Saturday, November 9, 2019

Trigonometric Graphs Workbook

I've decided to start sharing some of the workbooks I put together when I was at Hillhead, depending upon if people find them any use. This first one is a compilation of tasks focused around sketching and interpreting trig graphs, exact values etc. It's primarily focused at Scottish National 5 moving towards Higher.  

I normally teach this using unit circle, but that is done via boardwork. There are no specific references to the unit cirlce in the booklet. 

A fair chunk of this was based upon some old worksheets by Texas Instruments, which I collated together.   The rest of it is made up of tasks I wrote myself which are intended to draw focus upon key features of the graphs and be a basis for future learning.  

Download the booklet here

Monday, June 24, 2019

Building Blocks

This prompt is around the visualisation of prime decomposition.  I'm not convinced by the colours I've used, as in the last section this could be thought of differently.  I'd offer this as a prompt and then ask pupils to generate their own building block using division by primes.  This could then act as a lead into prime decomposition.  Further it may also provide a platform for new insights into multiplicative relationships for some pupils. Also links into area. 

Credit: @ChrisMcGrane84

Function Composition - Deeper Understanding


A simple twist on the typical exam question.  Do pupils appreciate that each point on a graph is an instance of (x, f(x)) ?  Do they realise that the composite function has coordinates (g(x), f(g(x)) ?

This graph could be presented to pupils.  Then have them select pairs of points to put in the table to notice the relationships and explore the underpinning principles of composition. 

Credit: @chrismcgrane84

Median - Working Backwards

This task is based upon a question from the 2018 SQA Higher Maths paper.  I've been considering how we can adapt existing questions to develop tasks for deeper mathematical thinking. The theme of freedom and constraint is prevalent here. 

Credit: @ChrisMcGrane84

Thursday, June 20, 2019

Fraction arithmetic

I don't envisage this task as being a worksheet to be issued to classes. Rather as the focus of a class discussion and investigation into the nature of fractions.  The task aims to link the number line representation with the procedures for adding and subtracting fractions.  Further it is about building fluency in counting with fractions.  All too often, I've found pupils aren't adept at mental approaches to this and overly rely on procedures.  I often write about developing "feel" for ideas.  The results of this task may not be immediately obvious, however, the discernments I hope may arise would be around familiarity with the family of fractions which is related to the twelfths, including halves, quarters, thirds and sixths.  Other possible extension is to extend the fraction line into improper fractions and or backwards into negative fractions.  

Credit: @ChrisMcGrane84

Straight Line - Everything we know!

This task from @MrsPhilipsMaths caught my eye.  I like the layout of this as a task template for other topics too.  It builds on the ideas of SSDD style problems, but could also very easily become a goal free problem.  "Create 8 questions that could be asked here", for instance. 

Credit: @MrsPhilipsMaths

Decimal Multiplication

Task and description from: @maths26772621

This could be extended in many different directions depending on the starting and target values. These could easily be set by the teacher or as I have done allow the pupils to select and in fact, I eventually had them create questions that their partner then had to solve which resulted in some tough questions. Quite quickly tweaks needed to be made like no multiplication by 1 but I wanted them to discover this “cheat”.  If the instructions potentially pose an issue, then you could set it up the way I have in the example but have them try to fill in the blanks for a few attempts?

Wednesday, June 12, 2019

Inquiry Prompts - Thinking deeper about Area and Trig

Part One

 Part Two

This task is designed to help pupils make connections between trig graphs (and the unit circle if taught this) and the application of trig in finding areas of triangles.  

I would suggest that we our ultimate aim should be to develop a culture where the pictures of the triangles would be enough for pupils to commence some form of mathematical activity.  The prompts I have bullet pointed underneath would be for the teacher to use with individuals or groups who weren't doing anything.  This task was written with inquiry and mathematical behaviour very much at the forefront of my mind. 

Part Two aims to deepen the understanding of what is happening.  A unit circle representation and the obtuse associated rotation would help pupils to appreciate this more.  

Credit:  @chrismcgrane84


Tom Carson recently shared this task with me.  I liked it so much that I asked to share it on the site.  So much going on with this one.  

A potential pedagogical action I have considered is to present only the table and ask the pupils to generate terms which would satisfy each cell.  Further prompting "generate one nobody else will have", "generate a really weird looking one" etc.  Some pupils might need a little exemplification, or to work in pairs to get started. 

Credit: @offpistemaths 

Horizontal Translations

This task is made up of a series of sheets.  It was designed to serve two purposes.   Firstly, it is about horizontal translations.  It is intended as a device with which to foster an awareness of why functions behave as they do when .  However, another key aim of this task was for pupils to enhance their understanding of functions and function notation.  In a recent conversation with John Mason he suggested to me that some pupils don't appreciate that any point on a function is a specific instance of (x, f(x)).  

Download the word file here

Credit: @chrismcgrane84

Order of operations Same/Less/More

Inspired by the Thinkers CPD delivered by John Mason in London last week, I have extended his idea of less/more grids into a 5x5 array, to take advantage of the "and another, and another" idea.  

John demonstrated a task using order of operations, which I have adapted and such that I think fits nicely into this framework. 

Credit: @chrismcgrane84/@Johnmoxford

Wave Function Representations

A task focused on wave function (combining waves to form a single trig function).  Multiple representations throughout and a bit of undoing on the bottom row.  

Download the word document here:

Wave Function Task

Credit:  @chrismcgrane84

Friday, May 17, 2019

Standard Deviation Representations

Click images to enlarge

This task builds on the tower representation which Don Steward utilises in some of his tasks on mean.  

This representation is used here to help pupils develop a sense of spread.  Later, pupils have to use their understanding to generate sets of data to satisfy the given criteria.  

Further, there is careful variation of the questions throughout with, mainly, one one attribute changing between each question.  This allows the effect on changes in either mean or standard deviation to be compared while the other remains invariant.  

Credit: @Chrismcgrane84 with thanks to @garyl82 for some suggestions. 

Tuesday, April 16, 2019

Clever Grouping: The Distributive Law

This series of tasks are the result of my recent thinking about how algebra emerges from number.  Algebra is the generalistion of number and number relationships.  Why is algebra often taught as something distinct from the number work pupils have already engaged with?

I would suggest that sort of task, in early secondary attempts to bridge the gap between the two.  Perhaps teaching this at the time of number work, but recalling it when later encountering the ideas in algebra.  This is something I am continuing to mull over.  I make no claims to having this "figured out".  

These tasks are starting points, each one is designed as a prompt to stimulate some mathematical activity.  I think there are a multitude of links here to prime factorisation, index laws etc, etc just waiting to be explored. One idea for a potential part (d) for task 3 might be "create a multi-layered factorisation of 12x^4". 

Credit: @chrismcgrane84

Thursday, March 7, 2019

Power Patterns - Negative Powers

This task was inspired by the work of Tony Gardner.  There is an assumption that pupils already know the law of indices for multiplication of terms and know the rule for power 0.  Like all tasks, the pedagogy around the task will determine the extent of the mathematical activity.  If issued as a worksheet to complete, without opportunities for conjecture, exploration and reflection then there may be less fruitful mathematical activity than if those opportunities were allowed for.  

The task aims to make connections between existing knowledge and this new 'rule' to be learned.  

The task can be downloaded from  here as a word document.

Credit:  @chrismcgrane84 and Tony Gardner

Monday, February 25, 2019

More/Same/Less: Gradient and y- intercept

Another task inspired by John Mason's framework of less, same and more.   Another possible use would be for pupils to draw a graph AND state a possible equation which satisfied the criteria.  

Credit:  @chrismcgrane84 and delegates at the Task design courses in Feb 2019

Monday, January 28, 2019

More/Same/Less: Standard Deviation and Mean

This task was inspired by a classic written by John Mason on area and perimeter.  

Pupils should calculate the mean and standard deviation (non calculator) of the data set in the middle cell.  They then have to create their own data sets which satisfy each of the conditions on the other cells.  E.g. the top middle cell has to have the same standard deviation but a greater mean the starting cell.  

Pupils really have to understand the effect altering the data set has on each of the measures.  A process of trial and error may be a starting point for some before more informed conjectures begin to emerge.  

Credit:  @chrismcgrane84

Standard Deviation and Mean: Deliberate Practice

This task was influenced by one written by my good friend @offpistemaths.  I really recommend that teachers work through the task first to get a sense of what is going on, and to understand the connections properly between each set of questions.  This task should be non-calculator as there is a nice moment with surds which comes into play towards the end.  

Credit: @chrismcgrane84

Saturday, January 26, 2019

Another angle chase

The last angle chase I published has been one of the most popular tasks on the site, so I've written another one.  

Thinking about a follow up task for quick finishers:

"The question gave you 8 angles initially.  Can you recreate the problem, with fewer angles given initially, such that it is still solvable?"

credit: @chrismcgrane84

Thursday, January 24, 2019

Sketching Quadratics from Given Conditions

This task can be downloaded in PDF format from this link.   This task is designed to focus attention on the key characteristics of quadratics and how the formula relates to the graph.  It is about deep understanding and appreciation of generalisation of the principles pupils often use numerically. 

Credit:  @chrismcgrane84

Friday, January 18, 2019

Sketching lines from given conditions

This task can be downloaded from here  in a pdf.  The idea, based upon a common National 5 question is to illicit evidence of pupils understanding of the general form of a straight line.

Credit:  @chrismcgrane84

Sunday, December 23, 2018

Factorisation: Variation of coefficients

This task is useful for helping pupils begin to see the underlying relationship between the coefficients of a quadratic, and the factorised version of the expression.  Particular attention should be drawn to the cases where the coefficient of x is 0.  This leads to a subsequent discussion of the special case of a difference of two squares.  Many teachers teach it as a separate and entirely disconnected from the general case of a quadratic, when in fact it is a natural progression. 

Credit:  @chrismcgrane84

Tuesday, November 27, 2018

Bracket connections: factorisation and distribution

A task which focuses on making connections between the distributive law and factorisation.  

Credit:  @chrismcgrane84

Minimally different common factors

A range of questions to build up proficiency of factorisation where powers are involved. 

Credit: @chrismcgrane84

Sunday, November 11, 2018

Algebraic Fractions: attending to simplification

This task uses the power of learner generated examples to draw attention to the constraints of when we can and cannot simplify algebraic fractions.  This is an area where many pupils make mistakes.  The first two tasks are deliberately designed so that learners chose from the small set of terms.  The next two sets give the learner opportunity to explore the example (and non example) spaces of these sorts of situations.

Credit:  @chrismcgrane84

Common errors when simplifying algebraic fractions

Credit:  @chrismcgrane84

Wednesday, November 7, 2018

Ordering Fractions - Probing Questions

These three questions are designed to illuminate the fact that procedural approaches aren't always the most efficient way of ordering fractions.  An appreciation of concept is much more important.  This task could be used as a formative assessment tool to illicit evidence of pupil understanding.  

Credit: @chrismcgrane84

Friday, November 2, 2018

Straight Line - practice of key questions

A revision task for the end of the straight line topic.  

Credit:  @chrismcgrane84

Wednesday, October 24, 2018

Straight Line Variation

I've had some success in using this task to focus pupils on appropriate strategy selection for finding the equation of a line.  There is deliberate variation built in which is useful for pupils have already studied various ways of finding equation of a line.  Line parallel to axes are a deliberate inclusion.  

credit:  @chrismcgrane84

Tuesday, October 23, 2018

Fraction equivalence and simplifcation

This task promotes the fluency of simplification and creating equivalent fractions.  It builds on knowledge of proportional reasoning.   

Some resulting board work is shown below, from use of this in a lesson.  

Credit: @chrismcgrane84

Tuesday, October 9, 2018

Multiple Representations of Quadratics

Quadratics is a big topic for our pupils to learn.  So many interlinked ideas.  There are many representations of the same idea.  This task is designed to help learners come develop an understanding of the equivalences of the representations.  

A potential use of the task is to present the first poster as it is, fully complete.  Then ask pupils to complete the first of the blank grids where only the expanded form of the quadratic is given.   There is much potential for collaborative work here.  The follow up tasks ask pupils to complete the grids based upon a different starting point.  It might be better to use A3 paper to print this task. 

Follow up tasks could be to give pupils one of the other representations and to work form there to complete the grid. 

Further follow up might be to use this as a means of introducing cubic functions - why this can't be expressed in completed square form is an interesting conversation.  

A PDF of the full task is available here:

Credit:  @chrismcgrane84