National curriculum content

• Use simple formulae
• Generate and describe linear number sequences
• Express missing number problems algebraically
• Find pairs of numbers that satisfy an equation with two unknowns
• Enumerate possibilities of combinations of two variables

Lesson objectives

1. 1-step function machines
2. 2-step function machines
3. Form expressions
4. Substitution
5. Formulae
6. Form equations
7. Solve 1-step equations
8. Solve 2-step equations
9. Find pairs of values
10. Solve problems with two unknowns

What we want children to know

Pupils should be introduced to the use of symbols and letters to represent variables and unknowns in mathematical situations that they already understand, such as:

• Missing numbers, lengths, coordinates and angles
• Formulae in mathematics and science
• Equivalent expressions (for example, a + b = b + a)
• Generalisations of number patterns
• Number puzzles (for example, what two numbers can add up to).

• For each number they put into a function machine, there is an output.
• Use simple algebraic inputs e.g. y. Using these inputs in a function machine leads them to forming expressions e.g. y + 4.
• Children are introduced to conventions that we use when writing algebraic expressions. e.g. y × 4 as 4𝑦.
• Children substitute into simple expressions to find a particular value.
• Understand that the same expression can have different values depending on what has been substituted.
• Use simple formulae to work out values of everyday activities such as the cost of a taxi or the amount of medicine to take given a person’s age.
• Use algebraic notation to form one-step equations.
• Solve simple one step equations involving the four operations.
• Progress from solving equations that require one-step to equations that require two steps.
• Use their understanding of substitution to consider what possible values a pair of variables can take.
• Find possible solutions to equations which involve multiples of one or more unknown.

What skills we want children to develop

Use knowledge to solve reasoning and problem solving questions such as:

True or false?

ab + b = 18

Mo says, ‘a and b must both be odd numbers’.

Is this true or false? Explain your decision.

Connected calculations

p and q each stand for whole numbers.

p + q = 1000 and p is 150 greater than q.

Work out the values of p and q.

Generalising

Write a formula for the 10th, 100th and nth terms of the sequences below.

4, 8, 12, 16 ………

0.4, 0.8, 1.2, 1.6, …….

Possible answers

a, b and c are integers between 0 and 5.

a + b = 6

b + c = 4

Find the values of a, b and c.

How many different possibilities can you find?

Vocabulary/Mathematical Talk

• What do you think “one-step function” means?
• Do some functions have more than one name?
• What is the output/input if ….?
• How many sets of inputs and outputs do you need to be able to work out the function? Explain how you know.
• How can you write + 5 followed by – 2 as a one-step function?
• If I change the order of the functions, is the output the same?
• If you add 3 to a number and then add 5 to the result, how much have you added on altogether?
• How can you write 𝑥 × 3 + 6 differently?
• Are 2𝑎 + 6 and 6 + 2𝑎 the same? Explain your answer.
• What does it mean when a number is next to a letter?
•  Is 𝑎 + 𝑏 + 𝑏 the same as 𝑎 + 2𝑏?
• What’s the difference between an expression and an equation?
• What’s the difference between a formula and an equation?
• Why do you think the equation is set up on a balance? What does the balance represent? How does this help you solve the equation?
• Why do you have to do the same to each side of the equation?
• Does the order the equation is written in matter?