The Tomato and the Bean
3 out of 3
At the beginning of May, Tom put his tomato plant outside. On the same day he sowed a bean in another pot. When will the two be the same height?
Problem
The Tomato and the Bean printable sheet
Tom's dad sowed some tomato seed in February. He gave Tom one of the tomato plants in a pot.
At the beginning of May, Tom put his tomato plant outside. On the same day he sowed a bean in another pot.
Ten days later the bean plant was just 1cm above the soil surface. Tom measured his tomato plant which was already 38cm tall.
Each evening Tom measured his two plants.
On the evening of the next day the little bean plant had grown another 2cm so it was 3cm high. Each day it grew double the amount it had grown the day before.
The tomato plant grew at a steady 5cm a day.
After how many days were the two plants the same height when Tom measured them in the evening? How high were they?
Getting Started
How tall will the bean plant be the next day?
How tall will the tomato plant be on that day?
How about the day after that?
Student Solutions
The solutions we had in, varied in the ways they were expressed.
We had many contributions from Crudgington Primary School in Shropshire, U.K.
Firstly, I drew two pictures one of a tomato plant and one of the bean plant. Then I made myself a little chart like this ...
... and so on. As you can see that on the section where I worked out the bean plant's results the first number I wrote was how much the plant would need to increase in size. I found that this helped me a lot. Then I wrote the height of the plant and finally how many days it took for the plant to reach that height. I did this same method for the tomato plant until the height was the same. So the answer is: the height is $63$cm after $15$ days of growing.
Tom
First I looked at the information and saw that the bean plant grew only $1$cm after $10$ days! So I started from $10$ days. And after the bean plant grew $1$cm the tomato plant had already grown $38$cm. And then after each day the bean plant doubled its current size and the tomato plant grew $5$cm. So after $15$ days both plants are $63$cm.
From Curl Curl North School in New South Wales we had two answers that at first looked different but were just being expressed from a different point of view. Christopher, Ashton, Grace, Charlotte, Kaden, Ben, Kate, Charlie, Dylan, Josh, Matthew and Mitchell's team, said;
$6$ days from the time the bean was first measured.
And from Sean, Emma, Emily, Amelia, Ed and Stacey's team we had:
$5$ days because you count from when the bean was $3$ cm tall.
I wonder if they thought the other team had the wrong answer?
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
This problem can help to give learners an understanding of the difference between a steady and an increasing rate of growth. It could be used when looking at number patterns, especially doubling. It puts patterns in a real context, and can be used to demonstrate the use of tables in problem solving. It will also be fascinating for you, as the teacher, to see how children represent the problem while they work on it.
Possible approach
You could start this problem as a whole class activity by introducing the growth rate of other plants as examples. (These plants could be real or imagined!) For example:
"There is a plant that is 3 centimetres tall. If it grows 4 centimetres a day how tall will it be the next day?" "And the day after that?" And so on ...
Then you can introduce a different type of example:
"There is a plant that is $5$ centimetres tall. If it doubles in height each day how tall will it be the next day?""And the day after that?" And so on ...
Then a new question needs to be asked to help in approaching the problem as written. For example:
"It starts at 5 centimetres and is 10 centimetres the next day. How much has it grown?"
"If it grows double that the next day, how tall will it be?"
You could then ask learners to do the given problem in pairs and encourage them to use anything they like to help them with the solution. Some may draw pictures and use numbers, some may make a list, some may use multilink to physically represent the plants, some will be able to make a table of the results, others may be happier if given this sheet with a table to be completed.
Key questions
At what height did the bean start?
How tall was it the next day?
How much did it grow by the next day? So how tall would it be that day?
And the day after that?
At what height did the tomato start?
How tall would it be on the second day? And the day after that?
Possible extensions
Learners can be encouraged to open out this activity further by considering plants that increase their rate of growing by trebling/quadrupling the growth each year. Each year's height could be calculated and learners can see what happens.
Doubling goes: 1, 3, 7, 15, 31, 63 ...
Trebling goes: 1, 4, 13, 40 ...
Quadrupling goes: 1, 5, 21, 85 ...
To open this problem out further, children may like to explore the digital roots of the numbers in their results. This sheet gives some suggestions as to how this might be done.
Possible support
Suggest using this sheet and filling in first the days (1-8) and the heights on the first day. The bean's growth is easier to work out than the tomato's so this can be tackled next.