**Why I love this problem:**

It’s accessible to anyone who can add, but can engage students of ALL ages. Plus, the problem starts with an open space where you can just explore BEFORE we ask an interesting question — what numbers can you make with an unlimited supply of 3 and 7 coins? This gives all students time to engage in the problem without fear or anxiety.

There are so many ways to solve this problem and I just love seeing how creative people are when attacking the problem of “What is the largest sum you can NOT make?” My first thought was there ought to be tons of numbers you can’t make. I was surprised when I finally solved it!

Finally, this problem can be adapted in so many ways! Change the values of the coins or add in another denomination. Be creative!

**Tip**: Try starting with a simple question like, “What are some sums that you can make with 3’s and 7’s? That will get students moving. Then you will see students slowly and naturally generalize to things like, “I can make all multiples of 3 and 7.” or “I can make all multiples of ten.”

**Grade Band**: 2nd – 8th

**Math Content**: Addition and place value.

**Math Standards**:

- 2.OA – Represent and solve problems involving addition and subtraction.
- 2.OA – Work with equal groups of objects to gain foundations for multiplication.
- 2.NBT – Understand place value
- 2.NBT – Use place value understanding and properties of operations to add and subtract.
- 3.OA – Represent and solve problems involving multiplication and division
- 3.OA – Solve problems involving the four operations, and identify and explain patterns in arithmetic.
- 4.OA – Use the four operations with whole numbers to solve problems.
- 4.OA – Generate and analyze patterns
- 4.NBT – Generalize place value understanding for multi-digit whole numbers.
- 5.OA – Analyze patterns and relationships
- 5.NBT – Understand the place value system

**Standards of Mathematical Practice**:

- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Attend to precision.
- Look for and make use of structure.

**Strategies to try**:

As you explore this scenario, you might want to try one or more of these strategies. Sometimes using more than one is best.

- Start by simply writing several amounts that you know you can make.
- Try to be organized and make the numbers one at a time starting from 1, 2, 3, …
- Think about how once you make a number, how does that number connect to the other numbers you can make.
- Choose a random number like 53, can you find an efficient way to make this number using 3s and 7s.

**Questions to explore**:

- You finish finding all the numbers that can’t be made with 3s and 7s, look for structure and see if you can relate the largest number you can’t make to 3 and 7.
- Repeat the two coins problem with coins that are worth 4 and 9.
- Repeat the two coins problem with coins that are worth 2 and 6.
- What do you notice about the two scenarios above?
- Try to find a rule to explain the largest number that can’t be made for any choice of the values for the two coins.
- Try this problem with 3 coins… maybe 6, 9, and 20. What happens now?

**Implementing online**:

You can use this Desmos activity with your students. Feel free to copy it and make your own modifications.

**Additional Information**:

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