Lesson 3

Factors

Est. Class Sessions: 2–3

Developing the Lesson

Part 3. Primes

Direct students to the Prime Numbers section of the Factors pages in the Student Guide. Ask students to read the definition for prime number as it is written. Refer students back to Lesson 1 that defined prime numbers as numbers that are larger than one and have only one rectangle. Display the two definitions and have the students compare them. Ask them to reflect on how there can be two different definitions.

Ask:

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  • What is the same in the two definitions? (A prime number is a number greater than one.)
  • What is different? (The first says a prime number has only one rectangle; the second says a prime number has only two factors.)
  • Do these two definitions say the same thing? How? (Yes, in the rectangles, the number of rows and the number in each row are the two factors. If there is only one rectangle that can be made, that means that there can be only two factors, one and the number itself.)

Ask students to work in pairs on Questions 14–16 and use the definition to list the primes between 1 and 25 (Question 15). Question 16 then asks students to consider which of 35, 27, and 41 are prime. They can solve this by searching for factors, either with calculators or by building arrays.

Encourage students to be systematic when they search for prime numbers. For example, when they search for the factors of 41, they should test the numbers in order: 41 ÷ 2 = 20.5, so 2 is not a factor (since 20.5 is not a whole number), 41 ÷ 3 = 13.67, so 3 is not a factor, 41 ÷ 4 = 10.25, so 4 is not a factor, and so on.

Some students will see pretty quickly that they do not need to test all the numbers up to 41 to be sure it is prime. Some students will see that they do not have to test any numbers greater than half of 41. Testing for factors up to half of a number is reasonable for students. Some students may realize they do not even need to test that far.

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Testing for Primes. When testing whether a number is prime, it is not necessary to look for factors any larger than the square root of the number. As in the previous Content Note, consider the list below of numbers whose product is 36. In each pair, there is always one number less than 6, the square root of 36.

 1 × 36
 2 × 18
 3 × 12
 4 × 9
 5 × 7.2
 6 × 6
 7 × 5.14...
 8 × 4.5
 9 × 4
10 × 3.6

When testing whether 41 is prime, for example, we have to test only numbers less than 7 (the square root of 41 is about 6.4). If 41 had any factors, at least one of them would be less than 7.

Student Guide:
Student Activity Book:
Master: