After October Half term, I joined Great Yarmouth Charter Academy. I was given the task of teaching Year 10 Set 5 out of 6. The first topic I was assigned to teach them was substitution to then go onto teaching expanding brackets. I planned two booklets to teach both topics. I then realised due to a lack of prior knowledge, what I had prepared was out of their reach. I went back to the drawing board to teach the topic of directed numbers. This blog post is a reflection on what I saw to be effective teaching of substitution purely because of how the prior knowledge required was sequenced before teaching substitution.

Teaching Directed Numbers

Directed Numbers is a high-leverage topic. This is because if the content of the topic is mastered then pupils are able to successfully learn other concepts with great ease. Pupils are able to substitute negative numbers into expressions if they are able to add/subtract/multiply/divide negative numbers without having to process what they need to do in their working memory. For example, when I planned my lessons on substitution, I broke my lessons down into this order:

1. Substitution of positive numbers
2. Substitution of negative numbers
3. Substitution of fractions
4. Substitution of decimals

Before I wanted pupils to be able to substitute negative numbers into different expressions I had to teach the following:

1. Adding and subtracting negative numbers (including large values)
1. -4 + 5 –> -41 + 51 (crossing over 0)
2. -4 + 3 (and 3 – 4) –> -41 + 32 (not crossing over 0)
3. -4 + 4 –> -329 – 329 (meeting at 0)
4. -5 + 6 – 9 (multiple terms)
2. Adding and subtracting negative numbers (double signs)
1. -4 – + 5
2. -4 – – 5
3. 4 – – 5
4. 4 + – 3
5. -4 – – 4
6. 4 + – 5
3. Multiplying and dividing negative numbers
1. If I multiply an even number of negative numbers then my result will be positive
2. If I multiply an odd number of negative numbers then my result will be negative
4. Raising negative numbers to different powers
5. Cube rooting negative numbers (including odd roots)

What worked?

Repetitive practice exercises

For the first lesson on teaching how to add and subtract negative numbers with no double signs, I had pupils completing exercises of one problem type. This was to help pupils recognise a pattern which I later shared. Pupils did lots of questions where they practised moving up and down the number line before hand.

E.g.

This was also the case when teaching pupils how to multiply and divide negative numbers:

Similarly, this also applied when asking pupils if the result when a negative number is held to an even or odd power would result in a positive or negative number.

Daily Recap and Questioning

After the starter of my lessons, the next 5-6 minutes I would ask them the same questions on a daily basis, change the order of the questions which was important to ensure that pupils were made to think. Similarly, I used the same problem types each day but would change the numbers.

My questions also became less scaffolded over the weeks to ensure that I was really testing the kids. I also specified the answer to the question. For example:

If the numbers are going in the same direction then what do I do?

This was because I wanted pupils to have absolute clarity over what they needed to do, and with a variation different phrases for the answer and being verbose would have confused them. Also, the repetition helped pupils see specific patterns e.g when we have the following problem types -5 – 6 we add the values of 5 and 6.

These are the questions that I asked them on a daily basis:

Questions for adding and subtracting negative numbers – no double signs:

• Are the numbers going in the same direction or different direction? ANS: Same/Different
• If they are going in the same direction, what do I do next? ANS: Sum the values
• If they are going in different directions, what do I do next? ANS: Find the difference
• When I have found the difference, will my answer be positive or negative? ANS: The larger value is negative so my answer is negative.
• What are the two things do I ask myself when I see this question?
1. Same direction? Sum the values
2. Different direction? Find the difference
• What is the first step?

Questions for multiplying and dividing negative numbers:

• How many negative numbers do I have? ANS: 2/3
1. Follow up: Is that an odd or even number? ANS: Even/Odd
2. If I have an even number of negatives then will my result be positive or negative? ANS: positive
3. If I have an odd number of negatives then will my result be positive or negative? ANS: negative
• Are there are an odd number or even number of negatives? ANS: negative
• If there is an even number of negatives then will my result be positive or negative? ANS: positive
• If there is an odd number of negatives then will my result be positive or negatives? ANS: negative
• Will my result be positive or negative? ANS: Positive/negative
• Is my power an odd number or even number? ANS: even/odd
• Will my answer be positive or negative? ANS: positive/negative

Questions for adding and subtracting negative numbers – with double signs:

• When I see two signs next to each other, what do I do? ANS: Circle the signs, and replace
• What is my first step? ANS: Circle the signs, and replace

Why was the re-teaching of directed number crucial?

Pupils were to attempt the following exercises.

Substituting negative numbers with calculations requiring knowledge of how to add and subtract negative numbers

Prior knowledge they needed that was taught:

a = -3

a + 11 = -3 + 11 = 8

1.Given that a = -3, b = -4, c = -5 and d = 4, evaluate each expression

Substituting negative numbers with calculations (double signs)

Prior knowledge they needed that was taught:

a = -3

11 – a =

11 – – 3 =

11 + 3 = 14

2. Given that a = -3, b = -4, c = -5, evaluate each expression

Substituting negative numbers where pupils will be multiplying and dividing negative numbers

Prior knowledge they needed that was taught:

a = -3

2a = 2 x -3 = -6

-a = -(-3) = +3

3. Given that a = -10, b = -4 and c = -6, evaluate each expression

Prior knowledge they needed that was taught:

4. Given that d = -12, e = -3, f = 16, g = -8 and h = -20

It is very early days but they are very comfortably learning how to substitute negative numbers. We will be learning how to multiply a constant and an expression, next week. The teaching of directed numbers will support the pupils particularly when attempting the following problem types.