Pattern sniffing with Decimal Subtraction

An idea for a mixed attainment class that came to me about 5 minutes before a lesson today:

  1. 3.4 – 3.04
  2. 5.2 – 5.02
  3. 7.8 – 7.08
  4. 8.2 – 8.02

Find other questions like this.  (The “weakest” student in the class told me the pattern before I’d even finished writing the fourth question on the board.)

What do you notice?  Why is the answer to Q2 the same as the answer to Q4?

Can you create a question with 0.54 as an answer?  How many different answers are there to these types of questions?

Then:

  1. 5.7 – 5.007
  2. 8.3 – 8.003
  3. 6.4 – 6.004
  4. etc.

These are more tricky and test the skills of column subtraction, something that should be secure by Year 7 but may not be. Maybe an opportunity for collaboration amongst students to show how.

And then finally, try these two calculations. Which is easier and why?

7 – 1.392

6.999 – 1.391

Show on a number line why this works and then try some more.  I think these questions are interesting to explore.  But I would hesitate to recommend it as a must-do method to solve e.g 8-2.5687.  Whether or not it is easier to turn it into 7.999-2.5686 or not is an interesting discussion and one which I would want my students to form their own opinion on, not be too swayed by mine.

 

 

 

The angle defines the ratio of side lengths in a right angle triangle

This post shows how to use Geogebra to demonstrate this fundamental truth in geometry and hopefully demystify Trigonometry to a certain extent.

As with all things Geogebra, I always try to start with a blank sheet (see other posts on this here and here).  This time, I’m not using the Geogebra app itself but just launching it from within a Chrome browser window which works pretty well.

1.gif

Once it is launched, I right click in the middle to remove the axes, but I am going to leave the grid on.

Then I create the triangle by constructing a line, a perpendicular line…

2.gif

…and a third point which I then join to create a triangle using the polygon tool.

3.gif

Next, measure the base angle of the triangle remembering the convention that angles are measured in an anti-clockwise direction.

4

The next bit is a tad fiddly. You need to right click on the line segment to change the label to “value”.  Then do the same for the other two sides of the triangle so that you now have one angle and all three side labelled.

5.gif

So far, this has taken about 2 minutes to create from a blank screen.  You could do it in advance of the lesson, but I think it is worth doing it in front of the class, maybe having practiced it a couple of times.  Using “something I created earlier” is less powerful – it looks like some sort of trick, somehow.

Now you have everything set up you can start moving the points as shown here.

6

I start by moving point B, thus keeping the angle fixed.  I would ensure students have calculators in front of them and ask them to calculate opposite divided by adjacent. Then move the triangle to get different values for side lengths. Then do the calculation again. The answer is the same, of course.  I might ask them how they could get that directly from the angle (tan angle).  Depending on where the discussion goes with that, I might then move on to look at sin and cos.

Finally, I always like to talk about how things were done in the old days, being careful to point out that I’m not that old and that I didn’t actually use these…crc_trig_tables.jpg

I explain that the sin button on your calculator is basically just looking up the values in the sin column of a table like this – not actually true, I know, but it helps understand what’s going on so that’s OK for me!

 

 

How do we report and measure attainment in Maths? And Why?

An interesting discussion in the Maths office this week has led to some musings.  When the English education system moved away from Levels, my school took the new GCSE (9-1) grades and “translated” the old Teacher Assessed Levels (TALs) into new Teacher Assessed Numbers (TANs) as shown here on the school website:

conversionchart785

As I teacher I am required to provide a TAN on each student I teach at various points in the year. As I was doing this earlier this week I started thinking about what these are used for.  Here I am not talking about the actual process of assessment. I am just thinking about why we collect that information and what we do with it. It seems to me that there are 3 key recipients of this information each with their own agenda.

1, The Students.  This is a form of feedback, albeit a very blunt, summative piece of feedback that basically tells the student how good they are at Maths (in my case) summed up in a single score. The student may be aware of their previous score(s) so they may also get a measure of their progress.  They may also discuss it with their peers so get a sense of their relative position in their year group.  But mainly, it is a single measure that tells them where they are at today.  Students used to have a good understanding of what Level 4, 5, 6, etc. meant.  In fact I still hear of Year 8s and even 7s asking what level they are. It wasn’t that long ago.  Inevitably they will take time to get used to a new scoring system. In our case it is linked to 9-1 GCSE grades. But the key difference of course is that these TANs are their teacher’s opinion (based on summative end of unit assessments) on how they are doing, rather than the external impersonal authority of the exam board.

Due to the simplicity of the summative score this feedback doesn’t actually tell the student how to improve other than “work more” or maybe even “work less” depending on that student’s disposition and level of ambition towards that subject.

2, The Parents.  All parents want to know how their child is doing at school. Through parents’ evenings and other contacts, we can provide much more nuanced information on this progress.  But I believe most parents like the clarity of some sort of score. The score needs to be understood in context, e.g. in our case it looks like a GCSE grade but it is not any sort of prediction, well not until KS4 anyway.  The question for me is what do parents then do with this information? Obviously a full range of responses exist here from nothing at all, to deciding to get a tutor in and putting additional pressure on the student to work harder in whatever way they see fit. Even though the parents may do nothing, the mere fact that the TAN is shared with the parents is likely to have some impact on the students’ engagement with school, positive or negative.

3, The school leadership. By having a regular school-wide, “score” for each student per subject the school can do all sorts of analysis of the attainment and progress of their student base.  What the school then do with this information is myriad: e.g. decide on classes/sets, plan intervention including deployment of support staff, provide support to teachers, evaluate teachers as well as track overall school improvement.  The data may be shared with Ofsted although my understanding is that this is not statutory.  These are pretty wide-ranging but basically boil down to helping the school focus their efforts and resources in the right place.

It strikes me that these are 3 quite different and potentially conflicting sets of objectives. For example the school may wish to collect data that is useful for analysing whole school performance but is not relevant or motivating to individual students. (A Twitter conversation here with @LaSalleEd highlights how their MathsAge system shares specific content objectives with the student, but calculates an overall score solely for school use)

The dynamic between students and parents varies as children get older.  I believe there is a case for parents of primary children having information that their children don’t see, but as pupils approach GCSE they need a realistic view of what they are aiming for which can prove an incentive to work hard.

Lots of questions, not many answers, I’m afraid. I would like to understand more about what other schools do. I understand many have adapted the old levels system by basically changing the scale but didn’t see a need to make a broader change to reporting.

Please leave comments below or get in touch on Twitter, @mhorley.  Thanks!

 

Ideas for better maths teaching