Lesson video

Hello, my name is Mr. Clasper and today we are going to be investigating upper and lower bounds and specifically looking at error intervals.

Let's have a look at this problem.

Our lower bound will be the smallest number which runs to 225 to three significant figures, and our upper bound will be the largest number, which will run down to 225 to three significant figures.

We need to think about place value to start this problem.

So in the number 225, the third significant figure is in the ones column.

So on a number line, we can place 225 as a midpoint, and either side of the number line will be one less and one greater than 225.

To find our lower bounds, if we look at the midpoint between 224 and 225, we have a value of 224.

5.

Any number less than this would run down to 224.

Therefore 224.

5 is my lower bound.

It's the smallest number which will round up to 225 to three significant figures.

If we look at the midpoint between 225 and 226, we get 225.

5.

However, this number would actually round up to 226.

Our actual upper bound is 225.

49 recurring.

As this is very, very close to 225.

5, we can still refer to the upper bound as 225.

5 To write this as an error interval, we use inequalities.

So given the inequality below, reading from left to right, this says that 224.

5 is less than or equal to x, which is less than 225.

5.

So using a less than symbol, means that we are not actually including the value of 225.

5.

Let's have a look at this example.

Which place value column is the third significant figure in? The third significant figure is represented by 5000s.

Because it's in the thousands column, I need a number line with x as my midpoint, and I need a number 1000 less, and 1000 greater on either side.

My lower bound will be the midpoint between 224,000 and 225,000 which would be 224,500.

Remember, the upper bound is just less than the midpoint.

This is because the midpoint would round up to 226,000.

However, referring to the upper bound as 225,500 is acceptable as it is very, very close to the actual upper bound.

To write this is an error interval, we need to make sure we use the correct symbols.

So, reading from left to right this means 224,500 is less than or equal to the value of x, which is less than 225,500.

So, again, in an error interval, using the less than symbol means we are not including the value 225,500.

Let's take a look at this example.

This time our third significant figure is in the 100,000th column.

This means my midpoint will be x and I need a number 100,000th smaller, and 100,000th larger on either side.

Again, my lower bound can be found here, and my upper bound, can be considered as the midpoint between 0.

00907 and 0.

00908.

And written as an inequality again, we have 0.

009065 is less than or equal to x, which is less than 0.

009075.

Here's some questions for you to try.

Pause the video to complete your task and click resume once you're finished.

And here are your solutions.

So for each of these numbers you need to quote two numbers, one being the lower bound, and the other being the upper bound.

Let's take a look at this problem.

The sidelines of an equilateral triangle is five centimetres to the nearest centimetre.

What is the least possible side length and what is the greatest possible side length? Well to the nearest centimetre would essentially mean we're rounding five to one significant figure.

Therefore my number line would have five as a midpoint, and we need to have one centimetre less and one centimetre greater on either side.

My lower bound is between four and five centimetres.

So this is the smallest possible length the equal lateral triangle can have.

My upper bound, is just less than the midpoint of five and six.

However as discussed, we can refer to the upper bound or the greatest length as being 5.

5 centimetres.

As an error interval, this will be written as 5.

5 centimetres is less than or equal to s, which is less than 5.

5 centimetres.

The side length of an equilateral triangle is five centimetres to the nearest centimetre.

What is the least possible perimeter and what is the greatest possible perimeter? Well, we know from the previous example that the smallest side length was 4.

5.

This means that we could have an equilateral triangle with three sides of 4.

5 centimetres.

This will give us a perimeter of 13.

5 centimetres.

This is the least possible perimeter.

And again from the previous example, we know that the greatest possible length of the triangle was 5.

5 centimetres.

Therefore, my perimeter would be 16.

5 centimetres, and this is the greatest possible perimeter my triangle can have.

Here are some questions for you to try.

Pause the video to complete your task and click resume once you're finished.

And here are the solutions.

So question two was fairly straightforward.

This is a worded problem similar to the ones that we solved in the last slide.

And for question three, be careful we need to find the error interval for the perimeter.

So once you've found the smallest possible length of the square, you need to find the perimeter of that square.

And likewise, once you've found the largest possible length of the square, you need to find the perimeter of that square.

And this should give you an error interval of 8.

2 centimetres is less than or equal to the perimeter, which is less than 8.

6 centimetres.

And that brings us to the end of our lesson.

So we've been looking at error intervals and investigating upper and lower bounds.

Why not try the exit quiz to show off your skills.

I'll hopefully see you soon.