Brian Bolt has written numerous resource books for teachers containing collections of rich mathematical problems, puzzles, investigations and games. Some are descriptions of classic problems and puzzles while others are new creations. I think these books are a great resource and I wanted to share three of my favourite problems from them.
Number the Sectors
This is problem #53 from Even More Mathematical Activities (1987) and problem #72 from The Mathematical Funfair (1989). Instead of starting by asking students to find a set of numbers that gives all the integers from 1 to 25, I like to create an example as a class and then challenge them to do better (get to a larger number). You can also ask them to prove what the maximum value is.
Bolt has an alternate version of this puzzle in A Mathematical Pandora's Box (1993) (#12 Can you Do Better), which has 5 sectors around a central circle. This version can be found is online at NRICH Maths as the Number Daisy.
How Large a Number Can You Make?
Make the largest number with just the digits 1, 2, and 3 once only and any mathematical symbols you are aware of, but no symbol is to be used more than once. The challenge is to see who can make the largest number. Here are some numbers to get the ball rolling:
This is problem #83 from Even More Mathematical Activities, (1987). I've given this as a warm-up problem for high school students and this often leads to a discussion of how to know which is bigger, 2^31 or 3^21?
Make a Century
By putting arithmetical signs in suitable places between the digits make the following sum correct:
1 2 3 4 5 6 7 8 9 = 100
There is more than one solution. See how many you can find.
This is problem #127 from Mathematical Activities (1982). I would start this challenge with students by asking them to make an expression using the numbers from 1 to 9 to make a value as close as possible to 100. I would then add on the challenge to try to find an expression exactly equal to 100. There is a very similar problem called Make 100 on NRICH Maths.
I saw an earlier version of this as problem #94 in Amusements in Mathematics (1917) by Henry Ernest Dudeney. In Dudeney's version, he includes an additional challenge to try to find a solution which "employs (1) the fewest possible signs, and (2) the fewest possible separate strokes or dots of the pen. That is, it is necessary to use as few signs as possible, and those signs should be of the simplest form. The signs of addition and multiplication (+ and ×) will thus count as two strokes, the sign of subtraction (-) as one stroke, the sign of division (÷) as three, and so on."
What are Your Favourite Problems?
Do you have a favourite problem or puzzle from one of Brian Bolt's puzzle books? Do you have other favourite collections of puzzles?
Sherman K. Stein, in the preface of his textbook Mathematics, The Man Made Universe, says "Mathematics is completely the work of man. Each theorem, each proof, is the product of the human mind. In mathematics all the cards can be put on the table. In this sense, mathematics is concrete, whereas the world is abstract." The order of operations is a mathematical convention that has been developed and refined over centuries by many mathematicians. Just about every student, at some point in their mathematical education, will run across the mnemonic BEDMAS (or PEDMAS if you live in the US). The usefulness of this mnemonic can be debated (Tina Cardone, the author of Nix the Tricks, wrote an article regarding this for the NCTM).
I recently chatted with a teacher who was looking for resources to help his grade 7 students practice using the order of operations with decimal numbers. I created a "row game" for his students to practice this skill. A "row game" is an activity for students to work with a partner to evaluate a series of expressions. The expressions are written in two columns. One student evaluates the expressions in the left column and the other student evaluates the expressions in the right column. The expressions in each row evaluate to the same value. If the students don't get the same answer, they can work together to determine where the error was made. There are two reasons that I like row games. The first is that students get instant feedback. They don't need to wait for a teacher to correct and return a sheet to know if they are being successful. The second reason is that it gets students to work together to find their mistakes. The teacher can then focus on providing assistance to students who are having real conceptual difficulties, not just making small computational errors.
Kate Nowak has created a shared Google drive folder where a large collection of row games are stored. A link to my row game for order of operations with decimals is below.
Confusion regarding the order of operations has lead to many debates and arguments on Facebook about the correct way to evaluate expressions. You might see expressions such as "6 ÷ 2(1 + 2)" or "48 ÷ 2(9 + 3)" discussed. These might be used to start a lively debate in math class. I really like Vi Hart's take on these types of problems. Take a moment to watch the video below.
Vi's contention is that expressions like these are ambiguous and it is incumbent upon the author of the expression to add brackets or other grouping symbols to make their mathematical expression clear. The placement of implicit multiplication (sometimes called multiplication by juxtaposition or simply putting symbols side by side) in the order of operations has not been settled by mathematicians such that is is part of the convention for the order of operations. Despite this, calculators and computers have to make a decision on how to interpret this. You will find calculators that give different answers for these types of expressions. The Math Forum writes, "I suppose I agree with you that it would be easier and perhaps more consistent to give multiplication precedence over division everywhere; but of course there is no authority to decree this, so the more prudent approach is probably just to recognize that there really isn't any universal rule. "
One method of avoiding this confusion is to write expressions is using Reverse Polish (RPN) notation. I'm old enough to have owned an HP calculator that used this notation. In RPN, the operator always follows all of its operands. For example, instead of writing "(1+2)÷(3+4)", you would write "1 2 + 3 4 + ÷". RPN often requires fewer key strokes to enter on a calculator because parentheses are not required. It is however more difficult to learn.
Nova Scotia Mathematics Curriculum Outcomes
Grade 9 N04 - Students will be expected to explain and apply the order of operations, including exponents, with and without technology.
Grade 7 N02 - Students will be expected to demonstrate an understanding of the addition, subtraction, multiplication and division of decimals to solve problems (for more than one-digit divisors or more than two-digit multipliers, the use of technology is expected).
I really like some of the questions found on the Openmiddle.com website. It is a great resource for questions that really get students thinking. They are often formatted so that there is a very low threshold for entry to the problem but they allow for enrichment and extensions. I created the problem below for a professional development session for Math 10 teachers. Students in the Math 10 course are near the following outcome in the yearly plan: AN03 Students will be expected to demonstrate an understanding of powers with integral and rational exponents.
Fill in the boxes with whole numbers 1 through 6, using each number at most once, so that the value of the expression is as large (or as small) as possible.
In order to verify that these were correct values, I wrote a short program in Python (see below) to check all of the 720 possible (6!) values. It was my personal Hour of Code activity. I`m just learning to use Python so my code could definitely be more efficient. If you can write some better code, let me know and I`ll post it here and give you credit.
largest = 0
smallest = 10000
for i in ValuesList:
for j in ValuesList:
for k in ValuesList:
for l in ValuesList:
for m in ValuesList:
for n in ValuesList:
if i!=j and i!=k and i!=l and i!=m and i!=n and j!=k and j!=l and j!=m and j!=n and k!=l and k!=m and k!=n and l!=m and l!=n and m!=n:
if z > largest:
largest = z
if z < smallest:
smallest = z
Update: I submitted this problem to the OpenMiddle.com website and it has been posted there.