Hands+On+Activities

= = == =**"Integer Sequences I: The Fibonacci Sequence"**= (Grades 2-3) This lesson could be subtitled "Guess my rule." I write a sequence of four or five numbers on the board. I then ask the students what number comes next. After the students have correctly added two or three numbers, I ask what rule they are using to figure out the next number in the sequence. In determining the rule, I remind them that the rule must work for every number in the sequence. I have taught this lesson to second and third graders.

Supplies
In addition to a chalkboard, you will need pieces of paper, about a quarter of a sheet, enough to give each student three pieces.

Lesson Plan
> It is important when discussing these explanations to take all suggested rules seriously, and to try to apply them to the pattern. I avoid labelling ideas as "right" and "wrong." Sometimes a student will come up with a new rule that fits the pattern. I tell the student that it is a good rule, but it is not the one I had in mind. > Have the students guess one more number (21). > Ask the students if they can figure out the rule. I've had a number of third graders and an occasional second grader get this one. To get the next number in the sequence, you add the previous two numbers, i.e. 1+1=2, 1+2=3, 2+3=5, and so on. This is called the //Fibonacci sequence.// = = =**"Gelatin Volcanoes"**= Magma leaves underground reservoirs through fractures in the surrounding rock. The fractures are either pre-existing or are created by the erupting magma. An active **//dike//** is a body of magma moving through a sheet-like, vertical or nearly vertical fracture. An important aspect of magma flow not dealt with in the gelatin activity is the heat lost during eruption. Magma, ascending as a dike begins to cool and solidify and the flow may become localized in the dike. Such localized eruption of magma over a long period of time produces a volcano. Stresses in the planet affect the orientation of dikes. Dikes open (widen) in the direction of least resistance. They propagate (grow longer and taller) perpendicular to the direction of opening. Hawaiian shield volcanoes are characterized by concentrated regions of dike injections, called **//rift zones//**. A series of experiments using gelatin models was conducted by researchers in 1972 to explain the growth and orientation of Hawaiian rift zones. The "Gelatin Volcanoes" classroom activity was inspired by this work. Gelatin, molded in bowls or bread pans, is used as transparent models of volcanic landforms. Colored water is used as the dike-forming magma. In this activity, dikes tend to propagate radially from the center of bowl-shaped casts of gelatin because the resistance to opening is the same in every direction. Dikes tend to parallel the long-axis of ridge-shaped (bread pan) casts of gelatin because the narrow dimension provides less resistance to opening than the long dimension. The dike opens in the narrow dimension and we see propagation in the long dimemsion. With a slow, steady injection rate, the colored water creates a dike and generally erupts from the flanks or ends of the gelatin casts. Edge-on, a dike appears as a line. When the gelatin cast is sliced through with a knife, dikes appear as red lines in the vertical, cut edges. Follow the directions listed on the student sheet for preparing the gelatin. Gelatin requires at least three hours of refrigeration to set. Use a warm water bath to free the gelatin from the bowl without getting water on the gelatin itself. Unflavored gelatin is ideal for this experiment because of its transparency. Sweetened gelatin desserts also work. If you prefer the dessert variety, then use a flavor that is easy to see through, such as lemon. Another alternative is agar. Agar hardens at room temperature, eliminating the need for refrigeration, but it must be made so it is easy to see through. Two-liter (or two-quart) capacity bowls work very well because the diameter allows enough space for multiple dike injections. This size is large enough for demonstration purposes. Smaller bowls, down to the size of margarine containers, have also been used successfully. Make sure a drip tray is placed under the gelatin to catch the colored water that drains out of the fractures. They will remain visible. Wear protective gloves to keep stains off hands. The colored water should not be injected too fast. Rapid injection drives the fluid straight up and creats an eruption but ruins the simulation of dike formation. When slicing the gelatin, choose a direction perpendicular to a dike to show its "line" shape on edge. Prepare gelatin in a bread pan and repeat the experiment. The original research by Fiske and Jackson used elongate models with triangular cross-sections. Fiske R. S. and Jackson, E. D., 1972, Orientation and growth of Hawaiian volcanic rifts: the effect of regional structure and gravitational stresses, //Proc. R. Soc. London, Ser. A, vol. 329//, 299-326.
 * 1) Write the numbers 1 3 5 7 on the chalkboard.
 * 2) Ask the students what number comes next. Usually a student will correctly guess 9.
 * 3) Ask for the next number in the sequence. Ask the student who answers how she or he knew that was correct. Students will offer explanations such as "You're skipping a number every time." If they don't bring it up themselves, point out that these are the odd numbers.
 * 4) Write the numbers 1 4 7 10 on the board. Ask for the next number (13). Ask for the number after that (16). Ask the students to explain the pattern.
 * 5) Write the numbers 1 2 4 7 11 on the board. Ask for the next number. It may take a few guesses for the students to come up with the correct answer of 16. Ask for the next number (22). Ask the students to explain the pattern.It may take several minutes for the students to figure out this pattern. Often they will say, "You're skipping two numbers." I respond by referring to the sequence and asking whether I am skipping two numbers between 1 and 2. I then ask what is happening between 2 and 4. As we proceed along the sequence a few students will guess the rule. They usually express it as "skipping one, then you skip two, then you skip three."
 * 1) Write the numbers 1 3 6 10 on the board. Ask the students what comes next. After they discover the next two numbers, 15 and 21, ask them to explain the rule. They usually figure this out pretty quickly, since it is the same as the rule for the preceding sequence.
 * 2) Draw three dots on the board, with one dot on top, and two dots in a row below so that all three form a triangle. Cover the lower two dots, and tell the students that here is one, then uncover the dots and ask them to count. Draw a row of three dots below the row of two, and ask the students to count the total number of dots. Add a row of four dots, then a row of five dots, counting each time. You are building a triangular array. Tell the students that the numbers 1 3 6 10 15 . . . are sometimes called the //triangular numbers.//
 * 3) Now it's time for the challenge. Write the numbers 1 1 2 3 5 on the board. Ask the students what comes next. You should get wildly varying responses. To involve all of the students, pass out squares of paper, and have each student write his or her guess in large digits so you can see it. Have the students hold their guesses up in the air. I read the guesses out loud. It generally comes out, "I see a 5, and a seven, and another 7, and an 8, and there's a 10, and more sevens . . ."Add the number 8 to the sequence, and ask the students to write a guess for the next number (13).
 * Background**
 * //Magma//** is molten rock, including crystals and dissolved gases, found at depth in a planetary interior. When magma **//erupts//** onto the surface, the volcanic products make distinctive landforms including lava plains and **//volcanoes//**, depending on the details of the eruption. One of the most interesting things to consider about magma is how it moves up from underground reservoirs, called magma chambers, to erupt as **//lava//** on planetary surfaces. Does it travel in natural tubes or pipes? Or along fractures? This experiment strikingly reveals the answer.
 * This Activity**
 * Preparation**
 * In Class**
 * Extension**
 * Reference**