Math Background

Lesson: Probability
Developing the Concept

Many problem-solving situations involve using probability. We can use probability to make predictions about events if we have data to analyze. Present the following problem to the class. Roll two 1-6 number cubes and find the sum of the two numbers. Roll again and find the product of the two numbers. Which roll has the greatest probability of getting an odd number for an answer, the roll with the sum or the roll with the product?

  • Say: Let's create some data so that we can make a prediction about the probability.
    Have a student fill in an addition table with the correct sums, which represents the rolling of the two number cubes. The table shows the 36 possible outcomes when rolling the two cubes.
    addition table

    Then have a student come to the board and circle all of the odd number sums.

    addition table
  • Say: Now let's create some data about multiplication.
    Then have a student come to the board and fill in the correct products in the multiplication table.
    Remind the students that the tables represent all the possible outcomes of rolling the two cubes and adding or multiplying the results. Six possible outcomes times 6 possible outcomes equals 36 total outcomes.
addition table

Have another student circle all the odd-number products in the multiplication table.

addition table
  • Say: Now that we have some data about sums and products from rolling our number cubes, we can make some predictions.
  • Ask: Who can answer the original question? Will the probability of rolling a sum that is odd be greater than the probability of rolling a product that is odd?
    Students should compare the circled odd numbers in the tables and conclude that finding an odd sum is more likely.
  • Ask: When we roll both cubes and add, how many outcomes are there? (36)
  • Ask: How many of those outcomes are odd numbers?
    (18 odd numbers)
  • Ask: What is the probability of the sum being an odd number?
    (eighteen-thirty-sixths = one-half)
  • Ask: How many total outcomes are there when we roll two cubes and multiply the numbers? (36)
  • Ask: How many of these products are odd? (9)
  • Ask: What is the probability that the product will be an odd number? (nine-thirty-sixths = one-fourth)
  • Ask: Compare the probabilities of the two events. Which is more likely, rolling an odd-number sum or rolling an odd-number product? Students should find that rolling an odd-number sum is more likely than rolling an odd-number product. They should point out that one-half > one-fourth.

    Have students make predictions about outcomes by answering the following questions, using data from this table.

Ramona's Music CDs
  1. If Ramona chooses a CD at random, what is the probability it has 50-54 minutes of music? (four-twenty-fourths = one-sixth)
  2. If Ramona chooses a CD at random, what is the probability it has less than 55 minutes of music? (six-twenty-fourths = one-fourth)
  3. If Ramona buys a new music CD, is it likely or unlikely to have at least 55 minutes of music? (likely)

Wrap-Up and Assessment Hints
Practice mental math probability activities with your students to strengthen their skills in probability and in fractions. For example, have them suppose they are choosing colored cubes or marbles from a bag. A bag contains 2 yellow, 3 black, 3 red, and 4 green marbles. Ask students questions that will lead them to find probabilities: What is the probability of choosing a red marble? (three-twelfths = one-fourth) a green marble? four-twelfths = one-third) a black or a yellow marble? (five-twelfths). Such activities will provide students with practice simplifying fractions and strengthen students' understanding of probability.

Houghton Mifflin Math Grade 5