How tall?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Problem
Here is part of the conversation between a group of children as they discuss a tall tree nearby:
"I wonder how tall it is?" says Linus.
"I think we could find out," replies Raj.
"It could be difficult as it's very high," says Toby.
I wonder how they each went about finding out the height of the tree?
I wonder how YOU would find out how tall a large tree in your surroundings is?
Getting Started
Could you use any other objects to help you?
Can you estimate how tall the tree is?
Student Solutions
We have had a good variety of thinking demonstrated in the responses received via emails. Here is a selection.
From Philippa at King's School Chester in England:
First of all, you need to wait until the sun is half way up the sky so the shadow is not too stretched and it is not mid-day so it is not right above the tree. When the time is right, you need to measure the shadow. I recommend a tape measure or a metre stick to do this. When you have the measurement you can say that this is the height of the tall tree.
From Dominic at Thomas Jolyffe School in the UK:
Measure a piece of wood and then stand it vertically on the ground and measure the length of its shadow. Quickly measure the shadow of the tree. Then work out the ratio of the length of the piece of wood to its shadow and this would be the same as the ratio of the height of the tree and the length of its shadow. Obviously this can only be done on a sunny day!
The Maths club from Chesham Prep School in England wrote:
We thought about using the tree's shadow but that would only work if
1. It was sunny
2. It was midday (otherwise the shadow would be shorter or longer, depending on the angle of the sun.
We knew how high $1$m was so we were able to estimate the height of the tree.
We knew how tall a member of the group was and he stood against the tree.
We then used his height to estimate the height of the tree.
We then used these estimates plus an imaginary diagonal line from the top of the tree to the ground. We then measured along the ground from that point to the tree.
Jiayuan from Harbord Collegiate Institute in Canada sent in:
Use a stick. When it's noon, put the stick straight and measure the width of
the shadow, and measure the width of the shadow from the tree.
Length of the stick : Width of the stick shadow = Height of the tree :Width of the tree shadow.
Holly from Guildford High School in England said she found the height without any danger. Here is the picture she sent in;
Other ideas came from:
Isobel and Harry at St. John the Baptist School in England:
I would measure something realistic against the tree for example a $6$ foot fence and compare it with that then stand back and calculate how many times taller it is going up in $6$ foot at a time.
Another idea from Abigail also at St. John the Baptist School:
Measure the tree on the screen in cm. Then make a key of $1$cm=$5$m. This will be an approximate height of the tree.
"Use angles and trigonometry" thought Alistair from Crosscrake School in England:
The children could calculate the height of the tree using simple trigonometry. What they should do is place an object somewhere nearby the tree, measure the distance between the tree and the object, and measure the angle of elevation of the tree from the rock. From this information, they could calculate the height of the tree with tan.
Well, here is a very different approach from Robyn at Brynhyfryd School in Wales:
I would climb the tree and send the very long measuring tape down to my friend.
St. Peter and Paul Primary School in England. Brian, Harshan and Edrich sent in these three separate ideas:
Brian - The shadow of the tree is the same length as the tree if it is sunny you can measure it in metres every time it is sunny it will be longer and in winter it will be smaller.
Harshan -Get a ladder and measure.
Edrich - First I will wait for a sunny day then I will measure its shadow. For example if I m $1.5$ meters my shadow will be $2.4$ meters so by calculating their space I can found the tree's length.
Finally Athena from EMS in the USA had another idea:
Use a yard stick to measure the tree by slowly sliding it or use ruler to measure the length of your hand in inches then slowly climb tree while vertically putting one hand on top of another on the trunk.
Well done! So many different ideas. Thank you for your efforts and for sending them in.
Teachers' Resources
How Tall?
Here is part of the conversation between a group of children as they discuss a tall tree nearby:
"I wonder how tall it is?" says Linus.
"I think we could find out ," replies Raj.
"It could be difficult as it's very high," says Toby.
I wonder how they each went about finding out the height of the tree?
I wonder how YOU would find out how tall a large tree in your surroundings is?
Why do this problem?
Possible approach
You could introduce the challenge by projecting the image of the tree and asking a child to read out the conversation. Give the class a few minutes of individual thinking time, then ask them to talk to a partner about how they might go about finding out the height of the tree. Explain that you're not expecting them to know straight away how to do it but encourage them to persevere and devise a way.
Give them plenty of time to plan what they would do and then ask some pairs to explain their proposed methods to the whole group. At this stage, it is important to allow a substantial length of time for discussion. What do the children think the advantages and disadvantages of the different approaches are? Invite pairs to decide on a method that they think is the "best" and encourage explanations as to why. It will be interesting to note those who opt for someone else's method rather than pursuing their own.
It may be appropriate for you (or them) to find a suitable tree in the school grounds for them to carry out their plans practically. It is possible that difficulties might arise as they try out their chosen approach so watch out for the ways in which different pupils overcome any challenges.
It would be very beneficial to have a final plenary following on from the practical experimentation. How will the group judge how well their method worked? This would also be a chance for the class to discuss whether they would use a different method if given a similar challenge on a future occasion.
Key questions
What do you think about what (so-and-so) said?
Do you see any ways of improving their method?
If you were asked to do something similar again what way do you think you would try?
Possible extension
Possible support