The June 2008 issue of the California Math Council ComMuniCator journal has posed the following problem in the student problem section:
“Use each of the numbers 3, 4, 5, 6, and 7, and any operations to express as many counting numbers as possible, beginning with 1. All five of the numbers must be used in each expression and each of the five numbers can only be used once in each expression.”
Many teachers are aware of the similar puzzle called “The Four 4’s”, in which only four 4’s may be used to create numbers from one through one hundred. It’s a challenging puzzle that develops lots of creativity and number sense in students. The puzzle is also great for introducing a puzzle that at first looks overwhelming, but becomes manageable after focusing on number strategies.
My students had previously worked with The Four 4’s and realized that the new puzzle from the CMC had many of the same features. At first they thought the new puzzle was harder since it had different numbers, but then they realized it was actually easier since they had five digits to work with.
This is a puzzle contest for students and the deadline for submission of entries is still open. Therefore I don’t want to supply any answers, but rather indicate the building blocks my students have found. Some of the discoveries they have made to date include:
4! = 1•2•3•4 = 24
5! = 1•2•3•4•5 = 120
sqrt (4) = 2, and therefore sqrt (7 – 3) = 2
6/3 = 2, and therefore 6/.3 = 20
The discoveries involving parentheses and exponents are too numerous to list, but students have become quite proficient and creative using these tools.
Some students originally look for a numerical expression equaling 1, then look for an expression equaling 2, then 3, etc. This normally is not the best and most efficient way to find solutions to this puzzle. I encourage students to “just play around” with the numbers and combine them in every way they can. When they do this, they are rewarded with “numerical treats” and get solutions for unexpected numbers.
We have worked on this puzzle for about a week. The first day I introduced it and discussed the rules, one of which is that using place value was not acceptable for a solution. In other words, the number 35 could not be used, but 5•7 could be used to achieve the same number. Squaring a number, such as 52, was not allowed since it involve using the digit 2, but 5(6-4) was allowed since there was not a 2 in the expression. We spent about 20 minutes on this puzzle in class, and the homework that night was to find expressions for the numbers from 1-50. The following days we spent about 10 minutes per day summarizing the discoveries made overnight. This allowed us to work with this puzzle while still continuing with our regular curriculum. After four days on this puzzle the students have expressions for all the numbers from 1-114. Our goal is to compute all the numbers from 1-150 by the end of the school year (we have 4 days remaining).
Opportunities for transitioning from number sense to algebra thinking abound with this puzzle. Order of operations is reinforced for almost every number created. Many times students substitute different, but equivalent expressions when trying to compute a number. Number properties show their strength in supporting algebra throughout this puzzle activity. Give it a try and watch the enthusiasm and skills grow in your students!
Backwards Math, Part 3 – Use Four 5’s to Create Expressions from 1 to 100
Backwards Math Extension- Four Fours Creating Many Equivalent Expressions
Backwards Math – An Activity for All Operations and All Levels of Students
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