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Preparation Steps
Practice 1: Maths Ideas and Thinking
Nature and Universe
Abstraction and Abstract Ideas
Paths, Length, and Distance
Dimensions of Objects
Volume of Objects
Limit of Objects (Surfaces)
Mass and Weight
Pluralities
Magnitudes and Quantities
Types of Magnitudes
Types of Quantities
Arithmetic I
Learning about Collections
Numeration
Other Numeral Systems
Preparation Steps
Roman Numerals
Equality and Inequality Relationships
Arithmetic Operations: Addition
Arithmetic Operations: Subtraction
Working with Additions and Subtractions
Complement Arithmetic
Arithmetic Operations: Multiplication
Working with Multiplications
Arithmetic Operations: Division
Working with Multiplications and Divisions
Real Numbers
Maths Ideas and Thinking
Nature and Universe
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Nature is everything that exists by itself, without being made by people.
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It includes all things that appear, grow, or happen on their own, without human help.
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Natural things follow their own rules and changes over time.
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Artificial things are made or changed by people.
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They do not appear or grow on their own.
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People use their ideas and tools to shape or build these things.
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The Universe is everything that exists — both natural and artificial.
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It includes all places and all things, near and far, whether made by nature or by people.
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Example 1: Elements of Nature
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Let’s look at some examples of things found in nature:
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Physical elements – like mountains, rivers, oceans, and landscapes.
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Living things – such as plants, animals, fungi, and tiny living things called microorganisms.
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Ecosystems – like forests, coral reefs, and deserts, where living things interact with their environment.
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Natural events – such as weather, earthquakes, volcanoes, and movements in space, like planets orbiting the Sun.
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Example 2: Artificial Things
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Let’s look at some examples of things made by humans:
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Buildings and structures: such as houses, bridges, and roads.
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Machines and technology: like computers, phones, and cars.
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Tools and equipment: like kitchen tools, medical tools, and farm tools.
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Factory-made goods: such as clothes, furniture, toys, and electronics.
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Cultural creations: like art, books, music, and other creative works.
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Exercise 1: Match the classifications in Column A with their correct elements in Column B.
Column A
Column B
Natural things -
Artificial things -
- A coconut tree
- A pencil
- A white paper pack
- The books on a shelf
- The rain
- The growth of a person
- The fall of a ball
- The construction of a building
- The process of making chocolate
- An earthquake
Abstraction and Abstract Ideas
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A physical object (or simply an object) is anything that takes up space — like a ball, a chair, or a tree.
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Simple generalisation is when you group objects that have something in common.
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Abstraction means looking at only one feature of an object, like its colour or size, and ignoring everything else.
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An abstract idea is a specific feature that allows us compare objects, like colour or shape.
Example 3: Examples of Objects
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Things on Earth – rocks, metals, water, oil, air, people, and gases like helium.
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Things in space – planets (Earth, Mars), stars (like the Sun), galaxies (like the Milky Way), and tiny particles (like atoms).
Example 4: Understanding Generalisation and Abstraction
Imagine a table with dishes, spoons, and forks in different colours.
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If you group all the red items, this is called generalisation.
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You are grouping things that have the same colour—in this case, the colour red.
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If you call all those red things simply "red objects," that’s abstraction.
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You focus only on the colour and ignore the shape or size.
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The colour becomes an abstract idea because it helps us compare the object.
Exercise 2: Link the items in Column A to their matching examples in Column B.
Column A
Column B
Simple generalisation -
Abstraction -
Abstract idea -
- Red pencils in a coloured pencil pack
- Height of an object
- Spoons on a table with several types of cutlery
- Objects inside a chest
- Black hair
- Blue objects inside a house
- Four-legged animals on a farm
- The legs that animals have
- Horses on a farm
Paths, Length, and Distance
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A path is a line that joins two or more points.
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It can be straight or curvy.
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Length is an abstract idea that helps us compare how long different paths are.
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The distance between a pair of points is the length of the shortest path that joins them.
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This path is called straight line segment.
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Note: Some segments can be curved, but here we focus on straight ones.
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Example 5: Comparing Paths by Length
You want to walk from City A to City B:
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One path goes through a jungle and takes a day and a half.
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Another follows a river and takes one day.
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The last path goes through mountains and takes half a day.
Comparing the Paths
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The shortest path is through the mountains, taking half a day.
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The path in the middle is along the river, taking a day.
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The longest path is through the jungle, taking a day and a half.
The Distance Between the Cities
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The distance between City A and City B is the length of the segment that links both cities.
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This distance can be the length of the shortest of the available paths (if this path is an straight line segment) or shorter than that.
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Exercise 3: Comparing Different Paths
Follow the instructions and answer the questions.
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Mark a Pair of Points
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Use your pen or pencil to mark a pair of points on your paper.
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These points are the start and end of the paths.
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Make Paths with Thread
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Use a thread to create a path between the points.
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Use other threads of different lengths to create more paths between the same pair of points.
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Arrange the Paths
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Compare the paths by the lengths of their threads.
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Write the paths in order from shortest to longest.
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Stretch a Thread
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Take a thread and stretch it straight and tight between the pair of points.
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Compare its length to the shortest path from earlier.
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What does the length of the stretched thread represent?
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Dimensions of Objects
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The dimensions are the abstract ideas that help us compare the size of different objects.
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The dimensions are called:
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Length
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It tells us how long something is.
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Width (or Height)
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It tells us how wide or tall something is.
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Depth
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It tells us how deep or thick something is.
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The segments that show the dimensions cross each other in a special way, called a perpendicular intersection.
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A perpendicular intersection of a pair of segments looks like a cross (𐠒), so it can be called a cross intersection.
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We group objects by how many dimensions they have:
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One-dimensional (1D) objects: Length only (like a line)
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Two-dimensional (2D) objects: Length and width (like a square)
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Three-dimensional (3D) objects: Length, width, and depth (like a box)
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Depth
Length
Width (or Height)
Example 6: Comparing the Dimensions of Objects
a) To compare the dimensions of objects, like a yellow and a red objects, follow these steps:
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Align the objects: Place the objects on top of each other, making sure their starting points match.
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Compare the dimensions: It’s easy to see that the red object is longer than the yellow one. These objects have length only, so they are 1D objects.
b) Now, let’s compare the dimensions of the same yellow object with the black object by following the same steps as before:
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Align the objects: Place the objects on top of each other, making sure their starting points of the yellow object matches the bottom-left corner of the black one.
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Comparing the dimensions: When we compare the yellow object and the black object, we see that the black object is longer and taller.
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But the yellow object has no height, so any 2D object will look taller than it.
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This means we can’t know how tall the black object is by using the yellow one.
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To find out, we need to compare the black object with another 2D object.
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<- Extra dimension (height)
<- Cross intersection
Example7: Comparing 2D Objects
Imagine you have the following objects:
To compare the dimensions of these objects follow these steps:
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Align the objects: Place the objects on top of each other, making sure their left edges line up.
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Find the dimensions of each object: Here’s how to find the length and height of the black and white objects:
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Black object:
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The bottom edge shows the length.
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There are no parts of the object at its left or at its right.
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The next edge (going up) shows the height.
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It crosses the length forming a cross intersection and no part of the object is taller than this edge.
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White object:
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The bottom edge shows the length.
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There are no parts of the object at its left or at its right.
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Draw a line from the top point straight down to the bottom.
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This line crosses the length forming a cross intersection and no part of the object is taller than this line, so this line shows the height.
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Compare the dimensions:
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When we compare the length, we see that the white object is longer than the black one.
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When we compare the height, we see that the black object is taller than the white one.
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<----- Height of the white object
Exercise 4: Compare the dimensions of the following pair of objects in their current positions:
a)
b)
c)
d)
Volume of Objects
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Volume is the space that a 3D object takes up.
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So, any 3D object has volume.
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Volume is the abstract idea that helps us compare how much space different objects take up.
Example 8: Objects with Volume (Three Dimensions)
Objects that have three dimensions have volume. Examples include:
Cube
Prism
Cylinder
Pyramid
Sphere
Exercise 5: Which of the following objects has volume? Explain why.
a)
b)
e)
f)
c)
g)
d)
h)
Limit of Objects (Surfaces)
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A surface is the space between an object and the space around it.
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It’s not part of the object, but it shows where the object ends.
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2D and 3D objects are surrounded by surfaces.
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A surface is the abstract idea that helps us compare the shape of objects.
Example 9: Examples of Surfaces
Surfaces are the boundaries or outer parts of objects. They can be flat or curved, and help us compare the shape of objects.
Examples of objects with surfaces:
Cube
Prism
Cylinder
Pyramid
Sphere
Exercise 6: Which of the following objects are surfaces? Why do you think so?
a)
b)
c)
e)
f)
g)
d)
h)
Mass and Weight
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Everything in the universe is made of matter.
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The amount of matter in something is its mass.
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All objects have mass.
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Mass is an abstract idea that helps us compare how much matter different things have.
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Weight is how strongly the Earth pulls on something.
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So, it is an abstract idea that helps us compare the pull of Earth on different objects.
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Heavier things usually have more mass.
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Example 10: Comparing the Weight of Objects
You have two prisms of the same shape:
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One is made of wood.
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The other is made of iron.
Hold one in each hand:
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The iron one feels heavier.
Why?
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Because the Earth pulls harder on the iron prism than on the wooden one.
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This means that the iron prism has more weight (or contains more mass) than the wooden one, even though they look the same.
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Exercise 7: Which of these objects has more weight? Why? What can you say about their masses?
a) A pencil or a mug of tea
b) A spoon or a pot of water
c) An empty bottle or the same bottle filled with water
d) A chair or a glass
Pluralities
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Plurality means having more than one object.
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It is an abstract idea that helps us compare groups of objects in different places.
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Example 11: Understanding Plurality
You want to know which room has more things (or more plurality) — your bedroom or your living room.
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Match the Items
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Try to match each item in your bedroom with an item in your living room. For example:
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A chair in your bedroom with a chair in the living room.
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A lamp in your bedroom with a lamp in the living room.
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Check for Extra Items
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If some items don’t have a match, that room has more things (or more plurality).
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What Did You Find?
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By matching items, you can figure out which room has more stuff (or more plurality).
Exercise 8: Which of the following collections has more objects (greater plurality)? Explain why.
A
a)
B
A
B
b)
A
B
c)
Magnitudes and Quantities
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Magnitudes are abstract ideas, like length and volume, that help us compare different objects.
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A quantity tells us how much of a magnitude there is at a certain time.
Example 12: Magnitudes and Their Quantities
Here are some magnitudes and example quantities:
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Length → the length of a street.
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Volume → the volume of a bottle.
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Surface → the surface of a ball.
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Mass or weight → the mass or weight of a person.
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Plurality → how many books there are in a library.
Exercise 9: Match the magnitudes to their correct quantities:
Magnitudes
Length -
Volume -
Surface -
Mass or weight -
Plurality -
Quantities
- The height of a building
- The space inside a boiler
- The outside of a banana
- The Earth’s attraction on a ball
- How many students are in a school
- The outer part of a bubble
- The length of a thread
- How many spoons are on a table
Types of Magnitudes
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Some magnitudes are smooth and have no clear parts.
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These are called continuous magnitudes.
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Some other magnitudes have parts you can clearly see.
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These are called discontinuous magnitudes.
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Scalars magnitudes have only size.
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Vectors magnitudes have size and direction.
Example 13: Continuous and Discontinuous Magnitudes
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Continuous magnitudes (no clear parts): length, volume, surface, mass, weight, temperature, sound, speed, force.
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Discontinuous magnitudes (with parts you can tell apart easily): like how many objects there are (plurality).
Also:
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Scalar magnitudes (only size): length, mass, temperature, sound.
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Vector magnitudes (size and direction): weight, velocity, force.
Exercise 10: Using the definitions of magnitudes, explain why the magnitudes in the previous example were classified this way.
Types of Quantities
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Continuous quantities match continuous magnitudes, and discontinuous quantities match discontinuous magnitudes.
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Scalar quantities come from scalar magnitudes, and vector quantities come from vector magnitudes.
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When we group quantities together:
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Homogeneous quantities are of the same kind.
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They measure the same type of magnitude, such as different lengths, different weights, or different volumes.
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Heterogeneous quantities are of different kinds.
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They measure different types of magnitudes, such as length and weight, or volume and temperature.
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Example 14: Examples of Quantities
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Continuous quantities (from continuous magnitudes):
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Length of a pencil
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Volume of a ball
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Surface of a spoon
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Weight of a person
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Temperature of food
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Sound of a piano
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Speed of a car
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Force from a spring
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Discontinuous quantities (from discontinuous magnitudes):
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How many students are in a classroom
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How many apples are in a pack
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How many books are in a bag
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For groups of quantities:
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Homogeneous quantities (same kind):
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The length of a pencil and the height of a building
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The weight of a person and the weight of a laptop
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Heterogeneous quantities (different kinds):
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The length of a street and the weight of a person
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