1. Material developed by Paul Dickinson, Steve Gough & Sue Hough at MMU
I Think of a Number
Material developed by Paul Dickinson, Steve Gough & Sue Hough at MMU

2. Thank you
Sue, Steve and Paul would like to thank all the teachers and students who have been involved in the trials of these materials
Some of the materials are closely linked to the ‘Making Sense of Maths’ series of books and are reproduced by the kind permission of Hodder Education

3. Note to teacher
This powerpoint introduces algebra and algebraic notation in a manner that makes sense to students. The need to write down an equation is created and this provides purpose to the topic. In trials, the dreaded question “what’s x?” was never asked and students were often relieved to see algebra written down after 20 minutes of having to rely on their memory. This unit develops algebraic notation and it is important that students use this precisely and correctly. The idea of mathematics as a language is strong here and the need to communicate in a consistent way should be emphasised.

4. Phase 1 – Doing and undoing (15 min)
Finding the original number after two (arithmetically accessible) operations . This should be done orally, perhaps with the addition of mini whiteboards
Say the following as ITOAN:
2x+1=17
3x+4=19
5x-3=27
2(x+1)=20
3(x-2)=21
5(x-2)=20
4(2x+1)=28
After the first 2 or 3, start to hear how students are working out their answers.
Listen carefully to their reverse methods.
This is a TEACHER SLIDE and should not be displayed to the class. The idea here is to introduce algebra to the class with mathematical purpose and without the class realising that they are “doing” algebra. The “I Think of a Number” (ITOAN) activity is designed as a way of introducing a letter to represent an unknown number as well as introducing algebraic notation and the concepts of inverse operations and order of operations.
The equations on this slide should all be given orally and, if you change the numbers, try to ensure that the operations are arithmetically accessible to the students. It will be important to hear how they arrived at their answer as some students may adopt a trial and improvement strategy. This strategy will be seen to be inadequate if more difficult numbers are selected and students hear the successful inverse strategy.
The equations at this stage have up to three operations and most students will be able to reverse the order without writing anything down on paper.
Begin to check the pupils work by starting with the value of x and then arriving at the answer.
Hear the class go forward and back (e.g. x2, +1, 17 ↔ 17,-1, ÷2)
Ask how one would reverse x7, ÷3, +2, -87, x3.142, +½, xβ etc.

5. Phase 2 – Communicating Algebra
Say fairly quickly: “ I TOAN, add 3, times by 2, take away 1, divide by 3, add 8 and the answer is 15.” What is my number?
I TOAN, take 3, divide by 2, multiply by 6, add 4 and the answer is 28
After being placed in a position of needing to write algebraically, students work on the consistent use of algebraic notation
This is a TEACHER SLIDE and should not be displayed to the class. Having gained confidence from the initial oral work, students are now given a ITOAN with multiple operations. They may grumble and complain (or even smile) but invariably someone will ask if you could write it down. This gives you the opportunity to introduce ‘x’ and formal written notation. Our experience in trials is that nobody asks “What is x?” at this point…they immediately recognise that x stands for an unknown number. Writing and saying the equations simultaneously is important in developing their ability to read and write algebraically.
Perhaps bring a student to the board to write the second example as you read it out. The rest of the class could then reverse the calculation on mini whiteboards. The teacher should then bring together the group’s responses by writing up the reverse statement using the correct notation.
Have a student write the second example on the board.
Have the rest of the class reverse the calculation on mini whiteboards. Teacher brings together the group’s responses by writing up the reverse statement using correct notation.

6. Complicated Equations?
Teacher reads these equations (as ITOAN) and the class write them down.
After each equation, the class reverse it to get the number the teacher was originally thinking of.
This is a TEACHER SLIDE and should not be displayed to the class. It is obviously important that the equations are correctly written down and so it might be a good opportunity to use mini whiteboards and allow the students to self-check that everyone has written down exactly the same. Insist on the same notation being used when the equations are being reversed.
Many students feel a real sense of achievement from successfully working on what appear to them as very complicated equations
Additional equations of a similar nature can be added as appropriate.
(Answers: Q1. x=5, Q2. x=7)

7. Write these equations in words (I think of a number..)
4x+3=35
6x-5=31
2(x+4)=22
3(x-8)=6
2x+8=48
3x+10=4
10(x+5)=40
67=6+3y
4=3x+5
12=4(y-7)
This activity is designed to encourage the students to think of equations as ITOAN and thus be able to solve them using a reverse operation strategy. Asking the students to return to the sentence from the abstract equation is a key skill being developed here. You may wish to miss out some of these questions as appropriate but it is important that students try questions 8, 9 and 10 as these may challenge some students and may require some class discussion.

8. Solve these equations To solve means to find the value of the letter
2x+1=15
3x-5=16
7z-3=18
2(x+3)=28
4(y-5)=20
3(x+7)=9
2(2x+3)=18
3(4x+2)=30
27=8x+3
41=9a-4
To solve means to find the value of the letter
This is the first time that the students have been told to solve the equations. Again they should be encouraged to say each equation as ITOAN and then reverse the operations.
(Answers: Q1. x=7, Q2. x=7, Q3. z=3, Q4. x=11, Q5. y=10, Q6. x=-4, Q7. x=3, Q8. x=2, Q9. x=4, Q10. a=5)

9. Say as ‘I think of a number…’
x+4=g
2x+h=6
5x-g=k
3x=t
2(x+w)=p
a(x-1)=h
mx+c=y
v = u+ax
T(x+u)=r
For these questions, ‘x’ is the number you are starting with.
These algebraic equations are more abstract them those previously encountered. The students should treat them in the same manner and here they are told to say each one as ITOAN, again moving from the more abstract equation to the more real sentence.
You might like to ask members of the class to take turns reading each one and then select 3 or 4 for the class to write down in their book.

10. Change each equation so it starts with the letter in the bracket equalling the rest
5x+1=w (x)
3x+t=r (x)
ax-b=h (x)
v=u+at (u)
y=mx+c (x)
C=2πr (r)
2A=(a+b)h (A)
d=a(b+c) (b)
This is called, making a letter the subject of the equation.
What you are doing is called rearranging an equation
It would be beneficial to model an example before they start by asking someone to read out the first equation as ITOAN with “x” as the number. More examples can be given as appropriate.
All equations where the subject of the equation appears only once can be done this way and allow students that normally find these questions difficult to answer successfully.

11. I Think of a Number (Summary)
Key Vocabulary: Solve (find the value of the number) Rearrange (change the way an expression is written) Subject of the equation (the letter that the equation is equal to)
4x+3=27
Say this as I think of a number
Reverse it (27…)
You should find x=6
C=xp-d
To make x the subject
Say this as I think of a number with x the number
Reverse it
You should find x=(c-d)/p