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]]>__What is the 20% of the input that contributes to 80% of the result?__

The 20% input is referred to as **Atomising**. This is the process where you** break down a topic into its sub-tasks**. This is a term coined by Bruno Reddy.

The 80% result this achieves is that it allows pupils to develop **flexible knowledge of the concept.**

**Atomising** avoids two common pitfalls. Firstly, it **avoids re-teaching**. I’ve certainly been in a position where I have taught a unit of work and then realised certain sub-tasks that I could have taught which would have helped my pupils develop a better understanding of the concept. Secondly, you **avoid missing out sub-tasks to teach**. When you breakdown a topic into its sub-task you avoid missing out certain aspects of a topic which really need to be taught explicitly. Due to the so-called ‘curse of knowledge’, teachers often forget that there are certain decisions and procedural knowledge that we have but which we forget to teach to the kids. Either it is so obvious that we think the kids already know it or, more commonly, we don’t realise which areas of the topic we are teaching pose difficulties for the kids. It could be that the difficulty is caused by a weakness the pupil may have in a different area of maths. For example, if a child is finding the missing side length of a shape, and the perimeter is a whole number, and the sides add up to a decimal, then the area of difficulty may be that a pupil doesn’t know (or remember) how to subtract a decimal from a whole number.

Here are the 12 sub-tasks that I used to break down the topic of perimeter. I chose perimeter deliberately because I think it is a perfect example of a topic which is perceived to be very easy for pupils to learn. However, there are certain aspects of perimeter that I will identify which **I don’t think are explicitly taught**. I think this because I have seen this to be the case when looking through several secondary maths textbooks.

__Atomising__

Perimeter:

1) Perimeter of Irregular shapes

2) Perimeter of rectangles

3) Perimeter of parallelograms

4) Perimeter of regular polygons

5) Manipulating the perimeter of regular polygons

6) Manipulating the perimeter of a rectangle – one side length given

- Given the perimeter, what is the perpendicular length
- Writing the possible side lengths of a rectangle given the perimeter

7) Manipulating the perimeter of an Isosceles triangle and an Isosceles trapezium

- Given the perimeter, what is the size of each equal side length

8) Perimeter of a compound shape

- What is a compound shape?
- Perimeter of a L-shape compound shape (irregular hexagon)
- Perimeter of non-L-shape compound shape

9) Manipulating the perimeter of a Compound shape

- Finding one missing length of a compound shape – not using the perimeter
- Finding one missing length of a compound shape – using the perimeter
- Finding two missing lengths (not connected) of a compound shape – using perimeter
- Finding two missing length (connected) of a compound shape – using perimeter

10) Perimeter of Compound shape – combining polygons

- Combining regular polygons to form a compound shape – using the perimeter of the original polygons
- Combining irregular polygons to form a compound shape – using the perimeter of original polygons

11) Perimeter: Increased, decreased or stayed the same

12) Drawing shapes on a square grid

The thinking time that went into breaking down the topic of perimeter took me 10 minutes. I think if I hadn’t gone through this process then I would have certainly missed many sub-tasks. In addition, it would have resulted in me having to do a lot of re-teaching.

*Please note that some images aren’t drawn to scale, unless stated*.

__Example Sets for Each Sub-Task__

1) **Perimeter of Irregular shapes**

I started to teach perimeter using explicit examples where pupils had to deliberately add all the lengths. They had questions where the side lengths were labelled using integers, decimals (same number as well different number of decimal places), multiples of 10, etc.

2) **Perimeter of rectangles**

I started teaching this with all side lengths labelled, then only three sides labelled, then only one pair of perpendicular lengths labelled. Then I included questions where pupils had to deliberately compare the perimeter of two shapes. The questions were designed so it wasn’t visually obvious which shape had the larger perimeter. Here is an example:

An example like the one above forces pupils to add the values because, even though the longest length has increased in size, the smaller length has decreased in size. This question is also designed so the shapes have the same perimeter.

3) **Perimeter of Parallelograms**

This sequence was very similar to the sequence for finding the perimeter of a rectangle:

- All side lengths labelled
- Three side lengths labelled
- One long and short length pair labelled, etc

4) **Perimeter of Regular polygons**

Here pupils are being taught to calculate the perimeter of a regular polygon using multiplication. They are taught that if you multiply one side length by the number of sides then you can calculate the perimeter.

The sequence of questions shown above demonstrates that we can get the same perimeter despite the polygon changing and its side length changing. I have also deliberately designed the questions to communicate this flexible understanding of perimeter. From the first question to the second, the side length changed. When you move to the third question then the shape and side length change, but the answer doesn’t change. For the fourth question, the side length and shape have changed but the answer doesn’t change.

5) **Manipulating the perimeter of a regular polygon**

Here pupils are given the perimeter of a regular polygon, and they know the number of sides each regular polygon has, so they need to find the side length of that regular polygon.

Here is an example set of questions:

Here the perimeter of the shape has stayed the same, but the shape has changed. Here pupils can see that **when the number of sides increases, then the side length decreases**, given that the perimeter hasn’t changed. They can also see examples where the side length can be a fraction.

I haven’t asked pupils to convert the improper fraction into a mixed number because a mixed number communicates size in a way that an improper fraction doesn’t. A mixed number would communicate more to a pupil about the side length of a square in relation to the side length of the pentagon in the previous question. However, the focus is not in understanding the magnitude of the square and heptagon. The focus here is **to get pupils to state the perimeter of a regular shape given the perimeter of that shape**. Pupils are practising the procedure of **dividing the perimeter** by the **number of sides of the regular polygon**, to find the side length.

Pupils are then asked questions where they are given the perimeter and they must state which regular shape would have the greatest side length. This helps pupils to see that a regular polygon with more sides will have a smaller side length compared to a regular polygon with fewer side lengths, given they have the same perimeter.

For example:

And here is an example where they aren’t given the perimeter at all:

Within this aspect of perimeter, I also created questions where pupils had to find the perimeter of a regular shape knowing that the **value of the side length is equal to the number of sides that polygon** has. Here pupils are using their square number knowledge to find the perimeter of a regular polygon.

6) **Manipulating the perimeter of a rectangle – one side length given**

Here pupils are given one side length of a rectangle and the perimeter of that rectangle. They now need to find the other perpendicular side length. Here they are practising the procedures of:

1) Double the side length given

2) Subtract from the perimeter

3) Divide that by two to find one of the missing side lengths

Here are some examples:

I deliberately put in an example where pupils were doubling an even number for the first example and an odd number for the second example. I deliberately gave pupils the longest length in the first example and the short length in the second example. This is because I want them to see that I have changed the side length but the procedure to find the other length hasn’t changed.

The third example uses a decimal because, **when you double it, you get an odd number**. When** you subtract from the perimeter, you get an odd number, so when you halve the odd number you then get a decimal.** I made the question arithmetically more difficult because **I knew that in the first two examples the answer would be an even number which is divisible by two**. I wanted to make a question where **pupils could see that you can get a rectangle with a side length which is a decimal**, and **not a whole number**.

I then moved on to asking pupils to state the different possible rectangles they could make for a specific perimeter. There are many possibilities. I restricted the possibilities by putting in a few conditions. Here is an example:

Here pupils can only give these answers, because the first value is A, and it must be larger than B’s value. In addition to this, I designed the question so pupils had to stop at 7 + 5 to avoid writing 6 + 6 because then the value of A and B would be the same. This is not what the question is asking.

7) **Manipulating the perimeter of an Isosceles triangle and an Isosceles trapezium**

Here pupils will practise finding the missing side length of an Isosceles triangle where they are practising the procedure:

1) Perimeter – base length

2) Divide by 2 to find the value of each equal side length

*The images aren’t drawn to scale.* Here are some examples:

Here I deliberately used 8 and 12 in the first two examples to get kids to actively use the procedure to avoid being in autopilot mode. The numbers are similar for this reason. Pupils can then see that as **the base length increases between the first and second question, the side length becomes smaller**. In the final example I use an odd number for the base number, to deliberately get each equal side length to be a decimal.

Pupils then apply a similar procedure to find the missing side lengths of an isosceles trapezium, but there is an additional step:

1) Add the known side lengths

2) Perimeter – total of side lengths

3) Divide by 2 to find the value of each equal side length.

*The images aren’t drawn to scale.* Here are some examples:

Pupils here also see that, **when the top parallel length increases from the first example to the second example, each equal side length decreases**. Similarly, from the second to the third example, when one of the parallel lengths go from 5cm to 4.5cm, each equal side length increases.

8) **Perimeter of a compound shape**

Pupils will learn what a compound shape is. They will then find the perimeter of a compound shape. I will use the L-shape compound shape because pupils will eventually learn that if you add the longest horizontal length and the longest vertical length, then double it, you will also find the perimeter of the shape.

Pupils will see questions where they are working with integers, decimals, both etc.

*The images aren’t drawn to scale.* Here are some examples:

They will then learn to find the perimeter of different types of compound shape. *The images aren’t drawn to scale.* Here are some examples:

9) **Manipulating the perimeter of a compound shape**

Here pupils will first be asked to find the missing length. They won’t be using the perimeter at all. This is because I want pupils to understand **which lengths are related to each other, and which aren’t**. In the second image, the vertical lengths are related because the longest length subtract the shorter length will find the value of A. These lengths are connected to each other.

Similarly, in the second image, the horizontal lengths are related because the two shorter horizontal lengths sum to give the size of the longest horizontal length. These lengths are related.

So, pupils will first practise finding the missing lengths without using the perimeter at all.

They will then be asked to find the perimeter of the shape once they have found the missing values for A and B.

*The images aren’t drawn to scale.* Here are some examples:

Here is another example which is rotated to get pupils to still identify the connected side lengths:

And another, finding a missing length for different compound shapes:

I then increase the difficulty in finding the missing length of a coumpound shape. **Here we can’t find the missing lengths unless we have the perimeter**. This is because **the third image has two conneted vertical lengths which are missing.**

*The images aren’t drawn to scale.* Here are some examples:

If the perimeter of the compound shape is 36cm, now I can find both missing lengths.

*The images aren’t drawn to scale.* Here are some examples:

Questions like these allow pupils to understand the following:

1) **If two connected lengths are missing, then I can only find these lengths if I have the perimeter**

2) Given the perimeter, I must follow this procedure.

I like these questions a lot because **I have made the concept more difficult by not combining other perimeter with other concepts**, or **by making the numbers more difficult**. I have taken the concept of perimeter with compound shapes and **changed the amount of information given**. **The difficulty is within the concept.**

10) **Perimeter of Compound shape – combining polygons**

**Combining regular polygons to form a compound shape – using the perimeter of the original polygons**

Here pupils are learning to combine different shapes to form a compound shape. Here they see a compound shape being formed and trying to understand **what lengths are required and which aren’t** required. *The images aren’t drawn to scale.* Here are some examples:

**Combining irregular polygons to form a compound shape – using the perimeter of original polygons**

Here we have moved from combining regular shapes to irregular shapes. I must structure the question in a way that one length of the first shape is equal to one length of the second shape. This gets pupil again recognising which side lengths of the two original shapes are required, and which aren’t, to find the perimeter of the new compound shape. *The images aren’t drawn to scale.* Here are some examples:

11) **Perimeter: Increased, decreased or stayed the same**

Pupils are moving onto identifying that the **perimeter is something that can change by increasing, decreasing or changing shape, but the perimeter can remain the same.**

Here they aren’t being asked to find the perimeter of a shape, but instead they are to say whether the shape has a perimeter which is greater than, smaller than, or equal to the perimeter of the original image. *The images aren’t drawn to scale.* Here are some examples:

This also allows pupils to see that we can change the image and that the perimeter can change, but it also may not change.

12) **Drawing shapes on a square grid**

Pupils are now at a point where they are taking their knowledge of drawing rectangles with a specific perimeter which has been taught in this sequence, but now drawing it on a square grid.

*The images aren’t drawn to scale.* Here are some examples:

Pupils are then equipped to match a specific perimeter to a specific image. They are applying what they have learnt rather than answering direct questions.

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]]>

Image: Mr Gabr teaching. This is from a different lesson. Not the lesson mentioned in the blog.

Mr Gabr’s million dollar smile greets children whilst he stands at the threshold of the Classroom. This is a Teach Like a Champion (TLACI) technique referred to as ‘Threshold’. One foot in the corridor to keep track of children walking towards the teacher. One foot in the classroom to ensure that pupils know that when they enter the room they are to be silent and start the work prepared on the whiteboard. Mr Gabr’s eyes are everywhere!

The first child enters the classroom, mirroring Mr Gabr’s smile. The pupil walks to his desk and starts thinking about the questions displayed on the board. This pupil silently recites the two definitions written on the board. Another two pupils enter and do the same. Every second counts and Mr Gabr is quick to start firing questions to individual pupils whilst they are organising their books and getting their equipment out, straight into a SLANT position, tracking Mr Gabr.

Mr Gabr asks one pupil a question which he answers incorrectly. Mr Gabr poses the same question to another pupil. The second pupil asked says the right answer, Mr Gabr then asks the first pupil to repeat the answer. The first pupil starts smiling, Mr Gabr reads his mind “See you did know the answer, you just spaced for a minute, happens to me too.” This is a lovely and slick demonstration of TLAC’s teaching technique referred to as “No opt out.”

All pupils have now entered the room, at their desk, book and equipment out, Mr Gabr tells all pupils to take a seat. He informs pupils to get writing into their books to answer the questions on the whiteboard. Great demonstration of TLAC’s ‘Do Now’, a speedy review for 5 minutes. Pupils are in a solid routine and start the work without any direction from Mr Gabr. To help pupils Mr Gabr reads out the question, asks for whole class choral response after 3 for the answer to the question.

**Mr Gabr:** *“What is the definition of a prokaryotic cell? 3 – 2 – 1″*

**Whole class:*** “A cell with no membrane-bound organelles.”*

**Mr Gabr:** *“Give me an example of what I mean by ‘no membrane-bound organelles’? 3 – 2 – 1″*

**Whole class:** *“No nuclei”*

**Mr Gabr:** *“What is the definition of a eukaryotic cell? 3 – 2 – 1″*

**Whole class:** *“A cell with Membrane-bound organelles.”*

**Mr Gabr:** *“Give me an example of what I mean by ‘membrane-bound organelles’? 3 – 2 – 1″*

**Whole class:*** “Cells with nuclei”*

**Mr Gabr:** *“Now only the boys!”*

**Boys only:** *“Cells with nuclei”*

Kids have finished writing. Pupils are asked to SLANT and track the teacher. Mr Gabr poses his next series of questions. A pupil responds with a mediocre answer where he used the term ‘egg’ instead of ‘ovum’. This is another learning opportunity for the kids. Mr Gabr then poses the questions “What is the scientific word for ‘egg’?”

There is a forest of hands up in the air. At Charter, we push to have as many pupils to have their hands up in the air to answer questions. The pupil who gave the mediocre response is asked to repeat his answer using the word ‘ovum’ rather than the word ‘egg’.

It’s been 10 minutes of teaching where every second has really mattered. He introduces what will be taught today and dives straight into looking at respiration’s chemical equation. He is very deliberate with wording, and he pre-empts possible misconceptions that pupils may have in advance. For example, he states that in a chemical equation we use an arrow to separate the reactants from the products, not an equal sign. Pre-empting that ‘equation’ is a term used in maths which refers to an equal sign or identity sign being present. He pre-empts that reactants of a chemical equation are on the left hand side of the arrow, and that products are on the right hand side of the arrow. He talks about how when we write carbon dioxide as one of the products in the chemical equation, we write dioxide right below carbon, we don’t go to the left of the margin to start a new line, as it may look like you wrote dioxide to be a reactant.

Image: Recreation of Mr Gabr’s board work

He explains that we write energy as a product but the term ‘+ energy’ must be in brackets, because energy is not matter. Chemical reactions do involve energy. He emphasised the importance of this and showed pupils how to write the equation with energy as a product, and the incorrect way of writing the equation with energy as a product.

I’ve only witnessed the first fifteen minutes of Mr Gabr’s lesson, but here is a summary of what I’ve witnessed to be great classroom practice from an experienced teacher:

- Warm welcome to the classroom
- Use of TLAC teaching techniques: “Threshold” “I say, you say” “No opt out” “Slant” “Do Now”
- Whole class choral response: all pupils are expected to participate, if they don’t know the answer then they are able to learn it from others around them
- Purposeful start to the lesson: task is prepared
- Improving pupil answers from mediocre answers to top quality answers
- Pre-empting misconceptions
- Precise and Concise teacher instruction

*At Great Yarmouth Charter Academy, we have an open door policy where teachers are welcomed to observe and learn from their colleagues. This invitation to visit extends to local members of the community, parents, teachers and Headteachers far and wide. We have had several visitors come and experience a day with us. This is also an invitation welcoming you to visit our school. Please feel free to get in touch with the school office to arrange a tour. *

*If you wish to know more about our school feel free to check out the blogs and tweets of our staff:*

*Head Teacher: Mr Barry Smith (@ BarryNSmith79)*

*Assistant Principal: Dr Anthony Radice (@ AnthonyRadice1)*

*Assistant Principal: Mr Darren Hollingsworth (@ DarrenHolly3J)*

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]]>16. Deciding whether to multiply by more than one or less than one

This is one of my favourite component skills taught in the Level D textbook series. In this exercise pupils needed to identify whether what we multiply the first integer by is more than 1, less than 1 or equal to 1 depending on the result of the calculation.

*If you multiply by more than one you end up with more than you start out with. *

*If you multiply by less than one you end up with less than what you start out with. *

Pupils did find this difficult to learn. I did breakdown the questioning further in addition to what was stated in the teacher presentation book.

1.Which number is bigger?

2.What number is smaller?

3.Am I going from a big number to a small number? Yes or No?

4.Am I going from a small number to a big number? Yes or No?

5.If I am going from a big number to a small number, then I am multiplying by less than 1.

6.If I am going from a small number to a big number, then I am multiplying by more than 1.

Pupils were also taught that if the value in the calculation is equivalent to the value in the answer then I have multiplied by 1. This was understood because of learning about equivalent fractions where you multiply by a fraction which can simplify to one.

The reason why I really enjoyed teaching this was due to the rigour of the exercises. The design of the exercise below is pure genius. It gets kids to think whether a fraction less than 1 is smaller than an integer, or which fraction is smaller out two fractions with the same denominator.

In regards to question (d), pupils are able to write seven-sevenths as an integer equal to 1. They are seeing that they are going from a small value to a big value so we are multiplying by more than

For question (e), pupils write the fraction of twenty-seven-ninths as an integer before they determine whether they multiply by more than 1, equal to 1, or less than one.

17. Multiple Representations

Pupils are identifying the relationship between multiplication and division by showing division calculations as a reverse multiplication calculation, a fraction and the bus stop method.

There are exercises which pupils are given a calculation in one form and they need to write it into the other format. I really value exercises like this because when pupils complete mathematics problems in their GCSE or A Level exam papers that they can write their answer as a fraction, or divide the top and bottom to write it as a decimal etc. I just recently marked my Year 11 mock papers and it was lovely to see that so many pupils were able to see that when solving to find a value that they saw it as a fraction, that they would then use the bus stop method to write the fraction’s value as an integer or decimal. This wasn’t the case in their paper workings from their first round of mocks in November.

18. Demonstrating equivalence

Pupils are now showing equivalence without images. Here they are given two fractions, and they need to determine what the first’s fraction top number is multiplied to get the second fraction’s top number. This is also done for the denominator. If they multiply the top and bottom by the same number then the fractions are equal, and they place an equal sign between the two. If the fractions are not multiplied by the same number then the fractions are not equal, and they place an not-equal sign between the two.

19. Number Family

A number family is a tool that pupils use to solve hard addition and subtraction problems. This has been blogged about here in a three-part series.

A number family is a sum between two small numbers presented above an arrow which has a large number at the end of the arrow. Pupils are taught that if you have a two small numbers then you add them to get the missing big numbers. Pupils are also taught that if you one big number and one small number then you subtract to get the missing small number.

Pupils are now taught how to apply this in problems where they need to demonstrate that the fraction of shaded parts and non shaded parts sum to 1.

On the number family, the fraction of shaded parts and the fraction of non shaded parts are the small numbers. They sum to 1, and pupils know this because they know that a fraction with the same top and bottom number simplifies to 1.

Exercises ask them to present a calculation between the shaded and non shaded parts.

Eventually pupils attempt exercises where they write a number family calculation from a sentence. They know that the number family’s big number will be 1. They write one as a fraction with the same denominator as the fraction of both small numbers.

This progresses in difficulty where pupils are given information not in terms of fractions but as whole numbers, they need to find the denominator and then write the information given as a fraction, then present in a number family structure.

Here are some examples of pupil work:

20. Writing a mixed number as a decimal

The final component skill looks as getting pupils to write a decimal as a top heavy fraction and then as a sum of an integer and a proper fraction. Pupils are told that the decimal point is the plus sign between the integer and fraction. This was a very smooth transition from working with fractions to writing decimals.

What’s next…

After looking in great detail into Engelmann’s teaching, specifically with fractions, I’m now attempting to create exercises which can be used in the classroom for mainstream teaching. Watch this space!

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]]>10. Introducing Mixed numbers

Mixed numbers are introduced as a sum between an integer and a (proper) fraction.

It is visually presented on a number line, *you go to the marker for the whole number. Then you count parts for the fraction.*

The skill is revisited where pupils write the addition sum between an integer and a fraction as a mixed number, without the number line. Pupils are then asked to do this again with three or four digit numbers.

Pupils also complete exercises where they:

- write an improper fraction as a sum between an integer and a proper fraction
- show the sum of an integer and fraction as a sum of two fractions

11. Multiplicative relationship between an integer and a fraction’s denominator

Pupils have been taught previously how to list a string of equivalent fractions equal to an integer, listed as skill 8. This was done through the use of their times table facts. Similarly, pupils are now taught how to find the missing numerator of an incomplete fraction equal to an integer.

12. Equivalent fractions from a diagram

Pupils are taught that:

*If fractions are equal, pictures of each fraction will have the same shaded area.*

*If the fractions are not equal, then the pictures do not have the same shaded area. *

**A teaching point to mention**, I got the pupils to **use a ruler to check to see if the shaded areas matched** to make them see the connection that equivalent fractions are equal in size, the equivalent fractions presented vary visually. This helped pupils to attempt multiple choice questions like the following:

This was also tested using equivalent improper fractions. In the image titled Part 9, I particularly like (c) and (e) because the parts are split horizontally and vertically. Engelmann here has varied the non-relevant aspects of each image: the number of units for each image, the shape being cut horizontally and vertically etc. What matter is the whether **the area between each image is the same**, **if so then those fractions are equivalent.** At this stage, pupils are not being asked to demonstrate equivalence using times tables between two fractions.

13. Multiplying fractions to demonstrate equivalence

Pupils are taught that **if you multiply an integer by a fraction which is equal to 1, then the result is the integer in the question**. Images are provided to show that you are taking two whole units, and splitting the shape into more parts in each unit.

Pupils are also learning to spot visually that when you multiply an integer by a fraction with the same numerator and denominator that it is equivalent to multiplying the integer by one.

Pupils are taught how to structure their working when multiplying fractions where they write the integer with a denominator of one, then multiplying the numerators, and multiplying the denominators.

14. Placing fractions on a number line (work on)

This was an exercise that pupils did find difficult. Pupils had to place an improper fraction on the number line. What pupils found the most difficult was picking an integer equivalent fraction which two-fifths came after. For example:

*I want to place two-fifths on the number line. I will go to the integer with a fraction which is just before two-fifths. I will go to zero-fifths, and count two parts to find two-fifths.*

*I want to place nineteen-fifths on the number line. I will go to the integer with a fraction which is just before nineteen-fifths. I will go to fifteen-fifths, and count four parts to find nineteen-fifths. *

*I want to place thirteen-fifths on the number line, I will go to the integer with a fraction which is just before thirteen-fifths. I will go to ten-fifths, and count three parts to find thirteen-fifths.*

Exercises were included where pupils were asked to also label fractions onto a number line where the integers, or the integers’ equivalent fractions aren’t labelled.

15. Equivalent Fractions – Multiplying a fraction by 1

In this exercises pupils are demonstrating equivalence between two fractions by writing the fraction to multiply the first fraction by to get the second fraction.

Pupils were using times tables to identify what the first fraction’s numerator and denominator are multiplied by. This fraction has the same value in the top and bottom of the fraction which then simplifies to one. **Equivalent fractions are multiplied by a fraction which simplifies to one.** This is also visually demonstrated.

Below is an example of an exercise, where pupils are asked to state the fraction for each image, and demonstrate equivalence by multiplying fractions.

A teaching point to mention: I did have to structure the working out for the pupils. I would say:

- I will write the first image’s fraction
- I will write a times sign
- I will draw a box for my missing fraction to show equivalence
- I will write an equal sign
- I will write the second image’s fraction
- What number goes in the top of the fraction?
- What number goes in the bottom of the fraction?
- Check that this fraction simplifies to one?
- Are these fractions equivalent? Yes or No?

It was important to include this because I wanted pupils to know exactly how to demonstrate their working out. The pupils in this group struggled to structure their understanding on paper in way which made sense to somebody reading their work. By sticking to the following, essentially sticking to a ‘script’ then pupils were able to demonstrate equivalence consistently in this type of exercise throughout the book.

Pupils were then given exercises where one fraction was equal to an incomplete fraction where they filled in the blank and the fraction that they were multiplying by. Here is an example of a pupil’s work.

In my next blog post, I will outline the remaining fraction skills that are taught in Engelmann’s sequence of lessons.

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]]>**Breakdown of each teaching point**

5. Simplifying a fraction to an integer

This is commonly taught. What made it so effective in Engelmann’s textbooks was the regular times table exercises (multiplication and division) which were included in every lesson.

Around Lesson 59, pupils were asked to simplify a fraction where **short division would be required**. Again, this was introduced only after pupils had learnt how to divide using short division.

Similar exercises were given where pupils were only asked to write the division problem before working out the problem. This was to ensure pupils were avoiding the common misconception of writing a division calculation were the divisor is written before the dividend.

6. Stating whether a fraction is an integer or not an integer

In this exercise pupils are asked to state whether a fraction will simplify to an integer, or will it not simplify to an integer. Note that pupils are stating a term for the positive case, and the ‘not’ term for the negative case. **The language of mixed number is not introduced**. This is because pupils will have to learn two separate terms for two positive cases. When you learn what something is and what it isn’t then you are learning where one concept is true in one instance and when it is false, and the case for when it is not true is the ‘not’ case. It is much easier for pupils to grasp than introducing two different terms for two positive cases.

7. Adding or subtracting fractions with like denominators

Pupils are taught that **when the denominators of a fraction are the same** then you **can add the fractions in they are written**. It is explained further that each fraction has each unit divided into the same number of parts, and this is why we can add fractions when they are written this way.

You can’t add these fractions the way they are written because the denominators are not the same.

The most important part of the wording is **‘the way they are written’.** Pupils are taught to visually spot when it is possible to add fractions by spotting the denominators being the same. Pupils are then given exercises to decide whether you can add these fractions the way they are written.

Engelmann does go into the reasoning behind why fractions with different denominators cannot be added or subtracted in the way they are written.

8. Listing multiple equivalent fractions for a specific integer

Since pupils practice their times tables in an exercise in each or every other lesson within the textbook series, pupils can quickly learn how to write multiple equivalent fractions for an integer. **Pupils learn the multiplicative relationship between the integer and the denominator of an equation. **Including ‘1’ as the denominator is important because I think it is sometimes overlooked.

9. Writing fractions from a sentence

Engelmann creates questions which force pupils to deliberately think. Here is a perfect example: pupils are told to write a fraction which meets certain conditions, such as the fraction being more than or less than one. Pupils are also using prior learning in this instance.

10. Introducing Mixed numbers

Mixed numbers are introduced as a sum between an integer and a (proper) fraction.

It is visually presented on a number line, *you go to the marker for the whole number. Then you count parts for the fraction.*

The skill is revisited where pupils write the addition sum between an integer and a fraction as a mixed number, without the number line. Pupils are then asked to do this again with three or four digit numbers.

Here is an example of a pupil’s work.

In my next blog post, I will outline the remaining fraction skills that are taught in Engelmann’s sequence of lessons.

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]]>**Connecting Maths Concept Textbook Series**

The CMC textbook series have multiple levels to select from to best suit the children who will be learning from the CMC textbook. I taught the Level D series. The textbooks served the purpose of acting as a remedial programme where pupils received 3-5 one-hour lessons per week on top of their mainstream maths lessons. There is a scripted teacher presentation book, a pupil workbook, pupil textbook and an answer key.

One note to mention on the scripted teacher presentation book. Many teachers do not like being given a teacher script because it seems unusual to teach using somebody else’s words. I felt this way initially. I then completely changed my mindset on this because the script was incredibly a**ccurate, deliberate and economical with text**.** I realised that Engelmann has written it better than anything else I had seen. **I found it to be a really humbling process because it made me realise that I had been very verbose with my teacher talk. I could say much more with fewer words. This is still a controversial topic in the Education world. However, I found that the script helped the children to articulate their understanding because they were able to repeat a precise explanation back to me, that was simply provided through the teacher script. This was something that @MrBlachford commented on too, on Twitter:

**A few overarching points**

The key theme that Engelmann demonstrates throughout his textbook series is using:

*teaching methods of future learning which never contradicts the teaching methods used for prior learning. *

There is total consistency throughout the teaching methods used and applied when learning a specific concept. This will be shown by outlining the teaching of 20 skills on the topic of fractions over a 100 lesson period. There is also a smooth track of **gradual complexity in the problem types that are used to test a pupil’s understanding. **The content that is taught looks deceivingly easy but pupils are taught teaching methods that can be applied in the most simplest of cases as well as the most complex of cases.

Engelmann’s lesson spread looks like this. The red sections are the teacher-led parts and the green sections are the parts where pupils work independently. Each lesson covers a range of 5-6 different topics such as adding, subtracting, fractions, times tables, ratio etc.

**Breakdown of each teaching point**

- Stating a fraction from a diagram

Engelmann introduces how to write a fraction from a diagram by using images where there was more than one unit, as shown below:

The wording to write a fraction was:

The top number is the total **number of shaded **pieces.

The bottom number is the **total number** of pieces in **one unit. **

The wording was precise, and fool-proof. I did alter the working to ‘the bottom number is the total number of pieces in **each unit’** simply because my pupils responded to that with more accuracy. The wording applied to cases where pupils were writing proper fractions, improper fractions as well as whole numbers. The questions that were selected which tested this also:

The wording also avoided the common misconception that pupils have in writing the denominator as the total number of pieces that they can see between the number of shapes shown. This is because the wording say ‘in one unit’.

2. Stating a fraction from a number line

The same wording was introduced. I did explain to pupils that **one unit** was **the gap between each whole number. **

There were exercises where pupils were practising both skill #1 and #2 side by side.

3. Stating whole numbers from a diagram or a number line

If you see a diagram or a number line where **a certain number of units** are **shaded** then the diagram or number line **represents a whole number.** The point is also mentioned that **if there are no leftover parts are shaded then we definitely have a whole number.** The opposite case is presented that if we do have leftover parts that are shaded then we don’t have a whole number.

The scripted nature of it states:* “3 units are shaded. So the fraction for this picture equals 3.”*

Similarly, there is a negative example of this where the script words what the teacher says as: *“There are more than 3 units shaded, but less than four.”*

The testing stage phrased the question as: **which picture shows a whole number?**

4. Fractions as integers from a diagram

For it to be the case that we are showing an integer visually then there is **only one part in each unit**, meaning that **each unit is one whole shape**. The number of shaded units is the top number. The number of parts in each unit is the bottom number. This **method is consistent with previous problem types of stating a fraction** from a diagram or number line. What I like the most about the sequence of questions is that the nuanced example is taught after the explicit examples. #4 is the nuanced example of stating a fraction from a diagram because its not as straightforward. Engelmann’s presentation book still uses the same wording to state a fraction from a diagram, which shows pupil that they are equipped to answer even the trickest of questions.

The wording to teach pupils how to state a fraction as an integer from a diagram goes as follows:

*There is one part in each unit. 4 parts are shaded. The fraction for the picture is 4 over 1. Here is the equation: 4/1 = 4.*

*There is one part in each unit. 6 parts are shaded. The fraction of the picture is 6 over 1. Here is the equation: 6/1 = 6*

I remember one of the boys in my intervention class, looking at the question and attempting it before I started going through it. He did say the correct answer, and he was really impressed with himself. I then asked him how he did it. His response was **‘using the way that you taught me’**. He was able to apply the method taught for previous problem types in this case too with no teacher guidance.

Pupils were then asked to state a fraction as an integer from a number line as well.

Pupils completed testing exercises where they had to state a fraction which could simplify to an integer where each unit had more than one part. These exercises also tested a pupil’s prior learning alongside new learning content.

The most difficult problem testing skill #4 was asking pupils to write the fraction for each image that shows a whole number. Pupils are now applying collection of separate component skills where pupils are to: read a number line, write a fraction from the number line, and state whether that fraction is a whole number.

The next degree of complexity introduced where now pupils need to state the fraction for each whole number on a number line.

Here is the wording used in the textbook:

*There are three parts in each unit. So the bottom number of each fraction is 3. The bottom number for each fraction is the same:*

*The top numbers are the numbers for counting:*

There are exercises further on in the series of lesson where pupils are given the same task but the number line doesn’t state the number of parts in each unit, instead the denominator is provided.

In my next blog post, I will continue to outline the remaining fraction skills that are taught in Engelmann’s sequence of lessons.

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]]>The post Resourcing: Applying Variation Theory appeared first on Conception of the good.

]]>**Variation Theory**

Variation Theory is posited on the view that **“when certain aspects of a phenomenon vary which its other aspects are kept constant, those aspects vary are discerned.”** (Lo, Chik & Pang 2006:3) It has been previously pointed out that **“to see or experience a concept in a certain way requires the learner to be aware of certain features which are critical to be the intended way of seeing this concept.”** (Lo and Marton 2012) Here is an example, I am attempting to teach pupils how to factorise a quadratic expression. The critical aspect I want pupils to focus on is **determining the factors of ‘c’ that sum to give the ‘b’ coefficient**. I have kept the ‘a’ coefficient the same, and I have kept the ‘c’ constant the same too. I want pupils to work on the aspect that the ‘b’ coefficient is always the sum of two factors of the constant. Here is a similar example:

Here is another example, but here I am focusing pupils’ attention on the critical aspect of **determining the factors that multiply to give the ‘c’ constant**. **The factors will always sum to 10 which is the ‘b’ coefficient.**

I have deliberately included the last example. Where pupils are taught that two numbers that add to give 10, but the product will be 0. I have included it at the end because it is a nuanced example of the set of expressions above. It is nuanced because its explicit features are not visible for pupils. It does not follow the structure of (a + __)(a + __) because the bracket of (a + 0) = a

What about introducing a practice exercise where you want pupils to focus on their negative mental arithmetic? The critical aspect for pupils to focus on is **determining the factors of -24 that sum to give the ‘b’ coefficient stated.**

What I really like about the above exercises is that pupils are experiencing different forms of variation from the previous set of expressions. However, in this practice exercise, I have changed only one aspect of each expression, which is the focal point of the intended concept being taught. **If I change both the ‘c’ and ‘b’ coefficients of a quadratic expression then I am asking pupils to make two decisions, essentially to focus on two different things, two numbers that sum to ‘b’ and the same two numbers that multiply to ‘c’.** This above exercises constrains a pupil’s thinking deliberately for two expressions at a time so the intended process of factorising a quadratic expression becomes clear. They can also see how the negative arithmetic changes the value of the ‘b’ coefficient.

I can now bring a practice exercise where pupils are now able to put their understanding of how to factorise a quadratic expression to the test:

A few key points about the design of this exercise set:

- I have
**only changed one aspect of each expression**from the previous expression. - I have focused on
**changing the surface of each expression**. The procedural process is the same. - Between (b) and (c) I have deliberately changed the focus from finding the sum of the constant’s factors to then finding the product of two factors of 12 that also give a sum of the ‘b’ coefficient.
- I have structured the calculation for the coefficients of ‘b’ and ‘c’ because I want pupils to explicitly see how they are determining the numbers that go into the factorised form. I wanted pupils to see the depth underneath the surface of each quadratic expression. I also want pupils to see the procedural relationship between the factorised form and its quadratic expression.

To summarise, this is simply an attempt to widely implement variation theory when resourcing. I have seen that pupils are able to draw the intended mathematical understanding being taught sooner. A greater proportion of the class are able to apply their understanding of the intended concept correctly. The design of this exercise also aids pupils to deliberately think and apply their understanding at each expression they had to factorise. The above exercise is an improvement from previous exercises I have made in the past, and can be improved further after seeing how pupils respond to it.

Paper used: http://www.cimt.org.uk/journal/lai.pdf

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]]>The post TLAC Show call: Using Mini-whiteboards in the Maths Classroom appeared first on Conception of the good.

]]>The use of mini-whiteboards in the classroom can go either way. It can be a really great resource to check pupil understanding, identify and close knowledge gaps etc. Or it can also be a nightmare unless pupils are taught how to use mini-whiteboards effectively in the classroom. This has been blogged about previously, which can be found here.

I use mini-whiteboards because it is the most effective way of using a teaching technique known as *Show Call**¸* from **Doug Lemov’s**, **Teach Like a Champion** 2.0 (TLAC).

**Craig Barton’s Podcast with Doug Lemov goes into further detail about Show Call. Highly recommend it.**

__What is Show Call?__

From TLAC *Show Call* is defined as a teaching technique which:

*“Create(s) a strong incentive to complete writing with quality and thoughtfulness by publicly showcasing and revising student writing – regardless of who volunteers to share.”*

Here is the context: I pose a question which is projected on the screen using a visualiser which I want pupils to complete. I **have provided an exemplary worked example**, **stating all the steps** to go from the question to the answer. I have emphasised the areas of the working out which pupils will struggle with or most likely make a mistake at a specific point. I have demonstrated **a piece of quality written work** that I want the pupils to aspire to. I then give a similar question for pupils to try on their mini-whiteboards.

At this point, pupils are scribbling away on their mini-whiteboard, I am circulating the room or watching pupils writing on their boards from the front of the classroom. I am looking for potential mistakes pupils are making that I may not have pre-empted, or spotting pupils who are greatly struggling or who are finishing of an exemplary answer, like the one projected on the screen.

I start the count-down. 10 seconds left. Start counting down from 5…4…3…2…1…and show. Pupils simultaneously show their whiteboards.

__What does Show Call achieve in this instance?__

Pupils are given an opportunity to put their initial thoughts and understanding of the concept being taught on a mini-whiteboard for me to then suggest improvements. **The public showcasing incentivises pupils to produce quality written work. **

I show off top-quality work which other pupils can see and aspire to. This creates an environment of **success and positivity balanced with an expectation that pupils need to develop and improve their work** if what they have produced isn’t parallel to what is being shown. The act of finding exemplary work is fully intentional.

Similarly, it gives me the teacher an opportunity to take a pupil’s piece of work which may present a mistake, or a common mistake that many pupils have made, or an area of improvement and start a conversation **giving pupils precise and actionable feedback**. The ability to give an entire class specific edits or points of revision empowers pupils to be constantly learning and improving their written work. At this point, **you see pupils thinking**, their eyes following the calculation on the board, and you see their eyes light up with a smile of satisfaction that they now know how they can go from **“good to great**”. Pupils are coming closer to making their work like the exemplary piece of work that has been celebrated. The process is **efficient** **and** **productive**.

__What type of revisions and edits is Show Call trying to achieve? __

When I started to use this technique in my second year of teaching, I didn’t use it correctly because I was focused on getting pupils to correct one-time mistakes rather than revising the structure of their work, or the content of their work and/or **focusing pupils to revise a replicable skill.**

The one-time mistakes **can be pre-empted by using very good worked examples** where a teacher goes through them and points out possible mistakes that pupils may make. These are mistakes that pupils need to avoid when attempting similar questions on their mini-whiteboards. The revisions that I needed to get pupils to get in the habit of could all be categorised under the umbrella of **‘applying a replicable skill to multiple concepts’**. E.g. when using the four operations when working with fractions, to always place a 1 in the denominator of a whole number. Or when expanding and simplifying an expression like the following. Being able to underline each expression with the number and sign which will multiply the entire expression in the bracket. Then drawing each table where we write -3 rather than 3. This is because pupils need to multiply each term of the expression by -3. (Image 1)

**OR **drawing a ‘boat’ around the two single brackets, including the sign on the left. This is not the case in the first image, but then achieved in the second image with a different example. ** (Image 2)**

**Image 1 and 2**

A constant push on making such revisions motivates **a pupil to actively listen to the teacher and implement the feedback provided.** *Lemov* calls the **act of reviewing work, as seen by champion teachers, as using the language of “check or change”**. Time does need to be provided for such revisions to be made, but with a sense of urgency so pupils are going at a productive pace. I have seen instances where pupils are taking ages because they lack that sense of urgency. It usually is the case that with a reminder that you have a minute left, along with a countdown then pupils pick up the pace.

__The Take and Reveal__

*Lemov* goes into detail about *how* the deliberate **act of taking** a pupil’s work and **the reveal** of the pupil’s work can increase the impact of *Show call. *

For example, when circulating the room, I’ll say **“I’m looking forward to picking the best piece of board work to show to the class so we can learn from it.”** Pupils then feel the desire for that piece of work to be theirs. Once I say, 3…2…1…and show! I will look across the room, and ask a pupil if I can show their piece of work so we can all learn from it. Sometimes, **I’ll take a pupil’s board work without saying anything**, in TLAC this is referred to as an **“unnarrated take”.** Giving pupils a sense that this is a normal thing to take the best piece of work to show to the class. Usually, I’m grinning from ear to ear over the best piece of work, and so is the pupil whose board I pick.

If I see a mini-whiteboard that demonstrates a common mistake, I may take the board without mentioning the pupil’s name, or I may ask the pupil **“Do you mind if I can take your board to share a common mistake that some others have made, so we can learn as a class to make everybody’s work better?” **I could hold the board up and say **“Can anybody tell me how this piece of work can be improved?”**

**Image 3 –** In this piece of work, the pupil’s working out is correct. The final expression is incorrect. They have identified that they need to sum the values of 4 and 2, but they have written +6c rather than -6c.

**Image 4** – Another pupil in the class made a similar mistake as demonstrated in **Image 3. **They were able to make the required revision in another question.

I then asked how the following answer could be written in a more elegant way, implying that we do not need to show the coefficient of b, as 1.

At this point, it is important to mention that sometimes this act can be pointless because the mistake may be too difficult to spot or to even understand, that it may make sense for you to go through it. Sometimes, I think it is **powerful** because **it gets pupils to deliberately spot a mistake**. The act of spotting a mistake and changing it **is a form of learning**, an effective one. At this point, I’ll see a couple of hands fly up in the air and then slowly the number of hands in the air will increase.

__What are the long-term benefits of Show call?__

The main benefit I have found is that **I can see pupils checking and/or changing their written work to the point of complete accuracy**. This then results in pupils completing written work in their books to be **thoughtful** and **well-constructed** work. I am not drowning in marking work with multiple mistakes that could have be corrected through the use of mini-whiteboards. The quality of pupils’ work in their books, then in their weekly quizzes, as well as their working out in the exam has drastically improved. The pupils are more successful and capable in the subject.

*The content of this blog is heavily borrowed from Doug Lemov’s Teach Like A Champiom 2.0. I would highly recommend reading this book to gain a greater understanding of Show Call, as well as other teaching techniques. *

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]]>The post Engelmann’s Faultless Communication in the classroom: Algebra appeared first on Conception of the good.

]]>*“rules out the possibility that the learner’s ability to respond appropriately to the presentation, or to generalise in the predicted way, is caused by a flawed communication rather than by learner characteristics.”*

For this to be the case, some assumptions have to be made about the learner:

*Assumption 1:*The capacity to learn any quality is exemplified through examples

*Assumption 2:*The capacity to generalise to new examples is on the basis of sameness of quality (and only on the basis of sameness)

**Structuring Examples**

The examples that are chosen demonstrate a quality of sameness of a concept, and such communication, as outlined in Engelmann’s *Theory of Instruction, *must meet these structural conditions:

- The examples present
**only one identifiable sameness in quality**– not more than one - The communication requires
**a signal for the quality of sameness that each example demonstrates.**A second signal is also required to identify the examples that do not share the quality of sameness. - The examples chosen must demonstrate
**the range of variation that typifies the concept**. Each positive example will be slightly different from each other but the examples will all share the quality that is to be generalised. - Negative examples, as well as positive examples, must be shown to show
**the limits of variation in quality that is permissible for a given concept**. - The communication must provide
**a test to check whether the learner has received the information**provided by the sequence of examples.

Here are a few examples of teaching sequences that I have used in the last week to communicate which terms are like terms:

In regards to the conditions stated above, this sequence meets the conditions because:

- It is communicating one quality of sameness – exactly the same unknown in each term
- There are two signals, one for the positive examples and one for the negative examples
- There is a wide variation that typifies the concept, negative terms, integer powers, fractional powers, negative powers etc. I could potentially include more examples with decimal powers etc.
- Negative examples have been included and positioned deliberately within the sequence to contrast with the example presented before.
- A test was given to follow.

**How did the lesson proceed?**

After getting the pupils to do a mini-quiz on whiteboards to identify whether the following group of terms within the expression were like terms or not like terms. I went through a series of teaching examples of how to collect terms to write one expression in its simplified form. In this lesson, pupils learnt how to:

- Differentiate between like terms and unlike terms within an expression
- Simplify an expression by collecting the like terms.

Here is the teaching sequence of examples used to teach the second point:

Pupils attempted to collect like terms on their mini-whiteboards using the following set of examples:

A few key points to mention here: I deliberately gave pupils examples where they had to collect negative terms. I taught pupils that the sign on the left side of the term was part of that term. For example, in regards to example 2, a pupil would say:

Pupil: The two like terms are a^2 and 4a

Pupil: The unlike term is -5a

I wanted pupils **to practise adding negative terms because that is what pupils struggle with the most when learning how to collect like terms.** My examples gradually escalated in difficulty but that difficulty is important for pupils to practice with the teacher.

In regards to example 6, I wanted pupils to see that when two terms give a result of 0, we don’t write 0a^2b and we don’t need to write + 0 to the simplified expression.

This was the second set of teaching examples that were used to collect like terms where a term had more than one variable:

There are some topics in Mathematics where it is easier to create an sequence of teaching examples in line with the structural conditions of Faultless Communication.

I attempted to apply the a similar structure when teaching Year 11 how to change the subject of a two step equation where the desired subject either has a power or root. Here is my attempt:

Make ‘f’ the subject:

- The quality of sameness I attempted to demonstrate was that when you list out the order of operations of what happens to ‘f’ we reverse that order. For example, I have ‘f’, I square it, then add 2. Now, I reverse the order of operations. Therefore, I first subtract 2 on both sides of the equal sign, and then square root both sides of the equal sign. The sameness was in the procedure that pupils used to change the subject of the equation.
- There is no signal that I used here because It wasn’t applicable for what I was intending on teaching – how to change the subject of an equation.
- The examples demonstrate that the if I add 2, subtract 2, multiply by 2, divide by 2 I will always reverse the order of operations of what happens to the desired subject.
- I didn’t have a set of negative examples because it wasn’t applicable for what I was intending on teaching.
- There was a test of similar examples to attempt

I do think it isn’t always possible to create a sequence of examples which meet all the structural conditions of Faultless Communication. The example where pupils learn how to rearrange the subject of an equation is a perfect example of this. However, what I have done is take some of the structural conditions which are applicable and applied that to create a sequence of examples. It is possible to design a better set than the one above where all conditions are met.

**Final Reflection**

Faultless communication has allowed me to design a sequence of teaching examples and questions for pupils to attempt on mini-whiteboards which allow pupils to infer the correct generalisation with the highest percentage of pupils understanding what has been taught on the first attempt. The intended concept has been taught and pupils have been able to take what they have learnt and apply it to an independent practice exercise. I am finding this to be successful with my classes of all abilities. This is simply an attempt of applying Engelmann’s work from his *Theory of Instruction *within the classroom. From what I have seen in terms of pupil work, it has been working. I hope this to be something I continue to use in my planning.

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]]>The post Typical Lesson (Substituting Fractions) appeared first on Conception of the good.

]]>Before I begin, I thought it would be useful to know that during the planning process I outline a progression model. I pick a topic that I want to teach, e.g. substitution, and I break down the topic into complex skills which I want to teach and form exercises which pupils will complete. Breaking down each skill into small component skills prevents pupils overloading their working memory during the initial stages of teaching new information.

**Recap**

The start of every lesson I give pupils a selection of questions to recap what has been taught over the last few days. Pupils do this on their mini-whiteboards. This is a quick exercise taking no more than 5-10 minutes. The questions have been prepared in advance because I want to ask questions which really test a pupil’s understanding. There is no point giving pupils a question to do on their mini-whiteboards which they will all more definitely get right. You want to tease out any confused understanding or misconceptions that pupils may have. This is because if a pupil proceeds to learn new content with misconceptions in mind then more misconceptions will form, or become consolidated.

**Recap: Required pre-requisite knowledge**

Recapping prior knowledge is essential. If pupils don’t have prior knowledge that future learning depends on then no successful learning will take place, and furthermore pupils will be cognitively overloaded from learning multiple skills. The core knowledge that needs to be recapped will be used again and again throughout the lesson when learning new content. Before I taught pupils how to substitute fractions into different expressions I recapped how to :

- Add and subtract fractions with common denominators
- Multiply fractions
- Multiply fractions and integers
- Divide fractions
- Add and subtract fractions with different denominators

**Initial Teacher Instruction: Faultless Communication**

I will think about the worked examples that I want to use to teach pupils how to substitute fractions in a manner which works for all problem types. Engelmann refers to this form of instruction as Faultless communication which I shall blog about soon. I try and design my teacher instruction so nothing that I say contradicts what I have taught in the past.

I want to teach pupils how to substitute fractions into different expressions. I plan my worked examples and I think about how I can design them so pupils can see that the process of how to substitute a fraction into an expression is always the same. The process is the same when I am substituting (adding and subtracting fractions with common denominators):

- two different fractions
- three fractions
- if the result after substitution gives me a fraction which I can simplify to an integer
- if the result after substitution gives me a negative result
- if the result after substitution gives me zero
- if the result after substitution gives me a mixed number

I will also pre-empt misconceptions:

- Question A: “When we add or subtract fractions with the common denominators. The denominators are the same, the denominator remains the same, add or subtract the numerators.”
- Question O: “I have ¼ – 6/4 they are going in different directions, I find the difference which is 5/4. The larger fraction is negative so my result is also negative
- Question G: “Here I have three terms that is ok. I can have questions like this…
- Question I: “I have eleven-tenths minus one-tenths and the result is ten-tenths which simplifies to 1. That is correct. That is possible.”
- Question P: “I am adding an integer to a fraction so my result is a mixed number. That is correct. That is possible.”

Not showing these examples means that when pupils try the following questions then they will be freaked out by getting answers which are negative or zero or mixed numbers. The process of substituting fractions where you have to add or subtract fractions with common denominators is the same regardless of the question set up or the answer you get.

**Show Call**

After going through my worked examples, I will give pupils a very similar set of questions to attempt on their mini-whiteboards. Here pupils are showing their written work to maximise the likelihood of pupils creating perfect answers and working out during independent work. I put a question on the board; give pupils 30 seconds to complete the work on their mini whiteboards. They hold their mini-whiteboards with the answer hidden so other pupils can’t copy each other. I then ask them to show me all at the same time – 3, 2, 1 and show. This lets me see all 32 pupils’ working out. I will give whole class feedback over mistakes that I see. I will show the class a pupil’s work if it is excellent. I will show a sub-optimal answer too and stay ‘Jonny, would have gotten a merit if he had done this…’ or ‘Jonny can improve this. Who can help him improve this? Thank you for letting us all learn from your work Jonny.’ Doug Lemov goes into more detail about the effectiveness of Show Call in his podcast with Craig Barton.

The purpose of this exercise is for me to fix any mistakes and close those gaps. Also, to identify the pupils who are still struggling and support them during the independent practice part of the lesson.

**Independent Work**

I will have designed a practice exercise of questions for pupils to attempt independently. I will circulate the room. Help pupils who are struggling. Glance at pupil work as I go around and fix any mistakes that I spot. At the end of this timed exercise I will read out the answers and get pupils to tell me how they did:

“Raise your hand if you got 10 or more questions correct. 15 or more. 20 or more. All correct so far.”

“If you made a mistake but you now understand that mistake raise your hand please.”

“If you made a mistake that you don’t’ understand and want me to go through it on the board then raise your hand. We can all learn from you.”

**No Plenary**

I don’t usually have plenaries in my lessons. I plan my lessons so pupils can complete all the exercises, even more so the hard staff at the end that I want pupils to have the time to complete. I don’t use Exit Tickets because I do a weekly quiz which informs me of the same information that an Exit Ticket does.

**Reflections**

I think this approach is similar to many teachers. Nothing here is extraordinary but ordinary. However, **what is most important to understand is that the typical lesson above includes features and details that allow pupils to learn, and more importantly it allows pupils to remember what they have learnt.** Everything that I have outlined is a combination of what I saw as good teaching from my own education but also from what I have witnessed and learnt from many exemplary teachers that have kindly let me watch them teach.

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