Magic cups
Stage 2 and 3 – a thinking mathematically targeted teaching opportunity focused on using a range of tools and strategies to record and communicate ideas.
Syllabus
Syllabus outcomes and content descriptors from Mathematics K–10 Syllabus (2022) © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2024.
Outcomes
- MAO-WM-01
- MA2-DATA-01
- MA2-DATA-02
- MAO-WM-01
- MA3-DATA-02
Collect resources
You will need:
- paper
- 3 different coloured markers, pens or pencils.
- 3 cups, or 3 of the same object (such as forks, spoons, markers or building blocks)
Magic cups – part 1
Watch Magic cups part 1 video (3:43).
These videos were developed with Kristy from Minchinbury PS, Stephanie from Parramatta PS and Alishia from Sylvania Heights PS.
[A title over a navy-blue background: Magic Cups. Below the title is text in slightly smaller font: MathXplosion. Small font text in the lower left-hand corner reads: NSW Mathematics Strategy Professional Learning team (NSWMS PL team). In the lower left-hand corner is the red waratah of the NSW Government logo.]
Speaker
Magic cups from MathXplosion.
[A title on a white background reads: You will need…
Bullet points below read:
- three cups (forks, spoons, markers or Lego blocks etc.)
- paper and three different coloured markers
Under the points is a row of four images. From left to right, the images are: 3 red thimbles, 3 silver forks, 3 silver spoons, 3 grey Lego blocks, 3 different coloured markers.]
Speaker
Before we start with the magic cups today, you will need three cups. You might want to use three forks or three spoons or markers or even three Lego blocks. It really doesn't matter what you choose as long as you have three of the same object. You'll also need paper and three different coloured markers.
[Text over a navy-blue background: Let’s play!
On a table, there is a white A4 paper in the centre. On the paper, 1, 2 and 3 are written on the left side. On the bottom of the paper are blue, green and red markers. On the left side of the paper are 3 red cups. The middle cup is facing down.]
Speaker
For this activity, you'll need three of the same type of object, and remember, the middle one…
[The speaker points to the middle cup.]
Speaker
…has to be facing down. Today, I'm actually going to represent my thinking…
[She circles the paper with her finger.]
Speaker
…using drawing. Mathematicians represent and organise their thinking to help them solve problems. The goal of the activity is to have all three…
[She points to each cup.]
Speaker
…cups facing down, and we've only got three steps to do it. We have to flip the cups in pairs, so we have to flip the cups two at a time. I'm going to start by representing how our cups are starting. The cup on the left is facing up.
[On the paper above the 1, using the blue marker, she draws an arrow pointing up.]
Speaker
The cup in the middle is facing down…
[On the paper next to the arrow, using the green marker, she draws an arrow pointing down.]
Speaker
…and the cup on the right is facing up.
[On the paper next to the second arrow, using the red marker, she draws an arrow pointing up.]
Speaker
OK, first flip.
[She flips the middle cup to face up, and the right cup to face down.]
Speaker
I'll represent what the cups look like now. The one on the left is still facing up.
[On the paper next to the 1, using the blue marker, she draws an arrow pointing up.]
Speaker
The one in the middle is now also facing up.
[Next to the arrow, using the green marker, she draws an arrow pointing up.]
Speaker
The one on the right has flipped and is now facing down.
[Next to the second arrow, using the red marker, she draws an arrow pointing down.]
Speaker
Second flip. Going to use the... move the outside cups.
[She flips the first cup to face down, and the right cup to face up.]
Speaker
So now my cup on the left is facing down.
[On the paper next to the 2, using the blue marker, she draws an arrow pointing down.]
Speaker
The cup in the middle has stayed facing up…
[Next to the arrow, using the green marker, she draws an arrow pointing up.]
Speaker
…and the cup on the right has flipped and is now facing up.
[Next to the second arrow, using the red marker, she draws an arrow pointing up.]
Speaker
One more flip to have all three cups facing down. Can you see which cups I need to flip? You got it.
[She flips the middle cup to face down, and the right cup to face down.]
Speaker
These two. So now, we have all three cups facing down…
[On the paper next to the 3, using the blue marker, she draws an arrow pointing down; using the green marker, she draws an arrow pointing down and using the red marker, she draws an arrow pointing down. She points to each number on the paper.]
Speaker
…and it took three steps.
[Text over a navy-blue background: Let’s investigate!
A white background.]
Speaker
Let's investigate. So when the guy from MathXplosion started the trick…
[In the upper left, an image appears of 3 red cups. Both cups on the end are facing up, and the middle one is facing down.]
Speaker
…the middle cup was always facing down. Now he made exactly three flips of two cups to turn them so that they all face down.
[On the bottom left, an image appears of 3 red cups arranged as above. There is a yellow piece of paper with a tick under the middle cup.]
Speaker
Hmm, could he have flipped the cups to all be facing down in exactly three flips…
[In the middle of the screen, a larger image appears of 3 red cups. The first cup is facing down, and the middle and third cup are facing up. There is a yellow paper with a question mark under the first cup.]
Speaker
…if he started with the cup on the left facing down? Hmm, what about if the cup on the right…
[On the right of the larger image, an appears of 3 red cups. The first and middle cup are facing up, and the third cup is facing down. There is a yellow paper with a question mark under the middle cup.]
Speaker
…was facing down?
[A title on a white background reads: Over to you, mathematicians!
Text below reads: Will it work in exactly 3 flips with the cups starting…
Under the text are the 2 larger images from the previous slide. Under the left image is text: Like this?
Under the right image is text: Or like this?]
Speaker
It's over to you now, mathematicians. See if you can have a go and get all of the cups to be facing down if you start with a different cup facing down.
Remember, you have to do exactly three flips of two cups each time. Record your attempts and see what you can figure out.
[Text over a navy-blue background: Over to you!]
[Over a grey background, the red waratah of the NSW Government logo appears amongst red, white and blue circles. Text: Copyright State of New South Wales (Department of Education), 2021.]
[End of transcript]
Instructions
Set up your three cups (or forks, spoons, etc.) So they are orientated like this:
- Notice how the middle cup is facing down and the two outside cups are facing up.
- Your challenge is to flip pairs of cups, in exactly three moves, to have them all facing down.
- Flip one pair of cups and record your move.
- Flip a second pair of cups and record your move.
- For the third time, flip a pair of cups and record your move!
- Once you've found one way of flipping pairs of cups in exactly 3 moves, see if you can find another way!
Magic cups – part 2
Watch Magic cups part 2 video (0:49).
[Text over a navy-blue background: What’s (some of) the mathematics? Small font text in the upper left-hand corner reads: NSW Department of Education. In the lower right-hand corner is the white waratah of the NSW Government logo.]
Speaker
What's some of the mathematics?
[A title on a white background reads: What's (some of) the mathematics?
Text below read: Mathematicians use a range of strategies and tools to record and communicate their ideas.
Speaker
So we know as mathematicians that we can use a range of strategies and tools to record and communicate our ideas.
[Below the text is an image of a piece of paper with a title consisting of 3 different coloured arrows; numbers 1-3 written down the left-side, and next to the numbers are 3 different coloured arrows. The row of arrows are pointing either up or down.]
Speaker
So here's one way we recorded our thinking. But we could have also shown what we did with the cups like this…
[On the left side of the image, an arrow appears. On the left side of the arrow is an image with the following text:
1) Up Up Down
2) Down Up Up
3) Down Down Down
The first column of text is in blue; the second in green and the last in orange.]
Speaker
…or like this.
[On the right side of the first image, an arrow appears. On the right side of the arrow is an image with a column of text referring the number of flips completed on the left and images of 3 red cups on the right, pointing either up or down.]
Speaker
I revised my first attempt…
[The images on either side of the first image disappears. Under the first image, a table appears with 4 column. The first column refers to the number of flips; the next 3 columns have either Up or Down and some cells were coloured red.]
Speaker
…clarifying my thinking more by putting it in a table. I color coded it to show which two cups were moved. So over to you again now mathematicians, try to use a table to record and communicate your ideas.
[Text over a navy-blue background: Over to you!
Over a grey background, the red waratah of the NSW Government logo appears amongst red, white and blue circles. Text: Copyright State of New South Wales (Department of Education), 2021.]
[End of transcript]
What to do next
Revise your recording by sharing your thinking in a table. You could use the ideas shared with you in the video.
Other ways to play
What does the starting position need to be to get the trick to work in exactly 4 moves using 4 cups? Or 5 cups in 5 moves?