space shape and measure – In Pursuit of Great Mathematics Teaching

Association of Teachers of Mathematics – Singapore Branch meeting

Last year we set up a branch of the Association of Teachers of Mathematics (ATM) in Singapore. The ATM is a UK professional organisation and I used to go to their meetings in the UK and in Hong Kong.

Last year, one of my colleagues in the Infant school decided to set up an ATM branch in Singapore. (Setting up a branch is free and astoundingly easy.) So far the ATM branch has met four times with about 50 teachers each time.

We had a meeting this week and it was brilliant to meet both primary and secondary colleagues from around Singapore to talk about maths teaching ideas. And eat cakes and fruit drink wine!

The theme for our meeting was shape and geometry teaching ideas. One good idea that was shared in the Secondary teachers chat was a gift-wrapped sphere with a note that said, “This Secret Santa present is yours if you can estimate its volume to within 50 cm2.” I made an estimate (without a calculator) but apparently I was 52 cm2 off the correct answer! Doh. I never did find out what was in that package.

I shared an idea for teaching about the surface area of a sphere using an orange that you unpeel. It’s described on this wonderful blog by William Emeny. (His blog’s name is eerily similar to my own!)

If you work in Singapore, please come along to our next meeting! We meet once a term (three times a year) and email invites are sent out.

If you don’t live in Singapore, consider joining a professional group near you. I highly recommend it for meeting new friends, hearing about other schools, and sharing ideas.

Five Superb Maths Lesson Ideas #2

1. Pythagoras and Trigonometry Revision Activity

I love activities that get students out of their seats. This task (designed by Steel1989) asks students to distinguish between Pythagoras and trig questions. Yet instead of a worksheet, the questions are designed to be printed out and stuck around the room on sheets of paper. Students get one to work on, answer it (in their book or on a mini whiteboard) and then write the answer on the back of the sheet. Then they put the sheet back up on the wall. When another student answers the same question, they check their answer with the one already written there. If the answers differ, they students need to talk to each other to discover which is correct.

pythagoras or trig

2. Polygraph Desmos Activity

Oh, wow, I’ve discovered a great one here and maybe you’ve heard the hype already. Desmos has introduced a teacher section that allows you to run class-side activities. I tired out the Polygraph: Lines activity with one of my classes. Have a look at the teacher guidance to learn more. Only you as the teacher needs to create an account; you give students a code to join the game. One student chooses a linear graph and their assigned partner has to ask yes/no questions to guess which graph it is. Meanwhile, as a teacher you can see all the questions and answer being given, who has been successful with the task (or not). I called one of my students over when I saw that she had typed “Does your graph go through the point y = 2x?”. I was able to clear up a misconception I didn’t even know she had until then.

The student’s view is shown in the screenshot below. Desmos is adding to the collection of class activities and I’m sure I’ll use them all in time!

polygraph lines.PNG

3. Tree Diagrams Challenge

A few of my year 11 students are ready to take on the challenge of those nasty tree diagrams questions that lead to quadratics. Fortunately, tonycarter45 has created this lovely sheet with probability extension questions. The sheet includes the answers.

Tony (who works at my school) has produced quite a few nice worksheets and you can see them on TES Resources. He specialises in thought-provoking questions. I like that his investigative worksheets often remove scaffolding parts as the questions progress.

tree diagrams

4. Two is the Magic Number worksheets

Three activities called “Two is the Magic Number” from Just Maths. Each one is a collection of cards solving a short problem, only two of which are done correctly. The rest show common errors and misconceptions. The cards generally cover number and algebra skills such as simplifying terms, using indices, and calculating with fractions. Depending on what you have taught your students, there may be a few topics that they haven’t learned, so check first. (My bottom set year 8 need to practice like terms, but they can’t do a conversion between meters squared and millimeters squared.) These sheets are great for checking students’ misconceptions.

two-is-the-magic-number-1

5. IB DP Maths Resource Collection

I have a former colleague, Andrew Clarke, who is a brilliant resource collector. He has now started three curated collections of maths teaching ideas for IB teachers. The one that is most relevant to me is Teaching Diploma Program Mathematics. He has collected all kinds of teaching ideas for Maths HL, SL, and Studies SL. One item that caught my eye is an investigation about using calculus to describe concavity, which is one topic I have never found a good way of introducing.

Andrew’s other two sites may interest you: Teaching MYP Maths and Teaching PYP Mathematics.

What superb lesson resources have you seen or used recently? Comment below or tweet me @mathsfeedback.

Five Superb Maths Lesson Ideas #1

1. Areas of Flags

Areas of Flags (from Owen134866 on TES Resources). One of my colleagues introduced me to this brilliant series of worksheets (and powerpoints) that use flags as a context for finding areas of rectangles, triangles, parallelograms, and trapeziums. There is also a further activity with circles.

areas of flags

2. BC Numeracy Tasks

I was browsing on the website of Peter Liljedahl from Simon Fraser University, Canada. (I was reading a paper of his about task design.) I discovered that he was on a team to develop tasks to assess students’ numeracy in British Columbia. They look as though they are lovely, well thought out tasks. However, there aren’t any solutions that I can see, likely because these are in use as assessment tasks in BC. I note that some of them are too Canadian, though! “Last week I went out crabbing with a friend. We took my canoe and paddled out to a point just off Belcarra Park and threw in our trap.” I’m not sure my city-dwelling, mostly expat students would know what to make of this. However, there are lots of great tasks here and I reckon I will try some of them out soon.

crab trap.PNG

3. GCSE Five a Day Sheets

These GCSE starter sheets, Five a Day, by Corbett Maths. Each sheet has five questions. They are available for numeracy, Foundation, and Higher, and answers are provided. One sheet for every day of the year. I have asked some of my students to use them at home on weekends, too.

4. “Think of a Number” Lesson for HCF and LCM

I’m planning to use this lovely lesson about highest common factor (HCF) and lowest common multiple (LCM) from the Mathematics Assessment Project. I like that it provides a pre-test (which could be used as homework) to help me plan the lesson. The main tasks are really well explained in the teacher notes and include a whole class discussion with mini whiteboard responses, and a card sorting activity. Then there’s a post-test to see what students have learned. All 100 of the lessons in this series are designed with a pre-test and a post-test; I love that it makes it easy to see how students have improved.

The only downside of this lesson is its American vocabulary. I am going to need to use white-out to correct greatest common factor (GCF) to HCF throughout!

hcf lcm shell map

5. Shakespeare and Numbers

Our Head of English has started talking about upcoming celebrations for the 400th anniversary of Shakespeare’s death (23 April 2016). I have been thinking about what we might do in maths to celebrate. So far I found this Numberphile video about the numbers in Shakespeare’s sonnets. I will continue hunting for some other things to use in lessons but this video (duration 4:36) will be a nice ender for lessons on that day.

What superb lesson resources have you seen or used recently? Comment below or tweet me @mathsfeedback.

Learning: Spotting Pattern Everywhere

I like this for a maths department motto: “Spotting Pattern Everywhere”. (Thanks to a former workplace for that one. I still love it.) Something I was reading today mentions that a lot of learning, not just in maths, is about spotting patterns. People love making meaning out of the things they see around them. The consciousness researcher Daniel Bor was interviewed in Time magazine talking about this. He calls the human brain “ravenous” for solving problems and making patterns. He says, “We get streams of pleasure when we find something that can really help us understand some deep pattern.”

Bor mentioned people who want answers to the question, “Why?” “The way I approach my job, it’s like trying to solve a really big fuzzy crossword puzzle and when you do put in that new clue and see the deeper pattern, that’s incredibly pleasurable.” I took this as encouragement to help students approach learning maths like a puzzle to be fit together. This could make learning more pleasurable for my students’ brains.

Bor was asked, “If our brains are hungry for information, then why do we tend to see learning as a chore and fail to recognize it as a huge source of pleasure?” He replied, “I don’t know. Obviously, more intelligent people get more pleasure from spotting these patterns, but I think almost every normal person does this. I think it’s a pretty pervasive thing but it’s almost as if we can’t notice it because it’s so pervasive.”

Ways to Make Maths Learning Like a Puzzle

1. Show a diagram and ask, What does this tell you?

I was about to teach about area and volume scale factors of similar shapes. I put this on the board and asked students to discuss what it might mean.

2. Ask students to watch you do a maths procedure in silence, then explain to their partner what it was.

Here’s a great example from the nrich website where you watch a very short video of someone summing a sequence, then students are asked to explain what just happened.

Thanks to Larry Ferlazzo for tweeting about this today and making me think more about it.

How do you make learning into a puzzle?

Circle Theorems Choice Board (A Differentiated Lesson)

I had just two 80-minute lessons and their homeworks to help my students learn about the circle theorems. So I put together this choice board for them (here’s a file for you). I had a (not very great) powerpoint with an overview of the circle theorems that they would use as a resource; you could also use a text-based resource or a video.

I told students about the first theorem, then said they were going to learn about the rest of them. They had to choose a line of three items to demonstrate their learning, horizontally, vertically, or diagonally. They could work individually or in a pair.

It seems that my students preferred the diagonal line from bottom left to top right, since many pairs picked those three items.

The treasure hunt is an activity in the pod outside my classroom with questions and answers scattered around that they have to find and follow in order. (It’s a paid-for item that my school has purchased from Mathsloops.)

The sorting cards are a collection of diagrams that need to be sorted according to which circle theorem applies, then the missing angles calculated. (Embarrassingly I can longer find the file from which I made them years ago. Here’s a card sorting activity I found online that’s a bit different but would work well here.)

2014-06-09 17.44.32

I had two great posters made by students and even had one short story. Two groups made games and one student tried a comic. Read the short story below!

The Life of a Circle

Once upon a time, in a land far away (called England) there lived a circle. This circle’s name was Kevin, his life was endless (pun intended) but his life was not as exciting as he wanted it to be. One day, when Kevin was rolling along the road he didn’t realize that there was a slope in his way. He started rolling down the hill very fast and couldn’t stop himself. “CRASH!!!” When Kevin looked up after recovering from his dizziness, he was amazed by the figure standing in front of him. He looked up to find that a beautiful tangent had stopped him from crashing into the brick wall just meters away from them. He finally spoke, asking her who she was. She replied kindly and softly saying, “My name is Tangentina Tangent, how about you?” Quickly, Kevin pulled himself together and gathered the words to tell her that his name was Kevin Circle and thanked her for saving his life. They both looked down nervously and realized that they had met at a 90° angle. Tangentine said, “You know, it’s a well known fact that circles and tangents become the very best of friends because they meet at a 90° angle. I think that this could be the start of a very very very long and wonderful friendship.” Kevin said, “Tangentina, I am very pleased to meet you but, you have to get your facts straight, it’s tangents and radii not circles. But yes, I am excited a the thought of having a new friend.” Tangentina smiled and took Kevin’s hand and they rolled and bounced off into the sunset.

THE END

(They live happily ever after)

I’m contributing this post to #made4math, a way for maths teachers to share projects in the classroom. It’s hosted by the blog Teaching Statistics.

Do you have a differentiation activity you like?

Lesson: Ferris Wheel Exam-Style Question

Sometimes I just want to remind myself and others that not every lesson has to be “special” or involve a game or video. Lessons that are successful are those where students learn to think better and experience mathematics at work. Here is one of those; it’s not flashy, just solidly successful.

It is time to review our work on trigonometric functions in my grade 11 class (IB Mathematics SL year 1). I made this document with an exam-style question closely modelled on recent exams. The question is about a Ferris wheel rotating at a constant rate.

The first page is the question, which I copied onto half sheets and gave out. These instructions were on the board.

I told my students to try to work in exam-style conditions for the first step, promising that under the black box on the slide there were other steps we were going to go through with this question. Then when it seemed like everyone had time to attack all the parts of the question, I moved the black box on the slide to reveal the next set of instructions. Students worked in pairs to create their best answers.

And finally, I gave out the mark scheme (the second page in the document) and we moved the answer papers around so every pair marked answers from someone else.

We then returned each student’s paper and they got to glue their corrected answer and the mark scheme into their notes.

This lesson mirrors the discussion idea called Think-Pair-Share and provides good exam practice. Students appreciate getting to see the mark scheme and how it applies to a their own and another’s answers.

My lessons are like this a lot of the time. I would say I teach a whizz-bang-special-game-or-video lesson once a week, or less when I’m tired. But I always try to get students talking, working together, and going deep into mathematics.

How much of the time do you teach whizz-bang-special lessons?

The Most Popular Spot in My Classroom: Who Tall Are You?

At the back of the room, near the back door, is this brilliant height chart, titled “Who Tall are You?” It’s the most popular place for students in my room. Sometimes when I go out for tea at break times, I’ll return to see some young people have slipped in and are crowding around it. Here’s a link to a good quality image of it. “I’m as tall as Beyonce!” one person squeals.

Every once in a while students will be as tall as someone they don’t know: some of the celebrities now seem a little dated (singer Charles Barkley, for instance) and others are just people my students haven’t heard of yet (such as Alexander Pope).

Since I bought it about five years ago I’m having trouble finding somewhere to buy another copy. But I was thinking today that students could make one of these for display. They could have one featuring their classmates (and teachers?) and heroes of their choice. I reckon my current classes would include more sports people than the original chart.

I’m as tall as George Clooney. If I was making a new chart I would be sure to include Jensen Button, since I am also the same height as him. If you know your height in centimetres, please leave a comment telling us “who-tall” you are!

Open-ended Question about Triangle Areas

This question is quite vague – a useful feature in my opinion! Describing the area of the triangle as “slightly smaller” means that there are many possible correct areas. Students are not intimidated and willing to try to make a triangle.

The task is also able to be differentiated to many levels. At the most basic, students can draw accurate triangles and count squares and parts of squares. Students who know how to find areas of right-angled triangles using the formula (1/2)bh can draw these. To find the final side length requires Pythagoras, though the teacher can decide not to request this. More advanced students can draw triangles and find their areas using the formula (1/2)ab sinC. Finding the third side length of these triangles requires the cosine rule.

This is great starter or ender to lessons. It can be used to consolidate or review area concepts.

Do you like vague questions?

A Pythagoras’ Theorem Open Question

An open question like this one:

is easy to pose,
is easy to understand,
can lead to lots of practice (if that’s your goal for the students),
can be kept simple or extended to suit lots of students,
can lead to an interesting discussion,
can lead to a generalisation.

I used this with a year 11 class that are revising for their Foundation IGCSE exam. I asked students to come up to the board and sketch some triangles that fit the criteria. Some triangles had 10 cm as the length of one of the shorter sides. Others had 10 cm as the length of the hypotenuse. Some students had chosen to draw isosceles triangles; other students drew scalene triangles.

This task can be made more simple by asking students to draw accurate triangles and measure the sides. Then they can extend this to checking using Pythagoras’ theorem.

On the other hand, this task can be extended by asking students to generalise what they have found. If 10 cm is length of the hypotenuse, what can be said about the other two sides? If the triangle is isosceles, there is only one answer. But if it is scalene, perhaps students will call the length of the second side x. Then they can come up with a formula for the third side in terms of x. And how does this change if 10 cm is not the length of the hypotenuse but of one of the other sides?

Giving Options to Students

I have been reading More Good Questions by Marian Small. One differentiation strategy she recommends is to give students 2 options. The questions are related and the follow-up discussion engages students who did either option. For the question above, we talked together after a few minutes.

What is the area of each of the full circles in your picture?
How can you test that the answer is reasonable?

I set these cheese questions for some year 11 students who are revising for their IGCSE Foundation exam. After a few minutes we talked about the two answers using these questions.

Which measurements did you use?
Did you look up any formula?
What units did you use for your final answer?

These two questions are available here.