FRACTION EQUIVALENCE
WITH
PATTERN BLOCKS
P
attern blocks provide an excellent concrete experience for understanding
fractional equivalence. When discoveries are recorded in the abstract
fractional form, they provide a bridge to the understanding of equivalence
at the abstract level.
Step 1—Exploration
If students have not used pattern blocks, they will need to be
given time for free exploration. They will want to make designs
and patterns.
Guided discovery can be encouraged when students are
familiar with the blocks. Ask the students to determine how
many of each of the types of pieces it takes to fill a yellow
hexagonal region. Have students trace their solutions. Also
have them record the number
and denomination of the
units as a frac tion of
the hexagonal region.
Ex am ple: For green
trangles they would record
6/6, and for red trapezoids they would record 2/2. There are
no solutions for the square and tan rhombi.
Step 2—Equivalent Names
Ask students to find all the ways to cover a fraction of a hexagon
with only one type of piece. Have them record their solutions by
tracing the pieces and recording the name as a fraction. Example:
A red trapezoid (1/2) can be covered by two brown trapezoids
(2/4), three green triangles (3/6), or six purple triangles (6/12).
Students should first explore unit fractions (1/2, 1/3, 1/4) before
being asked to move to other fractions (3/4, 2,3).
PROPORTIONAL REASONING
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© 2003 AIMS EDUCATION FOUNDATION
When students have recorded the names of the fractions, ask them to
determine a way to change one fraction name to an equivalent. The fractions
one-half and two-fourths are equivalent. To change from one-half to two-fourths,
both the numerator and the denominator are multiplied by two. When
looking at the pattern blocks, this two is represented by the two brown
trapezoids that are the same size as one red trapezoid.
Step 3—Changing One
A richer understanding of equivalence is gained when the value of one
whole is changed. Instead of the hexagon always representing a whole unit, a
dif ferent piece or combination of pieces is used to represent a whole.
If a red trapezoid represents one, then a brown trapezoid equals one-half
(1/2). This half can be covered with three purple triangles (3/6).
If a yellow hexagon and three green triangles represent a whole, then a red
trapezoid equals a third (1/3). The trapezoidal region can be covered with
three green triangles (3/9) or six purple triangles (6/18).
Have students justify their abstract equivalent fractions with tracings of
divisions of the whole with equivalent coverings.
Step 4—Student Generated Problems
Students will gain practice in equivalency by generating problems with
pattern blocks. Have them make a tracing of the pattern blocks that form
the region they want to use to represent the whole. Then direct them to
shade in a portion of the region and determine all the equivalent names for
the shaded region. When students are confident they have all the solutions
to their problem, invite them to exchange copies of their problem with a
partner to solve.
PROPORTIONAL REASONING
19
© 2003 AIMS EDUCATION FOUNDATION
PROPORTIONAL REASONING
20
© 2003 AIMS EDUCATION FOUNDATION
Make a record of all the equivalent fraction names you can find for the shaded
area of each pattern.
A.
B.
C.
E.
D.
H.
F.
PROPORTIONAL REASONING
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© 2003 AIMS EDUCATION FOUNDATION
Math
Measuring
linear
Proportional reasoning
Using statistics
finding correlations
averaging
Coordinate graphing
Writing equation
Topic
Body Proportions and Statistics
Key Question
If you found the hat of a student, how might you determine
that student’s height?
Focus
In this activity students determine if there is a correlation
between a person’s height to head circumference. Students
take measurements of the class members and analyze the
data with a scatter plot and averaging.
Science
Life science
human body
skeletal proportions
Guiding Documents
Project 2061 Benchmarks
• The expression a/b can mean different things: a parts of
size 1/b each, a divided by b, or a compared to b.
• Graphs can show a variety of possible relationships between two variables. As one variable increases uniformly,
the other may do one of the following: always keep the
same proportion to the first, increase or decrease steadily,
increase or decrease faster and faster, get closer and
closer to some limiting value, reach some intermediate
maximum or minimum, alternately increase and decrease
indefinitely, increase and decrease in steps, or do something different from any of these.
• The graphic display of numbers may help to show patterns
such as trends, varying rates of change, gaps, or clusters.
Such patterns sometimes can be used to make predictions
about the phenomena being graphed.
Integrated Processes
Observing
Collecting and recording data
Interpreting data
Comparing and contrasting
Generalizing
Materials
Tape measures, or string and meter sticks
Background Information
This activity explores one of the many ratios evident
in the human body. It also shows the deceptive nature of
estimating circumferences.
A mature adult’s height approaches three times the
head’s circumference. The comparison of height to head
circumference in early adolescents produces a ratio of about
2.8. This ratio decreases with age. Early school age students
have a ratio of about 2.2. The ratios are more consistent for
adults who have completed their periods of growth.
It must be established that there is a correlation between
two sets of data before one set can be used to predict the
other. Because there is a greater variance in ratios for
children, they are not as quick to recognize the correlation found in this activity. In order to help them recognize
this correlation and provide them with a way to verify any
correlation, they need to be introduced to the scatter plot.
A scatter plot is a statistical tool used to see if two sets of
data are correlated. On a coordinate grid, a point is plotted
for each corresponding pair of data. The closer this set of
points comes to forming a line, the stronger the correlation
is between the two sets of data. The more scattered this set
of points becomes, the weaker the correlation becomes.
Height to head circumference is a positive correlation.
It is called a positive correlation because as the head size
increases the height also increases. It would be a negative
correlation if one set of data changed inversely to the other
set. A negative correlation would exist if as head size got
bigger, the height decreased. When using a scatter plot, the
NCTM Standards 2000*
• Solve problems involving scale factors, using ratio and
proportion
• Select, create, and use appropriate graphical representations of data, including histograms, box plots, and
scatterplots
• Discuss and understand the correspondence between
data sets and their graphical representations, especially
histograms,stem-and-leaf
plots, box plots, and
scatterplots
• Make conjectures about
possible relationships
between two characteristics
of a sample on the basis of
scatterplots of the data and
approximate lines of fit
PROPORTIONAL REASONING
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© 2003 AIMS EDUCATION FOUNDATION
data’s plotted points for a positive correlation will form a
cloud of dots that drifts upward as it moves across the graph
to the right. The upward drift shows the positive correlation.
The “thickness” of the cloud communicates the strength
of the correlation. The narrower this band of points, the
stronger the correlation.
A line of best fit is a line that come closest to being in
the center of the cloud of plotted points. It is appropriate
for the middle-grade student to estimate and draw this
line on a scatter plot. The student can recognize that the
slope of this line is close to the average ratio of height-tohead circumference. This line will go up the average ratio
amount of units each time the line goes to the right one
unit. The equation for this line is: Height = (Average Ratio
X Head Circumference). This is the same as the common algebra
equation: y = ax, where y is the vertical axis, x is the horizontal
axis and a is the slope.
The scales on the graph for this activity have been set
up to help students recognize the positive correlation of the data.
Management
1. This activity is enhanced if before the activity the
teacher gets a hat that is sized to a specific person
whose height is known. An adjustable baseball cap
works well. This hat is used as an example in introducing
the activity. At the end of the activity it is used again
as students predict the height of the hat’s owner. If the
hat is sized for a student in kindergarten or first grade,
it will provide a discrepant event which can lead to
discussion and extension of the activity.
2. If measuring tapes are not available, string can be
wrapped around the head and measured with a meter stick.
3. Make sure all students are taking consistent head measurements. A good suggestion is to measure across the
eyebrows, above the ears, and across the bump on the
back of the head.
4. Allow two periods for this activity. One period is needed for gathering and sharing the data. (The graphing
can be done at home.) The second period is reserved
for analyzing and applying the data.
5. This activity works well with students working with apartner.
Procedure
1. Introduce the situation and discuss the Key Question.
2. Have students predict how they think their height
compares to their head circumference.
3. Direct students to measure and record their height and
head circumferences to the nearest centimeter.
4. Allow time for them to share their data with the class.
5. Direct students to make a scatter plot from the data.
6. Have students make the comparison of height to head
circumferences for all the individuals.
7. Ask students to calculate the averages for each type of data.
8. Take time to discuss what patterns they find in the scatter
plot and how the average ratio relates to the graph.
9. Have students determine a way to predict the height
of a student if the head size is known.
10. Direct students to record the average ratio in the second column on the prediction chart and use it to predict
the height for each head size.
PROPORTIONAL REASONING
11. Have them place an X on the scatter plot to represent
each head circumference and predicted height pair.
12. Discuss the significance of the line of best fit formed
by the Xs.
Discussion
1. What patterns can you identify on the graph? [Answers
will vary. Most notice the points line up in vertical columns. Depending on the data, some will recognize
that the data fall in a band.]
2. Why do the points never fall between the vertical lines
on the graph? [Measurements were taken in whole
centimeters.]
3. If someone told you their head had a circumference
of 55 cm, how would you predict their height from the
graph? [One can only tell the likely range of height.]
4. Have students trace a line around the outside of the
data points and ask them: What happens to the area
as it goes across the page to the right? [It goes up.]
5. If the area goes up as the head sizes get bigger, what
does that say about the height? [It also goes up.]
6. What does this graph tell us about the comparison of
height to head circumference? [As the head gets bigger,
a person gets taller. (Tell the students that this is called
a correlation.)]
7. How does the narrowness or thickness of the area of
plotted data on the graph affect how you can predict
the height of a person? [The narrower the band, the
more accurate the prediction will be.]
8. What do you notice about the decimal equivalents for
all the people? [They are close to the same.]
9. What does the average decimal ratio tell you? [The
middle ratio for the class. About how many times taller
a person is than their head’s circumference.]
10. How can you use this ratio to determine a person’s
height if you know their head size? How can you write
your explanation as a equation? [To get the height (H)
you multiply the ratio (R) by the head circumference
(C). H=(R) x (C)]
11. Predict the height of a person of each head circumference and plot an X on the graph to represent the
predicted height for that head size. What do you notice
about all the Xs? [They form a line that goes through
the middle of the data points.]
Extensions
1. Have students gather data of different aged students
to see if the ratio of height to head changes with age.
Direct students to make a scatter plot and find an average for each age. Students may find a box and whisker
plot provides a clearer comparison of all the data.
2. Have students determine the median ratio and the
mode ratio and see how they compare to the average
(mean) ratio. If there is a difference, does it produce a
more accurate prediction?
*
181
Reprinted with permission from Principles and Standards for
School Mathematics, 2000 by the National Council of Teachers
of Mathematics. All rights reserved.
© 2003 AIMS EDUCATION FOUNDATION
Class Results
Student
Name
Data
Head
Circumference
(cm)
Comparison
Height
(cm)
Ratio:
Height
Head
Decimal
Equivalent
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
Averages
PROPORTIONAL REASONING
182
© 2003 AIMS EDUCATION FOUNDATION
1. Record the head circumferences and heights of all the students in your
class on the Class Record page.
2. On the graph page, make a scatter plot with a point representing each student.
3. Calculate the Decimal Equivalent ratio for each student and find the class
averages.
4. What patterns do you see in the graph?
5. What does the Average Decimal Equivalent ratio
tell about the heights and heads of students in
your class?
6. How could you use the Average Ratio to determine
the height of someone if you knew his or her
head circumference?
7. Record the Average Ratio in the second column
of the chart to the right and use it to determine
the Predicted Height.
8. Put an X on the scatter plot to represent each
head circumference and its predicted height.
Head
Average Predicted
Circum.
Height
Ratio
(cm)
(cm)
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
PROPORTIONAL REASONING
183
© 2003 AIMS EDUCATION FOUNDATION
184
49
50
51
52
53 54 55 56 57
Head Circumference (cm)
What patterns do you see in the graph?
100
48
110
120
130
140
150
160
170
180
190
58
59
60
61
62
If you know the circumference of a person’s head, how can you predict his or her height?
200
Height (cm)
PROPORTIONAL REASONING
© 2003 AIMS EDUCATION FOUNDATION
Integrated Processes
Observing
Collecting and recording data
Interpreting data
Drawing conclusions
Applying and generalizing
Topic
Proportional Reasoning — Rates
Key Question
You have a chain of standard paper clips and your friend
has a chain of jumbo paper clips. If your friend told you the
length of objects in jumbo clips, how could you change that
number into standard clip lengths?
Materials
Jumbo paper clips, 12 per group
Standard paper clips, 19 per group
Focus
Students will measure objects with standard and jumbo
paper clips. They will then graph and analyze the data to
develop their understanding of proportional reasoning as
related to rates and graphic displays.
Background Information
Conversion rates are factors by which you multiply
one measurement unit to get a different type of unit. The
conversion rate is determined by using a ratio to compare
the quantity in one system to an equal value in the other
system. When the ratio is reduced to a unit rate, with a one
as the denominator, you have the conversion rate for the
two types of measures.
For paper clips the length of 9 standard clips is the same
as 6 jumbo clips. The conversion ratio would be 9 standard
clips to 6 jumbo clips (9/6). As a unit ratio this would be
1.5 standard clips for each (1) jumbo clip (1.5/1). The one
in the denominator is assumed and the rate is recorded
as 1.5 standard/jumbo. Multiplying the number of jumbo
clips required to measure a length by 1.5 gives the length
in standard clips.
The proportional nature of rates is shown graphically
on a coordinate graph with each of the axes representing
one system of measurement. When a number of different
lengths are graphed with each point at the intersection of
the length’s corresponding quantity in each system, a line
results. The slope of this line is the conversion rate between
the two systems.
Guiding Documents
Project 2061 Benchmarks
• Organize information in simple tables and graphs and
identify relationships they reveal.
• The graphic display of numbers may help to show patterns
such as trends, varying rates of change, gaps, or clusters.
Such patterns sometimes can be used to make predictions
about the phenomena being graphed.
NRC Standards
• Use appropriate tools and techniques to gather, analyze,
and interpret data.
• Mathematics is important in all aspects of scientif ic
inquiry.
NCTM Standards 2000*
• Understand and use ratios and proportions to represent
quantitative relationships
• Represent, analyze, and generalize a variety of patterns with
tables, graphs, words, and, when possible, symbolic rules
• Relate and compare different forms of representation for
a relationship
• Understand relationships among units and convert from
one unit to another within the same system
Management
1. This activity works well in groups of two with one partner
measuring objects with a jumbo clip chain and the other
measuring the same object with the standard clip chain.
2. To get more consistent results as a class, the teacher
may choose to pre-select the items to be measured by
students. In choosing items there needs to be a variety of
lengths from one jumbo paper clip to 12 jumbo clips.
3. The investigation is divided into two parts to allow for
flexibility. Classes with some experience with ratios will
want to start with Part B and use Part A only if students
experience difficulty or could benefit from clarification.
Less experienced classes would start with Part A, which
Math
Proportional reasoning
Patterns and functions
Measuring
length
Graphing
PROPORTIONAL REASONING
44
© 2003 AIMS EDUCATION FOUNDATION
may be the only part done. They could use the graph to
determine the conversion for unmeasured lengths. Part B
could be used when students become better grounded
in the concept of ratio.
Procedure
Part A
1. Distribute paper clips of both sizes and have student
groups make a chain for each size of paper clip.
2. Pose the idea that lengths could be measured by chains
of paper clips and have the students suggest what problems might arise if they used the chains they made.
3. Have students lay the chains side by side with one end
of each chain matched with the other chain.
4. Have students observe and record how long in standard
paper clips the given amount of jumbo clips are.
5. Have students make a coordinate graph of the data.
6. Discuss with students the patterns they see in the graph
and their relation to the data. Have them consider how
they can use the patterns to convert lengths in jumbo
paper clip units to standard paper clip units.
7. Use a jumbo chain to measure items that are not one of
the given lengths. Have students determine the items’
lengths in standard paper clips and check their answers
with a standard clip chain. Have them discuss the various
strategies they used to get their answers.
Part B
1. Pose the Key Question and have students discuss how
they might solve the problem.
2. Have student groups make a chain for both sizes of
paper clips.
3. Using Part B activity sheet, have students measure and
record the lengths of six objects in both standard and
jumbo paper clip units.
4. Have students make a coordinate graph of the data.
5. Discuss with students the patterns they see in the graph
and their relation to the data. Have them consider how
they can use the patterns to convert lengths in jumbo
paper clips to standard paper clip units.
6. Connect several jumbo chains and measure larger
objects in the room and have students use their patterns
to determine the length in standard paper clip units.
Several small chains can be connected and used to confirm the results. Have them discuss the various strategies
they used to get their answers.
Discussion
1. What patterns do you see in the chart? [ For every two
jumbo clips there are three standard clips.]
2. How many times bigger is the standard clip number than
the big clip number? [1.5]
3. What patterns do you see in the graph? [The points make
a straight line.]
4. How do the patterns you found in the chart show up on
the graph? [To stay on the line you go up three standard
clips and over two jumbo clips. To stay on the line you
go up 1.5 units for every one you go sideways.]
5. How could you use the patterns to determine the length
of something in standard clips if you know its length in
jumbo clips? [jumbo clips X 1.5 = standard clips, interpolate or extrapolate the line on the graph.]
6. What are some types of measurement conversions
that might be more practical than jumbo clips to
standard clips?
Extensions
1. Make conversion rates for other standard units such as
hex-a-link cubes, Unifix cubes, floor tiles.
2. Develop conversion rates to do estimations such as
feet/step, centimeters/hand span, yards/arm span.
*
PROPORTIONAL REASONING
45
Reprinted with permission from Principles and Standards for
School Mathematics, 2000 by the National Council of Teachers
of Mathematics. All rights reserved.
© 2003 AIMS EDUCATION FOUNDATION
Part A
Jumbo Clips
Long
Standard
Clips Long
Standard Clips for
Each Jumbo Clip
2
4
6
8
10
12
Standard Paper Clips
15
10
5
0
PROPORTIONAL REASONING
0
5
10
Jumbo Paper Clips
46
15
© 2003 AIMS EDUCATION FOUNDATION
Part B
Object
Measured
Jumbo Clips
Long
Standard
Clips Long
Standard Clips for
Each Jumbo Clip
20
STANDARD PAPER CLIPS
15
10
5
0
0
5
10
15
20
JUMBO PAPER CLIPS
PROPORTIONAL REASONING
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© 2003 AIMS EDUCATION FOUNDATION
Proportional Practice - Set 1
A. Shade in 1/2 of the square in at least six ways.
B. Shade in 1/3 of the rectangle in at least six ways.
C. Determine what fraction of each flag is shaded.
1.
2.
3.
4.
5.
6.
7.
8.
9.
PROPORTIONAL REASONING
10.
11.
29
© 2003 AIMS EDUCATION FOUNDATION
Proportional Practice Set 1
A. Tell what fraction the top line is of the bottom line.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
PROPORTIONAL REASONING
30
© 2003 AIMS EDUCATION FOUNDATION
Proportional Practice Set - 2
A. Give at least 4 names for what is shaded and show how you know it is right.
B. Give at least 4 names for what is shaded and show how you know it is right.
C. Give at least 4 names for what is shaded and show how you know it is right.
D. Give several names for what is shaded and show how you know it is right.
PROPORTIONAL REASONING
31
© 2003 AIMS EDUCATION FOUNDATION
Proportional Practice Set 2
Find as many equal fraction names for the fraction of the top to bottom lines.
Use the spaces below each set of lines to show each of the names is correct.
1.
2.
3.
PROPORTIONAL REASONING
32
© 2003 AIMS EDUCATION FOUNDATION
Proportional Practice Set - 3
A. Draw the whole rectangle from which the shaded region was made.
1.
1
4
5.
2.
3.
1
5
2
3
6.
5
9
3
8
7.
5
6
4
7
8.
4
9
4.
9.
10.
2
7
1
8
11.
3
16
PROPORTIONAL REASONING
12.
5
8
33
© 2003 AIMS EDUCATION FOUNDATION
Proportional Practice Set 3
Find the whole line from which the top was cut. Draw the whole on the bottom.
1.
2/3
2.
3/6
3.
3/4
4.
2/9
5.
7/8
6.
4/5
7.
4/7
8.
4/9
9.
5/11
10.
7/11
PROPORTIONAL REASONING
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© 2003 AIMS EDUCATION FOUNDATION
Proportional Practice Set - 4
A. Determine the percent of area each section is of the whole region.
b.
c.
1. a.
2. a.
3. a.
b.
c.
d.
b.
c.
4. a.
c.
5. a.
b.
d.
6. a.
b.
c.
b.
7. a. b. c.
c.
d.
8. a.
b.
9. a.
c.
d.
b.
c.
10. a.
b.
PROPORTIONAL REASONING
11. a.
b.
c.
35
c.
© 2003 AIMS EDUCATION FOUNDATION
Proportional Practice Set 4
Tell what percent the top line is of the bottom line.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
PROPORTIONAL REASONING
36
© 2003 AIMS EDUCATION FOUNDATION
Proportional Practice Set - 5
A. Shade the given percent of the whole region.
1.
2.
50%
3.
75%
33
45%
4.
45%
5.
68%
7.
PROPORTIONAL REASONING
1
3
%
45%
6.
68%
8.
37
© 2003 AIMS EDUCATION FOUNDATION
Proportional Practice Set 5
Draw the top line so that it is the correct percent of the bottom line.
1.
50%
2.
30%
3.
33 31 %
4.
76%
5.
48%
6.
6623 %
7.
64%
8.
150%
9.
275%
10.
36623 %
PROPORTIONAL REASONING
38
© 2003 AIMS EDUCATION FOUNDATION
Proportional Practice Set - 6
A. Draw the whole rectangle from which the shaded region was made.
1.
2.
3.
25%
20%
4.
5.
20%
6.
45%
35%
7.
8.
24%
75%
9.
175%
250%
10.
140%
PROPORTIONAL REASONING
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© 2003 AIMS EDUCATION FOUNDATION
Proportional Practice Set 6
Draw the 100% line on the bottom so that it makes the top line the
correct percent.
1.
84%
2.
30%
3.
75%
4.
70%
5.
24%
6.
60%
7.
80%
8.
150%
9.
175%
10.
125%
PROPORTIONAL REASONING
40
© 2003 AIMS EDUCATION FOUNDATION
Topic
Scaling
Key Questions
How can you use a chain of rubber bands to enlarge a
picture?
Focus
Students will explore how a chain of rubber bands can be
used to enlarge a picture. Then using their rubber band
enlargers students will make different scale enlargements
of a triangle to discover the numeric relationships of the
different scaled drawings.
Guiding Documents
Project 2061 Benchmarks
• The expression a/b can mean different things: a parts of
size 1/b each, a divided by b, or a compared to b.
• Mathematical statements can be used to describe how
one quantity changes when another changes. Rates of
change can be computed from magnitudes and vice
versa.
NCTM Standards 2000*
• Work flexibly with fractions, decimals, and percents
to solve problems
• Understand and use ratios and proportions to
represent quantitative relationships
• Solve problems involving scale factors, using
ratio and proportion
• Understand relationships among the angles, side
lengths, perimeters, areas, and volumes of
similar objects
Math
Proportional reasoning
scaling
Measuring
linear
Integrated Processes
Observing
Collecting and recording data
Applying
Generalizing
Materials
Rubber bands, #18-19,32-33 (see Management 1)
Butcher paper (or chart paper)
PROPORTIONAL REASONING
Scissors
Tape or glue
Rulers
Activity pages
Background Information
Rubber bands generally stretch proportionally along their
length. That means that if the halfway point is marked on
a rubber band, the mark will be in the middle whether it is
stretched or relaxed. This proportional relationship works
conversely as well, making the total length of the rubber
band always twice the length of halfway mark. This relationship is explored more fully in the investigation “Percent
Bands.”
By linking rubber bands of the same type together in a
chain, one can construct an enlarging instrument with the
knots marking unit lengths.
Fixing one end of the chain at a point, placing a pencil
through the rubber band at the opposite end, and stretching and moving the rubber band as the pencil draws a path
will cause each knot in the string to trace a path similar to
the pencil. More practically, if one uses the pencil to stretch
and move the rubber band so the knot traces the outline
of a drawing, the pencil will draw a scaled enlargement of
the original.
Initially it seems that the placement of the fixed position
could affect the size of the enlargement. Experimenting
with the positions of the fixed point will quickly confirm
that it only changes the positions of the enlargement.
The enlargements are always on the opposite side of the
original’s fixed point.
Further experimentation will reveal that the relationship
of the distance between the fixed and tracing positions and
the fixed and drawing positions determines the scale of
enlargement. If the first knot in a chain of four rubber bands
is used as the tracing position, the ratio of lengths is 4 to
1 and the resulting enlargement will be four times as large
as the traced original. If the second knot in a chain of five
rubber bands is used as the tracing position, the ratio of
lengths is 5 to 2 and the resulting enlargement will be 5/2
or 2.5 times as large as the traced original.
Management
1. Rubber bands are critical to the success of this investigation. Best results have been found with 3" - 3 1/2" long
by 1/16" (#18-19) or 1/8" (#32-33) wide bands. If these
are not available, experiment with the selection that is
available to find which size gives the most consistent
results. A minimum of five bands is need by each group
for the activity.
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2. It is suggested that the teacher link the rubber bands
together for each group before the investigation, minimizing management problems of linking and controlling
rubber bands. For the initial exploration each group
needs two bands, and for the numeric exploration five
bands are required.
3. The bands are linked together with the looping knot
illustrated below.
is not affected by fixed point’s position although more
distortion may take place as the fixed position relative
to the original becomes greater.]
3. What patterns do you see in the lengths of the triangles?
[Side A is about 3 cm longer each time. Side B is about
4 cm longer each time. Side C is about 5 cm longer
each time.]
4. How does the number of rubber bands used in the chain
relate to the size of the enlargement? [The number of
rubber bands links used is the number of times larger
the enlargement is than the original, the scale factor.]
5. How can you determine how big to make a rubber band
chain to get the size enlargement you want? [The ratio
of drawing position to tracing position is scale factor of
enlargement.]
Extensions
1. Have students use rubber band chains to make huge
enlargements with scale factors of eight to ten.
2. Have students explore and determine a way to make
enlargements of non-unit scale factors such as 2.5. They
will need to have a rubber band ratio equal to the scale
factor. For a scale factor of 2.5 they could use the second
knot as the tracing position and the fifth rubber band
as the drawing position.
4. The enlargements made with rubber bands are slightly
distorted due to inconsistencies in rubber band stretch.
For geometric shapes like the triangle, distortion can
be minimized by marking only the vertices and using a
straight edge to draw the sides. Before doing the activity
practice the technique in order to anticipate what to
expect in the classroom.
Procedure
1. Distribute a chain of two rubber band lengths to each
group and discuss the Key Question in reference to these
chains of bands.
2. Provide the other materials and have students follow the
instructions on Exploring Rubber Band Enlargements.
3. Have the class discuss the questions at the bottom of the
record sheet. Conclude the discussion with the question
“How does the length of a rubber band chain affect the
size of the enlargement?”
*
Reprinted with permission from Principles and Standards for
School Mathematics, 2000 by the National Council of Teachers
of Mathematics. All rights reserved.
Open-Ended Approach
After students have completed Exploring Rubber
Band Enlargements, have them discuss what might
change the position and size of the enlargement.
Have students develop investigations to confirm their
conjectures and then have groups report to the class
their findings.
4. Distribute chains of five rubber band links and have
students follow the instructions on Measuring Rubber
Band Enlargements.
5. Have students discuss patterns they found in measuring
the enlarged triangles.
Discussion
1. Describe the appearance of your enlarged drawing.
[general shape but with some distortion]
2. What happens to your enlargement when you move the
fixed position of the rubber band? [The position of the
enlargement changes. It is always on the opposite side
of the original’s fixed point. The size of the enlargement
PROPORTIONAL REASONING
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How can you use a chain of rubber bands to enlarge pictures?
1. Cut out the illustration in the lower left corner. Glue or tape it about halfway down
the large sheet of paper and 8 to 10 inches from the left side of the paper.
2. Mark a point to the left of the illustration that is more than one
rubber band length from the closest point on the illustration.
3. Have one student put the point of a pencil through the loop of one
of the rubber bands, and then place the point of the pencil on the
marked point on the paper to hold the end of the chain of rubber bands
at this place.
4. Have a second student put the point of a pencil through the loop
of the other rubber band and stretch the chain of rubber bands
until the knot is on top of a line on the illustration. Place the
pencil point down at this position.
5. While the first student
continues to hold the end of the chain of rubber bands in
place, have the second student trace the illustration with
the knot by moving the pencil along the paper to stretch
and move the chain of bands.
• How well did your rubber band chain enlarge the illustration?
• What do you think you could do to change the position and
size of the enlargement?
PROPORTIONAL REASONING
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© 2003 AIMS EDUCATION FOUNDATION
How does the length of a rubber band chain affect the size of the enlargement?
1. Cut out the triangle and glue or tape it
about halfway down the large sheet of
paper and 6 to 8 inches from the left
side of the paper.
2. Mark a point to the left of the
triangle that is more than one
rubber band length from the
closest point on the triangle.
3. Have one student put the point
of a pencil through the loop of the
last rubber band to hold the end
of the chain of rubber bands at the marked position.
4. Have a second student trace the triangle with the first knot while drawing an enlargement with the
pencil in the second rubber band. Draw three more triangles in the same way but move the drawing
pencil to the third, then to the fourth, and finally to the fifth rubber band.
5. Measure and record all the lengths of the sides of each triangle to the nearest centimeter.
Number of Rubber
Bands Used in
Enlargement
Length of
Side A
(cm)
Length of
Side B
(cm)
Length of
Side C
(cm)
3
4
5
Original (1 band)
• What patterns do you see in the
lengths of the triangles?
2 Rubber Bands
• How can you determine how big to
make a rubber band chain to get
the size enlargement you want?
3 Rubber Bands
4 Rubber Bands
5 Rubber Bands
Side C
de
Si
e
A
d
Si
B
PROPORTIONAL REASONING
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