Transcript Improving Math Rigor and Relevance through Interaction

Munching for
Meaning
Prepared for:
Florida Council of Teachers of Mathematics
October 12, 2007
Pam Ferrante, ED.S., NBCT
Donna Hunziker
Project CENTRAL
The University of Central Florida
Edible Activities
The Magic Circle
 Concept- Students explore the area of a
circle and the formula for it by decomposing
the circle into a rectangle.
Procedures

Concrete

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Cut eight diagonals across the pizza or tortilla, cutting the
pieces into approximately equivalent sizes.
Lay the pieces out horizontally alternating the pointed end
up, then down, then up, etc. (forming a “rectangular”
shape)
Apply the formula for area of a rectangle to this figure.
Examine the sides of the rectangle and discuss the
relationship of these lengths to the original circle’s
attributes
Procedures Cont. …

Representational


Make a sketch of the original circle and the
transformed “rectangle”
Label the relationships of the measurements on
each
?
r
radius
?
a
d
i
u
s
More with the Magic Circle

Abstract

Write the formula for the area of a rectangle and
use this formula to find a formula for the area of a
circle based on the sketches and measurements
taken on the “transformed” circle.
Area of rectangle = Length X width
Area of a Circle = ?
More Fun…
Fruity Cuts
 Concept-Students investigate the
relationship of angles formed by parallel
lines cut by a transversal.
Procedures

Concrete

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Roll out and flatten a Fruit Roll-Up ® square
Cut a pair of parallel lines across the Fruit Roll-Up ®
square
Cut a transversal across the parallel lines previously cut
Explore the relationship between the various angles
formed by the cuts

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Are any of the angles congruent? Which ones?
Are any of the angles complementary? Which ones?
Are any of the angles supplementary? Which ones?
Why do these relationships exist?
Procedures Cont.

Representational

Sketch and label the angles formed by your
parallel lines and transversal.
1
2
3
4
5
6
8
7
More with Fruity Cuts

Abstract

Write rules for the relationships that exist
between the angles examined in the fruit square
i.e.
Measures of angles 1 and 7 are congruent
and the measures of angles 2 and 8 are
congruent which means that alternate
exterior angles of two parallel lines cut by a
transversal are congruent.
Crunchy Corners

Concept- Students explore the relationship
between the angles of a triangle and a
straight angle having 180 degrees.
Procedures

Concrete


Give each student several triangular chips to examine.
Have students break off the corners of the chips and line
up together along the edge of a ruler to form a straight line
demonstrating that the sum of the angles of the triangle is
180 degrees.
Have students repeat the process with different size chips.


Does it make a difference what size triangle you use? Why
or why not?
Does it make a difference what size corner you break off?
Why or why not?
Procedures Cont. …

Representational

Make a sketch of the triangle’s transformation
and realignment on the straight line
Abstract

Write a rule about the relationship of the
angles of a triangle.

Does the rule apply to all triangles? (i.e.- right,
scalene, equilateral, etc.)
Fraction Fun

Concept-Students explore dividing fractions
and mixed numbers by fractions.
Procedures

Concrete
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Designate a denominator to be associated with each color of licorice (i.e.
red is fourths and black is thirds)
Provide students with 5 ropes of each color
Have students divide two ropes of each color into appropriate fractional
pieces (i.e.- red into 4 equal pieces, black into 3 equal pieces)
Have students use ropes to explore problems like 2 1/3 divided by ¼ and
4/3 divided by 2/3.
Provide additional problems for students to explore or have them create
their own using the given materials.
Have students reflect upon what patterns they see with the problems and
their answers

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When you divide by a fraction, what happens with your answer?
Is it a larger or smaller number? Why?
Will this always be true? Why or why not?
Procedures Cont. …

Representational

Sketch ¾ divided by 2/3
One
and 1/8
2/3 in 3/4
Abstract
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Write a rule about dividing fractions by
fractions
What do you notice?