# Mensuration

Philosophy of Mathematics |

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### Concept Map

### Additional Resources

#### Resource title

#### OER

- Web resources:
- Primary resources : Website gives printable resources for understanding measurements.
- Campusgate: An overview on mensuration is given in this website.
- Wikipedia: For general rules while writing units in measurements is described in this web page.

- Books and journals
- Textbooks
- Syllabus documents

#### Non-OER

- Web resources:
- Encyclopedia Britannica: Gives a brief on dimensions.
- Science HQ: The website lists formulas used in mensuration.
- WizIQ : An activity on surface area of cylinder.
- teachMathematics: Lesson on understaning prisms with activity.

- Books and journals
- Textbooks
- Syllabus documents (CBSE, ICSE, IGCSE etc)

The metric system is enormously powerful as a standard measurement system. In this video, you can explore from the very small to the very large and appreciate what a degree of ten means!

Slides with problems and options for solutions.

### Learning Objectives

- Recognizing informal units of measurements
- Differentiating standard and non-standard ways of measurements.
- Understanding importance of having standardized measurements.
- Justifying accuracy of Standard units of measurements
- Relating length,breadth,perimeter ,area
- Recognizing different units
- Classifying 2D and 3D figures
- Calculating the area, perimeter, volume or side of many different figures.

### Teaching Outlines

#### Concept #1. What is Mensuration ?

Measurements are an important part of our everyday life. Think about your day; you probably made some measurements. Perhaps you checked your weight by stepping on a scale,measuring shoes to fit your feet in the shoe store, or saw measuring up houses when they are doing renovations etc. If you did not feel well, you may have taken your temperature. To make some soup, you added 2 cups of water to a package mix. If you stopped at the Petrol bunk, you watched the petrol pump measure the number of litres of petrol you put in the car.

##### Activities

###### What and how to measure

Measurement is the act of determining a target's size, length, weight, capacity, or other aspect.

###### Concept #2.Informal units of measurements

The traditional ways of measurements are a type of measure which uses non-standard units such as hand spans, arm lengths, footsteps or pattern blocks to measure length, area, etc. Non-standard measurements are not always the same but vary from person to person.

##### Activities

###### Estimating distances

The teacher asks the students to gather information regarding earlier traditional measuring ways from their elders and have an initial discussion in the classroom

#### Concept #3. Standard units of measurements

A unit of measurement is a definite magnitude of a physical quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same physical quantity. Any other value of the physical quantity can be expressed as a simple multiple of the unit of measurement.

For example, length is a physical quantity. The metre is a unit of length that represents a definite predetermined length. When we say 10 metres (or 10 m), we actually mean 10 times the definite predetermined length called "metre".<br.
The definition, agreement, and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day. Different systems of units used to be very common. Now there is a global standard, the International System of Units (SI), the modern form of the metric system.

##### Activities

###### Hunting Treasure and Measuring - Part 1

Measuring things around us by using various instruments for understanding how parameters are measured differently.

###### Hunting Treasure and Measuring - Part 2

Exploring various modes of measurements and standard units for them.

#### Concept # 4. Scale drawing

A Scale drawing is a drawing that shows a real object with accurate sizes except they have all been reduced or enlarged by a certain amount (called the scale).

Since it is not always possible to draw on paper the actual size of real-life objects such as the real size of a car, an airplane, we need scale drawings to represent the size.

Drawing to scale is a tool that Engineers use for many different tasks. One key part of every scale drawing is the scaling factor. This number represents the degree to which our scale drawing or scale model has been reduced in size when compared to the original.

##### Activities

This activity explores representation of actual distances on paper using proportional distances.

###### Scale drawing - Part 2

Activity investigates how a blue print represents actual dimensions.

#### Concept #5. Measurements in solid figures

**Concept Map**

#### Concept #1 - Introduction to solid figures

Group activity for children to explore different dimensions in solids.

Introduction to 2D and 3D shapes

Activity 1- Visualising solid shapes

Activity 2- 3D shapes model making

#### Concept #2.Cube

A cube is a 3-dimensional figure having six congruent square faces joined along their edges. The three edges join at each corner to form a vertex. The cube can also be called a regular hexahedron. It is one of the five regular polyhedrons, which are also sometimes referred to as the Platonic solids. A cube has all edges the same length. This means that each of the cube's six faces is a square. The total surface area is therefore six times the area of one face.Surface area = , Where s is the length of any edge of the cube. Volume enclosed by a cube is the number of cubic units that will exactly fill a cube.The volume of a cube is found by multiplying the length of any edge by itself thrice. So if the length of an edge is 's' cm, the volume is

##### Activities

###### Building cubes

This activity explores various methods of making cubes

###### Surface area and volume of a cube

Cube is introduced and analysed to calculate the surface area.

#### Concept #2.Cuboid

A cuboid is a 3 dimensional solid having 6 rectangular faces. Opposite faces of a rectangle are congruent.

##### Activities

###### Surface area of a cuboid

Cuboid as a shape and its properties are examined.

###### Volume of a cuboid using unit cubes

#### Concept #3.Cylinder

A cylinder is a closed solid that has two parallel (usually circular) bases connected by a curved surface. It has two ends, called bases, that are usually circular. The bases are always congruent and parallel to each other. On 'unrolling' the cylinder one would find that the side is actually a rectangle when flattened out. The height h is the perpendicular distance between the bases. The radius r of a cylinder is the radius of a base. Axis of the cylinder is a line joining the center of each base. Hollow Cylinder is the one with open top and base. Ex. cylindrical tube.

##### Formulas

CSA=

Volume of Cylinder=

TSA of Cylinder=

Volume of Cylinder When h=h/2 is

##### Activities

###### Surface area of a cylinder

The shape can be thought of as a circular prism. Cylinder could be closed or open. Surface areas for a cylinder is investigated.

###### Paint and fill the Cylinder

The shape can be thought of as a circular prism. Cylinder could be closed or open. Surface areas for a cylinder is investigated.

#### Concept #4. Cone

A cone is a three dimensional solid with a circular base that narrows toward a point, called its vertex. It has a flat circular base, has 1 side which is a curved surface. Shapes which resemble a cone are said to be conical. A cone has a lateral surface area which is the area of its curved surface. It has a total surface area which is the sum of the area of its curved surface and its circular base.

##### Activities

###### Surface area of a cone

Areas related to cone are introduced with geogebra sketch.

###### Concept #6. Sphere

Sphere is a solid figure where all points are at an equal distance from the center point.It's a ball. It is perfectly symmetrical. It has no edges or vertices (corners). It is not a polyhedron. All points on the surface are the same distance from the center which is the radius "r" of the circular shape. Of all the shapes, a sphere has the smallest surface area for a volume. i.e it can contain the greatest volume for a fixed surface area.

##### Activities

###### Baseball and string activity to find the surface area of a sphere

This activity has been taken from the website ehow.com

#### Concept #7. Pyramid

A pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base. A triangular pyramid with all equilateral triangled faces becomes the regular tetrahedron, one of the Platonic solids. The base of a regular pyramid is a regular polygon and its faces are equally sized triangles. A pyramid with an n-sided base will have n + 1 vertices, n + 1 faces, and 2n edges. A right pyramid has isosceles triangles as its faces and its apex lies directly above the midpoint of the base.

##### Activities

###### Surface area of a pyramid

Investigating areas of surfaces in a pyramid is approached with this activity.

#### Concept #8.Prism

A prism is a polyhedron for which the top and bottom faces (known as the bases) are congruent polygons, and all other faces (known as the lateral faces) are rectangles. A prism is described by the shape of its base. For instance, a rectangular prism has bases that are rectangles, and a pentagonal prism has bases that are pentagons. When the sides are rectangles, the shape is known as a right prism, indicating that the lateral faces meet the sides of the base at right angles. A rectangular prism can also be called a cuboid. Parts of the prisms - faces, edges and vertices. The characteristics of a prism. Deriving formulae for surface area and volume of a prism..

##### Activities

###### Prism people

This activity has been taken from the website teachMathematics.

Difference between Prism and Pyramid

##### Solved problems/ key questions (earlier was hints for problems).

### Projects (can include math lab/ science lab/ language lab)

**Cylindrical Elephant**: Let us make an elephant using only cylindrical objects.

**Project -01-Cylinder**- Playing with shuttle cock case**

Materials required:

- Empty cylindrical shuttle cock case
- Scissors
- Instrumental box
- Papers
- Hard board
- Pins/Nails
- Gum
- Glitter pens

- Procedure:Take an empty cylindrical shuttle cock case, measure its height and radius of its base.Calculate its CSA,TSA and volume.Record these calculations in sheets. Separate the lid and then with the help of cutter take out circular base and top.By using scissors cut the cylinder vertically(also the portion attached to lid).Now calculate areas of these four portions separately. Add all the four areas and compare it with your earlier result of TSA of cylindrical shuttle cock case.

### Assessments - question banks, formative assessment activities and summative assessment activities

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