Scatter Graphs

This weblog is becoming a fleck statistics-heavy, which is ironic given that I don’t peculiarly similar instruction statistics at Key Stage three in addition to 4.  I much prefer instruction algebra. I’m also quite dandy on surds in addition to indices. And I similar trigonometry. In fact, apart from the strange annoying-to-teach topic similar loci in addition to constructions, I’d seat most information treatment topics correct at the bottom of my ‘things I relish teaching’ list. It never fails to amaze me how long roughly teenagers accept to plot a graph. What a faff.

Anyway, this ship service is all nigh ideas for instruction scatter graphs. And I intend to follow it upward amongst a ship service looking at applications of box in addition to whisker plots exterior the maths classroom.  But in addition to hence I’ll halt amongst all the statistics.

Practical activities

Unusually for a maths lesson, there's opportunities for roughly practical activities when nosotros innovate scatter graphs. The obvious watch is to receive got students plot pinnacle or manus bridge against shoe size. If you lot desire to acquire fifty-fifty to a greater extent than practical, endeavor a human scatter graph inwards the classroom or inwards the playground (using chalk axes).

Outliers 

I similar to purpose the ii scatter graphs shown below (stolen from a Ten Ticks worksheet) inwards a class give-and-take nigh outliers.  Ask your class what could receive got caused these extreme values in addition to you'll in all probability acquire roughly actually creative suggestions!

Line of Best Fit

I learn my students to depict a employment of best check past times trying to minimise the distances betwixt all the points in addition to the line. That’s what a to the lowest degree squares regression does. Contrary to pop belief, in that place doesn't receive got to live the same seat out of points to a higher house in addition to below the line. Nor does the employment receive got to become through whatsoever of the points. And the employment for certain doesn't receive got to become through the origin. Beware of dodgy methods taught inwards other subjects. This misconceptions activity past times frickard on TES is helpful - I've used this inwards a 'hands on caput if you lot retrieve it's true, hands on hips if you lot retrieve it's false' activity.

Impossible Predictions

'Extrapolation' from regression lines is something that comes upward at Influenza A virus subtype H5N1 level, simply the watch tin brand for interesting discussions at Key Stage three or GCSE. The cardinal shout out for is that fifty-fifty if nosotros bring out a linear human relationship inwards the information we're examining, nosotros can't assume that human relationship extends beyond the attain of the data. Here's a prissy instance for class discussion:

The graph below shows a negative correlation betwixt yr in addition to fourth dimension taken past times athletes to run 100m. It suggests that the speed of 100m runners has increased over time. Influenza A virus subtype H5N1 employment of best check has been drawn. Why produce you lot retrieve 100m runners are getting faster? Do you lot retrieve this tendency volition continue?

Possible reasons for the increment inwards speed include: advances inwards technology (ie to a greater extent than lightweight or streamlined wearing clothing in addition to trainers), query into optimal diets in addition to grooming plans, sponsorship important athletes receive got to a greater extent than fourth dimension to train, the introduction of starting blocks inwards the 1940s in addition to an improvement inwards the accuracy of timing the race.

If the tendency continues indefinitely (ie if nosotros depict the employment of best check into the future) in addition to hence at roughly shout out for people would live able to run 100m inwards null seconds, which is impossible. There must live a shout out for at which the fourth dimension volition plateau, due to limits inwards humans' physical capabilities. Although in that place is a linear correlation, nosotros must live careful non to brand predictions inwards the past times or time to come because the linear human relationship is over a express fourth dimension period.

The graph comes from this Minitab Blog post in addition to the watch was seat into my caput past times Kings College London PGCE tutor Chris Olley. 

"Wind is caused past times the rotation of windmills"

I love the causation vs correlation discussion.  It's a really of import learning shout out for for students who volition come upward across all sorts of dodgy statistics inwards everyday life.

doubt.it

Influenza A virus subtype H5N1 typical instance is shoe size in addition to reading age. The ii variables are positively correlated, simply apparently the size of someone's feet has no comport upon on their reading ability. For to a greater extent than ideas depository fiscal establishment check out this spurious correlations website.

If you're planning a class give-and-take on causation in addition to hence receive got a quick read of 'Correlation does non imply causation'on Wikipedia (not e'er the best source simply gives roughly skillful examples).

Correlation does non imply causation, simply it does waggle its eyebrows suggestively in addition to gesture furtively land mouthing "look over there". – xkcd 

Resources

Finally, here's a few resources that you lot could purpose inwards your lessons:

• I love this Bee Aware action from Don Steward. It could easily live made into a mini-project/display move nigh the drive in addition to comport upon of the demise of dearest bees.
• The fantabulous Mathematics Assessment Project gives us ii tasks - Scatter Diagram and Birds' Eggs
• There's no shortage of real-life applications of scatter graphs. This worksheet 'Shake It Up amongst Scatterplots' is i of many examples (here's the solutions).
• Nrich gives us this nice David in addition to Goliath activity.
• 'Foldables' seem to live a big affair inwards the US math lessons, simply I've never seen them inwards the UK. I retrieve my students would quite similar them. Here's an watch for correlation foldables from the weblog 'Everybody is a Genius'.
• I similar this online tool for checking the accuracy of lines of best check from Mark Ritchings
• Here's an interesting group project. I similar the watch of giving each pair/group ii options hence they tin pick out the variables they prefer, in addition to that everyone inwards the class is working on something different. Also, the 'assigned tasks' for each grouping fellow member (see lastly page) is an watch I haven't seen earlier for organising grouping work. These item tasks may non live suitable for a U.K. lesson because nosotros don't await students to calculate the equations of lines of best fit, simply the watch could live adapted.

I promise this ship service has been helpful.  Please comment if you lot receive got whatsoever ideas to share. 

By the way, this cool moving scatter graph is worth a look.

xkcd.com

5 Maths Gems #39

Hello as well as welcome to my 39^th gems post. This is where I percentage 5 maths teaching ideas I've seen on Twitter. This volition last my concluding post of the summertime holidays - I've been a busy blogger this summertime merely lesson planning takes priority this weekend. We all know how manic things larn at schoolhouse inwards September.

1. Bojagi
I'm excited close this. Dave Radcliffe (Gems 35) merely he intends to alter the motion painting every calendar week or fortnight, for illustration to display information close revision session, star students as well as thence on.

Five Things You Lot Mightiness Non Know Almost The Novel Gcse Content #2

I lately wrote a post about v changes to the content of maths GCSE. In today's ship service I listing v to a greater extent than changes that may non move widely known.

1. Invariance

When didactics shape transformations to Higher Tier students, you'll instantly demand to ensure that students are move able to pose invariant points. This is described inwards the specification every bit follows: "Describe the changes as well as invariance achieved past times combinations of rotations, reflections as well as translations".

We are given this instance enquiry from AQA:

"Write downwards the coordinates of a dot that is invariant when it is reflected inwards the business y = x".

Here, students volition demand to know that the points (1,1), (2,2) etc prevarication on the business y = x as well as would so move unchanged if reflected inwards this line.

There's a laid of resources for this topic inwards the Mind the Gap Maths Toolbox and Peter Mattock has shared a resources 'Invariance activeness sheet'.

2. Geometric Sequences
When I saw that geometric sequences instantly appear on the GCSE specification I assumed I'd move making usage of my AS score (C2) resources. This is non the case.

In C2 nosotros detect the nth term of a geometric sequence using the formula U_n = ar^n-1.  This gives us a 'position-to-term' rule. For instance if nosotros bring the sequence

3,  6,  12,  24,  48, ... 

so nosotros bring a = three (the start term) as well as r = 2 (the mutual ratio), so the nth term is U_n = 3(2^n-1).

Although geometric sequences create come upwards up at GCSE, there's 2 points from the specification that are worth noting when didactics this content:

1. Geometric sequences volition move inwards the cast rⁿ , for example:

2,  4,  8,  16,  ...  

3,  9,  27,  81, ...

√5,  5,  5√5,  25,  ...  (Higher Tier only)

2. Students are non required to detect expressions for the nth term of these sequences. They alone bring to create that for linear as well as quadratic sequences.

Examples of novel GCSE questions include:

The nth term of a sequence is (√3)ⁿ. What is the 5^th term of this sequence?

and...

U_n = 2U_n-1     U₁ = 2     Write downwards the start 4 price of this sequence.

In both examples students are using an nth term formula but non deriving the formula themselves. So when you lot learn this topic I propose that you lot focus on:

• recognising geometric sequences
• finding a mutual ratio as well as using this to proceed a sequence
• substituting into both position-to-term as well as term-to-term rules (including those using subscript notation). 

If you've spotted this enquiry inwards OCR's Practice Paper 5 you lot may intend that it contradicts what I've written hither most geometric sequences:

Here is a sequence.

2,  2√7,  14,  14√7,  ... 

a) Work out the adjacent term   (1)

b) Find the nth term   (3)

c) Find the value of the 21st term divided past times the 17th term.   (2)

I asked OCR for clarification as well as they sent a helpful reply:

"...the Assessment Objectives around making deductions, inferences as well as work solving hateful that some questions may involve taking known elements of content as well as taking them that fleck further, every bit seen inwards the question. Given that students at Higher tier bring to know how to detect the formula for the nth term of a quadratic sequence as well as too to 'recognise as well as usage sequences of... geometric progressions (r^n where n is a... surd) as well as other sequences', the enquiry is an instance of how higher powerfulness students powerfulness move expected to brand a deduction from known content. I don't await this to move a enquiry we'd regularly examine inwards alive assessment, but is the kind of affair we'd desire to include inwards sample assessment to hand an indication of how the content as well as Assessment Objectives are brought together inwards writing questions."

3. Scatter Graphs

There's a fleck to a greater extent than to scatter graphs now, though some teachers may bring already been roofing these things. Students volition demand to know:

• that correlation does non necessarily signal causation
• the deviation betwixt interpolation as well as extrapolation, as well as the dangers of extrapolation.

Although I volition usage the words interpolate as well as extrapolate inwards class, I incertitude these words volition move used inwards exams. Questions are probable to move inwards the cast of this instance from Edexcel:

You should non usage a business of best gibe to predict the let out of units of electricity used for heating when the exterior temperature is 30°C. Give 1 argue why. 

Students may too bring to pose an outlier from a scatter graph (note that this is informal identification of outliers - ie past times eye) as well as determine whether to ignore it when drawing a business of best fit.

The Scatter Graphs: True or False activeness from MathsPad covers all of this content as well as is absolutely excellent.

Don't forget that some statistics content has been removed from the novel GCSE, such every bit questionnaires as well as stratified sampling. It's worth looking at these previous post).

Also, inwards my ship service most real life graphs, I said that I was surprised to run into sketching cubics inwards my novel GCSE textbook - including identifying roots from factorised expressions.

Cubic graphs from Edexcel GCSE (9 - 1) Mathematics: Higher Student Book 

This is what the specification says most cubic graphs:

recognise, sketch as well as translate graphs of ... simple cubic functions ...

This is on both the Foundation as well as Higher Tier. The AQA Teaching Guidance provides farther details, stating that students should move able to:

draw, sketch, recognise as well as translate graphs of the cast y = x³ + k where k is an integer

There's for certain no advert of the type of cubic graphs that characteristic inwards my textbook, as well as I intend this is 1 of the things that volition likely move removed from subsequent editions.

A few to a greater extent than clarifications
I asked OCR what they are oftentimes asked most the novel GCSE. In improver to questions regarding grading as well as didactics time, they are often asked to clarify the following:

• whether students volition move expected to differentiate to detect the slope of a crease (no, they volition bring to approximate slope from a tangent at a point, or potentially a chord betwixt 2 suitable points) - run into my blog post most this. 
• whether Foundation Tier students are required to calculate turning points of quadratic graphs (no, at Foundation they’ll only demand to read from the graph. At Higher they volition detect turning points past times completing the square).
• if Higher Tier students bring to know most the equations for whatever circle (no, alone those centred around the origin).

This is actually helpful data from OCR.

So that's it - I promise you're instantly feeling to a greater extent than informed most the novel GCSE content. Do allow me know your thoughts as well as questions.