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MathBait™ Multiplication


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Substitution is one of the most powerful tools of a mathematician. In this activity, students develop an intuitive understanding of substitution as they engage in role-playing at a sandwich shop to help lay the foundation for prime factorization.


Resource Type

Warm Up

Primary Topic

Primes & Factoring







In Marco the Great and the History of Numberville, we present substitution as one of the most basic and powerful skills of a mathematician - because it is! Substitution is an amazing tool that has the ability to turn something complex into something much simpler. In this warm up, students will play with substitution.

Explain to students that at a very famous sandwich shop they offer a Classic Sub which includes:

  • White bread

  • Cheddar Cheese

  • Lettuce

  • Tomato

  • Onion

They charge $3.00 for this special which is less expensive than the Build Your Own with the same ingredients. May wants almost everything on the classic sub. She doesn't like onions. Instead, May would like either olives or avocado. Do you think they will allow May to swap out the onions for olives? What about swapping the onions for avocado?

Present the menu board to students and allow them about 5 minutes to determine their solution.

menu showing prices for each item.

Allow students to share their thoughts. Explain as needed that the shop will not allow May to switch out the onions for olives as onions are $0.25 and olives are $0.50. They certainly will not allow to switch onions for avocado. The shop will allow substitutions. A substitution is when you swap out an item for an item of equal value.

Ask students if there is anything on the menu she can substitute for the onions. Give them time and encourage creative solutions. Remind students May doesn't want onions.

Have students share their findings. Here are a few options:

  • If May is okay with no cheese, she could order the Classic, substituting the cheese and onions for avocado, as $0.75+$0.25=$1.00.

  • May could swap two items such as lettuce and onion or tomato and onion for olives.

  • May could order no onions and extra tomato (or lettuce).

  • May could swap lettuce, tomato, and onion ($0.75) for olives and update her cheese to Swiss.

If time permits, allow students to play more with this idea, creating their own menus and acting out orders and substitutions.

Conclude the activity by reinforcing the term substitution. Explain we can substitute anytime items have equal value. Tie back to the previous lesson, Centipede. It can be helpful to allow students to play a quick round of Centipede before moving forward. With substitution fresh on their mind, what do they notice?

In Centipede, we can eliminate segments by stating a multiplication fact. For instance, 56 can be expressed as 8×7. However, we can also use substitution. Since 8=2×4, we could eliminate the same segment using 2×4×7 as we replaced the value of 8 with an equivalent value of 2×4. In today's lesson, we will see how substitution can turn tricky problems into easier problems.

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