Dividing by a fraction

This post was prompted by conversations with April Brown and Leann Miller, teachers from the Peace Wapiti School Division in Alberta whom I met at our April 2018 SNAP Math Fair Workshop at BIRS.

Invert and multiply

Is there any real-life circumstance where the invert-and-multiply rule is naturally applied? Here are two instances when I used this rule without being aware that I was using it.

The rule I am talking about is this one:

where b and c are numbers and [A] is pretty well anything.

Red Deer to Edmonton

I'm driving from Red Deer to Edmonton. That's 144 km. I know from past experience that my average speed will be 90 km per hour. How many minutes will it take me to get to Edmonton?

Did you solve it like this:

144 divided by 90 is 1.6, so it takes 1.6 hours, which is one hour and 36 minutes, which is 96 minutes.

That's how I got the answer. Putting it in mathematical notation:

I could have obtained the answer by first converting the speed from km per hour to km per minute –– 90 km per hour is 90 km per 60 minutes which is 1.5 km per minute –– which gives this solution:

To travel 144 km at 1.5 km per minute will take 144 divided by 1.5, or 96 minutes.

In mathematical notation:

The two solutions are equally valid, which means that

So, dividing by 90/60 is the same as inverting the fraction and multiplying by 60/90.

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In this next problem, the numbers work out nicely. In the actual task, the numbers were not quite so friendly.

Building a shelf

You have been asked to construct a 28 inch wide shelf using 2 × 4 lumber.

The shelf will be 8 feet long (which is convenient because two-by-fours come in that length). The shelf will have the boards snug against each other, that is, no spaces are allowed between the boards. 

How many boards will you need? 

Since 28 ÷ 4 is 7, we will require 7 boards. But there's a catch: two-by-fours are not 2 inches by 4 inches. The two-by-fours that you buy at a lumber store are milled a quarter of an inch on all sides, so that their actual dimensions are 1½ inches by 3½ inches.

So the question becomes:

How many  3½ inch wide boards are needed to make a width of 28 inches? In other words, what's 28 divided by 3½ ?

If it were me, I would shamelessly bring up the calculator on my smart phone and punch in 28 ÷ 3.5 to get the answer: eight boards.

To avoid the awkward ½ in a mental calculation, you might note that two 3½ inch wide boards are together the same as one 7 inch wide board.  Since four 7 inch boards will span 28 inches exactly, this means that eight 3½ inch boards will do the job.

If you've followed this, we've just inverted-and-multiplied again. Using the fact that 3½ can be written as ⁷/₂ , this is what we did (after a slight rearrangement of the notation):

The expression on the left is 28 divided by 3½ while the expression on the right is 28 multiplied by the reciprocal of 3½.

Some additional comments

I learned the invert-and-multiply rule in grade 8. At that time, some of us had difficulty deciding what number we should use to divide or multiply by.  In the Red Deer to Edmonton problem, should one multiply by 60/90 or by 90/60?  Should the final calculation be

To overcome difficulties in situations like this, our physics teacher gave this useful advice (which today is sometimes offered on the web):

Always check that the units cancel properly!

When the units are included, the confusion about 60/90 versus 90/60 becomes:

Which of the following should we use

or, equivalently, which of these should we use

Here's what the physics teacher meant by the units cancelling properly:

In the expression on the left, the km units cancel, leaving us with an answer in minutes –– which is what we want. The units in the expression on the right don't cancel and so we are left with km²/minute, which doesn't make sense in this problem. 

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Finally, it is interesting to note that there is a connection between the invert-and-multiply rule and the rule for the distribution of the subtraction sign.

In general, without the usual fractional notation, the invert-and-multiply rule looks like this

a ÷ (b ÷ c) = a ÷ b・c.

Here's the rule for the distribution of the subtraction sign 

a – (b – c) = a – b + c.

In both rules, the sign outside the parentheses flips the signs of the operations inside the parentheses.

There are really a lot of math connections happening here. There is a tie-in with the BEDMAS convention for the order of precedence of operations. There is the potential lead-in to additive and multiplicative inverses. There are even hints about how to remove the mystery from rules like a-negative-times-a-negative-is-a-positive. 

Math fair workshop at BIRS

Every spring for the past 15 years there has been a 2-day math fair workshop at the Banff International Research Station. In 2018, this wonderful event runs from the evening of April 27 until noon April 29.

The workshop is all about how to run a math fair that emphasizes puzzle solving with lots of  interaction between the math fair visitors and the student presenters. The workshop participants will primarily be teachers, including some who have organized highly successful math fairs in their schools. More details about the workshop can be found here. I hope you will consider coming.

Some of the participants at the April 2017  workshop

The workshop sessions are held in the TransCanada Pipelines Pavilion. It's a superb venue, with the all the capabilities and facilities you would expect from a leading research institution. 

TransCanada Pipelines Pavilion. Photo Courtesy of The Banff Centre.

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Here are some examples of the sort of puzzles that we will be working with. These happen to involve arithmetic operations, but that is not obligatory –– all that is required is that the puzzles be mathematically based. 

Fill in the digits

(a)	(b)

(c)

(a) Put the digits 1, 2, 3, and 4 into the squares to make a correct sum. Use all four digits.

(b) Put the digits 1, 3, 6, and 8 into the squares to make a correct sum. Use all four digits.

(c) Put the digits 1 through 7 into the squares to make a correct sum. Use all seven digits.

The following two examples are both based on the same idea and show how puzzles can be adapted to challenge students at different levels.

Crosses and Sums

In the cross above, the numbers from 9 through 12 have been placed in the squares in such a way that horizontal and vertical sums are the same.

( 10 + 8 + 11 =  9 + 8 + 12. ) 

---

In each of the crosses below, the squares have to be filled with all of the digits from 1 through 5.

(a)	(b)	(c)

In each of (a), (b), and (c) put the remaining digits from 1, 2, 3, 4, 5 into the empty squares to make the horizontal and vertical sums the same.

Spokes

In the following figure, the digits from 1 through 7 have been placed in the circles so that the sums along the lines are the same:

1 + 7 + 6 = 2 + 7 + 5 = 4 + 7 + 3.

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In each of the following, the circles have to be filled with the digits from 1 through 7. In each case, three of the circles have already been filled.

(a) 

(b)

(a) Place the numbers 1, 2, 3, and 7 in the circles so that the sums along the lines are the same.

(b) Place the numbers 4, 5, 6, and 7 in the circles so that the sums along the lines are the same.

Both the Crosses and Sums and the Spokes puzzles are variations of the Spoke Sums puzzle from our math fair website, which is in turn a simpler version of a much older puzzle Henry Ernest Dudeney and/or Boris A. Kordemsky. I don't know who originated the puzzle –– during that era, puzzlers frequently "borrowed" from each other without giving credit.

For more examples of math fair puzzles, visit our SNAP math fair website.

If you are planning to come to this year's workshop and have some favourite math-based puzzles or games, please bring them and share how you have used them in your teaching.

Math fair workshop at Banff

The 14th annual math fair workshop at BIRS will take place over the weekend of May 6/7/8, 2016.
(BIRS = the Banff International Research Station.)

Right off, let me say that I have a pretty bad attitude about school science fairs.  You know — those competitions with poster sessions, baking soda volcanoes, and parent-created displays. The ones that end with an obligatory showcasing of a winner — a bright student who looks like he/she will go on to become the next Neil deGrasse Tyson, and who, for a short while, will be a poster-person for our education system.

OK, that's harsh, but it is still very much the norm to single out a winner.

How about having one that does not overly favour the highly talented? One that even a less confident student would enjoy and not end up feeling like a failure because he or she did not win a medal.

If you’re like me, you do not enjoy being tagged as a loser, and you would likely withdraw from a situation where that is liable to occur. Aviva Dunsiger touched upon this in her blog. Although her post is about phys-ed rather than mathematics, she paints a clear picture of the response to anticipated failure:

Yes, there were always strong athletes, but those that struggled (and I was one of them) wanted nothing to do with phys-ed. With my visual spatial difficulties, games like volleyball, basketball, and baseball were a tremendous struggle. I certainly never got picked for a team, and I couldn’t blame anyone. Why would I want to be physically active if I was only going to meet with failure?

[the emphasis is Aviva's]

Can we have a math fair where students can be mathematically active without the anticipation of failure?  One where students do not need a badge or ribbon to confirm that their efforts have paid off ?

Such math fairs do exist. They’re called SNAP math fairs because they are Student-centred, Non-competitive, All-inclusive, and Problem-based.

The fairs are built around math-based puzzles. The students first solve the puzzles^[1] and afterwards prepare the artwork and puzzle pieces that are required to display them.  

Visit such a fair and you will find students manning their puzzles. But, you will not see them exhibiting the solutions. Instead, they will invite you to try the puzzles yourself, and they will give you hints and help when you run into difficulty. The math fair is very interactive. It is much more than a poster-session.

* * *

Here are a couple of puzzles from past math fairs. The first one is for younger students to solve. 

Cats Pigs and Cows

A farmer has nine animal pens arranged in three rows of three. 

Each pen must contain a cat, a pig, or a cow. 

There is already a pig and a cat in two of the pens. 

The farmer wants you to fill the remaining pens so that no row or column contains two of the same animal.

The second puzzle is for older students.^[2]

The Sword of Knowledge

The dragon of ignorance has three heads and three tails. 

You can slay it with the sword of knowledge by chopping off all of its heads and all of its tails. 

With one stroke of the sword, you can chop off either one head, two heads, one tail, or two tails.

But the dragon is hard to slay !! 

• If you chop off one head, a new one grows in its place. 
• If you chop off one tail, two new tails replace it. 
• If you chop off two tails, one new head grows. 
• If you chop off two heads,  nothing grows.

Show how to slay the dragon of ignorance.

* * *

A SNAP math fair is remarkably adaptable to many different circumstances. If you are interested in learning about how you can incorporate a SNAP math fair into your own teaching environment, come to the BIRS workshop. You will meet teachers who have organized math fairs in their own schools. You will also meet a few mathematicians who have taught courses in which a math fair was key ingredient. 

As well, there will be math fair resources available, and the participants will be involved in puzzle-solving sessions.  

The BIRS workshop has room for about 20 participants, and it is oriented towards (but not limited to) K-9 teachers.

For more details about SNAP math fairs, visit the SNAP math fair site. And while you are there, take a look at the Gallery to see how students react.

For more information about the workshop, and who to contact, the link is here.

End notes

^[1] The solving part is a crucial element of the math fair.  Ideally, students solve the puzzle by themselves. They are surprisingly persistent.

^[2] I imagine the Sword of Knowledge puzzle would work with junior high or high school students. However, I once visited a SNAP math fair where two grade five students had solved it. Their teacher told me that they struggled with the problem but solved it after one of them grabbed a handful of pencils (tails) and erasers (heads).

Math fairs and puzzles and a weekend workshop at BIRS

Evensies 

(Puzzle number 21 in Boris Kordemsky’s Moscow Puzzles)

Erica doesn't like odd numbers, so the box of chocolates shown above meets with her approval. The problem is that she has to remove six chocolates from the box in such a way that she leaves an even number of chocolates in each row and each column.

Make a 4 by 4 grid, and using pennies or other tokens as chocolates, show how she can do this. There is more than one solution.

I belong to a group of teachers, mathematicians, and puzzle developers who advocate the use of math-based puzzles in the K-12 classroom. To this end, every April for over a decade we have held a weekend math fair workshop at BIRS. (BIRS =  Banff International Research Station). Although BIRS exists to aid research in mathematics and related disciplines, from its inception it has also supported educational initiatives, and our April workshops are an example of its continuing commitment. 

About 20 participants attend the workshops, many of them teachers and mathematicians who share a common interest in enhancing mathematics education. The participants are diverse in background and experience, and the workshops have quite a wide scope. Although the emphasis is on the use of mathematical puzzles and games, the freedom and informality of the workshops allows the discussion to veer off in related directions. For example, when teachers illustrate how they fit puzzles into their teaching, their presentations sometimes spark a discussion on how they have adapted to various other aspects of the math curriculum. 

At every workshop, we promote the use of a non-competitive puzzle-based math fair to elevate the students interest in mathematics. In a math fair of this kind, the students are in charge of booths which present puzzles to passers-by. The students are not there to demonstrate the solutions, but rather to give hints and suggestions to help the visitors solve the puzzle. The whole affair is very interactive. 

A visitor to the math fair might encounter the puzzle at the top of this post, where the students provide a 4 by 4 grid and sixteen removable “chocolates” for the visitors to work with. The students presenting the puzzle would have previously solved it themselves (without the help of parents or guardians) and would have mastered and practiced at least one solution. The students would also be expected to recognize whether or not a visitor has a solution. Depending upon the grade level, students could even be expected to answer a visitor who asks “Is zero an even number?” or “Do I have to worry about the diagonals?” In other words, the students will have become experts on the evensies puzzle. 

If you have tried the evensies puzzle you will know that is not instantaneously solvable. Most people will attack the puzzle using a “trial and error” or a “guess and check” approach—this is the way I first solved it, and, in fact, it is the way most of the participants at one of the recent workshops solved it. However, there are other ways to tackle the puzzle, and asking older students to find an approach that does not depend exclusively on trial and error would be a way to ramp up the mathematics involved. Perhaps one way to invoke a discussion about this would be to ask an entire class to solve the puzzle independently and display the different solutions they have obtained, and then ask them what they they notice. 

Perhaps someone will know a way to prompt the students to ask “What do you mean when you say two solutions are the same?” This is an important question but it may be difficult to answer because understanding “sameness” depends not only the context but also upon the student’s mathematical background. For example, most people agree that the numbers 1/8 and 0.125 are the same but many will balk when told that this is also true for the numbers 1.0 and 0.9999… (where the nines go on forever).  With regard to the evensies puzzle, a grade 6 student would not be expected to understand “sameness” in the same way as a grade 12 student. 

The evensies puzzle involves parity (the properties of even and odd numbers).  Even before solving the puzzle, students may have learned a few facts like “even + even is even” and “odd + even is odd”. The puzzle offers a meaningful context for a deeper investigation. One could ask students if it is possible to remove 5 chocolates and still have an even number in each row and column. Or if it is possible to remove a certain number of chocolates and end up with an odd number in each row and column. Or one could ask students to create a similar puzzle for a 3 by 3, or a 5 by 5, or a 6 by 6 grid. 



Some teachers have come to our math fair workshops specifically because they have either heard about or visited a puzzle-based math fair. Others have come because they have already used puzzles in the classroom, and are willing to share their knowledge. And some have come because they have learned about the workshops from other teachers. Whatever the reasons, feedback from the teachers has always been very positive.  

Some mathematicians have come to the workshop because they teach a math course for Education students who, as part of their course, will be presenting a math fair to local schools. Some mathematicians come because they are interested in K-12 education and will be involved in some way with their school district. If a mathematician is fond of recreational math puzzles, and if he or she is interested in K-12 education, the workshop is a great setting to merge both interests. In addition they will almost certainly encounter some math puzzles and games that they have not seen before, and that’s always interesting. 

The feedback from the mathematicians has also been positive. Some have told me that they were impressed by the quality and enthusiasm of the teachers at the workshops. They say that observing the teachers has made them reflect on their own teaching and assessment methods, and that it has provided ideas on how to help their own students take ownership of their learning. 

For some resources and information about non-competitive puzzle-based math fairs, I would urge you to visit the SNAP math fair website. If you want more information about the math fair workshop, please feel free to contact me at tjelewis@gmail.com.