Passy's World of Mathematics

Coordinates Bingo Game

Posted on October 26, 2013 by Passy

Graphics Sourced from Google Images

This fun game teaches students to get very good at plotting (x,y) points, as well as reading points on the Cartesian Plane, and it is lots of fun.

The game was developed here at Passy’s World by Passy and Jack, and has been trialed in classrooms with great results. (Thanks to Jack for coming up with most of the ideas on this one!).

There is a PPT Presentation for this Game which will be very useful for instructing the students on how to play the game.

Overview of Game

Students plot 20 points of their choice on a Cartesian X-Y Grid, which ranges from -6 to 6.

A special Positive / Negative Virtual Dice is rolled, to generate a “Bingo Coordinate”.

If a student finds they have that coordinate marked down, they circle it in red pen on their grid.

(Repeat drawing Bingo Coordinates by rolling the special dice)

Once a student has FIVE Coordinates circled anywhere on their Grid, they call out BINGO, and get their grid checked and win a prize.

Equipment Required for Bingo Coordinates

Grid Paper / Graph Paper (one per student)

PPT Presentation (See end of this lesson for Details)

Laptop PC and Data Projector (for Virtual Dice and PPT)

Lollipops or individually wrapped candy / lollies for Prizes.

Instructional PowerPoint

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Setting Up the Grid Paper

Students Rule up an X-Y Axis on their Grid Paper that forms a 6×6 Grid like the one shown below.

We need a 6×6 Grid because we will be rolling Dice to generate the Bingo Coordinates.

If you need some printable Grid Paper, then try the following links:

Paper Snake Free Graph Paper

Custom Graph Free Graph Paper

Four to a Page Cartesian Grids

Picking Points – Pre-Game Discussion

Students will need to plot 20 different random points of their choice onto their Grid.

We will be rolling dice to generate the Bingo Game Coordinates, and they will have to hope they get lucky, and some of the Dice Coordinates match their coordinates.

(We have found that 20 points makes for a game that goes for about 20 minutes and shopuld have at least 6 winners).

(You can also award some tasty treats for good answers during this pre-game discussion).

BEFORE THEY PICK THEIR POINTS – GO THROUGH THE FOLLOWING:

See if the students can figure out how many different points there are on the Grid.

ANSWER: The Grid is 12 x 12 = 144 possible points.

But what if we allowed Fraction and Decimal Coordinates ?

ANSWER: Infinite

For our game we are only using whole number integer coordinates – you can plot fractions or decimals if you like, but they will never come up on the dice!

There are certain whole number points on the Grid which should not be used – What are they ?

ANSWER: X-Axis points like (1,0) (-3,0) etc and Y-Axis Points like (0,2) (0,-5) etc as well as the Origin – because these coordinates have a zero in them, and zero can never come up on a Dice!

So how many usable points are there on our Grid for the Game ?

ANSWER: 144 – 6 – 6 -6 – 6 -1 (for the origin) = 144 – 25 = 119

So what are the chances of one of your coordinates being called as a Bingo Coordinate ?

ANSWER: 20 / 119 which is roughly 1/6 .

Note that the Game was not deliberately designed to make the chances 1/6 on the Grid, be the same as the Dice Chance; and if we played with only 10 coordinates, the chance for each one coming up would be 1/12.

What are the chances of a Bingo Coordinate coming up; AND then on the very next dice roll, the exact same coordinate coming up again ?

ANSWER: 1/6 x 1/6 = 1/36

Note from classroom trials, the game usually ends with no repeat Bingo coordinates happening during the whole 20 minute game.

Now get the students to mark on their grid their 20 chosen coordinates.

Their finished competition entry should look something like this:

Note the anime girl in the above image is upset because she just noticed that her friend plotted lots of her entry’s points on the X and Y Axis!

Positive and Negative Virtual Dice

The virtual dice which are needed to play the game can be found at the link shown below.

Two rolls need to be done for each coordinate, the first roll is the X-coordinate, and the second roll is the Y-coordinate.

The teacher, or a reliable student, needs to note down each coordinate which is rolled during the game, and also plot the Dice Rolled Coordinates on a Grid, for later answer checking.

Click the following link to access the virtual dice

Positive and Negative Virtual Dice App

Note that an alternative would be to use a Coin with a real Dice (Head = Positive, Tails = Negative)

Playing The Game

Project the Virtual Dice App onto a Screen at the front of class.

Roll the Virtual Dice once for the X-Coordinate, and one again for the Y-Coordinate,

Call out the resulting (x,y) Coordinate

(Teacher or an appointed student needs to note down the coordinate, and also plot it on the answer grid)

If a student has that coordinate they circle it in red pen

Keep rolling the dice until a student has 5 coordinates and calls out Bingo

Check the Student’s entry and award a prize

Keep playing for about 20 minutes until about 10 prizes have been awarded.

Positive Coordinates Game

Our Bingo Game can easily be adapted for beginners doing only Positive Coordinates.

Make a 12 x 12 Y-X Grid, which has (0,0) in the bottom left hand corner, and (12,12) in the top right hand corner.

Then use Two Dice to generate the coordinates.

There is a virtual Dice for this at the following link:

Click here for Virtual Double Dice App

Note that the game will be a little biased towards Coordinates containing 7’s, because 7 is the highest probability result for rolling two dice.

(Eg. 2 and 5, 6 and 1, 3 and 4, 4 and 3, 5 and 2, 6 and 1 = 6/36 = 1/6; whereas 12 is only from 6 and 6 which is a 1/36 chance).

See if this becomes apparent at all, as the game progresses.

Play the rest of the game as normal – Roll the two dice and add the result to get the X Coordinate, then roll them again to get the Y-Coordinate.

Note that Coordinates containing Zero or One will never come up in the Bingo Coordinates, because the lowest possible result from two dice is 1+1 = 2

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Posted in Algebra, Probability, Straight Line Graphs | Tagged Cartesian Plane, cartesian plane game, coordinates bingo, coordinates bingo using virtual dice, coordinates game, Passy's World coordinates bingo game, plotting points bingo game, plotting points game, virtual dice game, x y coordinates bingo game, x y coordinates game | Leave a comment

X and Y Intercepts

Posted on October 25, 2013 by Passy

Image Source: http://www.geograph.org.uk

In this lesson we look at Straight Line Graphs in relation to X and Y Intercepts.

This includes understanding what “Intercepts” are in terms of their similarity to Road and Rail Crossings and Intersections.

We then look at how to read Intercepts off Graphs, as well as finding Intercepts for a Straight Line Equation.

We then show how easy it is to make a sketch graph of a Straight Line using the values of its X and Y Intercepts.

First we need to understand exactly what “X and Y Intercepts” are, and how to read them off a graph.

X-Intercept and Y-Intercept

If we think of “Intercepts” as being like Intersections on roads, this might help us to grasp the concept.

In the opening photo for this lesson we have the Intersection of a Train Line with a Main Road.

Here is a map showing where a train line near where Passy works intersects with two main roads.

If we think of the Train Line as being our Straight Line, and the Two Main Roads being the “X-axis” and the “Y-axis”; then we have the following situation:

Note that in Coordinate Geometry the meeting points are called “Intercepts” rather than “Intersections”.

So when examining graphs, think of the X and Y Axises as being two main roads. (It might even help to draw over the top of them in different colours).

Think of the Straight Line as being either a Train Line, (or another major road), that intersects (or crosses over) the two main X and Y Roads.

The Point where the Straight Line crosses over the X-Axis is called the “X-Intercept”

The Point where the Straight Line crosses over the Y-Axis is called the “Y-Intercept”

X-Intercept and Y-Intercept Examples

Note that we can specify either the Axis Value of an Intercept, or its specific (x,y) coordinates.

Here is a second example to try yourself, the answer is shown further down the page.

Here is the Answer to Example One.

X-Intercept and Y-Intercept from Graphs Videos

This short one and a half minute video shows how to find the Intercepts for a basic Line Graph.

This next video also covers Horizontal and Vertical Lines, which only have one Intercept.

X-Intercept and Y-Intercept from Graphs Worksheet

The following Worksheet can be done as an Online Worksheet.

Questions 1 to 10 are on Y-Intercept, and Questions 11 to 20 are on X-Intercept.

Go to the bottom of the web page to check your answers.

X-Intercept and Y-Intercept from Graphs Worksheet

Calculating X and Y Intercepts from Equations

A lot of times we do not have a Graph of the Straight Line, and only have its Equation.

We could plot some points, and draw the graph, but this is a lot of work.

There is a much faster Algebra Equations Method for Calculating the Intercepts.

With this method we do not have to draw any graphs.

The Algebra Rule works like this:

To find the X-Intercept, substitute y=0 into the equation and solve for x

To find the Y-Intercept, substitute x=0 into the equation and solve for y

A lot of people get confused, and think for the X-Intercept, they should be substituting x=0 !

Try to remember that the substitution is the opposite way around to what you would think.

For the X-Intercept, we substitute y=0 because all points along the X-Axis have a y coordinate of zero.

For the Y-Intercept, we substitute x=0 because all points along the Y-Axis have an x coordinate of zero.

You may have to think about this for a while for it to start making sense.

Finding X and Y Intercepts – Example 1

In the following example we use zero substitutions to find the X and Y Intercepts.

Notice that whatever we substitute as zero, ends up as the zero coordinate in the final answer, as shown by the long arrows.

Once we have the two intercepts, drawing a sketch graph is very easy, as shown below.

Simply Plot the Two Intercept values onto a labelled X-Y Grid, and then connect them with a straight line with arrows on each end.

Also write the original Equation of the line onto the Grid.

Finding X and Y Intercepts – Example 2

In the following example we again use zero substitutions to find the X and Y Intercepts.

Complete the partly worked out example below.

(You can then scroll down the page to see the fully worked answer)

Here is the fully worked solution to Example 2.

Complete the Graph for Example 2 using the X-Y Grid Layout below.

Simply Plot the Two Intercept values onto a labelled X-Y Grid, and then connect them with a straight line with arrows on each end.

Also write the Equation of the Line onto the Grid.

(You can then scroll down the page to see the fully worked answer)

Finding X and Y Intercepts Videos

This first video shows how to find the X-Intercept and Y-Intercept using substitution of zero values.

This part of the video starts about two minutes into the video, the beginning is a review of locating the Intercepts on a Straight Line Graph.

This next two and a half minute video, shows how to find the Intercepts, and then goes the next step and Sketches the Straight Line Graph.

X and Y Intercepts Worksheets

This first Worksheet actually contain five sheets of questions with the answers at the bottom of each sheet.

Intercepts Worksheet 1

This second worksheet has graph Sketching Questions, (Questions 15 to 260, with Answers at the end of the sheet.

(Do not do Questions 7 to 14 of this sheet on Gradient Intercept, as we have not covered that work in this lesson).

Intercepts Worksheet 2

X and Y Intercept Online Calculator

Shown above is a great free Online App we can use for finding X and Y Intercepts.

Simply type your line equation into this calculator, Click the “Solve” button, and it will give you the coordinates of your X and Y Intercepts.

Click the following link to access this Intercepts Calculator:

Click here to use the X and Y Intercept Calculator

Related Items

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Posted in Equations, Straight Line Graphs | Tagged drawing graphs using intercept method, finding X and Y Intercepts, how to do X and Y Intercepts, how to graph X and Y Intercepts, how to X and Y Intercepts, linear functions, linear graphs, sketching graph using intercepts, sketching straight lines using intercepts, sketching using intercepts, straight line intercept, X and Y Intercepts, x y intercept videos, x y intercept worksheets, x y intercept worksheets with answers, x-intercept, y-intercept | 1 Comment

Gradient Slope Formula

Posted on October 22, 2013 by Passy

Image Source: http://pictures.4ever.eu

The Gradient Slope of the Snowboard Jump Ramp in the photo above, would need to be created at the correct incline to give the Snowboarder sufficient height to complete his aerial acrobatics.

Obviously some basic mathematical calculations would have been involved with building this jump ramp.

In a previous lesson we looked at Gradient and Slope, and how to calculate them by plotting points on a Grid, and drawing and measuring right traingles.

If you have not done this previous lesson, then you need to go through the material at the link below, before doing the “Gradient Formula” lesson.

Intro to Gradient and Slope Lesson

In our “Gradient Formula” lesson, we will be looking at finding the Gradient Slope between two points using an Algebra Formula.

The nice thing about the formula method, is that we do not actually need to draw the points onto a Cartesian Plane Grid.

This saves us a lot of time and work…. Nice!

Definition of Gradient

The Mathematical value of a Gradient or Slope involves comparing how far up we have gone, against how far across.

Original Image Purchased from Photozone.com

Change in Y and Change in X

Previously we have found Gradient Slope by drawing our points onto a grid and counting squares.

We can actually determine the UP and ACROSS squares counts by SUBTRACTING PAIRS OF X AND Y COORDINATES.

This observation led mathematicians to develop a Gradient Slope Formula which does the coordinate pairs subtractions.

Gradient Slope Formula

The Gradient Slope Formula involves labelling the x and y coordinates, and then subtracting the y’s and subtracting the x’s.

This gives us the “Change in Y” and the “Change in X”.

To get the Gradient we divide these two subtractions.

Applying this formula to our previous example points, we obtain the same Gradient Slope answer of “2”.

Gradient Slope Example 1

In this first example, we have plotted the points and graphed them on a Cartesian Grid.

HOWEVER, Graphing the points is not actually necessary, as we shall see in Example 2 later.

Have a go at substituting the required values into the Gradient Formula, and working out the final answer.

(You can then scroll down to see the fully worked solution).

Here is the fully worked solution to Example 1.

(It can be checked by counting squares on the Grid, and is Negative because from left to right it is a Downhill Line connecting the two points.

Gradient Slope Example 2

In this example we have not bothered plotting the points, because this is not necessary.

This is the great thing about using the Gradient Slope Formula…. it is so much faster to get to the answer!

Have a go at substituting the required values into the Gradient Formula, and working out the final answer.

(You can then scroll down to see the fully worked solution).

Here is the fully worked solution to Example 2.

Gradient Slope Videos

The following is a Gradient by Formula video, (10 minutes long), that shows the line plotted and how the formula works.

This next video talks about the change in y versus the change in x, using the Gradient Slope Formula.

The following is an excellent, but quite long YayMath video, which uses the Gradient Slope Formula.

The video even shows the calculation of zero and undefined gradients using the formula.

Straight Line Between Two Points Calculator

All we have to do is enter the (x,y) coordinates of any two points and click “Go” and this online interactive will calculate the slope, find the rule for the line, and even plot the line for us on an X-Y Grid.

It also tells us about Parallel and Perpendicular Lines.

Click the following link to use this interactive calculator.

http://www.mathsisfun.com/straight-line-graph-calculate.html

Gradient Worksheet

This worksheet does not have any answers; however the answers can be checked using the Gradient Slope Calculator available at the following link:

http://www.mathsisfun.com/straight-line-graph-calculate.html

Click the following link to get to the worksheet.

Click here for Gradient Slope Formula Worksheet

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Fractions on Both Sides Equations

Posted on October 19, 2013 by Passy

Image Source: http://www.blogspot.com

In this lesson we look at How To Solve Equations With Fractions on Both Sides.

Prior to doing this lesson, you need to know how to solve equations with letters and/or brackets on both sides.

If you need to find out how to do Variable Letters On Both Sides Equations, then go through our previous lesson on this at the following link:

Solving Variable Letter on Both Sides Lesson

The method for “Fractions Both Sides” is basically the same as solving equations with the letter on both sides, except that we need to do an extra cross multiplying step at the beginning.

Prior to going through our lesson on “Fractions Both Sides” :

We also recommend you check that you understand the material from the following previous lessons:

Cross Multiplying Ratio Equations Lesson

Expanding Brackets Using the Distributive Rule Lesson

Fractions Both Sides Equation

Here is a typical “Fractions Both Sides” Equation.

These types of Equations are done the same way as “Variable Letter on Both Sides” equations, but we have one extra beginning step at the start.

Fractions Both Sides Working Out Steps

The cross multiplying is the same as we do for Ratios.

Eg. The Ratio Equation 3/4 = 6/8 cross multiplied gives

8×3 = 6×4 which is a true equation,

and so cross multiplying is a legitimate method.

Fractions Both Sides – Example One

Here is how we do the Cross Multipying Step for a typical “Fractions Both Sides” Equation:

Note that we need to use Brackets to make sure ALLL the terms in the top line “numerator” get multiplied.

After we have completed the Cross Multiplying, we are left with a normal “Variable Letter Both Sides” equation to solve.

We solve this as follows:

If you do not know how to Solve for Variable Letter on Both Sides, then see our previous lesson on this at the following link:

Solving Variable Letter on Both Sides Lesson

Fractions Both Sides – Example Two

In this example, only one side has a fraction.

We do a little “trick” which involves putting the non-fraction side as being a fraction that is / 1

We can then do the Cross Multiplying Step.

Here is the example, showing the Cross Multiplying Step

Here are the remaining steps that are required to get to the final answer:

Videos About Fractions Both Sides Equations

These two videos show how to use the Proportions Cross Multiplying Method to Solve Equations.

And here is the second video you need to watch.

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Passy

Posted in Algebra, Equations | Tagged brackets on both sides. how to do fraction on both sides, equations with brackets, expanding brackets, fraction on both sides, Fractions Both Sides Equations, fractions on both sides worksheet, how to do brackets on both sides, how to solve equations, multi step equations, onion skins equations method, solving equations, Solving Equations with fractions on Both Sides, Solving Equations with ratios on Both Sides, variables on both sides video | Leave a comment

Variable On Both Sides Equations

Posted on October 18, 2013 by Passy

Image Source: http://oliveloafdesign.wordpress.com

When we have equations that start off with variable letters or brackets on both sides, we need to do some extra steps as part of solving them.

The reason we might want to solve equations like this, is to find the common “y coordinate” of the intersection point of two lines.

It can also be used to find when two items starting at different points will meet up with each other.

Letters both sides methods can also be used to compare the Economy (Cost versus Distance) of cars running on different types of fuels, or with different engine sizes.

In a Business, the method is used to find the position where enough sales dollars are generated, to cover the costs of running the Business.

This sales and cost balancing process is called establishing the “Break Even Point”, and is used a lot in Accounting.

In this lesson we look at how to solve equations which have variable letters and/or Brackets on both sides of the Equation.

Prior to doing this lesson, you need to know how to solve basic equations, as well as the “Distributive Rule” for expanding brackets.

If you need to review these concepts, then check out our previous lessons at the following links:

Solving Equations Using Onion Skins

Expanding Brackets Using Distributive Rule

Variable Letter On Both Sides Equation

Here is what a basic letters both sides situation looks like.

Before we can solve equations like this, we have to do some extra steps to remove the letter term from one of the sides.

This involves applying either SUBTRACTION OR ADDITION of a variable letter term value on both sides to the equation.

The steps we need to do are as follows.

Note that by identifying the smaller letter item and using that, it greatly helps reducing the prescence of negative signed items to deal with.

We can in fact reverse out and eliminate either letter item, but our work is easier if we eliminate the smaller item.

Solving Letter On Both Sides Equations

Here is how we apply the standard steps to solve a letters both sides equation.

Once we have completed getting the letter variable on one side, we can now use either “Onion Skin”, “Flowcharts”, or “Algebra Reversing”, to finish solving the equation.

Here is how to finish the question using the “Onion Skins” method:

If you are not familiar with Solving Equations using the “Onion Skins”, then you can review our lesson on this at the following link:

Solving Equations Using Onion Skins

Letters Both Sides Videos

This video from Mahalo Maths explains one example equation fully.

Here is another Video Example which has a fraction answer.

The following video has some fractions to deal with on both sides.

We deal with them by multiplying by the lowest common multiple.

This is a reasonably involved example, which takes a bit of working out:

Variables Both Sides Example

Variables on both sides do not always involve subtraction as the first step.

Note that -5h is smaller than -3h, because -5 is smaller than -3, and so we ADD to both sides to eliminate the -5h.

We can now finish off solving our equation using Onion Skins

Brackets Both Sides Equations

Sometimes questions have the letter term on both sides, and also have some brackets which we need to expand.

If you are not familiar with Expanding Brackets using the “Distributive Rule”, then you can review our lesson on this at the following link:

Expanding Brackets Using Distributive Rule

The following example shows how we FIRST EXPAND THE BRACKETS, and then after this perform the usual letter on both sides step.

Once we have completed expanding all brackets and getting the letter variable on one side, we can now use either “Onion Skin”, “Flowcharts”, or “Algebra Reversing”, to finish solving the equation.

Here is how to finish the question using the “Onion Skins” method:

Videos for Brackets on Both Sides

In this video, there are some brackets to expand using the distributive rule, and then after this we do letters both sides working out.

Here is another video about solving equations with brackets on both sides.

Summary

Here are the steps we always need to follow when solving equations with brackets and/or letter variables on both sides.

It is possible to occasionally get some strange answers for our letter variables on both sides questions, where either every number is an answer, or there is no number answer.

The following web page has some good examples of these abnormal types:

http://www.shmoop.com/basic-algebra/funky-equations.html

Worksheets on Equations

The following worksheet has a good mixture of questions and includes brackets questions:

Equations Worksheet 1

Here is another worksheet with similar types of questions to Worksheet 1

Equations Worksheet 2

This third worksheet lets you click to get the answers, and also allows you to generate a number of different questions to work on.

Equations Worksheet 3

Online Equations Quiz

The following Online Quiz also has worked solutions.

Note you need to page back using your web browswer back button after viewing a solution, to get back to the quiz.

Click the link below to do the Quiz

Click here to do Equations Quiz

Solving Equations Calculator

The following web page has an awesome calculator where you can type in your brackets and/or variable on both sides equation, and it will then show you a fully worked algebra solution that gets the answer:

The current Stats at the bottom of the Solution, indicate that this calculator has solved over 39 million equations since it first went onto the web in 2004!

Click the link below to use this Calculator / Solver

http://www.algebrahelp.com/calculators/equation/

Here at Passy’s world we love the above Equations Solver App, because it shows the Algebra the computer performed, as well as giving the final answer.

People who are able to take mathematical items like this, and turn them into Apps are really in demand at the moment, and hopefully in the future students will do more Applied Mathematics using Apps that are freely available.

They will do this instead of learning so much Algebra pen and paper manual working out for equations.

Solving Equations Millionaire Game

This free Online Game is a Millionaire type game, where the later questions have variables on both sides.

Play it at the following link:

http://www.quia.com/rr/380806.html

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