Intro:
In this session, we will focus on developing a conceptual understanding of ratios—
the types and language associated with ratios.
This session is designed to provide teachers with strategies for helping students
understand how to teach ratios at a deep level. In addition to watching this pencast,
we recommend you examine resources such as The Van de Walle Professional
mathematics series volume 3 chapter 6 for this task set in a real-world context and
additional extensions.
Previous:
Before we examine today’s task, let’s take a minute to discuss how ratios are
typically developed in textbooks and classrooms.
Often this concept is taught after fractions. Students would have prior experience
with fractions and simplifying a faction, which may help with the understanding of
ratios. Typically, students are introduced to a ratio by the definition and the forms
in which ratios are written. After discussing the types of ratios, the teacher or
textbook example may then have students’ practice simplifying ratios and writing
them the three different ways. Students may be led through an example like the one
written here.
A ratio is a comparison of two numbers by division. The ratio of the number 3 to the
number 5 can be written in these ways: 3 to 5, 3:5, or 3/5.
Students will be asked to express a ratio as a fraction in simplest form using one or
all of the three procedural ways to write a ratio.
While helping students develop procedural fluency is important, teachers need to
balance this with a strong conceptual understanding. If teachers teach ratios from a
strict procedural approach, students are learning the “how to” but do not fully
understand ratios. Students who do not have a conceptual understanding about
ratios are more likely to have a number of misconceptions, which might include not
knowing how to get started, that ratios are just fractions, and students who are
struggling with the abstract meanings of the types of ratios may also struggle with
proportionality.
Now let’s look at another way to introduce ratios. As a first step, let’s look at a chart
to analyze what type of ratios there are and really understand the language with
each ratio not just how to write a ratio as stated above.
Ratio
Comparing same type of measure
Part to part
part to whole
Comparing different types of measure
whole to whole (rate)
As we look at the chart we notice that there are two main types of ratios…those that
compare the same type of measures and those that compare different types of
measure. Breaking it down a little further we see that ratios under the same
measure are either comparing part-to-part or part to whole of the same item or
measure while the different measure compares whole to whole or the rate.
Next let’s take a real world example and decide what type of ratio we have by
categorizing the ratios into part to whole, part-to-part, or whole to whole/ (rate).
The example we will use is a test. We will look at three different ratios using the test
example.
The first ratio is the number of points earned to the number of possible points.
What type of ratio is that?
Please take a minute to pause and decide where that example belongs in the chart.
That is an example of a part to whole ratio. (The part being the number of points
earned and the whole being the total number of points) 21 to 25
The second ratio is the number of points earned to the number of points not earned.
What type of ratio is that?
Please take a minute to pause and decide where that example belongs in the chart
That is an example of a part-to-part ratio. (The part being the number of points
earned a part being the number of points not earned.) 21 : 4
The third ratio is the number of points earned to the number of students in the class.
What type of ratio is that?
Please take a minute to pause and decide where that example belongs in the chart.
That is an example of a whole-to-whole ratio. (The whole being the number of
points earned and the whole being the number of students in the class) which are
two different measurements (points and students or points per student) Total # of
points earned out of the total # of students taking the test.
Notice, that when these ratios are sorted that each fits into one area, part to part,
part to whole, or whole-to-whole.
Now it is important to note that when doing the task with students you would want
varied examples of ratios placed on cards and then allow the students to cut and
sort the ratios, as well as discourse with each other while completing the sorting.
As you have students do this task some key questions to pose are: “what do you
notice” and “what is your reason for placing a ratio in a certain area? ” That is where
emphasizing looking at the words and language of the ratio is important.
Teaching Implications:
This task is a great way to use a real world approach to examining what is a ratio by
sorting the different types of ratios. Students may also begin to see how the ratio
really describes the comparison between two quantities. While being able to model
with the ratio cards and reason abstractly students will be practicing some of the
skills for the standards of mathematical practice.
So, in closing, we believe the best way to introduce ratios is to provide students with
an opportunity to develop a conceptual understanding first before formalizing rules
and algorithms for fractional computation, for comparing ratios, and for solving
proportions. Students need to understand the different types of ratios in addition
to how to work with ratios. These way students develop a rich understanding of the
mathematics and are better prepared to reason proportionally.
This wraps up our session. Thank you and please join us for our other parts of the
Ratio pencast series.