be helpful to ask if the number were between 1 and 1000 ("Is the number greater than 500?"), if the number were a fraction ("Is the numerator less than the denominator?"), if the number were a decimal ("Does the number have more than one decimal place?"), and so on.

As an extension, change the range of the number or the number type, and continue playing. Also, you may want to have a student choose the number and answer the other students' questions.

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