Crossings
In this problem we are looking at sets of parallel sticks that
cross each other. What is the least number of crossings you can
make? And the greatest?
Age
7 to 11
Challenge level
Problem
In this problem we are looking at sets of parallel sticks that cross each other.
Image
|
Image
|
How many crossings do the five green sticks make?
Still keeping two sets of parallel sticks, this time with seven sticks in total, can you arrange them in another way, to get a different number of crossings?
What is the least number of crossings you can make?
What is the greatest number of crossings you can make?
Can you find all possible numbers of crossings with seven sticks?
What do you need to do to prove that you have them all or how could you show that you have them all?
Can you find the least/greatest number of crossings for ten sticks?
Can you find the least/greatest number of crossings for fifteen sticks?
Can you predict the least/greatest number of crossings for fifty sticks?
Can you predict the least/greatest number of crossings for any number of sticks?
Getting Started
Remember that you must have two or more sticks to make a set.
Try starting with a set of $2$. How many sticks would be in the set placed across these $2$? How many crossings would that make?
If you can't have a set of $1$ stick, what number could you try next?
Do you notice anything about the number of sticks in each set and the number of crossings? How do these numbers relate to the total number of sticks?
Student Solutions
Tobi and Charles from Luanda International School both sent in correct solutions to this problem. Some of you made the mistake of thinking that you could have one stick crossing all of the rest, but the question did say that there must be 2 sets of parallel sticks.
Tobi's solution is given below.
10 sticks:
To find the biggest number of crossings, put 5 sticks going down and 5 sticks going across so there will be 25.To find the smallest number of crossings, put 2 sticks going down and 8 sticks across so there will be 16.
15 sticks:
To find the biggest number of crossings for an odd amount, instead of splitting the sticks in half split them into the the closest numbers to half (e.g 7 and 8).
To find the smallest number of crossings do the same as you did before, put 2 going down and the rest going across.
Estimate for 50 sticks:
For the biggest amount put 25 down and 25 across (625).
For the smallest amount put 2 down and the rest across (96).
Estimate for any number:
To find the biggest for any number, split the amount in half and put half across and half down but if its an odd amount split the amount into the 2 numbers closest to half.
To find the smallest amount put 2 down and all the rest across.
Charles explains how to find the greatest number of crossings for an odd number of sticks a different way.
If the number of sticks is an odd number, then you will have
to divide the odd number of sticks by 2. Then add a half a stick to
the top layer of sticks and take half a stick away from the bottom
layer. After that multiply the two numbers together.
Teachers' Resources
Crossings
In this problem we are looking at sets of parallel sticks that cross each other.
Image
|
Image
|
How many crossings do the five green sticks make?
Still keeping two sets of parallel sticks, this time with seven sticks in total, can you arrange them in another way, to get a different number of crossings?
What is the least number of crossings you can make?
What is the greatest number of crossings you can make?
Can you find all possible numbers of crossings with seven sticks?
What do you need to do to prove that you have them all or how could you show that you have them all?
Can you find the least/greatest number of crossings for ten sticks?
Can you find the least/greatest number of crossings for fifteen sticks?
Can you predict the least/greatest number of crossings for fifty sticks?
Can you predict the least/greatest number of crossings for any number of sticks?
Why do this problem?
This problem gives learners a great opportunity to apply number facts and skills to a seemingly spatial problem. It is also a good context for them to give reasons for the patterns which emerge.
Possible approach
Many could benefit from having sticks or straws with which to investigate this problem, although it can be tackled purely by sketching sticks.
Key questions
What is the least number you may have in a "set of sticks"?
If you start with a set of $2$, how many sticks would be in the set placed across these $2$?
How many crossings would that make?
Can you think of a good way of recording what you have done?
Do you notice anything about the number of sticks in each set and the number of crossings?
How do these numbers relate to the total number of sticks?
If you have $10$ sticks how many could you put down to start with?
How many would be left to cross them?
How many crossings would this make?
Can you arrange them to make more/less crossings?
Can you generalise your ideas for any number?
Possible extension
Learners could go on to Divide it out.