Lesson video

Today, we'll be learning to use fractions to express proportions.

All you'll need is a pencil and piece of paper.

So pause the video now and get your things together if you haven't done so already.

Here's the agenda for today's learning.

So we're going to use fractions to express proportions.

You'll start with a quiz to test your knowledge from our previous lesson.

Then you'll describe shape patterns using fractions, then extend shape patterns before doing some independent learning and a final quiz.

So I'd like you to, first of all, look at this pattern and think about the three questions below.

How could you describe the pattern? What can you say about the number of squares and what can you say about the number of triangles? Pause the video now to make some notes.

So these are some of the ideas that you may have come up with.

The pattern goes triangle, two squares, triangle, two squares, triangle, and so on.

There are three triangles and six squares in total.

For every triangle, there are two squares and there are double the number of squares to triangles.

So now we're going to have a look at the pattern, just rearranged, so that we've got all of the triangles together and all of the squares together.

And you can see that the pattern has nine shapes altogether, three of which are triangles and six are squares.

And I'd like you to think about using your knowledge of fractions.

Can you express the proportion of the whole that are triangles and the proportion of the whole that are squares? So we know that three out of the nine shapes are triangles and that represents 3/9, six of the nine shapes are squares, which represents 6/9 as a fraction.

Now let's zoom in on one section of the pattern.

So Elizabeth looks at the repeating section of the pattern, which has three shapes.

One is a triangle and two are squares.

So now again, I would like you to use this section of the pattern, to use fractions, to describe the proportion that are triangles and the proportion that are squares.

Pause the video now to make some notes.

So we can see that there are three shapes altogether.

And two out of these three shapes are squares, which represents 2/3 of the shapes.

And one out of these three shapes is a triangle, which represents 1/3.

So if we look back at the original pattern, we said that the proportion of triangles is three out of the nine shapes, so that's 3/9.

And the proportion of squares is 6/9, six out of the nine.

In our zoomed-in section of the pattern, we said that 1/3 were triangles and 2/3 were squares.

Now, if you have a look at the whole pattern, this is three times greater than the section of the pattern because there's three of those repeated sections.

Therefore the number of squares and triangles is three times greater.

So we have one here, one triangle in the small section, but three in the whole pattern.

So that's three times greater than one and the same for the squares.

So that shows that the proportional relationship stays the same because 1/3 is equivalent to 3/9 and 2/3 is equivalent to 6/9, but it is the number of shapes within the pattern that changes.

And now I'd like you to have a go at describing this pattern, independently.

Think about the questions below.

How can you describe it? What can you say about the number of green beads and white beads, and then use fractions to express the proportion of green beads and the proportion of white beads.

Pause the video now to make some notes.

So we can say that the proportion of green beads is 9/15, so nine out of the 15 beads are green and the proportion of white beads is 6/15.

So if we zoom in on one section and using your fractions, can you express the proportion of green beads and white beads for this section of the pattern? So we would say that for this section of the pattern, the proportion that are green beads is 3/5, three out of five.

And the proportion that are white beads is two out of five, or 2/5.

And you can see that if we scale it up to the full pattern, the full pattern is three times this individual part of it.

And 3/5 is equivalent to 9/15, three times three is nine, five times three is 15.

And the same is true for 2/5 and 6/15.

So here we're looking at equivalent fractions, when we simplify it just into one section of the pattern.

So now we have our proportions as fractions.

I'd like you to try and express them using decimals and percentages.

Pause the video now and make some notes.

So we know that 3/5 of the beads are green and 2/5 are white, and we can look at an equivalent fraction in terms of tenths.

So 3/5 is equivalent to 6/10, which is equal to 0.

6.

So there it is represented as a decimal.

2/5 is equivalent to 4/10, which is equal to 0.

4.

So there you can see them expressed as decimals.

And we know that 0.

6 plus 0.

4 is equal to one whole and that represents a whole.

And then if we represent as equivalent fractions with 100 as the denominator, then we can easily express them as percentages.

So 60% of the beads are green and then 40% are white and make sure that those two percentages add up to 100%, which is equal to the whole.

Now we're going to look at extending patterns.

So here we have a pattern of parallelograms and hexagons and I'd like you to think about what proportion are parallelograms and what proportion are hexagons? And can you express these proportions, using fractions, decimals, and percentages? So for the parallelogram, we have one out of the five shapes represented as parallelogram.

And that is equal to 2/10, which is 0.

2 as a decimal, or 20% as a percentage.

So 20% of the shapes are parallelograms. And then we do the same for the hexagons.

Four out of the five shapes are hexagons.

That's equivalent to 8/10 if we scaled it up to have 10 shapes in the pattern, which is equal to 0.

8, or 80%.

Now let's think about another way of looking at this.

If we scale up our pattern even further.

So our initial one has five shapes in, but if we increase it to 15, 20, or 30, you can see the number of hexagons and the proportion of hexagons.

I want you to think about this column here.

What do you notice about the proportion of hexagons as the pattern increases? Pause the video now to make some notes.

So what we can see here is we'll look at the relationship between the first two patterns.

So in the first one, there's five shapes and 15 in the second.

So the second pattern is three times greater than the first pattern.

Therefore, the number of hexagons is also three times greater, four times three is equal to 12.

And then you can see there, four out of the five shapes are hexagons, or 12 out of the 15 shapes are hexagons in the second pattern.

So whatever the number of shapes in the pattern is multiplied by, the number of hexagons in the pattern is multiplied by the same amount.

So we can see between five and 20, five has been multiplied by four there and the same has happened between four and 16.

So the proportion of hexagons is always 4/5 of the whole.

Now you're ready to complete some independent learning.

So pause the video and complete your tasks and then click restart once you're finished and we'll go through the answers together.

So for question 1, you had a pattern of triangles and circles and you were asked to use fractions to describe which proportion are triangles and what proportion are circles.

So we know six out of the 12 shapes are triangles, which is equal to 1/2.

So it's the same for the circles then.

Six of the 12 are circles, which is equivalent to 1/2.

Then expressing them as decimals or percentages, each is represented by 0.

5, or 50%.

Part c you were asked to determine whether this statement is true or false.

The proportion that are triangles will always be 1/2 of whatever is the size of the pattern.

This is true because if you added another section to the pattern, so you added another two triangles and another two circles, the fraction of each shape would be 8/16, which is also equal to 1/2.

So if you're asked a true or false question, it's always good to give an example to show how you know that it's either true or false.

Part d, true or false? "In a pattern of 100 shapes, there will be 40 circles." Well, this is false because we know that the number of circles will always be half of the total number of shapes.

So in a pattern of 100 shapes, 50 would be circles.

Question 2, again, using fractions, you were asked to express the proportion of purple beads and white beads.

So we can see that eight out of the 14 beads are purple, which is equal to 4/7, therefore 3/7 are white beads, which is six out of the 14.

And we can see that 4/7 plus 3/7 is equal to 7/7, which is the same as one whole, which is what this represents.

Part b, true or false? The proportion that are white beads will always be 3/4.

Well, we've proven that to be false because we know that the proportion of white beads in its simplest form is 3/7.

Part a of question 3, "What proportion of the whole pattern is each different shape?" So the hexagons represent three out of the six shapes, So 3/6 or 1/2.

The triangle, one out to the six, so 1/6, and the circles, 2/6, or 1/3.

So one third of the shapes are circles.

And what if the pattern repeated to contain 30 shapes? So what you would need to do here is to find the equivalent fractions with the denominator of the whole of 30.

So the hexagons, 1/2 was equivalent to 15/30.

So if there were 30 shapes in the pattern, 15 would be hexagons.

Five would be triangles because 1/6 is equivalent to 5/30 and 10 of the shapes would be circles because 1/3 is equivalent to 10/30.

Great work today.

Well done.

Before you go, don't forget to complete your final quiz to test everything you've learnt today.