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I can explain that addition and subtraction of fractions are inverse operations.

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Licence

This content is © Oak National Academy Limited (2023), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Key learning points

You know that if 3 + 4 = 7 then 7 - 4 = 3
Facts about whole numbers can be applied to the numerators of fractions with the same denominator.
If 3 tenths plus 4 tenths is equal to 7 tenths then 7 tenths subtract 4 tenths is equal to 3 tenths.

Common misconception

Children might add or subtract the denominator as well.

Reinforce to children that the denominator denotes the unit fraction being used in calculation so does not need changing. Count aloud with physical objects to start with before moving onto counting in unit fractions.

Keywords

Additive - Additive equations involve addition or subtraction.
Inverse - Inverse means the opposite in effect or reverse of. For example, subtraction is the inverse operation of addition.
Numerator - A numerator is the top number in a fraction. It shows how many parts we have.
Denominator - A denominator is the bottom number in a fraction. It shows how many parts a whole has been divided into.

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Hi there, my name is Mr. Tazyman and I'm really looking forward to teaching you this lesson today from the unit, all about the composition of non unit fractions.

Some people can find fractions a bit tricky sometimes, but I hope, with what I'm about to show you, I can make sure that you finish this lesson feeling really confident and ready to tackle any fractions that come your way.

Okay, we're gonna get started.

Here's the outcome for the lesson.

By the end, we want you to be able to say, I can explain that addition and subtraction of fractions are inverse operations.

These are the key words that you might expect to see.

I'd like you to repeat them back to me, please.

So I'll say my turn, say the word and then I'll say your turn, and you say them back.

Ready? My turn, additive, your turn.

My turn, inverse, your turn.

My turn, numerator, your turn.

My turn, denominator, your turn.

Well done.

Let's check the meanings of all those words.

Additive equations involve addition or subtraction.

Inverse means the opposite in effect or reverse of, for example, subtraction is the inverse operation of addition.

A numerator is the top number in a fraction it shows how many parts we have.

A denominator is the bottom number in a fraction it shows how many parts a whole has been divided into.

Okay, this is the structure of the lesson.

We're gonna start by using inverse operations and then we're gonna look at additive fraction equations.

All of this will help us to explain that addition and subtraction of fractions are inverse operations.

We'll start with using inverse operations, then.

We've got some friends here, Jacob and Aisha.

These two are gonna help us by explaining some of the maths that we come across giving us some hints and tips and also through their own thinking, which will help ours.

Ready, Jacob? Ready, Aisha? Ready, everyone? Let's start.

Jacob and Aisha are performing a dance.

They start by jumping forward three times.

One, two, three.

They turn around and jump three times.

One, two, three.

Where do they end up? The dance is an example of the inverse.

You can use a number line to show this.

So we've added three.

They turn around and then we subtract three and they end up back where they started.

Okay, let's check your understanding so far, true or false? The inverse add 2 is subtract 2, pause the video here and decide what you think.

Welcome back, that was true, but what is the justification for that? There are two here and you need to choose which one you think is best.

Is it A, they're different operations so they can't be related? Or is it B, if you add and subtract the same number, the process appears undone.

Pause the video and decide which you think is best.

Welcome back.

B, was the best justification here.

If you add and subtract the same number, the process appears undone.

This time Jacob and Aisha complete diddy jumps one quarter of the length they did previously.

Tiny little jumps.

They complete three one quarter jumps, one quarter, two quarters, three quarters.

They turn around and jump three one quarter jumps back.

One quarter, two quarters, three quarters.

Where do they end up? The diddy jumps are another example of the inverse.

You can use a number line in a similar way to show this + 1/4, + 1/4, + 1/4, turn around, 1/4, - 1/4, - 1/4, and they've ended up back where they've started.

Okay, let's check your understanding again, true or false? The inverse of add one third is subtract one third.

What do you think? Pause the video and I'll give you the answer shortly.

Welcome back, that was true, but what was the justification for it? Well, here's two, and I want you to choose the best.

A, they both have a denominator of three or B, if you add and subtract the same fraction, the process appears undone.

Pause the video and decide which you think is correct.

Welcome back, B was the correct justification here.

If you add and subtract the same fraction, the process appears undone.

Ready to move on? Let's go for it.

Jacob and Aisha play a game using a number line.

Jacob is Amy Adder so always adds and the objective is to get the counter to 10.

Aisha is Stevie Subtract, so always subtracts and the objective is to get the counter to zero.

Stevie Subtract, with a subtraction symbol and Amy Adder with an addition symbol.

\"Start on five because it's halfway,\" says Aisha.

There it is a counter on 5.

\"I'll start.\" Jacob rolls a 2.

Add 2, it goes to 7.

\"So I subtract 2 to undo,\" back to 5.

\"Now I roll.\" Aisha rolls a 3.

Subtract 3, she lands on 2.

\"And I add 3 to undo,\" says Jacob.

Back to 5, \"Hmm, this isn't very fun.

We are using the inverse to undo.

Why don't we each have a dice? Then we might be able to do more than undo, but we will still be using the inverse operation.\" Remember the inverse operation means subtraction is the inverse operation to addition and vice versa.

\"Yes, great,\" says Jacob.

This time they each have a dice to use.

Can you work out where the counter will go after each of their inverse roles? \"I'll start,\" says Jacob, he rolls a 2.

Where do you think the counter's gonna go? It's gone to 7, because he is Amy Adder.

Aisha rolls a 5, she's Stevie Subtract so where do you think that counter's gonna end up now? It ends up on 2.

Jacob rolls a 6, it ends up on 8.

Stevie rolls a 1, where's it gonna end up? 7.

Jacob rolls a 3, where's it gonna end up? It's on 10.

\"I win,\" says Jacob.

Jacob and Aisha discuss the game, \"We were both acting as the inverse operation,\" says Aisha.

\"Yes, I was adding and you were subtracting.\" \"Could we play using fractions do you think?\" Oh, that's a good idea, Aisha, what do you think? Do you think they could use fractions here? \"The game we played was adding and subtracting using ones as the unit.

\"Why don't we use 1/10 as the unit instead of ones?\" \"Yes, let's try it.\" So they've changed the number line.

You can see the labels now are in tenths rather than just in ones.

They play again, \"I'll start this time,\" says Aisha.

She rolls a 3.

That's her numerator.

She's subtracting 3/10 because she's Stevie Subtract.

Where do you think it's going to end up? \"5/10 subtract 3/10 equal to 2/10.\" Jacob rolls 4 as the numerator.

So he's adding on 4/10.

\"2/10 add 4/10 is equal to 6/10.\" There it is, Stevie rolls a 5, we're subtracting 5/10.

So where will the counter end up? It's gone to 1/10, because 6/10 subtract 5/10 is 1/10.

Jacob rolls a 5.

Where will the counter end up? He's adding 5/10, 1/10 add 5/10 is equal to 6/10.

Aisha rolls a 6, she's taking away 6/10.

6/10 subtract 6/10 is equal to 0/10.

Aisha has won.

Jacob and Aisha discussed the game further.

\"Do we have to use 1/10? Could we use a different denominator?\" What do you think? \"Yes, we could, but we would have to change the number of intervals.\" \"Okay, should we try 1/5?\" There's a number line with 1/5.

\"I don't think they will work very well.

With a six sided dice, the game will be over quickly.\" \"That's true.

You've a strong chance of winning on your first go.

Also, there's no fair place to start from.\" Have a look at the number line.

There isn't a middle part, there isn't a middle tick.

\"We need a smaller fraction, which means a greater denominator.

Shall we do 1/16? They're small enough.\" Wow, look at that, they changed the number line so that it features sixteenths instead.

\"Yes, okay, we can start on 8/16.\" That's halfway.

Here we go, 5/16 for Amy Adder.

8/16 plus 5/16, 13 sixteenths.

2/16 for Stevie Subtract 13/16 subtract 2/16, that's 11/16.

4/16 for Amy Adder, 11/16 plus 4/16 is equal to 15/16.

4/16 for Stevie Subtract.

15/16 subtract 4/16 is equal to 11/16.

6/16 for Amy Adder, 11/16 plus 6/16 it goes off the scale, it goes off the number line.

\"I've gone past 16/16, that means I win.\" \"Well done,\" says Aisha.

Okay, let's check your understanding then, true or false? Addition and subtraction are inverse operations when using fractions as units.

Pause the video and decide whether you think that's true or false.

Welcome back, that was true, but let's look again at a justification.

You've got two choices here.

Is it A, changing the unit to a fraction doesn't change how addition and subtraction function or B, because you can always add and subtract anything.

Which of those do you think is a good justification for our statement that's true? Pause the video and have a think.

Welcome back.

The first one was the best justification here.

Changing the unit to a fraction doesn't change how addition and subtraction function.

Okay, it's time for your first task.

Very simply, you are gonna have a go at playing your own versions of the game.

Someone needs to be Amy Adder and someone needs to be Stevie Subtract.

Choose a denominator that is an even two digit number less than 30, such as 12 or 28.

Pause the video here and have a good go at playing that game and I'll be back in a little while with some feedback.

Enjoy.

Welcome back.

Did you have a go at playing? Aisha says they used several different denominators, \"At one point we used 1/26.\" \"We needed to use a 10 sided dice to be able to finish that game,\" says Jacob.

Did you find that? Did any of you think of swapping your 6-sided dice for something greater? Okay, that's the first part of the lesson done.

Let's move on to the second part now.

Additive fraction equations.

Aisha sets Jacob a challenge involving additive equations.

\"I've got a challenge for you,\" she says.

\"Here is a number line with an addition on it.

Can you write a pair of equations from this?\" \"Hmm, challenge accepted,\" says Jacob.

I'll start with the equation the number line shows.

It starts on 3 and adds 4, which is equal to 7.\" 3 + 4 = 7.

There's the equation.

Now I can use the inverse as well to create a subtraction.

He's put it on the number line and there's the equation.

7 - 4 = 3.

\"Brilliant.

Have you got a challenge for me?\" Says Aisha.

\"Yes.

I'm going to use 1/10 on the number line.\" Aha, so it's a bit like the game you played earlier.

Instead of using integers, they're gonna swap them for 1/10.

There they go, so 1/10 is now the unit we are using.

\"Your challenge, how does that change the equations?\" Really good challenge that, Jacob.

What do you think? How does that change the equations that Jacob came up with? Well, Aisha says, \"The numbers in your equations are the numerator.

The number line is 1/10, so they all need to have a denominator of 10.

There they are.

Okay, let's check your understanding then.

Can you write a pair of equations using the number line below? Pause the video and have a go at that.

Welcome back.

Let's reveal those equations then.

So we added on 3/5 on our number line.

So we needed to have 1/5 + 3/5 = to 4/5.

We can also use the inverse where we're taking away 3/5.

4/5 - 3/5 = to 1/5.

Are they the equations you found? Jacob decides to set another additive equation challenge this time using a bar model instead of a number line.

It's always good to use a range of representations to help to strengthen our understanding of what it is that we are learning about.

\"Here is a bar model with 2 parts and a whole.\" There it is.

\"Can you use it to write four equations?\" \"Okay, first I'll use 9 and 3 as addends.

So 9 + 3 = 12 Addends are commutative, so also 3 + 9 = 12.

This model also shows subtraction.

12 is the minuend, so 12 - 9 = 3.

The subtrahend could also be 3, which is 12 - 3 = 9.

\"Well done.

Can you challenge me now?\" \"What if we change the unit from ones?\" This is a bit like what they did last time.

\"What if we used 1/15 instead? \"Okay, the bar model and equations need to change.

The current numbers are the numerators.

They tell us how many 15ths they need a denominator of 15.\" And he puts the denominator in for all of them.

Your turn.

Let's check that understanding.

Can you find four additive equations from this bar model? And remember, additive means addition or subtraction.

Pause the video and have a go.

Welcome back.

Did you manage to find some equations that were additive? Here's one, 4/8 + 3/8 = 7/8.

3/8 + 4/8 = 7/8, that's using commutative there.

The addends have been switched over.

7/8 - 4/8 = 3/8 and 7/8 - 3/8 = 4/8 using that partitioning structure.

Here's the second task, then.

Use the number lines below and your knowledge of the inverse to find pairs of equations.

You can see you've got two different number lines there.

Both of them have got a jump on.

Just be careful, look at the direction of the jump.

I wonder which operation that is.

Here's number 2, from the bar models below write four additive fraction equations.

And here's number 3, use your understanding of the inverse to find the missing numbers in these equations.

\"What do you notice?\" Pause the video here and I'll be back in a little while to give you some feedback.

Welcome back.

Are you ready to mark? Well, let's look at 1, first.

A, we could have had 4/10 + 3/10 = 7/10, 7/10 - 3/10 = 4/10.

So we had a number line that showed addition to begin with, but we could then use the inverse to give us a second equation that featured subtraction.

Let's look at B, 10/16 - 9/16 = 1/16.

Then we can use the inverse and that tells us that 1/16 + 9/16 = 10/16.

Here's number 2, from the bar models below, write four additive fraction equations.

So we can see those bar models have got fractions in them.

In the first one, the denominator is 6, that means sixths.

We don't need to change that denominator, because of course denominators stay the same when you are adding and subtracting.

3/6 + 1/6 = 4/6, use that commutativity.

We can also say that 1/6 + 3/6 = 4/6.

4/6 - 3/6 = 1/6, 4/6 - 1/6 = 3/6.

So those are the four equations that you could get from that bar model.

Now let's look at B.

B featured ninths.

And again, the denominator needed to be the same across all four equations.

So we might have started with 2/9 + 5/9 = 7/9 addition is commutative, so we could also have 5/9 + 2/9 = 7/9, then we can use the inverse.

7/9 - 5/9 = 2/9, 7/9 - 2/9 = 5/9.

Okay, ready to move on? Here's number 3.

We've got some missing numbers here.

So on A we had 2/7 + 2/7 = 4/7.

For B, 4/7 - 2/7 = 2/7.

For C, 7/10 = 2/10 + 5/10.

And for D, 2/10 = 7/10 - 5/10.

For E, 5/11 = 2/11 + 3/11.

And for F, 2/11 = 5/11 - 3/11.

Hopefully you've got all of those, but Jacob asked, \"What did you notice?\" So you might have decided to compare and look at some of the differences and similarities there.

Aisha says, \"The pairs of questions are inverses of one another.\" What does she mean? Well, let's look at A and B.

We've got 2/7 + 2/7 = 4/7 and the inverse of that is that 4/7 - 2/7 = 2/7.

We've managed to undo by subtracting the same amount that we added.

That's the end of the lesson then, here's a summary.

Addition and subtraction are inverse operations.

This is also true when using fractions instead of whole numbers.

The inverse operations of addition and subtraction can be used to find other equations and known facts involving fractions.

The inverse operations of addition and subtraction can also be used to solve missing number problems involving fractions.

My name is Mr. Tazyman and I've really enjoyed learning with you today.

I hope to see you again soon in another maths lesson.

Bye-bye for now.

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\\frac{1}{7}

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\\frac{1}{8}

of the whole bag. How many marbles are there in the bag?" }, "questionType": "Multiple choice", "correct": [ { "text": "8" } ], "choices": [ { "text": "1" }, { "text": "8" }, { "text": "9" }, { "text": "10" } ], "id": "19d478b6-35a8-4c52-d7a6-3bd3919817d9", "lessonId": "d38e7808-7b7c-930d-16e9-b36b4929a0e2" }, { "type": "Question", "question": { "text": "A full box of eggs contains 12 eggs. First there were 9 eggs in the box, then 6 were used to make pancakes. Which equation matches this problem?", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1713527089%2Frbanmfv2kkvqqqgdvjxq.png&w=640&q=75", "imageKey": "4e96ac48-c9e8-413a-0369-5eba32a939dd/question/b83e209a88db45544dfc1503e952a902.png" }, "questionType": "Multiple choice", "correct": [ { "text": "

\\frac{9}{12}

−

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" } ], "choices": [ { "text": "

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" }, { "text": "

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\\frac{4}{12}

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\\frac{4}{10}

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\\frac{1}{10}

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" }, { "text": "

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−

\\frac{5}{10}

" } ], "choices": [ { "text": "

\\frac{9}{10}

−

\\frac{5}{10}

" }, { "text": "

\\frac{2}{10}

\\frac{3}{10}

" }, { "text": "

\\frac{1}{10}

\\frac{2}{10}

" }, { "text": "

\\frac{7}{10}

−

\\frac{5}{10}

" } ], "id": "c37328a3-97cc-66b7-4597-598f7dfc8450", "lessonId": "d38e7808-7b7c-930d-16e9-b36b4929a0e2" }, { "type": "Question", "question": { "text": "What are the possible values of the missing numerator?", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1713527091%2Fkvccbadd1u1nxarvqepa.png&w=640&q=75", "imageKey": "4e96ac48-c9e8-413a-0369-5eba32a939dd/question/ef29d1cd6ac61354a0022d6e10afe123.png" }, "questionType": "Multiple choice", "correct": [ { "text": "1" }, { "text": "2" }, { "text": "3" } ], "choices": [ { "text": "1" }, { "text": "2" }, { "text": "3" }, { "text": "4" }, { "text": "5" } ], "id": "eb4612b4-b8a3-7df8-202a-b90d1df1fdc5", "lessonId": "d38e7808-7b7c-930d-16e9-b36b4929a0e2" } ], "lessonId": "d38e7808-7b7c-930d-16e9-b36b4929a0e2", "meta": { "gradeStageId": "ks2", "subject": "Maths", "grade": "Year 3", "topic": null, "unit": "Composition of non-unit fractions: addition and subtraction", "lessonCount": "10", "age": [ 7, 8 ], "gradeId": "age_7_8", "gradeStage": "Key Stage 2", "lesson": "Explain that addition and subtraction of fractions are inverse operations" } }, { "id": "d099960c-9214-a344-f876-4f81616c015b", "type": "Resource", "resourceType": "Quiz", "title": "Explain that addition and subtraction of fractions are inverse operations", "quizType": "Exit quiz", "items": [ { "type": "Question", "question": { "text": "Which equation is the inverse of 5 + 3 = 8?" }, "questionType": "Multiple choice", "correct": [ { "text": "8 − 3 = 5" } ], "choices": [ { "text": "3 + 5 = 8" }, { "text": "8 − 5 = 3" }, { "text": "8 − 3 = 5" }, { "text": "3 − 5 = 8" } ], "id": "ef95a70a-5ab2-83b9-42e9-61b8ff1f7ef9", "lessonId": "d38e7808-7b7c-930d-16e9-b36b4929a0e2" }, { "type": "Question", "question": { "text": "Which equation does the number line show?", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1713527085%2Fvi0byvtnt3aaoejw3bha.png&w=640&q=75", "imageKey": "d099960c-9214-a344-f876-4f81616c015b/question/0338724b7ae1ada3a7e1e3d0e402d212.png" }, "questionType": "Multiple choice", "correct": [ { "text": "

\\frac{2}{10}

\\frac{5}{10}

\\frac{7}{10}

" } ], "choices": [ { "text": "

\\frac{7}{10}

\\frac{2}{10}

\\frac{9}{10}

" }, { "text": "

\\frac{7}{10}

−

\\frac{2}{10}

\\frac{5}{10}

" }, { "text": "

\\frac{2}{10}

\\frac{5}{10}

\\frac{7}{10}

" } ], "id": "7e39dc45-f071-22a1-ec6c-405cf324b7cb", "lessonId": "d38e7808-7b7c-930d-16e9-b36b4929a0e2" }, { "type": "Question", "question": { "text": "True or false.\nAddition and subtraction are inverse operations when using fractions as units." }, "questionType": "True or false", "correct": [ { "text": "True" } ], "choices": [ { "text": "True" }, { "text": "False" } ], "id": "884ea89a-8167-7860-861a-6d032abd4349", "lessonId": "d38e7808-7b7c-930d-16e9-b36b4929a0e2" }, { "type": "Question", "question": { "text": "What is the inverse of this calculation shown on the number line?", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1713527085%2Fvi0byvtnt3aaoejw3bha.png&w=640&q=75", "imageKey": "d099960c-9214-a344-f876-4f81616c015b/question/0338724b7ae1ada3a7e1e3d0e402d212.png" }, "questionType": "Multiple choice", "correct": [ { "text": "

\\frac{7}{10}

−

\\frac{5}{10}

\\frac{2}{10}

" } ], "choices": [ { "text": "

\\frac{7}{10}

−

\\frac{2}{10}

\\frac{5}{10}

" }, { "text": "

\\frac{7}{10}

−

\\frac{5}{10}

\\frac{2}{10}

" }, { "text": "

\\frac{7}{10}

−

\\frac{2}{10}

\\frac{2}{10}

" } ], "id": "10fe71eb-9391-5d30-35e1-6c2eec73c184", "lessonId": "d38e7808-7b7c-930d-16e9-b36b4929a0e2" }, { "type": "Question", "question": { "text": "Which equation is the inverse of

\\frac{3}{10}

\\frac{6}{10}

\\frac{9}{10}

?" }, "questionType": "Multiple choice", "correct": [ { "text": "

\\frac{9}{10}

−

\\frac{6}{10}

\\frac{3}{10}

" } ], "choices": [ { "text": "

\\frac{9}{10}

−

\\frac{6}{10}

\\frac{3}{10}

" }, { "text": "

\\frac{3}{10}

−

\\frac{6}{10}

\\frac{9}{10}

" }, { "text": "

\\frac{9}{10}

−

\\frac{3}{10}

\\frac{6}{10}

" } ], "id": "de5a09e1-8cbb-7734-7d06-257c9b8c95e3", "lessonId": "d38e7808-7b7c-930d-16e9-b36b4929a0e2" }, { "type": "Question", "question": { "text": "Which of the additive equations is incorrect?", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1713527084%2Fi1q3smzwm1t8i1fbjecg.png&w=640&q=75", "imageKey": "d099960c-9214-a344-f876-4f81616c015b/question/03aeb7bdf13c9d206483c1fd743049c2.png" }, "questionType": "Multiple choice", "correct": [ { "text": "

\\frac{9}{15}

−

\\frac{3}{15}

\\frac{12}{15}

" } ], "choices": [ { "text": "

\\frac{12}{15}

−

\\frac{9}{15}

\\frac{3}{15}

" }, { "text": "

\\frac{9}{15}

−

\\frac{3}{15}

\\frac{12}{15}

" }, { "text": "

\\frac{9}{15}

\\frac{3}{15}

\\frac{12}{15}

" }, { "text": "

\\frac{3}{15}

\\frac{9}{15}

\\frac{12}{15}

" } ], "id": "233cecc5-2844-e060-4ec1-338a7aef15e9", "lessonId": "d38e7808-7b7c-930d-16e9-b36b4929a0e2" } ], "lessonId": "d38e7808-7b7c-930d-16e9-b36b4929a0e2", "meta": { "gradeStageId": "ks2", "subject": "Maths", "grade": "Year 3", "topic": null, "unit": "Composition of non-unit fractions: addition and subtraction", "lessonCount": "10", "age": [ 7, 8 ], "gradeId": "age_7_8", "gradeStage": "Key Stage 2", "lesson": "Explain that addition and subtraction of fractions are inverse operations" } } ] }