Intersecting, Parallel, Perpendicular Lines

Background

We've talked about the drawing of a line. So what happens if we have a second line interact with that first one?

Question

Draw a line and a point not part of that line, labeled Point A. If we draw a line through Point A, in how many ways can the two lines interact? 

Answer

Two ways - they can not intersect (i.e. be parallel) or they can intersect. If they do intersect, there is one way they can form equal angles (i.e. be perpendicular).

Analysis

Let's draw the asked for elements. Point A will be in red:



If we draw lines through the red point, there are an infinite number of lines that will intersect the green line (either on the screen shot, or further off):



Of all the lines that can pass through the red point, only one will never intersect the green line:



This one red line that will never intersect the green line, we can say that the two lines are parallel.

Let's look again at a red line that intersects the green line:



Let's look at the blue arc and the black arc - they are two different lengths. There is only one intersection between the red and green lines where the two arcs are the same length:



The red and green lines are said to be perpendicular.

Vocabulary used:

For more information check out these links (comment to add your favourite link):

Where might you have come from?

Fact-orials Index

Geometry:

• Points, Line Segment, Line

Where might we go?

Graphing:

• X, Y Axes 

Geometry:

• Defining a Circle and Right Angles
• Perimeter and Area

Points, Line segment, Line

Background

We can follow paths off of The Number 1 to talk about more and more complex operations involving numbers. This path is going to follow paths involving shape and form.

Question

1. Graph a point. 
2. Graph another point. 
3. Graph a line segment between the two points. 
4. Extend the line segment in one direction. What's that called?
5. Extend the line segment in both direction. What's that called?

Answer

See below for the steps. A ray is a line segment extended in one direction. A line is a line segment extended in both directions.

Analysis

Without worrying about identifying where the point is put, let's just put a point on a page:



And now let's just graph another point:



Perfect. So now let's connect the two dots (we'll use the shortest distance possible, so no twisty lines or anything like that):



The green connector between the two dots is called a line segment.

Perfect! Now let's extend the line segment up and right so that it extends forever (so that if we had a large enough sheet of paper, the line segment would extend for as far as the paper reaches, and then continue even more - the only thing holding back the extension here is the limits of the image):



When we have a line segment that extends off into forever in one direction, this is called a ray.

And now let's extend the line segment in both directions:



A line segment extended in both directions into infinity is called a line.

At this point we might say "so what?". Why go through all this? Well, we've just worked through some of the work done by ancient peoples (the Greeks worked through this and other cultures might have as well. Euclid, in 300 BCE, as you can see in the link below, formalized what we just worked through - we just did the first two of his five geometrical postulates):

1. It is possible to draw a straight line from any point to any point, and
2. It is possible to extend a line segment continuously in a straight line

Vocabulary used:

For more information check out these links (comment to add your favourite link):

https://www.storyofmathematics.com/hellenistic_euclid.html
A pdf work (requires a download) at intellectualmathematics.com that takes Euclid's book, Elements, adds illustrations, and leads you through geometry: http://intellectualmathematics.com/geometry/

Where might you have come from?

Fact-orials Index

Numbers:

• The Number 1

Where might we go?

Geometry:

• Intersecting, Parallel, Perpendicular Lines
• Defining a Circle and Right Angles
• Perimeter and Area

Addition, Subtraction, and Whole Numbers

Background 

Now that we have learned about Whole Numbers, Addition, and Subtraction, let's work a problem (there's more problems in the Practice Problem section)

Question 

Find 5 + 2 - 3 - 1 + 6 - 0 + 0

Answer 

9

Analysis  

In prior entries, we've worked individual calculations, say like 5 + 2, both with and without the number line. We'll do the same here but we'll start on with the leftmost term and work our way through the different operations (the plusses and minuses).

We're evaluating 5 + 2 - 3 - 1 + 6 - 0 + 0

We can start with a number line and a red dot on the number 5:



So let's colour the 5 in our expression red: 5 + 2 - 3 - 1 + 6 - 0 + 0

Our first operation is plus 2 (+ 2). We move our point two numbers to the right, to 7 (I'll show that with blue):

5 + 2 - 3 - 1 + 6 - 0 + 0



Now starting at the blue dot at 7, we subtract 3 (- 3), or move 3 spots to the left. We'll land on the 4. I'll show that with green:

5 + 2 - 3 - 1 + 6 - 0 + 0



Starting at 4, we subtract 1 more to land on 3 (I'll show that in orange):

5 + 2 - 3 - 1 + 6 - 0 + 0



Now we add 6. Starting from the 3, we'll land on 9 (shown in purple):

5 + 2 - 3 - 1 + 6 - 0 + 0



And now we're asked to first subtract 0 and then add 0. Notice that when we add or subtract 0, the dot doesn't move! And so we leave our dot at 9.

This ability, to add and subtract 0 as many times as you want to any number and not change that number's value (like what we did with the 9 - we didn't change its value at all by adding or subtracting 0), gives 0 a special name - the additive identity.

Ok - so now let's work this same problem one more time, but this time without the number line. Here we go:

5 + 2 - 3 - 1 + 6 - 0 + 0

7 - 3 - 1 + 6 - 0 + 0

4 - 1 + 6 - 0 + 0

3 + 6 - 0 + 0

9 - 0 + 0

9

We start on the left and work our way to the right.

Vocabulary:

• Whole Numbers - The set of numbers that starts with 0 and increases by 1 (0, 1, 2, 3,...)

For more information check out these links (comment to add your favourite link): 
Where might you have come from? 

Fact-orials Index

Numbers:

• The Number 0
• Whole Numbers

Operations:

• Addition
• Subtraction

Graphing:

• Number Line

Where might we go?

Operations:

• Grouping Terms, Parentheses

Operations with different kinds of numbers:

• Addition, Subtraction, and Whole Numbers - Practice Problems
• Multiplication, Division, and Whole Numbers
• Addition, Subtraction, and Integers

Properties:

• Associative Property

Greater/Lesser Than

Background 

Now that we have a number line where we have numbers all in a line, we can start talking about numbers that are greater (i.e. bigger) and those that are lesser (i.e. smaller).

Question 

Is 30 bigger than 10? How do I express that using math symbols?

Answer 

Yes it is. 30 > 10

Analysis  

Let's look at a piece of a number line:



Let's just look at the numbers 10, 20, 30, and 40. Notice that as we move to the right along the number line, the numbers get bigger. This is true all along the number line, and so any number that is to the right of another number on the number line is greater. We can also say that any number that is to the left of another number on the number line is lesser.

Let's mark the 10 (in red) and the 30 (in green) on the number line:



The green 30 dot is to the right of the red 10 dot, so 30 is bigger than 10. We can also say it the other way, that 10 is less than 30.

So let's talk about how to write it using math symbols.

I'm going to write the two numbers and leave a place for a symbol for "greater than":

30 ___ 10

Teachers I've had would talk about a hungry fish that lurks between those two numbers. It can only eat one of them and it wants to eat the bigger of the two. The mouth can either be turned to the left > or to the right <. In our case, we have:

30 > 10

and we say "30 is greater than 10".

We can also write it the other way:

10 < 30

and we say "10 is less than 30".

For more information check out these links (comment to add your favourite link): 

https://www.themeasuredmom.com/less-than-greater-than-math-activity-using-toys/

Where might you have come from? 

Fact-orials Index

Graphing:

• Number Line

Where might we go?

Associated Operations:

• Absolute Value

Relations:

• Greater/Lesser Than and Rational Numbers
• Equality
• Inequality

Addition

Background 

Now that we have Counting/Natural Numbers and the Number Line, can we start to use them?

Question 

How do I add? What are some words associated with addition?

Answer 

See below for how to add and also for terminology

Analysis 

Let's start with the Number Line to see how we add:



Let's put a blue dot on the number 1:



That dot means we have 1 of something. Perhaps we have one paperclip or one book or one channel we watch on Youtube. It's one of something.

What happens when we get another of that same thing? We now have 2 of them. To express this as addition, what we do is start with 1, add 1 (and we can use the + sign to indicate that), and then say that it equals 2 (and we show equals with the = sign). Ok - let's write this out step by step:

1 plus 1 equals 2

1 + 1 = 2

On the Number Line, we can graph this. We start at the blue 1. Since we are adding 1 (+ 1) to the blue 1, we move one spot to the right of the blue 1 and end up at the orange 2. I'll show the movement from the blue 1 to the orange 2 with a line connecting the two:



See how the first number tells us where to start? And how the amount that we add by tells us the number of numbers to move to the right?

Let's try another. Let's start with 3 and add 2 more. We put a dot (it can be any colour) on the number 3 and then we move up (to the right) by 2 more spots:



So that's 3 + 2 = 5

Let's talk about addition terminology:

The numbers that we are adding together (so both the starting point and the number(s) we're adding) are called terms. (They can also be called addends). The number that we get to as a result of adding (like the 5 above) is called the total or sum).

In word problems, look for words like sum, total, add, increase, and plus to indicate that addition is needed.

For more information check out these links (comment to add a resource to the list!): 

Where might you have come from? 

Fact-orials Index

Numbers:

• The Number 1
• Counting/Natural Numbers

Where might we go?

Numbers:

• The Number 0

Operations:

• Subtraction
• Multiplication
• Grouping Terms, Parentheses
• Order of Operations

Operations with different kinds of numbers:

• Addition, Subtraction, and Whole Numbers
• Addition, Subtraction, and Variables

Properties:

• Commutative Property
• Associative Property

Relations:

• Equality
• Inverse Relations

Number Line

Background

Now we have words and symbols that refer to amounts of things, but those words and symbols, unless arranged into something useful, is just a jumble of stuff.

Question 

In what way can we arrange the Counting Numbers in a way that's useful? Something visual, perhaps.

Answer 

We certainly can - it's called the Number Line

Analysis 

Take a moment and look at the keyboard of the computer (or texting app on your phone) and notice that the letters, numbers, and symbols are all arranged in a set manner. This is what is called standardization. (It's why, no matter where in the world you go, the burger you get in a McDonald's is the same. The burger in New York City is the same as in Tokyo is the same as in Abu Dhabi - it's thanks to hamburger standardization).

For mathematicians, having a standard way of arranging the Counting Numbers is important. Ideally, it'd be great to lay them out in a row, the next number being 1 higher than the number below it.

Enter the Number Line!

How do we make it? First we draw a horizontal (which means left to right) line. Then we make marks on that line to indicate where on the line the numbers go, then we write the numbers above those little marks. Overall, it looks like this:



I know there's a whole bunch of boxes on this view but I'd like to focus on the red line going across - that's the actual number line. I've made little blue marks to help place where the Counting Numbers are along the line.

Keep in mind that the Number Line continues to the right into infinity - if we wanted to, we could find the number 1,000,000 - we'd have to scroll a long long way but it'd be there.

We're going to use the concept of the Number Line a lot as we move forward with more and more entries.

Vocabulary:

• Counting Numbers - The set of numbers that starts with 1 and increases by 1 (1, 2, 3, 4,...)

For more information check out these links (comment to add a resource to the list!): 

https://apps.mathlearningcenter.org/number-line/

Where might you have come from? 

Fact-orials Index

Numbers:

• Counting/Natural Numbers

Where might we go?

Numbers:

• Whole Numbers
• Negative Numbers
• Fractions and Decimals
• Real Numbers

Operations:

• Addition
• Subtraction

Operations with different kinds of numbers:

• Addition, Subtraction, and Whole Numbers

Related Operations:

• Absolute Value

Relations:

• Greater/Lesser Than
• Greater/Lesser Than and Rational Numbers
• Inequality

Graphing:

• X, Y Axes