In my prior post, I presented an interactive Web Sketchpad odometer that is a great tool for introducing young learners to place value.
Well, technology moves fast these days, and the latest odometers are more powerful than ever. While our prior odometer featured ‘+’ buttons above each digit, our newest innovation in number-tracking technology features ‘+’ and (gasp!) ‘–’ buttons.
I kid, of course, but the introduction of ‘–’ buttons really does change the scope of questions we can ask students. In the interactive Web Sketchpad model below, check what happens when you press the various ‘+’ and ‘–’ buttons.
When I use this model with students, I start by asking them how to reach 9. Students will typically press the ‘+’ button above the 0 in the units place nine times. That’s a fine method, but can they reach 9 in fewer button presses? I reset the odometer to 0 and ask them to try again.
With a little experimentation, they discover that pressing the ‘+’ in the tens place followed by the ‘–’ in the ones place lands them at 9 in just two button presses (10 – 1 = 9). Students are excited that they were able to improve their “score” by knocking 7 presses off their prior attempt.
I continue by asking students how they can reach 99, 999, and 9,999 as economically as possible. Students quickly see the pattern and realize that they can reach any of these values in just two button presses. They find it amazing that even 999,999 is just two stops away from 0.
I then give students a variety of numbers and challenge them to see who can reach each target value in the fewest button presses. I’ll typically start with numbers less than 100. It doesn’t take long for strategies to emerge: To reach 34 quickly, follow the sequence 0, 10, 20, 30, 31, 32, 33, 34. But to reach 37, it’s better to progress to 40 first and then count down: 0, 10, 20, 30, 40, 39, 38, 37. Why is that? Can this pattern be generalized to other numbers?
Numbers with 5 in their units place are interesting. Contrast the quickest routes for reaching 45 (0, 10, 20, 30, 40, 41, 42, 43, 44, 45) and 75 (0, 100, 90, 80, 79, 78, 77, 76, 75). In the case of 45, it’s best to round down to 40 and then count up to 45. For 75, it’s more efficient to round up to 80 and then count down to 75. Why is that? Can this pattern be generalized to other numbers?
As students hone their skills, they progress to numbers larger than 100. What is the fastest way to reach 172, or 836, or 2014, or 86,555? With any number between 1 and 1,000,000 as fair game, there are lots of puzzles for students to ponder!
After students have solved a variety of these odometer challenges, I ask them to think about the strategies they’ve developed and share them with their classmates. As a homework assignment, they write a strategy guide for the game, clear enough for a newcomer who has never played it. Some students might only have strategies that work for numbers up to 100. Other students’ insights might extend to much larger numbers. It doesn’t really matter—the process of teasing apart strategies, both simple and complex, is a perfect opportunity to put the Common Core Mathematical Practices into action. In particular, I’m thinking of these practices:
- Make sense of problems and persevere in solving them;
- Construct viable arguments and critique the reasoning of others;
- Look for and make use of structure.
If you have a chance to use this odometer game with students, I’d be very interested to hear about your experience!