Chapter 56: Area and Perimeter Relationships
Learn Grade 5 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Area and Perimeter Relationships with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.
Key Technical Terms
- Rectangle area review (a Grade 5 idea used in this chapter)
- Parallelogram area from rectangle relationships (a Grade 5 idea used in this chapter)
- Formula for parallelogram area (a Grade 5 idea used in this chapter)
- Triangle area from parallelogram relationships (a Grade 5 idea used in this chapter)
- Formula for triangle area (a Grade 5 idea used in this chapter)
- Area word problems (a Grade 5 idea used in this chapter)
- Perimeter review (a Grade 5 idea used in this chapter)
- Shapes with equal area and different perimeters (a Grade 5 idea used in this chapter)
- Finding missing dimensions (a Grade 5 idea used in this chapter)
- Area and perimeter problem solving (a Grade 5 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.
56.1 Rectangle area review
Rectangle area review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Rectangle area review?
- 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: Rectangle area review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Rectangle area review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Worked Example 2
Problem: What should you identify first before solving a problem about Rectangle area review?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Rectangle area review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Rectangle area review.
- 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: Rectangle area review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Rectangle area review 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: Rectangle area review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Rectangle area review.
- 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: Rectangle area review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Rectangle area review 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 supplementary angle to 35°.
- Supplementary angles total 180°.
- 180 - 35 = 145.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 145°
Worked Example 7
Problem: Find the supplementary angle to 48°.
- Supplementary angles total 180°.
- 180 - 48 = 132.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 132°
Worked Example 8
Problem: Find the supplementary angle to 67°.
- Supplementary angles total 180°.
- 180 - 67 = 113.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 113°
Worked Example 9
Problem: Find the supplementary angle to 72°.
- Supplementary angles total 180°.
- 180 - 72 = 108.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 108°
Worked Example 10
Problem: Find the supplementary angle to 110°.
- Supplementary angles total 180°.
- 180 - 110 = 70.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 70°
Practice Exercise
Create one new question about Rectangle area review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
56.2 Parallelogram area from rectangle relationships
Parallelogram area from rectangle relationships develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Parallelogram area from rectangle relationships?
- 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: Parallelogram area from rectangle relationships develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Parallelogram area from rectangle relationships develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Worked Example 2
Problem: What should you identify first before solving a problem about Parallelogram area from rectangle relationships?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Parallelogram area from rectangle relationships develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Parallelogram area from rectangle relationships.
- 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: Parallelogram area from rectangle relationships develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Parallelogram area from rectangle relationships 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: Parallelogram area from rectangle relationships develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Parallelogram area from rectangle relationships.
- 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: Parallelogram area from rectangle relationships develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Parallelogram area from rectangle relationships can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: For y = 2x + (1), find y when x = 3.
- Substitute x = 3.
- y = 2(3) + (1).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 7
Worked Example 7
Problem: For y = -1x + (4), find y when x = 3.
- Substitute x = 3.
- y = -1(3) + (4).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 1
Worked Example 8
Problem: For y = 0.5x + (-2), find y when x = 3.
- Substitute x = 3.
- y = 0.5(3) + (-2).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: -0.5
Worked Example 9
Problem: For y = 3x + (0), find y when x = 3.
- Substitute x = 3.
- y = 3(3) + (0).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 9
Worked Example 10
Problem: For y = -2x + (5), find y when x = 3.
- Substitute x = 3.
- y = -2(3) + (5).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: -1
Practice Exercise
Create one new question about Parallelogram area from rectangle relationships. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
56.3 Formula for parallelogram area
Formula for parallelogram area develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Formula for parallelogram area?
- 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: Formula for parallelogram area develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Formula for parallelogram area develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Worked Example 2
Problem: What should you identify first before solving a problem about Formula for parallelogram area?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Formula for parallelogram area develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Formula for parallelogram area.
- 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: Formula for parallelogram area develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Formula for parallelogram area 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: Formula for parallelogram area develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Formula for parallelogram area.
- 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: Formula for parallelogram area develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Formula for parallelogram area 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 supplementary angle to 35°.
- Supplementary angles total 180°.
- 180 - 35 = 145.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 145°
Worked Example 7
Problem: Find the supplementary angle to 48°.
- Supplementary angles total 180°.
- 180 - 48 = 132.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 132°
Worked Example 8
Problem: Find the supplementary angle to 67°.
- Supplementary angles total 180°.
- 180 - 67 = 113.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 113°
Worked Example 9
Problem: Find the supplementary angle to 72°.
- Supplementary angles total 180°.
- 180 - 72 = 108.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 108°
Worked Example 10
Problem: Find the supplementary angle to 110°.
- Supplementary angles total 180°.
- 180 - 110 = 70.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 70°
Practice Exercise
Create one new question about Formula for parallelogram area. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
56.4 Triangle area from parallelogram relationships
Triangle area from parallelogram relationships develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Triangle area from parallelogram relationships?
- 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: Triangle area from parallelogram relationships develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Triangle area from parallelogram relationships develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Worked Example 2
Problem: What should you identify first before solving a problem about Triangle area from parallelogram relationships?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Triangle area from parallelogram relationships develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Triangle area from parallelogram relationships.
- 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: Triangle area from parallelogram relationships develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Triangle area from parallelogram relationships 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: Triangle area from parallelogram relationships develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Triangle area from parallelogram relationships.
- 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: Triangle area from parallelogram relationships develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Triangle area from parallelogram relationships can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: For y = 2x + (1), find y when x = 3.
- Substitute x = 3.
- y = 2(3) + (1).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 7
Worked Example 7
Problem: For y = -1x + (4), find y when x = 3.
- Substitute x = 3.
- y = -1(3) + (4).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 1
Worked Example 8
Problem: For y = 0.5x + (-2), find y when x = 3.
- Substitute x = 3.
- y = 0.5(3) + (-2).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: -0.5
Worked Example 9
Problem: For y = 3x + (0), find y when x = 3.
- Substitute x = 3.
- y = 3(3) + (0).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 9
Worked Example 10
Problem: For y = -2x + (5), find y when x = 3.
- Substitute x = 3.
- y = -2(3) + (5).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: -1
Practice Exercise
Create one new question about Triangle area from parallelogram relationships. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
56.5 Formula for triangle area
Formula for triangle area develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Formula for triangle area?
- 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: Formula for triangle area develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Formula for triangle area develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Worked Example 2
Problem: What should you identify first before solving a problem about Formula for triangle area?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Formula for triangle area develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Formula for triangle area.
- 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: Formula for triangle area develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Formula for triangle area 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: Formula for triangle area develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Formula for triangle area.
- 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: Formula for triangle area develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Formula for triangle area can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: A triangle has angles 35° and 50°. Find the third angle.
- Triangle angles total 180°.
- 180 - 35 - 50 = 95.
Very beginner explanation: Subtract the known angles from 180°.
Answer: 95°
Worked Example 7
Problem: A triangle has angles 48° and 50°. Find the third angle.
- Triangle angles total 180°.
- 180 - 48 - 50 = 82.
Very beginner explanation: Subtract the known angles from 180°.
Answer: 82°
Worked Example 8
Problem: A triangle has angles 67° and 50°. Find the third angle.
- Triangle angles total 180°.
- 180 - 67 - 50 = 63.
Very beginner explanation: Subtract the known angles from 180°.
Answer: 63°
Worked Example 9
Problem: A triangle has angles 72° and 50°. Find the third angle.
- Triangle angles total 180°.
- 180 - 72 - 50 = 58.
Very beginner explanation: Subtract the known angles from 180°.
Answer: 58°
Worked Example 10
Problem: A triangle has angles 110° and 50°. Find the third angle.
- Triangle angles total 180°.
- 180 - 110 - 50 = 20.
Very beginner explanation: Subtract the known angles from 180°.
Answer: 20°
Practice Exercise
Create one new question about Formula for triangle area. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
56.6 Area word problems
Area word problems develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Area 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: Area word problems develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Area word problems develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Worked Example 2
Problem: What should you identify first before solving a problem about Area word problems?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Area word problems develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Area 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: Area word problems develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Area 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: Area word problems develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Area 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: Area word problems develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Area 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: Find the area of a rectangle 6 cm by 3 cm.
- Use A = length × width.
- A = 6 × 3.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 18 cm²
Worked Example 7
Problem: Find the area of a rectangle 7 cm by 4 cm.
- Use A = length × width.
- A = 7 × 4.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 28 cm²
Worked Example 8
Problem: Find the area of a rectangle 8 cm by 5 cm.
- Use A = length × width.
- A = 8 × 5.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 40 cm²
Worked Example 9
Problem: Find the area of a rectangle 9 cm by 6 cm.
- Use A = length × width.
- A = 9 × 6.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 54 cm²
Worked Example 10
Problem: Find the area of a rectangle 10 cm by 7 cm.
- Use A = length × width.
- A = 10 × 7.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 70 cm²
Practice Exercise
Create one new question about Area word problems. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
56.7 Perimeter review
Perimeter review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Perimeter review?
- 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: Perimeter review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Perimeter review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Worked Example 2
Problem: What should you identify first before solving a problem about Perimeter review?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Perimeter review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Perimeter review.
- 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: Perimeter review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Perimeter review 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: Perimeter review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Perimeter review.
- 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: Perimeter review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Perimeter review 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 perimeter of a rectangle 5 cm by 2 cm.
- Use P = 2(l + w).
- P = 2(5+2).
Very beginner explanation: Perimeter measures the total distance around a 2D figure.
Answer: 14 cm
Worked Example 7
Problem: Find the perimeter of a rectangle 6 cm by 3 cm.
- Use P = 2(l + w).
- P = 2(6+3).
Very beginner explanation: Perimeter measures the total distance around a 2D figure.
Answer: 18 cm
Worked Example 8
Problem: Find the perimeter of a rectangle 7 cm by 4 cm.
- Use P = 2(l + w).
- P = 2(7+4).
Very beginner explanation: Perimeter measures the total distance around a 2D figure.
Answer: 22 cm
Worked Example 9
Problem: Find the perimeter of a rectangle 8 cm by 5 cm.
- Use P = 2(l + w).
- P = 2(8+5).
Very beginner explanation: Perimeter measures the total distance around a 2D figure.
Answer: 26 cm
Worked Example 10
Problem: Find the perimeter of a rectangle 9 cm by 6 cm.
- Use P = 2(l + w).
- P = 2(9+6).
Very beginner explanation: Perimeter measures the total distance around a 2D figure.
Answer: 30 cm
Practice Exercise
Create one new question about Perimeter review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
56.8 Shapes with equal area and different perimeters
Shapes with equal area and different perimeters develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Shapes with equal area and different perimeters?
- 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: Shapes with equal area and different perimeters develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Shapes with equal area and different perimeters develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Worked Example 2
Problem: What should you identify first before solving a problem about Shapes with equal area and different perimeters?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Shapes with equal area and different perimeters develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Shapes with equal area and different perimeters.
- 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: Shapes with equal area and different perimeters develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Shapes with equal area and different perimeters 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: Shapes with equal area and different perimeters develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Shapes with equal area and different perimeters.
- 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: Shapes with equal area and different perimeters develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Shapes with equal area and different perimeters 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 area of a rectangle 6 cm by 3 cm.
- Use A = length × width.
- A = 6 × 3.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 18 cm²
Worked Example 7
Problem: Find the area of a rectangle 7 cm by 4 cm.
- Use A = length × width.
- A = 7 × 4.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 28 cm²
Worked Example 8
Problem: Find the area of a rectangle 8 cm by 5 cm.
- Use A = length × width.
- A = 8 × 5.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 40 cm²
Worked Example 9
Problem: Find the area of a rectangle 9 cm by 6 cm.
- Use A = length × width.
- A = 9 × 6.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 54 cm²
Worked Example 10
Problem: Find the area of a rectangle 10 cm by 7 cm.
- Use A = length × width.
- A = 10 × 7.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 70 cm²
Practice Exercise
Create one new question about Shapes with equal area and different perimeters. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
56.9 Finding missing dimensions
Finding missing dimensions is an important Grade 5 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 missing dimensions?
- 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 missing dimensions is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Finding missing dimensions is an important Grade 5 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 missing dimensions?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Finding missing dimensions is an important Grade 5 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 missing dimensions.
- 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 missing dimensions is an important Grade 5 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 missing dimensions 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 missing dimensions is an important Grade 5 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 missing dimensions.
- 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 missing dimensions is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Finding missing dimensions 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 missing dimensions 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 missing dimensions 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 missing dimensions 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 missing dimensions 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 missing dimensions 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 missing dimensions 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 missing dimensions 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 missing dimensions 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 missing dimensions 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 missing dimensions 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 missing dimensions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
56.10 Area and perimeter problem solving
Area and perimeter problem solving develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Area and perimeter problem solving?
- 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: Area and perimeter problem solving develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Area and perimeter problem solving develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Worked Example 2
Problem: What should you identify first before solving a problem about Area and perimeter problem solving?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Area and perimeter problem solving develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Area and perimeter problem solving.
- 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: Area and perimeter problem solving develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Area and perimeter problem solving 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: Area and perimeter problem solving develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 Area and perimeter problem solving.
- 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: Area and perimeter problem solving develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Area and perimeter problem solving 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 area of a rectangle 6 cm by 3 cm.
- Use A = length × width.
- A = 6 × 3.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 18 cm²
Worked Example 7
Problem: Find the area of a rectangle 7 cm by 4 cm.
- Use A = length × width.
- A = 7 × 4.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 28 cm²
Worked Example 8
Problem: Find the area of a rectangle 8 cm by 5 cm.
- Use A = length × width.
- A = 8 × 5.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 40 cm²
Worked Example 9
Problem: Find the area of a rectangle 9 cm by 6 cm.
- Use A = length × width.
- A = 9 × 6.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 54 cm²
Worked Example 10
Problem: Find the area of a rectangle 10 cm by 7 cm.
- Use A = length × width.
- A = 10 × 7.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 70 cm²
Practice Exercise
Create one new question about Area and perimeter problem solving. 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 Rectangle area review.
- Create and solve one original problem about Parallelogram area from rectangle relationships.
- Create and solve one original problem about Formula for parallelogram area.
- Create and solve one original problem about Triangle area from parallelogram relationships.
- Create and solve one original problem about Formula for triangle area.
- Create and solve one original problem about Area word problems.
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 Rectangle area review?
Answer: Rectangle area review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Q2. What is the key idea in Parallelogram area from rectangle relationships?
Answer: Parallelogram area from rectangle relationships develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Q3. What is the key idea in Formula for parallelogram area?
Answer: Formula for parallelogram area develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Q4. What is the key idea in Triangle area from parallelogram relationships?
Answer: Triangle area from parallelogram relationships develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Q5. What is the key idea in Formula for triangle area?
Answer: Formula for triangle area develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Q6. What is the key idea in Area word problems?
Answer: Area word problems develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Q7. What is the key idea in Perimeter review?
Answer: Perimeter review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Q8. What is the key idea in Shapes with equal area and different perimeters?
Answer: Shapes with equal area and different perimeters develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Q9. What is the key idea in Finding missing dimensions?
Answer: Finding missing dimensions is an important Grade 5 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 Area and perimeter problem solving?
Answer: Area and perimeter problem solving develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
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 5 problem?
Answer: Break it into smaller parts, solve one part at a time, and check each result before continuing.
Q26. How can you practise Area and Perimeter Relationships 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 Area and Perimeter Relationships 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 Area and Perimeter Relationships 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 Area and Perimeter Relationships 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 Area and Perimeter Relationships effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.