Passy's World of Mathematics

Basic Exponents and Indices

Posted on April 28, 2012 by Passy

Image Source for iPad: http://www.mydigitallife.info

What do you notice about the numbers:

2, 4, 8, 16, 32, 64 ?

They are all computer memory and storage numbers, like
16 Gig, 32 Gig, 64 Gig on an iPad.

Power of 2 exponents are the basis of all computing which is done in “Binary” or base 2 numbers like these.

In this lesson we will be learning the basics of Indices, Powers, and Exponents.

Here are some examples of Powers multiplication.

Introductory Video on Exponents

Watch the following video which goes through the basics of Powers and Indexes (Indices).

Base and Index

The main idea with Powers is that we have a normal number, and then a superscript (raised up in the air) smaller number.

These two values then combine together to create a line of multilplications that need to be done.

Squares Numbers

If we multiply a number times itself we get a “Square” number. These numbers are part of the normal times tables, and form the areas of square shapes.

Cubic Numbers

If we multiply a number together three times, then we generate the volume of a cube.

These numbers are called “cube numbers” or “Cubics”.

Index Expanded and Number Forms

There are three ways to write out Powers Terms.

It is important to read mathematics questions carefully, so that we understand the form in which the final answer needs to be written.

Some Examples To Try

Here are some basic Powers questions for you to try.

The answers to the above questions are as follows:

The Power of Zero

In the previous examples, we saw that the power of one, means we just take the base once on its own.

The Power of Zero is not so easy to understand.

However Mathematicians have used patterns to work out that any number to the power of zero always equals 1.

More Examples to Try

Here are some more Powers questions for you to try.

The answers to the above questions are as follows:

Algebra Powers

If we have Algebra letters instead of numbers in our question, it does not make much difference.

We still do Index Form and Expanded the Form the same way as we do for numbers.

However because we have letters, we cannot make a simplified whole number answer, so there is no “Numerical Form” for Algebra.

It is easy to start getting confused as soon as we work with Algebra.

Make sure you are very clear on the difference between the following types of Algebra Terms.

Interactive Online Exponents Lesson

This online activity does take a while to load in on the computer, but it is well worth the wait.

Be patient while the mostly blank screen loads the activity in.

This animated lesson on Exponents and Indices is excellent.

It goes through the basics, and even shows how to use the x to the power of y button on an interactive calculator to do powers questions.

Click the link below to do this interactive online lesson.

http://www.bbc.co.uk/schools/ks3bitesize/maths/number/powers_roots/activity.shtml

Videos About Exponents and Powers

Here is a great video by Mr Konst about Exponents.

The following video is part one of a quick general introduction to Exponent Form.

Here is the continuation part two of the previous video.

Worksheets on Exponents and Indices

Click the following links to access free worksheets on the Internet that cover basic Exponents questions.

http://www.ezschool.com/EZSheets/Algebra/Exponents/sheet14.html

http://www.ezschool.com/EZSheets/Algebra/Exponents/sheet15.html

http://www.ezschool.com/EZSheets/Algebra/Exponents/sheet6.html

http://www.mathworksheets4kids.com/exponent/find-missing-exponent-prime-base.pdf

http://www.mathworksheets4kids.com/exponent/find-missing-prime-base.pdf

Exponents Games

Try out each of these free online games to improve your exponent and powers skills.

Alien Exponents

In this game we click the spaceship which has the correct number answer on it to score each point.

If we get the question wrong, then the Alien makes it to earth and his team gets one point.

http://www.ezschool.com/Games/Exponents.html

Exponents Battleship

In this game we click on the right hand side grid to try and locate the enemy ships.

If we click where a ship is, then we are given an Exponents question to solve like this:

In this question, the ^ symbol means power.

1^4 means 1x1x1x1 and 3^1 means 3 so when we * multiply them together we have the correct answer of “3”.

Have fun playing this game by clicking the following link:

http://www.quia.com/ba/1000.html

Greater Than Less Than Game

In this game we pick and click the less than or greater than sign.

http://www.softschools.com/math/games/exponents_practice.jsp

Asteroids Game

This game is difficult at first as the screen is full of moving asteroids.

However as we get each question correct, it gets easier because there are less and less asteroids clogging up the screen.

http://www.mathdork.com/games/asteroidsexp3/asteroidsexp3.html

Exponents Matching Activity

In this activity we click on the left item, and then click on its matching right hand side item.

If we are correct, a mathcing line gets drawn. If we are incorrect, we have to try again.

http://www.studystack.com/matching-14392

Online Exponents Test

Now that we have watched all the videos, done some practice questions, and played all the games it is time to test our Basisc Exponents Skills.

Click the link below to do an exponents test online at the BBC Skillwise site.

http://www.bbc.co.uk/apps/ifl/schools/ks3bitesize/maths/quizengine?quiz=powers_roots&templateStyle=maths

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Passy

Survivor Algebra

Posted on April 25, 2012 by Passy

Image Source: http://i3.fc-img.com

Survivor Algebra is a fun way of doing Algebra with Mathematics Classes.

It involves dividing a class up into teams, where they outmath their opponents for prizes by their results on “Challenges”.

The idea was invented by Karen at the “Cool Math” website, and Passy has used this with his Mathematics classes and it works really well.

Just in case anyone doesn’t know what the “Survivor” TV show was all about, here is an amusing one minute clip from YouTube.

How Survivor Algebra Works

The class is divided up into seven teams with names as follows:

Algebros, Additrons, Decimos, Fractonons, Newtonicks, Calculars, and Pentanos.

In a class of 28 students, this works out as seven groups of four.

The groups need to be set up so they are as even as possible in ability.

Students work together in their teams during Mathematics classes when we are doing Algebra.

Immediately after every two or three exercises out of their text book, every student in the class does a “Challenge” individually under test conditions.

A challenge is a two page quiz containing questions on the relevant exercises. It takes about 20 to 30 minutes to complete, and is marked by the teacher and graded just like a normal test.

The scores on each of these individual “Challenges” all contribute to each student’s final grade for Algebra.

For each Team, the lowest challenge score is dropped, and the remaining team members scores averaged to create a Team Score for that challenge.

The Team Scores are progressively accumulated and recorded for each challenge on an A3 sized laminated scorecard on the wall of the classroom that looks like this:

If you would like a PDF of the standard score card to print out and enlarge on a color photocopier, then click the following link:

http://passyworldofmathematics.com/pwmPDFs/SurvivorScoreChartPDF.pdf

For each Challenge prizes can be given to the winning team, such as lollies, candy, or chocolates.

There can also be a grand prize at the end of the competition for the student in the winning team with the highest overall individual score, such as a free movie pass. ( This motivates the best student in each of the teams to encourage their fellow team members to achieve their very best on all challenges).

Cool Math Survivor Algebra Details

So far we have been talking about how we do Survivor Algebra in our classes.

We strongly recommend that teachers also read through the great information at “Cool Math” – the people who invented this game.

http://www.coolmath.com/Survivor-Algebra/

Introducing Some Randomness

We have found that it is fun to also introduce some Luck and Randomness into the competition.

This can be done by a variety of means such as rolling dice, picking from homemade cards, or spinning a homemade “Wheel of Misfortune”.

For our randomness, we use the following app we developed in “Scratch”.

(Note that Scratch uses Java, so this app will not run on Apple Devices)

This app selects one of our seven teams at random, and then assigns a Percentage Bonus to their Team Score which can range anywhere from 10% to 30%.

Click the Green Flag at the top right hand corner of the app to make it process through.

How the Scratch Random Picker Works

The other thing we like about using our Random Picker, is that it is an excellent example of how Algebra Variables and Mathematics is used in creating software apps.

There are two sprite objects in the app: the Cat and the Shark.

Here is the script code for the Cat.

And here is the script code for the Shark.

That’s it for Survivor Algebra, we recommend all mathematics teachers try this fun activity with their classes.

Feel free to link to this post, share it on social media sites, or use it on Learning Management Systems in Schools.

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Enjoy,
Passy

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Testing Out Scratch and Vimeo

Posted on April 25, 2012 by Passy

This is a very quick test to see if an iframe into a Scratch Project we just made works or not.

Here is a test of a Vimeo video embed code

Hereis the same video manually upsized from Vimeo’s 400×225 to 500×281

That’s all for now folks.

Posted in Uncategorized | Leave a comment

Substituting with Negative Numbers

Posted on April 23, 2012 by Passy

Image Source: http://d.yimg.com

Substituting negative numbers into expressions is the same as substituting positive numbers, except that we need to know the “Integers Rules”.

“Integers” is the mathematical word that refers to positive and negative numbers.

The main Integers Rules we need to remember are the ones for Multiplying, Dividing, and Subtracting.

Integers Revision

For Multiplying integers, we use the following set of rules.

If you need to review Multiplying Integers, then click the following link:

http://passyworldofmathematics.com/multiplying-integers/

The rules for Dividing Integers are the same as for multiplying positives and negatives.

If you need to review Dividing Integers, then click the following link:

http://passyworldofmathematics.com/dividing-integers/

For Subtracting Integers, we Add the Opposite.

Subtraction can also be done using the “Keep Flip Change” rule.

If you need to review Subtracting Integers, then click the following link:

http://passyworldofmathematics.com/subtracting-integers/

Videos About Substituting Negatives

Here is a video that involves substituting negative and positive values.

This next video shows how to substitute when brackets are in the question.

Substituting Negatives Examples

Values Tables Substitution

We sometimes need to use a supplied rule equation to complete a values table of “x” and “y” values.

These tables are used when we need to draw a graph for the rule. However we will not be drawing the graph in this example.

If you want to see an example where we make the graph from the completed values table, then click the lesson link below:

http://passyworldofmathematics.com/plotting-graphs-from-t-tables/

This next example is basically four separate substitution questions, where we are putting our four answers into an answer table.

We now need to work out “y” answers for the other three “x” values the same way.

At Passy’s World we find it easiest to do these questions by adding a middle column to the table where we do the working out.

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Enjoy,
Passy

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Olivers Hill Wall Collapse

Posted on April 21, 2012 by Passy

Could this be Melbourne’s worst million dollar mathematics mistake?

A huge Gabion retaining wall that was being built near where Passy lives crashed down in spectacular fashion a couple of days ago.

Here is a YouTube video of the aftermath.

The Olivers Hill Gabion Wall

We have been watching this wall being built over the last two months, and it looked to us like it was being built far too high without enough base thickness to support the weight of the rocks on top.

The angles, thickness, staggering, and terracing did not look to be correct. The fact that there was no stepping or terracing on such a high wall is very puzzling indeed.

Initially the wall was tilted back into the hillside at an angle, but we had noticed in the last week that the wall seemed to be getting more and more vertical as it got higher.

The wall was built several metres out from the cliff face, and the gap between the two was recently filled with soil that was brought in from elsewhere.

It appears that the weight of this filling soil may have caused a failure along the bottom of the wall, and the whole middle section then collapsed. Local residents said they saw the bottom sections of the wall beginning to buldge outwards several hours before the disaster. Miraculously nobody was killed or injured.

Here at Passy World, we suspect that “Internal Bearing Failure” in the wall itself is probably the root cause of the disaster.

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

The bearing capacity of the bottom rows of baskets was possibly not strong enough to hold the baskets stacked above them, along with the weight of the soil behind them.

If they do find out the exact cause the failure and the information is made public, we will supply an update.

Another serious issue we have with the Gabion Wall involves it being built about 500 meters away from the sea shore of Port Phillip Bay. What is the life expectancy of its basket wire openly exposed to such salty corrosive conditions ? If the wire rusts away, won’t the whole wall come tumbling down onto the adjacent seaside highway, in just a few short years time ?

The wall was being built by the State Government roads authority here in Melbourne, and so far has cost local taxpayers over one million dollars. A recent article on the Internet put the exact cost at 1.4 million dollars.

In this lesson we look at how Gabion Walls are supposed to be constructed, and the Mathematics of Gabion Wall Design.

About Gabion Walls

“Gabions” are rectangular stone filled baskets used to stabilize soil and prevent erosion. The baskets are stacked on top of each other to form retaining walls.

They have been used since Egyptian times and were first made of woven reeds and rock and used to stabilise the banks of the river Nile.

Here is what a typical empty Gabion basket looks like.

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

The Olivers Hill wall which collapsed was being built with the wire being meshed onsite, like in this picture of a wall being built elsewhere in the world.

Image Source: http://images.tradekool.com

Another method of building such walls is to use solid welded stainless steel baskets like in this example.

Image Source: http://www.greenfab-media.com

Here is a Gabion Wall in New York State USA.

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

Notice that this wall is angled back into the hill and stepped with terraces.

The Olivers Hill wall which collapsed was not built like this.

Here at Passy World we are very surprised that the bottom of the Olivers Hill wall was not thicker, and that the Olivers Hill wall was not stepped with terraces.

More information about Gabion Walls can be found at the following links:

http://en.wikipedia.org/wiki/Gabion

http://www.gabions.net/index.html

http://www.limitstate.com/geo/gabion-walls

Mathematics of Gabion Walls

There is a great design article about Gabion walls at the following link:

http://www.hy-tengabions.co.uk/technical_publications/design_guide.pdf

The mathematics and design is quite complicated because of the following factors:

The underlying ground and soil has to support the weight of the wall

The weight of the baskets needs to be able to hold back any unstable soil and rock geology that is behind the wall

The wall needs to be a stable self supporting monolith

Complex Geometry, Algebra, and Physics are involved with the successful construction of a Gabion Wall.

Here are some screen shots of some of the mathematics involved, taken from the “Hy-Ten Gabions” design document, and modified slightly by Passy’s World.

First of all we need to know the total weight of the wall.

( The funny “E” looking symbol is “Sigma” which means to add up all the courses or rows of the wall to a grand total.)