Arranging Integers in Order

Girl with Rocket Diagram
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An important skill is to be able to place positive and negative “integer” whole numbers into ascending and descending order.


Comparing Integers

A Number Line is very useful for working with directed number Integers.

Pic of a nbr line from the web
Image Source: http://www.green-planet-solar-energy.com

On a number line, NUMBERS TO THE RIGHT ARE BIGGER.

NUMBERS TO THE LEFT ARE LESS THAN ones to the right.

For example: 6 is bigger than 3, but -6 is smaller than 3.

We say that 6 is greater than 3, and we write this as 6 > 3

However, -6 is less than 3, and we write this as -6 < 3

Web pic of nbr line with explanantion
Image Source: http://elgee18.files.wordpress.com

Using Inequality Signs

> means greater than, or bigger than

< means less than, or smaller than. ( less than points to the left )

For the “inequality signs” < and >

THE POINTY ARROW ALWAYS POINTS TO THE SMALLER NUMBER.

Eg. 2 < 4, 8 > 5, -4 < -1, -2 > -6

The Fish Rule

Some people also like to think of the arrow sign as looking like the mouth of a fish.

GT and LT Fish Mouths
Image Source: http://illuminations.nctm.org

They then use the following rule:

“The hungry Fish always wants to eat the bigger amount”.

So the open “<” mouth of the fish always goes towards the bigger item.

Fish Mouths Example
Image Source: http://illuminations.nctm.org

Placing Inequality Signs Rules

When putting a < or > sign in between two integers:

If the first number is more to the right, put a > sign.

If the first number is more to the left, put a < sign.

This will ensure that the arrow always points to the smaller number.

Remember to use a number line, ( especially for negative numbers ), and that :

NUMBERS TO THE RIGHT ARE BIGGER.

NUMBERS TO THE LEFT ARE SMALLER.

THE POINTY ARROW ALWAYS POINTS TO THE SMALLER NUMBER.

Represent Order and Compare Integers

The following is a video about comparing Integers, eg. Greater Than and Less Than, and placing Integers into ascending order.

It has no soundtrack, but has good slides.

The one limitation is that we cannot go full screen.

Click on the image below to go to the web page containing this video.

Image Link to Qicktime Video

Using a Number Line to Order Integers

The following video from “School Tube” gives two examples of plotting Integers onto a Number Line.

We can then write the Integers out from left to right, which gives us their order from lowest to highest.

Click anywhere on the video below to launch the video full screen in a new window.

Schooltube fake embed one

Here is another video about ordering and comparing integers using signs.
[youtube http://www.youtube.com/watch?v=m3oOpSBAeGI&w=540&h=304]

Ordering Integers Worksheet

The clickable link below is to a PDF document which gives you practice questions to do, as well as an answer sheet.

Click here for Ordering Integers Worksheet

Ordering Integers Game

In the following game, use the mouse to place the three red integer numbers in order, onto the carriages and the train engine.

The biggest number always goes onto the steam engine.

It is very important to place the numbers exactly in the middle of the carriages and engine like this:

Pic of placing nbrs on train

If your answer is correct, the train should move to the right and off the screen.

Click anywhere on the image below to launch the game in a new window.

Note that you will probably have to page down the screen a bit to get to the actual game.

Integers Order Train Game

Ordering Integers Online Test

The clickable link below will take you to a web page from “Maths Goodies” that has a lesson you can read, followed by a short online test.

Click here to do Integers Online Quiz

Related Items

Introduction to Integers
Adding Integers Using Number Lines
Adding Integers Using Zero Pairs
Subtracting Integers
Multiplying Integers
Dividing Integers
Integers Order of Operations
Directed Number Integers Games
Integers in the Real World
Integers in Drag Racing

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Introduction to Integers

Drink bottles ice and thermometer
Image Source: http://www.odec.ca

Integers are positive and negative WHOLE numbers.

Decimals and Fractions are not “Integers”.

Zero is an Integer, but it is neither positive or negative.

Typical integers are:

-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5

Often the positive sign is left off positive integers.

Eg. 4 is the same as +4

Zero never has a positive or a negative sign on it.

Examples of Integers in the real world are:

-5 degrees below zero is a negative integer.

+4000 meters above sea level on a mountain is a positive integer.

The Inca ruins of Machu Pichu in Peru are located on a mountain that is 3800 meters above sea level.

Passy at Machu Pichu

Positive and Negative Situations

Integers are often called “Directed Numbers” because they indicate how big, as well as in what direction.

Direction is often down or up, or left and right.

The following table will help you determine whether the situation in a word problem should be represented as a “Negative” or a “Positive” number.

Positive and Negative Situations Table

Integers Slide Show

The following slide show will take you through the basics of Integers.

Click anywhere on the image below to open this presentation in a new window.

Intro Slideshow about Integers

Here is a YouTube about Integers that covers some of the basics.

[youtube http://www.youtube.com/watch?v=w_FQfJTPyrA&w=540&h=304]

Integers Music Videos

This music video gives a brief look at Integers in the Real World.

[youtube http://www.youtube.com/watch?v=ihSG9mbF3js&w=540&h=304]

Let’s finish up with an Integers Rap song from YouTube.

[youtube http://www.youtube.com/watch?v=BswlX1dXL_s&w=540&h=396]

Hope you’re feeling positive about the work we will be doing with Integers!

Related Items

Arranging Integers in Order
Adding Integers Using Number Lines
Adding Integers Using Zero Pairs
Subtracting Integers
Multiplying Integers
Dividing Integers
Integers Order of Operations
Directed Number Integers Games
Integers in the Real World
Integers in Drag Racing

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Plotting a Line Using a Rule Table

Vals Intro

In this lesson we look at plotting linear graphs using a values table.

We have to make our own values table from an Algebra Rule that is given to us.

The rule always tells us how to calculate any y value from any x value that we have chosen.

To keep the mathematics simple, it is best to choose very small “x” values, such as: -2, -1, 0, 1, and 2.

Algebra Substitution Method

As a first example, let’s take a look at the line for the rule: y = x + 2.

Vals 1

We could also work out other y values using Algebra substitution.

We would use the x values -2, -1, 1, and 2 like this:

y = -2 + 2 = 0

y = -1 + 2 = -1

y = 1 + 2 = 3

y = 2 + 2 = 4

We could then make these into a set of (x,y) pairs and graph the line:

For x = -2, we obtained y = 0, and so our first point is (-2,0)

For x = -1, we obtained y = -1, and so our second point is (-1,-1)

and so on.

However, there is an alternative way of doing this process, where we make a Values Table, and do all our working out in the table.

We are still doing the same thing, but using a Table to do the Algebra Substitution.

Values Table Method for y = x + 2

We set up the usual values table for plotting points, but put an extra working out row into it.

Because y = x + 2 is a “one step equation” we only need one working out row.

It is a “one step equation” because only one thing is being done to x, and that is adding the 2 by the “+ 2” next to x.

The working out row is shown in light blue below.

We also have a green row at the bottom for final answers.

Vals Two

Here is how we use the blue row for working out.

Vals Three

The next step is to do the working out for each box in the table, and write the answer each time into the purple “Y Value” row.

Vals Four

The final step to complete the values table is to write our numbers from the X and Y rows as (x , y) values.

Vals five

We are now ready to plot our (x , y) values onto the Cartesian Plane.

Vals six

We then extend the line, add some arrows to the ends, and create the finished straight line.

We also write the original “y = x + 2” rule next to this line.

Vals 7

y = x + 2 was a fairly simple example, and many of us could have worked out the y values in our head, and not needed the blue working out row.

However, let’s look at a more involved fractions example, where working-out rows are definitely required.

Values Table Fractions Example

Here is a more involved example of an equation with several steps in it that involves fractions and negatives.

Frac Vals Table 1

Because we have a three step equation, we need a working out row for each step like this.

Frac Vals Table 2

The first working-out row is for multiplying “x” by negative one.

Frac Vals Table 3

We now work on the second blue row, which is for the dividing by 2 step.

We take each answer we just got in blue row one, and divide it by 2, putting the answer into blue row two.

Frac Vals Table 4

We are now onto the third working out step which involves adding 3 to everything.

Frac Vals Table 5

Now we take our final worked out values from the bottom blue row, and copy them into the purple Y values row.

Frac Vals Table 6

The final step to complete the table is to combine the original “x” values with the “Y” values, to create a set of (x , y) points.

Frac Vals Table 7

Now we can start making our graph by plotting these points onto the Cartesian Plane.

Frac Vals Table 8

We then complete the graph and write the original rule for the equation onto the graph.

Frac Vals Table 9

Summary of Plotting from a Rule

Here are the steps required to plot any straight line graph from an algebra rule.

Frac Vals Table Summary

Values Table for Practicing Questions

Here is a blank Values Table you can print out or project onto a whiteboard to practice making Values tables from algebra rules.

Simply cross off and leave blank any working-out rows which are not needed for a question.

The full table of three working out rows should only be needed when we have an involved three step equation like y = 2x/3 + 5 .

(Clicking the Image should take you to an 800×594 pixels Table that can be projected or printed).

Blank 800 wide table

X-Y Grid for Practicing Questions

Here is a blank Cartesian Plane you can print out or project onto a whiteboard to draw the graph from your Values Table.

(Clicking the Image should take you to an 800×600 pixels X-Y Grid that can be projected or printed).

Blank 800 wide X-Y Grid

That’s it for this lesson. There can be a lot of working out to do, but if we follow the steps we should end up with a straight line graph each time.

Related Items

The Cartesian Plane
Plotting Graphs from Horizontal Values Tables
Plotting Graphs from T-Tables of Values
Real World Straight Line Graphs I
Real World Straight Line Graphs II

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Plotting Graphs from T-Tables

T-Tables Intro Picture

The above diagram contains a values table of (x,y) coordinates that is arranged vertically down the page.

This is called a “T-Table” because its shape with the x and y headings at the top resembles the letter “T”.

T Table Diagram

We can also represent the same values in a horizontal table like this:

Horizontal Values Table

When we compare the two table layouts, we believe that the T-Table is far superior.

The T-Table is better because it makes it easier to write out the set of (x,y) coordinates.

From these (x,y) coordinates we are going to make our graph.

If you are not familiar with plotting (x,y) points onto an X-Y Grid, then click the link below to learn about this before continuing any further.

Cartesian Plane Lesson

Graphing From a T-Table

To plot a graph using values from a T-Table we follow these steps:

Step 1) Write the table out as a set of (x,y) coordinates.

Step 2) Rule up an X-Y grid on graph paper.

Step 3) Plot the points onto the grid.

Step 4) If the points form a pattern, then use a ruler to join the points together.

Step 5) Extend the line to fill the grid, and then add arrows to both ends.

T Tables slide 2

Example Plotted Graph from T-Table

Here are Steps 1 and 2 completed.

T Tables slide 3

Notice how easy it is to write out the (x,y) values. This is because they are already set out in order within the T-Table.

Completing Steps 3 to 5 produces the following graph.

T Tables slide 4

Cartesian Plane Practice Grid

Here is a blank Cartesian Plane you can print out or Project onto a Whiteboard to practice plotting graphs.

Clicking the Image should take you to an 800×494 pixels grid that can be projected or printed.

Cartesian Plane Grid

That’s it for plotting points from X-Y values that are in “T-Tables”.

In the next lesson we will look at plotting points to make a line starting with an Algebra Equation “rule” like y = 2x + 1 .

Related Items

The Cartesian Plane
Plotting Graphs from Horizontal Values Tables
Plotting a Linear Graph using a Rule Equation
Real World Straight Line Graphs I
Real World Straight Line Graphs II

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