Lesson video

Hi, I'm Mrs. Wheelhouse, and welcome to today's lesson on further algebraic terminology.

This lesson is part of our unit on algebraic manipulation.

Now algebra can be incredibly useful, so let's get started, and see how we're gonna be using it today.

By the end of today's lesson, you'll be able to appreciate the difference between an equation, an identity, an expression, a term, and a formula.

A quick recap of a keyword here, and that's term.

Now, term is a single number, or letter, or the product of numbers and/or variables, and each term is separated by the operators, plus and minus.

Here are some examples of terms. Expressions contain one or more terms, where each term is separated by an operator.

An equation is used to show two expressions that are equal to each other.

An identity is an equation that holds true for all values of the variables, and we use this symbol so it's like equals, but with another horizontal line.

And we use it to show two expressions are equivalent, and form an identity.

A formula is a rule linking sets of physical variables in context.

A numerical coefficient is a constant multiplier of the variables in a term.

A coefficient in general is any of the factors of a term, relative to a given factor of the term.

And a constant is a term that does not change.

In other words, it contains no variables.

Our lesson has two parts today, and we're gonna begin by looking at algebraic terminology.

In mathematics, letters are often used to represent unknowns or variables.

Individual letters or numbers, or letters and numbers multiplied together are called terms. Here are some examples of terms. 2t, a, 12, negative 30, c squared, xyz, 2/5.

And here are examples that are not terms. So for example, 3x plus 2y is not referred to as a term.

Nor is 5 subtract a.

We've then got the addition symbol, well that's an operator.

And we can also see the equals sign.

Now we form expressions from multiple terms. Being able to use correct language allows us to be more precise when we write mathematically.

Below is an expression made up of three terms. So the entire thing is referred to as an expression, and here are the three individual terms. We have 3x, 5y, and 7.

The variables in this expression are x and y.

The coefficients of these variables are 3, and 5 respectively.

We would read that as the coefficient of x is 3.

The coefficient of y is 5.

Terms which have a fixed value because they don't contain variables, are called constants.

So for example, you can see 3, 8.

7, 1,970.

And one that you might not have realised is a constant, and that's pi.

But pi has a fixed value so it's definitely a constant term.

Some non examples, you can see we've got x, 4a.

All of these terms contain variables, so they are not constant.

The constant in the expression at the top of the screen is 7.

What are the coefficients of x and y in this expression? When we have exactly one lot of a variable, we do not write the 1, and that's because we're being efficient.

So x subtract 4y is the same as 1x subtract 4y.

So the coefficient of x is 1.

When we subtract a term, it is the same as adding the negative term.

So x subtract 4y is the same as saying 1x plus negative 4y.

And in that way we can see the coefficient of y is negative four.

What do you think the terms are in this expression? Well, we could refer to each of these as individual terms, so 2x is a term, 3y is a term, and we could refer to the 5 as a term.

However, we can write this fraction as follows.

And this is the same as saying 2/5 of x plus 3/5 of y.

And now we could say that there are two terms for this expression.

So 2/5 of x is one term, and 3/5 of y is the other term.

What are the coefficients of each variable now? Well the coefficient of x is 2/5, and the coefficient of y is 3/5.

We could treat the entire numerator like a term to allow us to manipulate expressions and equations.

For example, if I wanted to manipulate these, I could write this as follows.

This is in fact four lots of 2x plus 3y.

So in other words, we treated the 2x plus 3y as a term, and then we grouped like terms. Sam wants to write this expression as a single multiple of x.

What is the coefficient of x this time? Well, let's consider the expression.

2 negative 3y and 7 are the coefficients.

We can add the coefficients of x to write this as a multiple of x.

In other words, 2 subtract 3y add 7 lots of x.

But then I can collect the 2 and the 7 because they are like terms and write this more efficiently as 9 subtract 3y lots of X.

So the coefficient of x is now 9 subtract 3y.

Time for a quick check.

Fill in the blank please.

p plus 2ab subtract 6xy subtract 0.

6 is an example of an equation, an expression, a term or a variable? Pause the video and make your choice now.

You should have said it's an example of an expression.

You can see we've got multiple terms here, and we can see more than one variable.

And it's not an equation because it's not equal to anything.

In the expression p plus 2ab subtract 6xy subtract 0.

6, What I'd like you to do is to write down the constant term, the coefficient of p, the coefficient of ab, the coefficient of b and the coefficient of x.

Pause and do this now.

Welcome back.

Let's check and see how you got on.

Well, the constant term is negative 0.

6, because that's the term that contains no variable.

The coefficient of p is 1.

Remember, for efficiency we don't tend to write this, but we do need to know it's there.

The coefficient of ab is 2, and the coefficient of b is 2a.

The coefficient of x is negative 6y.

We can form expressions for scenarios using mathematical conventions.

Lucas is a years old.

In other words, we don't know how old he is so we're using a letter to stand for his age.

This year his dad is three times his age.

His granddad is double his dad's age, and his aunt is 50 years younger than his granddad.

His sister is half his aunt's age.

Gosh, that's quite a mouthful, but we can write this more simply if we use our algebraic notation.

Let's see what that looks like.

Well, his dad is three times his age.

If Lucas is a years old, his dad is 3 lots of a.

Now his granddad is double this so so 2 lots of 3a is equivalent to 6a.

His aunt is 50 years younger than his granddad.

Well that's 6a subtract 50.

And his sister is half his aunt's age.

Well that's 6a, subtract 50, then divide by 2.

Now you can of course simplify that to be 3a subtract 25 by halving both terms. When two expressions are equal, we can form an equation.

You may have seen equations such as these.

Equations can have multiple variables, and they can have exponents.

You may have seen equations for straight lines, or equations of parabolas.

Sometimes it is possible to find a solution to an equation, and some equations have more than one solution.

What do you notice about this equation? Any value of x can be substituted to make this equation balanced.

This means that x plus x and 2x are identical expressions.

They're just different ways of writing exactly the same thing.

Now, when two expressions are identical, regardless of the value of the variable, they form an identity.

This is an identity, so we should write it with the identity symbol.

That's better.

Let's do a quick check.

Andeep is p years old.

His dog is half his age, and a year younger than his sister.

Which expression represents the age of Andeep's sister? Pause and make your choice now.

Welcome back.

You should have picked A.

Remember his dog is half of his age, so that's p divided by 2, and the dog is a year younger than his sister.

So in order to get to the age of Andeep's sister, we need to add on 1.

True or false? x plus x plus x is equivalent to 2x plus 5.

In other words, that's an identity.

Is that true or false? Pause and make your choice now.

Well done if you said false, but it's now time to justify your answer.

So is it A, there is no value for x, which makes both sides of the equation equal, or is it B, this is not true for every value of x? Pause and pick which one justifies your answer of false.

Welcome back.

You should have said B.

There is indeed a value for x, which does make both sides equal, and that is when x is 5.

But therefore it's not true for every value of x.

So it's not an identity.

Time for your first task.

Please sort the algebraic statements into the boxes for equations, expressions, and identities.

When you've identified an identity, then you should rewrite it using the identity symbol.

Pause and do this now.

Welcome back.

This is where you should have grouped the following algebraic statements.

What I suggest you do is pause the video, so you can check your groups with the ones on the screen.

Pause and do this now.

Welcome back.

Did you get them right? Well done if you did.

It's now time for the second part of our lesson, and that's on mathematical formulas.

A formula is a rule linking two or more physical variables in context.

Formulas show how things are related to each other, and we can use formulas to calculate one variable given values for the other variable or variables.

What formulas do you know? Pause the video, and have discussion with the people around you.

Here are some you may have come across before.

Did you say any of these? Now you will see and use formulas in many subjects and situations.

So it's not just maths, in particular science.

You are gonna use formulas a lot there.

And there are a lot of jobs that use formulas to calculate various things such as, for example, electricians.

So these are our formula, and they calculate going down, average speed, volume of a cuboid, formula for the area of trapezium, formula for the area of a triangle, the circumference of a circle.

For right angle triangles we have here Pythagoras' theorem, so calculating the lengths of any of the sides.

And then the final formula is for converting degrees Celsius into Fahrenheit.

Now we can write a formula to generalise a range of situations.

For example, there are seven days in a week.

How could we write a formula for calculating the number of days in any given number of weeks? Well, let's think about this.

And I agree with Sofia here.

"I like to think about an example, and work out how I calculate that." So it's gonna help me write my formula.

"Okay," says Jun, "Let's say I wanna calculate how many days there are in five weeks." Sofia says, "That would be 35, and I would find this by doing 7 multiplied by 5." So there's the number of days in a week multiplied by the number of weeks.

So as a formula to find the total number of days I do 7 multiplied by the number of weeks, and that 7's never changing because there are 7 days in a week.

Ah, this is definitely a lot easier now to spot where my variables are going to go.

Number of days was d, and number of weeks was w.

So d is equal to 7w.

Jun says, "I've written a formula for the amount of money I spend in the school canteen." What's the problem with Jun's formula at the moment? We don't know what the variables represent.

We don't even know what units he's measuring cost in.

Is his cost in pounds, pence? We've got no idea.

Ah, Jun's now told us c is cost in pounds, m is meals, d is drinks and f is fruit.

Well, that's better, but Jun needs to be far more precise with what the variables represent.

For example, he said m is meals, but what does that mean? It would be better if he'd said, m is the number of meals he buys.

Ah, that looks better.

So what does the number 0.

4 represent? That it costs 40 pence for a piece of fruit.

Ah, that makes things a little easier.

Right so how much does a drink cost? Well, it costs £1 per drink because if we just have d written there, it means I have 1d, and therefore it must be £1 per drink.

Now we can write a formula for any scenario.

For example, let's consider the cost of booking a hotel.

How can we work out the cost of staying for 3 nights? Well, if I wanted to stay for 3 nights, I'd do 100 times 3 because it's £100 per night, and then add on the £80 booking fee.

What about if I wanted to stay for any number of nights? Well, it's 100 multiplied by the number of nights I stay for, so I'm gonna use n for that.

So a 100n and then add 80.

So in other words, the cost for the hotel is equal to 100 lots of n where n is the number of nights, and then add on 180 because 80 is that booking fee.

Let's do a quick check.

There are 60 seconds in a minute, and 60 minutes in an hour.

Which of the following is a formula for calculating the number of seconds in any number of hours? And Sofia points out, you can use an example to help you work out which of these would be the correct formula.

Pause the video, and do this now.

Welcome back.

Did you spot that it's D? We wanted to calculate the number of seconds.

So that told us it was gonna be s equals, and then I had to correctly work out whether it was going to be 120 lots of h, or 3,600 lots of h.

Well, 60 times 60 is 3,600.

So it's definitely the bottom one.

Time for your final task.

For Question 1, match the words at the bottom of the page to their correct definitions.

Pause and do this now.

Question 2, there are 24 hours in a day, which of the following shows the correct relationship between the number of days and the number of hours? Question 3, there are 52 weeks in a year.

Which of the following shows the correct relationship between number of weeks and number of years? And Question 4, the time to cook a chicken is 40 minutes per kilogramme plus 20 minutes.

Which of these is a formula for the cooking time for a chicken? So for each one you just need to select the correct formula from the options that are given.

Pause and do this now.

Question 5, electricity prices are calculated with a cost per day, and a cost per kilowatt hour.

Aisha's family have just changed energy suppliers, and the new company charges 28 pence per kilowatt hour, and 54 pence per day.

So for Question A, please write a formula for the total electricity cost.

In B, tell us what the variables represent in your formula.

And C, use your formula to work out how much 30 days of electricity would cost.

And we're gonna tell you that we've used 225 kilowatts per hour in that time.

And then D, in a year, 365 days, it costs Aisha's family £1,037.

05.

Write an equation using this information.

Pause and do this now.

Welcome back.

Let's go through our answers.

So 1A was an expression, B is an identity, C, an equation, D, a formula, and E, a term.

Question 2, you should have selected the formula, h equals 24d.

For Question 3, you should have selected w equals 52y, and then for 4, you should have selected that the cooking time is equal to 40 lots of a plus 20.

Question 5, you had to write a formula for the total electricity cost.

Now it's really up to you as to whether or not you chose to write this where we calculate the cost in pence, or if you calculate the cost in pounds.

If you went for c equals 28a plus 54b, then you're doing the cost in pence.

And if you went for c equals 0.

28a plus 0.

54b, then you went for calculating the cost in pounds.

Now, you could have used any letters at all, and it's absolutely fine if you picked different letters to me.

Part B, you had to state what the variables represented.

So because I chose to write c equals, that variable by itself that was the cost in either pence or pounds, and then whatever you multiplied 28 or 0.

28 by that was the amount of energy used in kilowatt hours.

And then the other variable had to represent the number of days.

For C, you had to use your formula to work out how much 30 days electricity would cost.

And it doesn't matter which formula you used, you should have reached that it was £79.

20.

In D, you had to write an equation using the information that in 365 days it would cost Aisha's family £1,037.

05.

So what you see here is me having used the formula where the cost was in pounds to write my equation.

So I had 0.

28a plus 197.

1 equals 1,037.

05.

It's time to sum up what we've learned today.

A term is a single number or letter or the product of numbers and or variables.

An identity is an equation which is always true regardless of the values of the variables.

A formula is a mathematical rule connecting two or more physical variables in context.

Expression, equations, identities, and formulae can be identified by key features.

Formulas can be written for mathematical relationships, and scenarios.

Well done.

Done a great job today learning the correct algebraic terminology to allow us to talk about the algebraic statements that we write when we're trying to generalise.

I look forward to seeing you for more lessons in our algebraic manipulation unit.