Lesson 21 Expressions and equations

# Episode 2

Reasoning Resources
Whole class preparation
In the familiar context of ‘think of a number’ problems, realising that the answer and the original number determine each other.

Exploring intuitive ways of solving this problem.

Understanding that the problem can be expressed as an algebraic expression and then as an equation.

Focussing on the differences between an expression and an equation, including that the expression is a variable, while the equation defines a value for a given unknown.

Understanding the differences in two methods of solving the simple equation.

Splitting, or ‘breaking up numbers to suit is an intermediate approach that allows ‘doing the same to both sides’ to remain visible. On the board, write: I think of a number, double it and then add 5, and the answer is ? [x2} [+5]. When you were younger you might have written this as a function machine.What might the answer be? What will the answer depend on? Use a variety of ‘answers’ to show that varying the ‘answer’ changes the value of the number I first thought of. How would you have solved it? ‘Using inverse, ’trial and improvement.

• Can we write the first bit of this puzzle as an expression using algebra? (2n + 5),
• Can we write the whole puzzle as an equation? (2n+5 =)
• What is the difference between an expression and an equation?

Reinforce that in the expression # is a variable which can take any value, but in the equation only one value of # will make it true.

Write 3a + 15 on the board. Take a few values for a and work out the value of the expression. What is the smallest/largest value the expression can have? Explain.

Write an equation on the board, e.g. 2q + 5 = 13. How would you solve this in your head? Hopefully both the following methods will be given. Show how they could be written.

Method 1: 13 is 8 and 5, so the ‘2q' must be 8. 2x 4 is 8 so’q' must be 4. 2q+5=13

2q+5=8+5 (creating a 5 in the right-side to match the 5 on the left)

q+q=4+4 (showing each side in single values added together). What shall we call this method? ‘splitting up’;‘breaking the bits up’

Method 2: Take away the 5 from 13. This leaves 8 for the ‘2q/ so‘1q'is half of 8.
2q+5=13

2q+5-5=(13-5)=8 (subtracting 5 from both sides)

q = 4 (dividing both sides by 2) What shall we call this method? ‘doing the inverse or the opposite; ‘reversing’ or ‘balancing.'
• Pair and small group work
Standard procedure of balancing, or ‘doing the same to both sides’ needs rehearsing with language.

Understanding a different way of thinking about an equation - splitting up the sides in helpful ways.

Exploring this new method for themselves.
Give out Worksheet 1 and allow pupils time to consider the equation 6b + 2=11. Demonstrate the splitting up method for 6b + 2 = 11 and discuss other ways of solving the equation.

Can you apply this new ‘splitting up’ method to the equation 4n + 20 = 32 (on worksheet 1)?

Explain that it’s useful to have a number of different methods so that you can select the most appropriate for a given equation.
Whole class sharing and discussion
Share solutions and methods. Discuss the meanings of the ‘equals’ sign in the different methods. In the reversing method the equals sign means that one side equals the answer which you can work backwards from; in the splitting up method the equals sign means that the two sides of the equation are equivalent.