Number sense and exploring calculation strategies

Place value

Graphs

Addition and subtraction

Length and perimeter

Multiplication and division

Deriving multiplication and division facts

Time

Fractions

Angles and shape

Measures

Securing multiplication and division

Exploring calculation strategies and place value

Fractions: parts and wholes

Reasoning with 4-digit numbers

Interpreting and presenting data

Securing multiplication facts

Decimals

Area and perimeter

Solving measure and money problems

2-D Shape and Symmetry

Position and Direction

Reasoning with patterns and sequences

3D Shape

Working with fractions

Taking fractions further

Reasoning with large whole numbers

Problem solving with integer addition and subtraction

Line graphs and timetables

2-D shape, perimeter and area

Fractions and decimals

Angles

Fractions, decimals and percentages

Transformations

Converting units of measure

Calculating with whole numbers and decimals

2-D and 3-D shape

Volume

Problem solving with whole numbers and decimals

Equivalent fractions

Integers & Decimals

Calculation problems

Missing angles and lengths

Coordinates and shape

Decimals and measures

Percentages and statistics

Proportion problems

Extending calculation strategies and additive reasoning

Ratio review (9.7a)

Constructing triangles and quadrilaterals

Shape and patterns

Constructions, congruence, and loci

Faces, shapes and patterns; lines and turns

Coordinates

Classifying 2-D shapes

Shape and pattern

Angles in polygons

Circles

Pythagoras's theorem

Transforming 2-D figures

Bearings

Area of 2-D shapes

Shape and sorting

Volume and surface area of prisms

Maths

Y6 Maths

Key Stage 2

Year 6

Geometry

THEME

UNIT

In this lesson, we will be studying angles. We will use algebra and bar models to represent problems and calculate missing angles.

Find the value of missing angles

﻿Welcome to the first lesson in this unit, Missing angles and lengths. Today we'll be finding the value of missing angles. First of all make sure you have everything you need. Just a pencil and a piece of paper for today's lesson. Pause the video and get your equipment if you haven't already. Okay, here's our agenda for today. So we're finding the value of missing angles beginning with a quiz to test your knowledge from the previous lesson. Then we're going to calculate angles on a straight line and then around a point before you do an independent task and a final quiz to test your learning. Great work. Now for our Do Now. Calculate the degrees of rotation for each example. In the first one a key is turned 40 degrees less than a right angle. In the second one the key is turned three times. And in the third one the key is turned half a turn. Pause the video and calculate the degree of rotation for each example. So for the first one we were dealing with a right angle. We know that a right angle has a value of 90 degrees and your key turned 40 degrees less than that. So all you needed to do here was 90 take away 40. And I know that 90 take away 40 is 50 degrees, so your key was turned 50 degrees. Okay, let me write that again. It looks a bit less like 500. There we go. That's more a degree sign. In your second example, the key was turned three times. We know that a full turn is 360 degrees. So if the key was turned three times we're doing 360 multiplied by three and you may have done a column multiplication. Zero times three is zero. Six times three is 18. And three is nine plus the one is 10. So your key turned 1,080 degrees. And in the final one the key turned half a turn. Well, if a full turn is 360 degrees, then half a turn is 360 divided by two. And I know that 36 divided by two is 18. So 360 divided by two is equal to 180 degrees. Now let's start to think about what we know already about angles. Look at this angle. What do you know about it and what key vocabulary could you use to describe it? That's right, this is a right angle. The symbol in the interior angle tells me this, so I know that this right angle has a value of 90 degrees. Now if I put two right angles together, think about what we already know about the size of a right angle, what can you tell me about the angle described by the blue arrow? Well, the blue arrow represents two right angles, so I know that two lots of 90 is equal to 180. And I used my related facts there. I knew that two lots of nine were 18. 90 is 180 degrees. So the value of this blue angle is 180 degrees. And we can see that these two angles here are on a straight line. So the angles on a straight line add up to 180 degrees. Now again, stick with this for awhile. We now know that angles on a straight line add up to 180 degrees. So we have got two angles here on a straight line. Let's think about what is known and what is not known. First of all, we know that the angles on the straight line add up to 180 degrees. And we can represent the whole using a bar model. So here I can show that 180 degrees represents the whole of the bar model. The whole is 180 degrees. In this example, it's split into two parts because we have two angles. One we know, which is 60 degrees and we can label it on the bar model. And one we don't know represented here as p. So let's show that on our bar model. So expressing this algebraically, and don't be scared by that word because you've been doing algebra since you were in year one and two, we know that 60, which is our known, plus p is equal to 180. We know that this part equals whole. Now in order to find the missing part we need to rearrange this so you get the inverse. So we're going to have a whole 180, take away known part, which is 60, equals unknown part, which is p. Now I can either do a column subtraction here or I can use my known facts. I know that 18 take away six is 12. Therefore I know that 180 take away 60, 120. Therefore I know that p is equal to 120 degrees and I can check it using the inverse. So going back to this part, 120 plus 60 should equal to 80 degrees, I mean 180 degrees, sorry, the whole. There it does, okay? So we have worked our missing part using an algebraic expression. Now it's your turn. Pause the video and find the missing angle, a. Don't forget to draw a bar model to represent the problem. So you will have drawn a bar model where the whole was 180 degrees split into two parts. The known was 106 degrees and the unknown part was a. Algebraically you know that 106 plus a is equal to 180. Rearranging these in the inverse, 180 subtract 106 is equal to a. So you probably did a column subtraction here where you regrouped from the 10s. Okay, so you can see that 180 take away 106 is 74. So a is equal to 74 degrees. Good job. Now on this example the straight line has been split into three angles. So we're going to approach it in exactly the same way. We're going to represent using the bar model first. We know that the value of the whole is 180 degrees. So we have our bar model, which is split into our three angles and the whole is 180 degrees. We know one part is 40, so we put it on the bar model. We don't know the value of the next angle, it's called a, so we label that on our bar model as a. The third angle hasn't been given to us, but this symbol here lets us know that is a right angle which we know is equal to 90 degrees, so that's on my bar model as well. So if I express this algebraically I know that with three parts, now part, part, part, whole, I know that 40 plus a plus 90 is equal to 180 degrees. Now we do this exactly the same way. We rearrange because we want to get the unknown by itself and we're going to be using the inverse. 180 subtract 40 and 90 is equal to a. Now you can do this in one of two ways. Either you can do repeated subtraction, so you can do 180 take away 40 is equal to 140 and then you do 140 take away the 90 which is 50. Or, so there you know a equals 50. Or you may challenge yourself to use your knowledge of big math to represent this a different way. Now you know that if I'm subtracting 40 and then 90 I'm subtracting 40 and 90 all together. So you could challenge yourself to represent this as 180 take away 40 plus 90 and using your knowledge of big maths you know that whatever is in the brackets must be done first. So I know that I'm doing 180 take away 40 and 90. Four plus nine is 13, so 40 plus 90 is 130 and now I know that that is 50. So it's up to you which strategy you use. You can either do repeated subtraction or you can represent using big maths, your choice. So it's your turn. Now you've got three angles on a straight line. Don't forget to draw a bar model to represent the problem. Pause the video now and find the missing angle. So you will have drawn a bar model where 180, hopefully a neater bar model than me, 180 is the whole. And this time it was split into three parts. One part was 46 degrees. One part was 52 degrees. And the unknown part was a. And there we have it much neater. So algebraically we know that 46 plus 52 plus a is equal to 180. And reordering it using the inverse, we're doing 180 take away 46 and take away 52 which gave you 82. So a is equal to 82 degrees. You may have done repeated subtraction or you may have used big maths. Now let's start to look at angles around a point. So let's think back to what we already know. We worked out that the value of the blue angle is 180 degrees because it was two lots of 90 degrees. What is the value of the red angle? Well you can see that the red angle is the size as the blue angle, 180 degrees as well. So both angles together they make a full turn. So 180 plus 180 is equal to 360 degrees. So a full turn is equal to 360 degrees. One full turn is made up of two 180 degree angles or, if I draw this down here, four 90 degree angles. And we know that four times 90 is also 360. So let's stick with this. We know that angles around a point add up to 360 degrees. So let's think about what is known and what is not known. First of all, we know that the angles around a point add up to 360 degrees. And we can represent this using a bar model. So here we have our whole bar and the whole, the total is 360 degrees. We know that one angle is 40 degrees, so we can label that one on the smaller part. And we have an unknown part, which we have called t, and we can add it to our bar model. So it's the same principle as we've just been practising . First we represent it pictorially. Now we go algebraically. We know that 40 degrees plus t is equal to 360 degrees. This part equals whole. Now we rearrange using the inverse so we have whole, 360 degrees, subtract part 40 degrees is equal to part, which is t. So 360 take away 40 is 320 degrees. So t is equal to 320 degrees. And if we just check we know that the whole turn is 360. So if t is 320 and unknown part was 40 they should add together to make the whole which is 360 degrees. It's always good to check your answer using the inverse. Okay, your turn. Pause the video and find the missing angle. Draw a bar model to represent the problem. So you will have drawn a bar model where 360 is the whole. I think I have one prepared already. There we go. So 360's the whole. And we know one of the parts was 110 and the part that we didn't know was a. So algebraically 110 degrees plus a is equal to 360 degrees. Rearranges in the inverse, 360 degrees take away 110 degrees is equal to a. And we know that 360 take away 110 is 250 degrees. So a is equal to 250 degrees. Right, this time we've got something slightly different. Sometimes you will be given questions where you have two unknowns and in this case you've got to do some detective work. So here we have f and g are both unknown. They're not equal, so there must be something else going on. Now if I look really carefully at this diagram I can see another known part. There's actually a straight line running through this diagram. And I know that angles on a straight line add up to 180 degrees. See this straight line running through? So f can now be worked out by using this angle here, nine. I know that the whole of this straight line is equal to 180. And I've been given one part. 180 subtract nine is 171 degrees. So f is 171 degrees. Now I can either use that same principle because the bottom part is also on a straight line. So this is 41 plus g is equal to 180 degrees and then rearrange it, 180 subtract 41 equals g. That's one way that I could do it. The other way is to look at the whole diagram as a whole, okay, where 360 is the whole and it's split into four angles. I know three of them because I worked out f. 41 degrees, nine degrees, and 171 degrees, and now I have to work out g by itself. So I can use the whole where I know that 360 degrees subtract 41, subtract nine, subtract 171 is equal to g. And I've done this calculation. I know that 139 is equal to g. So g is 139 degrees. And if you've done it that way we can check it by doing it the other way. 180 subtract 41 is also 139 degrees. So either way we do it we find g to be 139 degrees. Okay, now it's time for you to do some independent learning. Pause the video and complete the task. Restart the video once you've finished and we'll go through the answers together. Okay, let's look at question one together. You have to calculate the value of angle a. Hopefully you drew a bar model to help to see it and discern it pictorially where the whole is 180 and we subtract the known part which would give us a. 180 subtract 22 is 158. So a is equal to 158 degrees. For question two, you were calculating angle c and although it was in a slightly different orientation to what you may be used to seeing, this symbol told us that this angle was 90 degrees. So the whole is 180 because it's on a straight line. The known part of 51 and 90 and you're trying to find angle c. 180 subtract 51 and 90 is equal to c. And I know that 180 subtract 51 subtract 90 is equal to 39 degrees. So c is equal to 39 degrees. Question three, we're looking at a full turn. So you know that one full turn is equal to 360 degrees. Whole subtract part equals part. 360 subtract 78 is 282 degrees. So b is equal to 282 degrees. Question four, you're calculating angle f. The whole, again, is around a full turn or 360. And there are your two known parts and your missing part. So whole 360, subtract part 165, subtract part 68, is equal to f. And I know that this is 127 degrees. So f is equal to 127 degrees. Question five we've got two unknown angles and we can this straight line running all the way through. So I have two bar models to help me. I'll start out with f and I've got one known which is 45 degrees. So 180 subtract the known 45 degrees is equal to 135 degrees. So f is equal to 135 degrees. And then g is my other unknown. And I know that I'm doing a whole 180 subtract 85 here which is equal to 95. So g is equal to 95 degrees. Okay, now it's time to pause the video and complete your final quiz. Restart the video once you're finished. In our next lesson, we'll be learning to compare and classify triangles. See you then!

In this lesson, we will compare and classify triangles, looking at how to recognise the different types of triangles and the sum of the angles in a triangle.

Compare and classify triangles

﻿Welcome to our second lesson in the missing angle, in this unit. Today we'll be comparing and classifying triangles. You will need a pencil and a piece of paper. As always, you will also need a piece of paper cut into the shape of a triangle today. Pause the video and get your equipment together. In today's lesson, we're comparing and classifying triangles. We'll start with a knowledge quiz to test your knowledge from yesterday's lesson, then we will classify triangles. We'll look at the angles in a triangle, and then you'll do some independent work and a final quiz. So let's start with the knowledge quiz. Pause the video and complete the quiz. Great work. Now here's your Do Now. Match the picture of the triangle to its name and description. Pause the video while you work on this task. So your first triangle which had size of equal length was an equilateral triangle. All of the sides are equal, and all of the angles are equal Your second triangle, which you could see had a right angle is a right angle. And the right angle triangle can also be Isosceles or Scalene. We'll come on to this in more detail later. The third triangle which has these two lines here, that show you that the sides are equal length is called an Isosceles triangle. And an isosceles triangle is named from the Greek word of Iso which means same, and skelos which means legs, so the two equal sides are called the legs and the third is the base. So it has the same length legs. The last triangle is a scalene triangle. All of the sides are different lengths, and all of the angles are different. Now let's work on classifying some triangles. We want to know which type of triangles are scalene, which of these triangles are scalene and which are not. And we need to be able to look at the information given and determine the type of triangle from it, because they're not always drawn to scale. Now, I know that in a scalene triangle, none of the sides are equal. So let's do one together. For triangle a, all of the sides of equal length, they're all five centimetres therefore, it's not a scalene triangle. It is, in fact, an equilateral triangle. So I'll put a in that column. Now pause the video and classify the rest of the triangles. So we did a together. We worked out that a was an equilateral triangle. b, each of the sides are different lengths, therefore it must be scalene. c, again, they were different lengths. So c is scalene. In d, we have two of the sides are equal. Remember that the legs and the base is different. So that is an Isosceles triangle. So that goes into a not scalene column. e, we have three sides of different lengths. So it's scalene. And f, all sides are equal. So it is equilateral. Let's look at a different type of triangle. So which of these triangles are Isosceles? Pause the video and sort the triangles. So, you will have looked at a and seen that the legs were equal length fair. So this a is an isosceles triangle, b, all three sides are different lengths, so it's not isosceles. You're right, it's scalene. c, all sides equal, therefore equilateral, not isosceles. d, this is a right type of triangle. But these lines here tell us that these two sides are the same length. Therefore, it is an isosceles, that was meant to be a comma. It is an isosceles triangle, and it's also a right angle triangle. In a, two legs of the same length, isosceles an f, all sides, different lengths, not isosceles but scalene. Now, let's look at the Angles in a triangle. Take your triangular piece of paper, and rip off the corners, as shown here. If you put your corners together so that vertices touch, so these parts of them. What do you notice? I'm going to draw a line along here to help you think. Well, when we put the vertices together, they create a straight line. And we know that angles on a straight line add up to 180 degrees. Therefore, we can infer that the angles in a triangle always add up to 180 degrees. Let's look at this in the context of a triangle. So here we have an equilateral triangle, we know that in an equilateral triangle, all sides are equal lengths, and all angles are equal. We know the three interior angles that inside the triangle are equal. So we represent it pictorially. We know the whole is 180 degrees. And this is split into three equal parts, and we've labelled these a. So if we think about it algebraically, we know that a plus a plus a is equal to 180 degrees. And we also know that we can, instead of using repeated addition, because these are the same number, we can use multiplication, so a times three is equal to 180 degrees. Now we need to use the inverse to find the value of a, 180 divided by three, gives us a. And we can use all related facts. If we know that 18 divided by three is six, we know 180 divided by three is 60. Therefore, a is equal to 60 degrees, an equilateral triangle, the interior angles are always 60 degrees. And we know that 60 plus 60 plus 60 is equal to 180 degrees. Now we're going to look at finding missing angles, in triangles in lesson four. So what you need to remember from this, is that the angles inside a triangle add up to 180 degrees. Let's have a look at some reasoning around triangles, I want to know is this statement, always sometimes or never true? A triangle only has acute angles. So I know that an acute angle, is larger than zero degrees, and less than 90 degrees. But I do know that some triangles are right angle triangles. And that angle is equal to 90 degrees. So that's not an acute angle. I also know that there are some triangles that have obtuse angles like this one. This one has an obtuse angle which is greater than 90 degrees, and less than 180 degrees. So I think that this statement is sometimes true. And if I use my acute angle up here, and join it up to make a triangle, I've got three acute angles there. And in fact, I can prove that even further. If I have an equilateral triangle, like we learnt on the previous slide, each of the angles in an equilateral triangle was 60 degrees. And now they are all acute angles. So I have one that has acute angles, one that has a right angle, and one that has an obtuse angle. Therefore it must sometimes be true. So when you're asked to reason about an answer, it's really important to give multiple examples to show that you can prove what you're saying. Now it's your turn. Is this statement always, sometimes or never true? A triangle has a reflex angle. Pause the video, and make some notes to explain your answer. So we know that angles in a triangle add up to 180 degrees. Now what do we know about the reflex angle? Well, we know that a reflex angle is greater than 180 degrees. So if a triangle is made up of three angles, then none of them can be a reflex angle, because we can't have a triangle with one angle of 180 and two of zero that will just be a straight line. Therefore, this statement is never true. Now its time to do some independent learning. Pause the video and complete the task. Restart it once you're finished. In question one you were asked to match the triangle to the correct name. The first triangle has a right angle, therefore it must be a right angle triangle. It also has sides of different lengths, so it's scalene as well. So it's a scalene right angle triangle. The second one has got the lines to indicate that these two sides or legs are the same length, therefore it is isosceles. The third one has got sides which are all different lengths, so it's scalene. And the final one has the three lines to indicate that those sides are equal length, therefore, it is equilateral. So in question two, you were asked to identify the two isosceles triangles, which had two sides of equal length, and you could see that this was an d, and you could use the grid lines to help you out with that one. Question three, what type of triangle is shown? We looked at this earlier. And although this one doesn't have those on to help you, you know that this is definitely equilateral because an equilateral triangle has equal sides and equal angles and we know that the angles in an equilateral triangle are always 60 degrees each. In question four, what type of triangle is shown here? Well, you could have given this two different names. So first of all, although it wasn't drawn in the usual square way, this is a 90 degree angle. So this is a right angle triangle, but it also has two equal angles, and we know that an isosceles triangle has two equal sides, but also two equal angles, so it's a right angle isosceles triangle. In question five, this triangle was not drawn to scale. So the reason that I drew it in this way, was to try and trick you into thinking that it was a different type of triangle. The most important thing is that you looked at the values of the angle sizes. So you could see that these are three different angles. Therefore it is a scalene triangle. Scalene triangle has sides of different lengths, and angles of different sizes. Finally, Fatma says I have drawn a triangle with one acute angle, one right angle, and one obtuse angle. Why is she incorrect? Well, we know that the angles within a triangle add up to 180 degrees, and an obtuse angle is at least 90 degrees. A right angle is 90 degrees. So already that would add up to 180 degrees, so she cannot be correct. Great work, but it's time for your final quiz. Pause the video, and then click Restart once you're finished. Great work today. In tomorrow's lesson, we'll be comparing and classifying quadrilaterals. See you then!

In this lesson, we will classify different types of quadrilateral and learn about the sum of their internal angles.

Compare and classify quadrilaterals

﻿Welcome to our third lesson in this unit, finding missing angles and lengths. Today we're going to be comparing and classifying quadrilaterals. First of all, make sure that you have everything you need. Just a pencil and piece of paper today. Here's the agenda for the lesson. We'll be comparing and classifying quadrilaterals, starting with a quiz to test your knowledge from the previous lesson. Then we'll look at classifying quadrilaterals before looking at angles and quadrilaterals. Then you'll move on to some independent learning before doing a final quiz to test what you've learned in today's lesson. So let's start with the quiz, pause the video and complete the quiz. Restart once you've finished. Great work, now here's your do now, who you agree with and why? The first person thinks that all of the shapes are quadrilaterals because they all have four sides. The second person thinks that half of the shapes are quadrilaterals and half are not. Who do you agree with? Pause the video and make some notes. Sides, that there's something very important to remember is that quadrilaterals have four straight sides and four internal angles. So when I look at these shapes, I can see that this shape and this one, and this one have straight sides, but here we have a curved side and so on these ones as well. Therefore the second person is right. Half of the shapes are quadrilateral, because they have straight sides. Now we've got some new vocabulary. So we're going to be looking today at polygons. A polygon is a flat two dimensional shape with straight sides that is fully closed. So all of the sides join up together. Remember the sides must be straight. Polygons can have any number of sides. But a shape with curved sides is not a polygon. So here's our first bit of vocab. A regular polygon is a polygon in which all sides and all internal angles are equal. So here we have a pentagon, which is a five-sided shape that has equal sides, therefore it is a regular pentagon. Then we have irregular shapes. That's a polygon where not all of the sides and internal angles are equal. Here we have an octagon which has eight sides, and you can see that these sides are equal lengths and these sides are equal length, but they're not all equal. Therefore it's an irregular octagon. The next word is parallel. And here's an image of parallel lines. These are lines that will never intersect, which means to join. And they are equidistant, which means that every, each point of the lines is the same distance apart. They are equidistant, they're an equal distance apart. And I always remember what parallel means because inside the word parallel, there's a pair of parallel lines. The last, perpendicular, this is when two lines intersect or join at a 90 degree angle. And here is your image to represent that word. So you may want to pause the video and make some notes on these words, we'll be using them throughout the lesson. Now our first shape that we're going to identify is parallelograms and if you think back to our vocabulary, this must have something to do with those parallel lines, lines that are equidistant, they're the same distance apart. We're going to be looking at these shapes here and identifying which are parallelograms and which are not. So a parallelogram is a polygon that has two pairs of parallel lines, and it has two pairs of equal angles. And squares and rectangles are also parallelograms. So I'll start with A, I can see that A has got a pair of parallel lines, and we mark parallel lines with an arrow like this. These lines, no matter where you get to on, each point on these two lines, they are an equal distance apart, therefore they are parallel. However, the other two lines are not parallel. So this is not a parallelogram. In B, although this is an orientation that you may not normally be used to seeing. This is a parallelogram because it has got pair of parallel lines here and a pair of parallel lines here. Shape B, this is important for later on in the lesson, is also called a rhombus. It's a parallelogram, which has all equal sides. That's a rhombus. So it has equal sides but it also has two pairs of parallel lines. Let's look at one more together before you look at the rest by yourself. Shape C, I can already see that it's a square, and parallelograms include squares because these lines are parallel and these lines are parallel. So that is a parallelogram, okay? And remember we'll go back to this one. A rhombus is also a parallelogram. Now pause the video and have a look at the rest of the shapes. Which of them are parallelograms and which are not? So shape D is a parallelogram because these two lines are parallel, they are equidistant. And so are these. And this is actually a rectangle as well. Remember that rectangles are parallelograms. So that's a yes. E, well I can see a pair of parallel lines here, but the other two are not equidistant, at the top they are closer together than they are at the bottom, so that's not a parallelogram. F is, the top and bottom line are parallel. And the side two lines are parallel, so that's a yes. G is also a parallelogram. And you will usually see these signs marked on the shapes so that you can judge it yourself. And then H has got a pair of parallel lines here, although its difficult to see, sometimes you have to turn your page, but the other ones they're much closer there than they are up there, so that's not. I has got no parallel lines and neither has J. So remember, a parallelogram has two pairs of parallel lines and two pairs of equal angles. Let's look at trapezia. So trapezia is the plural of trapezium. So you have one trapezium and lots of trapezia. Trapezium has one pair of parallel lines. Now I remember from the previous slide that there were a few shapes on here with one pair of parallel lines. The first one was A, so I'll mark that on there. That's a pair of parallel lines, but the other two are not. So this is a trapezium. Now pause the video and figure out which of the others are trapezia. So I won't go through all of them because we remember from the previous slide that B, C and D were all parallel. E is a trapezium because it has one pair of parallel lines. The other two are not. So that is a trapezium. F was a parallelogram, so was G and H is a trapezium. And like I said before it's easier if you turn your paper around. So if I turn my paper, you can now see what each would look like if it was standing on its base. And you can see that these two lines, if I carry this one along like that, they are parallel lines, so that is a trapezium. And we said, I and J have no pairs of parallel lines, so they are not. Now other quadrilaterals, I and J we're the only two shapes that we didn't identify. I is a quadrilateral because it has four sides, but it has no parallel lines, so it's not a trapezium or a parallelogram. It's got two pairs of equal sides that are of equal length and they're connected at a point. And you may have seen this in a different orientation, looking like this. I is a kite. And J, this is a quadrilateral that has got two pairs of equal length sides. So those two are equal and these two are equal. It also has one internal reflex angle. This shape is called a Delta and it looks a bit like an arrow head. Now it's your turn, pause the video and match the shapes to their name. And we remember that parallelograms have two pairs of parallel lines and include squares and rectangles. So that B is a parallelogram. B, G, and H. Remember that this is called a rhombus, but it's still a type of parallelogram, it has all equal sides. I keep coming back to that one, because you're going to need to use that later. Trapezia have one pair of parallel lines. So that's C, and it's always helpful to mark them on there. A, E, and it's sometimes easy to turn your piece of paper or if you know you're doing this on the computer or the iPad, so to turn that screen around. And J. And then your other quadrilaterals were F. F was your Delta. And I, I was your kite. Now let's look at the angles in a quadrilateral. So here we have a square. Think about what you know about the angles in a square. Think about how I have drawn those on. So they're all right angles. And you know that a right angle has a value of 90 degrees. So in a square there are four 90 degree angles. So a square, the interior angles are four lots of 90 degrees. So either 90 plus 90 plus 90 plus 90 or four times 90, that gives us the interior angles in a square. If I knew that four times nine is 36, then I know that four times 90 is 360. So the interior angles in a square add up to 360 degrees. I wonder if that's the same for all quadrilaterals. Let's have a look. In a rectangle, it's the same as a square. We've got four lots of 90 degrees. And we know that four times 90 is equal to 360 degrees. So the angles in a rectangle, sorry, also add up to 360 degrees. Now, it's your turn, you're going to add up the interior angles in the quadrilaterals below and see what you notice. Pause the video while you do so. So that's absolutely right. All of the angles inside these quadrilaterals added up to 360 degrees. Therefore we can generalise that the internal angles in a quadrilateral always add up to 360 degrees. Now, one more thing to remember, this is an important piece of information, in a parallelogram opposite angles are equal. So you can see that this angle is opposite, mark it on that angle is opposite to this angle, and they are equal, they're the same. And this angle is opposite this angle, and they're the same. And if you add them all together, they would total 360 degrees. It's the same for a rhombus, which is a type of parallelogram, opposite angles are equal. A is equal to C and B is equal to D. I've told you this for a reason, which will help you in your independent test. So you may want to pause the video and make a quick note of that. Now it's time for some learning. Pause the video and complete the task and restart once you're finished so that we can go through the answers together. Great work, let's look at question one. Here you were given a square that was represented by being divided diagonally in half. What did this help you to reason about the sum of its internal angles? Well, we can see from this, that a square is made up of two triangles. We know that the angles in a triangle add up to 180 degree. And we know that 180 multiplied by two is equal to 360 degrees. So the angles in a square equal 360 degrees. And this helps us to see that that's two triangles, two lots of 180. And it's important for us to link these ideas together. Question two, you were working through a flow chart. So I asked you to start with the top shape and work your way down. So starting with the trapezium, is there at least one pair of parallel sides? Yes. Are all of the sides the same length? No. And are there two pairs of parallel sides in a trapezium? No. So your trapezium should go in here. Okay. On a parallelogram, remember two pairs of parallel sides and two pairs of equal sides. Is there at least one pair of parallel sides? Yes. Are all of the sides the same length? No, not always. Are there two pairs of parallel lines? Yes. Are there any right angles? No. So our parallelogram goes in here. And you have to excuse my handwriting. There we go. Now onto our rectangle. And as you did this you may have found it helpful to draw as well. Are there at least one pair of parallel sides? Yes. Are the sides the same length? No. Are there two pairs of parallel sides? Yes. And are there any right angles? Yes. So our rectangle goes in here. And then on to the kite, remember the kite had two pairs of equal sides but no parallel lines. The kite, are there at least one pair of parallel lines? No. Are there two pairs of sides of equal length? Yes. So our kite goes there. Onto the square. Is there at least one pair of parallel sides? Yes. Are all the sides the same length? Yes. Are there any right angles? Yes. Our square must go here. The rhombus, I kept going on about that earlier, let's see where this goes. Is there at least one pair of parallel sides? Yes. Are all the sides the same length? Yes. Are there any right tangles? No. Rhombus. So we've got one space left, but let's just check. So quadrilateral, no special properties, in my mind, I'm thinking of the Delta. Is there at least one pair of parallel sides? No. And other two pairs of sides of equal length? Well, actually there are in a Delta, so we think of a different quadrilateral that is an irregular quadrilateral. It would have no pairs of sides of equal lengths. So that is your last one. Quadrilateral with nothing special about it. No special properties. Great work, onto question three. So each of the quadrilaterals has a perimeter of 16 centimetres. Remember that's the distance around all of the sides of the shape. Complete the table to show the possible length of each side. So remember these are all quadrilaterals. So they've all got four sides. That's why we have four columns. In a square, they had to be equal. So they had to be four, four, four, and four. And I'm actually going to go down and do the same for my rhombus because they all had equal sides. So they must be four as well. For the rectangle, you've got some combinations, you needed to have two sides the same length, and then another two sides of the same length, but they weren't equals, they're not squares. So you could have had two, two, six and six. Or you could have had some different combinations. Let's look at the parallelogram. The same as the rectangle it's got two pairs of equal sides. You could have had three, three, five, five. And the kite, again, two pairs of equal sides. You could have had one, one seven, seven. So these could be in any order here for the rectangle, the parallelogram and the kite. You could have had any of these three, but they could be in any place. And the trapezium, you can have any combination adding up to 16, because the sides don't need to be equal length. As long as the total was 16. Question four, you're asked to name the shapes and list their properties. So the first one was a square and you may have written something like, it has four equal sides. It has equal angles that are right angles. And you could have maybe also said, it's a type of parallelogram. The second one is a trapezium, with one pair of parallel sides. And the third one is also a trapezium. In question five, you were to use your knowledge of quadrilaterals to find the value of the missing angles in the shapes where you could see that these two are right angles, so you know that A and B are both 90 degrees. You may have also noticed that this is a trapezium. And if I added 180, 90, and 90, then my total of the internal angles is 360 degrees. My second shape is a rhombus. Remember that the opposite angles in a rhombus are equal. So this angle is equal to this angle. So C is 80 degrees. This angle is equal to this angle. So D is a hundred degrees. And if you add 100 plus 80 plus 100 plus 80, your internal angles total 360 degrees. Okay, it's time for your final quiz. Pause the video and complete the quiz. Restart it once you're finished. Our next lesson, we'll be finding the value of missing angles in triangles. I'll see you then.

In this lesson, we will represent the angles in a triangle pictorially and algebraically before learning how to calculate missing angles.

Find unknown angles in triangles

﻿Welcome to our fourth lesson in this unit, finding missing angles and length. Today we're going to be finding the value of missing angles in triangles. All you'll need is a pencil and a piece of paper. Pause the video and get your equipment together if you haven't done already. Here's our agenda. So we'll be finding the value of missing angles in triangles today, starting with a quiz to test your knowledge from yesterday's lesson, then we'll represent angles pictorially, we'll calculate missing angles before you do some independent work and a final quiz. So let's start with the knowledge quiz. Pause the video to complete the quiz, and test your knowledge from yesterday's lesson. Great work, now we're looking today at the angles in a triangle. We know that the angles in a triangle add up to 180 degrees. In an equilateral triangle, where all of the sides are equal, the angles are always 60 degrees each. In an isosceles triangle where two of the sides are equal length, two of the angles are equal, and one is different. In a scalene triangle all of the angles are different sizes. Now we can represent the angles in a bar model pictorially. So the whole is 180 degrees. And the three angles added together total 180 degrees. As an algebraic expression we can represent that as a plus b, plus c equals to 180 degrees. In this case, as it's a scalene triangle angles a, b and c are all different. But what about in an isosceles triangle? What do we know about angles b and c? We know that angles b and c are equal. And we know that a is different and it looks smaller than b and c as well. So we can represent it using a bar model. Again a whole is 180. We've got our missing angle a which is different to the other two, and b and c are equal. So their bars are the same length. And then we know that a plus b plus c equals to 180 degrees, but in this case, b is equal to c, and we'll look at how we work with that with using actual numbers in a minute. Now it's your turn. Represent the equilateral triangle pictorially, and then algebraically. So you will have drawn a bar model with a whole as 180 degrees, and you have three bars of equal length. Algebraically a plus b plus c is equal to 180 degrees. And as it's an equilateral triangle, we know that a is equal to b which is equal to c. Now let's get on with calculating some missing angles. Here we have a scalene triangle where all of the angles are different. We want to know the value of angle a. Let's represent as always pictorially first, but we have our whole is 180 degrees, and we have two known angles, 30 degrees and 135 degrees. And our unknown angle is angle a. Represented algebraically we can see that 30 degrees plus 135 degrees plus a is equal to 180 degrees. Now in order to find the missing part, which is a, we need to rearrange the equation using the inverse. So we had part, plus part, press part equals whole. Now whole take away part, takeaway part is equal to part, the unknown part. Now we touched on this in the previous lesson, we have two ways of doing this. Either we can use repeated subtraction, or you may represent it using BIDMAS. So I'll start off with repeated subtraction. So 180 subtract 135. Regroup from my tens. So that's 45, and then I'm doing 45 subtract 30. So I can see that a is equal to 15 degrees. Now, if you were representing using BIDMAS, you would know that when I'm doing 180 takeaway 135, takeaway 30, what I'm doing is 180 takeaway both of these numbers together. I may have shown it like this 180 takeaway 135 and 30. And my brackets tell me to do this first. So you're doing 180 away 165, which is equal to 15 degrees. So it's your choice, how you do it. But in fact, sometimes this is the most efficient way to do it because you're really doing a step less than in the other strategy. Now it's your turn. Pause the video and represent the angles pictorially and algebraically to find the value of the missing angle, we call it a. So your bar model will look like this, 180 degrees is the whole. You knew this angle, although it wasn't written on there because you know the symbol, 90 and 70, with my degree sign in there, and a equal to 180. Algebraically, we know that 90 plus 70 plus a is equal to 180 and that needed rearranging. So you had to use your knowledge of the inverse to rearrange that to 180, subtract 90, subtract 70, is equal to a, or you may have done your brackets to do it using BIDMAS, 180 takeaway 90 and 70. And I can do this in my head. I know that nine and seven is 16. So 90 and 70 is 160. So a is equal to 20 degrees. Now we're going to calculate the missing angles in an isosceles triangle. We know that in this triangle angles b and c are equal, and we haven't been given the size of either of those angles. So fast here is our bar model. The whole is 180 degrees. The known is 32. B and c are the same size angle. B is equal to c. So we know that we're subtracting 32 from 180 to give us the value of b and c together. So I know that's 148, because that's 180 take away 32 is equal to b and c. So I know that b and c are equal to 148, and because they are equal, I need to divide 148 by two to find their value. So the value of b and c is 74 degrees each. Now it's your turn. Here's another isosceles triangle. Pause the video and represent the angles pictorially and algebraically to find the value of the missing angles a and b. So for this one, you will have drawn your bar model with your known 110 and your unknowns, a and b, where a is equal to b. So you know that 180 subtract your known angle gives you what is left over is a and b and that's 70 degrees. So now you know that this 70 needed to be split equally between don't know, sorry, why I've changed this to b and c there. That should be a and b, cause a and b equal 70 degrees altogether, and 70 divided by two is 35 degrees. Remember that you can always check your answer using the inverse. So you're adding 110, 35 and 35 that should equal to 180. 110 plus 35 is 145. Plus another 35 is 180. So I can check by going the opposite way to check that it still works. Now you're going to do another one. This is another isosceles triangle, represent it pictorially, and then algebraically to find the value of angle a. So in this one, we didn't know this angle. It just wasn't written on for you. You know that the whole is 180. The two known parts are 40 and 40. So subtracting those two from 180 is the answer to a. So a in this case is equal to 100 degrees. Remember the angles in a triangle add up to 180, so you can check it. 100 plus 40 is 140, plus another 40 is 180. So that's definitely correct. Now it's time for your independent learning. Pause the video and complete the tasks, and then restart once you're finished. Okay, here's your first question. Question one, you're asked to calculate the value of angle a in this scalene triangle. Here's your pictorial representation. The whole is 180. The two known parts are here. So you will write it algebraically. 180 takeaway 48, takeaway 62 equals a, or you may have used BIDMAS, 180 takeaway 48 plus 62. And you will have found that a is equal to 70 degrees. Question two your isosceles triangle. We know the two parts, 63 degrees. The missing part was a, so we're doing 180 takeaway 63, takeaway 63 equals a, or you may have done 180 take away 63 plus 63, even 63 times two. And you will have found that a was equal to 54 degrees. for question three you had a right angled triangle. So this one was 90 degrees. You know, I've got my letters all muddled up here. The angle was y here. So in fact, I'll change it to either. So you know that algebraically 180 takeaway 37, takeaway 90 is equal to a, and again you may have done that using BIDMAS to find that a is equal to 53 degrees. Question four, you have an isosceles with the two equal angles are unknown. So you know that 180 take away 32 is equal to a and b together. 180 takeaway 32 is 148. So a and b together 148 degrees, because they're equal we have to divide 148 by two, which gives us 74 degrees. So both a and b are equal to 74 degrees. In question five you were asked to calculate the angles y and z. And I gave you a hint, start off by finding angle z. Now you can see the angles z is on a straight line with 130. We know that the angles on a straight line add it to 180. So 180 subtract 130 equals z. 180 subtract 130 is 50 degrees. So now we're just looking at a triangle where we have two known parts and one unknown. So you're doing 180 takeaway 75 and 50, will give you y and that is equal to 55 degrees. Great work, now it's time for your final quiz. Pause the video and complete the quiz. Great work today. In our next lesson we'll be finding the value of missing angles in quadrilaterals. I'll see you then.

In this lesson, we will represent the angles in a quadrilateral pictorially and algebraically before learning how to calculate missing angles.

Find the value of missing angles in quadrilaterals

﻿Today is our last lesson in this unit of missing angles and lengths. Today we'll be calculating missing angles in quadrilaterals. All you'll need today is a pencil and piece of paper. Pause the video and grab your things if you haven't done already. So here's our agenda. We're going to find unknown angles in quadrilaterals. You'll start with your quiz to test your knowledge from the previous lesson, then we'll work on representing angles, pictorially before calculating missing angles. Then you'll do some independent learning, and a final quiz to test your knowledge from the whole of the unit. So we'll start with our knowledge quiz. Pause the video and complete the quiz, and then press restart once you're finished. Great work, now let's get straight on with it. We're going to be representing our angles pictorially. So we know that the angles in a quadrilateral add up to 360 degrees. We can use a bar model to represent this. So the whole is 360 degrees, and the angles, a, b, c, and d add together to make the whole. So as an algebraic expression, a plus b, plus c, plus d are equal to 360 degrees. Now it's your turn. Pause the video and represent this shape using a bar model and then an algebraic expression. Angles were marked as c because they were the same. In this case you actually know that those angles are 90 degrees. So our whole is 360. A plus b and c and c are equal to 360 degrees. Algebraically that's a plus b, plus c, plus c equals 360 degrees. Or you may have challenged yourself to use BIDMAS, and represent it using brackets. Two lots of c are added to b and a to get to 360 degrees. Now think about this shape. What shape do you think this is, thinking back to our third lesson in the unit, and what do you know about the angles in the parallelogram? So you know that opposite angles are equal. So we know that this angle is equal to this angle, and this angle is equal to this angle. Let's go for our pictorial representation. We know that a plus b plus c plus d are equal to 360 degrees. We also know that a is equal to c, and b is equal to d. We know that a plus b are equal to 180 degrees, and c plus d are equal to 180 degrees, and shown as a bar model, we can see that two lots of 180 degrees are equal to 360 degrees. This will make more sense when we start to introduce some actual numbers. So let's do that now. We're going to calculate missing angles in quadrilaterals. Here we have a quadrilateral. This is a trapezium because it has one pair of parallel lines. I can represent it using a bar model. The angles in a quadrilateral add up to 360 degrees. 360 degrees is my whole. And I know 45, 72 and 135 degrees are three of my parts. My unknown part is d. Algebraically that's 45, plus 72, plus 135, plus d is equal to 360 degrees. So to find the missing part, we need to rearrange the equation using the inverse. So 360 degrees takeaway off three known parts gives us our unknown part. Now, here is where we come into some difficulty, repeated subtraction, if I'm going to set it out like this, like a column subtraction, I'm going to find that I can't subtract five, two, and five from zero. I need to regroup from the tens. So one 10 into 10 ones. Now here I have a problem. 10 takeaway five is five, takeaway two is three. I can't take away another five. You can't subtract all of these numbers in one go. So we've got two options. And we talked about this yesterday. Now option one is to subtract in stages, okay? So here is my subtraction. I can either subtract in stages, so I can do 360 degrees subtract 45 degrees. And I'll show you my working. So it's 315. So then I do this 315 subtract the next known angle, which is 72 degrees. So now I'm on 243 degrees, and then I subtract my final known angle, 135 degrees. So then I can see the angle d is 108 degrees. Now my second option is to use my knowledge of BIDMAS, which can be more efficient in this case. So I know that what I'm actually doing here is 360 take away 45, 72, and 135. So I know that I'm doing 360 subtract, and I've added these altogether, 252, and that is equal to 108 degrees. So you can think about which strategy you prefer, okay? It's up to you, but it's important that you know all the strategies available. Let's do another one together. This is a delta. It has no equal sides. It has one internal reflex angle. So I can represent this pictorially. 360 degrees is the whole, and I've got these three known angles. The unknown angle is p. Remember that that's going to actually be a reflex angle. So it's going to be greater than 180 degrees. I can represent that algebraically. The three known parts plus the unknown parts give me the whole of 360 degrees. And then I rearrange it using the inverse to find the value of p. So I'm doing the whole, 360 subtract 21, 49 and 83, which gives me p. Remember, you can do repeated subtraction, or you can use BIDMAS. In repeated subtraction you would do 360 takeaway 21, and then 49 and then 83. And in BIDMAS you would do 360 takeaway 21, 49, and 83. And once you've calculated all of those, you will find that p is equal to 207 degrees. And I can sense check that. I know that that is a reflex angle. So that looks like it's possibly right. And then I can double check by adding the four parts back together, and they will give me 360 degrees. Now it's your time. Pause the video and represent this shape pictorially and algebraically. Then use your preferred strategy for calculating the missing angle. You may choose repeated subtraction, or you might choose BIDMAS. So your bar model will have looked like this. 360 degrees is the whole, and here you've three known parts and one unknown. This one was 90 degrees. So that's how I've known to put that one in there. Here's your representation algebraically. So the four part added together equal 360 degrees, and then rearranged using the inverse, 360 subtract the three known parts gives me the unknown. You may have done repeated subtraction in which you found that f is equal to 80 degrees, or you may have chosen to use BIDMAS, either way you find that f is equal to 80 degrees. Let's do one more together. This is called an isosceles trapezium. It's a trapezium because it has got one pair of parallel sides. It's an isosceles because it's got these two sides. The lengths are equal length, okay. We know that it has got two pairs of equal angles. So in this case, if we're only given one angle, we can work out the rest. We've been told the angle d is 78 degrees. Therefore we also know that angle c is 78 degrees. So we can see this pictorially. We know that 360 is the whole. We know that a and b are equal size. And we know two of the angles, when those two added together, give me 156 degrees. So I know that 360 takeaway the known angles gives me a plus b, and that is 204 degrees. So I know that a and b are equal. So I know that I must divide 204 by two to find the value of a and b. And they will be 102 degrees each. Now it's time for some independent learning. Pause the video and complete the task, and click restart once you're finished so that we can go through the answers together. Okay, for question one, you were asked to calculate the size of angle d. You will have drawn a pictorial representation because with 360 degrees as the whole. So algebraically 360 degrees subtract 45, 127 and 81 degrees gives us d, or you may have represented that using BIDMAS. And I know that those three angles added together gives me 253 degrees. And therefore I know that angle d is equal to 107 degrees. For question two, you were given two known angles or two unlabeled angles, but we knew this one was 90 degrees. So we know that these four added together give us 360 degrees. So 360 degrees subtract 144, subtract 79, subtract 90 are equal to f. We know that we could have used BIDMAS to represent this. 360 degrees subtract 144, 79 and 90 gives us f, and I've added those three together to give me 313. 360 degrees subtract 313 degrees is equal to 47 degrees. So f is equal to 47 degrees. In question three, we had a parallelogram. So we know that opposite angles are equal. So we know that angle d is 110 degrees. So we know that we have to do 360 takeaway 110, and another 110 will give us b and c, and b and c are equal. So 360 take away 220 is 140. So b plus c together gives us 140 degrees. If I divide 140 by two, then I get the answer to one of those, 140 divided by two is 70 degrees. So b is 70 degrees, and c the 70 degrees. In question four, you were asked to calculate the sizes of angles, x, y, and z. So starting with x, I can see that this is on a straight line with 84 degrees. I know the angles on a straight line add up to 180 degrees. So 180 degrees, the whole takeaway the known part, 84 degrees gives me x. And I know that 180 takeaway 84 is 96. So x is equal to 96 degrees. Now I look at the quadrilateral, I've got three known parts and one unknown, which is y. I know that 360, which is what the angles in a quadrilateral add up to, subtract, and I'm going to use BIDMAS, 138, 96, which we just worked out, and 67 will give me y. I know that those numbers added together gives me 301, and 360 takeaway 301 is 59 degrees. So y is 59 degrees. Now I can work out z, y and z are on a straight line. So the whole is 180, subtract 59 gives me 121. Z is 121 degrees. So that was a three step problem. Great work if you persevered on that one. Your final question, you were asked to find four unknown angles, or a four step problems. And I gave you the hint to work them out in alphabetical order. Let's start with a. A is on a straight line with 95 degrees. So a straight line is 180, subtract 95 degrees is equal to 85 degrees. So a is equal to 85 degrees. B is part of a full turn with 309 degrees. I know that full turn is 360 degrees. So 360 subtract 309 degrees gives me 51 degrees. So b is equal to 51 degrees. Now I'll get to c. I've just focused on my quadrilateral here, where I know that the whole is 360. So I'm doing 360 subtract, 35, 85 and 51. So 360 subtract these added together equal to 271. And that subtraction is 89 degrees. So c is 89 degrees. Onto my last one, c and d are on a straight line, which equals to 180 degrees. And I'm subtracting 89 to find c. And that is equal to 91 degrees. So to find d, so d is equal to 91 degrees. Amazing work, it's time for your final quiz, Pause the video and complete the quiz, and click restart once you're finished. In our next lesson, we'll be starting a new two week topic on coordinates and shape. I'll see you then.

Unit Overview: Missing angles and lengths

Lessons:

Find the value of missing angles

Compare and classify triangles

Compare and classify quadrilaterals

Find unknown angles in triangles

Find the value of missing angles in quadrilaterals