# Mathematics

## Nick's Classes

April 2, 2019

In this pointing & clicking and copying & pasting world we're losing the ability to construct logical arguments, whether they be written or oral. Without the continual practice with fine motor skills, that writing by hand once provided, students are now struggling with forming written characters. In number intensive courses such as math and physics this is a real problem, where word processing is not helpful. (If you've ever used an equation editor to write mathematical solutions you'll know what I'm talking about.) Our ability to form logical mathematical arguments is closely tied to our ability to write mathematical solutions. Without the ability to write a neat and well organized solution we adversely affect our ability to form logical arguments. Don't get me wrong...I'm not advocating for the end of all word processed work...our students have become much more critical thinkers, able to digest copious amounts of information. They are also much more aware of the world they live in and thus more socially conscious. I've also seen an improvement in their ability to think, "outside the box". However, all of these added abilities aren't much use if the students lack the ability to argue a point in a logical way.

May 17, 2018

Peter Taylor from Queens University gave a lesson on surfaces, functions of two variables, to the Calculus class at Delphi today. The surprising result of walking a straight path of y = x +/- b, on the surface below, is that z, the height above the x-y plane, changes linearly with respect to x...like walking up a ladder.

March 13, 2018

Peter has written an article for an Education Journal that deals with Dispelling Myths in Math Education. In the last paragraph he describes his experience in which he participated in our lesson on frequencies and the 12 note scale. I've attached it here. Give it a read.

February 25, 2018

The game that Peter Taylor introduced to the grade 9s was a modification of the game of Skunk. The game involves probability and lead to us mathematically developing a strategy for success. Essentially the game involves standing while a die is rolled. Unless a 1 is rolled then the number adds to your score. If a 1 is rolled then your accumulated score drops to 0. Players must decide to continue to stand and risk losing their points for that round or sit to hold on to the points for that round. The game is played 10 times and the winner has the highest accumulated score. Ultimately, there's a critical total beyond which on average it makes more sense to sit...and this critical value we were able to calculate. If you want to know what it is, you'll have to ask one of our grade 9s.

With the 11/12 Physics class we discussed and investigated how we distinguish between different frequencies of sound...multiplicatively rather than additively. We were able to demonstrate with the class that our ears are able to discern frequencies that differ by as little as half a percent...this is significantly better than any other sense! We also looked at how sounds/notes combine to create consonance or dissonance and then showed how the musical scale could be developed with just a couple constraints...one of those constraints being that as long as the notes that combine have small integer frequencies ratios then those notes will sound good together. It turns out that of the reasonably possible choices for numbers of notes, the 12 notes scale has the most instances of good combinations.

Thanks again, Peter, for taking time to engage with our students. You are welcome anytime.

February 21, 2018

The Google Classroom code for 9 Academic Math has changed to mbh6py.

Also...Peter Taylor from Queens University will be joining us again this Friday. He'll be in all my Friday classes. He's planned a probability game with the 9 Academic and split 9/10P Math classes. As well, he'll be working with me in the split 11/12 Physics class where we'll be looking at the mathematics and physics of music. Any students who don't have class last two periods and have an interest in music and math are welcome to attend.

February 13, 2018

From today I will be using Google Classroom to communicate evaluation dates and post assignments, as well as their due dates for all the courses I teach. All this information will go straight to your google calendar. Basic course info will remain on the Delphi course websites, including course outlines.

Here are the codes for each of the classes I teach this semester:

9 Academic Math: mbh6py this code is new....use it to gain access starting Feb 21.

9 Applied Math: ch2q1v

10 Applied Math: 4172q67

11 Physics: cfcppp

12 Physics: 92hayb

12 Calculus: yxd802

February 7, 2018

As a new semester is now underway I figure it's an important time to talk about achievement. Your achievement in the courses you are taking will have a direct impact on your level of achievement in any courses that follow. Although 50% is considered a pass and earns you the credit, probability of success in a course that follows, when your previous mark was below 60%, tends to be quite low. Reviewing course material or re-doing the pre-requisite course will increase your chance of success when you have just scraped by. To that end...it's up to you to do the best that you can now. Make use of all the resources at your disposal such as remediation work, extra help from your teacher or your classmates and online resources. Complete all your assignments and don't wait till the last minute to start. Starting early gives you the opportunity to get extra help before the work is due. All your teachers will be able to make time for you as long as you request help more than a day before the work is due. And remember...ask questions...someone else likely has the same question as you. When you ask the question, the answer or explanation has greater permanence for you.

November 1, 2017

Peter Taylor(Queens University) came to Delphi a couple days ago to watch me work with mathematics problems he's generated for high school students. Click here for more info on Peter's 9-12 Math Project. Grades 10, 11 and 12 were involved. The grade 12 problem involved modeling the position of a marker on a small wheel as it rotated around a wheel with twice the radius. Models were tested using Desmos online graphing calculator. The key to solving this problem involved recognizing that the smaller wheel actually rotates about its own centre, 3 times as fast as the centre of the small wheel rotates around the bigger wheel. Yes...that's a lot to digest...it helps to sketch the problem then establish equivalent distances along the arcs of both circles. Essentially, these wheels are like gears, so there's no slippage. Tracking the position of the circular marker on the smaller circle involved creating an equation for that circular marker. It's centre position is determined by summing the distances from the centre of the big to the smaller circle with the distances from the smaller circle to the circular marker. Here's the Desmos animation of the solution.

October 2, 2017

Global Math Week starts next Tuesday, October 10. Our school is now one of thousands from around the world that will be participating. To kick-off this event Sunil Singh will be giving a presentation at Delphi on Wednesday this week at noon. Sunil taught secondary mathematics in Toronto before he got involved with the Global Math Project. Sunil is also the author of the recently published book, "Pi Of Life". Click for more info about Sunil.

September 14, 2017

Please be patient with network access this time of the year as changes for the better are in progress. Also, the tool that is used to update school websites has changed. As such, websites may not display as intended. I'm currently learning the new software and hope to be up-to-speed in a week or two.

January 11, 2017

In 12 physics we've been studying the motion of charged particles in electric fields. Today, specifically, we were looking at the maintenance of the resting membrane potential of a cell. The potential voltage difference across a nerve cell membrane is 70mV, with the inside being negative with respect to the outside. Essentially the cell membrane acts like two parallel oppositely charged plates...the inner part of the membrane being negative and the outer, positive. Active pumping of 3 sodium ions(Na+) out of the cell and 2 potassium ions(K+) into the cell contributes to this negative potential. The active pumping has to be done because each ion has a concentration gradient that favours the opposite direction. A question we were examining today involved the mechanism by which an injection of potassium chloride is lethal. A solution of potassium chloride(K+ and Cl- ions) is used to euthanize animals, but also for capital punishment in some US states.

Essentially, the exposure of the nerve cell to K+ and Cl- ions must cause the membrane potential to become either more or less polarized. If more polarized then it would become more difficult for the nerve cell to generate an action potential(send a signal) as it would be more difficult for the action potential threshold to be reached. If less polarized then no new action potentials could be generated or transmitted. This latter result seems more reasonable...exposing the nerve cell to excess potassium could change the concentration gradient for potassium, thus preventing the slow leak of potassium from the cell, causing an excess of potassium, and thus positive charge, to build up in the cell. This could have the effect of raising the resting potential from -70mV to 0 or more. However, exposing the nerve cell to excess chloride ion could force those negative ions into the cell, down their concentration gradient. This could make the cell membrane more polar...hmmm...two reasonable possibilities...more research is needed. If you'd like to weigh in on this issue just send me an email(nick.nielsen@tdsb.on.ca).

Ultimately, it's pretty cool to think of the fact that our bodies operate electrically at the cellular level...after-all... electricity is just the movement of electrons.

In 11 math we've been studying exponential functions. Today our focus was exponential growth and financial mathematics. I brought to class a credit card(Visa) statement to take a close look at the information provided. On the statement an estimate is provided for the time to pay the balance if only the Minimum Payment($10) is made. This estimate, however, does NOT take into account the fact that Visa charges almost 20% per year(20%/12 per month) on the entire monthly balance when the balance is not paid in full. So...for a statement balance of about $1500 the estimated time to pay when paying on the minimum payment is indicated as about 12 years. At $10 per month for 12 years that's $1440...almost $1500. However, interest on $1500 for one month is about $25. That's $15 more than the minimum payment. So...making a minimum payment of only $10 means you're not even covering the interest. The result is your balance owing will grow every month. In fact, in 10 years your statement balance, provided you stopped using the card, would be about $7000! Many people continue to use their card, making only the minimum payment. Imagine what they might owe after only a few years if on average they spend about $1500 per month. Yikes.....

What's to be learned? If you need to use a credit card then use only if you can pay the entire balance before the statement is due each month. Making the Minimum Payment will NOT enable you to pay off the debt. Also, paying any fraction of the balance will result in added interest on the entire balance, not just what remained after your payment. For example: one month I made a mistake with my payment and was short a couple dollars. The next month's statement indicated that I owed interest on the entire balance from the previous month, even though I had paid more than 99% of the balance.

March 30, 2016

Current, as it relates to electricity, always seems to be a difficult concept for students to grasp. Physics textbooks provide a bunch of questions all related to the basic equation that relates current to charge(Q) and time, but very little emphasis is placed on what's actually happening in a wire as electrons move through it. Today with my Grade 11 Physics students, I decided to explore this in detail. The question I gave the students was, "How many electrons would be in 1 mm of copper wire that had a cross-sectional area of 1 mm^2. And how would this number compare to the amount of electrons moving in a 15Amp circuit. As most students who take physics, also take chemistry, the students needed to use what they learned in chemistry about atomic mass, molar mass, number of electrons and density to solve this problem. We then compared the amount of electrons in that short section of copper wire to the amount of electrons per second that would be represented by a current of 15A…the standard household circuit. When comparing these two results it was obvious that electrons must be moving quite slowly through a circuit, as in 1 mm of wire there are about 150 times more electrons than those needed for a current of 15A. That translated to a speed of a about 1/150 mm/s or 0.007 cm/s. The elementary charge is really small(1.6x10^-19), but when you've got around 2.5x10^21 electrons in 1 mm of wire, all those charges add up.

December 2015

Just finished another 2.5 hour tutoring session with a friend's son who's taking 12 chemistry. His experience in high school is similar to other students' experiences that I'm aware of…the learning takes a back seat to the curriculum. There is so much pressure for the teacher to cover the entire curriculum that little to no time is spent actually engaging the students in the learning. You'd think by now most people would understand the pitfalls of delivering a curriculum that is a mile wide and an inch thick. It's my belief that students will always be better prepared when the teacher is able to focus on the learning. Learning to understand has more permanence, relevance and value than learning to remember.

March 2015

It really bothers me that most of what gets taught in high school math is rarely seen again by most students and hardly, if ever, appreciated. The fact that it's not appreciated is more a function of the material and less a function of the teacher. For the most part, they are doing their best with material that has relevance only to those who plan to continue mathematics at the post secondary level. What's ironic is that more students are likely turned away from studying post-secondary mathematics because of the material they are fed in high school. The material(curriculum) needs to draw students in, not turn them away. An appreciation for the beauty of mathematics and the mathematical beauty of the natural world needs to be the focus of the curriculum. With that as the guiding principal I think more students would see their high school math experience as a positive one. More on this topic to come…stay tuned.

October 2014

Just a thought, but I wonder why in elementary school they don't teach the same rule for dividing fractions as for multiplying?

When multiplying fractions we multiply numerators together to get a new numerator and we multiply denominators together to get a new denominator.

When dividing fractions we divide numerators together to get a new numerator and we divide denominators together to get a new denominator.....try it!

Of course, if the denominators are different, then we may end up still with two fractions dividing. However, if denominators are the same then we'll just end up with the fraction in the numerator divided by 1. If, however, the denominator in the top fraction is an integer multiple of the denominator in the bottom fraction then we'll end up with a fraction divided by a number other than 1.

Of course, all that you need to do to avoid this is to find a common denominator before you divide. I guess that's a bit more work than just multiplying the numerator by the reciprocal of the denominator. :-P 2013-2014