By Jane M. Wilburne, Posted September 29, 2014 –
If you have not had a chance to engage your students in the What Is the Largest Number You Cannot Make? problem, you can find the task here. What interesting patterns did your students find? What strategies did they use? Please post a comment and share with others. We enjoy learning about how these tasks are used in classrooms.
Some students start the problem by making a list of the numbers from 1–20. Then they list the multiples of 4 and multiples of 7 and make combinations of both sets of multiples. As they combine sets, they check off the total number of nuggets from the list of numbers from 1–20. Students soon realize they needed to extend the list to 40. For example, 3 sets of 4 is 12, 4 sets of 7 is 28, so 12 + 28 = 40 nuggets total.
Other students might list the numbers from 1–30 and start with 1, then 2, and so on to determine whether they can buy that number of nuggets. For example, they cannot buy 10 nuggets; but they can buy 11 nuggets (pack 4 + pack 7), and they can buy 12 nuggets (3 sets of 4 nuggets). They cannot buy 13, but they can buy 14 nuggets (two packs of 7 nuggets), and so on. Students often continue with this approach, sometimes arguing why they could or could not buy a particular number of nuggets. For example, Josh could not figure out how to buy 22 nuggets, but Andre explained that if you could buy 11 nuggets, you could buy 22 nuggets by doubling the number of packs to make nuggets (2 packs 4 + 2 packs 7 = 22 nuggets).
Did your students prefer one of these strategies? Which other strategies did your students use?
In a fifth-grade classroom, the teacher asked her students how many successful combinations of nuggets they could buy before they determined they could buy any number of nuggets beyond it. Sharise shared that once she had four numbers in a row, she was able to make any number of nuggets by just adding four to some of the number of nuggets she could buy. She shared that she could buy 18 nuggets (1 pack of 4 nuggets + 2 packs of 7 nuggets), 19 nuggets (3 packs of 4 nuggets + 1 pack of 7 nuggets), 20 nuggets (5 packs of 4 nuggets) and 21 nuggets (3 packs of 7 nuggets). Therefore, by adding one more pack of 4 nuggets to 18 nuggets, you could buy 22 nuggets. Adding one more pack of 4 nuggets to 19 nuggets would allow you to buy 23 nuggets, and so on. She used repeated reasoning to see that she could buy every number of nuggets beyond 17 using this approach.
Which other mathematical practices were your students engaged in when working on this problem?
One extension to the problem would be to ask, “How many packs of 4 and 7 nuggets would you need to have 98 nuggets?” “How many different ways could you purchase 98 nuggets?”
What other extensions can you suggest?
Students could create their own similar problems. As long as the two numbers (a, b) you select for the packs of nuggets are relatively prime (they have no common factors other than 1), you can find the solution by
(a * b) – (a + b). Now you can create your own “Greatest Number You Cannot Buy” problems.
Jane M. Wilburne is an associate professor of mathematics education at Penn State Harrisburg. She teaches content and methods courses for both elementary and secondary mathematics teachers as well as graduate mathematics education courses. She is a co-author of Cowboys Count, Monkeys Measure, and Princesses Problem Solve: Building Early Math Skills Through Storybooks (Brookes Publishing 2011) and has published numerous manuscripts in Teaching Children Mathematics, among other journals. Jane began serving as a member of the Teaching Children Mathematics Editorial Panel in May 2014, and her term will continue through April 2017.