In this unit we take samples of blocks from a bag and use them to make predictions about the blocks that the bag actually contains. The students learn that although we can't be 100% certain we can use the information that we gather from sampling to make reasonable predictions.
 Take samples and use them to make predictions.
This unit considers a very simple proportional model for estimating sizes and structures of populations. By making the naive assumption that the population structure is reflected in a sample size, we try to estimate populations. When we take a sample from the population we can calculate the ratio number of a type of object: sample size. We then assume that this is a reasonable estimate of the ratio of the number of objects of that type in the whole population. If we know the size of the population, we can find an estimate of the number of objects of the given type in the population as a whole.
In this unit the population is made up of coloured cubes. This simple idea is similar to an elementary method for calculating population size. In the wild a subset of the population of a species is captured and marked in some way. This subset is released. Later another subset of the population is captured. Some of these will be marked and some won't. Let the ratio of the marked ones to the size of the subset be r. Then assume that r = the ratio of the size of the original marked subset to the population size. In the last sentence we know everything in the equation except the population size. This we can now count. Capturerecapture theory is an important and sophisticated part of modern statistics. It is now increasingly important to estimate population sizes. This may be to determine whether fish stocks are high enough to harvest or if logging has caused harm to a species of bird.
Number Framework Stage 7, Advanced Multiplicative (Early Proportional)
 Paper bags
 Graph paper
 Coloured cubes
 Copymaster 1: Predicting Beads
 Copymaster 2: Predicting Red
Session 1
In this introductory session we work together as a class to take samples of cubes from a bag of 50 cubes. We use our samples to make predictions about the 50 cubes that are in bag. This activity practices the use of simple ratios.
 Put 50 cubes in a paper bag using a mix of red, yellow and blue cubes (or beads or pieces of paper).
In this bag there are 50 cubes of three colours. Our task is to find out how many there are of each colour.
We could look inside the bag and work it out exactly but today we are going to use a technique called sampling. Statisticians use samples to make predictions about situations that are unknown.
 Invite a student to take 5 cubes from the bag and record these on the board.
 Ask the students to share what they know about the contents of the bag from that sample. Possibilities include:
"I know that two of the three colours are red and blue."
"I think that there aren’t very many of the third colour as it didn’t get selected."
"I think that there are more blue cubes than red cubes in the bag."
 Based on this first sample ask the students to each write down how many cubes of each colour there are in the bag. Remind them that there are 50 cubes and three colours.
 Share some of the predictions, encouraging the students to justify their predictions.
For example:
20 red
28 blue
2 of the third colour
"I know that there aren’t many of the third colour but that there are some so I guessed 2. There are more blue than red. I guessed 20 red because 2 of the 5 cubes were red so 20 of the 50 cubes might be red. And 2 + 20 = 22 which leaves 28 to be blue."
Discuss other possibilities with reasons.
 At this stage the students are making predictions based on very little information so expect a large amount of variability in their answers. The predictions should however total to 50 and include 3 colours, as this information is a certainty. Tell the students that they will get to update their predictions after more samples have been taken.
 Ask another student to take out 5 cubes without replacing the first five cubes. Once more record these on the board and ask the students to make statements about what they could now say about the 10 cubes.
 Ask the students to revise their initial prediction and to justify their new prediction to a partner.
 Discuss as a class the new information that was gained from this sample. The students may need to be encouraged to combine the information from the two samples if, for example,
"We now know that the third colour is yellow."
"We are more certain that there are more blue cubes than red or yellow."
 Ask the students to tell you what they know for certain about what is in the bag?
For example:"The three colours are red, blue and yellow."
"There are at least 3 red, 6 blue and 1 yellow."
 Encourage students to make predictions that relate not only to what they know for certain, but also to what can be predicted.
For example:3/10 of the cubes we have drawn are red. Three tenths of 50 would be 15.
6/10 of the cubes we have drawn are blue. Six tenths of 50 would be 30.
 Keeping the 10 cubes out of the bag ask another student to take out a further 15 cubes. Once more record these on the board.
 Ask the students to revise their predictions based on the information that they now have. Remind the students that they now know about 25 of the 50 cubes from the bag.
For example:
"9 of the 25 cubes are red so I predict that 18 of the 50 cubes are red."
"2 of the 25 cubes are yellow so I predict that 4 of the 50 cubes are yellow."
"14 of the 25 cubes are blue so I predict that 28 of the 50 cubes are blue."
Discuss: How sure are you that your prediction is correct?
 Reveal the contents of the bag and compare with the predictions made by the students.
 Discuss: How did your predictions change, as the sample got larger?
Session 2
In this session we work in groups to take samples from bags of beads. We use these samples to make predictions about the beads in the bag. At the end of the session we compare our predictions.
Teachers’ Note: In this and subsequent sessions we suggest that you divide your class up into groups of no bigger than four. We have assumed that this gives you eight groups. For each of these groups you will need a plastic bag containing beads. Below we suggest the contents of each of the bags.
bag 
red 
white 
blue 
green 
Bag A 
25 
25 


Bag B 
40 
10 


Bag C 
20 
20 
20 

Bag D 
30 
20 
10 

Bag E 
20 
20 
20 
20 
Bag F 
50 
20 
5 
5 
Bag G 
55 
5 


Bag H 
27 
27 
6 

Naturally you will be constrained by the objects you have at your disposal. But try to organise containers of objects in about the proportions suggested in the table. If you need more bags of beads because you have more groups, you could have 30 red, 30 white, 10 blue and 10 green in another. Label each bag and mark the bag with the number of beads it contains.
 Show the students the eight paper bags of beads (labelled AH). Tell the students that their challenge is to take samples from the bags and use them to make predictions.
 Give each group one of the bags. Ask them to take a sample of 10 beads from the bag and use these to predict the total of each colour bead in the bag. Show the groups how to record their predictions on the recording sheet (see Copymaster 1).
Predictions 

Red 
White 
Blue 
Green 

Bag A 
20 
30 

Bag C 
45 
5 
 When the group have made their predictions they replace the beads and pass their bag to another group.
 Repeat this until the groups have made a prediction for the eight bags of beads.
 Encourage groups to record their predictions using fractions. Eg. 6/10 of the beads I drew were red and I know there are 50 beads, so I guessed 6/10 of 50 which is 30.
 Give each group one of the bags and ask them to count the number of each coloured bead in the bag. Students record the "population" of each bag on the board.
 Ask the groups to compare their predictions with the actual contents.
 Did any group predict exactly what was in a bag?
 Which bags were easier to predict? Why? (Expect students to find this for the bags where there are fewer colours and an equal number of each colour bead.)
Session 3
In the previous two sessions we took samples in order to make predictions about the population. In today’s session we link these predictions to the likelihood or probability of events occurring.
Teachers’ Note: In this session we again suggest that you divide your class up into groups of no bigger than four. We have assumed that this gives you eight groups. For each of these groups you will need a plastic bag containing beads. Below we suggest the contents of each of the bags.
bag 
red 
white 
blue 
green 
Bag A 
15 
15 


Bag B 
35 
5 


Bag C 
10 
10 
10 

Bag D 
24 
3 
3 

Bag E 
10 
10 
10 
10 
Bag F 
15 
15 
5 
5 
Bag G 
20 
10 


Bag H 
16 
7 
7 

Naturally you will be constrained by the objects you have at your disposal. But try to organise containers of objects in about the proportions suggested in the table. If you need more bags of beads because you have more groups, you could have 5 red, 5 white, 5 blue and 25 green in another. Label each bag and mark the bag with the number of beads it contains.
 Show the students the eight paper bags (labelled AH) of beads. Tell the students that their challenge is to take samples from the bags and use these to make predictions.

Show the students a bag containing beads but this time don’t tell them either the number of beads in the bag or the colours of the beads.
 Pose the question: What is the chance of getting a red bead from this bag? Tell them that they can select one bead from the bag but must put it back after each turn.
 Ask the students for their ideas about how to work out the chance of getting a red bead. Possible ideas include:
"Open the bag and have a look." (Remind the students that they are not allowed to peek into the bag – say they are allowed to put in their hand and take one bead.)
"Take a sample of beads and then look at how many are red." (Remind the students that they are only allowed to take one at a time.)
"Take beads out of the bag one at a time and keep track of which ones are red and which are not red." (Follow up on this response by asking them how they plan to keep track.)
 Ask one of the students to take a bead from the bag. If it is red ask them to tell you what this tells them about the probability of getting a red bead and what it tells them about the beads in the bag.
"There is at least one red bead in the bag."
"We don’t really know anything about the chance of getting a red."
 Record on the board in both a tally chart and a bar chart the outcome of the first draw.
 Replace the bead and then ask another student to make a draw and record this on both the tally and bar chart. Ask the students to make statements about what their predictions are after two draws.
"There are blue and red beads in the bag."
"I think that the chance of getting a red bead is ½ because 1 of the 2 beads so far is red."
"I think there is ½ chance of getting a blue bead."
"I think there are only blue beads in the bag so the chance of getting a blue is 100 percent or 1."
 Continue to draw beads from the bag. After 10 draws discuss the likelihood of getting a red bead. Discuss the use of fractions for expressing the probability of getting a red bead:
Estimated probability of a red bead = Number of red beads selected/Total number of beads selected
For example if 4 red beads were drawn the estimation for the probability of a red bead is 4/10.
 Open the bag and count the beads. Compare the findings to the predictions.
Session 4
In today’s session we work in groups to predict the probability of getting a red bead from a selection of bags.
Teachers’ Note: For this session use the bags of beads from Session 2.
 Show the students 8 paper bags (labelled AH) of beads. Tell the students that their challenge is to predict the probability of getting a red bead from each of the bags.
 Tell them that they can select one bead from the bag but must put it back after each turn. They need to decide on how they are going to keep track of the draws (for example, tally chart or bar chart) and then use this information to predict the probability of getting a red bead. Tell the students that they can make their prediction at any time.
 Show the groups how to record their prediction on the recording sheet (Copymaster 2).
Predict the probability of getting a red bead 

Bag A 
3/10 
Bag B 
 When the group have made their prediction they pass their bag to another group.
 Repeat this until the groups have made a prediction for the 8 bags of beads.
 Give each group one of the bags and ask them to count the number of each coloured bead in the bag. Students record on the board the actual probability of getting a red ball from the bag.
 Ask the groups to compare their predictions with the actual probability. (Note: This assumes that students are able to compare fractions.)
 Discuss.
Reflecting
In the final session of the week we make predictions about certain characteristics of people based on samples taken from within our class.
 Tell the students that today we are going to make predictions about children our own age based on samples from within our class. List the following probability investigations on the board.
 What is the probability of being left handed?
 What is the probability of having brown eyes?
 What is the probability of having a sister?
 What is the probability of having a dog in your household?
 What is the probability of getting a ride to school?
(These questions could be extended to numerical ones such as how many left handed children there are in the school or in the country.)
 Ask for other ideas from the class and include these in the list.
 In groups the students select one of the questions to investigate.
 Encourage the students to plan ways to keep track of the information they gather as they will need this to support the prediction statement that they make as an answer to the question. Also ask them to think about how many students they should ask in order to make as accurate a prediction as possible (the whole class!).
 Ask each group to write a report on what they did and what they found out.
 Share written reports with the class.