Lesson video

Hi, everyone.

My name is Miss Coe, and I'm really happy to be learning with you today.

It's going to be a fun, interesting, and challenging lesson in parts, but don't worry, I am here to help.

You will come across some new keywords and maybe some keywords you've already come across before.

We're going to work really hard today, but I am here to help and we can learn together.

In today's lesson from the unit Arithmetic Procedures with Integers and Decimals, we'll be looking at problem solving with integers and decimals.

By the end of the lesson, you'll be able to use your knowledge of calculating with integers and decimals to solve problems. Now, there's a few key words that you may or may not know.

So let's look at the word integer.

Well, an integer is any positive or negative whole number or zero.

We'll also be looking at the word additive inverse.

Now the additive inverse of a number is a number that when added to the original number gives the sum zero.

We're looking at the word centi.

Now, remember, centi placed before the units means 1 over 100.

We'll be looking at the word milli.

Remember, milli placed before the unit means 1 over 1,000.

And we'll be looking at the word kilo.

Kilo placed before a unit means 1,000 times.

So in our problem solving lesson, we'll be breaking it up into two parts.

The first, we'll be looking at integers, negatives, and decimals in the real world.

And the second one, we'll be looking at measurements and accuracy.

So let's have a look at integers, negatives, and decimals in the real world.

Now, decimals and integers are used so much in the real world.

Can you think of some examples of some real life decimal numbers you've possibly seen today? Well, there's lots of different real life examples out there.

So here's just a few: money, price of petrol, weighing ingredients, and temperature are just some examples of how we've seen decimals and integers used in the real world.

It's important we can use our skills on addition, subtraction, multiplication, division in the real world, given the fact that they appear so much in real life.

It's equally important when we know which operation to apply given the context of the question.

And this is what some people usually struggle with.

When they look at a question, they're just not quite sure what to do.

So without using a calculator or calculating any answer, and we'd have a think, which operation do you think we apply in this question? Is it addition, subtraction, multiplication, or division? Here we have Lucas and Lucas goes camping with his family.

On Saturday morning, the temperature was negative 4.

7 degrees Celsius.

On Sunday morning, the temperature is 8.

9 degrees Celsius, and he wants to find the difference in temperature from Saturday morning to Sunday morning.

So which operation do you think we're applying in this question? Is there a key words that will help you identify which operation it is? Well, hopefully, you've spotted, its subtraction because we're using the word difference.

So this means our calculation is 8.

9 subtract negative 4.

7.

So now we know our calculation is 8.

9 subtract the negative 4.

7, we're going to use additive inverses.

So we have 8.

9 add the additive inverse of negative 4.

7 is positive 4.

7.

So I want you to work out the difference in temperature between Saturday and Sunday morning.

A nice method to work out the answer is using the column method, 8.

9 add our 4.

7 gives 13.

6.

So the key with problem solving questions is really to break it down into little stages.

So the final answer in the difference between Saturday and Sunday morning is 13.

6 degrees Celsius.

Let's have a look at another type of question.

Same again, I don't want you to calculate anything or work out any answers.

I just want you to think about which operation are we applying to this question, addition, subtraction, multiplication or division? Here a school uses a cashless thumbprint machine for the students to buy food and drink in a school, and Laura has 3 pound 76 credit in her account.

Now she spends 2 pound 34 and then she spends another 1 pound 27.

So which operation do you think we're applying here? Well done.

So hopefully, you spotted, its subtraction again as she's spending, which means our calculation is 3.

76 subtract 2.

34 subtract 1.

27.

So given the fact that we're using subtraction, let's use our knowledge on additive inverses again, 3.

76 add the negative 2.

34 add the negative 1.

27.

Now what I want you to do is see if you can work out how much money Laura has left in her account.

Show you're working out and you can use the column method if it helps.

Press Pause if we need more time.

Well done, so let's see what you've got.

Well, summing up those negatives, the negative 2.

34 and the negative 1.

27 gives us a final answer of negative 3.

61.

I've used the column method here.

And we know we have our positive 3.

76, so that means our calculation would be 3.

76 subtract 3.

61 which gives 0.

15.

So that means Laura only has 0.

15 or 15 pence left in her account.

This is a good question, but the key to problem solving is completing it in little stages.

Now let's have a look at another question.

Without calculating anything or working out the answer, I want to have a look at which operation we're using here.

The question says, an Oak Academy teacher fills up their car with 23 litres of unleaded petrol.

Which operations do you think we're applying here? Well, hopefully, you've spotted its multiplication and the calculation is 1.

56 times 23.

So remember there's a few different methods to multiply decimals.

For me, we're going to use the multiples of 10 strategy, so we can calculate 156 times 23 and then multiply by 0.

01.

So what I want you to do now is I want you to work out how much the Oak Academy teacher will pay for those 23 litres of unleaded petrol.

See if you can work it out and press Pause because you'll need more time.

Great work, so let me show you my working out.

Here, I've worked out 156 times 23 using this method here.

Then summing up my values, I have a final answer of 3,588.

But remember, I wanted 1.

56 times 23.

So knowing that 156 times 23 is 3,588, to make the calculation 1.

56 times 23, I'm going to multiply it by 0.

01, thus to give me 35.

88.

So that means the teacher paid 35 pounds and 88 pence.

Well done if you got that one right.

It's important to remember with problem solving questions, sometimes we're required to use a combination of operations to solve problems, and there are lots of different ways to calculate an answer.

Don't worry if your method is slightly different as long as the right answer is obtained.

A structured approach to problem solving questions is essential though.

So let's have a look at a question.

Here, a shop sells pens and pencils, and Izzy buys three pencils and four pens with a 10 pound note.

What operations do you think we're using to calculate how much change she receives? Have a little think of which operations we're using.

Well, hopefully, you've spotted, we're using multiplication and then we're using subtraction.

We have to use multiplication to find out the cost of the pencils and the cost of the pens and the subtraction will be the change from the 10 pound note.

So what I want you to do is show all your working out and calculate how much change Izzy receives from her 10 pound note.

See if you can give it a go.

Show all your working out and press Pause as you'll need more time.

Great work, so let's see how you've gone on.

For me, multiplying 3 by 0.

76 tells me how much those three pencils cost.

4 multiply by 1.

56 tells me the cost of those four pens.

I'm going to work out 3 multiply by 0.

76 as 0.

7 times 3 and 0.

06 times 3 to give me 2 pound 10 at 18 pence, which is 2 pound 28.

Remember, there's lots of different ways in which you can work it out.

This is just my choice.

Now, 4 times 1.

56.

I'm going to do 2 times 2 times 1.

56, which is the same as 2 times 3.

12, which is 6 pounds 24.

Remember, there's lots of different ways in which you can work out the answer to 4.

156.

I'm just choosing this way.

So that means 10 subtract the 6 pound 24 and the 2 pound 28 is 10 subtract the 8 pound 52, which gives me 1 pound 48.

So Izzy receives 1 pound 48 change.

Well done if you've got this one right.

And remember, don't worry if you're working out slightly different as long as you've got the right answer.

Now what I want you to do is have a look at this question.

Remember, I don't want you to work out the answer, I just want you to identify which operations are we applying to this question.

Sam wants us to work out the mean average time spent on Matt's homework, and he asks eight friends from across the world how many hours they spend on their homework each week.

One person says 8.

3 hours.

Another one says 3.

5 hours.

Another one says 1.

2 hours.

Another friend says 3.

22 hours.

Then we have 8 hours, 4.

5 hours, 4.

25, and 3.

75.

What operations do you think we're applying to work out the mean average time here? Well done, to work out the mean average time, it means we need to sum the values and then divide by the number of values.

So what we'd like you to do is work out the mean time spent on homework of Sam's eight friends.

Ensure you show all your working out.

Great work, so let's see how you've got on.

For me, I'm going to add up all my values using the column method.

So you can see my working out here.

I've carefully lined up all my ones, my decimal places, tens, a hundreds, so on and so forth.

That's given me 36.

72.

Now, with 36.

72, I've divided it by the 8.

So that gives me 4.

59.

I've used a short division approach here.

So that means the mean hours spent on homework is 4.

59 hours.

Well done if you've got that one right.

Now, let's have a look at your task.

And remember, when approaching problem solving questions, break it up into little parts, identify the operations you need to use, and then work out each part needed.

For question 1, Aisha has 13 pounds 45 credit on her school account.

She spends 2 pounds 67, then she spends 7 pound 89.

She then puts back 4 pounds 21 into her account.

How much does Aisha have in her school account now? So think about the operations you've gotta use in that question.

Question 2, Jun has a 20 pound note.

He buys four green pens, six pencils, and two orange highlighters.

The shopkeeper gave Jun 6 pound 16 in change.

Is this correct? And you have to show you're working out.

So think about those operations you'll be using.

See if you can give these a go and press Pause if you need.

Well done, so let's move on to the next question.

Question 3 says, there are two places in Antarctica where scientists want to investigate.

They look at the temperature in the morning over five days.

Location one has these temperatures and location two has these temperatures.

Now from here, we're going to use the mean average to decide which is the coldest location.

See if you can give it a go and press Pause if you need.

Really well done, so let's move on to question 4.

Question 4 states, Andeep is taking part in a local Maths competition.

And there are 10 questions.

Now each correct answer gets 8 points.

Each incorrect answer gets negative 6 points.

Each question not answered gets negative 2 points.

Andeep answers four correct questions, three incorrect questions, and does not answer two questions.

How many points does he have? See if you can give it a go and press Pause if you need.

Part B says, in a different competition, Lucas gets 36 points.

Now there were eight questions in this competition and for each correct answer, he gets 12 points.

Each incorrect answer, he gets negative 4 points, and each question not answered, gets negative 2 points.

We're asked to give some examples of how Lucas performed in the competition.

See if you can give it a go and press Pause if you need.

Great work, well done.

So let's see how you got on.

Well, for question 1, hopefully you realise, she starts with 13 pound 45 credit.

Then she spends, so that's a subtraction of 2.

67.

Then she spends 7 pound 89, so that's a subtraction of 7 pound 89, and then she puts back 4 pound 21.

So that's an addition of 4.

21.

Changing the subtraction to the addition of the negative inverses gives 13.

45 add negative 2.

67 add negative 7.

89 add 4.

21.

Summing our negatives gives us negative 10.

56.

And summing our positives, we have 17.

66.

Therefore, 17.

66 add the negative 10.

56 gives us 7.

1.

So she has 7 pounds 10 pence left.

Really well done if you got that one right.

For question 2, well, hopefully you've realised we need to work out 4 times 1.

56, which is 6 pound 24, 6 times 0.

76 is 4.

56, and 2 times 1.

27 is 2.

54.

Summing these together gives the total price 13 pounds 34 for all the pens and pencils and highlighters.

So subtracting that from 20 gives us 6 pounds 66.

So the change is incorrect as Jun should have got 6 pound 66 and not 6 pound 16.

For question 3, you had to sum all the negatives to give us minus 115.

75, and then we divide it by 5 to give us the mean temperature for location one to be negative 23.

15.

The mean temperature for location two is minus 23.

56.

So that means location two is slightly colder on average than location one.

Question 4A, states Andeep gets four correct, three incorrect, and misses two.

So how many points does he have? Well, 4 times 8 add 3 times negative 6 add 2 times negative 2 is the same as 32.

Add negative 18 add negative 4 which is 32.

Add negative 22 which is 10.

So Andeep has 10 points in this competition.

Question 4B, in a different competition, Lucas gets 36 points.

So how could he have got 36 points? One example would be he could get four correct, two incorrect, and miss two.

There are lots of different examples in there as long as you get 36 points.

Well, really well done if you got that one right.

Well done, everybody, so let's move on to measurements and accuracy.

Now we use decimals with measurements to increase the accuracy, and the level of accuracy is important in so many different fields of expertise.

For example, medicine, engineering, biochemistry, construction, so on and so forth.

When applying operations to measurements, it is sometimes necessary to have all the units the same, so knowing the prefixes can help convert to the same units.

Remember, centi placed before the unit means 1 over 100, milli placed before the unit means 1 over 1,000, and kilo placed before the unit means 1,000.

So let's have a look at a really short example.

We have three builders and they measured three different parts of the house.

The first builder measured the first section of the house to be 2.

56 metres.

A second builder measured the second section to be 245 centimetres, and a third builder measured the third section to be 3,890 millimetres.

Now the owner of the house wants the length of the house in metres.

So what do you think the builders need to do? Well, hopefully, you've spotted, they need to convert all their units into metres.

Well, the first builder, well, it's already in metres, so we don't have to do anything there.

The second builder needs to recognise 245 centimetres is 2.

45 metres, and the third builder needs to recognise 3,890 millimetres is 3.

89 metres.

So converting to the same unit is really important when we're making comparisons or we're going to do a calculation.

Now the owner still wants the length of the entire house in metres.

So what do you think we need to now do with these lengths? So hopefully, you've spotted, we can now add them all up.

We can add them all up because they're the same units.

So using the column method, lining up all ones carefully, tens, hundreds, et cetera, et cetera, we have 8.

9.

So we now worked out the house length to be 8.

9 metres.

Now, let's have a look at the same house and the house needs new guttering.

Now if you're not quite sure what guttering is, guttering runs along the bottom of the roof and it collects all the drain water.

And the guttering needs to be the same length of the house.

Now, the guttering will cost 32 pounds per metre.

What operation do you think we're using to work up the cost of the guttering? Have a little think.

Well, hopefully, you spotted, its multiplication.

In addition to this, we're just going to identify that guttering is sold in metre lengths.

So you cannot go into a hardware store and ask for 0.

9 metres of a guttering.

So see if you can work out the cost of guttering and ensure you show you're working out.

Great work.

Well done.

So hopefully, you spotted, the calculation is 32 multiplied by 9.

Now for me, I'm going to use the distributive law to work this out.

There's lots of different ways.

So 32 multiply by 10 subtract 1 is 320 subtract 32 which is 288.

So the cost of the guttering is 288 pounds.

Well done if you got that one right.

Now, let's have a look at your task.

Here's an example of two car rental companies and an Oak teacher knows he needs to rent a car for a 70-kilometer journey.

Work out the difference in prices between the two companies for a 70-kilometer car hire.

Remember to recognise which operation to use and carefully use those skills using decimals.

See if you can give it a go and press Pause if you need.

Great work.

So let's have a look at question 2.

Question 2 states that Sophia is helping her neighbour feed her cats using a feeding chart.

Now, the feeding chart states that the mass of a cat to the nearest 10 grammes has a certain quantity of food.

So if a cat is in between 1 to 2.

99 kilogrammes, it must receive 70 grammes of food a day.

If a cat has a mass of 3 kilogrammes to 4.

99 kilogrammes, it must receive 90 grammes a day.

If a cat has a mass between 5 to 5.

99 kilogrammes, it must receive 100 grammes a day.

And if a cat is more than 6 kilogrammes, the cat must get 115 grammes of dry food per day.

Now we know a 1.

5 kilogramme bag of dry food costs 9 pound 82, and her neighbour has three cats, one weighing 4.

5 kilogrammes, another one weighing 5.

1 kilogrammes, and another one weighing 6.

2 kilogrammes.

And we're asked to work out how much will it cost to feed all three cats for two weeks.

This is a great problem solving question.

Take your time, know which operations to use, and do it in stages.

Really well done.

See if you can give this one a go.

Great work, so let's see how you got on.

Well, for question 1, let's work out how much rent a car costs.

Well, you have to do 0.

46 times 70.

You can do this in all sorts of different ways.

I've chose 0.

46 times 70 is the same as 4.

6 times 7 which is 32 pounds 20.

Now for rent a blue car, we know 2,500 metres is 2.

5 kilometres.

And given the fact that we know the teacher only wants to travel 70 kilometres, 70 divided by 2.

5 is the same as 700 divided by 25, I'm choosing to divide by an integer, equals 28.

Thus working this out gives me a cost of 30 pounds 80.

So 32 pounds 20 subtract 30 pounds 80 is equal to 1 pound 40 difference.

This was a great question, lots of different ways to work it out.

Well done if you got this one.

For question 2, we had to work out how much it'll cost to feed all three cats for two weeks.

Well, if we know a 4.

5 kilogramme requires 90 grammes a day, and a 5.

1 kilogramme requires 100 grammes a day and a 6.

2 kilogramme cat requires 115 grammes a day.

If we add this up, that means we have 305 grammes of cat food is eaten per day.

Because Sophia is feeding them for two weeks, we have to multiply 305 by 14 to give us 4,270.

So that means if we know 1.

5 kilogramme bag is 1,500 grammes, we need 4,270 grammes, that means she needs three bags.

So it'll cost 9 pound 82 times 3, which is 29 pounds 46 needs to be spent on cat food.

This was a great question, so many stages, but it's all about taking your time and recognising which operation to use.

So in summary, it's important we can use our skills on addition, subtraction, multiplication, division in the real world.

It's equally important when we know which operation to apply given the context of the question.

And we use decimals with measurements to increase accuracy.

This level of accuracy is really important in so many different fields of expertise, including medicine, biochemistry, construction, and so and so forth.

So hopefully, you can see how important decimals are and how they've been used in the real world.

A huge well done.