# Little Man

The Man is much smaller than us. Can you use the picture of him next to a mug to estimate his height and how much tea he drinks?

## Problem

*This problem has been inspired by "The Man", a story by Raymond Briggs, which you might like to read.*

The Man is much smaller than you and me.

Here is a picture of him standing next to a mug.

Image

Can you estimate how tall he is?

Can you think of something that you have at school or home that is approximately twice as tall as the Man?

What about something that is about half as tall as the Man?

How tall do you think the Man's mug might be?

Can you estimate how many "Man mugs" of tea might fill one of our mugs?

## Getting Started

Perhaps it would help to find a mug that you can look at.

Can you think of anything you have which might be about the same size as the Man's mug? Can you estimate how much tea that might hold?

## Student Solutions

It is interesting to see the different ideas you had about this task.

Rahini from Emirates International School Meadows Dubai wrote:

- I estimate the man is 10 cm tall.

- I think that our small school books are twice as tall than the man.

- An eraser is half as tall as the man.

- The mug is approximately 10 cm.

Chloe and Candace from Ngatea Primary in New Zealand sent in their detailed explanation:

We used a ruler to measure him and he is 4 cm on the picture. We looked at stuff that was two times taller and smaller than the man around our class and outside:

We measured a coffee cup it was 10cm.

If he is 10 cm tall and he is eight times taller than his mug, then his mug is 1$\frac{1}{4}$cm.

Ruby from Pakefield Primary School wrote:

We looked at the size of the large mug and sized it to an object in my dad's office, which was a stationery holder.

We then looked at the little mug, this was approximately a thimble size. We estimated that there were 10 layers of small cups, each layer containing 20 cups, so the calculation was 20 x10 = 200 cups, but there would also be gaps in between of about a quarter of a cup = 50 more

So, the answer is 250 cups

Bryan from Maadi British International School in Egypt wrote by hand and sent in a picture :

Matias from Cheltenham Elementary in the United States said: I think he's an inch and a little bit.

I think a cell phone standing up is twice as tall.

I think a big tomato is half as tall.

I think his mug is half an inch.

I think it would take 50 of his cups to fill our cup.

Alara from FMV Erenkoy Primary School sent us a very full solution to this problem:

Estimation of how tall he is : - It is not possible to say an exact number or measurement for his height. There could only be some estimations. However, I can make an estimation using the ratio of the man's height to the mug's height.

Something at school or at home which is approximately twice as tall as the Man : - It depends on the mug's height; for example according to my milk mug's height when I checked, my big dictionary and my cd box is approximately twice as tall as the Man.

Something which is about half as tall? And again, as I have written above, according to my milk mug's height, my post-it papers and small address note book are about half as tall as the Man.

How tall the Man's mug might be? My mug's height is 10 cm and I am 150 cm tall. The ratio between me and my mug is 10/150, that is to say 1/15. The Man's mug might be 15 times smaller than his height.

How many "Man Mugs" of tea might fill one of my mug? There is a way to answer this question, if we take into considerationthe "ratio principles". Above, I identified that I'm 15 times as tall as my mug. So, let's assume the Little Man is as tall as my mug, so he is 10 cm tall. If we consider the same ratio between the Little Man and his mug, the Man Mug is 15 times smaller than him. 15 "Man Mugs" of tea might fill my mug.

Sam from Farndon Primary School and Harri from Pilgrim's School also estimated similar heights for the Man, although Sam thought that 79 "Man Mugs" of tea would fill one of our mugs.

Thank you for all these solutions. Whose do you agree with, and why?

Rahini from Emirates International School Meadows Dubai wrote:

- I estimate the man is 10 cm tall.

- I think that our small school books are twice as tall than the man.

- An eraser is half as tall as the man.

- The mug is approximately 10 cm.

Chloe and Candace from Ngatea Primary in New Zealand sent in their detailed explanation:

We used a ruler to measure him and he is 4 cm on the picture. We looked at stuff that was two times taller and smaller than the man around our class and outside:

- My pencil
- Step height
- Wheelie bin wheel

We measured a coffee cup it was 10cm.

If he is 10 cm tall and he is eight times taller than his mug, then his mug is 1$\frac{1}{4}$cm.

Ruby from Pakefield Primary School wrote:

We looked at the size of the large mug and sized it to an object in my dad's office, which was a stationery holder.

We then looked at the little mug, this was approximately a thimble size. We estimated that there were 10 layers of small cups, each layer containing 20 cups, so the calculation was 20 x10 = 200 cups, but there would also be gaps in between of about a quarter of a cup = 50 more

So, the answer is 250 cups

Bryan from Maadi British International School in Egypt wrote by hand and sent in a picture :

Image

Image

Matias from Cheltenham Elementary in the United States said: I think he's an inch and a little bit.

I think a cell phone standing up is twice as tall.

I think a big tomato is half as tall.

I think his mug is half an inch.

I think it would take 50 of his cups to fill our cup.

Alara from FMV Erenkoy Primary School sent us a very full solution to this problem:

Estimation of how tall he is : - It is not possible to say an exact number or measurement for his height. There could only be some estimations. However, I can make an estimation using the ratio of the man's height to the mug's height.

Something at school or at home which is approximately twice as tall as the Man : - It depends on the mug's height; for example according to my milk mug's height when I checked, my big dictionary and my cd box is approximately twice as tall as the Man.

Something which is about half as tall? And again, as I have written above, according to my milk mug's height, my post-it papers and small address note book are about half as tall as the Man.

How tall the Man's mug might be? My mug's height is 10 cm and I am 150 cm tall. The ratio between me and my mug is 10/150, that is to say 1/15. The Man's mug might be 15 times smaller than his height.

How many "Man Mugs" of tea might fill one of my mug? There is a way to answer this question, if we take into considerationthe "ratio principles". Above, I identified that I'm 15 times as tall as my mug. So, let's assume the Little Man is as tall as my mug, so he is 10 cm tall. If we consider the same ratio between the Little Man and his mug, the Man Mug is 15 times smaller than him. 15 "Man Mugs" of tea might fill my mug.

Sam from Farndon Primary School and Harri from Pilgrim's School also estimated similar heights for the Man, although Sam thought that 79 "Man Mugs" of tea would fill one of our mugs.

Thank you for all these solutions. Whose do you agree with, and why?

## Teachers' Resources

### Why do this problem?

This problem offers a novel context in which to think about relative size, and gives children chances to estimate quantities. The fictional nature of the situation might draw learners in and make them curious to investigate further.

### Possible approach

It might be appropriate to read the book "The Man" (by Raymond Briggs) with the class prior to working on this task.

It would be worth having a range of measuring equipment to hand, including access to a water tray, bucket or large bowl along with a variety of everyday items such as spoons, bottles, cups and jugs.

As you pose each question, give the class time to consider their estimations and invite learners to explain how they arrived at them.

You could then ask how we might go about checking whether their estimates are realistic. At this point, try to provide the equipment that children request and stand back, rather than imposing your own method. The more practical experience that learners have with measuring, the better their sense of relative size. Exploring length/height/capacity/volume in practical contexts will give children 'reference points' to draw on when estimating.

You could facilitate a discussion about which methods for estimating gave answers that were closest to measured values, and why that might be.

You could then ask how we might go about checking whether their estimates are realistic. At this point, try to provide the equipment that children request and stand back, rather than imposing your own method. The more practical experience that learners have with measuring, the better their sense of relative size. Exploring length/height/capacity/volume in practical contexts will give children 'reference points' to draw on when estimating.

You could facilitate a discussion about which methods for estimating gave answers that were closest to measured values, and why that might be.

### Key questions

How high do you think a mug might be?

How did you come up with your answer?

How could we check whether that is about right?

How could we check whether that is about right?

Can you think of anything we might have at home or school which you know the capacity of?

How much do you think a mug might hold?

### Possible extension

You could challenge children to articulate the relationship between the relative size of the man and one of them. Is the man about twice as small as us? Five times smaller? Ten times smaller? Will this relationship hold for everything?

### Possible support

Some learners might find it helpful to have a series of tasks which requires them to estimate and then measure concrete items in the classroom before going on to estimate measurements of items they are visualising.