Teaching Notes for Stacking Containers
Materials: graph paper, at least 5 stacking containers each of several varieties, metric rulers,
chart paper, markers
Day 1 – Students work in pairs to complete the worksheet “Stacking Containers”. Each pair will
work with two different types of containers. It is suggested that each pair have at least one set of
containers that have a well-defined lip such as a Styrofoam hot cup. The second set should have
a noticeable difference between the “lip” and the height of the cup. It is ok if there are groups
that have the same sets of cups two groups do not have the same two sets of cups. Alternately,
you can have sets of cups at the front of the room and students can select a set to use, measure
them and then return them for another group to use. You may want to have the cups labeled A, B,
C, etc.
Two sets of cups should not be assigned to any groups. These sets will be used in Part Two as
part of the whole class discussion.
After setting up the problem - Your class has been hired by a company that makes all types of
containers – Styrofoam hot cups, plastic cold cups, food containers, and more – of different sizes.
The company wants to know how high boxes should be to hold a stack of 50 cups for each type of
cup – students work on completing the worksheet. If they don’t complete the sheet in class, it
should be completed for homework.
Day 2 – Students work in pairs to graph one set of their containers from the previous day on
chart paper. The graph should not include any descriptors of their container but it should include
a table of their data for that container. When they are done, post the graphs around the room and
label them 1, 2, 3, etc. Have the sets of containers from the previous day set out in the front of
the room with labels A, B, C, etc. Students working with their partner walk around the room and
match the graph with the container. They should both agree and be prepared to share their
reasoning.
After pairs have completed the matching, gather the class and pick a random poster and ask the
class which container goes with the poster. Pick a container and ask which graph goes with the
container. Make sure to ask students how they know the poster and container is a match.
During this part of the activity, you want to make sure that students are connecting particular
features of the container such as the “lip” and the height of the container to the graph and the
table. Using color to show the connections as students share their reasoning will help students to
connect the different representations. This will set a foundation for the derivation of equations of
the form y = mx + b in 8th grade. It is not important that students understand slope and y-intercept
but it may come out in the discussion. Some students may have trouble making sense of the yintercept if students connected the dots because 0 containers should not have a height. This is an
instance of SMP 2: Reason abstractly and quantitatively where the context (quantitative) doesn’t
lend itself to making sense of the y-intercept, but when reasoned abstractly, one could talk about
a line and the significance of the y-intercept.
During the discussion, you may want to choose two graphs that either have the same y-intercept
or slope and have a discussion with the class about the similarities and differences between the
two graphs and how these show up in the containers. Vis-versa you could choose two containers
that either have the same “starting height” (height without the lip) or “lip”.
After having a discussion of how students matched the containers and graphs, pick one of the
containers that was not used in Part 1. Show it to the class and ask them the following questions:
•
•
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What would the graph of this container look like?
Can we tell with one container?
What is the minimum number of containers needed to describe the graph?
Make sure that students support their answers with reasoning.
Repeat with the other container.
Extensions:
Show students an area of a graph that doesn’t have a container represented in it and ask the
students to describe what type of container could create a representation in that area and how do
they know. This discussion would allow students to connect explicitly connect what they know
about the features of the containers and their graphical representations.
Have a discussion about whether or not the “footprint” of a container affects its graphical
representation. What are the relevant features in this activity and what features are irrelevant?
Repeat the matching activity with the graphs and the tables.
Show students two very different containers and have them sketch on the same coordinate plane
what the graph of each container would look like. Discuss what the difference between a sketch
and an actual graph is. It is useful for students to be able to sketch what a general graph would
look like. This helps them to focus on the distinguishing characteristics between the two
containers and how they appear in a graph (the lip, initial height).
Stacking Containers
Your class has been hired by a company that makes all types of containers – Styrofoam hot cups,
plastic cold cups, food containers, and more – of different sizes. The company wants to know
how high boxes should be to hold a stack of 50 cups for each type of cup.
With your partner, determine the height of 50 cups for the 2 different types of cups you are given.
1a) Description of Cup A (i.e., hot/cold, plastic/paper, size):
1b) Description of Cup B (i.e., hot/cold, plastic/paper, size):
2)
Number of Cups
Cup A Stack Height (cm)
3) Graph your data, using one color for Cup A and one color for Cup B.
4) What variables are being investigated?
Cup B Stack Height (cm)
5) Use words and symbols to give a rule for the height of a stack containing any number of cups
for each type of cup you investigated. Be sure to explain the real-world meaning for each part
of your rule.
Cup A:
Cup B:
6) Compare the two cups. What do you notice between the cups, data, and graphs? How are
they different? How are they similar?
7) Predict how tall a stack of 50 cups would be. Explain how you made your prediction.
Cup A:
Cup B: