NTI Feb. 5
Day 2
Activity Notes for facilitators.
These notes are intended to support implementation of the various activities that were included for the
presentation on mathematical modeling. The notes do not constitute a script for implementation of the
activities. Not all activities included were discussed in detail during the presentation on Day 2 as
adjustments were made to include a session on Assessment.
Information included here differs by activity. Some possible answers to modeling questions are included.
Mathematical solutions depend on models created by participants.
Because of the mathematics included and the fact that solutions may vary, someone who has facility
with both mathematical content and mathematical thinking should be available during the professional
development.
1.Thanksgiving Dinner Video: Level: 6th grade.
Questions to consider as you watch the video:
Is the attention to the real world realistic?
Is the modeling question clearly stated?
Is the modeling question phrased in a way that there value to answer in the minds of students?
How does the material in the video fit with the modeling cycle? (Refer to the modeling cycle graphic on
the handout.) Specifically,
 What work/information/evidence, if any, which is in the video would you put under
“formulation?” step in the cycle? Who is providing the information, the teacher or the
student? (Refer to the top right box in the graphic.)
 What work/information/evidence, if any, would put under the
“compute/process/deduce” step in the cycle? (Refer to the lower right box in the
graphic)
 What work/information/evidence, if any, would you put in the “interpret” step of the
cycle? (Refer to the lower left box in the graphic.)
 What work/information/evidence would you put in the “revise” step in the cycle. Which
box(es) would the revisions go in?
What content standards are evident in the student activity? Give evidence.
Possible answer: Content Standard: 6.G-1
What mathematical practices are evident as students work? Give evidence. Use the long form of the
practices and be specific.
Examples of mathematical practices that appear might include:
MP 1. Evidence: see all of the comments about the modeling cycle below. These require students to
analyze relationships and constraints; try special cases
MP 3. Justify their conclusions (If students do justify by exhausting cases) and communicate their
solution to others. Reason with concrete objects.
MP 5. Done. Note that the use of concrete models was teacher-directed.
MP 6. It may be suggested that could have been attended to better. For example, the teacher might
have facilitated nor explicit recognition of the fact that “size of the table” means area of the table top.
However, as portrayed, it is often the case that realization of the need to be more explicit appears
during the processing of the model. Here, the recognition that area of the table is a factor (to be added
to the model) is an example of this occurrence. On the other hand, there is an assumption somewhat
“hidden” that should be mentioned (and belongs in the formulation process): In order to even have a
maximum sized table, it must be stipulated that there is not excess “empty” space along a side of the
table. Making this explicit is attending to precision-in terms of “communicating precisely.”
MP 7. It might be suggested that the teacher might have facilitated more explicit mention of the use of
symmetry of the table so that consideration of a 2 ft. by 11 ft. table has the same area as an 11 ft. by 2
ft. table. So, you don’t have to consider both tables in separate cases. This might have been done—but
it is not shown.
MP 8. This might get mentioned if it is conjectured that the students do multiple cases. There is some
regularity in computing area and there is a short cut because of the symmetry in table possibilities —
but, the focus of this practice is that it leads to generalization. That doesn’t happen here.
Comment: Note the focus is on STUDENTS engaging in mathematical practices. The WAY OF TEACHING
should facilitate this.
E.g. Formulation of the model: Table is rectangular. There are 22 people sitting at the table. The
space every person takes up at a table is one yard. (Concrete lengths of paper are used to represent this
length.) Another assumption: overlapping (lengths along each side). As mentioned another assumption
is that the lined-up physical lengths represent the entire the table sides. All of these characteristics are
given by the teacher. Reaching the food in the middle of the table is not addressed in the model. It is a
characteristic of the situation, but it is not needed in the model: YOU DON’T INCLUDE EVERYTHING. The
student realization that the “biggest table” has to do with area occurred later during work on the
problem. As mentioned, this can be classified as a revision for precision in the model as it is a clearer
definition of size.
The Compute/process/deduce step that at least some student do to solve the problem is to compute all
possible table areas under the constraints of the problem and then select determine seating
arrangements with the largest and smallest areas–since are only a few cases.
The Interpretation step may have been done by “acting out” cases.
After the video:
Share answers to the above questions:
Also discuss:
Are the answer to the question(s) justified by the students.
Is there evidence of deepening conceptual knowledge? Possible answers include: area means more
than just length times width and the fact that rectangles with the same perimeter can have different
areas—an important mathematical realization that CONNECTS concepts.
Comment: So, modeling doesn’t JUST use knowledge that is known. Sometimes new mathematical
knowledge results from the problem. This happened here: Rectangles can have the same perimeter
but different areas.
Questions/ comments /concerns about implementation in the classroom.?
Activities: Fun, Fun, Fun
Overarching questions to be considered for all activities:
Is the attention to the real world realistic?
Is the modeling question clearly stated?
Is the modeling question phrased in a way that there value to answer in the minds of students?
How does the work on the activity fit with the modeling cycle? Be specific.
What content standards are evident in the student activity? Give evidence.
What mathematical practices are evident as students work? Give evidence.
Could you use or modify this problem for the grade level at which you teach?
Comment to participants: There may be more than one way (or one model) to address many of the real
world question posed. In fact, even though some of the problems are described as relating to a
particular level, think about ways you might approach a similar problem at your teaching grade level.
We will give wide berth to ways of tackling the modeling problems and formulating models. However, it
will be important that your work and conclusions conform to the model you develop.
Note: When students are given wide berth to bring the knowledge they know and understand to a
problem it supports the mathematical practice of making sense of and persevering in solving problems.
If you want students to use specific concepts and you give such students opportunity to share their
reasoning and critique the reasoning of others, at least one (group of) student (s)might suggest an
approach using the knowledge you wanted. Or, after students have shared their answers, you, as
teacher, might ask students if there is a model that uses the knowledge you want them to use in the
model. Such an approach often serves as a way of making mathematical connections between topics
and/or deepens conceptual understanding.
2. Estimating Crowds
The facilitator might note that common contexts and questions are served by mathematical modeling.
Also, many modeling problems do not use sophisticated mathematics—but may use fundamental
mathematics in sophisticated ways.
Comments and possible solution:
Formulate: One consideration would be how people were standing in the field. Let use assume that the
people in the field are facing side of length 100 ft. The context of the problem is not as frenzied as a
rock concert. So here, let’s assume the adults are standing in rows and columns. Other quantities of
interest include size of people in the audience and the size of “personal space” setting people apart
from one another—and how these are represented. Decisions for this model (which could be adjusted)
are made below in a sample solution. In addition, it is important to consider where the government
speaker is located (e.g. decide that the speaker is positioned at the center of one of the short sides of
the field.) A simplifying assumption is that the adults are standing in straight rows for this first iteration
of the cycle. A simplifying assumption might be to represent a standing adult (and surrounding
personal space with a rectangular areas on the ground. See below (see below).
Compute /process/deduce: (see below).
The figure above represents a person (inner rectangle) and personal space needed in front, in back and
on each side of that person—if the person is surrounded by people. This will create the outer rectangle.
The inner rectangle represents the space occupied by a person (two feet wide and one foot thick). The
teacher could have students determine the average size of classmates or find some characteristics on
the web—but it is harder to find average “thickness.”).
The outer rectangle is constructed from the inner rectangle by lengthening the base of the “person”
rectangle by six inches on both sides and lengthening the height adding nine inches on both the top and
bottom. So, dimensions of the outer rectangle are 3 ft. by 2 1/2 ft. These dimensions were determined
by experimentation: having the class line up in rows deciding on appropriated between people. It is
suggested that teacher facilitate the choice of measurements so that simple rational dimensions of the
outer rectangle. Of course, these could be adjusted in another loop through the cycle.
The next step might include reflecting on the progress made by applying what has been determined to
the drawing of a smaller “field” to get an idea of what computations to make for the given field:
In this arrangement the people in the “back row” touch the back wall. The people in the right side touch
the right wall. Here we might suggest adding additional assumption to the model: Assume is OK to be
closer to inanimate objects than people—so you are cutting down on the size of personal space of the
people near the boundary of the field.
Note that the personal spaces of people overlap in this figure. The extra space in front of one person
could overlap with the personal space behind another person. This needs to be taken into account when
answering the problem.
There are several ways to proceed. If needed, a teacher might suggest looking for ways to break up this
configuration into non-overlapping rectangles. Students might suggest using rectangles that look like
this:
Now, the problem can be solved by figuring out how many of these overlapping “building blocks” will fit
in rows inside the field. Using the dimensions decided upon above, the dimensions of the building block
1
3
3
are 2 2 ft. long by 1 4 ft. To find out how many rows will fit, divide 50 by 1 4 What do you do in the
context of the problem if this is not an integer? Here it was decided have to round down to the nearest
integer, 28 so that everyone has the same “personal” space in front of them. To find out how many
1
blocks (people) would fit in each column, divide 100 by 2 2. (Answer: 40). So, this solution leads to 28 x
40 people. That is, it is estimated that 1,120 people were at the event using this model
1
3
Interesting question: Couldn’t you just work with (100 x50) and divide by (2 2 X 1 4 ) instead of using the
method described in the above paragraph?
1
2
This is a mathematical question about rounding. If you
3
4
divide 5,000 by (2 X 1 ) and THEN round down to the nearest integer, you get a different answer:
1,142 people. Which approach is better in the context of the problem?
Interpret . Are these results satisfactory? What could be a concern? You might want to have the
participants consider whether or not people would be content standing directly behind each other. How
would you revise the model? You might want to include safety concerns or wider lanes for people
traffic. There are many revisions to the basic model that are possible.
Ask: Which content standards are addressed at the grade level you were considering?
Example content Standard at seventh grade: 7.G.-1; 7.G.-6; 7.NS-1.
What mathematical practices appear? Likely candidates are MP 1-4, MP 7
At a higher grade level, students might use different shapes the represent people and their personal
space. E.G. circles might be used (in eighth grade and after) or more sophisticated tilings representing
people placement with, say hexagons, etc. This would require more sophisticated counting techniques.
Students also might research statistical evidence that might lead to a better measure of the space a
person takes up. At the high school level, there might also be interest in generalizing this to other cases;
e.g. to an n ft. by m ft. field and describing a general algorithm for the solution.
Have participants share solutions and their analyses in terms of the modeling cycle.
Ask participants for questions or comments.
3. Traffic Jam
This problem is a rephrasing of one found at: http://www.illustrativemathematics.org/illustrations . In
the Illustrations page, search the title Traffic Jam and choose N-Q Traffic Jam from the options.
This is listed as a high school activity, but could be modified for a lower level. Ask participants to solve
this as adults and to reflect how it may be modified for the level of mathematics they teach.
Analyze work using the modeling cycle (graphic).
It could be that “interpret” and “revise” in the modeling cycle are either eliminated or done while doing
step 1 when the size of a vehicle is computed as a weighted average of vehicle sizes. Also, note that
N-Q.3 would be addressed if a “unit” of “length” was designated that included the vehicle size plus the
length of the gap between vehicles. Here (and in other modeling problems), the units used may be
based on the context of the model.
Content standards would be addressed at high school level
N-Q.1, N-Q.2 and N-Q.3. See the referenced example for a description of a possible solution.
A possible modification for lower grade levels might employ the use of simpler measurements. How
much you scaffold this problem for lower grades depends on student experience with the modeling
cycle.
Also, discuss:
What mathematical practices appear?
Possible answers may include MP 2 and (obviously) MP 4
Ask for other comments or questions?
4.) Manhattanville
The level of the activity as written is Algebra I.
The work on this unit focuses on formulation of and processing in the mathematical model. Question 5
involves interpretation. (A discussion of how the rest of the cycle then follows.) The activity is a
modification of a much longer unit from the comprehensive (four year) high school textbook series
Mathematics: Modeling our World, COMAP Inc., Lexington MA 2010-2012, which uses modeling
problems as a way to make connections between mathematical topics students know as well as to
introduce new mathematics in a way that facilitates the solution of some components of the problem.
The problem here concerns where to locate an important community service, namely a fire station.
[Based on recent events, an alternate context might be where to locate emergency supply distribution
center for resident victims of a city devastated by a hurricane.]
Questions 1 and 2 focus on the “formulation” step of the modeling cycle.
Complete work and answering the common questions on problem 1 before continuing on to problem 2.
Complete work and discussion on questions 1 and 2 before continuing on to questions 3.-5.
Question 1 of the activity asks for a broad list of criteria for the selection of a location. (For this talk, this
version of the activity already includes several criteria and simplifications that classroom students might
suggest using a more open-ended version.) Some of these might be: the number of people living in the
houses, the kind of houses, where the traffic lights are located; etc.
As just mentioned, the given problem contains several simplifications such as the streets are parallel or
perpendicular to each other and form a uniform grid and the streets are located at street intersections.
Agree that a model that includes these assumptions rather than other location of the houses, the
number of people in the houses, etc. It is suggested that you also include a simplifying assumption that
there is only one house at each intersection . (There are no houses across the street from one another.)
Adding the other criteria will be reserved for additional trips through the cycle (not considered, here)—
hopefully based on what is learned in this simpler model.
Ask participants to complete question 2 and focus on analyzing it in terms of the modeling cycle.
Question 2 asks for a more precise definition of “best” location. Ask participants to create a definition
or criteria that describe the term “best.” Have them share their responses. The most typical response is
that the best location would be one that minimizes the average distance to one of the houses. Let’s call
this B1. Ask for other ideas. If someone does not suggest it, suggest that an alternative definition would
be to minimize the maximum distance the fire truck would need to travel. Let’s call this definition B2. A
third alternative might be to minimize the sum of the distances. This will give locations equivalent to
those where the average distances from the fire station the houses is minimized. Another suggestion
might be to minimize the average distance from the fire station to each person living in one of the
houses. This could be done by adjusting the functions mentioned in the discussion below. But, one
criticism of this is that the number of people is less stable number that the number of houses.) There are
other good choices. But, for the work below, we are going to limit the discussion to choices B1 and B2.
Revision to simpler case by and additional simplification to one dimension: Formulation: Now, ask
participants make an additional simplification to one dimension. That is, the first “loop” of the modeling
cycle will assume all the houses lie on the same street. This replaces Manhattanville with the town of
Beatenpath.
Ask participants to answer questions 3.)-5.) using each of the definitions B1 and B2.
More formulation: The point of question 3.) is to name the units. The most obvious unit is a “city block.”
It also may be helpful to decide that the locations of the houses will be represented at integer
coordinates on a number line (where units represent city blocks.)
Compute/process/deduce: Question 4.) asks participants to try several cases and try to notice and
justify a general pattern.
Question 5.) asks for interpretation of these results in context.
The pattern for B1 depends on whether or not the number of houses in Beatenpath is even or odd. If
there are an odd number of houses, the location that minimizes the average distance is the middle
house (or, in answering question 5. below, maybe across the street from the middle house.) Note that
this is the median value of the coordinates locating the houses. If there are an even number of houses,
say 2r houses, the fire station could be placed at any integer value between (or across the street from)
the rth and the (r+1)th house. There is not a unique location. (This could be satisfied by the median of the
house coordinates if the median was an integer.) Reasons need to be given for these conclusions.
For B2, if the houses are represented by certain integer values on a number line, the location of the fire
station that would minimize the maximum distance the fire truck would have to travel would be halfway
between the houses at the extreme ends of the town. Let’s call this the midrange value.
For question 5. relative to B2: Note that the midrange value may not be an integer. So, where would
you put the fire station when you “interpret” these results in context where the fire station must be an
intersection? The closest corner (if there is one)? What if both corners are equidistant?
Interpretation: In finishing the discussion of question 5., ask collective participants to compare or
contrast the two results obtained from the two definitions of “best” E.g. The location(s) determined
using B1 are sensitive to the relative position of the houses—i.e. how the houses line up and how many
houses there are, but not to the particular values of the distances between houses (except when
specifying the location of the fire state using specific coordinates.) B2 is sensitive to the distance
between the two “endpoint” or “outlying” houses, but not to the relative positions or distances
between the houses in between the endpoints.
Content standards are address?:
Possible answer: N-Q .1 and N-Q.2
What mathematical practices are apparent?
Answers probably include MP 4, 6, and 8. MP 1, 2, and 3 are also possible choices.
One point of this activity is to show that modeling can lead to new mathematical concepts. So, include a
descriptive overview of what happens next in the unit.
The next section of the curriculum unit introduces the notion of absolute value of a number and how
this concept represents distance. In NYSCCLS, sixth grade students are exposed to the concept and
notation of the absolute value of a number, 6-NS.7.c. (In implementation of NYSCCLS in Algebra II this
might be extended so that students could interpret the expression |𝑥 − 5| as representing the distance
between the point x and the point 5 on a number line.) Students then go on to build knowledge about
properties and alternative representations of absolute value including piece wise representations of
𝑥 − 5, 𝑖𝑓 𝑥 ≥ 5
|𝑥 − 5|. That is |𝑥 − 5| = {
. They then build a function with independent variable x
5 − 𝑥, 𝑖𝑓 𝑥 < 5
representing the average distance from the point x to houses in Beatenpath. That is,
The function defined by 𝑓(𝑥) =
|𝑥−2|+|𝑥−7|+|𝑥−10|
3
represents the average distance from the point x to
the points 2, 7, and 10 on a number line. Using piecewise representations students can reason that the
graph of such a function would consist of straight line segments if the domain of the function is the real
line. See the next graphic.
A
5
x
0
B
C
5
d(x) =
xA
xx + xB
10
xx + xC
Graph of: 𝑓(𝑥)3=
15
xx
|𝑥−2|+|𝑥−7|+|𝑥−10|
3
representing the average distance
from point x to the points A, B and C
on an x-axis.
The minimum value on this graphrrsits over the location where average distance would be minimized.
However, note that this graph is defined where x is allowed to be any number between 0 and 15. In
context, only integer values of x are relevant. A more accurate model would limit the domain.
This new part of the activity specifically addresses NYCCSL modeling conceptual category standards
including:
20
Algebra: Seeing Structure in Expressions A-SSE:
Interpret the structure of expressions
1. Interpret expressions that represent a quantity in terms of its context.★
Functions
Building Functions B-F
Build a function that models a relationship between two quantities.
1. Write a function that describes a relationship between two quantities.★
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
Interpreting Functions F-IF
Interpret functions that arise in applications in terms of the context. (Although, this involves function
other than linear, quadratic and exponential.)
4. For a function that models a relationship between two quantities, interpret key features of graphs
and tables in terms of the quantities. Key features include: intercepts; intervals where the function is
increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end
behavior; and periodicity. ★
The next portion of the activity “revises” the model moving into two-dimensions and examining whether
or not the results of the one-dimensional case extend to the two dimensional case.
Revision back to two dimensions: In analysis of the two-dimensional case with B1, all the mathematical
ideas presented are useful in supporting students as they eventually provide an argument (MP 3) that
the results extend to the original two dimensional situation that was described in the problem . The
strategy to be justified is if one can consider the first and second coordinates of a potential fire station
separately and apply what was learned in the Beatenpath model. Using the B2 definition of best, the
results of the Beatenpath model can also be extended to the original two dimensional situation, but
implementation in the real world is more complicated and beyond the scope of the present discussion.
(The participant is referred to the reference for this problem.
Ask for additional questions or make comments about this example or its implementation.
5. Cash for Gas
Have participants work with this task online at: http://www.nctm.org/rsmtasks/ . Participants should
read the task document online and then analyze the task based on the modeling cycle. What part of this
task involves mathematical modeling.? Give evidence.
6. Ceramic Tile
As written, this is a middle school activity—possibly 6th grade.
You might say that this might be one of the first situations that students encounter on modeling
Ask participants to answer question one on this task as 6th graders might.
Have participants share their answer.
If students have had little exposure to the modeling cycle, a frequent answer to question 1 might be:
14x16 or 224 pieces of tile. But, this may not be enough in real life. (See below.)
The real focus of this activity is “interpretation” of an answer in the real world: You compute something
and it doesn’t work. The intent is to help build awareness in students of in the need for care in
“formulation” component of the modeling cycle.
Have participants analyze the work on question 2 and share their answers.
When students begin to think about it, in our terminology, they will be revising the model. To the
students, they may not realize they need to take care in the model. Answers to question 2 should
include: “You could break some tile.” Another, less obvious consideration could be “The outline of the
room is not a perfect rectangle.” Ask participants how they might facilitate student realization of this?
E.g. you could tile a “simpler” model, say a closet or some small area, where a physical model with the
property that that perimeter of the room was not a perfect rectangle was available and ask students to
tile it.”
So, in the revised formulation one would want to list an assumption about how much tile might be
needed because of breakage and then a plan for configuring the tile on the floor that would minimize
visualization that the tile lines did not match the wall lines. Ask for student suggestions about what
should be listed. For, example, they could go online to look for statistics or advice on how much extra
tile would be needed to cover potential breakage. (This advice is usually given in terms of a percentage.)
Configurations could involve shifting the tile so that, say, the first row of each tile along a wall was
(approximately) a half-tile. This can generate a good amount of classroom discussion. Students can also
ask expert tile installers how they layout the tiling job as aesthetics can also plays a role in the tile
layout. Many experts start laying tile from the middle of the floor rather that at a corner.
6th grade content standards addressed.
6-G.1
Mathematical Practices might be addressed
MP 4
Ask for other questions or comments
7. Flu epidemic
This is a high school activity. The level is Algebra II (at least if problem 2 is included).
Ask participants if this looks like a modeling problem? Can they see components of the modeling cycle
in it? What might make it a better modeling problem? Have them give evidence or reasons for their
answers. (Anyone should be able to answer this by now—independent of their particular mathematical
knowledge)
The opening paragraph and problem one look like a traditional application. But, we now have
perspective to position it in the modeling cycle:
Technically, the “formulation” portion is mostly “given”—except for building a function that will lead us
to a solution. One thing that would make it a better modeling problem would be to solicit the type of
data that might be needed from the students so that they could help formulate the problem. The data
values could then be given.
It is not the intent that participants work on this problem in this session. It requires considerable
content knowledge.
Here is a possible approach to problem 1: if we let S(n) represent the number of people that are still
susceptible to the flu at the beginning of day n and we start at n = 0. (Starting time at zero is often done
in modeling.) Then S(0) = (0.9)x 20,000. That is S(0) = 18,000. (Note: some people will get sick as day 0
progresses. Then, S(1) = (0.93) x S(0), S(2) = (0.93) x S(1),…, S(n+1 )= (0.93) xS(n). Thus, in closed form,
we have S(n) = (0.93)n18000, where n = 0,1, 2,….. To have the flu run its course, we need to know when
everyone who is susceptible actually has been infected. One approach would be to find the smallest
(positive) integer, n, such that S(n) < 1. This can be done by several methods including graphing, using a
CAS, or using logarithms and their properties. Each of the previous methods would use technology in
some fashion. The solution is n = 136. That is, at the beginning of the 136thday, no one was susceptible.
So, since we started at day zero, the last person would be infected on the 136th day. Depending on what
the definition of “run its course” is decided to mean, we may want to add 5 days to this total so that the
flu would be completely gone from the population. Obviously, this is a prediction is not being tested in
the real world.
Problem 2 is asking if an immunization program might be able to end the flu infection timeline in 30
days. This suggests a revision to the above model where we now assume, say, K people are inoculated
each day. Assume we start at day 0. That is, to the statistic given 0.5 % of these inoculated people will
stay in the susceptible population each day. So, now, we had a “new” expression for S(n) if we assume
K people got inoculated on day zero.
The pattern is: S(0) = 18000; S(n+1) = (0.93) x S(n) –K(0.995) for n ≥ 0.
So, S(1) = (0.93)18000–K(0.995), S(2) = (0.93)218000–K(0.995)(0.93) - K(0.995); S(3) = (0.93)318000–
K(0.995)(0.93)2 - K(0.995)(0.93) – K(0.995); S(n) = (0.93)n18000–K(0.995)[(0.93)n-1 + K(0.995)(0.93)n-2 +…+
K(0.995)]; Noticing the structure of a geometric progression, we can represent this as:
1−(0.93)𝑛
0.07
S(n) = (0.93)n18000–K(0.995)[
] We want to find the smallest value of K so that S(30) < 1. That is,
we want the smallest positive integer value of K such that (0.93)3018000–K(0.995)[
1−(0.93)30
0.07
] < 1. (Note
that this is a linear inequality with variable K.) Again, K can be found by several methods. The solution
is K = 175. So, we want to inoculate about 162 people per day. Note that on the thirtieth day of
inoculation (day 30) there actually will be less than 162 people left to inoculate. (Why? Couldn’t we get
away with inoculating 161 people per day?)
If we apply this model, do you think it would fit a reality? No, this is still a simplification of the
situation. E.g. not all people who can get inoculated do get inoculated. etc.
Content standards are addressed
F-FB.1.a, F-LE.1.c & 2, A-SSE.4.
Mathematical standards addressed
MP 1, MP 2, MP 4 and MP 7 are prominent.
Ask for other questions or comments
8. For the Birds
This activity is an abbreviated version of the activity of the same name that appears in the Teachers
College Columbia Univ. Mathematical Modeling Handbook, COMAP Inc., Lexington MA 2011. There is
much more relevant information in that source.
This is a high school algebra activity at level Algebra I for content.
Have participants work on problems 1.-4. They should also think about this in terms of how students
might approach these problems. Mention that classroom students would work on questions 1. and 2.
without seeing any of the following questions.
Problem 1 is open ended and there may be many suggestions as judicious choice of criteria for the
model hasn’t been asked for, yet. The point becomes to decide upon relevant criteria as one continues
this activity. Problem 1 has to do with the “formulation” step in the modeling cycle.
Here is one approach: By problem 4. possible relevant criteria suggested by students in the formulation
stage might include consideration of:







What different types of birds come to the feeder. Make the simplifying assumption that all
birds are the same.
How bird seed a bird eats. Make the simplifying assumption that all birds eat the same amount
of feed each minute. The exact amount does not matter, but a specific amount may be
suggested. Make this the unit for the problem. That is, the unit for the problem is the number
of ounces of seed a bird eats per minute.
The feed capacity of the feeder. The exact amount doesn’t matter, but a specific amount might
be suggested. In this sample answer, the capacity of the feeder is denoted by the constant C.
The capacity is measured in ounces.
The amount of time (in minutes) the birds are feeding at the feeder. This is an appropriate
variable for the problem. Denote this variable by t.
The amount of feed (in, say, ounces) in the feeder. This is also a variable for the problem that
depends on time. It could be denoted by, say, F. It might be noted that F is a function of t.
However, this is not used in the sample answer, here. Instead, amounts of feed in the feeder
amount are expressed in terms of the capacity of the feeder as needed.
Assume birds are at the feeder whenever they can get feed from the feeder.
The feeder is cylindrical. Therefore, there is the same amount of feed in the top half as there is
in the bottom half of the feeder.
Facilitating student (as opposed to teacher) articulation of such criteria may require careful
questioning by the teacher.
The most typical answer from students for problem 2. might be 45 minutes.
Questions 3 & 4 refer to the Compute/process/deduce step. The answer question 3 is that it takes
an additional 45 minutes to empty the feeder. (So, it takes 90 minutes for the birds to consume the
feed in the lower half of the feeder.) A verbal justification might be given such as: “When there were
four birds at the feeder it took 45 minutes to eat half the feed. After that there were half as many
birds at the feeder (2 birds) so it will take twice as long for the birds to eat the rest of the feed. That
means it will take another 90 minutes to empty the feeder.”
A response to question 4 might attempt to use mathematical notation to capture the above
statement. It might start with using symbols (numbers) to indicate that four birds took 45 minutes
to eat half the feed: 4x45 =( ½)C. In context, the expression 4x45 means 4 oz. of seed per minute
times 45 minutes. To find out how long it will take to empty the feeder, we want to find the value t
so that 2t = ( ½)C. There are many ways to find this answer. One way would be to use structure to
note that 4x45 = 2(2x45). So 2x90 = ( ½)C. So, since there is a unique solution to the linear
equation we created, we have that t= 90. Thus, a full feeder of this type would take a total of 135
minutes to empty under the assumptions made above.
Problems 5 & 7 refer to a revision to a new case and continuing through the cycle again. Problem 6.
involves interpreting results in the real world.
Problems 5&7. can be solved using similar reasoning—except one needs to find two values of t: the
time, t1, it takes for birds to consume seed in the middle third of the feeder and the time, t2 it takes
to consume the feed in the bottom third of the feeder. Problem 7 might include the following:
Repeating and extending the reasoning from problem 4 we have: 6X30 = (1/3)C ; Find t1 so that 4t1
= (1/3)C. To do this solve the equation 4t1 = 6x30 for t1. Then, t1 = 45. Find t2 such that 2t2 = (1/3)C.
To do this solve the equation 2t2 = 6x30. Then, t2 = 90. So, for this situation, it takes a total of 165
minutes to empty a feeder that is full. (Explanations should accompany the equations.)
Problem 6. Is the interpretation step in the modeling cycle.
Problem 8. concerns generalizing (revising) the model to other cases. This can be considered a
“revision” step as it suggests a general result.
The remaining problems suggest increasing facility in decontextualizing and comparing procedures
to see if learned techniques can be useful in other problems
So, all four of the modeling cycle appear.
Content standards are addressed.
N-Q.1 & 2, A-CED.4
Ask participants what mathematical practices might appear.
Certainly, MP 1-4, MP 8, and possibly MP 7.
Ask for questions and comments.
9. Railing
This activity is suitable for Geometry after students have studied topics about circles. It is based on a
true story.
Have participants work on this activity and think about what students might do. Ask them to work on
the problem as a modeling problem.
Have participants share their work and analyses. Make sure they have interpreted their results in
context. (See below.)
Which content standards were addressed?
G-C.2, G-MG.3.
Which mathematical practices were used? Most probably MP 2-4, MP 7 will be mentioned—possibly
MP 5 would appear in a classroom situation.
Note that the problem has already been formulated except for an assumption about the tools the
carpenter has. It probably would be safe to assume that the carpenter has a tape measure and a Tsquare (that makes right angles).
Below is an interesting scenario regarding what might happen.
The first strategy might be to locate the center of the circle and then
measure the radius. Students most likely know that the center of a circle
lies on the perpendicular bisector of any chord. So, the strategy would be
to use the tape measure and perhaps a T-square to create the
perpendicular bisectors of two distinct chords. Then, since the center of
the circle is on both perpendicular bisectors, it is the point of intersection
of these two lines. See the figure to the right. So, the carpenter simply has
to measure the distance from the center, C to some point on the arc. The
problem is that when you interpret these results in the context, it is
unlikely that the center lies above the ground. It is also likely to be a long
way away from the railing—possibly making it physically impossible to
accurately locate the center. Thus, a different solution strategy is needed.
C
A revised strategy can be determined:
The “failed” strategy indicates that we can find the center of the circle at least theoretically. Refer to
the figure below to the right. Let’s use this circle to find a new strategy that doesn’t involve the circle in
“practice.” Here is the theory part: Given the circle that contains part of the railing arc in its
circumference, construct a chord ̅̅̅̅
𝐴𝐵 and determine its
B
D
midpoint, M. Then, construct the diameter of the circle, ̅̅̅̅
𝐷𝐸 ,
M
using the line through M and C. Finally, construct chord ̅̅̅̅
𝐴𝐷 .
A
Note that Δ EAD and ΔAMD are similar as ∠ADE = ∠MDA. So,
𝐴𝐷
𝐷𝐸
𝐷𝑀
= 𝐴𝐷 . Thus, DE=
𝐴𝐷 2
𝐷𝑀
.
Thus, here is a procedure you can give the carpenter:
Use your tape measure to create a chord between two points,
A and B, on the railing arc. Measure and find the midpoint.
Use the T-square to find point D on the railing. Measure
lengths DM and AD. Then the length of the diameter of the
circle is
𝐴𝐷 2
𝐷𝑀
C
.
E
Here, you don’t need to locate the center, C , physically. Moreover, the amount of “room” you need to
use can be small because there is no restriction on the distance between points A and B. they can be any
points on the railing. The extra “room” you need is above the chord.
Do we need to interpret these results again? Yes. We need to explore the issue of precision of hand
measurements. That is for another time! You CAN take this question with you.