Why I love this problem: At first, students usually think it is impossible to beat the tax collector. After some trial and error, they figure out it is possible to beat the tax collector. By this time, they are highly motivated to try and find the best possible set of paychecks and to explore other extensions.
Tip: Start by playing the role of the tax collector and let students choose the paychecks. Play up the idea that the tax collector never loses. They get a big thrill when they discover they can win.
Grade Band: 4th – 6th
Math Content: factors, multiples, prime, and composite numbers
- 4.OA – Gain familiarity with factors and multiples.
- 5.OA – Analyze patterns and relationships.
- 6.NS – Compute fluently with multi-digit numbers and find common factors and multiples
Standards of Mathematical Practice:
- Make sense of problems & persevere in solving them
- Reason abstractly & quantitatively
- Construct viable arguments & critique the reasoning of others
- Attend to precision
- Look for & make use of structure
Strategies to try:
- Try this a few times. Keep track of your choices.
- Write down all the factors of each paycheck and search for patterns.
- Consider which is the best paycheck to take first.
- Think carefully about the order in which you take the paychecks.
Questions to explore:
- What is the most money you can get?
- Is there an optimal strategy?
- Try a different set of paychecks and see if you can still win? What about paychecks from 1 to 100?
- Are there any sets of paychecks where the tax collector always wins?
You can make a copy of this Jamboard to use with your students and let them explore their own strategies.
- One player chooses the paychecks and the other player collects the taxes.
- If possible, split students into breakout rooms with 2 or 3 students in each one. (If you place three students in a breakout, the third student can take notes and help the team develop better strategies)
- Direct each breakout room to a specific frame of the Jamboard so they can play without interfering with other students.