Ladybirds in the Garden
Ladybirds in the Garden printable sheet
In Sam and Jill's garden there are two types of ladybirds. There are red Seven-Spot ladybirds with 7 black spots and black Four-Spot ladybirds with 4 red spots.
Sam and Jill looked at a leaf with three ladybirds on it.
"One Seven-Spot ladybird," said Sam, "and two Four-Spot ones."
"That's 15 spots altogether!" said Jill.
"I wonder if we could find ladybirds whose spots add to other numbers. I know how to do 16."
"And 14 is easy too," added Sam.
How would you make 16 and 14 spots with the Seven-Spot and Four-Spot ladybirds?
What other numbers can you make with adding 4s and 7s?
Can you get lots of numbers from 4 to 35?
Are there some numbers you can't get?
We were sent this solution by Jacob, Luc and William at The Hall School in London:
We looked at the number of spots that could be produced when you have a four spotted ladybird or a seven spotted ladybird or a combination of both.
We were asked if it was possible to make 16 and we found it is, by using four of the four spotted ladybirds. We discovered that the smallest number of spots we could produce was 4.
We were then asked what number of spots between 4 and 35 could be produced. We started off by writing a list of those numbers that did and didn't work. Then we made a table to show how we made the numbers of spots that were possible.
We found seventeen different numbers could be made and three that could be made in two different ways. These were 28, 32 and 35.
We have drawn a table to show the only number of spots that can be made between 4 and 35. For each number of spots that it is possible to make we have shown the number of four spotted and seven spotted ladybirds that make up the number of spots.
Thank you all for sharing those ideas with us! I wonder if any of the numbers that are missing from that table are possible or if they're all impossible? If you find any more solutions to add to this table, please email us to let us know.
We also had some ideas sent in from Christian at Heronsgate School in England and Olivia from Risley Lower Primary School in England. From Australia we had a solution sent in from Maths Group 2 at Brunswick South Primary School.
Well done everyone!
Ladybirds in the Garden
Ladybirds in the Garden printable sheet
In Sam and Jill's garden there are two types of ladybirds. There are red Seven-Spot ladybirds with 7 black spots and black Four-Spot ladybirds with 4 red spots.
Sam and Jill looked at a leaf with three ladybirds on it.
"One Seven-Spot ladybird," said Sam, "and two Four-Spot ones."
"That's 15 spots altogether!" said Jill.
"I wonder if we could find ladybirds whose spots add to other numbers. I know how to do 16."
"And 14 is easy too," added Sam.
How would you make 16 and 14 spots with the Seven-Spot and Four-Spot ladybirds?
What other numbers can you make with adding 4s and 7s?
Can you get lots of numbers from 4 to 35?
Are there some numbers you can't get?
Why do this problem?
Ladybirds in the Garden is a good problem for children to practise addition, subtraction and possibly multiplication. It is a useful context in which to encourage learners to "have a go" and "play" with numbers, but then you can focus on having a system to find all the possible totals and giving reasons why some can't be made.
Possible approach
It would be good to have the two pictures of the ladybirds in the problem displayed for the children to see, either printed off or on screen. You could get them started simply by asking how many spots there are altogether on the two, then how many on the picture of the three. You might say that you're picturing some ladybirds on a leaf whose spots add to $16$ and ask children in pairs to think about which ladybirds they are.
You could bring their ideas together by asking them to write each different total, and how they made it, on a strip of paper. Gather these strips on the board or a wall and ask the children to arrange them in numerical order. Draw out a list of numbers which haven't been made and ask all pairs to check that they're sure they are impossible. The important point here is for pupils to try to explain why these totals can't be made. Ask pairs to convince each other and then invite some explanations for all to hear.
Key questions
Which totals have you found so far? Which ladybirds made these totals?
Possible extension
The article, Opening Out, suggests possible routes for further investigation in general, but this problem is also mentioned specifically. Children could be asked what combination of spots would be needed to be able to generate all the numbers under $30$. Would it be possible to do this without a one-spot ladybird? If we couldn't use a one-spot ladybird, what totals would be possible?
Possible support
Some children might find it useful to have a sheet with consecutive numbers from $1$ to $29$ written on so that they can fill in ways of making each total as they go along. Calculators might be available, or other apparatus to support the arithmetic, for example a hundred square, numberline, Cuisenaire rods or counters/cubes. You might want to print out a sheet of ladybirds and cut these into cards for children to choose from. To make the task even more accessible, you could choose and create ladybirds with $2$ or $3$ spots instead.