Lesson video

Hi there.

My name is Ms. Lambell.

You've made a really good decision to decide to join me today to do some maths.

Come on, then.

Let's get started.

Welcome to today's lesson.

The title of today's lesson is, "Checking and securing calculations with negative numbers," and this is within the unit, "Arithmetic procedures and index laws." By the end of this lesson, you'll be able to evaluate calculations involving positive and negative integers.

A couple of things that would be worth reminding you of, 'cause you may not have done negative numbers for it well, are additive inverse.

The additive inverse of a number is a number that, when added to the original number, gives a sum of zero, and also, absolute value.

The absolute value of a number is its distance from zero, so EG, five and negative five are both five away from zero, so they both have an absolute value of five.

Today's lesson is split into three learning cycles.

In the first one, we'll deal with addition and subtraction of negatives, in the second one, multiplication and division with negatives, and in the third one, we will look at a combination of any calculations involving negatives.

Let's get going with the first one, addition and subtraction with negatives.

When dealing with positive and negative numbers, we can use or visualize the double-sided positive and negative counters.

Each counter represents one.

Here are our counters.

The yellow side up has a value of positive one, and the red side up has a value of negative one, and the two together make a zero pair.

We also rewrite any subtraction as an addition of the additive inverse.

Eight add negative five.

Let's take a look at what this looks like in counters.

We've got eight positive counters, and we add five negative counters.

We can see, then, that actually, the first five are zero pairs, and so they cancel each other out to leave us with three positive counters.

Negative eight, add negative five.

Negative eight, there are eight negative counters, and we're adding negative five, so we add five negative counters.

The total counters we now have is negative 13.

The next one, eight, subtract five.

We're going to rewrite the subtraction as an equivalent addition.

We're going to rewrite this as eight, add negative five.

The calculation started off as eight, subtract five, so that's eight, subtract positive five.

The inverse of positive five is negative five, so we're going to add negative five.

With our counters, we've got eight positive counters, and we're adding five negative counters.

Again, the first five are zero pairs, so that leaves us with three.

Eight, subtract negative five, so again, we're going to rewrite the subtraction as an equivalent addition.

The inverse of negative five is positive five, so this is going to become eight add.

I could have put eight, add positive five, but I don't need to, so this is just eight add five.

There are my eight positive counters, and I'm adding five more positive counters.

I can now see that, in total, I have 13 positive counters.

I'd like you now to pause the video and match each subtraction to its equivalent addition, so any subtraction, we are going to write it as an addition of the additive inverse.

Pause the video.

When you've got your answers, come back.

Great work.

Well done.

Now, let's check our answers.

So negative 10, subtract seven is negative 10, add negative seven, 10, subtract negative seven is 10, add seven, 10, subtract seven is 10 add negative seven, negative seven, subtract 10 is negative seven, add negative 10, negative 10, subtract negative seven is negative 10, add seven, seven, subtract 10 is seven, add negative 10, and the final one, negative seven, subtract negative 10, is negative seven, add 10.

How did you get on with those? Great work.

Well done.

We can generalize this, generalizing the addition of two integers.

If both integers are positive or both are negative, we find the sum of the absolute values.

If they're both positive, our answer will be positive, and if they're both negative, the answer will be negative.

If one integer is positive and the other is negative, we find the difference of their absolute values.

If the absolute value of the positive number is larger, then it's going to be a positive answer, and if the absolute value of the negative number is larger, then our answer will be negative.

Let's see how this applies to some different questions.

0.

6, subtract 2.

8.

We're going to write the subtraction as an equivalent addition.

This subtraction is 0.

6, subtract positive 2.

8.

The inverse of positive 2.

8 is negative 2.

8, so we rewrite it as 0.

6, add negative 2.

8.

Here, we have one positive and one negative, therefore, we're gonna find the difference of the absolute values.

The negative number has a larger absolute value, meaning our answer is going to be negative, so our answer is going to be negative 2.

2.

The difference between 2.

8 and 0.

6 is 2.

2.

Now, let's take a look at this one, negative 0.

6, subtract 2.

8.

Again, we're going to rewrite that subtraction as an equivalent addition.

It's going to become negative 0.

6, add negative 2.

8.

This time, both values are negative, therefore, our answer will be negative, and we find the sum of the absolute values.

The answer, therefore, is 3.

4.

The sum of 2.

8 and 0.

6 is 3.

4.

Both values were negative, so my answer will be negative.

Let's take a look at this one, negative 0.

6, subtract negative 2.

8.

Rewrite the subtraction as an equivalent addition.

The inverse of negative 2.

8 is positive 2.

8, so this is going to become negative 0.

6, add 2.

8.

Here, we have one positive and one negative, therefore, we're going to find the difference between the absolute values.

The positive number has a larger absolute value, meaning that our answer is going to be positive.

The answer to this is 2.

2.

I'd like you now to have a go at applying that to these questions.

You've got five questions on the right hand side, and I know there are only four on the right hand side, so two of them will go to one answer.

Pause the video, no calculators, you need to be able to do this without a calculator, and I'll be here waiting when you get back.

Super.

Let's check those answers, so negative 100, subtract 30 is negative 130.

We would add negative 30.

Negative 100, subtract negative 30 is negative 70, negative 100, add 30 is negative 70, 100, add negative 30 is 70, and 100, subtract negative 30 is 130.

Now, you can have a go at task A.

You need to calculate the answers to the following and decode the words.

Please, no calculators.

Make sure that you change all of your subtractions into an equivalent addition.

When you've worked out what the words are, you're then going to write down as many calculations as you can that have an answer of negative 100.

Good luck with this, pause the video, and then when you come back, hopefully, you've revealed the mystery words.

Good luck.

Now, let's check those answers.

You should have revealed the mystery words were absolute value, so A was 84, B, negative 84, E, negative 60, L, 60, O, negative four, S, four, T, negative 48, U, 48, and B, negative 40, and there are some examples there of some calculations that you might have.

You've probably got different ones, so now, I will allow you to pause the video and check your answers on your calculator.

Let's move on now, then, to multiplication and division with negatives.

Let's have a look at these products.

I'm gonna give you a moment just to look through all of those products.

Andeep says, "I've noticed that if both numbers are positive or both are negative, the answer is positive, and if there is one of each, it is negative." Do you agree with Andeep? Yes, Andeep is correct.

Generalizing for products or quotient of two integers.

If both integers are positive or both negative, then the product or quotient is the absolute value of the product or the quotient, so for example, three multiplied by four equals 12, and negative three multiplied by negative four is also 12, and 24 divided by four is six, and negative 24 divided by negative four is six.

If there is one positive and one negative integer in the product or quotient, then the answer will be the negative of the absolute value, so for example, negative three multiplied by four would be negative 12.

Three multiplied by negative four would also be negative 12.

Negative 24 divided by four is negative six, and 24 divided by negative four is negative six.

If the signs are both the same, then it is going to be a positive answer.

If the signs are different, it's going to be a negative answer.

Here, I'd like you to match each question to the correct answer.

Again, no calculators, and some of the answers will be the answer to more than one question.

Pause the video, and when you've got your answers, come back.

Well done.

Now, we can check those answers.

First one, negative two multiplied by three is negative six, negative 24 divided by three is negative eight, negative 24 divided by negative four is six, four multiplied by negative two is negative eight, negative eight multiplied by negative one is eight, and negative 32 divided by negative four is eight.

So we look at the first one, we had different symbols, we had a negative number and a positive number, so my answer was negative, the second was negative, and then the third would be positive, so you're looking for those symbols.

Hmm, this is interesting.

What have we got here? Well, Sophia says, "The answer to this will be negative, as there is an odd number of negatives." Do you agree with Sophia? Jun says, "Can you decide based on the number of negatives?" Yes, when we're multiplying or dividing, an even number of negatives will give a positive answer, and an odd number of negatives will give a negative answer.

True or false? The answer to the following is negative.

All I want you to do is decide if the answer is positive or negative.

You don't need to work out an exact answer, so the answer to the following is negative.

Is that true or false? And then decide on what your justification is.

Pause the video.

When you've made your decision, you can come back.

What did you decide? The correct answer was false, and the justification was, there is an even number of negatives.

There's an even number of negatives.

We've got negative three, negative four, negative two, negative seven.

That's an even number, 'cause that's four, so therefore, our answer is going to be positive.

Now, I'd like you to identify all of the following that will give positive answers, so you don't need to work out the exact answer.

I just want to know which ones will give a positive answer.

Pause the video, and come back when you're ready.

What did you decide? Positive answers, A, because I've got two negatives.

B is not positive, C is not positive, but D is, so we are looking for any calculation that has an even number of negative numbers, so A has two, and D has four.

Now, you can have a go at this task.

You need to fill in the missing integers, and this is a times table grid, so remember, no calculators.

I'd like you, please, to pause the video, and then, when you've managed to fill in all of the boxes, you can come back.

Well done.

Now, pause the video and check your answers carefully.

Great.

Now let's move on, then, to our final learning cycle for today's lesson, and that is calculations involving negative numbers.

If A equals eight, B equals negative 10, we need to write the following in order from smallest to largest.

Let's substitute in our values into our four algebraic expressions.

B squared would be negative 10 squared.

Notice I've got my negative 10 in a bracket, because it's negative 10 that's being squared.

A over B would be eight over negative 10, which I would prefer to write as negative eight over 10.

B, subtract A will be negative 10, subtract eight, and A, subtract B will be eight, subtract negative 10.

Do we need to calculate the exact answer to order these? No, we don't.

We do not need to calculate the exact answers, and I'm gonna show you on the next slide how we can do that.

Here are each of our expressions with their values substituted in.

Firstly, we can decide which ones are positive and which ones are negative.

You make a decision before I go through it.

Which are positive, and which are negative? Let's have a look, so B, that's negative 10, multiplied by negative 10, it's the product of two negatives, so my answer must be positive.

Clearly, the next one is a negative number, because it's a negative fraction.

If we take a look at the next one, we've got negative 10, and we would be adding negative eight, so that is going to be a negative answer, and the final one is eight, and we'd be adding 10, because we're changing it to the additive inverse, remember, which is positive.

We now know which are positive and which are negative, so we just need to compare those pairs.

Here are my two negative values.

Which one is larger, negative eight over 10 or negative 10, subtract eight? And negative eight over 10 is larger.

We can see that negative eight over 10 is the larger because it is closer to zero, and that is how we decide if a number is smaller or larger when we're talking about negative numbers.

Let's take a look at our two positive ones, negative 10 squared and eight, subtract negative 10.

So negative 10 squared is clearly greater than eight, subtract negative 10.

Now, we can write them in order from smallest to largest, so we've got B minus A, A over B, that's my two negative ones, and then my positive ones would be A minus B and then B squared.

True or false? If A is a positive integer, negative A cubed is greater than negative A squared.

I'd like you to decide on true or false, but also be able to justify your answer.

Pause the video, and when you've got an answer, come back.

What did you decide? Well, the correct answer was false, and the correct justification was B.

If I've got negative A cubed, I would have an odd number of negatives, therefore, my answer is going to be negative.

If I'm squaring, I've got two negative numbers, my answer is positive, and we know that negative numbers are smaller than positive numbers, so the correct justification was B.

Now, we can start our final task for today's lesson.

Did you hear about the mathematician who was afraid of negative numbers? You need to make sure you place an apostrophe in the correct place.

You need to work out the answers and then find them in the grid and then write down the letter, and it will answer the joke.

When you're done, come back, and we will move on to question number two.

Pause the video now.

Hope you enjoy this, Good luck, and remember, no calculators.

And question number two, A is a positive integer, B is a negative integer.

B has a greater absolute value than A.

Write the following in order from smallest to largest.

A bit more challenging, this question, but I know that you can do it.

Pause the video, and then come back, and we will check our answers.

Well done.

Question number one, then.

Did you hear about the mathematician who was afraid of negative numbers? They'd stop at nothing to avoid them.

Hopefully, that was the answer you got, and we should have put an apostrophe between the Y and the D in they'd.

Question number two, they should be in this order, and I've also shown you which are positive and which are negative.

The smallest would've been B cubed, then B, subtract A, A over B, A, subtract B, B squared, and then B to the power of four.

We can now summarize the learning from today's lesson.

Calculations involving negative numbers will come up in many topics in mathematics.

It is important that you are secure in understanding how to calculate with them.

Here, we've got our two generalized versions, so remember, when we're adding, and we only need to consider adding, because we are going to change any subtraction to its equivalent addition.

If both numbers are positive or both are negative, we find the sum of their absolute values, and if they're both positive, the answer will be positive, and if they're both negative, the answer will be negative.

If we have one positive number and one negative number, then our answer is going to be the difference between the two.

If the absolute value of the positive number is greater than the absolute value of the negative number, your answer will be positive, and if your absolute value of the negative number is greater than the absolute value of the positive number, your answer will be negative.

And then, when we're looking at multiplying and dividing, we're looking to see whether our answers are positive or negative.

If they are both positive or both negative, then our answer will be positive, be the positive absolute value, and if there's one of each, if there's one negative, one positive, it will be the negative of the absolute value.

We also did consider that if you have an odd number of negative values in a calculation, your answer is going to be negative, and if you have an even number, it's going to be positive.

Well done on today's lesson.

I hope you laughed at the joke, but you probably didn't, let's be serious.

Look forward to seeing you again.

Take care.

Goodbye.