Bugs in the garden
Problem
In Sam and Jill's garden there are two sorts of ladybirds. There are red Seven-Spot ladybirds with $7$ black spots and shiny 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!" laughed 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 $4$s and $7$s?
Can you get lots of numbers from say $4$ to $35$?
Are there some numbers you can't get?
Student Solutions
From Jacob, Luc and William at The Hall School in London we had the following, very thorough solution sent in,
We looked at the number of spots that could be produced when you have a four spotted bug or a seven spotted bug 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 bugs. 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 bugs that make up the number of spots.
Well done everyone!
Teachers' Resources
Bugs in the Garden
In Sam and Jill's garden there are two sorts of ladybirds. There are red Seven-Spot ladybirds with $7$ black spots and shiny 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!" laughed 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 $4$s and $7$s?
Can you get lots of numbers from say $4$ to $35$?
Are there some numbers you can't get?