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SHOELACES PROBLEMS
A series of POWs to encourage students
to explore ratios, fractions, percents and
decimals
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Introduction: Students use
graphing skills to help store customers
choose the best shoelace length and
giftwrappers the best ribbon length.
Instructions:
You have been hired as an analyst
by "As the Shoe Fits," a local shoe
store. Your first assignment is to
experiment with methods that will
make it easy for customers to decide
which length of shoelace will best
fit their sneakers. It seems that
many customers have been unhappy with
the laces they've chosen in the past.
Most have chosen lace lengths which
are either so long they have to be
double knotted or too short to make
a decent bow. Your boss has asked
you to create a guide for choosing
shoelaces, and to show her a graph
with your progress ASAP (as soon as
possible). You went out and did a
survey of shoes in the neighborhood,
looking for shoes in which the laces
were the perfect length. You've counted
the eyelets and measured the length
of the lace (in centimeters) for each
of those shoes. Turn to Page 2 to
graph the points, find the line of
best fit and then use the line to
help customers choose shoelace lengths.
When you are done you should have
a graph that looks similar to one
of the graphs shown here. (Click on
the "2" at the top of this page to
get to the next page.)
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Where's the Math: The problem
encourages students to make predictions
and develop generalizations from data
represented on a graph; in part II
and III students explore issues of
scale and the meaning of y-intercepts.
Standards: Data
Analysis and Probability, Measurement,
Algebra,
Role of Components: The Shoelaces
problem uses the PEN
component overlayed on top of a SimCalc
graph
to provide scafolding to students
new to point plotting. Positive feedback
is given in the form of a green circle
that appears on the graph when the
student moves the mouse over a point
in a given list. The second part of
the EPoW used PEN with simple function
to fit a line to the set of plotted
points. Students experiment with the
y-intercept using a swing
slider. Graph instructions are
displayed via the HTML
viewer, and solutions are entered
in a
text entry box. Javascript
handles communication among components.
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Shoelaces 1 hhhShoelaces
IIhhhShoelaces
III
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Shoelaces 1:
posted March 1, 2000 as a Math Forum EPoW
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Reflections:
Most students were successful in their
answer of what shoelace length would be
best for a shoe with 18 eyelets. Where students
ran into difficulty is in describing how
to use the graph. We were fairly picky (my
students say VERY) about how the student
explained this, but as a result, got some
really good descriptions of how to use the
graph to look up shoe lace length given
the number of eyelets. We are sorry that
some classes had technical difficulties
with shoelace 1. It provided some good immediate
feedback for those without technical difficulties.
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In this simulation, you are
an analyst for a local shoe store. Use your
graphing skills to make predictions and
help customers choose the best shoelace
length.
Sample submitted solution:
From: Jeni,
age 12
School: Issaquah Middle School, Issaquah,
Washington
1. What length shoelace will best fit
a shoe with 18 eyelets?
170cm
2. You are going to display this graph
in the shoelace department of As the Shoe
Fits. Write clear directions for customers
which describe how to use the graph to determine
which length of shoelace to purchase. When
choosing your shoelace length, you can look
at this graph to help you find the perfect
length.
Take a look at your shoes. Count how many
eyelets you have on 1 of your shoes. Then,
look at the graph, and find your amount
under the "eyelets" axis. Then, continue
to go up the line of where your eyelets
is plotted, until you reach the line shown
on the graph. Once you stop at the line
shown, look at the horizontal line that
intersects with it. The number written next
to it under the "lengh" axis is the perfect
length for your shoelaces! I found this
works because it shows how many eylets you
are going to have to shoelace, and how much
lace you will need. It also gives you shoelace
left over to tie a decent knot.
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Shoelaces 2:
posted March 15, 2000 as a Math Forum EPoW
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Reflections:
In shoelace 2, we wanted
students to think about the qualities of
a good window for graphing. When graphing
with a graphing calculator, it is always
beneficial to students to think about what
a good window would be for viewing their
data. Similarly on paper, we want them to
think before they graph, having their scale
be determined by characteristics of the
data they're graphing rather than by the
characteristics of the paper.
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In this simulation, you are
going to apply your graphing skills to a
help workers choose ribbon lengths when
they gift wrap packages.
Sample submitted solution:
From:
Scott, age 13
Josh, age 12
School: Issaquah Middle School, Issaquah,
Washington
1. What length of ribbon
do you need for a box with a dimensional
sum of 3cm?
about 25cm
2. What length ribbon do
you need for a box with a dimensional sum
of 70cm?
about 168
3. Polly is willing to
wrap packages with dimensional sums ranging
from 3cm to 70cm. What maximum X and maximum
Y values would you use on a graph to help
her employees choose ribbon lengths for
packages? Explain how you chose this scale.
70 and 170 I put the max values she could
wrap on the x axis then i put the line of
best fit on. then i made it big enough to
see.
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Shoelaces 3:
posted March 27, 2000 as a Math Forum EPoW
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Reflections:
Most submissions were correct, and
two solution methods were used. One used
the graph by lining up the appropriate lines
and looking at the y-intercept. The other
did the computations. There were some arithmetic
errors (be careful out there!) and some
solutions were submitted without explaining
how they found the answers. The bonus seemed
to be harder, or perhaps not worded clearly.
We were looking for the equation for the
two solution lines that were found. Some
people only gave equations for the other
lines, and some were confused about what
the equation should look like. But some
got it right, too.
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In this simulation you will learn about
y-intercept and what it represents.
Sample solution:
From: Andy,
age 13
Tyler, age 14
Vasiliy, age 13
School: Issaquah Middle School, Issaquah,
Washington
1. You need to figure out what base
pay you should ask for, knowing that: a.
you can sell 50 pairs of shoes in a week,
b. you will earn $2 commission on each pair
you sell, and c. you want to earn the same
full salary as someone who gets no base
pay and earns $5 per pair (red line). What
base pay would you ask for? How did you
figure it out?
The Base Pay i'd ask for is $150. By
using the graph you see that if you have
no base pay you make $100 from the commission
of shoes alone at $2 per shoe, and you make
$250 from commission of shoe sales when
you get $5 per pair of shoes sold, and $250-$100
gives you the weekly salary.
2. This time figure out what base salary
you should ask for, knowing that: a. you
can sell 25 pairs of shoes in a week, b.
you will earn $2 commission on each pair
you sell, and c. you want to earn the same
full salary as someone who gets a $300 base
pay and earns $.50 per pair (blue line).
What base pay would you ask for? How did
you figure it out?
$312.50-$50 = Base Salary for you Base Salary
= $262.50
Bonus: Write the equations of the two
lines you used to answer questions 1 and
2. Explain the elements of the equations.
(No answer submitted)
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