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.
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 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.
- Ask the class:
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
- 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
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
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 bar graph 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?
By the end of this lesson, students will:
- Recognize equivalent ratios.
- Determine good and poor estimates.
NCTM Standards and Expectations
- Understand and use ratios and proportions to represent quantitative relationship.
Common Core State Standards – Mathematics
Grade 6, Ratio & Proportion
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, ''The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.'' ''For every vote candidate A received, candidate C received nearly three votes.''
Common Core State Standards – Practice
Make sense of problems and persevere in solving them.