Expanded Lesson for CW – Ch 19 p. 1
Lemonade
Based on Activity 19.5, p. 359
Grade Level: Seventh or eighth grade.
Mathematics Goals
•
To develop proportional reasoning (multiplicative as opposed to additive).
•
To develop conceptual strategies for comparing ratios.
Thinking About the Students
Students understand equivalent fractions. They are also familiar with the term unit
fraction (for example, 1/3, 1/4, 1/9). Students have had experiences with ratios. This is not a
good activity for students who have not had some experience with proportional reasoning. Some
students may have also used symbolic or mechanical methods for solving proportions in a prior
year but the assumption is that these methods are not understood and if used in this lesson they
must be able to explain them conceptually.
Materials and Preparation:
•
Prepare copies of the attached Blackline Master 76, “Lemonade,” and also make a
transparency.
•
Have blank sheets of acetate to place over the transparency so that multiple solutions can
be recorded—or—prepare at least five copies of the lemonade transparency.
Expanded Lesson for CW – Ch 19 p. 2
LESSON
BEFORE
The Task
Students are to decide which of two lemonade recipes (as illustrated on the worksheet)
will produce the strongest lemonade flavor or if they are equally lemony.
Brainstorm:
•
Show the transparency of the two lemonade pitchers. Explain that the dark squares stand
for lemon concentrate and the open squares for water. Most importantly, emphasize that
these are the recipes. The drawing does not imply that there are only 5 cups of mix in
one pitcher and 7 in the other. Assume that both pitchers are full.
•
Without doing any computations, ask students to decide which pitcher they believe is the
strongest or if they are the same. Gather a few reasons but do not dwell on any ideas,
right or wrong. The purpose here is simply to generate a few ideas for solving this
problem.
Establish expectations:
•
Pass out the paper with the lemonade task on it. Students should work in pairs or threes.
•
Students must use words and numbers, as well as drawings on the picture if appropriate,
to explain their reasoning.
•
Caution students that a purely computational solution is not acceptable if the numbers
cannot be connected to the two pitchers of lemonade.
•
Explain that groups will be asked to share their work on the overhead along with their
reasoning.
Expanded Lesson for CW – Ch 19 p. 3
DURING
•
This is a challenging problem, and there are many paths to a solution. Stop by different
pairs or groups to find out what kind of reasoning each is using. In your discussion you
want to be able to call on students with different answers and different approaches.
•
There may be students whose reasoning is more subjective than mathematical. For
example, “It just seems like they are going to be awfully close in flavor. We think they
are the same.” Explain that because this is a mathematics class, close is not good enough.
They either are equal or not, even if only slightly different. That is what they are trying
to decide. It is very tempting to offer a hint that would lead to one of the possible
solution strategies. Avoid this temptation.
AFTER
•
Begin by asking who would like to argue that the two pitchers taste the same. A very
common and compelling argument, albeit incorrect, is to match waters and lemons in
each pitcher and then note that each pitcher has one extra water. Since the other pairs of
lemon and water are the same, the pitchers are the same. Ask other students if they have
questions or comments on this approach but simply acknowledge the solution. Do not
evaluate it. If there are different arguments for the two being equal, they should be
shared.
•
Very few if any students will argue for the left pitcher, but, if there are any, they might be
asked to share next. Most likely many students will argue for the right-hand pitcher.
•
If you have been observant while students were working, you should be able to call on
groups that have used different strategies. It is useful to get as many strategies on the
Expanded Lesson for CW – Ch 19 p. 4
floor as possible since it is these various strategies that are the goal of this lesson. Here
briefly are a few possibilities:
o Share waters to each cup of lemon visually. In the right pitcher, each lemon gets
1½ waters and on the right each gets 1⅓ waters. Lemon can be shared with water
in a similar manner. These are compelling arguments and do not rest on
computation.
o Form and compare ratios of lemon to water (part-to-part ratios). In either case
common denominators (equal water) or common numerators (equal lemon) can be
used. While some students may do this, they may not be able to maintain the
lemon context within the computations. These same arguments can be made by
“cloning” each recipe; that is, the same arguments can be made by duplicating
each recipe until either there are equal amounts of lemon or equal amounts of
water.
o Part-to-whole ratios of either lemon to total or water to total can be made in a
similar manner. Again, common denominators or common numerators can be
used to solve the problem. The difficulty students will have is keeping the
context connected to the computations.
•
Especially if there are “camps” of students favoring different answers to this problem, be
sure to have those in one camp attempt to explain why they disagree with the argument(s)
of the other. In the unusual event that those who are incorrect (pitchers are equal) cannot
be persuaded by their peers, promise to return to this problem on another day.
Expanded Lesson for CW – Ch 19 p. 5
ASSESSMENT NOTES
•
Students who use the “one-water-extra” argument are not using a multiplicative form of
reasoning. These students need additional experiences with distinguishing between
additive and multiplicative reasoning.
•
When there are multiple correct methods used, check to see if students who have one
correct method can still see the validity of another. Without this ability to be flexible
with different arguments, students have not reached a desired level of proportional
reasoning. Avoid favoring only one argument or strategy.
Lemonade
Name___________________________
= 1 cup of water
= 1 cup of lemonade mix
Which pitcher will have the strongest
lemonade flavor, or will they taste the same?
Explain.
Lemonade—76
© Allyn & Bacon 2007