If Solving Simultaneous Equations with Matrices is the Headache…

I really struggle to think of catchy headings – so I stole this one from Dan Meyer!

My Year 10’s are currently learning Matrices – operations, determinants, inverse matrices, etc. Because this group of students “loves” forming groups and solving problems, I decided to flip the lesson (Japanese style) and start with a problem.

To give the reader some context, 3 out of 7 groups had solved the “Tourists & Guides” problem in about 30 minutes, which I found on Mike Lawler’s blog.

tourists and guides

So I wanted to give my students a big headache, before giving them the aspirin or perhaps paracetamol? Here is the problem:

Three students go into a shop and make purchases. Katie buys 3 packets of chips, 2 cans of drink and a chocolate bar and pays. $10.65. Mark buys 4 packets of chips, a can of drink and 2 chocolate bars and pays $12.60, and Tony buys 3 packets of chips and 3 chocolate bars and pays $10.95. Determine how much each item costs.

Within minutes I had a buzz of noise as students began the problem. From the students point of view:

  1. The problem did not look too difficult – money, chips, drinks and chocolate.
  2. It is real world – students often buy these items.
  3. A simple strategy – trial and error – can be quickly employed.

After 30 minutes, I started to hear groans and comments like “we are only 5 cents off”. After 45 minutes, many students gave up and went off task. At this stage I promised to show them how to use an Inverse Matrix (learned previous lesson) to quickly solve the problem (with the use of a ti-Nspire CAS calculator). Aspirin time!

aspirin-for-primary-prevention-clopidogrel-meta-analysis

Student B however, was determined to solve the problem on his own – which he did in a seperate room, taking about 90 minutes in total.

Notes:

  1. I don’t plan on giving students many problems that they can’t solve in future!
  2. Next lesson we will solve with algebra.
Posted in Pedagogy, Problem Solving, Teaching Ideas | Tagged | Leave a comment

Equations of Tangents and Normals

My Year 12 Maths Methods students are now applying their differentiation skills to applications. Student D. has requested a worked solution to this question:

For the function y = loge (2x+3) , find the equations of the tangent and normal at the point on the graph where x = 0.

Tangent

My students often ask why the text answer is not in the form {y=mx+c}. I explain that they adopt the convention of writing solutions without fractions or negative numbers where possible. I personally prefer the standard y=mx+c and my students do to.

Posted in Pedagogy, Uncategorized | Tagged , , , | Leave a comment

Resources ….

spirit duplicator

As a young first year teacher in 1980, I had many things to learn.

I started searching for and making resources to help my students learn. The smell of methylated spirits from using a spirit duplicator is still a vivid memory.

Then I had to decide how to file these valuable resources for future use?file-cabinet

My filing cabinet soon was overflowing with worksheets, tests, homework, etc. But finding that fractions test master could sometimes take a while. Ten minutes before 8C Maths and half the contents of my filing cabinet would be scattered all over the office floor!

GlobalInternetThe invention of the computer and photo-copier has sure made teacher’s preparation a lot easier. We can now collaborate and share with teachers all over the world. Teaching blogs and websites now provide teachers with a vast amount of resources.

One of my new favourites is “Resourceaholic” by Jo Morgan. She labels her posts “Maths Gems” and I highly recommend that you investigate her website here.

Jo Morgan

Upon checking the hard drive on my nearly worn out Lenovo L420 laptop, I find that I have over 28000 files in my ~MATHEMATICS folder!

Mathematics Directory

 To find a particular file, the search function is invaluable. However, I decided a logical folder naming system was important too. Here is a snapshot of it:

Maths Folders

I am also experimenting with using “Notebook” to organise my resources. How do you keep all your Maths teaching documents organised and easily retrieved?

Posted in Personal, Uncategorized, Worksheets | Tagged , , , | Leave a comment

Emotional Blackmail Backfires

I tried a new strategy to try to motivate my Year 9 Maths class. It went like this:

1. Friday morning. I meet my year 9’s at the door. As usual there are stragglers.

2. I tell the class that I am going to show them a video. “It is not directly about Mathematics. We will discuss it after the video finishes.”

3. Video: Anna Clendening sings Hallelujah on America’s got talent. (Latecomers are told to wait outside until the video finishes)

Anna Clendening

4. I wipe a tear from my eye. “Please give me a chance to recover – this video always makes me cry!”

5. Class discussion initiated: “This video always makes me very emotional – who knows why?”

6. Trent says “Because you are soft Mister!” (backfire)

mister-softee

7. Class erupts in laughter and I am lost for words but manage a smile.

8. Finally, I get myself under control. I pause and look each student in the eyes. I say (in a quiet, serious voice) ” I get emotional because Anna gave her best performance ever, after fighting depression and anxiety. As a teacher, I want you, my students, to give your best performance in our classes. I want you to be the best you can be.”

9. The students are quiet as I hand out the days work (basic numeracy skills) and ask them for their best work.

Posted in Pedagogy, Personal, Teaching Ideas | Tagged , | 1 Comment

Spaced Practice and Mathematics

There is plenty of support for the notion that spaced practice will help consolidate information into long term memory. For example John Hattie’s work where it rspam5anks 12th in his table of effect sizes with a value of 0.71.

 

So perhaps unlike the food product shown, Spaced Practice and Mathematics is good for students!

 

So based on this belief, I have begun to make up some S.P.A.M. sheets aligned with the Australian Curriculum.

You can download 10.01 here.

Please leave a comment about the usefulness or otherwise of this type of assessment.

Posted in Assessment, Teaching Ideas, Worksheets | Tagged , , , , | Leave a comment

SWPB and Yu-Gi-Oh Cards

Schools are generally much more positive environments than they used to be. Our school community (including students, teachers, parents, principal class, cleaners and visitors) has many opportunities for positive, constructive interaction.

6:1

will-press-lever-for-food1Our aim is a ratio of six positives to every one negative. Of course the verbal comment giving recognition to effort, work process and product, and good work habits are still our mainstay. But we now add to this postcards sent home and raffle tickets.

This token economy system uses operant conditioning to reward positive behaviour on the spot with a special stamped raffle ticket. Students may redeem these at the end of the week at a shop stocked with various goods.

Yu-Gi-Oh!

I wondered whether students would see “Dungeons and Dragons” type trading cards as positive rewards. Z became very excited telling about a card game called Yu-Gi-Oh. He said he would bring his cards in next lesson. I talked to a younger Maths teacher who knew all about the game and told me you could make your own Yu-Gi-Oh cards online. Here are some of the cards I made:

01 Accuracy Des Koala

Student T. had just finished five scale drawings and had measured lengths and angles accurately. I announced to the whole class that T. had been awarded the “Accuracy Des Koala” Yu-Gi-Oh card. T. put Des Koala proudly on her work desk with a big smile.

07 Goblin Worker

Student S. always works hard in class, asks questions, uses good manners and has excellent work habits. I announced to the class that S. had earned “Goblin Worker” I again got a positive reaction from the class and the Year 10 student concerned.

04 Hippopotenuse

Student J. had finally mastered identifying the sides of a right triangle. He was pleased to get the “Hippopotenuse” card.

Some brainstorming in the Maths staffroom resulted in the following card which we hope a lot of students will want because of its design, power, and the difficult? challenge to get it!

02 Mathematician

It is early days in trialling this reward system, but feedback from students has been great so far. Some of the students want to design their own cards. Other students have stories they want to tell about trades, games, best cards, etc.

I enjoy creating the cards. I hope they encourage students to improve their Maths skills and work habits. The cards I have designed so far are:

01  Accuracy Des Koala                   08 Homework Hedgehog

02  Mathematician                           09  LOL Cat

03  SohCahToa                                   10  OnDemand Test Magician

04  Hippopotenuse                          11  OnDemand Penguin Soldier

05  Fraction Learner                        12  Correction Des Kangaroo

06  Power Spell Maker                   13  Grapha, Dragon Lord of Dark World

07  Goblin Worker                            14  Punctual Possum

What do you think of this idea? What methods of positive reinforcement do you use?

 

Posted in Games, Teaching Ideas | Tagged , , , , , , , , | Leave a comment

A Short History of Mathematics Teaching and Learning Part 1

As I progress through my 36th year of teaching, I can’t help but reflect on the many changes that have taken place. As a secondary student I did not have a calculator until Year 12. And so I depended on:

Kaye and Laby, 1968. Four-Figure Mathematical Tables. Longman, Aust.

Columns and columns of logarithms, anti-logarithms, sines, cosines, tangents, reciprocals, squares, cubes and the standard normal distribution.

old math tools

My first calculator was a Novus Scientific (Mathematician) with LED numerals, a 9V battery and Reverse Polish Notation. After 40 years it is still in good working order!

Until Year 12 though I did pages and pages of calculations using logarithms and antilogs. My Year 10 Maths folder (1971) as shown above was meticulously set out.

I show my students these items in the hope that they will also do their best and make the most of their learning opportunities.

Posted in History of Mathematics, Personal, Show and Tell | Tagged , , | 1 Comment

Teachers should take risks

risk_sml

The following journal article:

Risk Taking: A Distinguishing Factor of Good versus Great Teachers
by Gayle A. Brazeau, PhD, finishes with the statement:

“Perhaps risk taking is what in the end distinguishes a good teacher from a great teacher”.

With this encouragement, I decided to take a risk at the end of the 2013 schoolReed Gillespie year. I ran my idea past our assistant principal who, like Mr. Gillespie, put his trust in me.

What was the risk you might ask?

A radical new lesson plan?

Outdoor maths?

An excursion to the local racetrack to investigate probability and gambling?

No, No, No. Bringing my 5 year old, chocolate brown Labradoodle (Gary) into my Friday afternoon Year 8 Mathematics class!

Hey, give me a break, I hadn’t gone completely insane? After all, there is some good examples of dogs used in the classroom. Dogs in the classroom can be used to calm fears, relieve anxiety and teach skills. For example, Morgan:

“A high school student sat at a work table, feeling extremely upset. Sensing the students anxiety, Morgan, a 3-1/2-year-old certified therapy dog, wandered over to her and put his head on her lap. After a little while, she felt better and was able to focus on learning again.

Morgan is a cross between a golden retriever and a poodle — a Goldendoodle. His silvery-black coat is more hypo-allergenic than the coats of most other dogs, which makes him a good choice to work with students at Palmyra-Macedon High School in Palmyra, New York. He is one of seven therapy dogs in this district of about 2,200 students.

Golden Doodles for Sale

When students show signs of stress or anger, Morgan joins them and helps them calm down.

Morgan gravitates toward kids who are loaners, and he really brightens their day.

He’s quite a dog, he really is, said Jim Blankenberg, Morgans owner. Morgan has the ability to sense stress, and he goes to the stressed-out student. He’s like a Teddy Bear students can hug”.

THE PLAN

usain-bolt-200-p1Introducing and getting to know Gary. Incorporate Gary into a lesson on measuring and converting speed. Essential question:

“Can Gary beat Usain Bolt in a 100m sprint”.

 

WHAT HAPPENED

GaryAll except one of the year 8 kids loved Gary. He got lots of pats and wagged his tail furiously. We took him out to the oval and marked out 100m with a trundle wheel. One student assigned to hold Gary at the start line. Another student ready to time Gary using their smart phone. I told Gary to stay and walked to the finish line. Gary sat patiently waiting for the “COME” command. Good dog! I yelled “COME GARY” with the usual hand actions.

Gary spent the first 3 to 5 seconds frolicking with his handlers before racing to me. Time: 15 seconds. Failure. Perhaps I should have used some bait?

Any comments? Have you ever seen or heard of dogs used in a classroom or school?

Posted in Personal, Show and Tell, Uncategorized | Tagged , , , , | 1 Comment

Game On!

I thought I had read about or played most games during the 20942 days I have lived (online age calculator). And then today I discovered ULTIMATE TIC-TAC-TOE at “Math With Bad Drawings“. Here is a picture of a completed game:

ultimate tictactoe

The rules are:

1. The first player may place an “X” in any cell within any mini-square on the board.
2. The selected cell position within this mini-square corresponds to the mini-square position within the greater-square where the second player must then place an “O”.
3. Thereafter, the two players take turns placing their mark in any unfilled cell within the mini-square dictated by the cell position marked by the previous player. For the first player, this mini-square will be outlined in red.
4. The first tic-tac-toe winner in a mini-square remains the winner in that mini-square for the remainder of the game.
5. If a player is sent to a mini-square that has already been won, or in which all the cells are already filled, then the player may next place his mark in any unfilled cell in any other mini-board.

It got me thinking about whether their is a high correlation between good chess playing and being a good Mathematician.

The evidence and support for this idea is easily found on the web. If you are interested check out Edutech Chess: Why Chess?

Many countries now include chess in their school curriculum. These include Brazil, China, Venezuela, Italy, Israel, Russia, Greece and Armenia.

Not convinced? Then I suggest you visit Grand Master Susan Polgar’s “Get Smart Through Chess” website.

Susan_Polgar

Make sure that you also watch the National Geographic video about her titled “My Brilliant Brain”. The video claims that Susan is the living proof that any child can be turned into a genius. It explains some of the differences between the male and female brain. The roles of visualisation and memory are also explained.

Susan also has free for download on her site:

chess training guide

 

So what is my opinion on playing chess in the Mathematics classroom? A definite yes – the research indicates yes, my intuition says yes, my students say yes, many of my teaching colleagues (eg. Adrian Camm) say yes too. But I would add, do not just limit games to chess!

Posted in Games, Visualisation | Tagged , , | Leave a comment

Garden Design

Create smallI love watching students get absorbed in a creative design task. As a Maths teacher this is satisfying for a number of reasons:

1. A rich task naturally differentiates; their are multiple entry and exit points

2. Creating is on the top of Bloom’s taxonomy; higher level thinking is taking place.

3. Their is plenty of discussion and questioning, much of it student initiated.

4. Applied Maths tasks reinforce that Mathematics is a very useful subject.

My year 8’s are currently completing their “Garden Designs”. It wasn’t too long ago that I was doing the same thing in my own garden. I decided to create an octagonal flower bed. Here it is shown in the photo below. Do you like it?

garden

A copy of my “Garden Design” assignment can be downloaded here.

Posted in Applications, Teaching Ideas, Worksheets | Tagged , , , , , , | Leave a comment