The goal of this warm up is to allow students to explore different ways to find familiar products.

Provide students with a multiplication fact such as 6×7. Explain that our goal is to make an expression (a number statement) as long as possible. To do this, we need to think about what two numbers we can multiply to make 6 or 7.

Many students will rush to 1×6 and 1×7. This is okay. In fact, if students do this, encourage them to continue. Write each statement below the previous:

6×7

1×6×1×7

Allow students to continue to append the product with ones (until you run out of space or are over it).

6×7

1×6×1×7

1×1×6×1×1×7...

Explain that 1 is a special number. It acts as a mirror in multiplication. Normally, when a number is multiplied it acts as a growth laser, enlarging the value; but 1 has the special ability to only reflect, like a mirror. This begins to build a strong foundation for fractions. MathBait™ uses fun-house mirrors as an approach to fractions which is not only very successful and impactful, but also enjoyable to students. Setting up the idea of 1 as a reflector (i.e. multiplicative identity element) will help students with many mathematical concepts to come.

Now, challenge students to do the same, but without our magical mirror 1. With a 1, we can extend our expression indefinitely as multiplying by 1 simply reflects what is already there, but can we extend our expression without this?

Guide students as needed to notice 6=2×3, allowing us to write this expression as 2×3×7. Note that the only way to make a 7 is 1×7, so 7 cannot be extended. If a student mentions "prime" this is a great chance to acknowledge 7 is prime. But it is not necessary to bring up prime numbers if students do not mention it. 

Present a challenge. Provide students with a multiplication table to allow them to see all the products they have worked with. Ask students to first pick any number on the table and expand it as much as they can without using 1. Students can swap with a partner to check their work when they are finished. The image below shows the standard multiplication chart along with a chart listing the number of prime factors for each value. Students may benefit from displaying the image without context and asking them to share what information they think the image is trying to convey.

Discuss some of the longest and shortest expansions students found. Prime numbers will be the shortest. There are 24 values with an expansion of length 2 (such as 4 with 2×2 or 10 with 5×2), 34 values with an expansion of length 3 (such as 8 with 2×2×2 or 20 with 2×5×2), another 24 with an expansion of length 4 (such as 24 with 2×2×2×3 or 36 with 2×3×2×3), only 8 with an expansion of length 5 (such as 80 with 2×2×2×2×5), and a single value with an expansion of length 6.

Have students work together in small groups to try to identify the only value with an expansion of 6. Invite students to share their strategies. Will they simply divvy up the table and find it through force? Will they use problem-solving strategies to try to determine how to build such a number?

Play around with this idea as time permits. You can provide students with a challenge to find an expansion of length 4, once they have found one write it on the board forcing the remaining students to find a different number. Encourage students to develop ways to quickly build such a number (for example, starting with 2's and multiplying is a great method because 2's are small and we can multiply a good deal of 2's and stay under 100).