# How Many?

## Problem

Have a look at this photo of some peaches:

How many?

How did you count them?

Now take a look at this picture of oranges:

How many?

How did you count them?

And one last photo:

How many this time?

How did you count?

*This task was inspired by Christopher Danielson's tweets. You might like to take a look at Christopher's X (Twitter) account.*

*You may be interested in the other problems in our Low Threshold High Ceiling Feature.*

## Getting Started

What did you do to count?

How would you check?

What made you decide to change your way of counting, if you did?

## Student Solutions

We had a good number of solutions sent in to this task. These first ones are from Meavy C of E Primary School.

Samuel and Lucy

The first one. I spotted that it is an array and then I worked it out by counting the top line and the side line. Then I multiplied them. Like 4 x 5 = 20.

The last one we know we have 6 tubs and they mainly have around 50 to 60. Then we multiplied to get around 350 to 360.

Amir and Grace

Peaches: We saw there were 5 rows of 4 and 4 x 5 is 20.

Raspberries: there isn't really an answer. Say there's 20 in a pot 6 x 20 is 120.

Edith and Joe

Oranges: It would be about 27 because 3 x 5=15 then add 6 which equals 21 then adding on to that add another 6 which equals 27.

Peaches: There are 20 peaches here because 5 x 4 = 20.

Aoife

It equals 20 because 5 x 4 = 20.I counted the rowes and on the side row there 5. On the bottom 4.

Samuel and Lucy

I made it symmetrical and counted each side and it turned out to be 5 + 6 + 5 + 6 = 22. I added the middle line so it was 22 + 6 = 28. Then I subtracted one from the current answer 28. So the answer was 27 (when you do 28 - 1 = 27).

Then we had these three solutions from Kuala Lumpur Alice Smith School in Malaysia:

Coco and Seraphina

For the first one, you count all of the peaches, and get 20, then for the oranges, you count all of them as well, and get 37. Finally, you estimate, and get thirty approx, and times it by six, since there are six boxes. In total, you get 180 raspberries. We estimated 30, since there are about ten in each layer, and there are three layers.

Mia

For the raspberries my estimate for 1 box is: 39. My method was to take a screen shot then count the raspberries with dots. I knew how to separate the raspberries by looking at the lines. Then I did 39 x 6 = 234.

Hannah

So for the first one I just counted vertically and horizontally and multiplied them to get 20.

For the second I counted from top to bottom and got 27.

For the last one I took screenshots and crossed off the raspberries. I found a reasonable half way point of the numbers I'm guessing it might be 35.

PS: I didn't go further than that.

Kayan from St. Martins School sent in the following:

1) 5 rows, each with 4 peaches so 5 x 4 = 20 altogether

2) We saw a pattern. Ignoring the hole, 3 columns of 6 oranges and 2 columns of 5 oranges.

So 6 x 3 = 18. 5 x 2 = 10.

18 + 10 = 28

Then we need to take 1 away for the hole. So 27 oranges.

3) We assume each box has the same number of raspberries and we count the berries in one box as 36.

So we need to do 6 x 36. We first do 6 x 30 = 180 and then we add on 6 x 6 = 36. 180 + 36 = 216

We had the following sent in from The International School of Brussels:

Simran from Maurice Hawk School USA sent in this last piece of work together with three pictures:

The peaches are arranged as an array of 5 x 4, so there are 20 peaches. We can also count them as 5 rows of 4 each or 4 columns of 5 each, which makes 20 peaches either way we count.

The oranges were an interesting case. I stared at them for a while, there are a few different ways to count them. In the first way, there are two outer columns of 6 each and three inner columns of 5 each, so 27 oranges (6 x 2 + 5 x 3 = 27).

In the second way, there are rows of 3, 4 and 5 oranges and the pattern repeats on the lower side of the crate in reverse order and there are 3 extra oranges.

In the third way, there are four rows of 5 oranges each, 4 oranges in each corner and 3 extra oranges in the middle of the arrangement.

Any way you count there are 27 oranges but I think the first way is best as it does not leave any extra oranges to count later (see the pictures below).

The raspberry containers looked like quarter pound boxes. When I searched how many berries could be there, the answer was anywhere between 30 to 40 berries. So my estimate is that all the boxes of raspberries displayed contain between 180 to 240 berries.

Thank you all for all the submissions that came in to us. It was good to read about the variety of ways you worked.

## Teachers' Resources

### Why do this problem?

This low threshold high ceiling task is accessible to everyone. It gives children the chance to share the way they picture (visualise) numbers and their methods of counting. One of the key features of this task is that it can be interpreted differently, depending on the image, so that children can decide for themselves whether they are counting individual
fruit, cartons of fruit... Therefore there may also be an opportunity for children to develop their estimation skills as well as appreciating different ways of counting.

### Possible approach

Ideally, share the first image of the peaches with the class and simply ask, "How many?". Try not to say anything else at this stage, but give learners a few minutes of individual thinking time before encouraging them to talk with a partner. Suggest that they describe to each other *how* they counted, not just their final answer.

After five minutes or so, bring the whole group together and invite them to share responses. It could be useful to ask children to annotate the image as they describe their method and it would be worth keeping these annotated images, if possible, as they may be useful to refer to later. Try not to offer your own view as learners share their thinking, but encourage the group to ask questions
and/or comment if they wish.

Next, show the second image of the oranges and suggest that learners have a go in pairs. Circulate as they work, listening out for different ways of counting. You could warn three or four pairs that you'd like them to share their thinking with the whole class in the next mini plenary. You could deliberately select a variety of different ways, some of which might have been discussed earlier and
some which are totally new.

After a second mini plenary, share the photo of the raspberries and invite pairs, or small groups, of learners to work together. In the final plenary, draw attention to those children who decided to count individual raspberries, in addition to punnets, and instigate a conversation about ways in which you could do that. Did children actually count, or did they use estimation as well or instead?
When might estimation be more appropriate than counting accurately?

(At any stage of the lesson, it may be helpful to give learners a printed copy of this sheet, which contains all three photos, with the rasperries on the second page.)

### Key questions

How many?

How do you know?

What did you do?

How would you check?

What made you decide to change your way of counting this time?

### Possible support

Having a print-out of the photos on this sheet may help some children access the task. You could recreate the pattern of peaches or oranges using real fruit, or counters for example, so that children could physically re-group them to help them count if they wished.

### Possible extension

You might like to take your own photos as a stimulus to the question "how many?". You could make the mathematics more challenging by including a few pieces of fruit which had been cut into fractional amounts e.g. half apples. Children might like to try How Would We Count?, which is a similar task but uses arrangements of dots.