Chapter 42: Tallies and Frequency Tables
Learn Grade 3 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Tallies and Frequency Tables with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.
Key Technical Terms
- Meaning of a tally mark (a Grade 3 idea used in this chapter)
- Making groups of five tallies (a Grade 3 idea used in this chapter)
- Reading tally charts (a Grade 3 idea used in this chapter)
- Writing frequencies (a Grade 3 idea used in this chapter)
- Building a frequency table (a Grade 3 idea used in this chapter)
- Comparing frequencies (a Grade 3 idea used in this chapter)
- Finding the most common category (a Grade 3 idea used in this chapter)
- Finding the least common category (a Grade 3 idea used in this chapter)
- Checking totals (a Grade 3 idea used in this chapter)
- Frequency-table word problems (a Grade 3 idea used in this chapter)
How to Learn This Chapter
Read the explanation first, follow the examples one step at a time, cover the answer and retry the problem yourself, then use the practice and review questions to check understanding.
42.1 Meaning of a tally mark
Meaning of a tally mark helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Meaning of a tally mark?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Meaning of a tally mark helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Answer: Meaning of a tally mark helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Worked Example 2
Problem: What should you identify first before solving a problem about Meaning of a tally mark?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Meaning of a tally mark helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Meaning of a tally mark.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Meaning of a tally mark helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Meaning of a tally mark problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Meaning of a tally mark helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Meaning of a tally mark.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Meaning of a tally mark helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Answer: Meaning of a tally mark can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Find the mean of [4, 6, 8].
- Add the values: 18.
- Divide by 3.
Very beginner explanation: The mean is the total shared equally among all values.
Answer: 6
Worked Example 7
Problem: Find the mean of [5, 9, 10, 12].
- Add the values: 36.
- Divide by 4.
Very beginner explanation: The mean is the total shared equally among all values.
Answer: 9
Worked Example 8
Problem: Find the mean of [3, 7, 7, 11].
- Add the values: 28.
- Divide by 4.
Very beginner explanation: The mean is the total shared equally among all values.
Answer: 7
Worked Example 9
Problem: Find the mean of [20, 25, 30].
- Add the values: 75.
- Divide by 3.
Very beginner explanation: The mean is the total shared equally among all values.
Answer: 25
Worked Example 10
Problem: Find the mean of [6, 8, 9, 12, 15].
- Add the values: 50.
- Divide by 5.
Very beginner explanation: The mean is the total shared equally among all values.
Answer: 10
Practice Exercise
Create one new question about Meaning of a tally mark. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
42.2 Making groups of five tallies
Making groups of five tallies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Making groups of five tallies?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Making groups of five tallies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Making groups of five tallies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Worked Example 2
Problem: What should you identify first before solving a problem about Making groups of five tallies?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Making groups of five tallies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Making groups of five tallies.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Making groups of five tallies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Making groups of five tallies problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Making groups of five tallies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Making groups of five tallies.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Making groups of five tallies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Making groups of five tallies can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Explain Making groups of five tallies in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Making groups of five tallies becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Making groups of five tallies and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Making groups of five tallies becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Making groups of five tallies using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Making groups of five tallies becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Making groups of five tallies problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Making groups of five tallies becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Making groups of five tallies could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Making groups of five tallies becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Making groups of five tallies. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
42.3 Reading tally charts
Reading tally charts is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Reading tally charts?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Reading tally charts is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Reading tally charts is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Worked Example 2
Problem: What should you identify first before solving a problem about Reading tally charts?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Reading tally charts is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Reading tally charts.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Reading tally charts is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Reading tally charts problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Reading tally charts is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Reading tally charts.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Reading tally charts is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Reading tally charts can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Explain Reading tally charts in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Reading tally charts becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Reading tally charts and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Reading tally charts becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Reading tally charts using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Reading tally charts becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Reading tally charts problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Reading tally charts becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Reading tally charts could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Reading tally charts becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Reading tally charts. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
42.4 Writing frequencies
Writing frequencies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Writing frequencies?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Writing frequencies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Writing frequencies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Worked Example 2
Problem: What should you identify first before solving a problem about Writing frequencies?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Writing frequencies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Writing frequencies.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Writing frequencies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Writing frequencies problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Writing frequencies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Writing frequencies.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Writing frequencies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Writing frequencies can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Explain Writing frequencies in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Writing frequencies becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Writing frequencies and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Writing frequencies becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Writing frequencies using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Writing frequencies becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Writing frequencies problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Writing frequencies becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Writing frequencies could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Writing frequencies becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Writing frequencies. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
42.5 Building a frequency table
Building a frequency table helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Building a frequency table?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Building a frequency table helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Answer: Building a frequency table helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Worked Example 2
Problem: What should you identify first before solving a problem about Building a frequency table?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Building a frequency table helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Building a frequency table.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Building a frequency table helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Building a frequency table problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Building a frequency table helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Building a frequency table.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Building a frequency table helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Answer: Building a frequency table can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Classify the data [4, 6, 8].
- These values are numerical.
- Decide whether they represent counts or measurements.
Very beginner explanation: Quantitative data are numerical values used for calculations or comparisons.
Answer: Quantitative data
Worked Example 7
Problem: Classify the data [5, 9, 10, 12].
- These values are numerical.
- Decide whether they represent counts or measurements.
Very beginner explanation: Quantitative data are numerical values used for calculations or comparisons.
Answer: Quantitative data
Worked Example 8
Problem: Classify the data [3, 7, 7, 11].
- These values are numerical.
- Decide whether they represent counts or measurements.
Very beginner explanation: Quantitative data are numerical values used for calculations or comparisons.
Answer: Quantitative data
Worked Example 9
Problem: Classify the data [20, 25, 30].
- These values are numerical.
- Decide whether they represent counts or measurements.
Very beginner explanation: Quantitative data are numerical values used for calculations or comparisons.
Answer: Quantitative data
Worked Example 10
Problem: Classify the data [6, 8, 9, 12, 15].
- These values are numerical.
- Decide whether they represent counts or measurements.
Very beginner explanation: Quantitative data are numerical values used for calculations or comparisons.
Answer: Quantitative data
Practice Exercise
Create one new question about Building a frequency table. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
42.6 Comparing frequencies
Comparing frequencies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Comparing frequencies?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Comparing frequencies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Comparing frequencies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Worked Example 2
Problem: What should you identify first before solving a problem about Comparing frequencies?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Comparing frequencies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Comparing frequencies.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Comparing frequencies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Comparing frequencies problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Comparing frequencies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Comparing frequencies.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Comparing frequencies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Comparing frequencies can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Explain Comparing frequencies in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Comparing frequencies becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Comparing frequencies and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Comparing frequencies becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Comparing frequencies using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Comparing frequencies becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Comparing frequencies problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Comparing frequencies becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Comparing frequencies could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Comparing frequencies becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Comparing frequencies. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
42.7 Finding the most common category
Finding the most common category is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Finding the most common category?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Finding the most common category is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Finding the most common category is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Worked Example 2
Problem: What should you identify first before solving a problem about Finding the most common category?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Finding the most common category is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Finding the most common category.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Finding the most common category is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Finding the most common category problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Finding the most common category is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Finding the most common category.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Finding the most common category is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Finding the most common category can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Explain Finding the most common category in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Finding the most common category becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Finding the most common category and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Finding the most common category becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Finding the most common category using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Finding the most common category becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Finding the most common category problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Finding the most common category becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Finding the most common category could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Finding the most common category becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Finding the most common category. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
42.8 Finding the least common category
Finding the least common category is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Finding the least common category?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Finding the least common category is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Finding the least common category is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Worked Example 2
Problem: What should you identify first before solving a problem about Finding the least common category?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Finding the least common category is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Finding the least common category.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Finding the least common category is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Finding the least common category problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Finding the least common category is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Finding the least common category.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Finding the least common category is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Finding the least common category can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Explain Finding the least common category in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Finding the least common category becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Finding the least common category and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Finding the least common category becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Finding the least common category using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Finding the least common category becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Finding the least common category problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Finding the least common category becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Finding the least common category could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Finding the least common category becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Finding the least common category. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
42.9 Checking totals
Checking totals is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Checking totals?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Checking totals is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Checking totals is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Worked Example 2
Problem: What should you identify first before solving a problem about Checking totals?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Checking totals is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Checking totals.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Checking totals is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Checking totals problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Checking totals is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Checking totals.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Checking totals is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Checking totals can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Explain Checking totals in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Checking totals becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Checking totals and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Checking totals becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Checking totals using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Checking totals becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Checking totals problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Checking totals becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Checking totals could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Checking totals becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Checking totals. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
42.10 Frequency-table word problems
Frequency-table word problems helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Frequency-table word problems?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Frequency-table word problems helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Answer: Frequency-table word problems helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Worked Example 2
Problem: What should you identify first before solving a problem about Frequency-table word problems?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Frequency-table word problems helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Frequency-table word problems.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Frequency-table word problems helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Frequency-table word problems problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Frequency-table word problems helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Frequency-table word problems.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Frequency-table word problems helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Answer: Frequency-table word problems can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Classify the data [4, 6, 8].
- These values are numerical.
- Decide whether they represent counts or measurements.
Very beginner explanation: Quantitative data are numerical values used for calculations or comparisons.
Answer: Quantitative data
Worked Example 7
Problem: Classify the data [5, 9, 10, 12].
- These values are numerical.
- Decide whether they represent counts or measurements.
Very beginner explanation: Quantitative data are numerical values used for calculations or comparisons.
Answer: Quantitative data
Worked Example 8
Problem: Classify the data [3, 7, 7, 11].
- These values are numerical.
- Decide whether they represent counts or measurements.
Very beginner explanation: Quantitative data are numerical values used for calculations or comparisons.
Answer: Quantitative data
Worked Example 9
Problem: Classify the data [20, 25, 30].
- These values are numerical.
- Decide whether they represent counts or measurements.
Very beginner explanation: Quantitative data are numerical values used for calculations or comparisons.
Answer: Quantitative data
Worked Example 10
Problem: Classify the data [6, 8, 9, 12, 15].
- These values are numerical.
- Decide whether they represent counts or measurements.
Very beginner explanation: Quantitative data are numerical values used for calculations or comparisons.
Answer: Quantitative data
Practice Exercise
Create one new question about Frequency-table word problems. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
Chapter Notes, Practice, and Common Mistakes
Chapter Notes
- Read the question before calculating.
- Represent the mathematics with a model, table, graph, diagram, number line, or equation when useful.
- Show the reasoning and check the final answer with a second method when possible.
Extra Practice
- Create and solve one original problem about Meaning of a tally mark.
- Create and solve one original problem about Making groups of five tallies.
- Create and solve one original problem about Reading tally charts.
- Create and solve one original problem about Writing frequencies.
- Create and solve one original problem about Building a frequency table.
- Create and solve one original problem about Comparing frequencies.
Common Mistakes
- Skipping the meaning and trying to memorize a rule only.
- Using the wrong operation because the question was not read completely.
- Ignoring units, labels, place values, or the context of the problem.
- Not estimating or checking whether the final answer is reasonable.
30 Review Questions and Answers
Q1. What is the key idea in Meaning of a tally mark?
Answer: Meaning of a tally mark helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Q2. What is the key idea in Making groups of five tallies?
Answer: Making groups of five tallies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Q3. What is the key idea in Reading tally charts?
Answer: Reading tally charts is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Q4. What is the key idea in Writing frequencies?
Answer: Writing frequencies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Q5. What is the key idea in Building a frequency table?
Answer: Building a frequency table helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Q6. What is the key idea in Comparing frequencies?
Answer: Comparing frequencies is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Q7. What is the key idea in Finding the most common category?
Answer: Finding the most common category is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Q8. What is the key idea in Finding the least common category?
Answer: Finding the least common category is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Q9. What is the key idea in Checking totals?
Answer: Checking totals is an important Grade 3 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Q10. What is the key idea in Frequency-table word problems?
Answer: Frequency-table word problems helps learners collect, organize, display, and interpret information carefully so conclusions are supported by evidence.
Q11. Why should you estimate before or after a calculation?
Answer: Estimation gives a reasonable range and can reveal place-value or operation mistakes.
Q12. Why are labels and units important?
Answer: They show what a number represents and help prevent mixing unlike quantities.
Q13. How can a diagram help solve a problem?
Answer: A diagram makes quantities and relationships visible before calculation.
Q14. How can inverse operations check an answer?
Answer: An inverse operation reverses the original operation and should recover the starting value when the work is correct.
Q15. Why should you show steps?
Answer: Showing steps makes reasoning clear and helps find where an error happened.
Q16. What should you do after making a mistake?
Answer: Find the step where the reasoning changed, correct it, and try a similar problem again.
Q17. How can a number line support reasoning?
Answer: It shows order, distance, relative size, fractions, decimals, and inequality solutions visually.
Q18. When is a table useful?
Answer: A table organizes related values so patterns and comparisons are easier to see.
Q19. When is a graph useful?
Answer: A graph makes trends, comparisons, locations, or data patterns visible.
Q20. How do you decide which operation to use?
Answer: Use the meaning of the situation: combine, compare, find a difference, form equal groups, or find how many groups fit.
Q21. Why should you check place value?
Answer: A digit or decimal has a different value depending on its position.
Q22. What makes an answer reasonable?
Answer: It fits the context, has the expected size and units, and agrees with an estimate or second method.
Q23. How can you explain mathematical reasoning clearly?
Answer: Name the information, the rule or operation, the steps, and why the final answer fits the problem.
Q24. Why can more than one strategy be correct?
Answer: Different valid strategies can represent the same mathematical relationship and lead to the same result.
Q25. How should you approach a difficult Grade 3 problem?
Answer: Break it into smaller parts, solve one part at a time, and check each result before continuing.
Q26. How can you practise Tallies and Frequency Tables effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.
Q27. How can you practise Tallies and Frequency Tables effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.
Q28. How can you practise Tallies and Frequency Tables effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.
Q29. How can you practise Tallies and Frequency Tables effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.
Q30. How can you practise Tallies and Frequency Tables effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.