In this Mathematics Teacher article, Stephen Caviness (University of Virginia) shares what he calls a quintessential low-floor/high-ceiling math task – one that is accessible to every student and has multiple, creative solutions to explore. “Such tasks,” he says, “support students in embracing uncertainty, engaging in productive struggle, and expressing creativity in doing mathematics.”
He’s had success using this task – Close to 1,000 – with students from second grade to college level:
Using the digits 1 to 9 exactly once each, place a digit in each box to make the sum as close to 1,000 as possible.
Caviness encourages students to start filling the boxes without a strategy to see what happens – perhaps 123 + 456 + 789. Common mistakes are using a digit more than once and using zero, which is not an option. “From here,” he says, “students will have to think about place value as they make small iterations, narrowing in on sums closer and closer to 1,000. Ultimately, the closest that one can get to 1,000 is 999, but there are many ways to get there.”
As students find correct answers, Caviness suggests a number of ways the task can be extended to get students thinking about the math in deeper ways:
- Why do we regroup? What does it mean to “carry the one”?
- Which sum is furthest from 1,000? What would be the “wrongest” answer?
- What if you were allowed to use a zero?
- Are there any values as close to 1,000 as 999?
- Why can’t we add to 1,001?
- What patterns do you notice about our closest solutions?
- How many ways are there to add to 999? (There are 180.)
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