Lesson video

Hello there.

How are you today? My name's Ms. Coe, and I'm really excited to be learning with you today in this unit on using knowledge of part-whole structures to solve additive problems. If you are ready to get started, let's get going.

In this unit, we are looking at part-whole relationships and additive structures, and by the end of this unit, you will be able to say that you can explain how a combination of different parts can be equivalent to the same whole.

In this lesson today, we have one keyword, I'm going to say it and I'd like you to say it back.

Ready? Equivalent.

Great job.

Let's think about what that word means.

The word, equivalent, means they have the same value, so two expressions are equivalent if they have the same value.

This lesson today where we are explaining how a combination of different parts can be equivalent to the same whole is split up into two cycles.

In the first cycle, we're going to be comparing structures, and in the second cycle, we're going to be combining structures.

If you are ready, let's get started with the first cycle of this lesson.

In this lesson today, you are going to meet Sam, Lucas, Laura and Jun, and they are going to be helping us with our thinking and asking us some questions as we go along.

So Sam and Lucas are exploring using number rods.

Now you may have seen these in your classroom and you may have used them before, but they are different coloured rods that you can see on the screen here.

Sam wonders how many different ways we can combine the rods so that they are equal to the length of the orange rod.

So you can see the orange rod there, and she's wondering about how we can combine different rods so that they have a length that is equal to that of the orange rod.

So you can see that we have the brown rod there and we have the red rod and we can see that the brown rod and the red rod is equal to the length of the orange rod.

Lucas has found that pair and he says that the rods are equivalent to the length of the orange rod, they are equal in length.

So we can write that as an equation.

We can say that the orange rod is equal to the brown rod plus the red rod, and you can see that we've used the initials of the colours there to write our equations.

That's all it means.

O is equal to orange, BR is equal to brown and so on.

Remember, we can also write the equation like this, brown plus red is equal to orange, they mean the same thing.

Let's think about different combinations that we can use to equal the length of the orange rod.

Here's one here.

This is another pair of rods that is equivalent to the orange rod.

Hmm, I wonder if you can think about the equation that Sam might write to show this relationship.

That's right, we can write the orange is equal to the dark green rod plus the pink rod.

Laura has taken this one step further.

She has found three rods that are equivalent to the orange rod.

Should we have a look and see what she's done? There we go.

We can see that those three rods have an equal length to that of the orange rod, so we can say that the three rods are equivalent to the orange rod.

Jun's wondering how would we write that as an equation? I wonder if you can have a think about how you might write that as an equation now that we have three parts.

So that's right, we could write the orange rod is equal to the red rod, plus the light green rod, plus the yellow rod, and that's the equation that we'd write if we were using the initials of these coloured rods.

So Jun is gonna challenge Laura and he's asking whether we can find a way using only one type of rod.

So whether we can find a way that uses one type of rod only that is equal in length to the orange rod.

Hmm, I wonder what you think.

Laura suggests this.

What do you think, is that equal to the orange rod? Are those rods the same length as the orange rod? Well, yes they are, and we can write that as an addition equation.

I wonder what that would look like.

That's right, we can write the orange rod is equal to the red rod, plus the red rod, plus the red rod, plus the red rod, plus the red rod.

We have five red rods, so we can write that as an addition equation.

"What about a multiplication equation," asked Laura.

Hmm, well I can see a repeated addition equation here, so I think we can probably write it as a multiplication equation.

What do you think? Five lots of red or five lots of R is equal to the orange rod or to O.

So we can write that as O is equal to five times R, five groups of the red rod.

Time to check your understanding.

I'd like you to tick the equation that represents the image.

So you can see some rods on the screen and you have some different equations.

Which one represents that image? Take a moment to have a think.

Welcome back.

So let's have a think.

We can see the orange rod and we can see that there are three rods that are equal in length to the orange rod, the black, the red, and the white.

So we can see that all three of these equations represent the image because we have the black, plus the red, plus the white is equal to the orange.

Well done if you've got any of those and well done if you've got all three.

Let's do another quick check of your understanding.

This time I'd like you to tick the equation that represents this image.

Take a moment to have a think Welcome back.

So this time, we have the orange rod and it is equal to a number of rods of equal length, the white rods.

How many rods do we have there? Well, we have 10 white rods that are equal in length or equivalent to the orange rod.

So we could write this as 10 multiplied by W is equal to O, so 10 groups of white is equal to orange, or we can write that as a repeated addition equation.

So you can write W 10 times to show the 10 parts and this is equal to the orange.

Well done if you said both of those.

Okay, time for your first practise task.

Use the number rods to find equivalent sets of rods to the dark green rod.

So find the dark green rod in your set of number rods and find equivalent sets of rods.

Write the equation for each of your examples.

So here's an example that Laura came up with to set you up.

We can see that the yellow rod, plus the white rod is equal to equivalent to the dark green rod.

So we can write that as an equation using the initials of the colours.

I wonder how many different solutions you can find.

For question two, I'd like you to choose your own rod as the whole and write as many equations as you can to show sets of equivalent rods.

So Laura has chosen the blue rod as the whole, but you don't have to do that, you can choose a different rod.

I wonder how many solutions you can come up with for this one.

Pause the video here and have a go at those two tasks.

Welcome back.

How did you get on? I wonder which rod you chose for the second task.

For the first task, we asked you to think about the dark green rod and there are a few different combinations that you could have found that are equivalent to the dark green rod.

For example, the yellow rod plus the white rod is equivalent to the length of the dark green rod.

So you could have written an equation like this.

We could also have the pink rod plus the red rod, or two light green rods.

Now remember, we can write this equation as a repeated addition, or we can write it as a multiplication.

Now remember, for the second task, you may have chosen a different rod as your whole, but if we chose the blue rod, then there are lots of different combinations that we could have written.

And you can see these here, and you can see that we've written some as addition equations such as the brown rod plus the white rod, but we've also recognised that you can use equal parts.

So three lots of the light green rod is equal to the blue rod.

I wonder how many different combinations you came up with.

Laura found six different ways, but could she have found more? Let's move on to the second cycle of our lesson, which is combining structures.

So the children at Oak Academy are continuing to look at the equivalence to this orange rod.

The orange rod is the whole, and Sam says that she has found a different set of rods that are equivalent to the orange rod.

"Ooh," said Lucas, that looks a bit interesting.

Let's see what she's got.

Well, we can see that we have three red rods and a pink rod and they are equivalent to the length of the orange rod.

Lucas is wondering how we'll write this as an equation.

I wonder what you think.

Well, that's right Sam, we have three red rods and one pink rod and they are equivalent to or equal to the orange rod.

We can write this using addition.

So we can write O is equal to R, plus R, plus R, plus P.

Remember those initials represent the colours of the rods.

But because we have three red rods, we can write that part of the equation as multiplication as well.

Remember we have three equal groups of red or R.

So we can write O is equal to three multiplied by R plus P, three, lots of red plus the pink.

So that's right Lucas, we can combine addition and multiplication into one equation, we don't have to write just addition or just multiplication.

Can you create your own example with a number odds? Lucas has made a different example.

Can you see the whole and the parts there? Think about how you might write this as an equation.

Would you use addition or would you use a combination of addition and multiplication? Sam says this is a great example.

Well done, Lucas.

And let's write the equations.

So we have one black rod and three white rods which are equivalent or equal to the orange rod.

So we can write this using addition.

We can say that the orange rod is equal to the black plus the white rod, plus the white rod, plus the white rod, we can use the initials again, or, "Because we have one black rod and three white rods," says Lucas, "We can write this as a combination of addition and multiplication." "Nice work," says Sam.

We can see that we can write this as black plus three lots of white or three multiplied by white.

What about this example here? What can you see? What is equivalent to the orange rod? Lucas says, "That's an interesting one." I wonder why he says that.

Well this time, we have two light green rods and one pink rod that are equivalent equal to the orange rod, and we can write this using addition.

So we can say the orange rod is equal to the light green rod plus the pink rod, plus the light green rod.

Now even though the rods aren't next to each other, we can see that we have two light green rods.

So we can combine these to say that we have two lots of light green plus one pink rod, and we can write this using a combination of multiplication and addition.

Time to check your understanding.

Tick the equation that represents this image.

So we have a whole which is the orange rod, and we have some parts.

Which of the equations there represent this image? Take a moment to have a think.

That's right, it's B and C.

Let's have a look at why.

We have two red rods and a dark green rod, so we can write that as addition, the orange rod is equal to the red rod, plus a red rod, plus a dark green rod.

But because we have two of the same parts, we can write that bit as a multiplication.

We can write two times R or two lots of red plus the dark green rod.

Well done if you saw both of those.

Another check of your understanding.

This time I'd like you to draw a bar model or use number rods if you have them to represent the equation there.

Take a moment, have a think.

Welcome back.

So this time we have three multiplied by W plus P is equal to black, the black rod.

So remember that we can have a combination of multiplication and addition.

So three times W means three lots of white rod.

So we need three lots of the white rod, plus the pink rod, which is equal to the black rod.

Well done if you used number rods to make that or if you drew a representation of that.

Time for your second practise task.

For question one, I'd like you to write an equation using both addition and multiplication to represent these images.

So you can see that we have the blue rod as the whole for each example and we have different combinations of parts.

Look carefully where are the equal parts.

So you're writing a mixed addition and multiplication equation for each of these images.

For question two, I'd like you to create the following equations using number rods.

Remember, if you don't have number rods, you can draw a bar model.

So for A, for example, we have three multiplied by W, white, plus R, red is equal to Y, yellow.

So can you make those combinations using number rods? Good luck with those two tasks, and I'll see you shortly for some feedback.

How did you get on? Did you enjoy combining the different rods to find these different equations? So for question one, we asked you to write an equation for each of these images.

So for A, we can see that we have the blue rod as the whole and we have a combination of parts.

Now I can see that some of those parts are equal in size, so I could have written the light green rod, plus six lots of or groups of the white rod is equal to the blue rod.

So I can write LG plus six, multiplied by W is equal to blue.

For B, I can see that the red rod are my equal parts there, so I could have written three times R, three multiplied by red, plus light green is equal to blue, and finally, where are my equal parts here? That's right, they're the pink rods even though they're not next to one another.

So I have two lots of, two groups of the pink rod there.

Well done if you wrote those three equations.

For question two, you had to create the equations using number rods.

So let's look more closely at A.

We had three multiplied by W plus R equals Y.

Now I know that the W represents the white rod and I have three lots of whites, plus the red is equal to the yellow.

So the yellow is my whole, and I should have three white parts and one red part.

Remember, it doesn't matter which way round those go.

So the yellow could be on top for you, that's absolutely fine.

For the second one, we had three lots of red, plus the white is equal to black.

The next one was a little bit more complicated because we had two lots, two sort of groups of equal parts.

So we had two lots of the light green rod and three lots of the white rod.

Now remember, it doesn't matter the order you put those rods together.

So you might have started with a light green rod, that's absolutely fine.

And then finally, we had a really complicated one.

We had two lots of the blue rod, so we had two lots of the blue rod for the whole is equal to four lots of the light green rod, plus six lots of the white rod.

And you can see here that we've arranged these in any order.

It doesn't matter as long as we had six white rods and four light green rods.

Well done if you got all of those.

We've come to the end of the lesson and I'm really, really proud of all the work that you've done and how you've really thought carefully about those parts and wholes and the difference when we think additively or multiplicatively, so thinking about addition and multiplication.

Wholes can be composed of different parts, this can be recorded additively.

Wholes can be composed of the same parts and this can be recorded additively using addition, and multiplicatively using multiplication.

Wholes can also be composed from a combination of different parts and the same parts, and we can record this using both addition and multiplication in the same equation.

Thank you so much for all your hard work today, and I look forward to seeing you in another math lesson soon.