Showing posts with label problem solving. Show all posts
Showing posts with label problem solving. Show all posts

Saturday, December 22, 2012

Speed, Ratios, Unit Conversion and a Scalextric track

I was chosen ("volunteered") to work with students transitioning from year 7 to year 8 this year and needed a hands on lesson to position kids into seeing math as interactive and engaging.  With 30 students of varying levels of engagement that I didn't know well, it can be a little daunting.  In previous years I have chosen algebra or working mathematically, but for a change I chose a measurement topic this year.

My daughter Kensie has a Scalextric track (a common 1/32 slotcar racing system) and I've wondered how fast the cars actually go around the track.  I also had 10m of string, a 1.6m lump of wood and some stopwatches.

First we discussed speed itself and how it is encountered in the real world.  We used the example of travelling on the freeway.  Travelling at 120km/hr, they knew was too fast.  They knew that the value and units (speed) described how fast I was travelling.  We then discussed distance and time.  Students stated that we moved 120km if we travelled for one hour.

We then thought about how it related to our Scalextric track.  I suggested that we build a track long enough that the cars could travel for an hour.  The students then said we could go round the same track for an hour if we knew how long a lap was and then multiply the distance by the number of laps.

I gave a 1.6m ruler to the yr 10 helpers and they tried to measure the track.  The yr 7's laughed and said use the string to determine the exact length of the track.  They lined up around the track and held it in place until the string was in the slot all the way round.  They then removed the string and measured it against the 1.6m ruler.  They tended to take the ruler to the string rather than the string to the ruler which made it a bit awkward (the 1.6m ruler is quite a heavy bit of wood with measurements manually marked on).

We started the cars around the track and discovered that we didn't have enough time for the cars to travel for an hour (it was a 40min lesson) and that it was hard to keep the cars on the track for the whole time.  At the board we then looked at the speed measurement again

Firstly we converted hours to seconds

120 km per hour = 120 km per 1 hr
                           = 120 km per 60 minutes
                           = 2 km per minute (divide the distance by 60 for the distance travelled in 1 min)
                           = 2 km per 60 seconds
                           = 1 km per 30 seconds

Then we converted km to m

                           = 1000 m per 30 seconds (multiply the distance by 1000 to convert km to metres)
                           = ~33 m/s (divide the distance by 30 for the no. of metres travelled in 1 second)

By doing the reverse process we could work out the speed of the cars.

We timed the cars around the track and had a range of answers from the stopwatches timing a lap around the track.  Students suggested averaging the results.  We also discussed doing more than one lap and finding the average lap time.

This left us with a speed of 6m per 4.3s

This became 1.39m /sec and about 5km /hr (repeating the process above in reverse).

.. and no mention of 3.6 anywhere (to all you Physics heads!).  There's another lesson here for another day.


I'd like to continue this in our after school classes with my 11's and 12's for those that find related rates or kinematics difficult.

(This is the worst post for the year, drawing a lousy 3 visitors.. not sure if it is a poor idea or just the time of the year.  It's a shame as it is a good lesson.)

Saturday, May 2, 2009


This year I was asked to assist the yr 11/12 students with life skills once a fortnight. The idea is to give the students some understanding of the skills required to succeed post school. Last week was a course on memory.

I started out by asking the students to listen to 15 two and three digit numbers. I then waited ten seconds and asked them to write down as many as they could remember. The frist time varied between 4 numbers and seven numbers.

We then talked about different ways of remembering things

a) chunking (eg it is easier to remember 9456 1426 than 9 4 5 6 1 4 2 6)
b) rhyming (During the depression I felt fine, back in old '29. or creating concentration cards)
c) acronym (NATO - North Atlantic Treaty Organisation)
d) pictorial (see below)
e) Look Cover Write Check
f) multi-modal delivery (Hearing, Writing, Reading)

The pictorial one was interesting as it struck a chord with many students. I drew a picture with a guy jumping off a waterfall, a Teddy bear, some stick figures lying on the ground, a guy jumping out of a three story window, an arrow pointing down. Then I asked students to complete the picture with other images of the great depression. They could see how interesting pictures could help them remember.

We then talked about how getting information into STM was not enough, STM information decays rapidly. For information to be recalled from long term memory reliably it has to be input many times to prevent decay. We discussed that we could apply the number test to learning.. If you hear 15 points in a class but don't attempt to remember them your brain will just forget them! If you spend some time trying to learn and recall the information you will have less decay of information and better recall. Revision of the same topic multiple times over multiple days is important. (I really like Saddler's miscellaneous exercises for this in mathematics!)

Over learning was also discussed. I often say to students we go through three phases when learning.

...duh?.............I get it!.............. I know it!

When students are in the 'duh?' phase they don't have a clue and nothing makes sense. If they try, they may enter the 'I get it!' phase where they can follow the teachers and do some work independently. To reach the 'I know it!' phase they have to practice and experience a range of examples and scenarios integrating their knowledge with other areas of discipline.

Overlearning a topic comes after this when knowing when to implement skill or knowledge occurs to the point of automaticity (instant recall without thinking). This can only happen when a student learns the skill and then actively seeks deep understanding of the topic, mastering the skill to the point where they will never forget through constant practice well after the 'I know it!' phase.

Interference was discussed and how Ipods and the like can be beneficial if used to block out background noise (eg with a song that is well loved but does not require active listening) as opposed to a new song that would "interfere" with the learning process.

I then asked the student to listen to the 15 numbers again. After the ten second wait they again wrote down the numbers.

I was astounded, 5 students had all 15 numbers correct. I've run this test a number of times to test transferral of information from working memory to short term memory(STM) but never with these results.

Some clever cookies here!

Here is another article on the topic.

Alternatives to chess club

Chess club has always been a good way to get students (typically boys) to think ahead before making a decision or committing to a particular path of investigation. Unfortunately it is seen as the forefront of nerddom. With some students nerdiness is seen as a badge of pride, but students today are very socially conscious and if we seek to capture students with ability in lower years we need alternatives to foster this skill.

There are a range of alternate games, not as elegant as Chess, but have similar outcomes. The ones that I have been investigating are Caracasonne, Ticket to Ride, Portabello market, BattleLore and Small World.

The last two BattleLore by FFG and Small World by DoW seem to have the most promise as they are infinitely replayable (like Chess) but have a different level of appeal. The main issue I am having is that they require a permanent home as a game tends to take longer than 45 mins.

BattleLore is a fantasy war game that takes about 30 mins to learn and up to two hours to play. It runs through different missions and lends itself well to a leader board type scenario. The downside is that it is only played by two players at a time. This is the main factor I rejected it as a possibility for the entry point game.

Small World is different in that it has up to 5 players and takes between 40 minutes and 80 minutes to complete a game. Its humorous and requires thinking ahead and is quick to learn (less than 5 minutes)

We have created a web server and found six desktop machines. We aim to create a mathematics lab for key senior school topics. One of the kids is formatting the boxes. Maybe we could even use my personal cals for AOE or RON to increase a session size to 10-15 students!

Sunday, November 9, 2008

Developing problem solving, reading and comprehension skills

My little challenging group of year 10's can be quite difficult to engage at times. A real issue with their maths is getting them to read the question effectively. Completely out of character, they have loved exercises in the book Logic Mysteries by Jane Molnar. Although it misleadingly states grade 3-5 on the cover, the year 10 students have loved the idea of reading these problems and solving them. When I first introduced it, I abandoned the rest of my normal planned lesson as I had not seen these students this enthused and engaged since the algebra topic.

Each mystery has a story and is solved by eradicating options that do not exist. A grid is set up to record the findings as they go through the mystery.

Many great mathematical concepts can be investigated. For instance complementary events become obvious, if she has a bird - all the options in the bird column that are boys can be eradicated. Inequalities can be investigated through clues like Jane's age is less than Mary's. Sets can be investigated through concepts like Mary's item fits in a school bag... and so on..

The main thing is that it requires the students to read the clues that are not necessarily in order, requiring reading and re-reading until they are all done.

A similar book Quizzles or More Quizzles by Wayne Williams has proven very successful with my upper class of year tens. These logic puzzles are multi dimensional and can be quite difficult so be warned!

For these to be successful I invited students to attempt them themselves for 5 minutes then modelled how to complete a problem. Then the following day I gave another problem at the start of class.

Either way, improving comprehension and reading ability is more and more important in mathematics (the temptation to enter into a diatribe as to why we need to teach English in maths here is near on irresistible - I shall try though!). These three books have been some of the more enjoyable methods of developing literacy skills thus far.