Two Classroom Activities

These classroom activities are designed to allow the students to use mathematical reasoning to solve a problem and at the same time provide an opportunity for the teacher to share some math history with the class. These activities are designed to fit the needs of a teacher who may not have the class time to complete manipulative activities in class. As a teacher, I find that these activities work well as a "Problem of the Day" kind of activity that can be used as a "springboard" to "jumpstart" the day's lesson. The first activity revolves around "Euler Circles" and the second centers on the idea of taking steps, much like those purposed by George Polya to solve a word problem. It is an attempt to improve student's critical thinking skills by studying a problem and implementing strategies to find a solution. It is also a way to bring a form of logic into the classroom.

ACTIVITY 1: A Trip to the Candy Store

Jennifer likes to snack. On a recent trip to the candy store she bought some "mix and match" candy. She has 16 pieces of candy in all. Nine contain caramel; 10 contain coconut; and 6 contain both caramel and coconut. How many of the candies contain neither caramel nor coconut.

One approach to this problem is to use what is now called "Venn Diagrams." However, this idea of using circles to illustrate a problem was first purposed by the Swiss born mathematician, Leonhard Euler (1707 - 1783). He is credited with many accomplishments, especially in the areas of calculus and number theory, but he is also known for many other accomplishments one of which he left to another mathematician to receive credit. Instead of taking credit for the use of circles to illustrate statements and describe relationships among sets and calling them "Euler Circles", Leonhard Euler left it up to John Venn to expand his ideas and to this day they are called "Venn Diagrams."

Now, back to our problem. The first step in order to determine the solution to any problem is to make sure that it is understood. One way to help in the understanding of a problem is to use illustrations. Venn Diagrams are a good way to illustrate problems. O.K., let C represent the number of caramel candies, then n(C) = 9. Let Co represent the number of coconut candies, then n(Co) = 10. Finally, let (CCo) = 6 (where the caramel and coconut candies overlap).

It is obvious that if all the candies were either caramel or coconut, there would be a total of 19 pieces of candy. However, from the problem, Jennifer has only 16 pieces. From the given information we can construct a "Euler Circle", actually a Venn Diagram.

With the use of real candy this activity can become a class favorite. Of course, since coconut is not the favorite candy of many students, another flavor can be substituted. It may also be beneficial to make sure that the students understand the basic notation in working with Venn Diagrams and sets.

ACTIVITY 2: Knees and Trunks

On a recent safari to the deepest, darkest jungles of Africa , a group of tourists or "jungle explorers" and elephants contained 100 knees and 100 trunks. If each tourist carried three trunks, how many jungle explorers and elephants went on safari?

This activity is designed to get the student to critically think about the process of mathematics and approach a word problem, one step at a time. It is also the opportunity to insert a historical note about one of the leading thinkers in problem solving, George Polya (1887-1985). Polya, who grew up in Hungary and because of Hitler's rise to power, came to the U.S. where much of his work was done in mathematical research. In the 1950's he began writing a series of books and insightful articles on problem solving. From his four steps, many problem solving strategies have evolved. One problem solving strategy that is used is one that the Haysville Alternative High School implements across the curriculum and includes the following:

• Explore the problem
• Examine the details
• Experiment with an equation
• Execute your calculations
• Evaluate the solution

This method is time consuming because if you make a poor guess, you may be guessing for some time. It may not be the most efficient way to solve a problem, but it is nonetheless a problem solving method. Once a guess is taken the total number of knees and trunks can be counted to see if the requirements have been met for the safari. If the requirements are met-good guess. If the requirements are not met then keep on guessing.

A second approach to this problem is to use an equation. Try the following, where E represents elephants and H represents humans, K represents knees and T represents trunks. Once again, there are some assumptions to be made:

• An elephant has 4 knees and 1 trunk.
• A human has 2 knees and 3 trunks

Now we simply set up an equation and solve for one of the variables, then once we have one of the variables, we substitute into one of the equations and solve for the other variable. Lets begin with the equation, 4E + 2H = 100

4 (100 - 3H) + 2H	=	100	by substitution
400 - 12H + 2H	=	100	by the distributive property
400 (+ - 400) -12H + 2H	=	100 (+ - 400)	additive identity
- 10H	=	- 300	by substitution

and H = 30 We have now determined that we have 30 humans on safari. Now, simply substitute the number of humans into one of the equations to determine the number of elephants.

By taking the time to critically think through the problem, the student is able to find a solution and thus "solve a story problem" and implement the basic problem solving structure as introduced by George Polya.

Although these activities seem more like problems, they still allow the teacher the opportunity to work with the students in the area of basic problem solving using at least two different methods. Even though their appears to be a lack of creativity, these problems do work as "Problem of the Day" type questions because I have used them and know that they work. If you are able to find a place to use them in your class, feel free!

This activity is designed to help students of all levels review mathematical concepts and terminology for exams. Topics to incorporate vary from addition and subtraction to calculus calculations to historical figures to definitions.

Procedure:

Divide the class into two groups. The groups do not necessarily need to be equal in size but they need to incorporate multiple levels of student abilities.

Materials:

• Two hand bells
• Scratch paper
• Pencils
• Calculators (depending on magnitude of problems)
• Two sets of identical flash cards with problems and definitions

Activity:

Stack sets of flash cards in the same sequence on a table along with one hand bell in front of each stack.

Have two students from each group come to the front of the room and take the top flash card back to the group. The group may not communicate verbally, they must write or find another means of communication but they may not speak so that the other team may over hear their discussion of the problem.

Give the students a time limit appropriate to the problem that should not exceed 5 minutes per problem since this is primarily for review and you want to review all concepts.

Once time has expired the students must present their work. The group that gets the answer first presents first. Their work must be legible and complete. The group with the correct answer gets two points and the other group gets one point for attempting the problem. Continue with the next problem in the same manner as before.

The whole group must actively participate in the solutions, if not the team loses points, this hopefully encourages teamwork

Modifications:

You can have students answer the questions individually and race between groups to finish the set of problems first. Each time they get a correct answer they get a point. The team with the most points wins.

Lesson Objective: The student will identify points on a coordinate plane.

Historical Connection: Rene Descartes created the Cartesian Plane.

Supplies needed:

• Make a classroom set of laminated coordinate planes. Make these by gluing a sheet of graph paper, which already has the coordinate plane, in a folder, then laminate.
• This way the student can use the flap of the folder as a barrier so the other student can not see his “needles”. These folders can be used for other activities, also.
• Vis-à-Vis markers (1 per student)
• Damp paper towels or baby wipes

• Each student needs a partner. They should be facing each other.
• Each student needs to draw five line segments on their coordinate plane (haystack)
• One segment (needle) should have 3 coordinates, two “needles” should have four coordinates and two “needles” should have five coordinates.
• The students take turns calling out coordinates, in an attempt to find their opponent’s “needles”.
• When students are calling out coordinates, they need to be sure they mark the coordinates they have already called.
• If a student repeats a coordinate they have already called, they may not call another coordinate until it is their turn again.
• When a student has found all coordinates to one of their opponent’s needles, the opponent MUST tell the student that they found one of their needles.
• The game ends when one student has found the needles in the haystack!

Ancient History: During the summer of 1971 in the Powder River Basin of Wyoming we ran across a surveyor’s note on a land plat that listed a certain property marker as being at a certain bearing and at a distance of “two cigarettes on a mule.”

The following activities are designed to help math and science teachers introduce students to concepts such as area, proportion, and inferences.

Objective: To learn about using instruments such as balances and clocks to verify various measurements and math calculations.

Purpose: To gain an understanding that there can be several ways to arrive at the same information regarding area or volume; to enjoy mathematics.

Area by mass.

1. Draw several simple shapes on the same type of paper and then compare these figures by measuring the length of key dimensions and calculating their areas.
2. Carefully cut out the areas and weigh them on a centigram balance. Compare their calculated areas to their masses.
3. Using the same kind of material as in steps 1 & 2, cut out an irregular shape and weight it. Predict the area based upon the mass. Use a grid overlay to calculate the area of the irregular shape. Compare the two measurements of area and discuss which would be more accurate.
4. Use mass to demonstrate area identities. Triangles --
   Parallelograms --
   
5. Demonstrate c² = a² + b² by using squares and the Pythagorean triple values.

1. For a set duration of time and at as constant of a rate as possible, draw a non-overlapping line.
2. Transfer the line length to a string. Take care to avoid stretching the string. Measure the string length with a ruler or other appropriate measuring device.
3. Try to work backwards by again drawing lines but predicting the durations of time based upon the length of the lines drawn. Discuss the measurement of knots.
4. Discuss the repeatability of the experiment as well as error and ease of measurement.

1. Pipe or tubing thickness based upon mass.
2. Volume of container based upon fill time.
3. Color based upon temperature.
4. Temperature based upon conductivity (Michael Farraday).

Ancient History: In the summers of the middle 1950’s, mothers would make us take naps because rested, healthy children were supposedly less susceptible to poliomyelitis, which was running rampant in those days. At seven or eight, it was a battle to get us to rest in the afternoons before swimming. One afternoon, instead of taking a nap, we were given the option to rest quietly and write our numbers from one to one thousand. (This was a wonderful trade but I soon found that writing the numbers neatly required more time than taking the nap.) When the task was finished, we presented our lists to our mother. She scrutinized them then said that she had given us this assignment to show us how big a million is. She reminded us of how hard it was to write from one to one thousand then explained that we would have to write a thousand lists, each with a thousand numbers, in order to reach one million. The mathematical power of 1,000 x 1,000 = 1,000,000 became real.

The following activities are designed to help math and science teachers introduce students to concepts such as powers, sequence, counting integers, and proportion.

Objective: To learn about squared numbers, counting by adding powers of two, and changes in power as dimensions change.

Purpose: To gain an understanding of how numbers increase exponentially and to have tangible evidence that demonstrates the effects of changes in dimension; to enjoy mathematics.

Powers of 2 to count integers.

1. Introduce:
2. Assignment: Count from 1 through 25 by using only 2⁰, 2¹, 2², 2³, 2⁴ and addition.

The graph of power.

1. Doubling in 2-dimensions:

   12 = 1 x 1 = 1 22 = 2 x 2 = 4

2. Tripling in 2-dimensions:

   12 = 1 x 1 = 1 32 = 3 x 3 = 9

3. Doubling in 3-dimensions:

   13 = 1 x 1 x 1 = 1 23 = 2 x 2 x 2 = 8

4. Tripling in 3-dimensions:

   13 = 1 x 1 x 1 = 1 33 = 3 x 3 x 3 = 27

The Tower of Hanoi.
from Edouard Lucas (1842 - 1891)

The board has three pegs and a stack of different sized disks, five to seven in ascending order is common. The object of the game is to get the entire stack of disks relocated to another peg and also in ascending order. Only one disk at a time may be moved and a larger disk may not be placed upon a smaller one. Edouard Lucas claimed that the real Tower of Hanoi is attended by mystical monks who move only one disk each minute. The real tower also has sixty-four disks and as soon as the last disk is placed in the new stack the earth will collapse. The challenge to students is to determine how long it will take the earth to collapse. (Moving the disks without error is assumed.)

1. Introduce the pegboard without rings and tell the Tower of Hanoi story.
2. Predict number of moves needed to move 1 disk. Move one disk. List the number of moves on chart.
3. Predict number of moves needed to move 2 disks. Move 2 disks. List the number of moves on chart.
4. Predict moves required when the number is doubled to 4 disks and redoubled to 8 disks. Discuss 64.
5. Move 3 disks and list number of moves on the chart.
6. Move 4 disks and list number of moves on the chart.
7. Help students discover the sequence pattern. Discuss the moves needed for zero disks.
8. Use the power of two lessons to examine the equation 2ⁿ - 1.