Ima's Dilemma is an extended activity that students should be allowed to think about for several days. Distribute the Ima's Dilemma Activity Sheet, and read Ima's Dilemma aloud to students. Explain that
their job is to help Ima find a solution, but also explain that they do
not need to find a solution immediately. In particular, complete the
other activities in this lesson to give students the background they
will need to attack the problem.
Give students a deadline by which they should complete their work on
Ima's Dilemma, and allow at least 2‑3 days for them to think about it.
It works well to use Ima's Dilemma as a problem of the week; post the
problem on Monday, and require solutions to be submitted by Friday.
Splitting Beans Activity Sheet
Splitting Beans Overhead
For the main part of the day's lesson, allow students to work in
pairs. Each group should have at least one calculator and 40–50 beans
in a paper cup. Be sure to prepare the cups ahead of time. Distribute
the Splitting Beans Activity Sheet and use the Splitting Beans Overhead to complete a sample row. Have students gather 23 beans and put them into groups of 5. Then ask:
- How many whole groups do you have? [4.]
- How many beans are left over? [3.]
Explain that the 3 beans form 3/5 of a group, because they were
dividing the beans into groups of 5. Then show students how they can
use the calculator to find the same information: 23 ÷ 5 = 4.6.
Because 0.6 is equivalent to 3/5, it must be the case that 3 beans are
leftover; the other 20 beans were divded into 4 groups of 5.
After completing the example, tell students to complete the activity
sheet on their own. It works well to allow students to work on the
first few problems individually for several minutes. Then, allow them
to form pairs and work together to complete the rest of the activity
sheet. Misconceptions are often clarified if students are given an
opportunity to discuss the problems with a classmate.
Leftovers Activity Sheet
When students finish all questions on the activity sheet, ask, "How
does the fraction column relate to the calculator column?" [The
calculator column has the decimal form of the fraction.] Discuss
Questions 2 and 6. Students should recognize that the number leftover
is not the only factor influencing the decimal part of the quotient.
Students should realize that the number leftover and the group size
together influence the decimal portion. Use the Leftovers Activity Sheet to reinforce this concept. Allow students to use a
calculator to determine the number of whole groups and the number
Randomly select pairs to present each problem, discussing their
solution methods. Summarize the solution methods before returning to
Ima's Dilemma. As noted above, allow several days for students to find the solution.
They should include a thorough description of their solution method.
Students usually work this problem by guess and check, systematic lists
or a combination. If using guess and check, they try numbers (usually
on their calculators) and use the ideas from the Splitting Beans Activity Sheet to determine the remainders. If they use systematic
lists, they identify the possibilites for each clue and find one that
is on all the lists. If they use a combination of the strategies, they
generate their lists by guessing and checking
Splitting Beans Answer Key
Leftovers Answer Key