# Achieving QTS

## Student Resources

# Chapter 4 – Mathematical language, reasoning and proof

Consider the following statements and click to reveal the answer.

1. (a) Prove that the sum of the interior angles of a quadrilateral is always equal to 360°. (You may assume the sum of the angles of a triangle.)

(b) Suggest an approach to this in which children use inductive thinking.

**Answer:**

*(a) A quadrilateral can always be divided into two triangles, e.g.:*

*But, the four angles of the quadrilateral are also the sum of these six angles.*

*(b) Children could either draw several quadrilaterals, measure the angles and find the sum, or, tear off the corners and show they fit together to make one rotation.*

2. Prove that (a) the product of two even numbers is even, and (b) the product of two odd numbers is odd.

**Answer:**

*(a)Our two even numbers can be written, 2a and 2b, where a and b are integers. Multiplying gives: *

*2a × 2b = 2(2ab)*

*2ab is the product of three integers and must also be an integer (an assumption). Hence 2(2ab), the product of two even numbers, must be even.*

*(b) Writing our odd numbers as 2a + 1 and 2b + 1 and multiplying gives: *

*(2a+1)(2b+1) = 4ab + 2b + 2a + 1 *

*The final unit makes it impossible to divide this expression by 2, so it is not even. Therefore, the product of two odd numbers is odd..*

3. Prove that the product of three consecutive numbers is even.

**Answer:**

*Three consecutive numbers must contain at least one even number of the form 2a, where a is an integer. The product of 2a and any other two integers gives a number divisible by 2, hence the product is always even.*

4. Prove by exhaustion that there are only six ways of listing the letters a, b and c.

**Answer:**

*Undertaken systematically, the possibilities are:*

*abc bac cab*

*acb bca cba*

5. Find counter-examples to refute these assertions:

(a) Multiplying makes a number larger.

(b) If a shape has rotational symmetry, it also has reflective symmetry.

(c) If a and b are both

**Answer:**

*(a) 0.1 × 60 = 6 or 12 10 5×= would do.*

*(b) This shape or variations on it would refute*

*(c) 8 and 6 are both factors of 24, but 8 is not a factor of 6 and 6 is not a factor of 8.*

6. Prove that for any chord, AB, angles â and bˆ are equal. List any assumptions you make.

**Answer:**

*Assuming the truth of the theorem that the angle subtended at the centre is twice that at the circumference, we can argue:*

7. List three standard methods of proof that could be used in primary schools.

**Answer:**

*Three standard methods of proof that could be used in primary schools are:**deductive proof;**disproof by counter-example;**proof by exhaustion. *

8. What are the three ways in which we can represent mathematical ideas?

**Answer:**

*The three ways in which we can represent mathematical ideas are: *

*language describing using words, phrases and sentences;**through the use of pictures/objects including number lines and mathematical apparatus;**using specialist symbols.*

9. What do the following abbreviations mean?

(a) OE

(b) ME

**Answer:**

*The following abbreviations mean:*

*(a) OE Ordinary English;*

*(b) ME Mathematical English.*