BUILDING SHAPES FROM NUMBERS
Grades 5-8
From this collection of activities, the students will
experience
how to illustrate some of the many connections between number and
geometry.
Since arithmetic and geometry are closely related, having the opportunity
to explore the patterns that underlie one area of mathematics can help
the
students to understand other areas. Through this 2-3 day focus on square
numbers and triangular numbers, the students will be able to make
connections
among mathematical topics. Their skills in reasoning and problem solving
will be strengthened by linking those skills with the basic arithmetic of
multiplication and addition, and with combinations of essential geometric
shapes.
Math Talk #120: What Shape Is Your Number?
Students will be able to:
1. explain and illustrate square and triangular numbers.
2. demonstrate the number sequences that result in square and triangular
numbers.
3. make and use a table to organize data.
4. conduct investigations into how the ideas from this lesson can be
applied
to other geometric shapes.
Texas Assessment of Academic Skills (TAAS), Grade 8
Math Objectives:
#3: Demonstrate an understanding of geometric properties and
relationships.
#8 and 9: Use multiplication and division.
#11: Determine solution strategies.
#12: Determine solution strategies using mathematical representation.
Science Objectives:
#4: Interpret scientific data and/or information.
#7: Draw conclusions about the process(es) and/or outcome(s) of a
scientific
investigation.
NCTM Standards for Grades 5-8
Standard 1: Mathematics as Problem Solving
Standard 3: Mathematics as Reasoning
Standard 12: Geometry
Per student:
- bag of 36 two-color counters
- set of 15-21 cards from a deck of playing cards or blank index
cards
(numbers not necessary)
- copy of Pascal's Triangle
- highlighter pens
- set of construction paper or tag board shapes about 3 inches high
(1 triangle, 1 rectangle, 1 pentagon, not necessarily equilateral)
Day 1
Using a set of 10 cards, model the procedure for playing Bulgarian
Solitaire.
Divide the 10 cards into two (not necessarily equal) piles. A play
consists
of taking one card from each pile and making a new pile. Playing the
cards
does not need to be in any certain order. To model play, continue taking
one card from each pile making a new pile each time until you find a
pattern.
Specifically, play continues until you know exactly what will happen
every
time. For example, if a player started with two piles with 8 cards in one
pile and 2 cards in another, in the next play you would have 7 cards, 1
card, and 2 cards. On the next play, you would have 6 cards, 0 cards, 1
card, and 3 cards. On the next play, you would have 5 cards, 0 cards, 2
cards, and 3 cards. On the next play, you would have 4 cards, 3 cards, 2
cards, and 1 card...etc. Allow students to play Bulgarian Solitaire as
partners.
Remind them to continue play until they find a pattern.
As the students play the game, circulate among them. Note the students
that
are using particular recording methods to solve the problem, but allow
them
to find their own method of recording. When the groups have detected the
pattern (the piles will repeat a 4-3-2-1 pattern), ask, "Do you
think
you will get the same result when you start with two piles, 5 and 5? Can
you find any predictable patterns in your completed record that tell you
how many steps each combination takes to get to 4-3-2-1?"
Have students explain their pattern and write summary statements
explaining
why they believe their identified pattern is correct. Ask, "How does
the game change if you start with only 9 cards? What would you predict
will
happen?" If there is time, have the students try the patterns found
using 9 cards instead of 10 cards. If the game is played with 9 cards,
instead
of a fixed end point, a cycle occurs. The starting number is called
"Nice"
if the period of the cycle at the conclusion of the game is 1, that is,
we have a fixed point. Hence, 10 is a nice number and 9 is not. The
question
to investigate is to find whether other nice numbers exist and/or if
there
is a predictable pattern to their occurrence. Nice numbers turn out to be
triangular numbers.
Illustrate a square and a triangular number on the board. A number that
can be shown as a square array of dots is called a square number. One can
also think of a square number in terms of multiplying a number by itself
(3 x 3, for instance).
. . .
. . .
. . .
Similarly, a number that can be shown as a triangular array is called a
triangular number. Note that the shape of the triangle is not important.
There are two ways of arranging 10 dots.
[Note: It might be a better use of time to teach Bulgarian Solitaire on
a day previous to the day the video segments are to be shown. This would
allow ample time for investigations of the patterns inherent in the game,
as well as give more time for the students to complete the Day One
post-viewing
activities. In this case the pre-viewing activities would start with the
paragraph beginning, "Illustrate a square and a triangular number on
the board."]
Each student should have a bag of 36 two color counters and
paper and pencil at his/her desk. Tell the students that they will watch
two segments of a video where square and triangular numbers are both
illustrated
and used. The students' responsibility will be to use 2-color counters to
build successive triangular numbers and to complete the arithmetic
sequences
of squares and triangle with paper and pencil. Ask the students to listen
for specific definitions of square and triangular numbers.
BEGIN the Math Talk video at the beginning. Maria
Lopez
will say, "Welcome to Math Talk." PAUSE on the screen
that
says "Masterworks Theater Presents: The Trojan Pie." Ask the
students
to recall the introduction they have had to square and triangular number
sequences. Have two volunteers draw a simple diagram on the board of a
sequence
of squares and a sequence of triangles. RESUME the video.
PAUSE
after the soldier says, "It looks like a huge slice of blueberry
pie!"
Ask, "What shape is the piece of pie? What do you think this plan
will
involve?" Make note of the students' predictions.
RESUME the video. PAUSE after the peasant says, "If we
put three dots up like this, they form a triangle." Have students
use
two-color counters to start the triangle shape on their desks.
RESUME
the video. PAUSE after the peasant adds the row of three dots.
Have
students do the same. RESUME the video. PAUSE after the
soldier
says, "So 3, 6, and 10 are triangular numbers because they
form..."
Ask a student to tell what shape they form and continue their formations.
RESUME the video. PAUSE after the soldier says,
"...unless
you're telling me that 36 is a triangular number." Ask for a vote on
whether 36 is a triangular number.
RESUME the video. PAUSE on the graphic illustration. The
left
side of the screen shows a 3-dot triangle and a 4-dot square. The right
side of the screen says, "1 + 2 +" on top and "1 + 3
+"
on bottom. Ask the students to watch for how the sequences are formed.
RESUME.
PAUSE when the right side of the screen says, "1 + 2 + 3 + 4
+" and "1 + 3 + 5 + 7." Ask a student to explain what the
next term in the sequence forming triangular numbers will be and record
predictions on the board. Ask another student what the next term in the
square sequence will be and record predictions. Tell the students to
check
the predictions after the
video continues.
RESUME the video. STOP the video when the orange triangle
is at the top of the screen, the yellow square is at the bottom, and the
numbers in between show consecutive counting numbers and consecutive odd
numbers. Ask the students to copy and complete these two sentences:
Triangular numbers are found by adding ___________.
Square numbers are found by adding________________.
Ask for any observations on other patterns the students
notice.
On the overhead place this drawing representing a ball
rolling
down a ramp 25 meters long.
Tell the students that it takes 5 seconds for the ball to roll from the
top to the bottom of the ramp. The dots represent the positions of the
ball
after each second. Have the students write a number sequence representing
the total distances traveled by the ball in the first second, the first
2 seconds, the first 3 seconds, the first 4 seconds, and the first 5
seconds.
Ask what this sequence of numbers is called. (square numbers) Have the
students
write a number sequence representing the distances traveled by the ball
in the first second, second, third second, fourth second, and
fifth
second. Ask for the name of this sequence. (odd numbers)
Draw the following triangular arrays on the overhead:
Ask the students if they can think of any everyday objects that are
arranged
in triangular arrays. Give clues that two such arrays are related to
games
they may have played. (10 bowling pins ; 15 numbered balls in pool) Have
them copy the five terms (numbers) of the sequence from the overhead.
Continue
the sequence to show the next five terms. Instruct the students to now
add
each pair of consecutive terms of the sequence as shown below to make a
new sequence.
Have them write a sentence describing what they notice about the
resulting
sequence. This is a good place to stop the lesson for Day One.
pre-viewing activities-day 2
Introduce the Handshake Problem. Present to the students this question:
"If everyone in this room shook hands with every other person in the
room, how many handshakes would there be?" Brainstorm with the class
to create a list of possible strategies that might help solve the
problem.
If no one volunteers the strategy of acting out the problem, ask if any
students think such a strategy would be helpful.
focus for viewing
Review the sequence of triangular numbers with the students and remind
them
of the arithmetic sequence that forms them (sums of consecutive whole
numbers).
Tell the students that they will be viewing a third segment of the video
that was begun previously called "The Wide World of Sports No
One Has Ever Heard Of." Tell them that this segment features a
handshake
contest in which contestants try to shake hands with every other member
of their team in the shortest amount of time. Ask the students to watch
carefully for any appearance of triangular numbers. They should use
reasoning
to attempt to predict the number of handshakes necessary for any number
of people. Tell the students that they will be acting out the handshake
problem. They will be expected to organize the data they gather in a
table.
viewing activities
Fast forward the video from yesterday's stop point to the blue and green
screen with the "Wide World of Sports You've Never Heard Of"
logo.
Begin the video. Pause after the red-haired sportscaster says,
"...by
drawing a diagram." Ask if anyone in the class would like to
demonstrate
their idea for a diagram of this problem on the board. After a student
does
so, ask the other students if anyone would like to make any
modifications.
Tell them to compare their diagrams with the one about to be shown on the
screen. Resume the video. Pause after the redhead says, "That's why
they're called triangular numbers." Review knowledge of the
triangular
number sequence by asking the students what the next two triangular
numbers
will be. Resume the video. Pause after the redhead says, "That's a
lot to cram into 17.4 seconds." Review with the class the
mathematical
formula for finding the nth triangular number. Have them record in their
notes that 7 x 6 represents the number of people in the video problem who
must shake hands with each other. Stress that the result of 7 x 6 must be
divided by 2 in order to show that two people may only do one handshake
together. Show the students how to change the numerical formula they have
just learned into a variable formula which will allow them to solve the
problem for any amount of people. The formula (7 x 6) / 2 becomes n x (n
- 1) / 2. Resume the video to view the actual handshaking. Stop the video
when it has concluded.
post-viewing activities
Ask three students to come to the front of the room to begin the
handshaking
sequence. Instruct all students to select the triangle shape from their
set of polygons. As each student shakes hands with the other two, ask the
class to count the number of handshakes needed. (three) Record three dots
on the board in the shape of a triangle. Now ask them to imagine that
each
vertex of their paper triangles represents one of the three students.
Tell
the students to label the three vertices A, B, and C. Then say, "If
A shakes hands with B, represent the handshake by drawing a connecting
line
between A and B. Now draw a line representing B shaking hands with C. Do
the same thing to represent the handshake between A and C. Have all
possible
handshakes been represented?" Answer any questions that arise.
Invite a fourth student to join the others. Ask, "How many more
handshakes
will be needed?" Since the new member will have to shake hands with
each of the three original students, three more handshakes will be
needed.
Add three dots to the triangle on the board. Then direct the students'
attention
to their paper rectangles. Have them label the vertices A, B, C, and D.
Let them again simulate the handshaking process by drawing lines to
connect
every vertex of the rectangle with every other vertex. Let a volunteer
count
aloud the connecting lines he/she has drawn to verify that everyone has
drawn all necessary lines.
Continue the handshaking process by adding a fifth person to the team.
This
person will have to shake with the four other people, requiring four more
handshakes. Add four dots to the triangle on the board. The students at
their desks should then label the vertices of their pentagon shapes.
Instruct
them to follow the same process as before to connect the vertices. Ask if
anyone in the class is ready to describe how the triangular numbers are
being formed. Guide the students to make a table showing the information
already gathered. Ask them to look for any patterns in the table that
might
help them predict how many handshakes would be necessary for n number of
people. Have the students record the formula they have learned at the
bottom
of the table. The students' tables should look similar to this:
Now that the students have had some experiences with square and
triangular
numbers, show them that these numbers can be combined in a variety of
interesting
ways. For example, if you start with a square number and draw that number
of dots in a square pattern, those dots can then be grouped into two
triangular
arrays, like this:
Challenge the students to test the theory that every square number larger
than one can be written as the sum of two triangular numbers-in fact,
consecutive
triangular numbers: 36 = 15 + 21, 100 = 45 + 55, and so forth.
Examine the patterns found using even triangular numbers. Such can be
broken
up into four triangular numbers, one smaller than the other three. For
example,
36 = (3 x 10) + 6, as the picture illustrates:
Give the students a copy of Pascal's Triangle. Tell them to study the
numbers
contained in the triangle to see if they can locate the sequence of
square
numbers. After five or six minutes, ask a volunteer to come to the
overhead
projector and trace the numbers he/she has located on an overhead
transparency
of the students' sheet. Then instruct the students to look for the
triangular
numbers in the triangle. Again, have a volunteer trace them on the
overhead.
Save the copies of the triangle to be used with other number sequences
later
in the year. [Note: Three of the four example sheets following this
lesson
demonstrate how these are found.]
Have the students write letters or e-mail to students at a nearby high
school
who take physics. They should describe what they have learned about
square
and triangular numbers and ask the physics students whether they have
conducted
any experiments in class similar to the ball rolling down the ramp. If
they
have performed experiments that demonstrate relationships between square
numbers or triangular numbers, perhaps the middle school students could
arrange a visit by the high school students to demonstrate the
experiments.
Students can contact high school geometry and/or chemistry teachers who
would speak to the class. They can discuss some applications of square
and
triangular numbers in their high school classes, such as patterns present
in the periodic table of the elements.
Math: Conduct a systematic study of the number of ways
things can be combined. Include problem-solving situations from other
subject
areas. Students can also conduct investigations into how the ideas in
this
lesson can be applied to other geometric shapes. For instance, pentagonal
numbers can be represented by arrays of dots in pentagon shapes, as
hexagonal
numbers can be represented by arrays of dots in hexagon shapes. If
T-tables
are set up for the numbers of dots in the arrays, patterns may be
explored.
Social Studies: Make list of combinations of clothing possible for
certain world climates.
Art: Have the students create collage displays using only squares
and triangles.
Science/Technology: Have the students research some of the
technologies
used in making movies or television programs. Several science-fiction
shows
have been based on the theme of human beings changing size. If people
could
become larger or smaller, their physical characteristics would not change
at the same rate. The strength of their bones, for example, would vary
with
Click here to view the
worksheet associated with this lesson.
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