WEBVTT

00:00:00.086 --> 00:00:06.786
[Music] Welcome to the overview on
Developing Proportional Reasoning.

00:00:08.045 --> 00:00:09.576
Mr. Palm's unit on ratios

00:00:09.576 --> 00:00:12.076
and proportional reasoning
hasn't been going well.

00:00:12.896 --> 00:00:16.136
He introduced his students to the
cross-multiplication procedure,

00:00:16.366 --> 00:00:18.246
which they seemed to pick up quickly.

00:00:19.026 --> 00:00:21.696
Over time, however, Mr. Palm notices

00:00:21.696 --> 00:00:25.576
that his students are not developing
a solid understanding of how and when

00:00:25.576 --> 00:00:28.406
to use cross-multiplication
to solve problems.

00:00:29.226 --> 00:00:33.376
Even his most diligent students are
often puzzled when confronted with real-

00:00:33.376 --> 00:00:35.946
world problems that involve proportions.

00:00:37.076 --> 00:00:40.146
Proportional reasoning is
critical for success in algebra,

00:00:40.506 --> 00:00:44.686
so Mr. Palm asks the district math
coach to help him restructure the unit.

00:00:45.586 --> 00:00:49.246
The coach urges him to spend time
developing students' understanding

00:00:49.246 --> 00:00:52.926
of proportional relations before
teaching cross-multiplication.

00:00:53.716 --> 00:00:57.356
Students need to understand that even
though ratios are often expressed

00:00:57.356 --> 00:01:01.676
as fractions, ratios and fractions
are not always the same thing.

00:01:02.476 --> 00:01:05.586
Students often struggle with
this common misconception.

00:01:06.866 --> 00:01:08.566
Mr. Palm needs to build on

00:01:08.566 --> 00:01:11.916
and strengthen students'
informal proportional reasoning

00:01:11.916 --> 00:01:16.636
about relationships between quantities
and help them use multiple strategies,

00:01:16.816 --> 00:01:19.386
using different types of
visual representations,

00:01:19.626 --> 00:01:21.556
to solve real-life problems.

00:01:22.536 --> 00:01:27.196
Students need lots of practice with the
multiplicative relationships of ratios,

00:01:27.196 --> 00:01:31.956
rates, and proportions before they learn
cross-multiplication as an algorithm.

00:01:33.026 --> 00:01:35.946
They need to understand that
proportion is a relationship

00:01:35.946 --> 00:01:38.296
of equality between two ratios.

00:01:39.026 --> 00:01:40.906
In a proportion, the ratio

00:01:40.906 --> 00:01:45.316
of two quantities stays constant while
the values of the quantities change.

00:01:46.456 --> 00:01:49.986
Fortunately, multiplicative
relationships occur all the time

00:01:49.986 --> 00:01:53.886
in daily life, and most students have
an intuitive understanding of how

00:01:53.886 --> 00:01:55.606
to answer questions about them.

00:01:56.726 --> 00:01:59.316
How can you adjust this
recipe to serve more people?

00:02:00.156 --> 00:02:01.846
Which car gets better mileage?

00:02:02.606 --> 00:02:04.466
Which of these is the best buy?

00:02:05.786 --> 00:02:08.985
Teachers can demonstrate a
buildup strategy to show reasoning

00:02:08.985 --> 00:02:10.606
through questions such as these.

00:02:11.556 --> 00:02:14.676
The buildup strategy involves
creating equivalent ratios

00:02:14.676 --> 00:02:18.526
by the repeated addition or
partitioning of the numbers in a ratio.

00:02:19.816 --> 00:02:23.236
For example, two pounds of
fish will serve five people;

00:02:23.666 --> 00:02:26.166
if we expect 15 people
to come for dinner,

00:02:26.416 --> 00:02:28.256
how many pounds of fish should we buy?

00:02:29.356 --> 00:02:31.586
Students might initially
tackle that problem

00:02:31.586 --> 00:02:33.856
with counters or by drawing a diagram.

00:02:34.976 --> 00:02:38.566
By using a buildup approach,
students work up to the solution

00:02:38.566 --> 00:02:40.386
by repeatedly adding numbers.

00:02:41.626 --> 00:02:45.016
Students can use a ratio table
to organize their thinking

00:02:45.016 --> 00:02:48.856
and record the relations in a
proportion problem, helping them see

00:02:48.856 --> 00:02:52.966
that multiplication leads to the same
solution as the buildup strategy.

00:02:53.976 --> 00:02:57.016
The ratio table provides
a visual representation

00:02:57.016 --> 00:03:02.446
of how multiplying both numbers in the
2-to-5 ratio by 3 reveals the number

00:03:02.446 --> 00:03:04.816
of pounds of fish needed for 15 people.

00:03:06.026 --> 00:03:10.236
Once students have a good understanding
of buildup strategies and ratio tables,

00:03:10.496 --> 00:03:14.276
teachers can present more challenging
problems in which there is an advantage

00:03:14.276 --> 00:03:16.536
to using a unit ratio strategy.

00:03:17.726 --> 00:03:22.316
In a unit ratio approach, the quantities
in a ratio are multiplied or divided

00:03:22.316 --> 00:03:25.926
by the same factor to maintain
the proportional relationship.

00:03:26.936 --> 00:03:31.926
This strategy is best shown by solving
for x in a problem involving ratios

00:03:31.926 --> 00:03:33.436
without an integral relation.

00:03:34.326 --> 00:03:40.016
For example, take the problem
3 is to 9 as x is to 21.

00:03:40.846 --> 00:03:44.686
The unit ratio approach involves
reducing the known ratio to a form

00:03:44.686 --> 00:03:50.346
with a numerator of 1, that
is, changing 3 is to 9 to 1/3.

00:03:51.196 --> 00:03:55.676
The next step is to determine the
relationship between 1/3 and the ratio

00:03:55.676 --> 00:03:57.456
that includes the unknown factor.

00:03:58.176 --> 00:04:03.406
In this case, multiplying both numerator
and denominator of 1/3 by a factor

00:04:03.406 --> 00:04:07.456
of 7 gives us the value
of x, which is 7.

00:04:08.446 --> 00:04:12.256
The unit ratio strategy also may
be recorded in a ratio table.

00:04:13.186 --> 00:04:16.796
Experts recommend that students
get practice with applying buildup

00:04:16.946 --> 00:04:20.216
and unit ratio strategies
to real-world problems

00:04:20.216 --> 00:04:25.106
such as Comparing the unit prices
of various objects, Enlarging

00:04:25.106 --> 00:04:29.456
or reducing drawings or photos,
Adjusting quantities in recipes,

00:04:29.806 --> 00:04:34.286
or Calculating the time, speed, and
distance traveled by different vehicles.

00:04:35.306 --> 00:04:39.876
The goals for students are to
Identify the features of problems

00:04:39.876 --> 00:04:44.736
that involve ratios and proportions,
Notice the key information in a problem,

00:04:44.736 --> 00:04:48.276
and   Create diagrams or
tables to track their thinking.

00:04:49.406 --> 00:04:52.766
Students need many opportunities
to explain their reasoning

00:04:52.766 --> 00:04:55.686
and compare their solutions
to those of other students.

00:04:56.676 --> 00:04:59.976
Teachers should encourage use
of a variety of strategies

00:04:59.976 --> 00:05:03.056
and visual representations,
including drawings,

00:05:03.296 --> 00:05:05.796
ratio tables, and double number lines.

00:05:06.976 --> 00:05:09.696
Once students demonstrate
that they can reason their way

00:05:09.696 --> 00:05:12.466
through problems involving
ratios and proportions,

00:05:12.746 --> 00:05:15.936
they can then be taught
cross-multiplication as a procedure

00:05:15.936 --> 00:05:19.846
that works regardless of the
complexity of the numbers within ratios.

00:05:21.006 --> 00:05:24.236
Cross multiplication is based
on this principle:

00:05:24.236 --> 00:05:26.976
When two equal fractions
are converted into fractions

00:05:26.976 --> 00:05:30.976
with the same denominator,
their numerators are also equal.

00:05:31.746 --> 00:05:35.176
Consider the ratios of
4 to 6 and 6 to 9.

00:05:35.956 --> 00:05:42.176
Multiply 4 over 6 by 9 over 9 (which
is the same as multiplying by 1);

00:05:42.806 --> 00:05:47.056
then multiply 6 over 9 by 6 over 6.

00:05:48.116 --> 00:05:50.656
Calculate and check that
the denominators are equal.

00:05:51.176 --> 00:05:53.816
If two equal fractions
have the same denominator,

00:05:54.076 --> 00:05:56.126
then the numerators are equal as well.

00:05:57.226 --> 00:06:01.436
It's very important that teachers show
how the cross-multiplication algorithm

00:06:01.656 --> 00:06:04.676
leads to the same answer
as do reasoning strategies

00:06:04.676 --> 00:06:06.916
such as buildup and unit ratios.

00:06:07.946 --> 00:06:11.936
Give students a chance to solve
problems using all types of strategies,

00:06:12.156 --> 00:06:14.606
and discuss which methods
are most efficient

00:06:14.606 --> 00:06:16.206
for different types of problems.

00:06:17.306 --> 00:06:19.806
Once they have learned
about cross-multiplication,

00:06:20.046 --> 00:06:23.256
students may attempt to apply
the procedure without regard

00:06:23.256 --> 00:06:25.326
for whether a problem
is set up correctly.

00:06:26.006 --> 00:06:30.816
Also, by using cross-multiplication
exclusively, students may fail

00:06:30.816 --> 00:06:34.096
to recognize more efficient
ways to solve a proportion.

00:06:35.216 --> 00:06:38.346
It is important to continue
to focus on problem structure,

00:06:38.606 --> 00:06:42.226
labeling of key information,
and visual representations

00:06:42.396 --> 00:06:45.446
so that students develop
good problem-solving habits.

00:06:46.296 --> 00:06:50.586
Students benefit when teachers point
out the connections across problems

00:06:50.586 --> 00:06:53.776
with similar structures but
from different contexts.

00:06:54.896 --> 00:06:58.156
As always, if students are
to develop the critical skill

00:06:58.156 --> 00:07:01.296
of proportional reasoning,
remember what they've learned,

00:07:01.296 --> 00:07:03.706
and apply their knowledge
to solve problems,

00:07:04.016 --> 00:07:06.286
they need a strong conceptual
understanding

00:07:06.286 --> 00:07:07.976
of what they are doing and why.

00:07:09.586 --> 00:07:12.426
To learn more about Developing
Proportional Reasoning,

00:07:12.426 --> 00:07:16.916
please explore the additional resources
on the Doing What Works website.