Difference between revisions of "Mensuration"

Latest revision as of 06:59, 10 August 2023

ಕನ್ನಡದಲ್ಲಿ ನೋಡಿ

The Story of Mathematics

Philosophy of Mathematics

Teaching of Mathematics

Curriculum and Syllabus

Topics in School Mathematics

Textbooks

Question Bank

While creating a resource page, please click here for a resource creation checklist.

Concept Map

[maximize][fold]

Additional Resources

Resource title

Mensuration

OER

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

Books and journals
Textbooks
1. NCERT 10 Textbook Chapter 13-Surface Areas and Volumes
Syllabus documents

Non-OER

Web resources:
1. Encyclopedia Britannica: Gives a brief on dimensions.
2. Science HQ: The website lists formulas used in mensuration.
3. WizIQ : An activity on surface area of cylinder.
4. teachMathematics: Lesson on understaning prisms with activity.
Books and journals
Textbooks
1. Karnataka text book for Class 10, Chapter 16 - Mensuration
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

Scale drawing - Part 1

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

[maximize][fold]

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 = $6s^{2}$ , 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 $s^{3}$

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= ${2}{\pi }{r^{2}}{h}$

Volume of Cylinder= ${\pi }{r^{2}}{h}$

TSA of Cylinder= ${2}{\pi }{r}{(r+h)}$

Volume of Cylinder When h=h/2 is ${\frac {{\pi }{r^{2}}{h}}{2}}$ ${\frac {1}{3}}{X}{\frac {22}{7}}{7^{2}}{14}$

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).

Frustum of Cone

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

Categories will be: (Subject), (Topic), (Class 8), (Class 9), (Class 9), (Concept Map), (Question banks), (Assessments), Formative, Summative

@@ Line 1: / Line 1: @@
+<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#ffffff; vertical-align:top; text-align:center; padding:5px;">
+''[http://karnatakaeducation.org.in/KOER/index.php/ಕ್ಷೇತ್ರಗಣಿತ ಕನ್ನಡದಲ್ಲಿ ನೋಡಿ]''</div>
 <!-- This portal was created using subst:box portal skeleton  -->
 <!--        BANNER ACROSS TOP OF PAGE        -->
 {| id="mp-topbanner" style="width:100%;font-size:100%;border-collapse:separate;border-spacing:20px;"
 |-
-|style="width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|
+| style="width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " |
-[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story of Mathematics]
+[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story of Mathematics]
-|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Philosophy Philosophy of Mathematics]
+| style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " |[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Philosophy Philosophy of Mathematics]
-|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|
+| style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " |
-[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics]
+[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics]
-|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|
+| style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " |
-[http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Curriculum_and_Syllabus Curriculum and Syllabus]
+[http://www.karnatakaeducation.org.in/KOER/en/index.php/Maths:_Curriculum_and_Syllabus Curriculum and Syllabus]
-|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|
+| style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " |
-[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Topics Topics in School Mathematics]
+[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Topics Topics in School Mathematics]
-|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|
+| style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " |
-[http://karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks]
+[http://www.karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks]
-|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|
+| style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " |
-[http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Question_Papers Question Bank]
+[http://www.karnatakaeducation.org.in/KOER/en/index.php/Maths:_Question_Papers Question Bank]
 |}
 While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''].
-= Concept Map =
+=== Concept Map ===
-<mm>[[Mensuration.mm|flash]]</mm>
+__FORCETOC__
-= Textbook =
+[[File:Mensuration.mm|flash]]
+===Additional Resources===
-[http://www.ncert.nic.in/ncerts/textbook/textbook.htm?jemh1=13-14 NCERT 10 Textbook Chapter 13-Surface Areas and Volumes]
+==== Resource title ====
+[http://www.mathopenref.com/tocs/solidtoc.html Mensuration]
-=Additional Information=
+====OER====
-==Useful websites==
+#Web resources:
-#[http://www.cimt.plymouth.ac.uk/projects/mepres/allgcse/bs7act1.pdf Download PDF].  This is a good website for interesting activities on mensuration.<br>
+##[http://www.primaryresources.co.uk/maths/mathsE1.htm Primary resources] : Website gives printable resources for understanding measurements.
-#For standard measurements :       http://www.primaryresources.co.uk/maths/mathsE1.htm<br>
+##[http://www.campusgate.co.in/2011/11/areas-and-mensuration.html Campusgate]:  An overview on mensuration is given in this website.
-#http://www.campusgate.co.in/2011/11/areas-and-mensuration.html
+##[[wikipedia:International_System_of_Units#General_rules|Wikipedia]]: For general rules while writing units in measurements is described in this web page.
-#For general rules while writing units ://en.wikipedia.org/wiki/International_System_of_Units#General_rules<br>
-#For teacher reference on dimension. http://www.britannica.com/EBchecked/topic/163641/dimension<br>
-#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!
-{{#widget:YouTube|id=0fKBhvDjuy0}}
-==Reference Books==
+#Books and journals
+#Textbooks
+##[http://www.ncert.nic.in/ncerts/textbook/textbook.htm?jemh1=13-14 NCERT 10 Textbook Chapter 13-Surface Areas and Volumes]
+#Syllabus documents
+====Non-OER====
+#Web resources:
+##[https://www.britannica.com/science/dimension-geometry Encyclopedia Britannica]: Gives a brief on dimensions.
+##[https://www.sciencehq.com/math-formulas/mensuration-formulas.html Science HQ]: The website lists formulas used in mensuration.
+##[https://www.wiziq.com/tutorial/197452-surface-area-of-cylinder-an-activity WizIQ] : An activity on surface area of cylinder.
+##[http://www.teachmathematics.net/page/10938/prism-people teachMathematics]: Lesson on understaning prisms with activity.
+#Books and journals
+#Textbooks
+##[http://ktbs.kar.nic.in/New/Textbooks/class-x/english/maths/class-x-english-maths-chapter16.pdf Karnataka text book for Class 10, Chapter 16 - Mensuration]
+#Syllabus documents (CBSE, ICSE, IGCSE etc)
+{{#widget:YouTube|id=0fKBhvDjuy0}}
-= Teaching Outlines =
+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!
-==Concept #1. What is Mensuration ?==
+{{#widget:Iframe |url=http://www.slideshare.net/slideshow/embed_code/43064607" |width=450 |height=360 |border=1 }}
-===Learning objectives===
-#units and measurement
-# Mensuration is the branch of Mathematics dealing with measurement of angles, length, area, and volume.
-# There are standard and non-standard ways of measurements.
-# Importance of having standardised measurements.
-# The length of the total boundary of a figure is called its perimeter. The Metric unit of perimeter is same as the unit of length - Metre.
-# The amount of surface covered by an object is called it area. The Metric unit of area is square metre.
-# The capacity of an object to hold is called its volume.
-# They should develop the ability to calculate  the area, perimeter, volume or side  of many different figures.
-===Notes for teachers===
+Slides with problems and options for solutions.
-Source: http://depts.washington.edu/chemcrs/bulkdisk/chem120A_sum06/handout_CH%201-4%20TIMBER.pdf<br>
-Summary : Measurement 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.
-===Activity No #1===
-What to measure?<br>
-*Estimated time: 30 minutes<br>
-*Materials required: Old news papers, scissors,measureing tape, 5 different size dolls,paper and pen.<br>
-*Prerequisites/Instructions:<br>
-)Measure the doll<br>
-)Before cutting the news paper decide the shape of shirt.<br>
-)Neither waste paper nor throw it.<br>
-*Multimedia resources :<br>
-*Website interactives/ links/ / Geogebra Applets<br>
-*Process:<br> Students are divided into 5 groups. Each group is given old news papers, scissors,measureing tape, 1 different size doll, paper and pen. They are asked to cut news paper in shirt shape for their doll size. At the end of the activity each group leader should have to present their doll's shirt and its measurements.<br>
-*Developmental Questions:<br>
-. What happens if shape is not decided?<br>
-. What is the importance of measuring? <br>
-*Evaluation:<br>
-How size is related to shape?<br>
-*Question Corner:<br>
-What were the early traditional measuring modes used.
-===Activity No #2===
+===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.
-==Concept #2.Informal units of measurements ==
+=== Teaching Outlines ===
-===Learning objectives===
+====Concept #1. What is Mensuration ?====
-#  The traditional ways of measurements are a type of measure which uses non-standard units such as hand spans, armlengths, footsteps or pattern blocks to measure length, area, etc.
+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.
-# Non-standard measurements are not always the same but vary from person to person.
-===Notes for teachers===
+=====Activities=====
-# The teacher can ask the students to gather information regarding earlier traditional measuring ways from their elders and have an initial discussion in the classroom.
-===Activity No # 1. Estimating distances===
+====== [[What and how to measure]] ======
-{| style="height:10px; float:right; align:center;"
+Measurement is the act of determining a target's size, length, weight, capacity, or other aspect.
-|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
-''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
+====== 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.
-*Estimated Time :1 hour
-*Materials/ Resources needed : Sticks, ropes, writing pad, pencil.
-*Prerequisites/Instructions, if any
-# They should have the ability to measure and document their findings accurately.
-*Multimedia resources
-*Website interactives/ links/ / Geogebra Applets
-*Process:
-# The teacher can ask the students to determine the distances of library, principal's room, playground, dining hall, entrance gate from their classroom using various non-standard measuring methods.
-# The children can decide which method to use - whether foot, sticks or ropes.
-# The task can be done in groups of 3 children.
-# Document and compare the results.
-# Discuss regarding the length of distance.
-# Reiterate that it is very important to use same measuring modes to facilitate comparisons.
-*Developmental Questions:
-# Which point would we mark as the point of reference for measuring our classroom.
-# Similarly what are the points of reference for other places.
-# How do we mark them.
-# Which measuring unit have you chosen ?
-# How will you document the findings ?
-# How can we tabulate our findings on board for comparisons ?
-# What would be our report back time ?
-# What are our findings ?
-# What conclusions can we draw ?
-*Evaluation:
-# What have we learnt so far about measuring object?
-*Question Corner:
-# How can we measure curves?
-# What are the problems with non-standard measuring units?
-# Can we think of methods to measure so that measures taken by anyone would always be same for a given object or distance.
-===Activity No # ===
+===== Activities =====
-{| style="height:10px; float:right; align:center;"
-|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
-''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
-|}
-*Estimated Time
-*Materials/ Resources needed
-*Prerequisites/Instructions, if any
-*Multimedia resources
-*Website interactives/ links/ / Geogebra Applets
-*Process/ Developmental Questions
-*Evaluation
-*Question Corner
-==Concept #3. Standard units of measurements==
+======  [[Estimating distances]] ======
-===Learning objectives===
+The teacher  asks the students to gather information regarding earlier traditional measuring ways from their elders and have an initial discussion in the classroom
-# The ability to obtain accurate measurements and communicate those measurements is a key requirement for progress.
-# Standardised measuring units ensure uniformity in meaurements.
-# The standard unit for length is metre, for weight id kilogram, for time is second, for temperature is Kelvin and for amount of substance it is mole.
-===Notes for teachers===
+====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.<br>
 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.
-===Activity No # 1.Hands-on activity in using standard measuring instruments ===
-{| style="height:10px; float:right; align:center;"
-|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
-''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
-|}
-*Estimated Time : 1 hour
-*Materials/ Resources needed : Weighing balance, measuring tape, spring balance, digital scale, postal scale, thermometer, litre and millilitre cans.
-*Prerequisites/Instructions, if any
-*Multimedia resources
-*Website interactives/ links/ / Geogebra Applets
-*Process:
-# Let the students measure different ingredients using suitable measuring instruments and record their observations in a tabular form.
-*Developmental Questions
-# What did you measure ?
-# What measuring instrument did you select ? Why ?
-# What was your reading ?
-# which unit do you use ?
-*Evaluation
-# Were the students able to comprehend which measuring instrument should be used for particular items ?
-# Were they able to document their observations appropriately in a tabular form.
-*Question Corner
-#  Write a note on why standard measuring units are very important.
-===Activity No # 2. Hunting treasure and measuring===
-{| style="height:10px; float:right; align:center;"
-|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
-''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
-|}
-*Estimated Time : 1 hour
-*Materials/ Resources needed:
-# Weighing pan with measures of 1 kg, 500gm, 250gm and 100gm and 50gm.
-# Digital scale
-# Postal scale.
-# Measuring cans of 1 litre, 500ml, 250 ml, 100ml,
-# Measuring tape, Ruler.
-# Watch or stop clock.
-# Paper and pen.
-*Prerequisites/Instructions, if any
-# The students should have been introduced to the concept of formal measurements by conducting an activity on using standard measuring instruments.
-# They should understand the importance of standard measurements and distinction between standard and non-standard units of measurements.
-# They should be well versed with standard units and subunits.
-# They should know unit conversions.
-# They should have the skill of measuring and recording accurately.
-# They should be aware that comparisons can be made only between similar measuring units.
-*Multimedia resources
-*Website interactives/ links/ / Geogebra Applets
-*Process:
-# The idea here is to inculde several measuring units for practise.
-# Initially instruct the students regarding the activity and its purpose.
-# Group students into 3 in each.
-# The teacher shall make packets of things as treasure and hide them in the school premises.
-# The students shall pick chits containing clues to hunt and find the packets.
-# 30 minutes is given to search.
-# The students search packets and come back to classroom.
-# The packets may contain cloth, ribbons, ropes, water, fruits, ingredients , vegetable , books , juice, stamps,envelops or any such measurable items.
-# The students have to quickly decide which measuring mode they are going to use, measure and record their findings.
-# They also have to measure the distance at which they found each and record the time taken to search.
-*Developmental Questions:
-Ask the students to write on
-# What treasures did they find ?
-# At what distance did they find ?
-# How much time did they take to find ?
-# What measuring instruments did they use ?, why ?
-# How will they tabulate their results ?
-# Can you list the purpose of each ?
-*Evaluation
-# The teacher should analyse the students skill of documentation.
-*Question Corner:
-# What are standard the units of length ?
-# What are the standard units of weight ?
-# How are liquids measured ?
-===Activity :Paint and fill the Cube   ===
-{| style="height:10px; float:right; align:center;"
-|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
-''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
-|}
-*Estimated Time: 20 minutes
-*Materials/ Resources needed
-#instrument box
-#Hard paper
-#gum tape
-#paints-green and red
-# brush
-*Prerequisites/Instructions:
-#Students are asked to cut paper into rectangular shape of any length (l) and breadth(b).
-#Students are asked to cut paper into two congruent circles of circumference=length of rectangle(l)
-#Students are asked to make a cylinder using rectangle and two congruent circles.
-#Students are asked to use Red paint for painting single side of rectangle and circles and green for filling cylinder
-*prerequisite knowledge:
-#What is cylinder?
-#Area of cylinder
-#volume of cylinder
-*Multimedia resources
-*Website interactives/ links/ / Geogebra Applets
-*Process:
-#Students are grouped (each group has 4 students). They are asked to calculate the area of rectangle  painted and area of circles painted.
-#Students are asked to find out the quantity of green paint used to fill the cylinder.
-* Developmental Questions:
-#which shape the rectangular paper takes when rolled?
-#What is the circumference of the circle?
-#What is the relation between length of rectangle and circumference of the  circle?
-#what is the area of rectangle?
-#what is the area of circle?
-*Evaluation
-*Question Corner
-==Concept # 4. Scale drawing==
-===Learning objectives===
-# The process of representing actual distances on paper using proportional distances is called a scale.
-# 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.
-===Notes for teachers===
-===Activity No # 1.(Part A): Representing distances on paper-Introduction to scale drawing ===
-{| style="height:10px; float:right; align:center;"
-|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
-''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
-|}
-*Estimated Time : 1 hour
-*Materials/ Resources needed :
-# Scale, pencil, graph paper.
-*Prerequisites/Instructions, if any
-# Skill of measuring and tabulating accurately.
-# Knowledge of ratio and proportions.
-*Multimedia resources
-*Website interactives/ links/ / Geogebra Applets
-*Process:
-# This activity can be done in groups by grouping 3 students in each.
-# The students should measure distances of library, dinning hall, principal's room, entrance gate, playground etc from their class room using a measuring tape in metres.
-# The teacher can mark reference points for each and explain to the students.
-# The students should accurately measure and note down.
-# After coming to the classroom let the teacher tabulate all distances on the blackboard.
-# Next ask the students to represent their observation of distances by drawing their school model on paper.
-# Ask them how will they go about it so that the distances are represented proportionately.
-# The teacher can then elicit from the students that a proper equivalent unit has to be used for the purpose.
-# After this activity she can spell out the term scale and discuss about the blue print of a house, mapping distances, atlas maps, model constructions and so on.
-# This activity can be done to introduce the students to the concept of scale drawing.
-*Developmental Questions
-# What did you measure ?
-# What measuring instrument did you select ? Why ?
-# What was your reading ?
-# which unit do you use ?
-*Evaluation
-# Were the students able to comprehend which measuring instrument should be used for particular items ?
-# Were they taking measurements accurately ?
-# Were they able to document their observations appropriately in a tabular form ?
-*Question Corner:
-# Discuss the applications of scale drawing ?
-# What is a scale factor ?
-===Activity No # (PART B) : Estimating Tile costs===
-{| style="height:10px; float:right; align:center;"
-|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
-''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
-|}
-[[File:Scale drawing activity.jpeg|300px|left]]
-*Estimated Time :45 minutes
-*Materials/ Resources needed:
-Graph sheets, scale, pencil,
-*Prerequisites/Instructions, if any
-# The students should have knowledge about scale drawing.
-# They should know about scale factor.
-# They should understand that a scaling factor should be maintained as constant throughout the sketches.
-# Students should be able to read and understand a scaling factor.
-# They should be able to find a scaling factor and create a scale drawing.
-# The activity on using standard measuring instruments should be done prior to this activity.
-# The students should have been introduced to area, calculations and unit conversions.
-*Multimedia resources
-*Website interactives/ links/ / Geogebra Applets
-*Process:
-# In the above blue print each individual unit is 4 feet.
-# Explain the meaning of 1 unit.
-# Each ceramic tile costs Rs 80 per square foot.
-# How much would it cost to tile the bathroom ?
-# How much would it cost to tile Kitchen, Living room and bedroom?
-*Developmental Questions
-# Along with knowing the length and width of the scale model, what additional information do you need to know ?
-# What is the scale factor here.
-*Evaluation:
-# What is 1 unit in the given blue print.
-# What does '1 unit = 4 feet mean' ?
-# What are the actual dimensions of each room ?
-*Question Corner:
-# What all factors do you need to know to estimate the painting costs for each of the rooms.
-# Write down the sequential steps for the calculations.
-= Hints for difficult problems =
-==Mensuration==
-Exercise 16.1<br>Problem 7.
-)Craft teacher of a school taught the students to prepare cylindrical pen holders out of card board. In a class of strength 42, if each child prepared a pen holder of radius 5 cm and height 14 cm, how much cardboard was consumed?
-Statement of the problem: How much cardboard was consumed to  prepare 42 cylindrical pen holders each of radius 5 cm and height 14 cm.
-*Interpretation;
-Case 1)Pen stand which is open on top and  closed base.
-Then we have to calculate Curved surface area of cylinder and area of one circle.
-*skills
-visualising the cylinder into rectangle and circle
-*Assumptions:
-Concepts used
-#basics of circles -radius
-# area of circles
-# addition and multiplication of fractions
-#unit of area
-knowledge to be used
-#knowledge of Polygons -Rectangles
-# knowledge of measurements
-#Formula of CSA of cylinder
-#substitution
-#computing
-#Value of Л
-Concepts to be taught
-#basics of circles -radius
-# area of circles
-#relation between circumference of circle and length of rectangle in a cylinder
-# addition and multiplication of fractions
-#unit of area
-solution: r=5cm, h=14cm, Л=<math>{\frac{22}{7}}</math><br>Curved surface area of cylinder and area of one circle= CSA of cylinder+area of circle<br>
-=<math>2Лrh+Лr^2</math><br>
-=<math>2X{\frac{22}{7}}X5X14+{\frac{22}{7}}X{5^2}</math><br>
-=440+78.5<br>
-=518.5cm
-Cardboard required to prepare 42 penstands= 42X518.5<br>
-=<math>21777{cm^2}</math><br>
-=<math>2.1777{m^2}</math><br>
-Case 2:-  Hollow pen stand: Pen stand which is open both the sides i.e., top and base.
-Then we have to calculate Curved surface area of cylinder.<br>
-Case 3:-Pen stand which is closed both the sides i.e., top and base.
-Then we have to calculate total surface area of cylinder.
-Activity
-Making use of 2200 cmcardboard sheet how many hollow cylinders of radius 7 cm and height 5 cm can be prepared.
-===solved problems on cone Example 10===
-#Find the weight of a solid cone whose base is of diameter 14 cm and vertical height 51,
-cm, if the material of which it is made weighs 10gm/
-*Statement of the problem:What is the weight of a solid cone whose base is of diameter 14 cm and vertical height 51cm, if the material of which it is made weighs 10gm/
-*Interpretation;
-Calculating the weight of solid cone by calculating its volume.
-density of a material is given. it is related to volume and weight.
-* Assumptions:
-Concepts used
-#basics of circles -radius
-# area of circles
-#  multiplication of fractions
-#unit of volume,density and weight
-#density
-knowledge to be used
-#radius=diameter/2
-#density=mass/volume
-# knowledge of measurements
-#Formula of volume of cones
-#substitution
-#computing
-#Value of Л
-Concepts to be taught
-#density
-#units conversion
-solution:<br>
-d=14cm    r=7cm, h=51cm<br>
-Volume of cone=
-=<math>\frac{1}{3}{Л}{r^2}h</math><br>
-=<math>\frac{1}{3}{X}{\frac{22}{7}}{X}{7^2}{X}51</math><br>
-=<math>2618{cm^3}</math><br>
-To calculate weight of the solid cone we have to use density of the given material.<br>
-density=<math>10gm/{cm^3}</math><br>
-=<math>{\frac{10Kg}{1000cm^3}}</math><br>
-Weight of solid cone=volumeXdensity
-=<math>2618{cm^3}{X}{\frac{10Kg}{1000cm^3}}</math><br>
-=26.18Kg<br>
-====problem on combination of solids example 01====
-Problem-03<br>
-#A Cylindrical container of radius 6cm and height 15cm is filled with ice cream. The whole ice cream has to be distributed to 10 children in equal cones with hemispherical tops. If the height
-of the conical portion is 4 times the radius of its base , find the radius of the cream cones.<br>
-[[Image:combinations of solids.png]]
-Solution<br>
-# Statement of the problem<br>
-#A Cylindrical container of radius 6cm and height 15cm is filled with ice cream. The whole ice cream has to be filled in 10  equal cones with hemispherical tops. and the height
-of the conical portion is 4 times the radius of its base.<br>.
-Assumptions
-#Student should know volume of cone,and volume of Hemisphere
-# Student should know the value of л=22/7
-# Student should know the difference between radius and height
-# Student should know the proper substitution simplification
-Concepts to be taught
+=====Activities=====
-#Let us consider the volume of a cone having hemispherical top =2л
-#Volume of Cylindrical container is equated to volume of cone having hemispherical top
+====== [[Hunting Treasure and Measuring - Part 1]] ======
+Measuring things around us by using various instruments for understanding how parameters are measured differently.
-Gaps
+====== [[Hunting Treasure and Measuring - Part 2]] ======
-# Comparision between the volumes of cylinder and (cone+Hemisphere)
+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).
-Skills
+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.
-# To imagine a cone , Hemisphere,and cylinder
-# To imagine a cone having Hemispherical top
+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=====
+'''[[Scale drawing - Part 1]]'''
+This activity explores representation of  actual distances on paper using proportional distances.
-Algorthem<br>
+====== [[Scale drawing - Part 2]] ======
-Part 1:<br> To derive the volume of a cone with hemispherical top<br>
+Activity investigates how a blue print represents actual dimensions.
-#volume of a cone with hemispherical top<br>=<math>{\frac{1}{3}}л{r^2}h+{\frac{2}{3}}л{r^3}</math><br>
-						=<math>{\frac{1}{3}}л4r +{\frac{2}{3}}л{r^3}</math>(on simplification)<br>
-						=<math>2л{r^3}</math><br>
+==== Concept #5. Measurements in solid figures ====
+'''Concept Map'''
-Part 2 :<br>To calculate the volume of 10 cone with hemispherical top<br>
+[[File:measurements_in_solids.mm|Flash|link=http://karnatakaeducation.org.in/KOER/en/index.php/File:Measurements_in_solids.mm]]
-										=<math>10X2л{r^3}</math><br>
-										=<math>20л{r^3}</math><br>
-	To  calculate the volume of ice-cream in cylindrical container<br>
-								=<math>л{r^2}h</math><br>
-								= л X6X6X15<br>
-								=540лc<br>
-Part 3 :<br>To apply condition given in the problem<br>
+====Concept #1 - [[Introduction to solid figures]]====
-	volume of 10 cone with hemispherical top= volume of cylindrical container<br>
+Group activity for children to explore different dimensions in solids.
-			<math>20л{r^3}</math> =540л<br>
-		<math>{r^3}</math>=<math>{\frac{540л}{20л}}</math><br>
-				=27<br>
-				r=3cm<br>
-Hence the radius of cream cones=3cm
+[https://karnatakaeducation.org.in/KOER/en/index.php/Introduction_to_2D_and_3D_shapes?venotify=created 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 = <math>6s^2</math>, 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 <math>s^3</math>
+=====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=<math>{2}{\pi}{r^2}{h}</math>
+Volume of Cylinder=<math>{\pi}{r^2}{h}</math>
+TSA of Cylinder=<math>{2}{\pi}{r}{(r+h)}</math>
+Volume of Cylinder When h=h/2 is<math>{\frac{{\pi}{r^2}{h}}{2}}</math> <math>{\frac{{1}}{3}}{X}{\frac{{22}}{7}}{7^2}{14}</math>
+=====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  [http://www.ehow.com/info_7838171_classroom-activities-surface-area-sphere.html 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 [http://www.teachmathematics.net/page/10938/prism-people teachMathematics].
-=
+[[Difference between Prism and Pyramid]]
-= Project Ideas =
+=====Solved problems/ key questions (earlier was hints for problems).=====
-Projects Given for the Different Groups<br>
+[[Frustum of Cone]]
-Students Will be divided into different groups and each group is named as cylinder,cone,pyramid etc<br>
-'''Group-01(cylinder)<br>'''
-Students of this group will be asked to make different cylinders say Hallow cylinder,Solid cylinder,cylinder with a base without top etc using cardboard.<br>
-Now Students should have to write the following<br>
-# Different properties of Cylinder<br>
-# Different formulae (LSA,TSA,VOLUME of Cylinder)
-# Procedure:<br>
-*Materials used in making cylinder<br>
-*Different cuttings that they made <br>
-*Measurements of those cuttings<br>
-NOTE:<br>
-The Students of other groups will be asked to follow the same method mentioned above<br>
-= Math Fun =
+===Projects (can include math lab/ science lab/ language lab)===
+*'''Cylindrical Elephant''': Let us make an elephant using only cylindrical objects.
-'''Usage'''
+*'''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===
+Categories will be:  (Subject), (Topic), (Class 8), (Class 9), (Class 9), (Concept Map), (Question banks), (Assessments), Formative, Summative
-Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template
+[[Category:Class 8]]
+[[Category:Mensuration]]

Difference between revisions of "Mensuration"

Latest revision as of 06:59, 10 August 2023

Concept Map

Additional Resources

Resource title

OER

Non-OER

Learning Objectives

Teaching Outlines

Concept #1. What is Mensuration ?

Activities

Concept #2.Informal units of measurements

Activities

Concept #3. Standard units of measurements

Activities

Concept # 4. Scale drawing

Activities

Concept #5. Measurements in solid figures

Concept #1 - Introduction to solid figures

Concept #2.Cube

Activities

Concept #2.Cuboid

Activities

Concept #3.Cylinder

Formulas

Activities

Concept #4. Cone

Activities

Concept #6. Sphere

Activities

Concept #7. Pyramid

Activities

Concept #8.Prism

Activities

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

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

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

Navigation menu

Search