Teaching ideas

•   How much area? (Teaching resource sheet 93)

This link displays some spinners coloured in different ways (not necessarily by sectors) and asks students about the probability of landing on colours.

Discuss the fact that it is the area that each colour covers that determines the probability.

•   How do I know? (Teaching resource sheet 94)

This link displays cards for you to print, laminate and cut out. There are eight identical singers with microphones, five identical people playing keyboards, three identical people playing drums and nine identical people playing guitars.

1.     Count the total number of cards and the number of each different type of picture card.

a  Shuffle all the cards together and pick a card at random.

b  What proportion of the cards are the same as the one you picked? What was the chance that you would pick this card?

    (The chance of getting a card with a singer is 8/25, 32% or 0.32 because this is the proportion of cards that are singers,

    with a keyboard player 5/25, 20% or 0.2 because this is the proportion of cards that have keyboard players, with a drum
    player 3/25, 12% or 0.12 and with a guitar player 9/25, 36% or 0.36).

c  Replace the card and repeat steps a and b several times.

2.     Randomly remove five of the cards and count the total number of cards and the number of each different type of picture
        card in the remaining cards.

a  Repeat parts 1a and 1b several times.

3.     How are the proportions of picking a particular type of card different in 1 and 2? Can you explain what has caused
        the change?

4.     If you randomly removed another 2 cards, describe how you would work out what the proportion might be.

5.     How does using the proportion of a particular card as the chance of getting that card, if you picked a card at random,
        relate to the following formula?

Probability of an event = number of ways of getting an event divided by the total number of outcomes.

• What if it doesn’t happen?

1.     Write down on a piece of paper ten different sandwich fillings that you could get on a school camp.
        (Examples, tomato, cheese, pickle…).

2.     Put the pieces of paper into a hat.

3.     Ask “What is the probability that I will draw out ‘tomato’ filling?" (1/10)

4.     Ask “What is the probability that I will not draw out ‘tomato’ filling?” (9/10)

5.     Ask ”What can you say about the probability that I will either draw out ‘tomato' filling or I won’t draw it out?”
        (That is certain and has probability of 1.)

6.     Ask “What does 1/10 + 9/10 equal?”

7.     Ask “What is the probability of these?”

             I won’t get a 6 when I roll a die

             I won’t get a 3 or a 4 when I roll a die

             "If I put your five names (name five students) into a hat, I won’t get ??? when I pull out one name."

• Coin and die

1.     Tell students you are going to toss a coin and roll a die.

2.     Ask “What is the probability I will get a number less than 5 and a head?” Students are unlikely to be able to tell you.

3.     Discuss how helpful it would be to list the outcomes on a table or diagram. Ask them to do this on mini whiteboards or
        in their books.

4.     Repeat question 2 and discuss the answer.

Teacher notes

• Students will usually give fractional answers to theoretical probability questions. Sometimes these need to be converted to percentages, decimals or ratios. You might need to revise with the students how to change fractions into percentages and thus
  into decimals.
• It is helpful for students to work with practical materials to assist them in understanding the probability formula, 
  Probability of an event = number of ways of getting the event divided by the total number of outcomes. 
• It is important that students understand that some events are impossible and have a probability of 0 and that some are certain and have a probability of 1. The probability scale helps with this understanding.

• Check that students understand why probability has to be between 0 and 1, with 0 being impossible and 1 being certain.

• Encourage students to think of probability as a proportion. For example, if there is a greater proportion of yellow counters in a bag then it is more likely that a yellow counter will be drawn out.
• Encourage understanding of the answer rather than routinely using a formula to calculate it.
• Emphasise that to calculate probability using the formula Probability of an event = number of ways of getting the event divided
  by the total number of outcomes events must be equally likely to occur. For example, if there are two yellow beads, one red
  bead and one green bead in a bag then it is not equally likely that you will get either yellow, green or red because there are more yellow beads than red or green. Although there are three colours the total number of outcomes is four because there are four
  beads in the bag so the probablility of getting yellow is not 1/3. The probablilities of getting each colour are red 1/4, green 1/4
  and yellow 2/4 or 1/2.
• It is easier to see the outcomes which satisfy the event for which the probability is being calculated in a table or diagram.