Lesson video

Hello, my name is Mr. Tazzyman, and today I'm gonna be teaching you a lesson from a unit that is all about solving problems featuring two unknowns.

You might have come across these kinds of problems previously, but in this unit we are really going to look at comparing their structures and trying to solve them using things like trial and improvement and maybe bar models where we can.

So, make sure you're ready to learn and let's go for it.

Here's the outcome for today's lesson, then.

By the end, we want you to be able to say, I can explain how to balance an equation with two unknowns.

These are the key words that you might expect to hear during the lesson, difference and subtrahend.

I'm gonna say them and I want you to repeat them back to me.

I'll say my turn, say the word and then I'll say your turn and you can repeat it back.

Ready? My turn.

Difference.

Your turn.

My turn.

Subtrahend.

Your turn.

But what do these words actually mean? Well, let's make sure that we know that.

The results of subtracting one number from another number is known as the difference.

The value used to subtract from the whole is known as the subtrahend, and there's an example equation at the bottom there.

5 subtract 4 is equal to 1.

In that equation, 5 is called the minuend.

4 is the subtrahend and 1 is the difference.

This is the outline for today's lesson, then.

We're gonna start by finding equal addition expressions.

Then we're gonna move on to finding equal subtraction expressions.

Sofia and Laura are gonna join us today.

They're gonna be discussing some of the maths prompts to help us to learn.

Hi Sophia.

Hi Laura.

Okay, everyone ready? Let's get learning.

How could you solve this problem? We've got a missing number added to 55 is equal to a missing number added to 25.

Sophia says, "We have two unknowns and there is no context to limit what solutions we could use, so we could use whole numbers, decimal numbers, or even negative numbers." Laura says, "I can see that the known addends have a difference of 30.

That means the second addend must be 30 more than the first addend to make the expressions equal.

So if the first addend was A, then the second addend would have to be A plus 30." "So if the first addend was 10, then the second addend would be 40." "Let's check," says Laura.

10 added to 55 is equal to 40 added to 25.

65 is equal to 65.

It's correct.

It works.

"There we go," says Laura.

What if our problem changed to this? What do you notice now? So you can see we've got the problem that we just thought about and if we moved it onto a missing number added to 60 is equal to a missing number added to 30, what would be different? "Both the known addends have increased by the same amount.

They have both increased by 5." "Does that mean we need to add 5 onto both of the unknown addends too?" "No, because the difference between the two known addends is still the same.

The difference between 60 and 30 is the same as the difference between 55 and 25." "Oh, I see.

So again, if we called the first unknown addend A, then the second addend would still be A plus 30." How could you solve this problem? "If the first addend was 10, then the second addend would be 40 again." "Let's check again." 10 added to 60 is equal to 40 added to 30.

70 is equal to 70.

"There we go.

We have applied the same difference rule to help us.

In the original equation, we had a difference of 30 between the second addends.

We can show that on our number line here." So you can see that we've got 55, subtract 30 is equal to 25.

"The second equation also had a difference of 30 between the second addends.

We can see that the difference remains the same, it just moves up on our number line." Okay, let's check your understanding of what we learned so far.

Change the second set of addends in this equation so that the difference would remain 30.

Okay, pause the video and have a go at that.

Welcome back.

"I changed the 60 to 82, which is an increase of 22, so I would need to increase the other second addend by 22 as well." There you can see it's changed to 52.

What if our problem changed to this? What do you notice now? "The second addend in the first expression has increased by 5 from 55 to 60.

But this time the second addend in the second equation has decreased by 5." "How will that change the difference?" asks Laura.

Hmm? I wonder.

"In the original equation, we had a difference of 30 between the second addends.

We can show that on our number line here." So there it is.

"The second equation, however, had a difference of 35 between the second addends.

So the difference has increased in size." Okay.

How could you solve this problem then? "Well, so if we called the first addend in the first expression A, we would need to call the first addend in the second expression A plus 35." "So if we use 10 as the first addend, the other addend would need to be 5 more than what it was before, so 45.

Let's check." Well done, Laura.

Always important to check.

10 added to 60 is equal to 45 added to 25, 70 is equal to 70.

"There you go." Okay, your turn again.

This time you're gonna find the difference between the second two addends and then fill in the missing value.

Pause the video and give it a go.

Welcome back.

Let's see what Sofia thought about this.

"The difference between the second addend has increased by 20, so if the first addend was A, then the first addend of the second expression would be A plus 50." That means that the missing number was 60.

10 plus 70 is equal to 60 plus 20.

80 is equal to 80.

We've checked it.

Okay.

It's time for your first practise task.

Find the difference.

Then give three sets of solutions for each example.

For number two, you've got to find the following, A, a solution using whole numbers, B, a solution using decimals, C, a solution using negative numbers and D, a solution using whole numbers and negative numbers.

You can see that we've got the equation at the top there.

We've got an unknown added to 20 is equal to an unknown added to 35.

Pause the video here and give those questions a go.

Good luck.

Welcome back.

Let's do some feedback then.

Be ready to mark.

Here's three sets of solutions for the first example, 1 and 11, 2 and 12, 3 and 13.

And for the second, you've got the same set of solutions.

For the third, you've got 1 and 13, 2 and 14, 3 and 15.

Slightly different.

For the fourth, 1 and 15, 3 and 16, 3 and 17.

And for the last one, 1 and 19, 2 and 20, 3 and 21.

Now of course we know that there might be many more solutions than this.

Pause the video here if you need to discuss any of the other ones that you've got.

Let's look at number two then, now.

26 and 11 would be a solution using whole numbers.

20.

5 and 5.

5 would be a solution using decimals.

Negative 20 and negative 35 would be a solution using negative numbers.

And 5 and negative 10 would be a solution using whole numbers and negative numbers.

Okay, these are just one solution for each of these.

You might have some different numbers, but they could still be the correct solution.

The important difference is that the difference between them is 15.

Pause the video here if you need to do some more marking.

Okay, let's move on to the second part of the lesson, finding equal subtraction expressions.

How could you solve this problem? An unknown takeaway 55 is equal to an unknown takeaway 25.

Sophia says, "We have two unknowns again, but this time each expression is a subtraction expression." "Let's start by using numbers that would make each expression a positive whole number." Okay, we've got a table drawn out here.

Value of the first missing number is the first column.

Value of one side of the equation is the middle column and value of second missing number is the last column.

"Let's start with 60 as the first missing number." So they plunk 60 into the table and into the equation.

"That would leave the value of the first expression to be 5." "We now need the second expression to be equal to 5 too.

That means the second missing value needs to be 30." They've got one row of their table completed.

"Let's try a different number now.

Let's try 100." In goes 100.

"That would leave the value of the first expression to be 45." "We now need the second expression to be equal to 45 too.

That means the second missing value needs to be 70." "Why don't we try a negative number? Let's try a negative five.

Negative 5 subtract 55 is equal to negative 60." "We now need the second expression to be equal to negative 60 too.

That means the second missing value needs to be negative 35." Okay, so what do we notice then when we look at this table of calculated values? "I think I've noticed something," says Sofia.

60 subtract 30 is equal to 30.

100 subtract 70 is equal to 30.

Negative 5 subtract negative 35 is is equal to 30.

"The difference between the first missing number and the second missing number is always 30." "Of course.

That's because the difference between the subtrahends is also 30." You can see there the subtrahends have been circled.

"55 subtract 25 is equal to 30." Okay, let's check your understanding then.

Choose a pair of values that would balance this equation.

This is the same equation that we've just been working on.

Pause the video and give that a go.

Welcome back.

Sofia chose 200 and 170.

Those two values have a difference of 30.

Whereas Laura chose 75.

25 and 45.

25.

Again, those two numbers have a difference of 30.

"Whatever the first missing value is, the second missing value must be 30 less," "So if the first value was A, the second value would be A take away 30." What if our problem changed to this? What do you notice now? So they've taken that problem that we've just been discussing and changed it.

It's become an unknown subtract 60 is equal to an unknown subtract 30.

"The subtrahend in the first expression has increased by 5 from 55 to 60.

And the second subtrahend has also increased by 5.

How will that change the difference?" "Just like earlier, there will be no change in the difference as both subtrahends have increased by the same value.

The difference is still 30.

Let's check it out." 75 subtract 45 is equal to 45 subtract 25, 20 is equal to 20.

Those two values work for that first equation.

Let's put the same values into the second equation then, because we think that this will work as well.

75 subtract 60 is equal to 45 subtract 30, 15 is equal to 15.

"The equations balance." "And the best bit is you can choose any two unknowns with a difference of 30, and they will always work for these subtrahends." Okay, let's check your understanding again, then.

Find two subtrahends that would also have a difference of 30 to balance the equation.

Pause the video and give that a go.

Welcome back.

We've chosen 50 and 20.

"Here's an example I've come up with," says Laura.

All right, it's time for your first practise task.

Find the difference then give three sets of solutions for each example.

For number two, find three different pairs of subtrahends that would balance this equation.

Pause the video and give those a go.

Welcome back.

Let's look at number one, then.

We've got some example solutions for each.

You could have had 100 and 90, 99 and 89, 98 and 88.

Or you could have had on the second one, 100 and 90, 99 and 89, 98 and 88.

You can see that those sets of solutions are identical for the first two.

That's because the subtrahends have increased by the same amount in both expressions.

50 to 52, an increase of 2, and 40 to 42, an increase of 2 as well.

Here are some example numbers for the third one, making sure that the difference is 12.

The second number needed to be 12 less.

100 and 88, 99 and 87, 98 and 86.

For the fourth one, 100 and 86, 99 and 85, 98 and 84.

And for the last one, 100 and 82, 99 and 81, 98 and 80.

Pause the video here if you need extra time to discuss any of your solutions.

Here's number two, then.

Sofia's got some examples of different pairs that would balance this equation.

"The difference between the minuends is 20, so we need two subtrahends with a difference of 20." Here are three examples, 50 and 30, 45 and 25, 20 and 0.

Okay, that brings us to the end of the lesson, then.

Here's a summary.

You can use your knowledge of the structure of addition and subtraction to help you balance equations.

You can apply the same difference rule to help you find unknown values in equations that require balancing.

My name's Mr. Tazzyman.

Hope you enjoyed that today.

I did.

Maybe I'll see you again soon.

Bye for now.