# Starting Points

A collection of tasks and starting points to help teachers plan for richer and more effective learning experiences. All of the tasks include suggested teaching points and questions for discussion with learners. The tasks are a collection of ideas for learner generated examples, some rich tasks, intelligent practice, some examples of variation theory and various other curiosities. Contributions mainly from teachers in Scotland.

## 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

Credit: @chrismcgrane84

## Sunday, November 11, 2018

### Algebraic Fractions: attending to simplification

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

Credit: @chrismcgrane84

## Friday, November 2, 2018

## 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

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:

https://www.dropbox.com/s/vczk5e1th88bqjl/quadratics.pdf?dl=0

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

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

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

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

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 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 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

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

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**

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

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

### Complex Conjugates

A simple little task for Advanced Higher pupils. Formal proof is a topic in the course, but further, a number of experienced teachers have suggested that building in opportunities for direct proof or use of counterexamples throughout the teaching of topics prior to formal methods of proof teaching can be beneficial. This task also highlights how effective task design can be used at any level of the curriculum.

@chrismcgrane84

## Monday, September 3, 2018

### Units of length

This prompt can be used in many ways... below is some very messy boardwork from a lesson. I have listed out the actions from the lesson below. The focus of this lesson was not on memorising conversion rules, but in getting a sense of the contentedness of the units of measurements, primarily via developing a sense of proportionality. 1km = 1000m, so 2km = 2000m etc.

Possible Sequence of tasks

1. Write down 1km = 1000m, 1m = 100cm and 1cm = 10mm

2. Ask pupils to label the top line using metres

3. Select some in between values such as 8.5km, 0.3km and ask pupils to write them in metres.

4. Go through a similar process of metres, asking learners to convert each of them into cm. Select some in between values for conversion to cm.

5. And the same again for centimetres into millimetres.

6. Now go back up the way and look at, for instance 9m. We'd previously written it is equal to 900cm. Convert this answer to mm. Ask pupils to do the same for some other places on the metre line.

7. Look at, for instance 7km. We'd previously written it as 7000m. Convert this to cm and mm. Ask pupils to do the same for some other places on the km line.

credit: @chrismcgrane84

### m = tanÎ¸ sorting task

The aim of this sorting task is to draw learners attention towards the relationship between angles and gradient. Further, the inclusion of the worded "for every x boxes along go y boxes up" and the diagrams is to help learners consider gradient's meaning from a geometric perspective. Often it can appear that learners see gradient as being a number with no inherent meaning.

Credit: @chrismcgrane84

A PDF version is available by request. Either tweet or email!

Credit: @chrismcgrane84

A PDF version is available by request. Either tweet or email!

## Friday, August 31, 2018

### Ascending Trig

This task is designed to encourage learners to attend to the y coordinates of the points on the trigonometric graphs. Common errors have, in experience, included ordering simply by the magnitude of the angle. Suggesting that learners draw a sketch of each graph in order to appreciate the concept may help.

Extension could be to make use of the periodicity of the graph and add (or subtract) mutliples of 360 from the values in the questions above.

Credit: @chrismcgrane84

## Thursday, August 30, 2018

### Chain Rule Challenge

This task has been used with Higher pupils for stretch and extension, and for Advanced Higher pupils who need to sharpen their chain rule skills before embarking upon calculus at that level.

A nice follow up is to ask learners to generate examples of chain rule with 2 layers, 3 layers, 4 layers etc.

Credit: @chrismcgrane84

### One Triangle, Many Problems

This task was first used with a class who had been working on right angled trig, but hadn't done anything in context yet. It was used as a task to focus on mathematical thinking and problem solving skills. If pupils have never solved a problem involving an isosceles triangle using trig then this task can be used to make significant steps in their understanding.

The framework this particular class have developed together is (in no particular order of doing):

1. Make a sketch

2. Fill in anything we know, or that we can work out easily.

3. Mark in what we want to know

4. Write down any formulas that might be useful.

5. Consider making a simpler sketch

All of these strategies can be utilised in this task.

Some possible task stems:

The framework this particular class have developed together is (in no particular order of doing):

1. Make a sketch

2. Fill in anything we know, or that we can work out easily.

3. Mark in what we want to know

4. Write down any formulas that might be useful.

5. Consider making a simpler sketch

All of these strategies can be utilised in this task.

Some possible task stems:

- Find the missing sides and angles of this triangle
- Or less leading would be... Find the perimeter of the triangle
- Find the area of the triangle
- Or simply, find everything that you can about this triangle

Credit: @chrismcgrane84

## Wednesday, August 29, 2018

### Target Integers

A very simple task idea for learner generated examples. One example is given, but this is easily extendable for other cases. "How many ways can you generate 0, using 3 different numbers from: -3, -2, -1, 1, 2, 3?"

Credit: @chrismcgrane84

Credit: @chrismcgrane84

### Integers on a number line

Possible Introductory task

This task is designed for the patterns inherent in the additive relationships between integers to become more apparent. It could be used to help learners further develop their "sense" of negative ordering and value. A possible follow up task might be to either give learners a set of negative number questions, which follow no obvious pattern, and ask them place them on a number line. Some questions which give the same solution might be nice! An alternative approach might be to ask learners to generate their own set of questions and then to place them.

Credit: @chrismcgrane84

## Monday, August 27, 2018

### Trig and Pythagoras Decisions

This task assumes that students can find either side of a triangle using the Tan ratio, find the angle using the Tan ratio and know Pythagoras. It is designed to help students focus their attention on the subtle differences in the algebra of calculating the opposite or adjacent sides. The inclusion of Pythagoras is designed to help pupils assimilate trig and Pythagoras in their mental schema and deal with the redundant information given.

I have used it before introducing Sin and Cos ratios.

Credit: @chrismcgrane84

I have used it before introducing Sin and Cos ratios.

Credit: @chrismcgrane84

### Powers and Roots

A little task looking at the interconnections of powers and roots of the same number. Two examples are given of a task that is easily adaptable to support the development of these key ideas.

Credit: @mpcopland

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