Volume and Capacity

SUBJECT: MATHEMATICS

CLASS: BASIC FIVE / PRIMARY 5

TERM : SECOND TERM

WEEK : WEEK 10

TOPIC : Volume and Capacity 

  • Measurement of volume in cubes and cuboids using unit cubes
  • Measurement of volume in cubes and cuboid using formula
  • Comparing volume of spheres and cuboids
  • Discovering the relationship between litre and cubic centimeter
  • Real life problems

Importance

  • It is useful in Science and catering services
  • It helps in the measurement of quantity eg groundnut oil, palm oil, kerosene, water etc

Learning Objectives :

Pupils should be able to

  • Use units to find the volume of cube and cuboids
  • Use formula to find the volume of cuboid
  • Find the relationship between litres and cubic litres and vice versa
  • Students will be able to understand the concept of volume and capacity.
  • Students will be able to measure the volume of different shapes using unit cubes and formulas.
  • Students will be able to compare the volume of spheres and cuboids.
  • Students will be able to understand the relationship between liters and cubic centimeters and convert between the two units.

Learning Activities :

Pupils in pairs

  • are taken for a gallery work round the school. They observe different work round the school. They observe different storage materials that have their volumes and capacities written on them. For example water storage tank, water bucket, keg, water bottles, etc. They then compare the differences between all various types of materials with their respective volumes and capacities, arranging them in their increasing sizes

Embedded Core Skills

  • Critical thinking and problem solving skills
  • Communication and Collaboration
  • Student Leadership skills and Personal Development

Learning Resources

  • Flash cards
  • Formula of perimeters of shapes
  • Cardboards to cut to different shapes
  • Unit cubes
  • Measuring tape or ruler
  • Graduated cylinder or pipette
  • Various shapes (e.g. cubes, cuboids, cylinders, spheres, triangular prisms)
  • Handouts with formulas and conversion tables
  • Maggi cubes
  • Dice

Content

Measuring the volume of a cube or cuboid using unit cubes means breaking down the larger shape into smaller, equal-sized cubes called “unit cubes.” The total number of unit cubes used to fill the larger shape is equal to the volume of the shape.

For example, a cube that measures 5cm x 5cm x 5cm has a volume of 5 x 5 x 5 = 125 cubic cm. This can be visualized by filling the cube with 125 unit cubes, each measuring 1cm x 1cm x 1cm.

Similarly, a rectangular shaped box that measures 10cm x 8cm x 6cm has a volume of 10 x 8 x 6 = 480 cubic cm. This can be visualized by filling the box with 480 unit cubes, each measuring 1cm x 1cm x 1cm.

You can also use unit cubes to measure the volume of cylinders, spheres, triangular prism and other 3D shapes. The concept is the same as breaking down the shape into smaller unit cubes and counting the number of unit cubes required to fill the shape.

 

  1. A cube shaped box that measures 5cm x 5cm x 5cm has a volume of 5 x 5 x 5 = 125 cubic cm because each side measures 5cm and a cube’s volume is found by multiplying the length of one side by itself three times.
  2. A rectangular shaped box that measures 10cm x 8cm x 6cm has a volume of 10 x 8 x 6 = 480 cubic cm because each side measures 10cm x 8cm x 6cm and a rectangular shaped box’s volume is found by multiplying the length x width x height.
  3. A cylinder that measures a radius of 3cm and a height of 6cm has a volume of 3.14 x 3 x 3 x 6 = 113.09 cubic cm because to find the volume of a cylinder you need to use the formula (pi) x (radius)^2 x height.
  4. A sphere that measures a radius of 4cm has a volume of 4/3 x 3.14 x 4 x 4 x 4 = 268.08 cubic cm because to find the volume of a sphere you need to use the formula (4/3) x (pi) x (radius)^3.
  5. A triangular prism that measures 8cm x 6cm x 4cm has a volume of 8 x 6 x 4 = 192 cubic cm because a triangular prism’s volume is found by multiplying the base x height x length

 

 

Measurement of volume in cubes and cuboid using formula

Measuring the volume of a cube or cuboid using a formula is another way to find the amount of space inside a 3D shape.

  1. The formula for finding the volume of a cube is V = s^3, where s is the length of one side of the cube. For example, a cube that measures 5cm on each side has a volume of 5^3 = 125 cubic cm.
  2. The formula for finding the volume of a cuboid is V = l x w x h, where l is the length, w is the width, and h is the height. For example, a rectangular shaped box that measures 10cm x 8cm x 6cm has a volume of 10 x 8 x 6 = 480 cubic cm.
  3. The formula for finding the volume of a cylinder is V = πr²h, where r is the radius of the base, h is the height and π is a mathematical constant. For example, a cylinder that measures a radius of 3cm and a height of 6cm has a volume of 3.14 x 3² x 6 = 113.09 cubic cm.
  4. The formula for finding the volume of a sphere is V = 4/3 x π x r³, where r is the radius of the sphere and π is a mathematical constant. For example, a sphere that measures a radius of 4cm has a volume of 4/3 x 3.14 x 4³ = 268.08 cubic cm.
  5. The formula for finding the volume of a triangular prism is V = b x h x l, where b is the base, h is the height, and l is the length. For example, a triangular prism that measures 8cm x 6cm x 4cm has a volume of 8 x 6 x 4 = 192 cubic cm.

Evaluation

  1. What is the formula for finding the volume of a cube? a) V = s^3 b) V = l x w x h c) V = πr²h d) V = 4/3 x π x r³
  2. What is the formula for finding the volume of a cuboid? a) V = s^3 b) V = l x w x h c) V = πr²h d) V = 4/3 x π x r³
  3. How do you find the volume of a cylinder? a) V = s^3 b) V = l x w x h c) V = πr²h d) V = 4/3 x π x r³
  4. How do you find the volume of a sphere? a) V = s^3 b) V = l x w x h c) V = πr²h d) V = 4/3 x π x r³
  5. How do you find the volume of a triangular prism? a) V = b x h x l b) V = l x w x h c) V = πr²h d) V = 4/3 x π x r³
  6. What is the volume of a cube with side length of 4cm? a) 12 cubic cm b) 16 cubic cm c) 64 cubic cm d) 128 cubic cm
  7. A rectangular shaped box measures 8cm x 6cm x 4cm, what is its volume? a) 24 cubic cm b) 32 cubic cm c) 48 cubic cm d) 192 cubic cm
  8. A cylinder has a radius of 2cm and a height of 3cm, what is its volume? a) 4π cubic cm b) 6π cubic cm c) 12π cubic cm d) 18π cubic cm
  9. A sphere has a radius of 5cm, what is its volume? a) 25π cubic cm b) 50π cubic cm c) 100π cubic cm d) 125π cubic cm
  10. A triangular prism has a base of 4cm, a height of 6cm and a length of 8cm, what is its volume? a) 32 cubic cm b) 48 cubic cm c) 64 cubic cm d) 96 cubic cm

Comparing volume of spheres and cuboids

Comparing the volume of spheres and cuboids can be done by using the formulas for finding the volume of each shape.

A sphere’s volume is found by using the formula V = 4/3 x π x r³, where r is the radius of the sphere and π is a mathematical constant. This means that the volume of a sphere increases as the radius increases.

A cuboid’s volume is found by using the formula V = l x w x h, where l is the length, w is the width, and h is the height. This means that the volume of a cuboid increases as any of its dimensions (length, width, or height) increases.

In general, spheres have a smaller volume than cuboids of the same size. This is because a sphere’s volume is dependent only on its radius, while a cuboid’s volume is dependent on its three dimensions (length, width, and height). A cuboid can have larger dimensions than a sphere of the same volume, making it appear larger.

However, it’s worth noting that the volume of a sphere and a cuboid can be equal if the cuboid is a cube and the sphere’s radius is equal to the cube’s edge length.

It’s also worth noting that while spheres and cuboids are different shapes, spheres are very efficient in terms of volume to surface area ratio. This makes spheres useful in design and manufacturing of everyday objects like pressure vessels, tanks, and even footballs

 

Moore Explanation

 

 

  1. A sphere with a radius of 2cm has a volume of approximately 33.51 cubic cm. A cuboid with the dimensions of 2cm x 2cm x 2cm has a volume of 8 cubic cm. The cuboid has a larger volume than the sphere, even though they have the same radius.
  2. A sphere with a radius of 5m has a volume of approximately 523.60 cubic m. A cuboid with the dimensions of 5m x 5m x 5m has a volume of 125 cubic m. The cuboid has a larger volume than the sphere.
  3. A sphere with a radius of 1km has a volume of approximately 4188.79 cubic km. A cuboid with the dimensions of 1km x 1km x 1km has a volume of 1 cubic km. The cuboid has a smaller volume than the sphere.
  4. A sphere with a radius of 3cm has a volume of approximately 113.10 cubic cm. A cuboid with the dimensions of 2cm x 2cm x 3cm has a volume of 12 cubic cm. The sphere has a larger volume than the cuboid.
  5. A sphere with a radius of 0.5m has a volume of approximately 0.52 cubic m. A cuboid with the dimensions of 0.5m x 0.5m x 0.5m has a volume of 0.125 cubic m. The sphere has a larger volume than the cuboid.

It’s important to note that while a cuboid can have a larger volume than a sphere, spheres are more efficient in terms of volume to surface area ratio, which makes them useful in design and manufacturing of everyday objects like pressure vessels, tanks, and even footballs. Also, it’s important to notice that the volume of a sphere and a cuboid can be equal if the cuboid is a cube and the sphere’s radius is equal to the cube’s edge length.

 

Discovering the relationship between litre and cubic centimeter

Liters (L) and cubic centimeters (cm³) are both units of measurement for volume. However, there is a difference between the two units in terms of scale.

1 liter is equal to 1,000 cubic centimeters. This means that if you have a container that holds 1 liter of liquid, it will also hold 1,000 cubic centimeters of liquid. To convert from liters to cubic centimeters, you would multiply the number of liters by 1,000.

For example, if you have 2 liters of liquid, this would be equivalent to 2,000 cubic centimeters.

On the other hand, 1 cubic centimeter is equal to 0.001 liters. To convert from cubic centimeters to liters, you would divide the number of cubic centimeters by 1,000.

For example, if you have 2,000 cubic centimeters of liquid, this would be equivalent to 2 liters.

In everyday use, the liter is more commonly used for measuring larger quantities of liquid such as fuel, water, and other beverages, while cubic centimeters is more often used for smaller quantities such as medicine doses, laboratory samples, and engine oil.

It’s important to note that liters and cubic centimeters are part of the metric system and are internationally accepted units of measurement for volume

 

Real life Problems involving volume and capacity

  1. A construction company needs to calculate the amount of concrete needed to pour a slab for a new building. They measure the area of the slab and the depth of the slab, and use the formula volume = area x depth to determine the amount of concrete needed.
  2. A farmer needs to calculate the amount of water needed to fill a irrigation tank. They measure the dimensions of the tank and use the formula volume = length x width x height to determine the amount of water needed.
  3. A scientist needs to measure the volume of a liquid sample in a laboratory. They use a graduated cylinder or a pipette to measure the volume of the sample, which is typically measured in milliliters or cubic centimeters.
  4. A homeowner needs to calculate the amount of paint needed to paint a room. They measure the dimensions of the room and use the formula volume = area x height to determine the amount of paint needed.
  5. A gas station needs to calculate the amount of fuel needed to fill up a tanker truck. They measure the volume of the tanker truck and use the formula volume = length x width x height to determine the amount of fuel needed

Evaluation

  1. How many cubic centimeters are in 1 liter? a) 100 b) 500 c) 1000 d) 5000
  2. How many liters are in 5000 cubic centimeters? a) 0.05 b) 5 c) 50 d) 500
  3. A container holds 2 liters of liquid, how many cubic centimeters is this equivalent to? a) 2000 b) 200 c) 20 d) 2
  4. A container holds 500 cubic centimeters of liquid, how many liters is this equivalent to? a) 0.5 b) 5 c) 50 d) 500
  5. How do you convert liters to cubic centimeters? a) Multiply by 1,000 b) Divide by 1,000 c) Add 1,000 d) Subtract 1,000
  6. How do you convert cubic centimeters to liters? a) Multiply by 1,000 b) Divide by 1,000 c) Add 1,000 d) Subtract 1,000
  7. A tank holds 250 liters of water, how many cubic centimeters is this equivalent to? a) 250000 b) 25000 c) 2500 d) 250
  8. A container holds 1 liter of oil, how many cubic centimeters is this equivalent to? a) 1000 b) 100 c) 10 d) 1
  9. A cylinder has a radius of 5cm and a height of 20cm, what is its volume in liters? a) 0.314 b) 3.14 c) 31.4 d) 314
  10. A container holds 0.5 liters of liquid, how many cubic centimeters is this equivalent to? a) 500 b) 50 c) 5 d) 0.5
  11. What is the difference between volume and capacity? a) Volume is the amount of space an object takes up, while capacity is the amount of liquid an object can hold. b) Volume is the amount of liquid an object can hold, while capacity is the amount of space an object takes up. c) Volume and capacity are the same thing. d) Volume is the weight of an object, while capacity is the amount of space an object takes up.
  12. What is the formula for finding the volume of a cube? a) V = s^3 b) V = l x w x h c) V = πr²h d) V = 4/3 x π x r³
  13. How can you measure the volume of a cylinder? a) By using unit cubes b) By using the formula V = πr²h c) By using a graduated cylinder d) By using a ruler
  14. What is the relationship between liters and cubic centimeters? a) 1 liter is equal to 1 cubic centimeter b) 1 liter is equal to 10 cubic centimeters c) 1 liter is equal to 1000 cubic centimeters d) 1 liter is equal to 100 cubic centimeters
  15. How do you convert from liters to cubic centimeters? a) Divide the number of liters by 1000 b) Multiply the number of liters by 1000 c) Add 1000 to the number of liters d) Subtract 1000 from the number of liters
  16. How do you convert from cubic centimeters to liters? a) Divide the number of cubic centimeters by 1000 b) Multiply the number of cubic centimeters by 1000 c) Add 1000 to the number of cubic centimeters d) Subtract 1000 from the number of cubic centimeters
  17. What is the volume of a cube with a side length of 6cm? a) 36 cubic cm b) 216 cubic cm c) 6 cubic cm d) 1296 cubic cm
  18. A container holds 2 liters of liquid, how many cubic centimeters is this equivalent to? a) 2000 b) 200 c) 20 d) 2
  19. A cylinder has a radius of 3cm and a height of 10cm, what is its volume in liters? a) 0.028 b) 0.283 c) 2.83 d) 28.3
  20. A container holds 0.5 liters of liquid, how many cubic centimeters is this equivalent to? a) 500 b) 50 c) 5 d) 0.5
  21. The amount of space an object takes up is called __________.
  22. The formula for finding the volume of a cube is __________.
  23. The formula for finding the volume of a cylinder is __________.
  24. The formula for finding the volume of a sphere is __________.
  25. The formula for finding the volume of a triangular prism is __________.
  26. To convert from liters to cubic centimeters, you would __________ the number of liters by 1000.
  27. To convert from cubic centimeters to liters, you would __________ the number of cubic centimeters by 1000.
  28. A container holds 2 __________ of liquid.
  29. The relationship between liters and cubic centimeters is that __________ is equal to 1000 cubic centimeters.
  30. The volume of a cube with a side length of 6cm is __________ cubic cm.

Lesson Presentation

Previous lesson

 

Introduction (10 minutes):

  • Introduce the concept of volume and capacity to the students.
  • Show examples of everyday objects and ask the students to estimate the volume/capacity of each object.
  • Discuss the difference between volume and capacity and the units used to measure them (cubic cm, cubic m, liters, etc.).

Direct Instruction (20 minutes):

  • Demonstrate the use of unit cubes to measure the volume of different shapes.
  • Provide students with the formulas for finding the volume of cubes, cuboids, cylinders, spheres and triangular prisms.
  • Provide students with examples of how to measure the volume of different shapes using unit cubes and formulas.

Guided Practice (20 minutes):

  • Provide students with various shapes (e.g. cubes, cuboids, cylinders, spheres, triangular prisms) and have them measure the volume of each shape using unit cubes and formulas.
  • Have students work in pairs or small groups to compare their results and discuss any discrepancies.
  • Have students use the graduated cylinder or pipette to measure the volume of small liquid samples.

Independent Practice (20 minutes):

  • Provide students with a set of problems to solve individually, related to measuring volume of different shapes, comparing volume of spheres and cuboids and converting between litres and cubic centimeters.

Closure (10 minutes):

  • Review the key concepts covered in the lesson.
  • Ask students to share their understanding of volume and capacity and how they can measure it.
  • Provide students with a set of problems as homework related to the topic

Assessment:

  • Observe students during the independent practice to assess their understanding of the concepts covered in the lesson.
  • Review and grade the homework problems related to the topic.
  • Have students complete a quiz or test that covers the key concepts discussed in the lesson.
  • Ask students to complete a hands-on project that involves measuring the volume of different shapes, comparing the volume of spheres and cuboids and converting between litres and cubic centimeters.
  • Assess students’ participation and engagement during class discussions and group activities.

 

Note:

  • During the lesson, make sure to provide students with plenty of opportunities to practice and apply the concepts they are learning.
  • Encourage students to ask questions and share their understanding.
  • Differentiate instruction as needed to meet the needs of all students.
  • Provide students with feedback on their progress and areas where they need improvement.
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