TIMS Laboratory Investigations
Area
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(ISBN 0-7872-4071-0, primary): The students pick 24 one-square-inch tiles, are asked to divide them into 4 equal piles, and then make, using certain rules, 4 different-shaped surfaces. We teach the children how to draw these shapes, first using a grid, then using only dots, and finally with no external guide. The two variables are surface shape and area, and we want the children to see that the shape does not determine the area. The children then measure the length and width of each figure and explore the role length and width might have in determining area. In this case, the answer is “none.”
Picture This—Area I
(ISBN 0-7872-4072-9, primary): The students use tiles to find
the area of different shapes by counting square inches. We go
beyond simple rectangles and learn to find the area of odd-shaped
surfaces. The areas of the tiles are 1-sq-in squares, 1\2-sq-in
cut along a diagonal, 1\2-sq-in cut parallel to one side, and
1\4-sq-in squares. All areas are found by placing the tiles on
the figures and then “counting” sq in by piecing together the
tiles to make unit squares or fractions of unit squares. The shape
of the object is the manipulated variable, the area the responding
variable. We again explore what role length and width might have
in determining the area. In this case, the answer is again “none.”
Spreading Out I
(ISBN 0-7872-4073-7, primary & intermediate): Students use eyedroppers
to make spots of water on paper towels. The number of drops is
fixed. The type of towel is the manipulated variable, and the
area of the spot is the responding variable. The children cut
out the irregular-shaped spot, trace it onto centimeter graph
paper, and find its area by counting square centimeters. The children
learn how to add fractional square centimeters to make whole square
centimeters. They make bar graphs of the data and are challenged
to read and interpret bar graphs in the Comprehension section.
Using their data, the children estimate how many drops it takes
to cover one sheet of each type of towel. They learn how symmetry
can be used to make this counting easier. We also probe critical
thinking skills by asking the children to compare the towels’
ability to absorb and to decide which towel is the best absorber
and which might be the best buy.
Picture This—Area II
(ISBN 0-7872-4074-5, primary & intermediate): This lab is a repeat
of Picture This: Area I, only now we graduate to units of square
centimeters. Instead of tiles, the students use a 1-sq-cm grid
to define 1 sq cm, 1\2 sq cm (diagonal cut and parallel cut),
and 1\4 sq cm. Four shapes are laid out on a sq-cm grid. The goal
is to determine the area of these increasingly complex shapes
by counting sq cm. We again measure the length and width of each
shape, and see if there is any relationship between these variables
and the corresponding area. Challenging area questions involving
adding and subtracting large numbers of sq cm are presented.
Spreading Out II
(ISBN 0-7872-4075-3, intermediate & middle): This is a variation
of the investigation Spreading Out I. An eyedropper is used to
place drops of water on a paper towel. Here the children control
the type of towel, and see what happens to the area of the spot
when they systematically increase the number of drops. The children
determine the area of each spot by counting square centimeters.
The plot of number of drops vs. area is a point graph with a best-fit
line through (0,0). This allows us to go into interpolation, extrapolation,
and proportional reasoning, along with counting square centimeters
and adding fractions that make whole square centimeters. In this
lab we introduce the idea of how to make a quantitative comparison
between predicted and measured values of a variable. With different
teams using different towels, the children learn which brand is
the best absorber and which is the best buy. As the children will
agree, being a wise shopper is important, too.
Surface Area vs. Shape
(ISBN 0-7872-4076-1, intermediate & middle): This important investigation
has several goals. First, to have the children become familiar
with the idea of surface area. Second, to find surface area by
counting square centimeters. Third, to build up children’s spatial
perceptions by having them draw figures made up of Cube-O-Grams.
Fourth, to determine surface area and volume from Cube-O-Gram
drawings. Fifth, to confront the idea that a given volume may
have different surface areas depending upon its shape. And finally,
to explore how nature uses this last property as a strategy of
survival, as well as seeing this effect in action in their everyday
lives. This time we draw the six shapes the children are to make.
All have the same volume. The children determine the surface area
and plot the results on a bar graph. They explore the relationship
between surface area and shape and then using inductive logic,
apply their results to the world around them, from microbes to
planets.
Surface Area vs. Height
(ISBN 0-7872-4077-X, intermediate & middle): The surface area
and height of towers made from Cube-O-Grams are found. The cross
section of the towers is controlled. The relationship between
surface area and height is linear but not proportional, since
the curve does not go through (0,0). This latter idea helps dispel
the notion that every lab has a graph that is a straight line
through the origin. In spite of this difficulty, we show the children
that they can still interpolate and extrapolate to make predictions.
Besides the notion of a cross section, we introduce the idea of
a limit. That is, when the height of the tower goes to zero, the
cross section stays the same. The children use this idea to explain
their graph, and then generalize all the ideas in the investigation
to questions about towers of different cross section.
What Big Feet You Have
(ISBN 0-7872-4078-8, intermediate & middle): This is an open-ended
investigation where the students are asked to collect data to
see if there is a relationship between weight and footprint size.
Size is not defined. Each student chooses a footprint variable
(length, width, area) that they think best correlates with weight
and then takes measurements of their classmates. To obtain the
best correlation data, the student should collect information
from adults, too. We expect each student to organize the effort
as they have done with previous investigations: to draw a detailed
picture, set up appropriate data tables, collect data, graph the
results, and draw conclusions. We have outlined a story, “The
Road to Survival,” which you can use to preface the lab.
Area vs. Perimeter
(ISBN 0-7872-4079-6, intermediate & middle): In this investigation
the children determine the relationship between the perimeter,
P, and the area, A, of a group of figures that have the same shape.
In the first part the children examine a series of squares, in
the second, a series of equilateral triangles. For the latter,
the children find the area by counting square centimeters. The
best-fit curve, in either case, is not a straight line. This brings
up the problem of making accurate interpolation and, especially,
extrapolation predictions. Having made their predictions, the
children construct figures, check their predictions, and calculate
percent differences. As an optional Advanced TIMS Topic, you have
an opportunity to straighten out the curve and find the analytical
relationship between A and P. What with the concept of similarity,
learning how to deal with a curve that is not a straight line,
and possibly the relationship between A and P, the laboratory
contains enough new and challenging ideas to keep the children
intellectually busy for a week.
Hand Area vs. Height
(ISBN 0-7872-4080-X, intermediate & middle): Similar to Arm Span
vs. Height. Many variations are possible. For example, foot areas
can be plotted against hand areas. In this investigation, data
is also gathered from several different grades and from adults.
The result is a hand area vs. height curve that is nonlinear but
passes through (0,0). The comprehension questions explore the
reasons for this and bring out the important concept of scaling
as it applies to biological systems.
Surface Area vs. Length—Cylinders
(ISBN 0-7872-4081-8, intermediate & middle): This is a follow-up
investigation to Surface Area vs. Height, only now the students
use cylinders of constant circular cross section to find the relationship
between surface area and length. The results will be the same,
a linear relationship that does not pass through (0,0), but the
geometry is more difficult and the children have to bring to bear
different skills than in the earlier investigation in order to
find these results. Together the two labs illustrate the important
concept of generalization. Unlike the investigations with rectangular
towers, we extend the analysis to include an analytic treatment
of the data resulting in two-step logic problems.
Moldy Bread
(ISBN 0-7872-4082-6, intermediate & middle): Spores are collected
on bread, and the resultant fungi allowed to spread over the surface.
The students determine the area of the mold vs. time. Time is
the manipulated variable, area is the responding variable, and
the type of bread is controlled. The curve of area vs. time is
not a straight line. The children make predictions based on these
nonlinear results, and percent differences between prediction
and experiment are calculated. Different types of bread are compared.
Counting Out πR2
(ISBN 0-7872-4083-4, middle): Using cans or jar lids, the children
study the relationship between the radius of a circle and its
area. In Part I they determine A and R independently, plot the
relationship, and study how to interpolate and extrapolate a nonlinear
curve. The children make predictions of the area of given circles,
check their predictions by counting square centimeters, and calculate
percent differences. In Part II, the children learn how to straighten
out the curve. Using their data, they measure the value of π from the slope of the curve. The children are asked to compare
their value of π with the precise value, to check their prediction from Part I
with the more accurate data from Part II, and to solve lots of
A = πR2 problems. Rich in proportional reasoning, taking squares and
square roots, using multiple step logic, and finding a value for
pi, hopefully this investigation will turn out to be a real piece
of cake for your students.
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Copyright © 1997 by Kendall/Hunt Publishing Company
Copyright © 1999 Institute for Mathematics and Science Education.
All rights reserved.
UIC—University of Illinois at Chicago