Students will learn how to maximize the product of two integers, given their integer sum.

Watch these videos:

- Sum of 32 and Maximum Product Quadratic Application by Anil Kumar
- Find Two Positive Real Numbers Whose Product is a Maximum (PreCalculus) by Mario's Math Tutoring

Also cover the case of how to maximize the product of two integers, given their sum. It's only slightly different from the general case.

Conclude by giving your students these challenges:

- Shear Magic by NRICH
- Two Points Plus One Line by NRICH
- 2004 AMC 8, Problem 24

Go here and complete levels 1-3. After that, conjecture an algorithm that works for levels 1-3, but that you also believe will work for levels 4-5. Then test your algorithm on levels 4-5.

Here are the solutions:

- Level 1: \(52 \cdot 431\)
- Level 2: \(631 \cdot 542\)
- Level 3: \(742 \cdot 6531\)
- Level 4: \(8531 \cdot 7642\)
- Level 5: \(9642 \cdot 87531\)

Prove the algorithm correct, at least informally.

Go here and complete levels 6-10. After that, conjecture an algorithm that works for levels 6-10, but that you also believe will work for levels 11-20. Then test your algorithm on levels 11-20.

Here are the solutions:

- Level 6: 3, 3
- Level 7: 3, 4
- Level 8: 3, 3, 2
- Level 9: 3, 3, 3
- Level 10: 3, 3, 4
- Level 11: 3, 3, 3, 2
- Level 12: 3, 3, 3, 3
- Level 13: 3, 3, 3, 4
- Level 14: 3, 3, 3, 3, 2
- Level 15: 3, 3, 3, 3, 3
- Level 16: 3, 3, 3, 3, 4
- Level 17: 3, 3, 3, 3, 3, 2
- Level 18: 3, 3, 3, 3, 3, 3
- Level 19: 3, 3, 3, 3, 3, 4
- Level 20: 3, 3, 3, 3, 3, 3, 2

Prove the algorithm correct, at least informally.