Ratio with Bar Modelling

I currently run support classes for numeracy for years 4, 6, 8, 9 and 11.

Many of the students I see struggle to remember techniques, particularly when they are seemingly abstract.

Two main bits of inspiration brought me to re-teach ratio through bar modelling

(1) Concrete - Pictorial - Abstract
Research suggests students grasps concepts best if initially they can "play" with objects, then sketch it pictorially, before moving to the abstract technique.

(2) White Rose Maths Hub
With so many examples of bar modelling on their twitter feed @WRMathsHub, I visited their website and found a few bar modelling resources, here.

Ratio with Bar Modelling
Within the first 5 minutes the students appeared confident and comfortable with the new technique, and commented that it was much easier for them to understand what was going on.

Here is a little of their finished work:

These questions in their template can be found here.

What is the Question?

I'm trying to foster problem solving early with my year 7 class... here's what we've been trying for decimals.

Inspired by the Rio Olympic 100m swimming finals, they had about 6 standard questions to give some scaffolding first, such as "how much faster did Pieter swim 100m that Simone?"

Then we moved on to the outstanding questions below (questions are normally split up into good, excellent, outstanding/mastery).

Estimation

While tutoring a new students last week, I was surprised, but perhaps shouldn't have been, that they didn't have any appreciation for how much a gram was, or a kilogram, or how many pens would fit in my pencil case. Time for some research of how to best to engage and educate... I came across the game above and although it's a leap too far right now, it could be easily adapted to a range of objects that the student would be more aware of, and perhaps even things they could round up in the home after doing the task to check their answer.

Meanwhile for whole classes as well as tutoring, Andrew Stadel has created a fantastic website called Estimation 180 where you input your estimation into a google form, can see other peoples' responses and also get an answer, for all those that require a sense of resolution. In his blog he suggests using it as a start of lesson activity with students recording their guess and confidence level in the back of their book.

Inspired by Dyslexia Series: A Fraction

To me, a fraction is a specified number of pieces, where the pieces are equal segments of a whole (I'll probably revise this many times during my career). 

So the equal segments part of the meaning is represented by this:

But to show the specified number of pieces part, I feel a video might be required. Also the picture, does not define but does imply, that a fraction must be less than one. Hmm.

An Insight from Canada

During the Easter Department week, we were graced by John Mason, who shared a wealth of ideas and thoughts with us on attention - here are just some of the few I managed to jot down.

Our first task of the morning was this:

We all initially considered it individually, then discussed our thought processes as a group. One person started looking at the overview first before focusing down, another looked for a particular corner first before the rest of the shape, and another was mesmorised by a certain part of the diagram!

When we were given the answer, it was noted that a variety of emotions: delighted / satisfied / depressed / cheated were felt, depending upon how far through the problem a person had worked.

Parking

One of the key ideas of the session was to develop pupils' ability to park an idea; when they had a good idea, temporarily put it to one side and see if there is a more efficient way of doing it, rather than just ploughing on through.

Just

Having said I'd aim not to use this word, I realise I've written it in the sentence above. "Just" implies a simple step, in the eyes of the speaker, but it may not be so in the eye of the listener, so should be avoided.

If a student is attending to a problem in a different way, they may not be able to hear what you're saying, unless you give sufficient detail to show how you are attending to it prior to any further development.

Group Discussion

Throughout this session, John elicited ideas from us and exemplified classroom techniques. The two I have taken away from this are:

Please

In PDP sessions in my S1 school, we were advised to use thank you rather than please as a behavioural technique. However, although this is effective in terms of showing pupils that a behavioural pattern is expected, I had almost forgotten where please could be used to equal effect/

As John called upon us to offer our opinion, he said please, inviting us to talk and demonstrating that he was interested in our comments.

I think I want to modify my conjecture

A helpful alternative to "bummer, that doesn't make any sense"

Say What You See (SWUC)

Possibly stolen from catchphrase this was really helpful in allowing pupils of all abilities to participate, such that everyone can succeed - it lowers the threshold. This was implemented with the image below.

Update 22/05/2013: Used this with a Year 9 class who weren't particularly vocal but came out with some good ideas, particularly when we allowed the large blue block to be worth £300 and determined how much each of the other blocks would be worth.

However, one comment: "I notice it's a fraction wall cos I did it in Year 8" was followed by further discussion, where the student in question knew it was a fraction wall but couldn't tell me any fraction that the wall could represent. Perhaps not so memorable, or not enough discussion in Year 8?

A Selection of Activities

Without time to comment on further, here are some of the other ideas that appeared in the session...

Bimbles, Gloops and Flobs

Today's OUDE session was spent at Oxford Spires Academy, which has a high EAL population and hence was our focus for the day. The day consisted of a mixture of observations, EAL discussions sessions and a debrief with Anne.

Nonsense Questions
By far one of the most useful parts of the day for me was the idea of nonsense questions. In simplest form, all non-essential mathematical language is replaced with random words, such as:

A question like this puts native speakers and EAL students on a level playing field, where no-one knows what bimbles, gloops or flobs are, but should be able to do the mathematical calculations required to answer the question.

A further extension to this might be to give them the original question and then highlight the words that were unimportant to the understanding of the question.

Fractions
We were advised to use the Oxford Primary Maths Dictionary for EAL students, as it has useful pictures and descriptions of mathematical key words, e.g.

Fractions: the bottom part (denominator) tells you the number of equal parts. The top part (numerator) tells you the number of those part you are dealing with.

However, this definition made no logical sense when considering 8 over 5 for example, or when doing algebraic fractions. Instead, the division definition might be more appropriate:

On a slight slide note, one of the teachers at the school found a memorable way to introduce top heavy fractions, with a little help from Katie Price...

Jordan Fractions: memorable top heavy fractions

Interesting Teaching...
Although not directly related to EAL, I saw a few examples of interesting teaching today as well.

Speed, Distance Time Graphs
Rather than asking the pupils to answer a list of question on SDT graphs, the teacher instead presented them with graphs and asked the pupils to tell their story.
I give the following as a pupil example:

"I was on my way to Nottingham to see my friend Georgia. It took me 3 hours to get there and my speed was 60mph. I then arrived and spent 1 hour having a poo. Then I took Georgia's dog for a walk at a speed of 5mph. My average speed for the whole day was 38mph."

Units and Measure
In preparation for lessons on unit and measure, pupils were asked to answer the following questions:

• How much coca-cola is in a can?
• How far is it from Oxford to London?
• How tall are you?
• How long is your fingernail?
• How much do you weight? 

Misconceptions

By listening for what you want you don't hear what they say.

Read What is a Fraction. Ask what the denominator is telling us.