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Basic 8 · Mathematics · Term Revision

Term Revision in Mathematics

In this topic you will recall and summarise the term's work on numeration, sets, operations and indices, and practise answering end-of-term questions.

By the end of these notes you should be able to:

  • Recall key ideas on large numbers, standard form and significant figures
  • Recall key ideas on sets, operations and decimals
  • Recall the laws of indices and answer revision questions

What you already know

  • All topics covered in Mathematics this term

Copy everything that has a blue line on its left into your notebook. The other parts are for reading.

Lesson 1

Strand
Revision
Sub-strand
Term Revision

Introduction

This revision covers place value, comparison, standard form and accuracy. These skills are used in every topic.

1. Key Ideas

Numeration summary
TopicKey points
large numbersperiods up to trillions; expanded form with powers of ten
skip countingequal steps; find the rule by subtraction
comparing and orderingcount digits; compare from the left
standard formA × 10ⁿ, where 1 ≤ A < 10; negative n for small numbers
significant figurescount from the first non-zero digit
decimal placesdigits after the point
word problemsread, choose the operation, calculate, check

2. Revision Examples

Mixed revision

(a) Write 4,560,000,000 in standard form. (b) Correct 7,862,510 to 3 s.f. (c) Write 0.00037 in standard form.

  1. (a) 4.56 × 10⁹.
  2. (b) Next digit 2, so 7,860,000.
  3. (c) 3.7 × 10⁻⁴.

Answer: 4.56 × 10⁹; 7,860,000; 3.7 × 10⁻⁴

Students revising mathematics together
Figure 1: Revision group Make a mind map of the numeration topics.

3. Common Mistakes

Learners often write 45 × 10⁶ as standard form, forget zeros in place values, or confuse significant figures with decimal places. Check that A is between 1 and 10, that each period has three digits and that you have read the question carefully.

4. How to Revise Numeration

Do not only read your notes. Cover the answers to the worked examples and try them again yourself. Practise saying large numbers aloud, because reading and writing numbers go together. Write ten numbers from the news or a textbook in standard form. Then correct each one to two significant figures, and check your work with a friend.

New words

revision
Studying again what has been learned.
standard form
A number written as A × 10ⁿ.
period
A group of three digits.
mind map
A diagram linking ideas.

Summary

  • Large numbers are read in periods and may be written in standard form.
  • Compare by digits and place value, and order accordingly.
  • Significant figures and decimal places express accuracy.

Class activity: Numeration Stations

  1. Rotate through stations on place value, standard form and accuracy.
  2. Solve two problems at each.
  3. Check with the answer card.

You need: Bottle tops, Number cards, Graph paper

Exercises

Answer in your exercise book first, then check your answers.

  1. 1. 600,000 in standard form is…

    • A. 6 × 10⁵
    • B. 6 × 10⁶
    • C. 60 × 10⁴
    • D. 0.6 × 10⁶
    Show answer
    Answer: 6 × 10⁵Five places.

  2. 2. 45,678 to 2 s.f. is…

    • A. 46,000
    • B. 45,000
    • C. 45
    • D. 46
    Show answer
    Answer: 46,000Next digit 6.

  3. 3. One trillion has twelve zeros.

    • A. True
    • B. False
    Show answer
    Answer: True10¹².

  4. 4. 0.003 = 3 × 10 to the power ______.

    Show answer
    Answer: −3Negative index.

  5. 5. Arrange in ascending order: 5.2 × 10⁴; 4.9 × 10⁵; 8 × 10³.

    Show answer
    Answer: 8 × 10³; 5.2 × 10⁴; 4.9 × 10⁵Index.

  6. 6. Write the digits 9, 0, 4, 2 to make the smallest four-digit number.

    Show answer
    Answer: 2,049Cannot start with 0.

Lesson 2

Strand
Revision
Sub-strand
Term Revision

Introduction

Sets organise numbers, and operations let us solve problems with them. Let us review both.

1. Key Ideas

Sets and operations summary
TopicKey points
perfect squaresn × n; √ is the inverse; halve the indices
setsmembers, ∩, ∪, Venn diagrams
n(A ∪ B)n(A) + n(B) − n(A ∩ B)
HCF and LCMlargest common factor; smallest common multiple
powers of ten× moves digits left; ÷ moves digits right
mental strategiesproperties, compensation, percentages
operationscolumn methods, long multiplication and division
decimalsline up points; count places; unit rates

2. Revision Examples

Sets problem

In a class of 45, 30 take Art, 25 take Music and 15 take both. How many take neither?

  1. n(A ∪ M) = 30 + 25 − 15 = 40.
  2. Neither: 45 − 40.

Answer: 5 learners

Decimals

Find the cost of 2.5 m of cloth at GH₵18.40 per metre.

  1. 18.40 × 2.5 = 18.40 × 5 ÷ 2.
  2. 92 ÷ 2.

Answer: GH₵46

Students working with calculators in class
Figure 2: Using calculators Draw a Venn diagram for a survey of your class.

3. Linking the Topics

Sets of factors connect to the HCF and LCM, which are used to simplify fractions. Mental strategies help you to estimate and check long calculations. Decimals link to money and measurement. Noticing these links makes mathematics easier to remember.

4. Choosing the Right Method

Before you calculate, decide what the question is really asking. Words such as "both" and "neither" point to sets. "Share equally" points to division, and "how much more" points to subtraction. "Largest number that divides" means the HCF, while "when will they next meet" means the LCM. Underlining these key words helps you to choose the correct method.

New words

Venn diagram
Overlapping circles showing sets.
HCF
The highest common factor.
LCM
The lowest common multiple.
unit rate
The amount for one unit.

Summary

  • Perfect squares and square roots are inverses.
  • Sets and Venn diagrams solve counting problems.
  • Operations on large numbers and decimals are used in real-life problems.

Class activity: Revision Relay

  1. Each member solves one step of a multi-step problem.
  2. Problems cover sets, operations and decimals.
  3. The group checks the final answer.

You need: Bottle tops, Number cards, Graph paper

Exercises

Answer in your exercise book first, then check your answers.

  1. 1. √144 =

    • A. 12
    • B. 14
    • C. 72
    • D. 24
    Show answer
    Answer: 1212 × 12.

  2. 2. The LCM of 6 and 8 is…

    • A. 24
    • B. 48
    • C. 2
    • D. 14
    Show answer
    Answer: 24Smallest common multiple.

  3. 3. The union contains members of either set.

    • A. True
    • B. False
    Show answer
    Answer: True∪.

  4. 4. 0.45 × 100 = ______.

    Show answer
    Answer: 45Two places.

  5. 5. Calculate 1,234 × 25.

    Show answer
    Answer: 30,850123,400 ÷ 4.

  6. 6. n(A) = 20, n(B) = 15, n(A ∪ B) = 28. Find n(A ∩ B).

    Show answer
    Answer: 720 + 15 − 28.

Lesson 3

Strand
Revision
Sub-strand
Term Revision

Introduction

The laws of indices are a major part of the term. Let us review them and practise examination questions.

1. Key Ideas

Indices summary
Law or ideaExample
aᵐ × aⁿ = aᵐ⁺ⁿ3² × 3⁴ = 3⁶
aᵐ ÷ aⁿ = aᵐ⁻ⁿ7⁵ ÷ 7³ = 7²
(aᵐ)ⁿ = aᵐⁿ(2³)² = 2⁶
a⁰ = 19⁰ = 1
a⁻ⁿ = 1/aⁿ3⁻² = 1/9
same base, equal indices2ˣ = 2³, so x = 3
PEDMASbrackets, exponents, division and multiplication, addition and subtraction

2. Sample Questions

Practice question 1

Simplify 2⁷ × 2² ÷ 2⁵.

  1. 2⁷⁺² = 2⁹.
  2. 2⁹⁻⁵ = 2⁴.

Answer: 16

Practice question 2

Solve 3ˣ⁺¹ = 27.

  1. 27 = 3³.
  2. x + 1 = 3.

Answer: x = 2

Practice question 3

Evaluate 4² + 3 × (6 − 2)².

  1. Brackets: 4² = 16.
  2. 3 × 16 = 48.
  3. 16 + 48.

Answer: 64

A student writing a mathematics examination
Figure 3: Mathematics examination Write a model answer to one practice question.

3. Examination Tips

Show all your working, since marks are given for method. Estimate before you calculate. Include units. Check that your answers are reasonable and that you have answered every part. Leave time to review your paper.

4. Common Mistakes with Indices

Learners often multiply indices when they should add them, as in writing 2³ × 2⁴ = 2¹². Others multiply the base as well, as in writing 3² × 3³ = 9⁵. Remember that the base stays the same. Another common error is thinking that a⁰ = 0, when it equals 1. A negative index gives a reciprocal, not a negative number, so 2⁻² = ¼ and not −4.

New words

working
The steps in solving a problem.
method mark
A mark given for correct steps.
simplify
To write in a shorter form.
evaluate
To find the value of.

Summary

  • Know the laws of indices and the order of operations.
  • Solve exponential equations by matching bases.
  • Show your working and check your answers.

Class activity: End-of-Term Quiz

  1. Answer ten questions on all the term's topics.
  2. Mark your work with the model answers.
  3. List two topics to practise further.

You need: Bottle tops, Number cards, Graph paper

Exercises

Answer in your exercise book first, then check your answers.

  1. 1. 5² × 5³ =

    • A. 5⁵
    • B. 5⁶
    • C. 25⁵
    • D. 10⁵
    Show answer
    Answer: 5⁵Add.

  2. 2. 2⁻² =

    • A. ¼
    • B. −4
    • C. 4
    • D. ½
    Show answer
    Answer: ¼Reciprocal.

  3. 3. Marks are given for correct working.

    • A. True
    • B. False
    Show answer
    Answer: TrueMethod.

  4. 4. If 2ˣ = 16, then x = ______.

    Show answer
    Answer: 42⁴.

  5. 5. Simplify (3²)³ ÷ 3⁴.

    Show answer
    Answer: 3² = 9Laws.

  6. 6. Evaluate 2³ × 5 − (4 + 2)² ÷ 3.

    Show answer
    Answer: 40 − 12 = 28Order of operations.

Homework

  1. Make a summary sheet of the term's rules and formulas.
Term Revision in Mathematics – Basic 8 Mathematics Notes | sirbus