Fraction Talk Compilations from has a wonderful website devoted to visual fraction puzzles that build proportional reasoning while embedding in geometric shapes, real life and whimsy. Here are some teasers. Their website have explanations on how to use them as individual fractions talks. These three pages are designed to lure teachers into seeing their power to increase fractional understanding and logical and geometric thinking.

Enjoy the pdf  Link:      Triangle Fraction Setups

Also follow on twitter  @FractionTalks & the author @NatBanting

And check out their websites:     and


Visualizing As A Tool For Building Number Sense

Below are the materials that I utilized at the Northwest Math Conference 2015 in Whistler. In the presentation, we looked at various aspects of visualization and how they empower a greater sense of number, problem solving ability and enhances communication and reasoning. Student visualizations give us access into their thinking which reveal their strength of grasp or the misconceptions that are interfering with their progress.

Visualizing as a number sense tool


Visualizing Divisibility Tests to Strengthen Numeracy

Visualizing Divisibility to Strengthen Numeracy

By Fred Harwood, SFU grad student, educational consultant, and (semi)-retired math educator

I am currently studying elementary and middle school numeracy at SFU. As a recently retired secondary math teacher, these studies give me new perspectives to look at how students develop their mathematical understanding. In secondary schools, we often lose sight of the underpinnings of numeracy and utilize patterns or rote memorization.

Marc Garneau provided me with a beautiful perspective that was very empowering. In a 2014 NCTM conference presentation he showed how the divisibility test for nine could be visualized. Picture the number 3213. For decades I taught the test of adding the digits to see if the total is divisible by 3 or 9 as this is what works. Here 3+2+1+3 = 9 so both 3 and 9 would divide into 3213. Marc said to look at the place values of 1000’s, 100’s, 10’s and 1’s.


You can think of 10x10x10 cubes, 10×10 squares, 10 strips and leftover 1’s. One less than any power of 10 is obviously divisible by 9.

Taking the one cube away from the 1000 cubes leaves 999 cubes. If students don’t see this is divisible by 9 then look at the fact there are 99 tens and 1 nine. Taking one from each of the 99 tens give us 99 nines and the 99 ones can be made into 11 sets of 9. This gives us 111 sets of 9.

If we take 1 piece from each massed set we would end up with groups that are all products of 9. In our number we would see three sets of 999, two sets of 99, one set of 9 and then all the leftover pieces loosely gathered that total 9 pieces. Every set of pieces is divisible by 9 so the whole number 3213 is also divisible by 9.

In the example 64 236, we could take 6 singles, one from each of the ten-thousands, a single from each of the 4 thousands, 2 singles from the hundreds, 3 singles from the tens and amass them with the 6 leftover pieces. There are now six sets of 9999, four sets of 999, two sets of 99, three sets of 9 and 6 + 4 + 2 + 3 + 6 = 21 left over pieces. All the complete sets are divisible by 9 and also 3 except for the 21 leftovers. Therefore 64 236 is not divisible by 9 but would be divisible by 3.

Now consider other divisibility tests. Powers of 10 are divisible by 2, 5 and 10. For example, 100 is 50 twos, 20 fives or 10 tens. Since every place value is based on a power of 10 then every place above the ones place is already divisible by 2, 5 and 10. Only the leftover 1’s place is left to check for divisibility. In 2 352, the two thousands, three hundreds and five tens are each divisible by 2, 5 and 10. So we check the two leftover 1’s. Only 2 goes into this so 5 and 10 won’t go into 2 352 but 2 will.

Which is numerically more powerful: Even numbers end in 0, 2, 4, 6, or 8 or seeing these numbers as one 2, two 2’s, three 2’s, four 2’s or five 2’s by rearranging little blocks?

Testing for 4 or 25 (or 100) we can have students memorize looking at the last two digits because 4 is 2 squared and 25 is 5 squared OR we can have them visualize that all powers of 10 greater than one are divisible by 4 and 25. For example, thousands and hundreds are both divisible by 4 and 25 so we only need to check the tens and ones to see if they too are divisible.

Eleven has many divisibility tests. Thinking about what numbers close to a power of 10 is divisible by 11, we see that 11 is 1 more than 10. 99 is one less than 100, 1 001 is one more than 1 000 and 9 999 is one less than 10 000 etc. Therefore the scenario alternates between having a place value needing one more for each holder or one less. Taking 12 837 as an example, we need to subtract 1 from the 10000, add 2 ones o the two thousands, subtract eight 1’s from the hundreds, add three 1’s to the tens to make each of these groups divisible by 11. If we were using flip chip colouring, we would have 1 + 8 + 7 whites and 2 + 3 reds making a total of 11 once the zero pairs are deducted. The alternating of digits positive and negative and summing to test for 11 is now obvious that it works. 872 146 822 would have 8 + 2 + 4 + 8 + 2 of one colour and 7 + 1 + 6 + 2 of the other. 24 – 16 = 8 so this big number could not be divisible by 11.

Having student teams develop their own tests for divisibility using their understanding of powers of 10 place values would be very powerful and involve much good mathematical reasoning together. Manipulatives could aid this reasoning and visualizing for numbers under 10 000. Given an existing divisibility test, trying to prove why they work instead of just demonstrating that they do would also be a valuable number theory activity for students from intermediate to post-secondary.


Why I do discovery based learning

A journal reflection inspired by Lockhart’s Lament, Stewart’s Letters to a Young Mathematician, and Van de Walle’s What Does It Mean To Do Mathematics in response to a listserve thread:

A math professor writes regarding why we should be teaching the long division algorithm:

… there is a difference between being convinced, and grasping the validity of a proof.  And it is a critical one.  Allow me to illustrate again, with something from the WNCP framework, which from a mathematical perspective is quite alarming.

The outcome: Gr 7 NS 4

  1. Demonstrate an understanding of the relationship between positive repeating decimals and positive fractions, and positive terminating decimals and positive fractions.

The Achievement Indicators:

Predict the decimal representation of a given fraction using patterns,

e.g., 1/11=0.09 both repeating        2/11=0.18 both repeating       3/11=?…

Match a given set of fractions to their decimal representations.

Sort a given set of fractions as repeating or terminating decimals.

Express a given fraction as a terminating or repeating decimal.

Express a given repeating decimal as a fraction.

Express a given terminating decimal as a fraction.

Provide an example where the decimal representation of a fraction is an approximation of its exact value.

Now, the associated textbook materials (MMS and Math Focus) I’ve seen address this outcome are hardly better than this.  So let’s just assume that one develops a lesson based largely on what is shown here.  What does it show?

Students are to develop a BELIEF that fractions (of whole numbers) always give repeating decimals.

What is the basis of that belief?  “using patterns”.   Where do these patterns come from?

Well, they don’t come from long division — because long division (and any specific precursors necessary for its proper instruction) does not appear anywhere in WNCP Math, so it cannot be assumed.

This is a serious problem for many reasons but, specifically to this point, it is the very reason why decimal expansions of rational numbers are repeating.  When we cover geometric series in our calculus and analysis classes I often use repeating decimals as an example, and as an aside I’ll ask the class if they know why this works.  . . .

It does not take long to get to 1/17.  If you don’t know the outcome to that one, pick up a calculator and try it.  Unless your calculator has a screen that displays over 30 digits you will not see a “repeating decimal.   And neither will the bright child in the class”.

My Response:  It sure doesn’t take long since I can do it in my head now. Below is how I (and many of my students) arrived with this ability.

No, you won’t see a repeating pattern for one division on a calculator. However, if a cluster (I call the process chunking) were put up at one time . . . for them to explore what their calculator was telling them and to then look for patterns over the families of fraction conversions and to connect the patterns between these families and to formulate hypothesis and to test them with other fraction families and then to publish/share their discoveries so they can discuss with other students what they are seeing and to re-hypothesize and to re-test and to pose questions and to . . . THEN they are becoming mathematicians who believe in the dynamic nature of mathematics and they believe that they can contribute to growing that body of mathematics. Yes there is incredible elegance in the masters and the beauty of their algorithms, concepts and proofs but so many of them are dead. Mathematics is so much more than what my grandparents or parents learned. I don’t want to just copy them. I want to discover things anew.

I will wager that long division using secondary students would never have discovered that there are two-digit fibonacci patterns in the 18 digit repeating period of the 19th fractions? I bet they didn’t notice that every digit except for 0 & 9 are repeated twice in the pattern and thus can be used to predict accurately any 19th fraction’s 18 digit pattern. I’ll further wager that they would never notice that you can double any 19th fraction by just moving the 18th digit of the period to the front of the period (after discounting any whole number effect). NOW that they believe the patterns exist and are helpful, would they ask why it works? [How ready are these students for Modular mathematics????]

Would they have seen 19 goes into 100 five times with a remainder of 5 and that the repeating period is an expansion of 19 going into 1.000…. 0.05r5 and by multiplying by 5 they could generate the 18 digit pattern by doing the multiplication repeatedly and incorporating the carry overs

0.05 25 ¹25 625 ³¹25 . . . note the sequence is 5, 25, 125, 625, 3125 etc then carry over the hundreds & thousands.         1/19 = 0.0526315789473681 all repeating                                                                               

Now this pattern was discovered after a student noticed that, in the 1/17th fraction, it began with 0588 and four times this is 2352 and if you keep multiplying by 4 (incorporate any carry-overs) you generate the 1/17th 16-digit period whose first half adds to the second half to make eight 9s. 1/17 = 0.05882352 94117647 repeating and that all the other 17th fractions use the same digits in the same order but obviously start with one of these 16 digits. His discovery led to the students questioning if there were other similar patterns in other repeating periods. They had already seen that 1/7 was 14 doubled to give 14, 28, 56, 112, 224, 448, 896, 1792 etc. and by overlapping these carryovers outside of the two digits that they generated the 0.142857 repeating period for 1/7 (and this might have triggered Novid for discovering the 4 digit x4 pattern for the 17th period [entitled the Super Novid]). What followed was a flurry of discoveries with each morning 4 or 5 new ones being submitted by excited students. Two different students wrote programs to help with the testing and verifying of these discoveries. One was a long division program that gave a new set of 20 digits on the display for every enter key. The other allowed for the entry of any seed set of digits and then to apply a multiplier.

Into this climate of discovery, the algorithm was introduced on a scaffold of understanding to a group of students completely ready to see that a comparison of their discoveries for an over-riding pattern meant that they could generate any decimal number from a fraction by considering what the remainder was for dividing it into any power of 10. Eg. since 7 goes into 100 fourteen times with a remainder of 2 then 7 goes into 1.00 –> 0.14 and apply the remainder 2 as a multiplier.

The Super Novid: 17 goes into 10000 –>588 r 4 so 17 into 1.0000 goes 0.0588 and apply 4 as the multiplier

Many of these discoveries are shown on my website’s right side (New Discoveries link)

Most math teachers will not be able to instill the passion for the history of math that you can since students read our authenticity completely. Liston’s Lure of Learning, Stewart’s Letters to a Young Mathematician and Lockhart’s A Mathematician’s Lament all speak to instilling a passion and sense of wonder into our teaching. Chunked discovery, patterns, question posing, testing, proofing and disproving, creating their own algorithms are more likely to instill this passion in students because it will be obviously instilling itself into the teacher at the same time.

If 1/89 utilizes a Fibonacci pattern of 0.011235 8 (13) but carry over the 1 etc. to create the 44 digit repeating period and1/19 utilizes a two digit seed of 0.05 26 31 57 88 (145) but carry over the 1 etc. to create the 18 digit period then what fraction uses a three digit Fibonacci pattern? Hmmm???

Now I should have been in bed by now or I should have been writing a reflective journal response to Lockhart’s Lament and Letters to a Young Mathematician for my grad class tomorrow. I think I just did do an adequate job of the reflective journal by revealing the passion created in me by my student’s (and my own) discoveries using the very methods you decried in your last post.

I am thoroughly enjoying the exchange. Discovery based and problem based learning have kept me young and growing so that I retired from teaching to take another Masters degree.

We both want the same outcome in students . . . for them to be blessed with the wonder, awe and beauty but we also want them to be so much more than procedural users of someone else’s algorithm.

PS – One of my bright grade 9’s invented integral calculus to solve the question of why was it 4/3rds in the volume of a sphere formula. His learning was always anchored in discovery and problem based learning and he believed that there must be a way to figure it out. He didn’t end up a mathematician but he is one of Canada’s leading doctors of internal medicine.



Retired Richmond BC math educator and current SFU student

“Teaching/learning are not my job, they are my passion! Getting better at them, that’s my job.” edited from a Jose Popoff quote

“Give someone a fish, you feed them for a day. Teach someone to fish, you’ll feed them for a lifetime. Teach someone to love fishing, you’ll feed whole generations!”

“Teach with passion!”