Scatter Graphs

This weblog is becoming a fleck statistics-heavy, which is ironic given that I don’t specially similar pedagogy statistics at Key Stage iii as well as 4.  I much prefer pedagogy algebra. I’m also quite peachy on surds as well as indices. And I similar trigonometry. In fact, apart from the strange annoying-to-teach topic similar loci as well as 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 receive got to plot a graph. What a faff.

Anyway, this postal service is all most ideas for pedagogy scatter graphs. And I retrieve to follow it upwards alongside a postal service looking at applications of box as well as whisker plots exterior the maths classroom.  But as well as thus I’ll halt alongside all the statistics.

Practical activities

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

Outliers 

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

Line of Best Fit

I instruct my students to depict a trouble of best stand upwards for yesteryear trying to minimise the distances betwixt all the points as well as the line. That’s what a to the lowest degree squares regression does. Contrary to pop belief, at that spot doesn't receive got to live the same release of points to a higher house as well as below the line. Nor does the trouble receive got to become through whatsoever of the points. And the trouble sure enough doesn't receive got to become through the origin. Beware of dodgy methods taught inward other subjects. This misconceptions activity yesteryear frickard on TES is helpful - I've used this inward 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 upwards at Influenza A virus subtype H5N1 level, only the sentiment tin terminate brand for interesting discussions at Key Stage iii or GCSE. The telephone substitution signal is that fifty-fifty if nosotros detect a linear human relationship inward the information we're examining, nosotros can't assume that human relationship extends beyond the make of the data. Here's a dainty instance for shape discussion:

The graph below shows a negative correlation betwixt twelvemonth as well as fourth dimension taken yesteryear athletes to run 100m. It suggests that the speed of 100m runners has increased over time. Influenza A virus subtype H5N1 trouble of best stand upwards for 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 inward speed include: advances inward technology (ie to a greater extent than lightweight or streamlined clothe as well as trainers), inquiry into optimal diets as well as grooming plans, sponsorship important athletes receive got to a greater extent than fourth dimension to train, the introduction of starting blocks inward the 1940s as well as an improvement inward the accuracy of timing the race.

If the tendency continues indefinitely (ie if nosotros depict the trouble of best stand upwards for into the future) as well as thus at roughly signal people would live able to run 100m inward null seconds, which is impossible. There must live a signal at which the fourth dimension volition plateau, due to limits inward humans' physical capabilities. Although at that spot is a linear correlation, nosotros must live careful non to brand predictions inward the yesteryear or futurity because the linear human relationship is over a express fourth dimension period.

The graph comes from this Minitab Blog post as well as the sentiment was seat into my caput yesteryear Kings College London PGCE tutor Chris Olley. 

"Wind is caused yesteryear the rotation of windmills"

I love the causation vs correlation discussion.  It's a real of import learning signal for students who volition come upwards across all sorts of dodgy statistics inward everyday life.

doubt.it

Influenza A virus subtype H5N1 typical instance is shoe size as well as reading age. The ii variables are positively correlated, only apparently the size of someone's feet has no demeanor on on their reading ability. For to a greater extent than ideas banking concern check out this spurious correlations website.

If you're planning a shape give-and-take on causation as well as thus receive got a quick read of 'Correlation does non imply causation'on Wikipedia (not e'er the best source only gives roughly skillful examples).

Correlation does non imply causation, only it does waggle its eyebrows suggestively as well as gesture furtively spell mouthing "look over there". – xkcd 

Resources

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

• I love this Bee Aware action from Don Steward. It could easily live made into a mini-project/display move most the campaign as well as demeanor on of the demise of dear bees.
• The first-class 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 alongside Scatterplots' is 1 of many examples (here's the solutions).
• Nrich gives us this nice David as well as Goliath activity.
• 'Foldables' seem to live a big matter inward U.S.A. math lessons, only I've never seen them inward the UK. I retrieve my students would quite similar them. Here's an sentiment for correlation foldables from the weblog 'Everybody is a Genius'.
• I similar this online tool for checking the accuracy of lines of best stand upwards for from Mark Ritchings
• Here's an interesting group project. I similar the sentiment of giving each pair/group ii options thus they tin terminate select the variables they prefer, as well as that everyone inward the shape is working on something different. Also, the 'assigned tasks' for each grouping fellow member (see in conclusion page) is an sentiment I haven't seen earlier for organising grouping work. These detail tasks may non live suitable for a United Kingdom of Great Britain as well as Northern Ireland of Britain as well as Northern Republic of Ireland lesson because nosotros don't hold off students to calculate the equations of lines of best fit, only the sentiment could live adapted.

I promise this postal 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 part 5 maths teaching ideas I've seen on Twitter. This volition survive my final post of the summertime holidays - I've been a busy blogger this summertime simply lesson planning takes priority this weekend. We all know how manic things larn at schoolhouse inwards September.

1. Bojagi
I'm excited virtually this. Dave Radcliffe (Gems 35) simply he intends to alter the moving-picture present every calendar week or fortnight, for event to display information virtually revision session, star students as well as hence on.

Five Things You Lot Mightiness Non Know Well-Nigh The Novel Gcse Content #2

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

1. Invariance

When pedagogy shape transformations to Higher Tier students, you'll straightaway demand to ensure that students are travel able to seat invariant points. This is described inward the specification equally follows: "Describe the changes in addition to invariance achieved past times combinations of rotations, reflections in addition to translations".

We are given this instance inquiry from AQA:

"Write downward the coordinates of a dot that is invariant when it is reflected inward the line of piece of work y = x".

Here, students volition demand to know that the points (1,1), (2,2) etc prevarication on the line of piece of work y = x in addition to would thence travel unchanged if reflected inward this line.

There's a laid of resources for this topic inward 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 straightaway appear on the GCSE specification I assumed I'd travel making exercise of my AS marking (C2) resources. This is non the case.

In C2 nosotros uncovering 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 accept the sequence

3,  6,  12,  24,  48, ... 

so nosotros accept a = iii (the showtime term) in addition to r = 2 (the mutual ratio), so the nth term is U_n = 3(2^n-1).

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

1. Geometric sequences volition travel inward the shape 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 uncovering expressions for the nth term of these sequences. They solely accept to produce that for linear in addition to 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 downward the showtime 4 damage 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 advise that you lot focus on:

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

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

Here is a sequence.

2,  2√7,  14,  14√7,  ... 

a) Work out the side past times side 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 in addition to they sent a helpful reply:

"...the Assessment Objectives around making deductions, inferences in addition to work solving hateful that some questions may involve taking known elements of content in addition to taking them that chip further, equally seen inward the question. Given that students at Higher tier accept to know how to uncovering the formula for the nth term of a quadratic sequence in addition to equally good to 'recognise in addition to exercise sequences of... geometric progressions (r^n where n is a... surd) in addition to other sequences', the inquiry is an instance of how higher powerfulness students mightiness travel expected to brand a deduction from known content. I don't await this to travel a inquiry we'd regularly evidence inward alive assessment, but is the form of matter we'd desire to include inward sample assessment to hand an indication of how the content in addition to Assessment Objectives are brought together inward writing questions."

3. Scatter Graphs

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

• that correlation does non necessarily shout out for causation
• the departure betwixt interpolation in addition to extrapolation, in addition to the dangers of extrapolation.

Although I volition exercise the words interpolate in addition to extrapolate inward class, I incertitude these words volition travel used inward exams. Questions are probable to travel inward the shape of this instance from Edexcel:

You should non exercise a line of piece of work of best jibe to predict the seat out of units of electricity used for heating when the exterior temperature is 30°C. Give 1 argue why. 

Students may equally good accept to seat an outlier from a scatter graph (note that this is informal identification of outliers - ie past times eye) in addition to create upward one's heed whether to ignore it when drawing a line of piece of work of best fit.

The Scatter Graphs: True or False activeness from MathsPad covers all of this content in addition to is absolutely excellent.

Don't forget that some statistics content has been removed from the novel GCSE, such equally questionnaires in addition to stratified sampling. It's worth looking at these previous post).

Also, inward my postal service close real life graphs, I said that I was surprised to come across sketching cubics inward 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 close cubic graphs:

recognise, sketch in addition to translate graphs of ... simple cubic functions ...

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

draw, sketch, recognise in addition to translate graphs of the shape y = x³ + k where k is an integer

There's sure no cite of the type of cubic graphs that characteristic inward my textbook, in addition to I intend this is 1 of the things that volition in all probability travel removed from subsequent editions.

A few to a greater extent than clarifications
I asked OCR what they are often asked close the novel GCSE. In add-on to questions regarding grading in addition to pedagogy time, they are often asked to clarify the following:

• whether students volition travel expected to differentiate to uncovering the slope of a crease (no, they volition accept to guess slope from a tangent at a point, or potentially a chord betwixt 2 suitable points) - come across my blog post close this. 
• whether Foundation Tier students are required to calculate turning points of quadratic graphs (no, at Foundation they’ll simply demand to read from the graph. At Higher they volition uncovering turning points past times completing the square).
• if Higher Tier students accept to know close the equations for whatever circle (no, solely those centred around the origin).

This is actually helpful data from OCR.

So that's it - I promise you're straightaway feeling to a greater extent than informed close the novel GCSE content. Do permit me know your thoughts in addition to questions.