Fractals:
In 1982 Mandelbrot defined fractal as follows:"A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole."
if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer and is in general greater than its conventional dimension
Patterns and functions are used in many real-life applications, including:
About Pythagoras
Pythagoras of Samos (c. 570 – 495 BCE) was a Greek philosopher and mathematician. He is best known for proving Pythagoras’ Theorem but made many other mathematical and scientific discoveries.
Pythagoras tried to explain music in a mathematical way and discovered that two tones sound “nice” together (consonant) if the ratio of their frequencies is a simple fraction.
He also founded a school in Italy where he and his students worshipped mathematics almost like a religion, while following a number of bizarre rules – but the school was eventually burned down by their adversaries.
Pi Day
Around the world, maths fanatics celebrate Pi day every year.
Pi day falls on 14 March, because π ≈ 3,14.
Some people prefer to celebrate on 22 July, because π ≈ 1⁄2
.
When the wheel has turned halfway around, it reaches a point 3 and a bit units along the number line. When the wheel has made a whole turn, it is double this length, 2 x (3 and a bit).
So the circumference of the circle is the radius of 1 unit x 2 x (3 and a bit). The extra bit is always the same and it is approximately equal to 0,14 units. The measure of 3,14… is given the name “pi” and represented by the symbol p.
Circumference = 2 x radius x p = 2pr.
Do the same experiment with many different sizes of wheels or circles. Measure the radius and use that length as your unit measure for the circumference. We keep finding the same answer for the circumference, no matter how big or small the circle is.
Note: In our calculations, we use a good approximation of p as 
or 3,14.
Remember that while probabilities are great for estimating and forecasting, we can never tell what actually will happen. The more information we gather, the more accurate the predictions will be.
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Whole numbers:- Mental calculations |
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Exponents:- Mental calculations |
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Integers:- Counting, ordering and comparing integers |
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Common Fractions:- Ordering, comparing and simplifying fractions |
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Decimal Fractions:- Ordering and comparing decimal fractions |
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Numeric and geometric patterns:- Investigate and extend patterns |
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Functions and relationships:- Input and output values |
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Algebraic expressions:- Algebraic language |
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Algebraic equations:- Number sentences |
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Graphs:- Interpreting graphs |
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Geometry of 2D
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Geometry of 3D
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Geometry of
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Transformation
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Construction of geometric figures:- Measuring angles |
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Area and perimeter of 2D shapes:- Area and perimeter |
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Surface area and volume of 3D objects:- Surface area and volume |
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Collect, organize and summarize data:- Collect data |
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Represent data:- Represent data |
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Interpret, analyse, and report data:- Interpret data |
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Probability:- Probability |
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Whole numbers:- Mental calculations |
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Exponents:- Mental calculations |
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Integers:- Counting, ordering and comparing integers |
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Common Fractions: - Calculations with fractions |
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Decimal Fractions:- Ordering and comparing decimal fractions |
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Numeric and geometric patterns:- Investigate and extend patterns |
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Functions and relationships:- Input and output values |
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Algebraic expressions:- Algebraic language |
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Algebraic equations:- Equations |
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Graphs:- Interpreting graphs |
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Geometry of 2D
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Geometry of 3D
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Geometry of
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Transformation
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Construction of geometric figures:- Constructions |
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Area and perimeter of 2D shapes:- Area and perimeter |
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Surface area and volume of 3D objects:- Surface area and volume |
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The Theorem of Pythagoras:- Develop and use the Theorem of Pythagoras |
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Collect, organize and summarize data:- Collect data |
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Represent data:- Represent data |
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Interpret, analyse, and report data:- Interpret data |
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Probability:- Probability |
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Whole numbers: -
Properties of numbers |
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Exponents: - Comparing and representing numbers in exponential form |
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Integers: - Calculations with integers |
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Common Fractions: - Calculations with fractions |
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Decimal Fractions: - Calculations with decimal fractions |
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Numeric and geometric patterns:- Investigate and extend patterns |
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Functions and relationships:- Input and output values |
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Algebraic expressions:- Algebraic language |
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Algebraic equations:- Equations |
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Graphs:- Interpreting graphs |
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Geometry of 2D
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Geometry of 3D
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Geometry of
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Transformation
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Construction of geometric figures:- Constructions |
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Area and perimeter of 2D shapes:- Area and perimeter |
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Surface area and volume of 3D objects:- Surface area and volume |
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The Theorem of Pythagoras:- Solve problems using the Theorem of |
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Collect, organize and summarize data:- Collect data |
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Represent data:- Represent data |
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Interpret, analyse, and report data:- Interpret data |
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Probability:- Probability |
Calculations and solving problems
Circles
or 3,14 are approximate rational values of π in everyday use.The Grade 7 to 9 school classroom for Mathematics on the STEMulator gives you a window into the irrational numbers, with a focus on pi (π).
Numbers are fascinating!
Remember:
N = {1; 2; 3; …} is the set of natural numbers.
N0 = {0; 1; 2; 3; …} is the set of whole numbers. It is all the natural numbers and zero.
Z = […; -3; -2; -1; 0; 1; 2; 3; …} is the set of integers.
Q is the set of rational numbers.
All rational numbers can be written in fraction form 
, with integers in the numerator and in the denominator.
Some rational numbers don’t look like fractions, but you can change their form:
Q’ is the set of irrational numbers.
Irrational numbers cannot be written as fractions. The decimal places of an irrational number continue infinitely with no repeating patterns.
Our number system consists of all real numbers, including rational numbers and irrational numbers.
Can we show p on a number line?
All positive common proper fractions fit in the space between 0 and 1 on this number line:
This number cannot be written as a fraction. It is an irrational number, with an infinite number of decimal places. We know it is between 0,9 and 1. We even know it is between 0,94 and 0,95, but we cannot pinpoint it on the number line.
There are an infinite number of numbers between 0 and 1!
Some are rational and some are irrational!
Likewise, looking at the space between 3 and 4 on the number line, we don’t know exactly where to place the number p, but we do know that it is between 3,14 and 3,15; or even between 3,141 and 3,142.
Look for the answers to these questions on the STEMulator:
Test your knowledge: (Grade 8 and 9)
Here are two challenging questions that put your knowledge of pi, circles and their circumferences and areas to the test.
AD and BC are diameters of semicircles with centre O.
BC is half of AD. Show that the shaded shape is exactly 
of the semicircle with diameter AD.
The shape below is made up of 1 large semicircle and 4 smaller identical semi-circles. Prove that the perimeter of the shape is equal to the circumference of a circle with a diameter AB.
Let the learners explore the STEMulator page in the Grade 7 to 9 classroom by themselves, or with a partner. They can click on the book on the table or find the page through the curriculum books on the shelf.
Describe the real number system by recognising, defining and distinguishing properties of:
Learners should recognize the following distinguishing features of the number systems:
or 3,14 as rational number approximations for π in calculations.The Grade 7 to 9 school classroom for Mathematics on the STEMulator gives you a window into the irrational numbers, with a focus on pi (π).
Numbers are fascinating!
Remember:
N = {1; 2; 3; …} is the set of natural numbers.
N0 = {0; 1; 2; 3; …} is the set of whole numbers. It is all the natural numbers and zero.
Z = […; -3; -2; -1; 0; 1; 2; 3; …} is the set of integers.
Q is the set of rational numbers.
All rational numbers can be written in fraction form , with integers in the numerator and in the denominator.
Some rational numbers don’t look like fractions, but you can change their form:
Q’ is the set of irrational numbers.
Irrational numbers cannot be written as fractions. The decimal places of an irrational number continue infinitely with no repeating patterns.
Our number system consists of all real numbers, including rational numbers and irrational numbers.
Can we show p on a number line?
All positive common proper fractions fit in the space between 0 and 1 on this number line:
This number cannot be written as a fraction. It is an irrational number, with an infinite number of decimal places. We know it is between 0,9 and 1. We even know it is between 0,94 and 0,95, but we cannot pinpoint it on the number line.
There are an infinite number of numbers between 0 and 1!
Some are rational and some are irrational!
Likewise, looking at the space between 3 and 4 on the number line, we don’t know exactly where to place the number p, but we do know that it is between 3,14 and 3,15; or even between 3,141 and 3,142.
Look for the answers to these questions on the STEMulator:
Test your knowledge: (Grade 8 and 9)
Here are two challenging questions that put your knowledge of pi, circles and their circumferences and areas to the test.
AD and BC are diameters of semicircles with centre O.
BC is half of AD. Show that the shaded shape is exactly of the semicircle with diameter AD.
The shape below is made up of 1 large semicircle and 4 smaller identical semi-circles. Prove that the perimeter of the shape is equal to the circumference of a circle with a diameter AB.
Let the learners explore the STEMulator page in the Grade 7 to 9 classroom by themselves, or with a partner. They can click on the book on the table or find the page through the curriculum books on the shelf.
Investigate and extend patterns
Investigate and extend numeric and geometric patterns looking for relationships between numbers including patterns:
Describe and justify the general rules for observed relationships between numbers in own words or in algebraic language
Input and output values
Determine input values, output values or rules for patterns and relationships using:
Equivalent forms
Determine, interpret and justify equivalence of different descriptions of the same relationship or rule presented:
The journey from patterns to graphs
Look at how the same pattern can be represented with matchsticks, with numbers, in a table. in a flow diagram and even with a straight line graph.
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GEOMETRIC PATTERN
Use the pattern to find next diagram or “term” of the pattern:
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NUMBER PATTERN The number pattern of the number of sticks used in each step is 3; 5; 7. Note: |
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Table 
The pattern in words: “Start with 3 matchsticks and keeping adding 2 more” |
Rule 
The pattern that links the triangle to the number of matchsticks is |
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Flow Diagram The same pattern can be represented as a flow diagram, with input values and output values: 
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Graph 
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Go to the Nature page and click on the spiderweb to see the symmetrical pattern being spun by a spider!
https://www.stemulator.org/Nature.html
A flow diagram with input steps and output steps is used for the steps of creating electricity from coal, the sun, water, wind and even nuclear power on the energy page of the STEMulator: https://www.stemulator.org/Energy.html
Look for the many ways patterns and functions (input is worked on to get output) are used on the following STEMulator pages:
https://www.stemulator.org/Hospital.html Click on the ultra sound machine and in the laboratory.
https://www.stemulator.org/Factory.html Click on the transport system and the packaging system.
https://www.stemulator.org/Body.html
https://www.stemulator.org/Agriculture.html
https://www.stemulator.org/Car.html
https://www.stemulator.org/Plane.html
https://www.stemulator.org/Science.html
Look for the answers to these questions on the STEMulator:
Test your knowledge: (Grade 8 and 9)
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Let the learners explore the STEMulator page for Patterns and Functions in the Grade 7 to 9 classroom by themselves, or with a partner.
Investigate and extend patterns
Investigate and extend numeric and geometric patterns looking for relationships between numbers including patterns:
Describe and justify the general rules for observed relationships between numbers in own words or in algebraic language
Input and output values
Determine input values, output values or rules for patterns and relationships using:
Equivalent forms
Determine, interpret and justify equivalence of different descriptions of the same relationship or rule presented:
The journey from patterns to graphs
Look at how the same pattern can be represented with matchsticks, with numbers, in a table. in a flow diagram and even with a straight line graph.
|
GEOMETRIC PATTERN
Use the pattern to find next diagram or “term” of the pattern:
|
NUMBER PATTERN The number pattern of the number of sticks used in each step is 3; 5; 7. Note: |
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Table
The pattern in words: “Start with 3 matchsticks and keeping adding 2 more” |
Rule
The pattern that links the triangle to the number of matchsticks is |
|
Flow Diagram The same pattern can be represented as a flow diagram, with input values and output values:
|
Graph
|
Go to the Nature page and click on the spiderweb to see the symmetrical pattern being spun by a spider!
https://www.stemulator.org/Nature.html
A flow diagram with input steps and output steps is used for the steps of creating electricity from coal, the sun, water, wind and even nuclear power on the energy page of the STEMulator: https://www.stemulator.org/Energy.html
Look for the many ways patterns and functions (input is worked on to get output) are used on the following STEMulator pages:
https://www.stemulator.org/Hospital.html Click on the ultra sound machine and in the laboratory.
https://www.stemulator.org/Factory.html Click on the transport system and the packaging system.
https://www.stemulator.org/Body.html
https://www.stemulator.org/Agriculture.html
https://www.stemulator.org/Car.html
https://www.stemulator.org/Plane.html
https://www.stemulator.org/Science.html
Look for the answers to these questions on the STEMulator:
Test your knowledge: (Grade 8 and 9)
|
Let the learners explore the STEMulator page for Patterns and Functions in the Grade 7 to 9 classroom by themselves, or with a partner.
Investigate and extend patterns
Investigate and extend numeric and geometric patterns looking for relationships between numbers including patterns:
Describe and justify the general rules for observed relationships between numbers in own words or in algebraic language
Input and output values
Determine input values, output values or rules for patterns and relationships using:
Equivalent forms
Determine, interpret and justify equivalence of different descriptions of the same relationship or rule presented:
The journey from patterns to graphs
Look at how the same pattern can be represented with matchsticks, with numbers, in a table. in a flow diagram and even with a straight line graph.
|
GEOMETRIC PATTERN
Use the pattern to find next diagram or “term” of the pattern:
|
NUMBER PATTERN The number pattern of the number of sticks used in each step is 3; 5; 7. Note: |
|
Table
The pattern in words: “Start with 3 matchsticks and keeping adding 2 more” |
Rule
The pattern that links the triangle to the number of matchsticks is |
|
Flow Diagram The same pattern can be represented as a flow diagram, with input values and output values:
|
Graph
|
Go to the Nature page and click on the spiderweb to see the symmetrical pattern being spun by a spider!
https://www.stemulator.org/Nature.html
A flow diagram with input steps and output steps is used for the steps of creating electricity from coal, the sun, water, wind and even nuclear power on the energy page of the STEMulator: https://www.stemulator.org/Energy.html
Look for the many ways patterns and functions (input is worked on to get output) are used on the following STEMulator pages:
https://www.stemulator.org/Hospital.html Click on the ultra sound machine and in the laboratory.
https://www.stemulator.org/Factory.html Click on the transport system and the packaging system.
https://www.stemulator.org/Body.html
https://www.stemulator.org/Agriculture.html
https://www.stemulator.org/Car.html
https://www.stemulator.org/Plane.html
https://www.stemulator.org/Science.html
Look for the answers to these questions on the STEMulator:
Test your knowledge: (Grade 8 and 9)
|
Let the learners explore the STEMulator page for Patterns and Functions in the Grade 7 to 9 classroom by themselves, or with a partner.
Example of solving problems using the Theorem of Pythagoras (Grade 8):
Example of solving problems using the Theorem of Pythagoras (Grade 9):
Classifying 2D shapes
Geometry of 2D shapes
Solving problems
The Grade 7 to 9 school classroom for Mathematics on the STEMulator has a deeper investigation of the famous theorem and some inspiring examples of its uses.
Important points about Pythagoras’ Theorem
Theorem statement:
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In a right-angled triangle,
(hypotenuse)2 = (side 1)2 + (side 2)2
Hypotenuse: the name given to the side of a right-angled triangle that is opposite the right angle.
The square of the hypotenuse: the length of the hypotenuse times by itself.
So the area of the square on the hypotenuse is the same as the sum of the areas of the squares placed on the other two sides of the triangle.
In this diagram, the red square and the green square can be cut up and put into the yellow square - the cut pieces will fit exactly into the yellow square.
On the STEMulator, you can see how a knowledge of triangles is essential to construction at https://www.stemulator.org/Construction.html.
Go to the house STEMulator_House and click on the solar panels to see the solar energy page.
In South Africa, these solar panels have to be built at an angle of between 21 and 34 degrees.
What Is the Best Angle for Solar Panels Installation in South Africa? (ecoflow.com)
The hyperlinks below take you to different proofs of Pythagoras’ Theorem, more Pythagorean triples, interesting information and some problem solving.
https://mathigon.org/course/triangles/pythagoras takes you to an interactive site where you can test and prove Pythagoras in different ways using your own created triangles. Sign up for free!
https://learn.olico.org/course/view.php?id=237§ion=2 takes you to questions to solve, that progress from solving triangles to solving more complex problems, still using Pythagoras. Sign up for free!
Look for the answers to these questions on the STEMulator:
Test your knowledge: (Grades 8 and 9)
Look for the answers to these questions on the STEMulator page about Pythagoras.
Let the learners explore the STEMulator page in the Grade 7 to 9 classroom by themselves, or with a partner.
Example of solving problems using the Theorem of Pythagoras (Grade 8):
Example of solving problems using the Theorem of Pythagoras (Grade 9):
Classifying 2D shapes
Geometry of 2D shapes
Solving problems
The Grade 7 to 9 school classroom for Mathematics on the STEMulator has a deeper investigation of the famous theorem and some inspiring examples of its uses.
Important points about Pythagoras’ Theorem
Theorem statement:
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In a right-angled triangle,
(hypotenuse)2 = (side 1)2 + (side 2)2
Hypotenuse: the name given to the side of a right-angled triangle that is opposite the right angle.
The square of the hypotenuse: the length of the hypotenuse times by itself.
So the area of the square on the hypotenuse is the same as the sum of the areas of the squares placed on the other two sides of the triangle.
In this diagram, the red square and the green square can be cut up and put into the yellow square - the cut pieces will fit exactly into the yellow square.
On the STEMulator, you can see how a knowledge of triangles is essential to construction at https://www.stemulator.org/Construction.html.
Go to the house STEMulator_House and click on the solar panels to see the solar energy page.
In South Africa, these solar panels have to be built at an angle of between 21 and 34 degrees.
What Is the Best Angle for Solar Panels Installation in South Africa? (ecoflow.com)
The hyperlinks below take you to different proofs of Pythagoras’ Theorem, more Pythagorean triples, interesting information and some problem solving.
https://mathigon.org/course/triangles/pythagoras takes you to an interactive site where you can test and prove Pythagoras in different ways using your own created triangles. Sign up for free!
https://learn.olico.org/course/view.php?id=237§ion=2 takes you to questions to solve, that progress from solving triangles to solving more complex problems, still using Pythagoras. Sign up for free!
Look for the answers to these questions on the STEMulator:
Test your knowledge: (Grades 8 and 9)
Look for the answers to these questions on the STEMulator page about Pythagoras.
Let the learners explore the STEMulator page in the Grade 7 to 9 classroom by themselves, or with a partner.
Circles
The Grade 7 to 9 school classroom for Mathematics on the STEMulator provides you and your learners with some ways to investigate the properties of the circle and to see how valuable circles are in the real world.
Parts of a Circle
Circle: A circle is a set of all points at a fixed distance (radius) from a given point (centre of circle).
Radius: The distance from the centre of a circle to any point on the ‘perimeter’ (circumference) of the circle. The plural of radius is radii.
So any straight line from the centre to the circumference is a radius and all the radii of a circle are equal.
Diameter: A line joining any two opposite points on the circumference and going through the centre of the circle is called a diameter. A diameter is double the length of a radius of the circle.
Circumference: The circumference of a circle is the distance around the circle. In most other shapes, we call this distance the perimeter.
Look for the answers to these questions on the STEMulator:
Let the learners explore the STEMulator pages in the Grade 7 to 9 mathematics classroom by themselves, or with a partner. They can click on the yellow cylinder or find the page through the curriculum books on the shelf.
Perform simple experiments where the possible outcomes are equally likely and:
Vocabulary:
Examples:
Relative frequency
If you toss a coin (this is the event) 20 times, you can expect it to land on heads about 50% of the time, right?
Wrong! Try this experiment for yourself. The actual number of heads you throw will be different for each experiment you do. We call this the relative frequency of the outcomes.
If you toss the coin 1 000 times, the relative frequency of throwing heads will be closer to 50% than if you only toss the coin 10 times.
The hyperlink below takes you to the body tile of the STEMulator.
Look carefully at the sites on the STEMulator and you will find plenty of use of probabilities! Here are just three examples:
https://www.stemulator.org/Factory.html
Many production operations are not 100% guaranteed, but they have found the point where the most likely outcomes are successful.
https://www.stemulator.org/Science.html
Probability helps to determine the possibility of extinction of the African Penguin.
Predictions based on fossil evidence tell us about when and how dinosaurs roamed on earth.
https://www.stemulator.org/Nature.html
The behaviour of future weather can be predicted based on past and present weather patterns observed.
You want to work out what is the most likely sum of two dice thrown at the same time.
Here are all the possible outcomes in this table:
Use the table to answer the following questions:
Let the learners explore the STEMulator page for Probability in the Grade 7 to 9 by clicking on the red die in the classroom by themselves, or with a partner.
Consider a simple situation (with equally likely outcomes) that can be described using probability and:
Vocabulary:
Examples:
Relative frequency
If you toss a coin (this is the event) 20 times, you can expect it to land on heads about 50% of the time, right?
Wrong! Try this experiment for yourself. The actual number of heads you throw will be different for each experiment you do. We call this the relative frequency of the outcomes.
If you toss the coin 1 000 times, the relative frequency of throwing heads will be closer to 50% than if you only toss the coin 10 times.
The hyperlink below takes you to the body tile of the STEMulator.
Look carefully at the sites on the STEMulator and you will find plenty of use of probabilities! Here are just three examples:
https://www.stemulator.org/Factory.html
Many production operations are not 100% guaranteed, but they have found the point where the most likely outcomes are successful.
https://www.stemulator.org/Science.html
Probability helps to determine the possibility of extinction of the African Penguin.
Predictions based on fossil evidence tell us about when and how dinosaurs roamed on earth.
https://www.stemulator.org/Nature.html
The behaviour of future weather can be predicted based on past and present weather patterns observed.
You want to work out what is the most likely sum of two dice thrown at the same time.
Here are all the possible outcomes in this table:
Use the table to answer the following questions:
Let the learners explore the STEMulator page for Probability in the Grade 7 to 9 by clicking on the red die in the classroom by themselves, or with a partner.
Consider situations with equally probable outcomes, and:
Vocabulary:
Examples:
Relative frequency
If you toss a coin (this is the event) 20 times, you can expect it to land on heads about 50% of the time, right?
Wrong! Try this experiment for yourself. The actual number of heads you throw will be different for each experiment you do. We call this the relative frequency of the outcomes.
If you toss the coin 1 000 times, the relative frequency of throwing heads will be closer to 50% than if you only toss the coin 10 times.
The hyperlink below takes you to the body tile of the STEMulator.
Look carefully at the sites on the STEMulator and you will find plenty of use of probabilities! Here are just three examples:
https://www.stemulator.org/Factory.html
Many production operations are not 100% guaranteed, but they have found the point where the most likely outcomes are successful.
https://www.stemulator.org/Science.html
Probability helps to determine the possibility of extinction of the African Penguin.
Predictions based on fossil evidence tell us about when and how dinosaurs roamed on earth.
https://www.stemulator.org/Nature.html
The behaviour of future weather can be predicted based on past and present weather patterns observed.
You want to work out what is the most likely sum of two dice thrown at the same time.
Here are all the possible outcomes in this table:
Use the table to answer the following questions:
Let the learners explore the STEMulator page for Probability in the Grade 7 to 9 by clicking on the red die in the classroom by themselves, or with a partner.
