1. Grade 10 Mathematics
Rational Numbers

2. Curriculum Statement Algebraic Expressions
Understand that real numbers can be rational or irrational.
Establish between which two integers a simple surd lies.
Round off to an appropriate degree of accuracy.
Multiplication of a binomial by a trinomial.
Factorisation – including: HCF, DOTS, trinomials, grouping, Sum and difference of 2 cubes.
Simplification of algebraic fractions with denominators of cubes (limited to sum and difference of cubes)

3. Unit 1 Identify rational numbers
A rational number (Q) is a fraction and can be either recurring or terminating.
It may be a common fraction or a mixed number e.g or
Fractions can be written as a decimal fraction and vice versa e.g.
and
Terminating decimals are those that ends e.g. 0,5 or 1,25 or 0,678.
A recurring decimal is a decimal fraction which repeats itself or part of it e.g. 0,33333……. or
Irrational numbers do not terminate (end) or recur

4. Unit 1 SAMPLE Number Systems
Represent number systems using Venn - diagrams

5. Unit 1 Convert between terminating and common fractions.
Examples:
A terminating decimal is a ending decimal:

6. Unit 1 Converting between recurring decimals and common fractions.
Convert 0,33… to a common fraction
To convert a recurring decimal into a common fraction:
Multiply it by appropriate powers
of 10
Then use subtraction to eliminate
the repeating part.
Then simplify the fraction.

7. Question Time 1) 2) 3) 4) Question 1 Which of the following numbers
are rational : 1) 2) 3) 4)

8. Question Time 1) 2) 3) 4) Question 2 Write the following decimals as a
common fraction: 0,125
Possible Answers 1) 2) 3) 4)

9. Unit 1 Rounding Off Numbers to Certain Decimal Places
The rules for rounding off numbers to a certain number of decimal places are as follows:
Count to the number of decimal places after the comma that you
want to round off to.
Look at the digit to the right of that number:
- if it is < 5, drop it and all the numbers to the right of it.
- if it is ≥ 5, add one more digit to the previous digit, and then drop it
and all the numbers to the right of it.
- keep all zeros (place holders) where required

10. Question Time Question 1
Round off the following to two decimal places:
3,2562 Answer: 3,26
1, Answer: 1,89
2, Answer: 2,21

11. Unit 1 Representing Numbers on a Number Line
SET BUILDER NOTATION
Read as: x is greater than or equal to four.
Read as: x is greater than -5, and less than or equal to 3.
INTERVAL NOTATION

12. Unit 1 Multiplication of Algebraic Expressions
PRODUCT OF TWO BINOMIALS
Firsts, Outers, Inners, Lasts (FOIL)
SQUARING A BINOMIAL
Square the firsts, Twice the product, Square the last
PRODUCT OF A BINOMIAL AND A TRINOMIAL
Multiply each term in the binomial by every term in the trinomial

13. Question Time
Question 1: Simplify the following 1. 2. 3.

14. Question Time
Answers
Question 1: Simplify the following 1. 2. 3.

15. Unit 1 Factorising Taking out a Highest Common Factor: (HCF)
- Always do this first. Look for a numerical or variable factor that is
common to each term in the polynomial.
Difference of Two Squares: (DOTS) - When you see a “ ”
Quadratic trinomial:
- when you have a trinomial in the form ax² + bx + c

16. Unit 1 Factorising (cont.)
4. Grouping
- when you have 4 terms – group into two terms and then take out a
HCF from each.
- then take out a Highest common Bracket
Sum and Difference of Two Cubes
- when you have “ ” or “ ”
- cube root each term to give you your fist bracket
- then: square first, square last, product change the sign for your second bracket

17. Unit 1 Algebraic Fractions
Multiplication and Division
Factorise all parts of the expression. Cross cancel where applicable.

18. Unit 1 Algebraic Fractions
Addition and Subtraction:
Factorise all parts of the expression.
Find a LCD and simplify

19. Unit 1 Simplify expressions using the laws of exponents.
A product to n powers: a.a.a.a.a……….. is defined as :

20. Unit 1 Simplify expressions using the laws of exponents.

21. Unit 1 Simplify expressions using the laws of exponents.
A surd is the square root of a whole number which produces an irrational number.
A surd is an irrational number ( you need a calculator to determine the numerical value of the surd e.g which is a never ending, non – recurring decimal.
The surd lies between which can be written
as:
are surds, but we can simplify them further.
Note:
We may never root over a plus or minus sign e.g.

22. Unit 1 Simplify expressions using the laws of exponents.
To use the ladder method to determine the factors of a number by using prime factors:
Prime numbers: Numbers with only two factors i.e. 1 and itself: e.g. 3 = 3.1
E.g. 2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; 37 …
We divide the smallest prime number into the number until we get 1:
Worked Example:
Use common factor here e.g.

23. Question Time
Question 4
Simplify 1) 2) 3)
4) None

24. Question Time
Question 5
Simplify 1) 2) 3) 4)

25. Question Time Question 6 Without using a calculator and by showing all
workings, determine between which two integers lies
1) Between 6 and 7 2) Between 5 and 6
3) Between -6 and -7 4) Between 7 and 8

26. Question Time Question 7 Determine the product of
1) ) ) ) -2

27. Unit 1 Working with numbers
f(x) is the function value or the y value.
If f(-1) = 2, it means that when x = -1, then y = 2, and we then have the co-ordinate pair (-1;2).
{(-2;3); (0;2); (3;2); (5;6)}
Check to see if any of the x values are repeated. No ordered pair in the function has a first value that are repeated. It is a function
If a graph is given, it is easier to run a vertical line, (use your ruler) from top to bottom and see if there is more than one point of intersection. If at all times we have only one point on the graph intersecting with our ruler, this is a function.
Example:
Given: {(1;2); (2;1); (2;2); (3;1); (3;2); (4;4)}…..
Note: 2 and 3 are first components in two co-ordinate pairs!
Domain: {1; 2; 3; 4}
Range: {1; 2; 4} Not a function

28. Question Time Find: g(4) a if g(a) = 25 the domain of f
A function is defined by g = {(3;2); (4;5); (8;25)}
Find:
g(4)
a if g(a) = 25
the domain of f
the range of f.

29. Solutions
Answer:
g(4) = 5
a = 8