## Bean Counting and Ratios

• Lesson
6-8,9-12
1

By using sampling from a large collection of beans, students get a sense of equivalent fractions, which leads to a better understanding of proportions. Equivalent fractions are used to develop an understanding of proportions.

This lesson can be adapted for lower-skilled students by using a more common fraction, such as 2/3. It can be adapted for upper grades or higher-skilled students by using ratios that are less instinctual, such as 12/42 (which reduces to 2/7).

Scaffold the level of difficulty in this lesson by going from a simple ratio such as 2/3 to more complicated ratios such as 2/7 or 5/9.

Students may need a review of

• Converting a fraction to a decimal
• Rounding to three decimal places

This lesson is an application of equivalent fractions. Having a concept of equivalent fractions is important for simplifying fractions. The number-sense of recognizing equivalent fractions is useful when students study slope and proportions.

Pairs (or groups of) students use a cup of beans to find ratios to express the number of marked beans in the cup compared to the total number of beans in the cup. Theoretically, each sampling ratio should be essentially the same. The decimal representation of each ratio confirms that the ratios are, indeed, approximately equivalent.

Before Class

Prepare 1 cup of beans that has 20 marked beans and 20 unmarked beans for a total of 40 beans. This cup is for you to use in the whole-class example.

Prepare student cups of beans. Each cup should have 45 beans. Mark 30 of the beans on both sides with a marker, and leave the other 15 unmarked. This number of beans is consistent with Activity Sheet A.

Activity Sheet B Prepare enough of these cups so that each pair/group has their own.

If your students are comfortable with fractions, you may use the less intuitive ratio 12 beans marked out of 42 (which is 2/7). In this case, use Activity Sheet B.

With the Class

Demonstrate the activity that students will do on their own with the following:

• Show the class the cup of 40 white beans and tell them you’ve marked 20 of them with a colored marker.
Suppose I pour 30 of the beans into my hand. How many of them do you expect to be marked? Why?

[15, because the ratio of marked to total is 1/2. Therefore, we expect half of them to be marked]

Suppose I scooped out 12. What do you predict the number of marked beans will be? Why?

[6, because we expect half of them to be marked]

• Shake the cup and pour some beans onto the table. Tell the class how many beans are on the table. Ask how many they expect to be marked and why. Ask a student to count the marked beans and discuss the results. Most likely, there won't be exactly half the beans marked. Talk about how the result of an experimental trial may not perfectly reflect the expected value, but it should be close. Put the beans back in the cup and do another trial.
• Put students in their pairs/groups and distribute the activity sheets. Be clear with students that the cups you are distributing are not the same as the one you used for the demonstration.
• Draw the table from the worksheet or use this overhead to emulate filling in the data for the activity.

Tell students how many beans are in the cup and how many of them are marked. Fill in the top row of data as a class to remind students that this cup is different from the demonstration cup

Make sure to emphasize that students must put the beans back into the cup after each trial so they always start with the same number of beans. Encourage them to take both small and larger handfuls for the trials.

Then, allow students time to complete the additional trials.

Some pairs/groups may need further assistance with the bottom section of the table on page 1 of the Activity Sheet, where they determine what values produce the expected ratio. Ask students to find patterns, or guess-and-check to find the missing values. Proportions are also an option.

Students should find that most of the decimals they calculate for each trial in the last column of the table are close together. However, some trials may have too few or too many marked beans. This will enrich the conversation at the end of class, guided by the Questions for Students section below.

Discussing Variation in Experimental Results

Consider asking students to describe the variation in the decimal representations. The class could make a bargraph of their decimal values to enrich or drive questions such as:

• What is the mean of the decimal representations?
• How many of the decimal representations are within a certain absolute value of the mean?
• What would be an acceptable range for a "good" calculated answer? Why?

Assessments

1. Ask students to predict outcomes for these examples:
In this class there are 28 students. One half of them are female. How many of them do you expect to be female? If 9 students are called to the office, how many are likely to be girls? [4 to 5]

A florist used 25 flowers to make a bouquet, Five of the flowers are daffodils. If 10 flowers are removed from the bunch at random, how many of them do you expect to be daffodils?

2. Write another example of your own.
3. Given a pair of ratios, how can you tell if they’re equivalent?

Extensions

1. How can you use this concept of equivalent ratios to determine how many beans are in a cup if you don’t know how many I give you at first?
2. Distribute another set of cups, and tell students how many beans are in them. Ask them to use sampling to determine how many of the beans in the cup have been marked.
3. A factory puts 150 raisins and 100 peanuts in each package of peanuts and raisins. In a sample with 75 pieces, how many pieces do you expect to be raisins? How is this question different from the other questions we have explored in this activity?
[In this example, the total number of beans isn't explicitly given. The total is 250, so the part-to-whole ratio is 150/250 or 3/5 are raisins. The part-to-part ratio (raisins/peanuts or 150/100 = 3/2) can also be used.]

Questions for Students

1. How did your results for the trials (your experimental values) compare to the expected value? Why do you think these differences occur between an experimental value and the expected value?
2. Say you have a sample that contains only marked or only unmarked beans. How would that affect the inference you make about the population the sample was taken from?
3. Explain the method(s) you used to fill in rows 7—10 on the Activity Sheet.

Teacher Reflection

• How did you address the definition of ratio for students who weren’t familiar with the term?
• What methods for finding the missing values did students come up with?
• What common errors did students exhibit in trying to find the missing values?
• What examples would you use to help students understand the errors in their reasoning that led to incorrect results?
• How did student pairs divide up the work?
• How will you change if you use it again?

### Learning Objectives

By the end of this lesson, students will:

• Recognize equivalent ratios
• Determine good and poor estimates