# 2.3 Estimations, equations and variables (algebraic equations)

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## Memorandum

14. (a) 100

(b) 12

(c) 124

(d) 8

15. (a) 10 99 75 5

(c)

 ______ 8 ______ 21 39 ______ 74 ______

16.2 (a) 5 х + 7 = 22

(b) 8 х - 10 = 46

(c) 5 + 9 х = 59

(d) х - 13 = 6

16.3 (a) 21

(b) 17

(c) 34

18 (a) 72 (j) 30

(b) 12 (k) 5

(c) 7 (l) 57

(d) 141 000 (m) 9

1. 900 (n) 9 987
2. 47 (o) 125

(g) 135

(h) 336

(i) 7

20. (a) х >10

(b) y <2 000

(c) ( c +8)>6

(d) y <50

(e) k – ( k ÷ 2)<20

## Activity: estimations, equations and variables (algebraic equations) [lo 1.7.2, lo 1.9.1, lo 1.10, lo 2.5]

14. Now replace the letters with the correct numbers in the following flow diagram:

15. Puzzles!

a) Can you solve the following?

I am thinking of a particular number. When I multiply this number by 7, then subtract 6 and divide the answer by 8, the quotient is 8. What is the number that I have in mind?________________________________________________________

I am starting with 9. When I multiply this by 12, than add 2 and subtract 11, the answer is_____________________________________________________________

b) We can compile a table that will help us with problems like these.

 Number 9 7 5 Number × 12 + 2 – 11 99 75 51

c) We could replace the word "number" with any letter of the alphabet. Can you fill in the missing numbers in the following table?

 k 6 ________ 13 _________ (k × 5) + 9 49 114

16.1 Did you know?

The statement (k x 5) + 9 = 49 is referred to as an algebraic equation. “Algebra” is the “study of number sentences”.

16.2 Write the algebraic equation for the following:

a) A particular number x 5 + 7 = 22

____________________________________________________________________

b) 8 x a number – 10 = 46 ______________________________________

c) 5 + (9 x a number) = 59 _______________________________________

d) When 13 is subtracted from a bigger number the difference is 6.

_____________________________________________________________________

16.3 Solve the following equations: (You may use your pocket calculator).

a) 49 x a – 29 = 1 000

_____________________________________________________________________

b) (b + 15) x 6 = 192

_____________________________________________________________________

c) 16 x c – 15 = 529

_____________________________________________________________________

17. Did you know?

The letters that are used in place of numbers are called variables.

18. Let's first see how you do in your next mental arithmetic test.

a) 9 x 8 = ___________________________

b) ___________________________ x 4 = 48

c) 4 x ___________________________ = 28

d) 6 x 235 x 100 = ______________________

e) 25 x 9 x 4 = _________________________

f) 16 + 17 + 14 = ______________________

g) 104 + 15 + 16 = _____________________

h) Triple: 112: = _________________________

i) 42 ÷ 6 = ___________________________

j) ___________________________ ÷ 6 = 5

k) 35 ÷ ___________________________ = 7

l) (7 x 3) + (4 x 9) =_______________________

m) 4 x 9 + ___________________________ = 45

n) 10 000 – 13 = ___________________________

o) 5 to the power of 3 = _______________________

15

19. Did you know?

When we use>and<-signs in numbers sentences, e.g. when we

want to say that a number divided by 4 is smaller than 5, we write

it like this:

y ÷ 4<5

If we, for example, wished to say that a number multiplied by 5 is greater than 16, we would write it like this:

b x 5>16

Number sentences like these, which do not have the = sign, are called

inequalities.

20. Write the following word sentences as inequalities:

a) The number of sweets that I have is more than 10.

_____________________________________________________________________

b) The number of learners in our school is greater than 2 000.

_____________________________________________________________________

c) A number increased by 8, is greater than 6.

_____________________________________________________________________

d) There are less than 50 learners in our class.

_____________________________________________________________________

e) If I give away half of my marbles, I shall have fewer than 20.

_____________________________________________________________________

## Assessment

Learning Outcome 1: The learner will be able to recognise, describe and represent numbers and their relationships, and to count, estimate, calculate and check with competence and confidence in solving problems.

Assessment Standard 1.7: We know this when the learner estimates and calculates by selecting and using operations appropriate to solving problems that involve:

1.7.2: multiple operations with integers;

1.7.7: exponents;

1.10: uses a range of strategies to check solutions and judges the reasonableness of solutions.

Learning Outcome 2: The learner will be able to recognise, describe and represent patterns and relationships, as well as to solve problems using algebraic language and skills.

Assessment Standard 2.1: We know this when the learner solves or completes number sentences by inspection or by trial-and-improvement, checking the solutions by substitution (e.g. 2 x - 8 = 4).

Assessment Standard 2.1: We know this when the learner investigates and extends numeric and geometric patterns looking for a relationship or rules, including patterns;

Assessment Standard 2.3: We know this when the learner represents and uses relationships between variables in order to determine input and/or output values in a variety of ways using:

2.3.2: flow diagrams;

2.3.3: tables;

Assessment Standard 2.5: We know this when the learner solves or completes number sentences by inspection or by trial-and-improvement, checking the solutions by substitution (e.g. 2 x - 8 = 4).

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