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Chapter 54: Probability Foundations

Learn Grade 8 Math from beginner foundations through advanced Grade 8 problem solving with detailed explanations, at least five examples per topic, practice exercises, and review questions.

Grade 8Beginner Friendly5+ Examples Per TopicPractice30 Q&A
Estimated reading time0% read
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What this chapter covers

This chapter contains 13 topics. Technical terms are followed by plain-language meanings where they first appear. Each topic includes at least five worked examples. Coding is included only where it naturally supports the Grade 8 coding expectations.

54.1 Probability

Probability measures likelihood from 0 to 1, or 0% to 100%. In this section, the focus is Probability.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find the probability to roll an even number on a fair die.

  1. Favourable outcomes = 3.
  2. Total equally likely outcomes = 6.
  3. P = 3/6.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 2

Problem: Find the probability to flip heads on a fair coin.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 2.
  3. P = 1/2.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 3

Problem: Find the probability to draw red from 5 red and 3 blue marbles.

  1. Favourable outcomes = 5.
  2. Total equally likely outcomes = 8.
  3. P = 5/8.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 5/8

Worked Example 4

Problem: Find the probability to choose a vowel from A,B,C,E.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 4.
  3. P = 2/4.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 5

Problem: Find the probability to spin section 1 on four equal sections.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 4.
  3. P = 1/4.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/4

Worked Example 6

Problem: Find the probability to roll a number greater than 4 on a die.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 6.
  3. P = 2/6.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/3

Worked Example 7

Problem: Find the probability to draw blue from 2 blue and 6 green counters.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 8.
  3. P = 2/8.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/4

Worked Example 8

Problem: Find the probability to choose an odd number from 1–10.

  1. Favourable outcomes = 5.
  2. Total equally likely outcomes = 10.
  3. P = 5/10.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 9

Problem: Find the probability to flip tails.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 2.
  3. P = 1/2.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 10

Problem: Find the probability to choose a multiple of 3 from 1–12.

  1. Favourable outcomes = 4.
  2. Total equally likely outcomes = 12.
  3. P = 4/12.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/3

Practice exercise

Create one new Grade 8 problem involving Probability. Show the important steps, include units when needed, and explain how you checked the answer.

54.2 Outcome

Outcome is an important Grade 8 concept in Probability Foundations. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find the probability to roll an even number on a fair die.

  1. Favourable outcomes = 3.
  2. Total equally likely outcomes = 6.
  3. P = 3/6.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 2

Problem: Find the probability to flip heads on a fair coin.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 2.
  3. P = 1/2.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 3

Problem: Find the probability to draw red from 5 red and 3 blue marbles.

  1. Favourable outcomes = 5.
  2. Total equally likely outcomes = 8.
  3. P = 5/8.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 5/8

Worked Example 4

Problem: Find the probability to choose a vowel from A,B,C,E.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 4.
  3. P = 2/4.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 5

Problem: Find the probability to spin section 1 on four equal sections.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 4.
  3. P = 1/4.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/4

Worked Example 6

Problem: Find the probability to roll a number greater than 4 on a die.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 6.
  3. P = 2/6.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/3

Worked Example 7

Problem: Find the probability to draw blue from 2 blue and 6 green counters.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 8.
  3. P = 2/8.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/4

Worked Example 8

Problem: Find the probability to choose an odd number from 1–10.

  1. Favourable outcomes = 5.
  2. Total equally likely outcomes = 10.
  3. P = 5/10.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 9

Problem: Find the probability to flip tails.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 2.
  3. P = 1/2.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 10

Problem: Find the probability to choose a multiple of 3 from 1–12.

  1. Favourable outcomes = 4.
  2. Total equally likely outcomes = 12.
  3. P = 4/12.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/3

Practice exercise

Create one new Grade 8 problem involving Outcome. Show the important steps, include units when needed, and explain how you checked the answer.

54.3 Event

Event is an important Grade 8 concept in Probability Foundations. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find the probability to roll an even number on a fair die.

  1. Favourable outcomes = 3.
  2. Total equally likely outcomes = 6.
  3. P = 3/6.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 2

Problem: Find the probability to flip heads on a fair coin.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 2.
  3. P = 1/2.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 3

Problem: Find the probability to draw red from 5 red and 3 blue marbles.

  1. Favourable outcomes = 5.
  2. Total equally likely outcomes = 8.
  3. P = 5/8.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 5/8

Worked Example 4

Problem: Find the probability to choose a vowel from A,B,C,E.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 4.
  3. P = 2/4.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 5

Problem: Find the probability to spin section 1 on four equal sections.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 4.
  3. P = 1/4.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/4

Worked Example 6

Problem: Find the probability to roll a number greater than 4 on a die.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 6.
  3. P = 2/6.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/3

Worked Example 7

Problem: Find the probability to draw blue from 2 blue and 6 green counters.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 8.
  3. P = 2/8.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/4

Worked Example 8

Problem: Find the probability to choose an odd number from 1–10.

  1. Favourable outcomes = 5.
  2. Total equally likely outcomes = 10.
  3. P = 5/10.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 9

Problem: Find the probability to flip tails.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 2.
  3. P = 1/2.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 10

Problem: Find the probability to choose a multiple of 3 from 1–12.

  1. Favourable outcomes = 4.
  2. Total equally likely outcomes = 12.
  3. P = 4/12.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/3

Practice exercise

Create one new Grade 8 problem involving Event. Show the important steps, include units when needed, and explain how you checked the answer.

54.4 Sample space

A sample is a smaller group selected from a population. In this section, the focus is Sample space.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Classify this data set: [4, 6, 8].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 2

Problem: Classify this data set: [5, 9, 10, 12].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 3

Problem: Classify this data set: [3, 7, 7, 11].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 4

Problem: Classify this data set: [20, 25, 30].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 5

Problem: Classify this data set: [6, 8, 9, 12, 15].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 6

Problem: Classify this data set: [2, 4, 4, 5, 20].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 7

Problem: Classify this data set: [10, 11, 12, 13, 14].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 8

Problem: Classify this data set: [1, 3, 5, 7, 9].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 9

Problem: Classify this data set: [8, 8, 8, 9, 10].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 10

Problem: Classify this data set: [15, 18, 21, 24, 27].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Practice exercise

Create one new Grade 8 problem involving Sample space. Show the important steps, include units when needed, and explain how you checked the answer.

54.5 Certain event

Certain event is an important Grade 8 concept in Probability Foundations. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find the probability to roll an even number on a fair die.

  1. Favourable outcomes = 3.
  2. Total equally likely outcomes = 6.
  3. P = 3/6.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 2

Problem: Find the probability to flip heads on a fair coin.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 2.
  3. P = 1/2.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 3

Problem: Find the probability to draw red from 5 red and 3 blue marbles.

  1. Favourable outcomes = 5.
  2. Total equally likely outcomes = 8.
  3. P = 5/8.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 5/8

Worked Example 4

Problem: Find the probability to choose a vowel from A,B,C,E.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 4.
  3. P = 2/4.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 5

Problem: Find the probability to spin section 1 on four equal sections.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 4.
  3. P = 1/4.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/4

Worked Example 6

Problem: Find the probability to roll a number greater than 4 on a die.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 6.
  3. P = 2/6.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/3

Worked Example 7

Problem: Find the probability to draw blue from 2 blue and 6 green counters.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 8.
  3. P = 2/8.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/4

Worked Example 8

Problem: Find the probability to choose an odd number from 1–10.

  1. Favourable outcomes = 5.
  2. Total equally likely outcomes = 10.
  3. P = 5/10.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 9

Problem: Find the probability to flip tails.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 2.
  3. P = 1/2.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 10

Problem: Find the probability to choose a multiple of 3 from 1–12.

  1. Favourable outcomes = 4.
  2. Total equally likely outcomes = 12.
  3. P = 4/12.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/3

Practice exercise

Create one new Grade 8 problem involving Certain event. Show the important steps, include units when needed, and explain how you checked the answer.

54.6 Impossible event

Impossible event is an important Grade 8 concept in Probability Foundations. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find the probability to roll an even number on a fair die.

  1. Favourable outcomes = 3.
  2. Total equally likely outcomes = 6.
  3. P = 3/6.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 2

Problem: Find the probability to flip heads on a fair coin.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 2.
  3. P = 1/2.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 3

Problem: Find the probability to draw red from 5 red and 3 blue marbles.

  1. Favourable outcomes = 5.
  2. Total equally likely outcomes = 8.
  3. P = 5/8.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 5/8

Worked Example 4

Problem: Find the probability to choose a vowel from A,B,C,E.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 4.
  3. P = 2/4.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 5

Problem: Find the probability to spin section 1 on four equal sections.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 4.
  3. P = 1/4.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/4

Worked Example 6

Problem: Find the probability to roll a number greater than 4 on a die.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 6.
  3. P = 2/6.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/3

Worked Example 7

Problem: Find the probability to draw blue from 2 blue and 6 green counters.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 8.
  3. P = 2/8.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/4

Worked Example 8

Problem: Find the probability to choose an odd number from 1–10.

  1. Favourable outcomes = 5.
  2. Total equally likely outcomes = 10.
  3. P = 5/10.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 9

Problem: Find the probability to flip tails.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 2.
  3. P = 1/2.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 10

Problem: Find the probability to choose a multiple of 3 from 1–12.

  1. Favourable outcomes = 4.
  2. Total equally likely outcomes = 12.
  3. P = 4/12.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/3

Practice exercise

Create one new Grade 8 problem involving Impossible event. Show the important steps, include units when needed, and explain how you checked the answer.

54.7 Likely event

Likely event is an important Grade 8 concept in Probability Foundations. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find the probability to roll an even number on a fair die.

  1. Favourable outcomes = 3.
  2. Total equally likely outcomes = 6.
  3. P = 3/6.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 2

Problem: Find the probability to flip heads on a fair coin.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 2.
  3. P = 1/2.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 3

Problem: Find the probability to draw red from 5 red and 3 blue marbles.

  1. Favourable outcomes = 5.
  2. Total equally likely outcomes = 8.
  3. P = 5/8.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 5/8

Worked Example 4

Problem: Find the probability to choose a vowel from A,B,C,E.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 4.
  3. P = 2/4.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 5

Problem: Find the probability to spin section 1 on four equal sections.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 4.
  3. P = 1/4.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/4

Worked Example 6

Problem: Find the probability to roll a number greater than 4 on a die.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 6.
  3. P = 2/6.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/3

Worked Example 7

Problem: Find the probability to draw blue from 2 blue and 6 green counters.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 8.
  3. P = 2/8.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/4

Worked Example 8

Problem: Find the probability to choose an odd number from 1–10.

  1. Favourable outcomes = 5.
  2. Total equally likely outcomes = 10.
  3. P = 5/10.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 9

Problem: Find the probability to flip tails.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 2.
  3. P = 1/2.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 10

Problem: Find the probability to choose a multiple of 3 from 1–12.

  1. Favourable outcomes = 4.
  2. Total equally likely outcomes = 12.
  3. P = 4/12.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/3

Practice exercise

Create one new Grade 8 problem involving Likely event. Show the important steps, include units when needed, and explain how you checked the answer.

54.8 Unlikely event

Unlikely event is an important Grade 8 concept in Probability Foundations. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find the probability to roll an even number on a fair die.

  1. Favourable outcomes = 3.
  2. Total equally likely outcomes = 6.
  3. P = 3/6.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 2

Problem: Find the probability to flip heads on a fair coin.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 2.
  3. P = 1/2.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 3

Problem: Find the probability to draw red from 5 red and 3 blue marbles.

  1. Favourable outcomes = 5.
  2. Total equally likely outcomes = 8.
  3. P = 5/8.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 5/8

Worked Example 4

Problem: Find the probability to choose a vowel from A,B,C,E.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 4.
  3. P = 2/4.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 5

Problem: Find the probability to spin section 1 on four equal sections.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 4.
  3. P = 1/4.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/4

Worked Example 6

Problem: Find the probability to roll a number greater than 4 on a die.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 6.
  3. P = 2/6.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/3

Worked Example 7

Problem: Find the probability to draw blue from 2 blue and 6 green counters.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 8.
  3. P = 2/8.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/4

Worked Example 8

Problem: Find the probability to choose an odd number from 1–10.

  1. Favourable outcomes = 5.
  2. Total equally likely outcomes = 10.
  3. P = 5/10.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 9

Problem: Find the probability to flip tails.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 2.
  3. P = 1/2.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 10

Problem: Find the probability to choose a multiple of 3 from 1–12.

  1. Favourable outcomes = 4.
  2. Total equally likely outcomes = 12.
  3. P = 4/12.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/3

Practice exercise

Create one new Grade 8 problem involving Unlikely event. Show the important steps, include units when needed, and explain how you checked the answer.

54.9 Probability as fraction

A fraction (a numerator over a denominator) can represent part of a whole, a quotient, a ratio, or a point on a number line. In this section, the focus is Probability as fraction.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Write 1/2 as a decimal.

  1. Divide numerator 1 by denominator 2.
  2. 1 ÷ 2 = 0.5000.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.5

Worked Example 2

Problem: Write 2/3 as a decimal.

  1. Divide numerator 2 by denominator 3.
  2. 2 ÷ 3 = 0.6667.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.6667

Worked Example 3

Problem: Write 3/5 as a decimal.

  1. Divide numerator 3 by denominator 5.
  2. 3 ÷ 5 = 0.6000.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.6

Worked Example 4

Problem: Write 5/8 as a decimal.

  1. Divide numerator 5 by denominator 8.
  2. 5 ÷ 8 = 0.6250.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.625

Worked Example 5

Problem: Write 7/10 as a decimal.

  1. Divide numerator 7 by denominator 10.
  2. 7 ÷ 10 = 0.7000.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.7

Worked Example 6

Problem: Write 4/9 as a decimal.

  1. Divide numerator 4 by denominator 9.
  2. 4 ÷ 9 = 0.4444.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.4444

Worked Example 7

Problem: Write 5/6 as a decimal.

  1. Divide numerator 5 by denominator 6.
  2. 5 ÷ 6 = 0.8333.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.8333

Worked Example 8

Problem: Write 3/4 as a decimal.

  1. Divide numerator 3 by denominator 4.
  2. 3 ÷ 4 = 0.7500.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.75

Worked Example 9

Problem: Write 2/5 as a decimal.

  1. Divide numerator 2 by denominator 5.
  2. 2 ÷ 5 = 0.4000.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.4

Worked Example 10

Problem: Write 7/12 as a decimal.

  1. Divide numerator 7 by denominator 12.
  2. 7 ÷ 12 = 0.5833.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.5833

Practice exercise

Create one new Grade 8 problem involving Probability as fraction. Show the important steps, include units when needed, and explain how you checked the answer.

54.10 Probability as decimal

A decimal uses place value to represent parts of one, including tenths, hundredths, thousandths, and smaller values. In this section, the focus is Probability as decimal.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Which is greater: 3.6 or 0.4?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 3.6

Worked Example 2

Problem: Which is greater: 8.25 or 1.5?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 8.25

Worked Example 3

Problem: Which is greater: 12.75 or 2.4?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 12.75

Worked Example 4

Problem: Which is greater: 0.96 or 0.3?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 0.96

Worked Example 5

Problem: Which is greater: 5.04 or 1.2?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 5.04

Worked Example 6

Problem: Which is greater: 14.8 or 2?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 14.8

Worked Example 7

Problem: Which is greater: 7.125 or 0.5?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 7.125

Worked Example 8

Problem: Which is greater: 20.05 or 3.75?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 20.05

Worked Example 9

Problem: Which is greater: 2.4 or 0.06?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 2.4

Worked Example 10

Problem: Which is greater: 100.5 or 4.02?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 100.5

Practice exercise

Create one new Grade 8 problem involving Probability as decimal. Show the important steps, include units when needed, and explain how you checked the answer.

54.11 Probability as percent

Percent means per hundred, so 35% means 35 out of 100. In this section, the focus is Probability as percent.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find 10% of 80.

  1. Convert 10% to decimal 0.1.
  2. Multiply 0.1 × 80.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 8

Worked Example 2

Problem: Find 25% of 60.

  1. Convert 25% to decimal 0.25.
  2. Multiply 0.25 × 60.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 15

Worked Example 3

Problem: Find 15% of 120.

  1. Convert 15% to decimal 0.15.
  2. Multiply 0.15 × 120.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 18

Worked Example 4

Problem: Find 5% of 250.

  1. Convert 5% to decimal 0.05.
  2. Multiply 0.05 × 250.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 12.5

Worked Example 5

Problem: Find 13% of 75.

  1. Convert 13% to decimal 0.13.
  2. Multiply 0.13 × 75.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 9.75

Worked Example 6

Problem: Find 35% of 140.

  1. Convert 35% to decimal 0.35.
  2. Multiply 0.35 × 140.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 49

Worked Example 7

Problem: Find 8% of 500.

  1. Convert 8% to decimal 0.08.
  2. Multiply 0.08 × 500.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 40

Worked Example 8

Problem: Find 60% of 45.

  1. Convert 60% to decimal 0.6.
  2. Multiply 0.6 × 45.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 27

Worked Example 9

Problem: Find 2.5% of 320.

  1. Convert 2.5% to decimal 0.025.
  2. Multiply 0.025 × 320.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 8

Worked Example 10

Problem: Find 125% of 64.

  1. Convert 125% to decimal 1.25.
  2. Multiply 1.25 × 64.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 80

Practice exercise

Create one new Grade 8 problem involving Probability as percent. Show the important steps, include units when needed, and explain how you checked the answer.

54.12 Probability scale

Probability measures likelihood from 0 to 1, or 0% to 100%. In this section, the focus is Probability scale.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find the probability to roll an even number on a fair die.

  1. Favourable outcomes = 3.
  2. Total equally likely outcomes = 6.
  3. P = 3/6.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 2

Problem: Find the probability to flip heads on a fair coin.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 2.
  3. P = 1/2.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 3

Problem: Find the probability to draw red from 5 red and 3 blue marbles.

  1. Favourable outcomes = 5.
  2. Total equally likely outcomes = 8.
  3. P = 5/8.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 5/8

Worked Example 4

Problem: Find the probability to choose a vowel from A,B,C,E.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 4.
  3. P = 2/4.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 5

Problem: Find the probability to spin section 1 on four equal sections.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 4.
  3. P = 1/4.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/4

Worked Example 6

Problem: Find the probability to roll a number greater than 4 on a die.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 6.
  3. P = 2/6.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/3

Worked Example 7

Problem: Find the probability to draw blue from 2 blue and 6 green counters.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 8.
  3. P = 2/8.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/4

Worked Example 8

Problem: Find the probability to choose an odd number from 1–10.

  1. Favourable outcomes = 5.
  2. Total equally likely outcomes = 10.
  3. P = 5/10.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 9

Problem: Find the probability to flip tails.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 2.
  3. P = 1/2.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 10

Problem: Find the probability to choose a multiple of 3 from 1–12.

  1. Favourable outcomes = 4.
  2. Total equally likely outcomes = 12.
  3. P = 4/12.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/3

Practice exercise

Create one new Grade 8 problem involving Probability scale. Show the important steps, include units when needed, and explain how you checked the answer.

54.13 Complementary events

Complementary events is an important Grade 8 concept in Probability Foundations. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find the probability to roll an even number on a fair die.

  1. Favourable outcomes = 3.
  2. Total equally likely outcomes = 6.
  3. P = 3/6.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 2

Problem: Find the probability to flip heads on a fair coin.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 2.
  3. P = 1/2.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 3

Problem: Find the probability to draw red from 5 red and 3 blue marbles.

  1. Favourable outcomes = 5.
  2. Total equally likely outcomes = 8.
  3. P = 5/8.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 5/8

Worked Example 4

Problem: Find the probability to choose a vowel from A,B,C,E.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 4.
  3. P = 2/4.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 5

Problem: Find the probability to spin section 1 on four equal sections.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 4.
  3. P = 1/4.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/4

Worked Example 6

Problem: Find the probability to roll a number greater than 4 on a die.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 6.
  3. P = 2/6.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/3

Worked Example 7

Problem: Find the probability to draw blue from 2 blue and 6 green counters.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 8.
  3. P = 2/8.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/4

Worked Example 8

Problem: Find the probability to choose an odd number from 1–10.

  1. Favourable outcomes = 5.
  2. Total equally likely outcomes = 10.
  3. P = 5/10.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 9

Problem: Find the probability to flip tails.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 2.
  3. P = 1/2.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 10

Problem: Find the probability to choose a multiple of 3 from 1–12.

  1. Favourable outcomes = 4.
  2. Total equally likely outcomes = 12.
  3. P = 4/12.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/3

Practice exercise

Create one new Grade 8 problem involving Complementary events. Show the important steps, include units when needed, and explain how you checked the answer.

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Chapter 54 Review Questions and Answers

Q1. What is important to remember about Probability?

Answer: Probability measures likelihood from 0 to 1, or 0% to 100%. In this section, the focus is Probability.

Q2. What is important to remember about Outcome?

Answer: Outcome is an important Grade 8 concept in Probability Foundations. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q3. What is important to remember about Event?

Answer: Event is an important Grade 8 concept in Probability Foundations. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q4. What is important to remember about Sample space?

Answer: A sample is a smaller group selected from a population. In this section, the focus is Sample space.

Q5. What is important to remember about Certain event?

Answer: Certain event is an important Grade 8 concept in Probability Foundations. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q6. What is important to remember about Impossible event?

Answer: Impossible event is an important Grade 8 concept in Probability Foundations. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q7. What is important to remember about Likely event?

Answer: Likely event is an important Grade 8 concept in Probability Foundations. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q8. What is important to remember about Unlikely event?

Answer: Unlikely event is an important Grade 8 concept in Probability Foundations. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q9. What is important to remember about Probability as fraction?

Answer: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, a ratio, or a point on a number line. In this section, the focus is Probability as fraction.

Q10. What is important to remember about Probability as decimal?

Answer: A decimal uses place value to represent parts of one, including tenths, hundredths, thousandths, and smaller values. In this section, the focus is Probability as decimal.

Q11. What is important to remember about Probability as percent?

Answer: Percent means per hundred, so 35% means 35 out of 100. In this section, the focus is Probability as percent.

Q12. What is important to remember about Probability scale?

Answer: Probability measures likelihood from 0 to 1, or 0% to 100%. In this section, the focus is Probability scale.

Q13. What is important to remember about Complementary events?

Answer: Complementary events is an important Grade 8 concept in Probability Foundations. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q14. Why should you show your steps?

Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.

Q15. Why is estimation useful?

Answer: Estimation helps you judge whether a calculated answer is reasonable.

Q16. Why do units matter?

Answer: Units tell what a value measures and prevent incompatible quantities from being mixed.

Q17. When should you round?

Answer: Usually round near the end unless the question specifically asks for earlier rounding.

Q18. How can you check an answer?

Answer: Use an inverse operation, substitution, estimation, a graph, or a second method.

Q19. What should you identify first in a word problem?

Answer: Identify what is known, what is unknown, and the mathematical relationship connecting them.

Q20. What makes a final answer complete?

Answer: Give the value, correct units when needed, and a brief interpretation in context.

Q21. What should you do if an answer seems unreasonable?

Answer: Re-read the question and check copied values, signs, units, formulas, and calculations.

Q22. Why are diagrams useful?

Answer: A labelled diagram can reveal relationships among lengths, angles, areas, and coordinates.

Q23. Why are tables useful?

Answer: Tables organize values and help reveal patterns, rates, relationships, and missing information.

Q24. Why should you explain your reasoning?

Answer: Reasoning shows why a method works, not just what answer was obtained.

Q25. How do mistakes help learning?

Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent repetition.

Q26. When is a calculator useful?

Answer: A calculator helps with lengthy arithmetic after the mathematical setup is understood and estimated.

Q27. Why compare more than one strategy?

Answer: Different strategies may make a problem easier and provide a way to verify the result.

Q28. What is mathematical communication?

Answer: It is presenting ideas clearly with words, symbols, diagrams, tables, graphs, and justified steps.

Q29. What is mathematical modelling?

Answer: It is representing a real situation with mathematics, testing the model, and revising it if needed.

Q30. Why should assumptions be stated?

Answer: Assumptions show the conditions the model depends on and help readers judge whether it is reasonable.