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

Basic Logarithm Practice, and Guided Discovery of Key Rules

This exercise is designed with a couple of aims in mind.  Firstly, to provide some basic practice of evaluation of logarithms and secondly, to lead pupils into some key laws of logs.  

Question 16 offers pupils opportunity to generate their own examples and demonstrate/consolidate/develop a deep understanding of logarithms.

Question 18 provides an element of challenge, but is designed to help learners attend to the key underlying idea of logarithms as being connected with powers. 

credit:  @chrismcgrane84 

Monday, October 8, 2018

Introduction to exponentials

This task was inspired by an article in the ATM journal "Mathematics Teaching", issue 116 by Tasos Patronis titled "Exploring Exponentials".  

He makes the point that many pupils do not have a sense of the types of phenomena that can be captured by mathematical functions.  I feel this task helps to illustrate the key property of exponential growth and how this differs from linear growth.  

The exponential graph arises as a matter of course from this.  Pupils can be asked to speculate on the values where x = -1 and x = -2.  They could be asked to suggest an equation for this curve. 

Some may feel this task is too scaffold-ed.  It might be nice to offer the initial problem without the table of values and the graph as learners own ideas may prompt interesting dialogue.  

Credit: @chrismcgrane84 and Tasos Patronis 

Tuesday, October 2, 2018

Chasing Angles

I really enjoy doing angle chase problems.  I've used variations of this task with classes for years and have often found that they enjoy the sense of accomplishment that comes from completion.  Initially it appears like there is too little information, however, they soon realise that a journey to completion is possible.  It's a nice recap of a lot of work covered in a typical unit on angles. 

Credit:  @chrismcgrane84

Monday, October 1, 2018

Straight Line Matching Activity (1)

A sorting task based upon the various representations of a straight line equation.  The blank cards are for pupils to write the equation of the line in the standard form of y = mx + c.  Alternative versions of this resource will follow.

Credit: @chrismcgrane84

Friday, September 28, 2018

Number Venn

Classifying numbers using Venn diagram.  Useful for separating different, but similar/overlapping ideas.  For instance this sort of task could be used for classifying which type of factorisation is required for various expressions.  

Credit:  @chrismcgrane84

Tuesday, September 25, 2018

Functions Matching Activity

Many learners make errors during initial learning of composite functions.  Substituting the wrong way around etc.  This task aims to help draw attention to the core idea.  While normal procedurally focused exercises are important, this sort of task is complementary to those and can help to move the learning forward. 

A nice extension would be to have pupils come up with a pair of functions of their own and then make up some cards for the various composition permutations of them.  

Credit: @chrismcgrane84

Thursday, September 20, 2018

Collecting terms - missing coefficients task

Credit:  @chrismcgrane84

Collecting Like Terms - Sorting Terms

Pupils can be offered this set of terms to be cut up into individual terms.  

You could ask pupils to group them together in any way they like.  Some might go straight to like terms, others may have a pile of numbers, a pile of letters.  Others might do positive or negative.  This is an opportunity to explain likeness and have pupils group the like terms together. 

The first task that can be performed is asking pupils what they get when they combing the terms.  From experience, I would encourage pupils to physically lay the terms out for each question - side by side.  

There is then opportunity to probe their understanding a bit further in this next couple of tasks. 

Credit: @chrismcgrane84

Tuesday, September 18, 2018

Digits of the year problem

This task focusing on basic number skills:  order of operations, powers roots etc provides learners with the opportunity to strengthen their understanding of these ideas by generating their own examples.  A nice idea is to have a wall of the class dedicated to this challenge and have the class collectively work towards generating every number from 0 to 100.  

Credit: @chrismcgrane84  

Friday, September 14, 2018

Addition of Logs Inquiry Prompt

A little inquiry task to arrive at the addition rule for logs.  It is nice to let pupils attempt each of the tasks and then let them direct a whole class discussion.  This approach allows more mathematical thinking than simply telling them the rule.  A suggested next step would be to have pupils explore "why" the rule works.  Encouraging pupils to consider the exponential form of the expressions is a useful starting point. 

Credit:  @chrismcgrane84

Log Pyramid Task One

An opportunity for learners to practice their evaluation of logs. 

Credit: @chrismcgrane84

Log Expressions

A little task designed to develop fluency and understanding of log expressions.  Pupils have the opportunity to generate their own examples - a good test of understanding.  

Credit:  @chrismcgrane84

Thursday, September 13, 2018

Functions Pyramids

Straightforward composition of functions followed by a more challenging thinker!

Credit:  @chrismcgrane84

Monday, September 10, 2018

Exploring Log Base 2

This task may be best used with a class who could evaluate log expressions and solve equations using them, so as to ensure there is some familiarity with the basic ideas. The task can be a lead in to looking at logarithmic graphs.  In experience when asked what is a log many pupils respond with " the graph", which may demonstrate a lack of understanding.  

This task is designed to help learners make associations between different representations of logarithms.  Written explicitly as a log statement, written as an exponential expression and also as a table of values for the graph of log base2.  

Some nice follow up tasks would be for the following table to be completed.  

The first task is looking at "nice" values on the logarithmic curve, this one then looks at how a small subset of those values look in comparison to their nearby values.  Pupils could be encouraged to write exponential and log expressions for these values too.  
Information from both tables can be combined to plot the graph.

Credit:  @chrismcgrane84

Friday, September 7, 2018

Units of length conversion

This task is designed to help learners develop their confidence and fluency at converting between different units of length.  Rather than trying to remember a list of confusing rules (x 10 for cm to mm, divide by 100 for cm to m etc) this task aims to support learners at internalising the connections between the different units and making connections with place value work. 

Credit: @chrismcgrane84

Thursday, September 6, 2018

Complex Numbers - Polar multiplication inquiry

This task is intended as a lead in to the multiplication of complex numbers in polar form.  Some pupils may make the connection automatically, others may need some prompting to notice what is happening.  Experience of using the task has suggested it helps learners to develop their appreciation of the equivalence of the two representations of complex numbers, and also have a sense of the graphical interpretations.  Further, for those learners who do not make the conceptual leaps of their own accord, the exercise is still worth while practice.  

Credit:  @chrismcgrane84

True or False - Surds

True or false
This prompt can be used as a good discussion prompt. One approach would be to have pupils work in pairs, discussing, before each pair shared their ideas with the class.  The task could be used at many levels - for instance with a class who are working on surds, this might be a formative check that they understand sqr(5) + sqr(5) = 2sqr(5).  In other contexts it may be used to deepen understanding and appreciation of how irrationals sit on the whole number line.  

The results of a discussion in my S1 class are shown below:

@blatherwick_sam suggested the following follow up questions on Twitter:
Could you get an upper and lower bound then of what sqr(5)+sqr(5) could be? What about sqr(3)+sqr(3)? Is sqr(x) + sqr(x) always bigger than sqr(2x)? Why? How about cube roots?oh and this one - sqr(x) + sqr(x) = sqr(10), what is x? and at the top of the board there is in effect a number line... where would 2.5 fall on that number line?

Credit: @chrismcgrane84

Wednesday, September 5, 2018

Similarity SSDD

A similarity task aimed at encouraging attention on the wording of questions.  

Credit: @mpcopland