USES OF ALGEBRA IN OUR DAILY LIFE

Determining probability

How do we determine probability? Let's say there are 12 socks in your dresser drawer. Five are red and 7 are blue. If you were to close your eyes, reach into the drawer, and draw out 1 sock, what is the probability that it would be a red sock? Five of the 12 socks are red, so your chances of picking a red sock are 5 out of 12. You can set this up as a fraction or a percentage that expresses the probability of picking a red sock:

Your chances of picking a red sock are 5 out of 12, or 5 divided by 12, which is about 42%. Not bad, as odds go.

Imagine you're choosing between 2 colleges, 1 in California and 1 in Massachusetts. You decide to flip a coin. Heads, you'll go to California. Tails, you'll go to Massachusetts. When you toss the coin, what is the probability that the head side of the coin will be facing up once the coin hits the floor? There are 2 sides to a coin, and 1 of them is heads, so your odds are 1 out of 2. In other words:

One divided by 2: that's a 50% chance of heads, and therefore also a 50% chance of tails. The odds are equal. You're as likely to go to California as to Massachusetts if you base your decision on a coin toss.

Even if you don't frequent casinos, you probably play the odds all the time. You might invest in the stock market. You might buy auto, health, and life insurance as a hedge against the costs of damage or injury. In many cases in which you are trying to predict the future, you're using the mathematics of probability.

Do you avoid gambling on the stock market or at a casino because you fear heavy financial losses? You may be surprised to hear that you're just as likely to lose money because of your everyday banking decisions. Many people collect only 1 to 3% interest on money in a savings account while simultaneously paying rates as high as 18 to 20% on credit card balances. Over time, this can mean some pretty heavy losses.

With some math smarts and an understanding of simple and compound interest, you can manage the way your money grows (and ideally keep it from shrinking). The principles of simple and compound interest are the same whether you're calculating your earnings from a savings account or the fees you've accumulated on a credit card. Paying a little attention to these principles could mean big payoffs over time.

Understanding the basics

When you put money in a savings account, the bank pays you interest according to what you deposit. In effect, the bank is paying you for the privilege of "borrowing" your money. The same is true for the interest you pay on a loan you take from the bank or the money you "borrow" from a credit card.

Interest is expressed as a rate, such as 3% or 18%. The dollar amount of the interest you earn on a savings account is figured by multiplying the money you deposit (called the principal) by the rate of interest. If you have $100 in an account that pays only 1% interest, you'll only earn $1 in interest. If you shop around for an account that pays 5% interest, you'll earn five times that amount.

In banking, interest is calculated and added at the end of a certain time period. You might have a savings account that offers a 3% interest rate annually. At the end of each year, the bank multiplies the principal (the amount in the account) by the interest rate of 3% to compute what you have earned in interest.

Interest on interest: Compounding

There are two basic kinds of interest: simple and compound. Simple interest is figured once. If you loaned $300 to a friend for one month and charged her 1% interest ($3) at the end of the month, you'd be dealing with simple interest. Compound interest is a little different. With compound interest, the money you earn in interest becomes part of the principal, and also starts to earn interest. If you loaned that same friend $300 for one month but charged her 1% each day until the end of the month, you'd be using compound interest. At the end of the first day, she would owe you $303. At the end of the second day, she would owe you $306.03. At the end of the third day, she would owe you $309.09, and so on.

Compound interest is what makes credit cards and loans so difficult to pay off. The rules of interest are the same ones that increase your savings over time, only with credit and debt, they're in the bank's favor—not in yours. With some rates as high as 21%, collecting interest on credit card loans can be a lucrative business.

Know the Terms: How to Manage Your Credit Cards

APRs. Annual fees. Finance charges. What do all these hidden fees mean, and how can you stay on top of them? Credit cards seem to have a language of their own. Once you understand this lingo, you can learn how to save hundreds or even thousands of dollars in unnecessary fees.

The first credit card concept that you should know about is the minimum payment, or the minimum amount you are required to pay each month. The amount of your minimum payment is usually equal to 2% of your average balance due. So if your average daily balance for a certain month is $500, your minimum payment for that month will be $10. This probably sounds like a good deal until you remember that you're also paying interest on that $500. If you're being charged an interest rate of 18% on the balance of $500, that $10 won't make a dent in paying off your balance. This is how most credit card companies make their money. By requiring that you pay only very small amounts each month, they know that it will take you much longer to pay off your debt, and that you'll end up paying a lot more interest over time.

Finance charges are a murky area, since each credit card company can determine its own method of computing these fees. The finance charge is the interest you pay on the money you owe the credit card company. Many companies compute monthly finance charges by determining what your average daily balance was for the month, then charging interest on that average figure. Other credit cards figure finance charges based on the average daily balance plus any new purchases you make. To avoid paying additional finance charges, find a credit card that figures fees based only on your average daily balance for the current month—not past balances or new purchases. Better yet, pay off your entire balance each month to avoid these fees.

The Annual Percentage Rate (APR) is another important credit card concept. The APR is the finance charge (which can vary month to month) expressed as an annual percentage rate. The APR is normally somewhere between 5% and 21%. One of the things to look for when comparing the APRs offered by two credit card companies is whether the rate is fixed or variable. A variable APR can fluctuate. It might follow the national interest rate, for example. A fixed APR stays the same unless changed in writing by the credit card company. A fixed APR will normally offer the best value over time, especially if you can find a low fixed rate.

You will also need to know what the grace period is for your card. The grace period is the period from when you make a purchase to when the credit card company begins charging you interest for that purchase. Without a grace period, you'll start accumulating interest charges immediately. Your best bet is to find a card with a grace period of 25 days or more and pay off the balance before that period ends.

Getting the Picture: Communicating Data Visually

According to U.S. census estimates, the population of Texas grew from 17,045,000 people in 1990 to 18,378,000 in 1994. The population of Massachusetts grew from 6,018,000 people in 1990 to 6,041,000 people in 1994.

If the population figures above were difficult for you to read and absorb, you're not alone. Reading about data can be awkward. When it's presented like this, it's hard to grasp the essential information and to see the important messages that may be behind the numbers. If this information were presented as a chart or, better yet, as a picture, it would be much easier to understand.

Ratios: Relationships between quantities

That ingredients have relationships to each other in a recipe is an important concept in cooking. It's also an important math concept. In math, this relationship between 2 quantities is called a ratio. If a recipe calls for 1 egg and 2 cups of flour, the relationship of eggs to cups of flour is 1 to 2. In mathematical language, that relationship can be written in two ways:

Both of these express the ratio of eggs to cups of flour: 1 to 2. If you mistakenly alter that ratio, the results may not be edible.

Working with proportion

All recipes are written to serve a certain number of people or yield a certain amount of food. You might come across a cookie recipe that makes 2 dozen cookies, for example. What if you only want 1 dozen cookies? What if you want 4 dozen cookies? Understanding how to increase or decrease the yield without spoiling the ratio of ingredients is a valuable skill for any cook.

Let's say you have a mouth-watering cookie recipe:

This recipe will yield 3 dozen cookies. If you want to make 9 dozen cookies, you'll have to increase the amount of each ingredient listed in the recipe. You'll also need to make sure that the relationship between the ingredients stays the same. To do this, you'll need to understand proportion. A proportion exists when you have 2 equal ratios, such as 2:4 and 4:8. Two unequal ratios, such as 3:16 and 1:3, don't result in a proportion. The ratios must be equal.

Going back to the cookie recipe, how will you calculate how much more of each ingredient you'll need if you want to make 9 dozen cookies instead of 3 dozen? How many cups of flour will you need? How many eggs? You'll need to set up a proportion to make sure you get the ratios right.

Start by figuring out how much flour you will need if you want to make 9 dozen cookies. When you're done, you can calculate the other ingredients. You'll set up the proportion like this:

You would read this proportion as "1 cup of flour is to 3 dozen as X cups of flour is to 9 dozen." To figure out what X is (or how many cups of flour you'll need in the new recipe), you'll multiply the numbers like this:

Now all you have to do is find out the value of X. To do that, divide both sides of the equation by 3. The result is X = 3. To extend the recipe to make 9 dozen cookies, you will need 3 cups of flour. What if you had to make 12 dozen cookies? Four dozen? Seven-and-a-half dozen? You'd set up the proportion just as you did above, regardless of how much you wanted to increase the recipe.

Meters and Liters: Converting to the Metric System of Measurements

Most of the world uses a standard system of measurements called the metric system. This system is based on a unit of measurement called the meter, which gets its name from the Greek wordmetron, "a measure." One meter is equal to 1 ten-millionth of the distance from the equator to the North Pole. It's a standard for measuring length that is derived from the planet we live on.

The metric system has been around for 300 years. France was instrumental in its creation and in 1795 was the first country to adopt it (though in the early 1800s, the emperor Napoleon briefly set it aside in favor of the old system of measurement). The United States remains one of the few countries that has not yet adopted the metric system as the standard for measurement.

Converting to metric values

Remember that mouth-watering chocolate chip cookie recipe? What if you wanted to send it to a friend in Portugal? You could send him the recipe with the measurements given in cups and teaspoons and hope it worked out for the best. Or you could convert the recipe to metric values, guaranteeing that the cookies would taste as delicious in Portugal as in the U.S.

To convert from cups to the appropriate metric measurement, liters, you need to know how many cups are in a liter. The table on this page shows some common conversion values. You can see that 1 cup is equal to 0.24 liters. To convert your 1 cup of flour to liters, you'd multiply 1 by 0.24. The chocolate chip cookie recipe calls for 0.24 liters of flour.

What about sugar? In the original recipe, you need 1/3 of a cup of sugar. To make the conversion easier, convert the fraction 1/3 to a decimal: 0.33. Now multiply 0.33 by 0.24. Your friend will need 0.08 liters of sugar to make your cookie recipe.

7 WAYS MATHS CAN SAVE THE WORLD

Can numbers, algebra and trigonometry save the planet? This was the question put to experts during a panel discussion at Imperial hosted by the Grantham Institute and the Mathematics of Planet Earth CDT. CDT students Paula Rowińska and Tom Bendall report back on seven ways that mathematicians are already working towards securing our planet’s future.

From meteorology to economics, a wealth of scientific research will be necessary to improve our understanding of climate change, its impacts and what we can do to prepare for them. Scratch beneath the surface and you’ll find mathematicians doing their bit to save the planet in a multitude of ways:

1. Designing better weather forecasts and climate models

Accurate weather forecasts predict when and where extreme weather may strike, whilst climate projections are key to identifying weather patterns changing on a longer time scale. Our ability to predict weather and climate has advanced in leaps and bounds in the last few decades, thanks to maths. Modern weather forecasts rely on computers to solve the complex equations that simulate the atmosphere’s behaviour – from global processes that influence the flow of the jet stream down to local rain clouds.

Mathematicians play an important role in this process, working with a set of equations that describe the atmosphere, taking into account temperature, pressure and humidity. Global Circulation Models (GCMs) describe the interactions between oceans and atmosphere to look at what the average conditions could be in decades to come.

2. Getting ‘bang for buck’ out of supercomputers

The computers used to model weather and climate get more powerful every year – but sheer processing power isn’t everything. Maths makes these computers far more effective both through contributing to technological improvements in areas like quantum computing, and by rethinking the algorithms used in computer programs. For instance, new research allows the computer to automatically zoom its attention in on areas where the weather is particularly interesting, such as around storms.

Optimising computers’ performance can also reduce their energy demand. For example, the Met Office’s Cray supercomputer runs on 2.7 MW of electricity, so even modest efficiency gains could have a massive impact on its overall energy consumption.

3. Making the most of renewable energy sources

Optimising the layout of wind turbines enables them to harvest more energy

Renewable energy sources lie at the heart of a low-carbon world. By choosing optimal locations for wind or solar farms and designing the most effective layouts for tidal and wind turbine arrays, mathematicians ensure that these technologies harvest the maximum energy as efficiently as possible.

Mathematicians contribute to research into energy supply and demand that ensures networks incorporate higher proportions of weather-dependent energy sources such as wind or solar power, making sure that the lights stay on in years to come.

4. Preparing for change

The effects of climate change will be felt on many levels, and knowledge is key to safeguarding human health and livelihoods as we adapt to changing circumstances. Mathematicians use their understanding of probability and uncertainty to advise policymakers on the likelihood of heatwaves, floods or other changes in weather patterns, and help them to plan accordingly.

Businesses also need detailed information on how climate change might affect them. The food industry for example is highly dependent on agriculture, and could use advance warning of an upcoming drought for instance to prepare themselves for smaller yields. Mathematicians try to predict who might be at risk so they can prepare for the future.

Moreover, mathematical simulations are a valuable tool for estimating the possible consequences of specific actions, by playing out different scenarios. This too can help policymakers choose one course of action over others. By presenting the hard numbers, mathematicians with an environmental conscience can seek to influence the ways businesses operate.

5. Making sense of ‘big data’

Collecting billions of pieces of data in environments, from ice sheets to cities, can deliver precious insights into our planet’s physical processes, human behaviour and everything in between. Climate scientists rebuild the history of our planet’s atmospheric composition by analysing the tiny bubbles trapped in ice records, in order to anticipate the scope of future changes. But without the statistical methods that mathematicians bring to analyse this data and assess its reliability, the information has less value.

6. Developing new technologies

Mathematical modelling is key to the development of new technologies such as CCS

New technologies are key to a low carbon future. carbon capture and storage (CCS), for instance, could safely lock away greenhouse gases emitted by fossil fuel-fired power stations, and is likely to play a key role in averting dangerous levels of global warming. Detailed mathematical models make this research possible by using sophisticated logistics methods, network analysis, statistical modelling and many other mathematical tools.

7. Making maths accessible to everyone

Crucially, maths can’t save the planet on its own. Many of the global challenges we face are multi-disciplinary: overcoming them requires mathematicians to collaborate with scientists and engineers in different fields. And although the basic science behind climate change is well understood, convincing the general public and decision makers to take action to reduce carbon emissions is very much a work in progress. With their firm grasp of concepts such uncertainty and probability, Mathematicians are uniquely placed to communicate the science, data and forecasts, and ensure that this information is meaningful to the people who need it.

For maths to have a real impact on our planet’s fate, mathematicians therefore need to communicate the importance of their work clearly and effectively, knowing when to swap complicated equations for persuasive story-telling, pictures, games or genuine interaction. Opening up maths up for the world to understand might just be the best way that we can come to our planet’s rescue.

10 EVERYDAY REASONS WHY STATISTICS ARE IMPORTANT

Statistics are sets of mathematical equations that are used to analyze what is happening in the world around us. You've heard that today we live in the Information Age where we understand a great deal about the world around us. Much of this information was determined mathematically by using statistics. When used correctly, statistics tell us any trends in what happened in the past and can be useful in predicting what may happen in the future.

Let's look at some examples of how statistics shape your life when you don't even know it.

1.Weather Forecasts

Do you watch the weather forecast sometime during the day? How do you use that information? Have you ever heard the forecaster talk about weather models? These computer models are built using statistics that compare prior weather conditions with current weather to predict future weather.

2. Emergency Preparedness

What happens if the forecast indicates that a hurricane is imminent or that tornadoes are likely to occur? Emergency management agencies move into high gear to be ready to rescue people. Emergency teams rely on statistics to tell them when danger may occur.

3. Predicting Disease

Lots of times on the news reports, statistics about a disease are reported. If the reporter simply reports the number of people who either have the disease or who have died from it, it's an interesting fact but it might not mean much to your life. But when statistics become involved, you have a better idea of how that disease may affect you.

For example, studies have shown that 85 to 95 percent of lung cancers are smoking related. The statistic should tell you that almost all lung cancers are related to smoking and that if you want to have a good chance of avoiding lung cancer, you shouldn't smoke.

4. Medical Studies

Scientists must show a statistically valid rate of effectiveness before any drug can be prescribed. Statistics are behind every medical study you hear about.

Genetics

Many people are afflicted with diseases that come from their genetic make-up and these diseases can potentially be passed on to their children. Statistics are critical in determining the chances of a new baby being affected by the disease.

6. Political Campaigns

Whenever there's an election, the news organizations consult their models when they try to predict who the winner is. Candidates consult voter polls to determine where and how they campaign. Statistics play a part in who your elected government officials will be

7. Insurance

You know that in order to drive your car you are required by law to have car insurance. If you have a mortgage on your house, you must have it insured as well. The rate that an insurance company charges you is based upon statistics from all drivers or homeowners in your area.

8. Consumer Goods

Wal-Mart, a worldwide leading retailer, keeps track of everything they sell and use statistics to calculate what to ship to each store and when. From analyzing their vast store of information, for example, Wal-Mart decided that people buy strawberry Pop Tarts when a hurricane is predicted in Florida! So they ship this product to Florida stores based upon the weather forecast.

9. Quality Testing

Companies make thousands of products every day and each company must make sure that a good quality item is sold. But a company can't test each and every item that they ship to you, the consumer. So the company uses statistics to test just a few, called a sample, of what they make. If the sample passes quality tests, then the company assumes that all the items made in the group, called a batch, are good.

10. Stock Market

Another topic that you hear a lot about in the news is the stock market. Stock analysts also use statistical computer models to forecast what is happening in the economy.

HOW TO MAKE MATHEMATICS CLASS INTERESTING

Make it meaningful

Many math courses suffer from the following issues:

1. The teachers don't know why they are teaching particular math topics, and they often don't know what else the students are learning in other subjects.
2. As a result, the students don't know why they learn those math topics, either. The common question, "Why do we have to learn this?", is a reasonable one. Do you have a good answer, beyond "It's in the exam" or worse, "Because it's good for you"?

Some possible ways to fix this:

• Find out where the students will use each math topic you teach (it may be in their science class, or some engineering subject). It's great when you can use actual examples from those other subjects and let the students know that's where they'll use each math topic.
• Help students make connections between the math topic and the "real world". If you're not sure how it's used in the real world - do a search!

Start with concrete examples - leave the abstract concepts to later

Math is largely about abstraction. Mathematicians for centuries have thought about real problems and come up with practical ways to solve those problems. Later, they have generalized the process, usually presenting the solution using algebraic formulas.
When students have no idea what the original practical problems actually mean, how can they be expected to understand the abstractions of those problems (using the formulas)?

Use watermelons to teach calculus

Instead of starting each topic with a formula, start with concrete examples of the problems that were originally solved using that math. Then, help the students see how the math theory can help to solve such problems by showing them the thinking behind the solution.

Start with an interesting, real-world problem (preferably localized)

Most math lectures start with "Here's the new formula for today, here's how you plug in values, here's the correct answer."
Problem is, there's no attempt to motivate the learners.
It is good to pique curiosity with a photograph, a short video, a diagram, a joke, or perhaps a graph. This trigger should outline an interesting problem in your local area (so students can relate to it better and feel more ownership).

Where you can, use computers to do the drudge work

Many math courses seem to be more about calculation rather than concepts. These days, it doesn't make sense for humans to spend hours learning how to calculate using complicated algebra.
To quote John Allen Paulos:
Mathematics is no more computation than typing is literature.
And then this one, usually attributed to Albert Einstein:
Computers are incredibly fast, accurate, and stupid. Human beings are incredibly slow, inaccurate, and brilliant. Together they are powerful beyond imagination.
For the vast majority of your students (who will not eventually become mathematicians), it's more important they understand the concepts and which process to use when confronted with different real problems. They then should learn how to use computer algebra systems (or graphics calculators) to solve such problems.
Some examples:

1. I observed a lesson recently where the math teacher got his students to calculate the standard deviation of a set of scores. He started with the formula then allowed the students to use Excel. He wanted them to apply the formula (rather than let Excel do it directly using STDEV, which is what many students actually did).The problem was, the students didn't even know what "standard deviation" meant. They were just plugging in numbers with little idea of what it meant. Finally someone remembered the standard normal curve and could explain what they were all doing. (See Probability Distributions - Concepts)
   The teacher should have made sure the students had the general idea first, including the general meaning of "distance from the mean".
2. When finding the length of a curve using calculus, it's actually not possible to find the answer using ordinary integration (because the integral does not exist). But using a computer algebra system, we can easily find the answer for such problems and spend more time understanding the problem and the solution given. (See Arc Length of a Curve.)
3. Creativity and ownership

Many math students feel very little ownership for what's going on.
They have little say in what the topics are (that's usual for most formal education) and the exact same assignments are given to everyone. It's not surprising there is little enthusiasm for such "one size fits all" approaches.

Computer-generated flower

We are all creative, and we all enjoy being creative, but in most school systems creativity is discouraged. 
There are many ways we can encourage creativity in math. Technology is one avenue - get students to use creative means to describe a mathematical concept (it could be a video, an animation, a diagram or perhaps a concept map).
Such individualized assignments get them thinking about the bigger picture, encourages creativity, and is more likely to generate feelings of ownership than the normal mass-produced assignment.

TEN CREATIVE WAYS TO TEACH MATHEMATICS

Here are some activities for your classroom to add a bit of sparkle and creativity. As children work, ask critical questions such as "Did you try this?" "What would have happened if?" "Do you think you could?" to enhance children's understanding of mathematical ideas and vocabulary.

1. Use dramatizations . Invite children pretend to be in a ball (sphere) or box (rectangular prism), feeling the faces, edges, and corners and to dramatize simple arithmetic problems such as: Three frogs jumped in the pond, then one more, how many are there in all?
    
2. Use children's bodies. Suggest that children show how many feet, mouths, and so on they have. When asked to show their "three arms," they respond loudly in protest, and then tell the adult how many they do have and show ("prove") it. Then invite children to show numbers with fingers, starting with the familiar, "How old are you?" to showing numbers you say, to showing numbers in different ways (for example, five as three on one hand and two on the other).
    
3. Use children's play. Engage children in block play that allows them to do mathematics in numerous ways, including sorting, serializing, creating symmetric designs and buildings, making patterns, and so forth. Then introduce a game of Dinosaur Shop. Suggest that children pretend to buy and sell toy dinosaurs or other small objects, learning counting, arithmetic, and money concepts.
    
4. Use children's toys. Encourage children to use "scenes" and toys to act out situations such as three cars on the road, or, later in the year, two monkeys in the trees and two on the ground.
    
5. Use children's stories. Share books with children that address mathematics but are also good stories. Later, help children see mathematics in any book. In Blueberries for Sal, by Robert McCloskey (Penguin, 1993), children can copy "kuplink, kuplank, kuplunk!" and later tell you the number as you slowly drop up to four counters into a coffee can.
6. Use children's natural creativity. Children's ideas about mathematics should be discussed with all children. Here's a "mathematical conversation" between two boys, each 6 years of age: "Think of the biggest number you can. Now add five. Then, imagine if you had that many cupcakes." " Wow, that's five more than the biggest number you could come up with!"
    
7. Use children's problem-solving abilities. Ask children to describe how they would figure out problems such as getting just enough scissors for their table or how many snacks they would need if a guest were joining the group. Encourage them to use their own fingers or manipulatives or whatever else might be handy for problem solving.
    
8. Use a variety of strategies. Bring mathematics everywhere you go in your classroom, from counting children at morning meeting to setting the table, to asking children to clean up a given number or shape of items. Also, use a research-based curriculum to incorporate a sequenced series of learning activities into your program.
    
9. Use technology. Try digital cameras to record children's mathematical work, in their play and in planned activities, and then use the photographs to aid discussions and reflections with children, curriculum planning, and communication with parents. Use computers wisely to mathematics situations and provide individualized instruction.
    
10. Use assessments to measure children's mathematics learning. Use observations, discussions with children, and small-group activities to learn about children's mathematical thinking and to make informed decisions about what each child might be able to learn from future experiences. Also try computer assessments. Use programs that assess children automatically.

ALGEBRA TEACHING TIPS

Solving Equations

After explaining the basic concepts in solving equations, I show students a simple equation to solve such as 2x + 4 = 8.  I tell students that the goal is to get x by itself on one side of the equation. So the first step is to subtract 4 from both sides, and we are left with 2x = 4. Most are able to follow this procedure. Then when I ask them what should be the next step, some students want to subtract the 2 from the 2x. So I explain that 2x means 2 multiplied by x, so to get x by itself we need to divide by 2. This satisfies some students but others remain confused.

I have achieved good results by turning this equation into a word problem: Two cantaloupes = $4. I ask “How much is one cantaloupe?” and they all answer $2. Then I ask “How did you figure this out?” This helps them realize that they had to divide both sides by 2 in order to get the answer.

Collecting Like Terms
I use a similar strategy when I am explaining how to add like terms. For example, if we have an expression such as 6x² + 2x⁴ - 2x² + 7 + 3x² - 3 and the goal is to combine like terms, I first ask three students to name their favorite fruit. If I get answers such as apples, oranges and bananas, I first rewrite this expression as 6 apples + 2 oranges – 2 apples + 7 bananas + 3 oranges - 3 bananas. Then I ask them to evaluate how many fruits of each kind are left. Most can figure this out as 4 apples + 5 oranges + 4 bananas.

Then I tell students that in math we also have different kinds of fruits but they have mathematical names such as x⁴, x² and just plain numbers. They are then usually able to make the connection and get the right answer.

Factoring Polynomials
Most algebra textbooks introduce factoring of polynomials before describing the purpose of factoring. I prefer to start by showing my students a graph of a quadratic equation which is a parabola, and explaining that the solutions of this quadratic are the x intercepts (they are already familiar with the concept of intercepts from linear equations). Then I indicate that there are three ways of solving this type of equation: 1) The quadratic formula, 2) completing the square and 3) factoring (when possible). Then the reason for factoring becomes more understandable in view of the ultimate goal.

Systems of Equations
Before showing the two methods of solving a system of two equations (substitution and elimination), I first write this equation on the board: x + y = 4.

Then I ask my students “What is x and what is y?” After the common answers of 2 and 2, 3 and 1 and 4 and 0, most of them soon realize that an equation in two variables does not have a unique solution, because there are an infinite number of possible answers. Then I write two equations like this:

x + y = 4
x - y = 2

I ask “What is x and what is y?” Most are able to figure out that x is 3 and y is 1. This makes the explanation that we need two equations to solve for x and y more understandable.

Finding the LCM (Least Common Multiple) of Rational Expressions
This topic is often quite challenging for students. What I try to do is clearly demonstrate the analogy with finding the LCM of numbers, which they have studied in an earlier chapter and know fairly well by the time we get to rational expressions.

In order to find the LCM of  12, 15 and 18, break each number down into its prime factors:

12:  3,2,2
15:  3,5
18:  3,3,2

So the LCM is 3 x 3 x 2 x 2 x 5 = 180

To find the LCM of x² - 4, x² + 4x + 4, 2x - 4, break each expression into its prime factors (after defining a prime factor for an algebraic expression):

x² - 4:  (x + 2) (x - 2)
x² + 4x + 4:  (x + 2) (x + 2)
2x - 4:  2(x - 2)

Using the same procedure we used for numbers, we pick each prime factor to the maximum number of times it appears in any of the three expressions:

So the LCM is 2(x + 2) (x +2) (x - 2)

10 POINTS FOR EXPLORING MATHS CREATIVELY

1. Empowers pupils to take ownership of their learning as active learners

It is no longer about the teaching but rather about the learning. The distinction reflects the important notion that pupils should be actively engaged in the learning process. Pupils are given the opportunity to enquire, investigate and choose from a variety of resources. They can direct the focus of their learning according to their interests and prior knowledge. Pupils’ motivation and expectations increase and so does their confidence in engaging with maths skills and concepts. 

2. Promotes investigative and problem-solving skills

Creative mathematics is all about developing problem-solving skills which enables pupils to solve unfamiliar mathematical problems creatively. Pupils realise that there might be more than one possible solution to solving a given situation and learn how to adopt diverse strategies towards problem-solving which best suit their learning styles, capabilities and situation. Pupils are also given the time, space and resources to explore mathematical skills and concepts and can devise their own path to a solution. 

3. Establishes connections to real life making learning more relevant

The notion of a classroom has been subject to strong competition with the real world beyond its walls, as well as the instantly accessible virtual world. In today’s information-based and highly globalised society, it is simply absurd to teach without acknowledging real data that is surrounding and bombarding us every second. Teaching and learning should be ever more connected and contextualised in real life circumstances. We cannot have pupils ask; “Why are we learning this?”. The more we establish links between learning and real life, the better can pupils apply their knowledge and skills, and regard the learning as valuable and relevant. Learning tasks should be more based on real life situations, enabling learners to tap into their prior knowledge whilst becoming more engaged with the task at hand. Such examples of realia include; menus, TV schedules, informative websites, transport information, published newsletters, promotional leaflets, sports websites, etc. 

4. Presents opportunities for collaborative learning and communication

Creative learning tasks entail the exploration of diverse learning modes which include collaborative group work. Pupils learn to work with other learners who have different learning abilities and together attempt to find a strategy on how to produce something or solve a given task. Throughout this process, the pupils are actively engaged in dialogue. They learn to verbalise their mathematical thinking and to consolidate their use of mathematical vocabulary. Pupils learn to enage in self-assessment and they evaluate their best capabilities, and assign different tasks of the project to specific members of the group in order to reach their final goal. Such tasks allow pupils to develop their social and communication skills, which prepares them for the future. It is the teacher’s responsibility to form functional group clusters by having diverse learners grouped together. Groups should be kept small, so that every student remains engaged and feels important to the rest of the team. One important tip is to assign specific roles within the student’s abilities, and whose duty is necessary in order for the group to reach their final objective or produce their desired outcome. 

5. Fosters initiative, innovation and creative thinking

Much focus is being placed upon the terms initiative and creativity, as part of the list of transversal skills required in the 21st century. Society needs citizens who are able to take initiative, who are good decision-makers, problem-solvers and who are able to be creative and think outside the box. Exploring mathematics creatively involves providing open-ended opportunities for our pupils to work collaboratively and to design innovative strategies and solutions to a given situation. This practise allows pupils to foster such important skills which allow them to thrive and to be better equipped for tomorrow’s world. 

6. Explores maths through technology

Our children are constantly surrounded by technology especially mobile touchscreen devices. From a very young age they seem to hold an instinctive disposition to interact with screens and to respond to visual cues. This exposure is enforced both in households as well as in other locations outside the home, such as shops, restaurants, shopping centers, etc. Technology has revolutionized the concept of education and has shifted the learning process to one which is more self-directed, creative and also game-based. Exploring maths creatively acknowledges and values the potential and vast resources which technology can provide us. Through technology, pupils learn various skills such as language, creativity, social skills, mathematical thinking, and problem-solving. Technology also provides models and opportunities for pupils to explore their learning through an appealing and relevant medium. One great example of creative maths through technology is the soaring use of coding, which can be carried out either using apps (such as online, mobile or tablet apps) or else through floor robots such as the Pro-Bot. Pupils have the possibility to learn how to programme, design, engage in content creation and problem-solve whilst exploring different skills related to maths such as shape, measure, fractions, angles and position. Coding skills are considered as a language in itself and it forms the basis of logical reasoning. Pupils can be given the opportunity to create their own problem-solving tasks which they can then share with their peers or other pupils from other schools. 

7. Supports pupils with diverse abilities

Adopting this approach towards exploring maths in real life, also serves to cater for the different students who are diverse in terms of learning abilities and preferences. Such an approach can be considered inclusive and through the continuous representations of mathematical situations as drawn from real life, pupils will have the chance to explore maths from different perspectives while the learning becomes more appealing. The variety of learning modes, enables most students to participate and to remain engaged on the task. 

8. Nurtures mathematical thinking and reasoning

Fundamental to mathematical learning is the ability to think and reason mathematically. It is important to present opportunities whereby the students are able to explore the process of problem-solving through mathematical thinking and reasoning. These situations also enable the learner to become better at communicating their thinking and in finding the appropriate vocabulary to explain their reasoning. Students are also able to observe that there might be more than one possible and reasonable solution to a given problem. Mathematical reasoning can be exemplied verbally, visually or through models.

9. Blurs the boundaries among different curricula areas

One very positive aspect about exploring maths creatively is that it does not only establish a more dynamic relationship among the teacher and the students, but it also establishes links with other learning areas, creating a multi-disciplinary approach to learning. Teachers can collaborate into providing project-based learning scenarios whereby students work in groups and explore a given situation or location and require a myriad number of skills in order to reach their goal or final product. Students learn to establish connections between maths and language, art, history, science, technology, physical education and other aspects of the curricula. 

10. Heightens understanding and retention

Hands-on experiences allow students to apply their learnt skills and concepts in practice. Such opportunities provide pupils with a repertoire of experiences which they can recall and allows them to become more confident at applying their knowledge in the future. Through hands-on practice, learners are able to self-assess where they require further support. These activities provide a meaningful context to learning and promotes retention of learnt maths skills and concepts.

Throughout my past years as a maths support teacher, I have had the possibility to visit many primary schools in my country. These visits implied opportunities to collaborate with class teachers and together explore innovative mathematics pedagogies, which appeal to different students and which attracts them towards participating in engaging maths activities. I have gained a lot of insight into the type of quality education which our pupils require not only during their young age but also to increase their participation in tomorrow’s world. We as teachers realize, that we are preparing our learners for a world which is rapidly changing, and for future jobs which are yet to be created. Our teaching has to remain abreast with what technology presents to us and valid to today’s rising generations. It remains, however, our duty to instill in our pupils a sense of love for learning and creating, and to recognize the relevance, inter relatedness and beauty of education in the real world beyond the classroom.