1. Use algebra to construct proofs.
Grade 8
Algebraic Proof
Use algebra to construct proofs.
If you have any questions regarding these resources or come across any errors, please contact

2. Lesson Plan Lesson Overview Progression of Learning
Objective(s)
Use algebra to construct proofs
Grade 8
Prior Knowledge
Algebraic notation and manipulation
Duration
30 minutes
Resources
Print slides:
Equipment
Progression of Learning
What are the students learning?
How are the students learning? (Activities & Differentiation)
The concept of proof
Using slide 4 and 5 introduce the concept of proof. 5
Standard grade 8 proofs
Demonstration a standard GCSE algebraic proof that the sum of two consecutive square numbers is an odd number. Students to then attempt 4 further proofs using the concept understood from the example. 10
Algebraic proofs in exam questions (from specimen papers)
Give students slide 15. This includes 4 exam questions related to objective. Students need to use notes from lesson to answer the questions. Ensure that all steps are shown. Relate to mark scheme to show how the marks are allocated. 15
Next Steps
Assessment
PLC/Reformed Specification/Target8/Algebra/Proof

3. Key Vocabulary
Factorise Simplify Quadratic Linear Collect

4. What is proof?
A proof is where you are asked to show that something is true using mathematics, in this case algebra.
It could be simple:
Or more complicated
But you have to take the key information from a question and show what you are asked to.
Prove that the area of a rectangle is 30cm2:
5cm
6cm
Prove that the difference of two consecutive square numbers is an odd number.

5. Constructing a proof Looking at the simple example:
Area = length x width
= 6cm x 5cm
= 30cm2 as required
Showing a fact and how the information you have fits that fact is a proof. The fact and the details must be stated.
Prove that the area of a rectangle is 30cm2:
5cm
6cm

6. Constructing a proof
Looking at the complicated, grade 8 example:
A number = n That number squared = n2
The next number = (n+1) That squared =(n+1)2
The sum of the squares is n2+(n+1)2
You have now assembled the information you need to complete the proof.
Prove that the sum of two consecutive square numbers is an odd number.

7. So, the sum of the squares is n2+(n+1)2
Expand brackets and simplify: =n2+(n+1)(n+1)
=n2+n2+2n+1
=2n2+2n+1
Sentence to reason: 2n2 is even
2n is also even
1 is odd
even +even + odd = odd
Hence you have proven that the sum of consecutive
squares is odd.

8. Practice
Construct the algebraic proof for the following:
Prove that the sum of 3 consecutive even numbers is always a multiple of 6.
Show that (4n+1)2-(4n-1)2 is always a multiple of 16 for positive integer values of n.
Prove that the difference between the squares of two consecutive integers is always equal to the sum of the two integers.

9. Answers 2n+(2n+2)+(2n+4) are 3 consecutive even numbers. 6n+6
6n is a multiple of 6, as is 6, therefore all combinations of three consecutive even numbers will be multiples of 6.
2) (4n+1)2 = 16n2+8n+1
(4n-1)2 = 16n2-8n+1
If the second is subtracted from the first we are left with:
16n. This is a multiple of 16 for integer values.
(n+1)2 - n2 is the difference of 2 consecutive integers.
n2+2n+1 -n2
(2n+1) is the sum of n and n+1, hence proven

10. Exam Question – Specimen Papers

11. Exam Question – Specimen Papers

12. Exam Question – Specimen Papers

13. Exam Question – Specimen Papers

14. Constructing a proof DEMO PRACTICE Student Sheet 1
Prove that the sum of two consecutive square numbers is an odd number.
Prove that the sum of 3 consecutive even numbers is always a multiple of 6.
Show that (4n+1)2-(4n-1)2 is always a multiple of 16 for positive integer values of n.
Prove that the difference between the squares of two consecutive integers is always equal to the sum of the two integers.
Student Sheet 1

15. Exam Question – Specimen Papers
Student Sheet 3