NARRATION:Back in the deepest, darkest depths of time, there lived a great mathemagician called Hypatia. Numbers fell under her spell one by one. With them, Hypatia could do anything. Now Hypatia is looking for people to share her powers with. She has forged a mathematical maze. Only true mathemagicians can solve the puzzles and find their way out. Join two young explorers, Oliviaand Hassan, to solve the puzzles, escape the maze and become the greatest mathemagicians of all time.
Multiplying and dividing fractions.
OLIVIA:Hassan?
HASSAN:Olivia?
OLIVIA:Hassan?
ARCHIMEDES:Archimedes!
HASSAN:Ah! Ohhh.
ARCHIMEDES:We made it to the Crystal Cave. Grab some gems.
HASSAN:Wow, that's amazing! Wow!
OLIVIA:Wow!
HYPATIA:The cave holds many secrets.Perhaps this equation will help your escape. 2/5 times 5/8.
OLIVIA:Hmm, I'm sure we can work something out.
HYPATIA:Hmm, I'll be curious to see.
ARCHIMEDES:Let's take a look at an example. We multiply the top numbers, the numerators, together. Then we multiply the bottom numbers, the denominators, with each other. We can see that 15/63 is a smaller fraction than either of our two original numbers.
HASSAN:Hey, look at those grids! When we are multiplying fractions, we are essentially multiplying a portion by a portion, creating something even tinier.
ARCHIMEDES:Exactly. And 15/63 is difficult to work with, so by using the common factor of three, we can simplify our product to 5/21.
HYPATIA:When multiplying fractions by integers, by whole numbers, things are slightly different, as you will see.
ARCHIMEDES:Yeah, we take our grid and copy it four times. 3/4 times four is the same as 3/4 multiplied by 4/1. Three times four is 12, and four times one is four. 12 divided by four is three. Or, if you want to think of it this way, we take our grid and copy it four times. Then we add all these sections together, resulting in three completed grids, three completed wholes.
OLIVIA:So our answer's simply three?
ARCHIMEDES:Exactly.
OLIVIA:Hmm.
ARCHIMEDES:Take 1/6 divided by five. We're breaking up 1/6 of a whole into parts that are five times smaller. Through doing that, the total number of parts that make up the whole is increasing, so to reflect that, we now actually need to multiply the denominator of our number by the whole number. And six times five is 30, giving us 1/30.
Oh, nice outfits! Shall we solve the equation?
HASSAN:Oh we can't move. These are really heavy.
OLIVIA:But we have a present for you guys.
HYPATIA:Oh wow
ARCHIMEDES:Thank you.
HASSAN:We found 15/5 of magic gems and were able to push them together to make three magic gems.
ARCHIMEDES:Oh so, what maths were you using there?
OLIVIA:We were using maths?
HASSAN:Oh, we were multiplying fractions together!
OLIVIA:Oh!
HASSAN:Oh, and that must be the same for our magic gems. 15 magic gems is the same as 15/1 which we multiply with 1/5, to make 15/5.
OLIVIA:15/5 is equal to three, because 15 divided by five is three.
HYPATIA:You guys are getting good at this. But how about my puzzle? Can you solve it? Or have you decided to stay here forever?
OLIVIA:Hmm, I think the answer is 2/8.
HASSAN:And I think the answer is 1/4.
HYPATIA:You are both right. We can divide both the numerator and the denominator of 2/8 by two, to make 1/4, meaning they are equivalent.
ARCHIMEDES:When we have multiple options for simplifying, we always go for the one that results in the smallest digit for the denominator.
OLIVIA:Woo!
HASSAN:Ha!
OLIVIA AND HASSAN:Thank you.
ARCHIMEDES:Was that the only exit?
HYPATIA:Uh-oh!
ARCHIMEDES:Oh no.
HYPATIA:Now young mathemagicians, it is your turn. Follow in Hassan and Olivia's footsteps and become mathemagicians. Something divided by four is 7/24. 5/20 divided by something is 5/80. What are the missing fractions and integers?
Video summary
In a mysterious crystal cave, Olivia and Hassan get to work dividing and multiplying fractions in order to find their way out.
Archimedes has ²⁄₅ of an emerald and asks how much he would have left if he gave ¹⁄₂ to Hypatia.
After working out the division of this fraction, Olivia and Hassan get to work to divide and multiply fractions.
With a little help from Archimedes, Olivia and Hassan then learn how to multiply and divide more complex fractions.
This short animated film is from the BBC Teach series, Hypatia's Mathematical Maze.
Teacher Notes
Before watching the film:
Prior to this lesson you may wish to introduce pupils to other relevant topics, for example:
- Multiplication and division
- Comparing and ordering fractions
- Simplification of fractions
During watching the film:
Depending on the focus of your lesson, you may wish to pause the film at certain points to check for understanding, asking questions such as:
- What is the difference between an integer and a fraction?
- How else can you represent the fractions?
- How would we simplify the fractions that Olivia and Hassan have?
- What different stages will you use to answer Hypatia’s final question?
Final question:Something divided by four is seven twenty-fourths. Five twentieths divided by something is five eightieths. What are the missing fractions and integers?? ÷ 4 = ⁷⁄₂₄
5⁄₂₀ ÷ ? = 5⁄₈₀
Answer to the final question:7⁄₆ ÷ 4 = ⁷⁄₂₄5⁄₂₀ ÷ 4 = 5⁄₈₀
Following on from the film:
- Use bar models to represent dividing the fractions into further pieces.
- Use examples where the pupils have to explain their thinking in terms of factors, multiples, commutativity and equivalent fractions e.g. Which is larger? ⅘ ÷ 3 or 4⁄₃ ÷ 5? What is larger, ⅔ ÷ 5 or 4⁄₆ ÷ 5?
- Play true or false by exploring the relationship between dividing fractions by integers and multiplying fractions by other fractions, e.g. Which of these statements are true?
- ¹⁄₂ ÷ 2 = ¹⁄₂ x ¹⁄₂
- ¹⁄₂ ÷ 4 = ¹⁄₂ x ³⁄₄
- ¹⁄₂ ÷ 3 = ¹⁄₂ x ¹⁄₃
- ¹⁄₂ ÷ 5 = ¹⁄₂ x ¹⁄₅
This short film is suitable for teaching maths at KS2 in England, Wales and Northern Ireland and 2nd Level in Scotland.
Adding and subtracting using mental methods. video
In a mythical temple full of obstacles, Olivia and Hassan learn simple mental maths and apply it to work out the passcode to open the door.
Adding and subtracting using written methods. video
Having been transported to a floating library, Olivia and Hassan's ability to solve large equations using written methods is tested.
How and why we round numbers. video
Olivia and Hassan are taught how to round to the nearest 1000, 10,000, and 100,000 when a disgruntled magic carpet demands money from them.
Using addition and subtraction in multi-step problems. video
Inside a giant computer created before the age of mathemagicians, Olivia and Hassan are faced with a multi-step problem to reach the labyrinth’s next stage.
Multiplying by 10, 100 and 1000. video
When Olivia and Hassan find themselves on the roof of an enormous skyscraper they learn how to multiply by 10, 100 and 1000 and work out the number of windows on the building.
Mental multiplication. video
While exploring a jungle, Olivia and Hassan learn the true nature of square numbers as being created when two of the same number are multiplied together.
Factors, multiples and primes. video
Trapped in a desert together with Archimedes, Olivia and Hassan learn what common factors, multiples, and prime numbers are.
Multiplying using written methods. video
Finding themselves in space, Olivia and Hassan watch mathemagician Hypatia perform long multiplications in the sky by bending the stars.
Dividing using written methods. video
Olivia and Hassan learn how to perform long division in order to free gridlocked traffic on a bridge.
Recognising and comparing fractions. video
Olivia and Hassan are enjoying themselves in a world made out of pizza and learn how to work with numerators and denominators in fractions.
Adding and subtracting fractions. video
Amongst the ruins of an ancient city, Olivia and Hassan learn how to add and subtract fractions.
