What's in the bag?

Purpose

In this unit we experiment with cubes to make predictions about likelihood based on our observations. With these experiences the students find out that with probability there is no way of knowing for sure that will happen.

Specific Learning Outcomes
  • make predictions based on data collected
  • identify all possible outcomes of an event
  • assign probabilities to simple events using fractions (1/2, 1/6 etc)
Description of Mathematics

In this unit the students gather information in order to make predictions about what is likely and unlikely based on what they have observed. As the students explore the meaning of likely and unlikely and perform experiments that involve making predictions, they begin to form a more numerical sense of probability. The students are also introduced to ways of identifying all possible outcomes of an event (the event or sample space). As they play games, record results and use the results to make prediction they find out that with probability they can never know exactly what will happen next, but they get an idea about what to expect. At Level 2 Students can handle simple fractions and at Level 3 they build on this (Number knowledge AO4). Assigning simple probabilities provides them with an interesting and useful application of these numbers. As a result, we would expect students to be able to see that the probability of getting a blue cube when there is a red and blue cube in the bag to be ½. Given there were three different coloured cubes in the bag we would also expect them to see that the probability of getting any one of the colours is 1/3, and so on. The explanation would be that there are three equally likely events and that one of them has to happen. Hence, over the long run, you would expect the chance of getting a particular colour is one chance out of three, or 1/3.

Required Resource Materials

cubes of different colours

paper bags

Key Vocabulary

 probability, prediction, possible outcome, likely, unlikely, certain, combinations

Activity

Today we make predictions about the cubes that are hidden in a bag. We find out that even when we can’t peek in the bag we can still make a good guess about what is in it.

  1. Put 4 cubes in a bag (3 red and 1 blue).
    Here is a bag with 4 cubes. The cubes are either red or blue and we’re going to try to find out what they are by peaking at one at a time.
  2. Shake the bag and ask a student to select one and show the class. Record the colour on the board and get the student to put the cube back in the bag.
    (Note: Each time a student takes a cube it must be returned before the next student draws a cube. Other wise, the probabilities will change.)
  3. Ask another student to select a cube.
    What colour have you got?
    If it is the same colour as the one previously drawn ask: Do you think that it is the same cube? Why or why not?
  4. Ask a third student to draw a cube but this time get them to predict what the cube might be.
    Why did you guess that?
  5. Ask a fourth student to draw a cube.
  6. Look at the result of the 4 draws.
    Do you think that we have seen all the cubes?
    Do we know what the 4 cubes are? Why or why not?
    Would we find out more if we had more turns?
  7. Let another 4 students select a cube and record these on the board. Each time ask the student to predict the colour of the cube.
  8. Before we look in the bag I want us to think about all the possible combinations for the cubes.
    Record these on the board.
      
    Possible combinations for 4 cubes
    RedBlue
    13
    22
    31
      
  9. Ask the students to decide which combination they think is likely.
  10. Look inside the bag and check the cubes. Discuss.
  11. Put the cubes back in the bag and ask:
  12. I am going to draw a cube. Which colour do you think it will be?Why?
    Can we be sure that I will get that one? Why?

Exploring

Over the next three days we work in pairs to make our own bags of cubes. We swap them with our friends to see if they can guess "what’s in the bag?"

 

  1. Give each pair of students 10 cubes each of 2 colours. Ask them to put 10 cubes in their bag using any combination of the 2 colours.
  2. Swap bags with another pair and tell the students that they are to predict how many cubes there are of each colour in the bag by taking turns drawing cubes from the bag. Remind them to put the cube back in the bag after each draw. Tell the students that they can make as many draws as they like before they make their guess.
  3. Ask the students to make a recording of their draws and to write about why they made the prediction.
  4. At the end of each session let the pairs share their predictions and then open the bags. Discuss.
    What did you predict? Why?
    Were you surprised when you looked in the bag? Why?
    How many times did you draw from the bag?
    Do you think that the more times you draw from the bag, the more you know about what’s inside?
  5. Over the 3 days you could vary the number of cubes but limit the choices to 2 colours.

Reflecting

In this final session we toss 2 coins and see if we can guess how they might land.

    • If I toss one coin how do you think it will land? Why?
      What if I toss 2 coins?
      What are the possible combinations if I toss 2 coins?
    • List these on the board. (2 heads, 2 tails, 1 of each)
    • Which one do you think is most likely?
    • How could we find out?
    • Give each pair of students two coins and ask them to toss them 20 times keeping track of the results. Share and discuss the results as a class.
    • Ask the pairs to toss the coins another 20 times. Let the pairs circulate around the class looking at the results of others.
      Did you get the same results? Why do you think that happened?
      Does one of the ways of landing seem to come up more often than the other?
      If you had to guess which way it was going to happened which would it be? Why?
      Can you be certain about what is going to happen? Why?

(Note: You expect about ½ of the tosses to be "1 of each" with a ¼ for "2 heads" and ¼ for "2 tails". The fraction of times that you get "1 of each" gets closer to ½ as the number of tosses increases. Of course, we do not expect students at this age to know this, but the students can begin to develop a sense of this as the tosses increase.)


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