Conception of the good

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UK Maths Challenge

On Friday, I spent a couple of hours with a pupil in Y8 who has been selected to go through to the Junior Kangaroo challenge. Over half term, he was given past papers to attempt. We went through his answers and we discussed the questions he found difficult to do. In the process of going through some questions I realised how important knowing specific factual and procedural mathematical knowledge is crucial to a pupil’s success in the UK Maths challenge.

I am a huge fan of the UK Maths challenge. Why? Simply because each question is created to test ‘powerful’ knowledge. Daisy Christodoulou and Michael Young refer to ‘powerful’ knowledge as certain forms of knowledge which allows humanity (or in this context specifically pupils) to advance in some way to communicate more accurately. This blog will include questions which explore the ‘powerful’ knowledge that the Maths challenge tests, and how teaching such knowledge allows pupils to be successfully in the  challenge.

Here is a list of the topics that I am referring to as ‘powerful knowledge’ that allows pupils to become incredibly flexible in applying their knowledge to such questions:

  • Divisibility Tests
  • Prime factor
  • Identifying Square numbers
  • Angle facts (vertically opposite, interior angles of regular polygons)
  • Common Fraction/Decimal/Percentage facts 3/8 à375 à 37.5%
  • Double the radius for the diameter, Halving the diameter for the radius
  • Formulas for the area of a rectangle, triangle, trapezium
  • Algebraic expressions for the area and perimeter of a rectangle/square e.g. 2(a + b)

Procedural knowledge:

  • Decimal multiplication
  • Order of Operations (GEMS and BIDMAS)
  • Applying exponents and roots
  • Four Operations applied when using fractions
  • Doubling and halving
  • Formula manipulation
  • Forming and Solving Expressions

Each UKMT question requires pupils to have basic knowledge facts at the fore front of their mind e.g. a square number has an odd number of factors, the first 6 rows of pascal’s triangle etc. More importantly, it is the interleaving of different concepts in each question which is why I think the questions are intelligently designed.

If pupils are quick in identifying the completing basic procedures within the following topics they can apply that knowledge to a wide selection of UKMT questions. Here are a few examples of questions which evidence this. The questions below also demonstrate how pupils’ knowledge of multiple concepts are being tested too:

UKMT 1 UKMT 2 UKMT 3 UKMT 4 UKMT 5 UKMT 6 UKMT 7 UKMT 8 UKMT 9 UKMT 10

Arithmetic Average: Reflecting on Property 5

In my last post, I discussed whether an instructional programme could be created to teach a deeper concept of mean. I mentioned five different properties.

After a conversation with Kris Boulton about the blog post, he spotted something interesting about property 5.

Property 5: The average is only influenced by values other than the average.

At the end of my previous post, I explained property 5. It is because when we add the average value to the data set then the resulting sum is divisible by the mean. When we add a value that is not the previous average, the resulting sum is not divisible by the new denominator. However, the second sentence is incorrect.

Image 1

If I do not add the average value of the original data set, I can still have a resulting sum which is divisible by the new denominator. The third example shows this.

Example 1: Original data set with a mean of 5.

Example 2: Add the average value to the new data set – resulting sum is divisible by the new denominator. The mean has not changed.

Example 3: Not adding the average value to the new data set – resulting sum is still divisible by the new denominator. The mean has changed.

Is there another way to communicate this property?

Image 2

This is an idea from Kris which he discussed as a potential second attempt to communicate property 5. It communicates property 5 nicely. It also has non examples too. If I do not add the average value then the mean is not 5.  Below is another example that can go along with the example above because the average value (19) is not a number in the original data set. Pupils are then seeing an example where the average value can be present in the data set, and where the average value will not be a number in the data set.

Image 3

 

I have added the average value to the original data set.

The mean has not changed.

 

 I added the average value twice to the original data set.

The mean has not changed.

 

I have not added the average value to the original data set.

The mean has changed.

I think both examples are important to use because they communicate the same property. The second example communicates the message in a more explicit fashion. The first example should be included because it is a nuanced example compared to the second. This may or may not be that important, but telling a pupil that the average value can be a number in the original data set is knowledge a pupil should know.

I wanted to convey an example where the procedure of dividing by the number of values was visible to pupils. However, at the same time, what is the most important aspect is pupils seeing that we have added the average value and the new average hasn’t changed. The division isn’t adding the value that I initially thought it was. The new examples communicate a quality of sameness that can clearly communicate the property effectively. Also, showing the features of the new data set which are permissible to be a positive example, and the non examples show the features of the data set which are not present to be an example for property 5.

Arithmetic Average: developing conceptual understanding

Arithmetic Average: developing conceptual understanding

Many pupils are taught how to calculate the arithmetic average (the mean.)  They are also taught a shallow understanding of the concept: it goes along the lines of “the mean is a calculated ‘central’ value of a set of numbers.” Or pupils have a very loose understanding of ‘average’ as colloquially referred to as ‘on average’. This is a start, but can an instructional programme be created to teach a deeper concept of the mean, accessible even to young pupils (8-14)?

I believe it can.

If so, it will aid future understanding of the weighted average at GCSE level.

Furthermore, it will lay the groundwork to understand how the trapezium rule approximates the area beneath a curve, at A Level.

I’m going to cover:

  1. Properties of the mean
  2. Selecting which properties to teach
  3. How to teach them
  4. Why it’s important

Fundamental properties of the mean

In Strauss and Efraim (1988) seven properties of average are outlined and these properties were chosen to be discussed because “they are fundamental, and tap into three aspects of the concept”.

I’m going to discuss what I think are the five most important.

Property 1: The average is always located between the extreme values

The average value of a set of data cannot be smaller than the minimum value or larger than the maximum value of the said data set.

This seems like common sense to an adult but expertise induced blindness underestimates the difficulty for pupils to understand this property of the mean.

How can this be communicated to pupils?

By example, show that the average value is never smaller than the minimum value or larger than the maximum value.

average pic 1

Then, test pupils’ understanding of the property using the following type of question:

For each question the average value could be true or definitely false, state whether the average value is true or false for the corresponding data set:

average pic 2

The following true or false questions are testing a pupil’s understanding of the property rather than their procedural knowledge of calculating the mean.

Property 2: The average is representative of the values that were averaged.

More technically: “the average is the value that is closest to all of the others in the set of values that are being averaged.”

Therefore, the average value represents all the values within the data set.

This property ties in nicely with the third property.

Property 3: The sum of the deviations from the average is 0.

Each value in the data set is a certain distance away from the average value which is clearly understood from a visual example. If we total the distances between each value and the average value, that total will equal to 0. This helps pupils to visualise that the average value is central to all the values. This highlights that the concept of the average value is again the central value to values within the data set. Furthermore, this highlights that the average value represents all the values that were averaged (property 2).

How do we communicate this to pupils?

  1. Outline how to find the distance between each value and the average value.
  2. Emphasise that I do (value – average value) and that the distance can be negative, if the value is smaller than the average value. However, the pupils need to picture the distances.
  3. Total the distances, also tell pupils that we call this sum the sum of deviations

average pic 3

 

average pic 4

average pic 5

There are two questions that can be asked to develop understanding around property 3:

  1. Show for each data set that the sum of the deviations from the average is 0 (Qs 1 – 3 only)
  2. For each set of data, the average value is either true or false. Determine which of the following average values for the corresponding data set are true or false using your knowledge of the following property:

The sum of the deviations from the average is 0.

The first question is asking pupils to apply their knowledge of finding the sum of the deviations for the average value. The second question is asking pupils to apply this knowledge but then decide which of the following data sets has the correct average value.

average pic 6

Property 4: When one calculates the average, a value of 0, if it appears, must be taken into account.

Dylan Wiliam has an excellent hinge question that deals with this property very well.

average pic 7

Pupils often think they don’t have to include a 0 value when calculating the mean.

A good way to overcome this is to simply include it in one of your examples.

Seeing this in a concrete context also allows pupils to see why 0 must be included as a value in the data set.

image 5a

Concrete context:

Sarah collected some money from her three siblings to raise money for charity. Hannah donated £5. Adam donated £7. Clare donated £9. Sarah did not donate any money. What was the average amount of money each of her siblings and Sarah donated?

Property 5: The average is only influenced by values other than the average.

average pic 9

This is a really nice point to make about arithmetic average:

Adding a value to a data set which is equal to the average value of the current data set does not influence the new average.

Why not?

Because when we add a value that is the previous average, the resulting sum is divisible by the new denominator. How can we communicate this to pupils?

The worked examples above show how the mean does not change when you add the average value of the previous data set (going from example 1 – 2 and 2 – 3). The last example shows that the mean does change when a value which is not the average of the previous data set is added.

To communicate this I would ask pupils to determine if the number added would change the average value. For example, the question series is designed for pupils to do the following

  1. calculate the average for each data set
  2. Decide if the mean has changed by adding a value.
  3. Explain that we have added a value which is equal to the average value. Adding the average value does not influence the values being averaged.

average pic 10

At the end, I would explain how when we add the average value to the data set (5), we get a sum which is divisible by the denominator (20/4 = 5). When we add a value that is not the previous average, the resulting sum is not divisible by the new denominator.

average pic 11

In summary, these properties of arithmetic average can be taught effectively with the correct worked examples and problem exercises which communicate each property at one time. It gives pupils a spatial understanding of the mean which is above and beyond the procedural calculation of calculating the mean. I think it is highly powerful knowledge that can lend itself for more complex understanding of the mean when learning about the difference between simple mean and weighted average mean.

 

#Mathsconf9 – Automatising Procedural Knowledge for A Level study

On Saturday, I delivered a workshop which looked at seven specific high-leverage procedures that should be automatised for pupils to successfully learn certain concepts taught in AS Mathematics. The purpose was to raise awareness of the amount of procedural fluency required for different sub-procedures in large calculations such as when completing the square, or simplifying index notation etc

Here is a bit of context behind what inspired me to deliver this workshop. I was tutoring a friend’s daughter in the summer holidays (2016) before she started learning AS Mathematics after achieving a A in her maths GCSE. My friend’s daughter is intelligent, enjoys studying mathematics and is able to learn new concepts very quickly but she struggled to make the correct decisions to complete each sub-procedure in long calculations. She would struggle to multiply an integer and a fraction, because she didn’t know what decision to make i.e. write the integer as a fraction over one.  So I spent a lot of time creating questions within drills and deliberate practice exercises for her to gain the procedural knowledge to a degree of fluency where she wasn’t struggling to complete different sub-procedures.

I am going to look at one high level procedures that I mentioned in my workshop; multiplying fractions.

Multiplying fractions.

Historically, when teaching how to multiply fractions we teach how to multiply fractions where there are two terms within the equation. However, there are several different problem types that can be taught:

  • Multiplying two terms
  • Multiplying more than two terms
  • Multiplying terms that share factors that can be cross-simplified
  • Multiplying a fraction and an integer
  • Multiplying a mixed number and a fraction
  • Multiplying mixed numbers
  • Multiplying decimals (that can be converted into fractions) and fractions.
  • Multiplying an improper fraction with and/or proper fraction

fraction 1

When teaching my tutee index notation she struggled to attempt the following problem because she wasn’t able to identify how to multiply an integer with a fraction. I told her explicitly how to multiply 2 and a third quickly, because that is what I do, and all pupils do this if and when they spot the pattern (but not applicable when the numerator of the fraction is not 1).

fraction 2

fraction 3Later on in another session, my tutee struggled with the following question because she didn’t realise that a fraction can be written as the product of an integer (numerator of the fraction – 2) and a fraction (unit fraction with the same denominator – 1/3 ) e.g. 2/3 equals to the product of 2 and 1/3.

fraction 4Furthermore, this piece of procedural knowledge is necessary to attempt the following question:

fraction 5And again, when a question gets more complex like the one below because the base number is a mixed number, rewriting 5/2 as a product of 5 and ½ is a necessary step to evaluate the question.

fraction 6However, what the take-away here is is that multiplying fractions is a high leverage procedure. A part of the procedure is to rewrite a fraction as a product of an integer and a fractions. So how can pupils learn this  to successfully automatise their procedural knowledge of multiplying fractions? This can be achieved through an understanding of multiple representations of the following possible problem types that arise when evaluating the base number/term held to a power which is a fraction.

fraction 7

The emphasis of exploring different problem types available within the topic of multiplying fractions, and recognising that a fraction is a product of an integer (numerator of the fraction) and a fraction (unit fraction using the same denominator), and then seeing the multiple representations of this enables a level of proficiency to allow successful learning of new content. My tutee wasn’t really learning how to simplify indices because she was focusing on learning how to complete each sub-procedure. I think this proficiency needs to be achieved in KS3.

In my next post, I shall give another example – halving!

Deciding the first step – Different decisions for different problem types

Deciding the first step is a type of question where pupils are only asked to decide explicitly which step to perform. A pupil reviews the problem type and identifies what steps he/she needs to take. After acquiring knowledge of a particular step then automatically the end of each step triggers the start of the next one. I will be looking at some questions which test whether pupils can decide on the correct step to perform, and compare them to more commonly asked questions which make the decisions for pupils.

Compare these three questions being asked by a teacher to a set of pupils

Decision Blog

What is 4 times 2?

What two numbers do I multiply first?

What is the first step?

The correct answer to all the above questions are the same. However, the first question being asked makes the decision for the pupil because they are told that they are to multiply, and which numbers to multiply. The second question is testing whether the pupils know that we multiply the numerators, and then multiply the denominators, but the decision is being made for the pupils. It is possible that a pupil will give a wrong answer by stating two numbers that we do not multiply (4 x 11). The final question is specifically asking a question to test if a pupil knows what step to perform, compared to the first two question.

Compare these two question being asked by a teacher to a set of pupils

Decision Blog 2

What do I cross-simplify first?

What is the first step?

Again, the first question makes the decision for the pupils compared to the second question. The second question is testing if a pupil can recall the first step in answering this question. This problem type is different from the one above because you can cross simplify. And I would consider cross-simplifying the first step to the procedure but simplifying the product of both fractions can happen after a pupil has multiplied the two fractions.

Compare these two questions being asked by a teacher to a set of pupils:

Decision blog 3

What is the first fraction as an improper fraction?

What is the first step?

Again, the first question makes the decision for the pupils. Furthermore, the first step of the previous two problem types cannot be applied here because there is a necessary step before both fractions can be multiplied.

The decisions pupils have to make to attempt the following calculations, vary:

Decision Blog 4

The following question – “What is the first step?” not only helps pupils to learn explicitly which step to perform first but it also allows pupils to distinguish between different problem types. This is because a pupil is then attaching their knowledge of what decision to take depending on the make-up of the question. They are acquiring surface knowledge of the problem type – If I see mixed numbers when I am multiplying fractions then I must convert them to Improper fractions first.

This form of questioning allows pupils to develop mathematical reasoning around different procedural calculations. For example, a pupil recognises that to multiply mixed numbers we must convert them into improper fractions because we cannot multiply 1 with 1 and 4/5 with 2/11 because 1 and 4/5 is not 1 x 4/5 but 1 + 4/5 which equals to 9/5. This also consolidates pupils existing knowledge of mixed numbers and their equivalent forms as improper fractions.

Another example which is interesting is making decisions about the first step when comparing two negative fractions. Here are the different problem types and the first step to each one. The question posed for all the problems below is “Which is greater?”

Decision Blog 6

Even though the problems may look the same to pupils, through teaching pupils explicit decision making around the problem types the pupils are doing two things. They are identifying the first step they need to take, and they are attaching their knowledge of the first step by identifying the features of each problem type.

For the most competent pupils their mathematical reasoning will make the features of each problem type and the first step they need to take seem obvious. For pupils who’s mathematical understanding isn’t as fluid they greatly benefit from being asked such questions, because they are identifying the features of each problem type which makes them different from each other, and thereby helping them know what is the first step they need to take.

 

Algebraic Circle Theorems – Pt 2

Last week, I explored different number based circle theorem problems that can test (a) a pupil’s ability to identify the circle theorem being tested and (b) problem types where a pupil has to find multiple unknown angles using their circle theorem knowledge as well knowledge of basic angle facts.

In this blog, I’m displaying a few different problems within the topic of circle theorems where each angle is labelled as a variable or a term. I am interleaving lots of different knowledge:

  • Forming and simplifying algebraic expressions
  • Forming algebraic equations
  • Equating an algebraic expression to the correct circle theorem angle fact
  • Equating two algebraic expressions which represent equivalent angles and solving for the value of the unknown. Furthermore, using the value of the unknown to find the size of the angle represented by the algebraic expression.

I have interleaved fractional coefficients into a couple of questions to add some arithmetic complexity to the questions. Enjoy!

A triangle made by radii form an Isosceles triangle

Image 0The angle in a semi-circle is a right angle

Image 1
The opposite angles in a cyclic quadrilateral add up to 180 degrees (the angles are supplementary)

Image 2 Angles subtended by an arc in the same segment of a circle are equal

The questions for this circle theorem differ in nature from the problem types shown above. Here you are equating two algebraic expressions which represent equivalent angles. We are no longer forming a linear expression and equating it to an angle fact like 180o.

Image 3

The angle subtended at the centre of a circle is twice the angle subtended at the circumference

In these problems types the key mistake that a pupil may make is equating the angle subtended at the centre to the angle subtended at the circumference without doubling the angle at the circumference. This can be pre-empted by asking pupils a key question of “What is the first step?” The answer I would be looking for after going through a few worked examples would be “you need to double the angle subtended at the circumference.”

Image 4Tangents to circles – From any point outside a circle just two tangents to the circle can be drawn and they are of equal length (Two tangent theorem)

Image4a

Alternate Segment Theorem – The angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment.

Image 5

I would be keen to hear any thoughts or feedback. Please don’t hesitate to email me on naveenfrizvi@hotmail.co.uk.

Creating Problem Types – Circle Theorems Part 1

Last summer, I made as many different problem types for the topic of Circle Theorems. I looked through different textbooks and online resources (MEP, TES, past papers). I did this because when I last taught circle theorems at my previous school there weren’t enough questions for my pupils to get sufficient deliberate practice. This was a two fold issue. Firstly, I would find a practice set of questions which would not provide enough questions for a pupil to practise one particular problem type. Secondly, the sequencing of questions in terms of difficulty would escalate too quickly or not at all. Here I will outline the different problems types I created (using activeinspire) and then explain the thinking behind them. I have been very selective with the problems I have included here; I have made more questions where certain problems types are more complicated which I shall discuss at the end. I shall more in the following posts.

I made two different categories of problems for each circle theorem. The first type would explicitly test a pupil’s understanding of the theorem to see if they could identify the circle theorem being tested.

The second type would be testing two things. Firstly, such a problem type would be testing their ability to determine the circle theorem being applied in the question. The second aspect of the problem type would be testing related geometry knowledge interleaved which can be calculated as the secondary or primary procedure in the problem e.g. finding the exterior angle of the Isosceles triangle.

One common theme in these questions is that procedural knowledge applied is executed in a predetermined linear sequence. Hiebert and Lefevre wrote that “the only relational requirement for a procedure to run is that prescription n must know that it comes after prescription n-1.” Multi-step problems such as the ones that you will see show that procedures are hierarchically arranged so that the order of the sub procedures is relevant. Here are the different problem types for each circle theorem where I explain how many items of knowledge is being tested in each question, and what each item of knowledge is.image-0Figure 1: A triangle made by radii form an Isosceles triangleimage-1

Figure 2: The angle in a semi-circle is a right angleimage-2

Figure 3: The opposite angles in a cyclic quadrilateral add up to 180 degrees (the angles are supplementary)image-3Figure 4: Angles subtended by an arc in the same segment of a circle are equal

image-4
Figure 5: The angle subtended at the centre of a circle is twice the angle subtended at the circumferenceimage-5Figure 6: Tangents to circles – From any point outside a circle just two tangents to the circle can be drawn and they are of equal length (Two tangent theorem)image-6aFigure 7: Alternate Segment Theorem – The angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment.

To conclude, there are many different problems types for the topic of circle theorems and the complexity of the problem can be addressed in many ways such as:

  • arithmetic complexity
  • Orientation of the problem
  • Multiple representations of the same problem type
  • multiple subprocedures to determine multiple missing angles
  • Interleaving the application of multiple circle theorems.
  • Interleaving the use of basic angles facts
    1. as a necessary step in the procedure to find other angles
    2. as an independent step in the procedure where finding one angle is not necessary to find another angle.

I would be keen to hear any thoughts. Please don’t hesitate to email me on naveenfrizvi@hotmail.co.uk.

Teaching Imaginary numbers – Part 2/2

In my last post I outlined how I developed pupils’ knowledge of rational numbers and simplifying surds in preparation for pupils to solve the following equations.

pi1

I asked them if we could have any number that could be squared to result in a negative number. This was the point in the session that I introduced the idea of imaginary numbers. I gave them a brief history referring to the work and discussions by Rene Descartes and Leonard Euler.

Now, “If I can square a number to get a negative result then it is an imaginary number because when we square a positive number we get a positive result, and when we square a negative number we get a positive number. This is how we display the square of an imaginary number”:

pic2

I told the kids that this is a fact that we are going to accept. Now let’s move on. “We are now going to square root both sides to see what  is equal to. We can’t square root a negative number but I am going to present it like this, and we can present it like this, again we are going to accept this for now and move on.”

pic3

These two knowledge facts are the foundation in solving the next few problems which I reiterated again and again to the kids. I then showed how we can evaluate the two calculations shown at the start where the square of an imaginary number will result in a negative integer answer. I explicitly outlined each step, one by one, and I kept each line of the algorithm between the equation and the solution consistent for each question I demonstrated.

pic4

“I am going to square root both sides, and I am going to include the positive and negative sign in front of the square root. We must include this.”

pic5

“I am going to separate √-25 as a calculation of √25 and √-1 because I can then square root positive 25.”

pic6

“I am going to evaluate √25.”

 pic7

“How handy! I know that √-1 is equal to . We are using our knowledge facts which I showed you midway into our session,”

 pic8

“This is my final answer.”

I demonstrated another example, and then asked pupils to attempt a selection of questions by themselves.

I then told the pupils, “We are now reaching the peak of our lesson where we are going to combine our knowledge of finding the solutions of a negative number, but this time the number that is going to be square rooted will not be a square number, this is where our knowledge of simplifying √24 and √500   will help us here.”

I modelled the next example with the following teacher instruction:

Here is my problem:

pic9

“I am going to square root both sides, include the positive and negative sign in front of the square root, -72 will be within the square root”

pic10

“I am going to separate -72 where it is shown as a product of -1 and 72, both values will be within a square root.”

pic11

“I am going to simplify the square root of 72 where I have an integer and a non rational number, because the square root of 72 is positive, no longer negative!”

pic12

“I am going to replace the square root of -1 with .

pic13

“I am going to rewrite this so i is next to the integer, you will do the same for all your answers too. We are finished.”

pic14

This is an example of a session that was delivered where the goal at the end was to have pupils being able to solve equations where they had to combine their understanding of rational numbers, and imaginary numbers. Pupils who attend this session are learning about different concepts in a succinct and limited manner to then apply their understanding to specific problem types selected by myself and the department. The questions the kids did to apply their understanding were carefully crafted by me to ensure that pupils were deliberately practising what I had modelled on the board. I think it worked quite well. If anything It made me realise that through a strong foundational understanding of number and times tables and thought out instruction, we can teach something as abstract as imaginary numbers. Furthermore, to get the delivery of the worked example to be as tight and accurate as possible I did script this lesson and rehearse it as well.

The idea of imaginary numbers is huge! There is so much that can be taught but I narrowed the focus to ensure that pupils could achieve the goal required. I have attached a few images of pupils’ work, and a video of a pupil dictating his understanding to me. I stop him because he did not simplify the surd where we had the greatest possible k integer. The pupil was correct in the method that he was using as it had two steps, but I wanted pupils to simplify the surd using the largest factor.

This post  explained how I taught pupils the procedure of solving equations where the result has an imaginary number. The video posted shows the outcome of the session where two pupils solve one equation. Enjoy the video available at this link with a selection of photos of the pupils’ work.

https://1drv.ms/f/s!AjSSrbwXqTsRhFjuw0ViPISNYBns

Think this sounds interesting? Come visit! We love having guests. It challenges our thinking and it boosts the pupils’ confidence to have people come in to see them.   

Think it sounds wonderful? Apply to join our team of enthusiastic maths nerds! We are advertising for a maths teacher, starting in September (or earlier, for the right candidate). Closing SOON: https://www.tes.com/jobs/vacancy/maths-teacher-brent-440947 

Teaching Imaginary numbers – Part 1/2

At Michaela Community School we run a selection of extra curricular activities after school which complement pupils’ mainstream learning. In the Maths Department, we have a club called Mathletes. This club entails an additional hour of learning taking place once a week after school and it is specifically for one class that I teach (Top Set Y8).

This week I taught a session on imaginary numbers whereby pupils were able to solve the following equations independently:

Pic1007abThis blog is split into two parts. The first post will explain how I developed the pre-requisite knowledge pupils need to solve the equations above, with much focus placed on rational numbers because of the second equation. The second post will outline how I introduced a limited understanding of imaginary numbers, enabling pupils to solve both equations.

During my planning, I think of the pre-requisite knowledge pupils need in order to access the new topic that I will teach in the upcoming session. Before I started introducing the idea of imaginary numbers, I introduced the idea of rational numbers in a very limited sense. This was done deliberately – something which I will explain later on in this post.

A rational number is the square root of a square number which results in an integer answer. I introduced the concept like this.

This is not a rational number √2

This is not a rational number √3

This is a rational number √4

This is not a rational number √5

This is not a rational number √6

This is not a rational number √7

This is not a rational number √8

This is a rational number √9

This is not a rational number √10

This is not a rational number √11

This is not a rational number √12

This is not a rational number √13

This is not a rational number √14

This is not a rational number √15

This is a rational number √16

The above methodology was inspired by Kris Boulton’s talk on “The genius of Siegfried Engelmann”, delivered at The Maths and Science ResearchED conference in Oxford, and the National Mathematics Conference 7 in Leeds.

I asked pupils to raise their hands when they noticed the pattern. This was my introduction, because I wanted pupils to recognise that we can categorise the square root of a square number where the result is an integer as a rational number. I told them, for now, that this is a fact that you are going to accept and adopt, and that this is ok.

I am the classroom teacher of the pupils that attend the session and so I know how quick they are in recalling their square numbers and square roots. I also know they have memorised their times table facts from doing Bruno’s Times Table Rockstar programme, as rolled out in year 7 successfully by Bodil Isaksen, Head of Maths.

My next step was to pose the following question: “Now, we can spot which numbers are rational numbers, and which are not rational numbers. We are going to look at how we can rewrite the following  in the form k where k is the largest possible integer, and √a  is not a rational number.”

This is how it was laid out: “We are going to look at the number inside the square root. We are going to split it into two numbers each square rooted, where I have one square number, and one number which is not a square number.”

√72 = √36 x √2

“I am going to identify my square number which I can evaluate to get an integer.”

√72 = 6 x √2

“And I am going to rewrite it like this because we can”:

√72 = 6 x √2 = 6√2

Here k is 6 and  √a is √2.

I also reiterated that I deliberately chose the largest square number (HCF) that can divide 72 because it is more efficient. I left it there, because I didn’t want to go into more discussion which would deviate pupils’ attention away from what I had shown on the board.

I then asked the pupils the following questions, before I asked them to simplify a selection of surds. I didn’t use the word surd because my focus for the session was not “what is a surd” or “how do I simplify surds” but because the point of this task was to develop sufficient pre-requisite knowledge in order to understand how we were planning to solve problems such as x2 = -72. This is why I previously mentioned that I introduced the idea of rational numbers in a limited sense.

Before pupils committed pen to paper to simplify the following surds I asked these questions which pupils answered by raising their hands.

“How can I rewrite √98  as a calculation where I am multiplying a rational number and a non-rational number, the rational number is a square number being square rooted?”

√98 = √49 x √2

“How can I rewrite √242  as a calculation where I am multiplying a rational number and a non-rational number, the rational number is a square number being square rooted?”

√242 = √121 x √2

I demonstrated a complete worked example of evaluating a surd and then I rephrased my questions:

“How can I rewrite  √48 as a calculation where I have an integer and a non rational number? Say Step 1 then Step 2”

Step 1:   √48 = √16 x √3

Step 2: √48 = 4√3

“Why is the integer 4?”

One pupil responded, “because the square root of 16 is 4, Miss.”

Twenty questions later and we were moving on. These were the questions pupils were asked to complete to check they were able to simplify the following surds into the form k√a where k is an integer.

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After the exercise, I mentioned that we had previously learnt that when we square root a positive number we get a positive  solution, for example √9 is equal to 3. We have also learnt that when finding the solutions of  x2 = 9 then we have two solutions -3 and 3. I then posed the question: “Is there a number which I can square where I get a negative answer? If you think so, then on your mini whiteboard have an attempt, and if you think it is not possible then write it down your thoughts in your exercise book.”

Some pupils were determined to find a number that they could square to get a negative result. Some pupils were already light years ahead with an explanation.

In my next post, I shall explain how I introduced the idea of imaginary numbers and structured my teaching in order to enable pupils to solve the two equations outlined at the start using the pre-requisite knowledge showed here. There will be a video and some pictures of pupils’ work too.

Thoughts from #Mathsconf7: Adding and subtracting unlike fractions: generalising and investigating different problem types

When are teachers to generalise a procedural calculation to apply to all problem types? When are teachers to display all the different problem types for a particular generalised procedure? I always jump between generalising and displaying different problem types, including not stating that one technique is better than the other but stating that in some instances one technique can be prioritised over the other.

In Maths there are many generalised procedural calculations which are true for all problem types of a concept. For example, listing a step by step process for pupils to follow which exists for all problem types regardless as to whether the problems include integers, decimals, percentages, numbers in index notation and square roots etc. in the problem type. We can try to create all the possible visual forms of a problem type, where generalising a procedure is a great teaching technique because it allows pupils to demonstrate efficiently their understanding by outlining the algorithm between the problem and the solution – step by step.

Here is an example of generalising the step by step method in order to add fractions with unlike denominators:

  1. Find the lowest common denominator.
  2. Form the equivalent fractions.
  3. Add/subtract the numerators.

This generalised algorithm is also applicable in instances of adding or subtracting like fractions; but pupils already have the lowest common denominator, and the fractions being added or subtracted are in their equivalent form, and so only step 3 is applicable. However, adding like and unlike fractions are seen as two distinct procedures, when a question with like fractions is a similar problem type to where step 1 and 2 are already completed.

A generalised procedure to adding and subtracting fractions is applicable for the following problem types where we have improper fractions, mixed numbers and a mixture of the two, or with calculations using more than three terms. Even through each example looks visually different, distinct and more or less challenging than the other, the generalised procedure is applicable.

Generalised procedures are great for pupils to develop their procedural understanding. Further, by discussing, in addition, the different problem types of adding fractions it can empower pupils further. The reason why I say this is because this generalised form of adding fractions neglects the different problem types that exist for this topic and many others. Here are the three different problem types I am referring to:

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  1. Fractions where the dominator are alike (LCD already present)
  2. Fractions where the denominator are co-prime (share a factor of 1)
    1.  Where the LCD is the product of the denominators
  3. Fractions where the denominator are not co-prime (share a factor greater than 1).
    1.  Where the LCD is provided as one of the denominators
    2.  Where the LCD is not the product of the denominators.

Exploring different problem types leads to a greater development of knowledge and understanding which can be applicable in instances such as the problem types below:

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‘A’ builds on the knowledge of the problem type where the denominators are not co-prime and where the LCD is provided as one of the denominators.

‘B’ builds on the knowledge that the denominators are co-prime and therefore the LCD is a product of all three denominators.

‘C’ builds on the knowledge where the denominators are not co-prime but the LCD is not the product of the denominators.

Can the teaching of problem types alongside the generalised procedure be beneficial in the other realms of mathematics such as algebra, geometry or data? Possibly, here is an example of expanding double brackets in a generalised form as well as exploring the different problem types.

The generalised form can be summarised into the following steps:

  1. Draw a 2 by 2 grid.
  2. Write each term of the expression in each section of the grid.
  3. Multiply all the different terms.
  4. Collect and simplify.

Furthermore, this generalised form allows even the most visually complex problem types to be solvable for pupils, as per the below:

  1. (2x + 3)(3 + 2x)
  2. (ab + cd)(2ab – fg)

The intention of generalising is not to make model algorithms for pupils to replicate. The point is to empower pupils to be able to look at the features of a problem, identify the problem and identify the route between the problem and the solution for every possible problem, regardless of how complex or simple the problem is.

The generalised procedure is applicable in all four different examples of multiplying two binomials to form a quadratic expression, where the expressions multiplied are in the following forms (a and b represent different values):

  1. (x + a) (x + b)
  2. (x – b) (x – a)
  3. (x – a) (x + b)
  4. (x – a) (x + a)

Discussions can then take place where pupils can begin spotting patterns, that:

  1. the product of two binomials where the constants are positive, the second term will have a coefficient which is the sum of the constants. (x + 2)(x + 1) = x2 + 3x + 2.
  2. the product of two binomials where the constants are both negative will have a constant which will be positive because the product of two negative numbers results in a positive number (x – 2)(x – 3) = x2 – 5x + 6.
  3. the product of two binomials where one constant is negative and the other is positive then the ‘c’ term will be negative.
  4. a difference of two squares will result in no ‘b’ term because the terms will result in 0 and the ‘c’ term will always be the square number of the constant in the binomial (x – 2)(x + 2) = x2 – 4.

To conclude, generalising a procedural calculation for all problem types, and exploring different categories of problem types for a concept, can be incredibly valuable because pupils start spotting patterns which can increase their confidence when learning. Primarily, enables pupils to be able to identify the problem and the correct generalised procedural calculation required to find the solution. Now, I believe that pupils will learn when to apply the generalised procedural calculation to a problem type if they are shown the different problem types that can exist. Yes, for some topics there are hundreds of problem types that can be explored, but then generalising the procedural calculation is even more important; for a select few topics the practice questions can be categorised into different problem types.

I am still getting my head around it all, but would love to hear people’s thoughts.

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