In Unit 1, we focused on skip counting. Students determined if a value was a multiple of a given number by asking "would we say this value when skip counting?". Now, we want to amp up this idea by introducing our understanding of prime numbers.

Remind students of the first activities they completed (such as The Whisper Game, Skip Count Pop, and BuZZ) if time allows. We recommend allowing students 10-15 to play BuZZ as a group or individually. Ask students to consider the difficulty as they play, is it easier than their first experience?

Multiplication is simply identifying the value we stop at when skip counting by a certain number. Ask students how we might be able to determine if 2 is a prime factor of a number. If needed, help guide students to realize that if 2 is a factor, this means the number is one we would say when counting by twos. As all numbers with 2 as a factor, or alternatively all numbers said when counting by twos, are even, all even numbers have 2 as a prime factor.

Later, students will learn about divisibility testing. They have essentially already learned this testing for prime factors of 2 and 5 (any even number is divisible by 2 and any number with a 0 or a 5 in the one's place is divisible by 5). Because we use a base-10 system, there is a nifty way to determine if a number is a multiple of 3 by using a remainder clock. You can learn more about this test on our Instagram and Facebook pages. Unfortunately, there is no good rule for 7's. While there is a divisibility test for 7 and 11, it is complex and not well-suited for young mathematicians. As students are not yet familiar with division, rather than divisibility testing, lean on what they do know - known multiplies and distribution.

Provide students with the following numbers and ask them to determine if each value is a multiple of 2, 3, 5, and/or 7.

• 48

• 54

• 99

Allow students to share their findings and explain their reasoning.

• Both 48 and 54 must have a prime factor of 2 as they are even and thus multiples of 2. 99 is odd and therefore doesn't have a 2 factor.

• All three numbers have a prime factor of 3. This may be difficult for students to identify. See the explanation below to add a new strategy to help identify prime factors.

• None of these values have a prime factor of 5 as all multiples of 5 have a zero or a five in the one's place.

• None of these values are multiples of 7 as 7×7=49 (skipping 48) and 7×8=56 (skipping 54). It may be more difficult to determine if 99 has a factor of 7. Starting at 7×10=70, we can skip count to find 77, 84, 91, and 98 to see counting by 7's will not stop at 99.

Identifying a factor of 3 above was challenging, next, students will be given larger values which may seem daunting. Discuss the following strategy before presenting students with their next round of values.

Announce to students you believe that if you add any two multiples of a number, the sum is also a multiple. Ask students to prove or disprove your belief. Place students in small groups and allow time to discuss. It can be beneficial to provide an example to solidify your statement. For instance, we know 15 is a multiple of 3 and 30 is a multiple of 3, so 15+30=45 must also be a multiple of 3.

Students may recall The Wizard activity played previously in MathBait™ Multiplication. By using their substitution skills, they can determine 15+30=3×5+10×3. This shows us a 3-wizard has cast a spell enlarging everyone on the field. If we reverse the spell we have 3×(5+10). Give students time to confirm 3×(5+10)=5×3+3×10. This tells us 3×15=15+30=45! Not only have we confirmed the sum is also a multiple of 3, we have also determined what times 3 will produce 45. While one example does not prove the rule, using an example can help students better see an abstract concept. While a formal proof is likely too advanced for students of this age, a more informal argument is simply to remember that multiplying can be thought of as repeated adding, computing something like 3×5 is the same as finding 3+3+3+3+3. Thus, when we have 3×5+3×10 (or adding any two multiples of the same number) we are adding together 5 threes and then adding 10 more which is the same as adding 15 threes. Rectangle diagrams can also be helpful in demonstrating this idea. Conclude by stating the statement once more - the sum of any two multiples of a number is again a multiple of that number.

Students can now use this strategy to help them determine the remaining values. Encourage students to decide if each of the primes under ten (2, 3, 5, and 7) can be skip counted to reach the number. Have students share their results and their reasoning.

• 138

• 140

• 148

• 210

• 287

Students should find the following:

• 138, 140, 148, and 210 all have a factor of 2 as they are even.

• 138 and 210 have a factor of 3. One way to determine this is by noticing 210=3×7×10. This is more challenging for 138. Students may determine since 60 is a multiple of 3, 120 is also (as 60+60=120). Adding another multiple of 3, 18, will land them at 138.

• 140 and 210 have factors of 5 as each is a multiple of 10 and 10=2×5.

• 140, 210, and 287 all have factors of 7. Students can note 140=14×10=2×7×10 and thus a 7 will appear in the factorization, similarly 210=3×7×10. While 287 may be more difficult to notice, students can determine 287=280+7 and as 280=7×40 and 7=7×1 both are multiples of 7 and therefore their sum is as well.

Conclude the activity by explaining that knowing if a number is a multiple can be very helpful! It is also powerful to be able to find the DNA of a number. In the remaining activities in this lesson, we will explore how to find all the prime factors of a given number. Breaking down numbers helps us to be able to build numbers as well and strengthens our multiplication skills. We want to now focus on what traits a number must have to help us to identify its factorization and its prime factorization. A good example to share with students is LEGO. If we are interested in building something new, it can help by taking apart existing sets and seeing how they work. This is what engineers often do. By studying how to build values, we are learning the inner-workings of numbers, which will allow us to build anything we can imagine!