### Prompt Cards

These two group activities use mathematical reasoning - one is numerical, one geometric.

### Exploring Wild & Wonderful Number Patterns

EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.

### Worms

Place this "worm" on the 100 square and find the total of the four squares it covers. Keeping its head in the same place, what other totals can you make?

# Buckets of Thinking

##### Stage: 2 Challenge Level:

There were a number of solutions that came in and here are those that had some extra information about how they went about it.

Shendy

First, I used letters to show the amount of liquid in each bucket.
R= The amount of liquid in the red basket.
B= The amount of liquid in the blue basket.
Y= The amount of liquid in the yellow basket.
One of the sentences states that:
Y$\div2 = 2$R
Multiply both sides by $2$.
Y$=4$R
As Y must be smaller or equal to $5$, R can only be $1$
So, the red bucket contains $1$ litre of liquid.
Y$=4$R
Y$=4\times1$
Y$=4$
The yellow bucket contains $4$ litres of liquid.
Also, $x$ is the amount of liquid poured from bucket A
As the amount of liquid in bucket B plus the amount of liquid poured from bucket A equals to the amount of water in bucket C,
B$+x=5$
As the red bucket contains only $1$ litre of water, $x$ can only equal to $1$
So,
B$+1=4$
B$=4-1 =3$
The blue bucket contains $3$ litres of liquid.

Jemima, Jasmine, Stephen and Olly used a trial and improvement approach:

red $1$ litre, blue $3$ litres, yellow $4$ litres
We tried yellow as $5$ litres and that didn't work because you're not allowed half litres.
But if yellow was $4$ and red was $1$ plus blue as $3$ it would work because if red was poured into blue it would make $4$ like yellow and twice the amount in red is $2$ and half of $4$ (yellow) is $2$.

From Ania:

Let the volume of liquid in the red bucket be N.
Let the volume of liquid in the blue bucket be P.
Let the volume of liquid in the yellow bucket be Q.
N, P and Q are given to be whole numbers.
Each volume is less than $5$ litres, therefore N, P, Q are less or equal to $5$ (litres).
We are also given:
N+P=Q and $\frac{1}{2}\times$Q$=2\times$N
Re-arranging the last equation we get Q$=4\times$N
As Q cannot be bigger than $5$ and both Q and N are whole numbers we must take
N$=1$ (if N$=2$ than Q$=8$, which is too much!)
Therefore Q$=4\times$N$=4\times1=4$
As P=Q-N we get P$=4-1=3$
So, the red bucket contains $1$ litre, the blue bucket contains $3$ litres and the yellow bucket contains $4$ litres.

From Nur:

Because half the liquid in the yellow bucket is the same as twice that in the red bucket, this means that there is a quarter of the amount in the red compared to the yellow.
Since they must be whole numbers and they can't be bigger than $5$, there must be $1$ litre in the red and $4$ in the yellow.
So then the blue bucket must have $3$ litres.

Finally, a solution that came in right at the end of the month from Ollie

I was working systematically. First I did the red bucket as $1L$, the blue bucket as $2L$ and the yellow bucket as $3L$. $1L$ add $2L$ equals $3L$ so that's ok but half of $3L$ equals $1.5L$ and $1$ times $2$ isn't $1.5L$ so I knew that was wrong so I went on to the next one. The red bucket is $1L$, the blue bucket is $3L$ and the yellow bucket is $4L$. I saw that $1L$ add $3L$ equals $4L$. And $4L$ divided by $2$ equals $2L$ and $1L$ add $1L$ equals $2L$, so I knew it was right. That's how I solved thiis problem.

Well done all of you and others who sent in the correct solution as well. I hope that those of you who did not send anything in but worked on it enjoyed the thinking that was necessary.