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).
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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.