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Solids Generated by Cross-Sections on a Circular Base
Allows students to examine solids generated by cross-sections on a circular base. Each model is hand-made of hardwood and has a base diameter of approximately 2.75." Models #102 and #103 are made of alternating, contrasting-color woods with a removable section. In addition to the removable section, each has sections embedded so that they may be clearly seen as part of the geometric solid. Model #102 is made up of 7 sections in 3 pieces. Model #103 has 5 sections in 3 pieces.  Models #102A and #103A are 1-piece models made of 50-60 sections of laminated hardwood so that a large number of sections may be viewed. These sections cannot be removed. Model #102 (Triangle Cross-Section) $42.00 + $5.00 Shipping. Model #102A (Solid Triangle Cross-Section $31.00 + $5.00 Shipping. Model #103 (Square Cross-Section) $47.00 + $5.00 Shipping. Model #103A (Solid Square Cross-Section) $31.00 + $5.00 Shipping.	Model #102 [Photo]	Model #102A [Photo]
	Model #103 [Photo]	Model #103A [Photo]
TIP: Have the students show that the side view of each model is elliptical with a and b in terms of the base radius r.

Solids of Revolution

Method of Cylindrical Shells
These 6" diameter hardwood models are created by revolving a near-parabolic section about the y-axis. Model #105 consists of 7 nested cylinders, all of which may be removed for inspection. Model #105A is the same shape as #105, but is a one-piece solid. Model #105 (Shells) $60.00 + $5.00 Shipping Model #105A (Shells Solid) $31.00 + $5.00 Shipping	Model #105 [Photo]	Model #105A [Photo]
TIP: If you flip the model over so that the flat base is up, you don't notice the actual curvature of the sides and will see what appears to be nested concentric right circular cylinders. TIP: Make or buy a moist cake made in a tube pan (like those for Angel Food cakes). In class, divide the students into groups and provide them with a ruler and some paper towels. Have each group cut a shell about 3/8" thick from the cake--carefully lay it out as a rectangle on the paper towels, measure and calculate the volume, record the volume, eat the piece. Sum the volumes to get the volumes of the cake. Don't worry if the calculations are not correct--the evidence has been consumed. Clean up the mess. If you want your students to never forget the Method of Shells, use a slice from a large red onion.

Method of Washers
This 6" diameter hardwood model is the solid created by revolving about the y-axis the section bounded by 2 curves and the x-axis. It is made up of five pieces: three 1/2" thick, and two 1/8" thick "washer" sections. Model #104 (Washers) $60.00 + $5.00 Shipping.	Model #104 [Photo]
TIP: If you hold the base parallel to the y-axis, both the inside and outside curvatures can be represented as a function of x. TIP: Using a flexible drafting curve, pipe cleaner, or wire, bend it to fit one of the curves, then transfer the curve to graph paper. Do the same for the other curve. Take data points about every 0.2," enter them into a graphing calculator, and perform a curve fit. Now you will have the equations representing the solid in hand.

Method of Discs
This model is created by revolving a parabolic section about the y-axis. It consists of 20 acrylic plastic discs, each of which may be removed. The discs alternate in contrasting colors for easy viewing and the pieces may be reversed to show a concave-down parabolic solid. The model is also particularly useful in discussion of Work problems--it's easy to see each disc being moved a particular distance in order to remove it from the center peg. Model #106 (Discs) $46.00 + $5.00 Shipping	Model #106 [Photo]
TIP: If you have time, pass out each disc and have the class calculate the volume of each disc for summing to the model volume. TIP: Using the measured radii and thicknesses of the discs, data points can be created for entering into a graphing calculator for a curve fit of the outside curvature. TIP: If you cover Work problems, this model is very useful if you used it during your instruction on the Method of Discs--it's something that they have already seen, and will relate to it quickly.

CALCULUS PACK

Model #108 Includes all 8 of the models shown above: #102, #102A, #103, #103A, #104, #105, #105A, & #106 Model #108 (Calculus Pack) $330.00 + $15.00 Shipping [Photo]

Intersecting Cylinders Model
The 5-piece wooden model of the right-angle intersection of two circular cylinders includes a one-piece solid intersection of 17 sections and four hollow wooden cylinders. The intersection is a square as viewed from above, and the 17 sections may be viewed from the ends of the 2" diameter cylinders. All pieces are removable for inspection, and grooves in the 6" square base allow the pieces to be displayed in correct orientation. (Information for classroom instruction is included.) NOTE: THIS MODEL HAS BEEN DISCONTINUED. IF YOU REALLY NEED ONE, LET ME KNOW AND I WILL GIVE YOU A QUOTE.	Model #109 [Photo]
TIP: If the solid intersection is turned on its side so that the square sections are parallel to the y-axis, it's easy to see that if all of the sections were to slide down to the same plane (like releasing a deck of cards), the resultant geometric figure would be that of square cross-sections on a circular base.

Cross-Sectional Solids

These models are designed for hands-on use by students. Each is a one-piece solid with a thin cross-section embedded either on the axis of symmetry of the major curve or parallel to the axis of symmetry. To find equations for the curves that define the base, students should trace the base outline onto a sheet of graph paper, collect data points, and enter the data into a graphing calculator then to fit equations for each curve. Students then can calculate arc lengths, base area and , given the shape of the cross-section, the volume of the model.

Each model is made of hardwood and is approximately 2.5" x 2.5" x5" long.

Applications

• CALCULUS: Curve fitting, Arc length, Area, and Volume
• PRECALCULUS: Curve fitting, Volume Approximation (using Riemann Sums)
• ALGEBRA: Curve fitting for parabolas

Each of the models in this section includes a Teaching Guide which covers:

• Method for taking the data.
• Potential errors associated with the data taken.
• 3-Dimensional drawings showing the relationship of the section to the coordinate system.
• Approximate equations for each curve.
• Volume calculations and numerical answers for each model.

Model #111a: The base is the section between a parabola and a line which is not perpendicular to the axis of symmetry. The embedded triangular cross-section is located on the axis of symmetry of the parabola. Model #111a Price: $38.00 + $5.00 Shipping

[Photo]

Model #111b: The base is the section between a parabola and a line which is not perpendicular to the axis of symmetry. The embedded square cross-section is located on the axis of symmetry of the parabola. Model #111b Price: $38.00 + $5.00 Shipping

[Photo]

Model #111c: The base is the section between a parabola and a line which IS perpendicular to the axis of symmetry. The embedded square cross-section is located on the axis of symmetry of the parabola. Model #111c Price: $38.00 + $5.00 Shipping

[Photo]

Model #111d: The base is the section between two parabolas of different curvatures but with parallel axes of symmetry. The embedded triangular cross-section is located between the axes of symmetry of the two parabolas. Model #111d Price: $38.00 + $5.00 Shipping

[Photo]

CROSS-SECTIONAL SOLIDS SET

Model #112 Includes all 4 of the models shown in this section #111a, #111b, #111c, #111d Model #112 Price: #148.00 + $10.00 Shipping [Photo]

Conics Model
Investigate the effects of a plane intersecting a solid. The handmade string model is continuously variable from a cylindrical shape to that of a conical surface, and can be locked in any position. Using the projection slit (included) on your overhead projector, your can project a plane of light passing through the strings on the model, and by varying the angle of the slit, conics and degenerated conics can be demonstrated. 35mm slides (not included) may also be used to demonstrate the effects of planes passing through the string model. Easy to put up and to take down. Stores in shipping box 12" x 12" x 4" high. Model #101 (Conics) $81.00 + $6.00 Shipping.	0 Degrees Rotation	Partial Rotation [Photo]	180 Degrees Rotation
	Projection Slit #101A(1) Included with Model #101
TIP: This model takes the place of your projection screen--focus the light on the strings in order to get a sharp image plane. TIP: Instead of having the model, overhead projector, and students all in a line, turn the overhead projector 90 degrees to project onto the model--this allows the students to see the sections better. TIP: If you have the time, allow your students to manipulate the projection slit--each one will want to see a different effect.

Large Conics Model: This model is the same as #101, except for size. The model is 14" in diameter and 36" high, making it easier to see parabolic and hyperbolic sections. Includes a single projection slit #101A(1). Model #101L (Large Conics Model) $93.00 + $7.00 Shipping.

Projection Slits
The single slit, #101A (1), is included with Model #101, but may be ordered separately if you already have a conics model. The double slit, #101A(2), is for projecting parallel planes. Please state which one on the order form. Model #101A (Projection Slit) $12.00 each + $4.00 Shipping	Single Slit #101A(1)	NO LONGER AVAILABLE   Double Slit #101A(2) NO LONGER AVAILABLE

Intersecting Planes
The 11" square model includes a wooden base and clear acrylic planes with a hole at the center suitable for inserting a "vector" (pencil, "D" Sticks, etc.) . Water-soluble overhead projector pens may be used for temporary marking of points, projections, and lines on the planes. Since all 8 octants are visible, classroom instruction can include discussion of the negative areas of the coordinate system. The planes fit into grooves in the base, so the model is easy to handle. The model is easy to assemble, and may be stored assembled, or flat in the shipping carton. Model #107 (Intersecting Planes) $50.00 + $6.00 Shipping.	Model #107 [Photo]
TIP: You can show additional planes intersecting this coordinate system by placing drafting squares or pieces of cardboard onto the planes. TIP: The base can be used as another plane--it's parallel to the clear horizontal plane. TIP: You can show that the intersection of two planes is a line by simply marking the intersection with an overhead pen. TIP: If you want to make some semi-permanent marks on the plastic, use permanent overhead markers--the marks can be cleaned up with rubbing alcohol.

Unit Circle Set

This is a class set of fourteen 1/2" thick wooden discs with a radius of 1 decimeter (10 cm). Each disc has a hole at its center and is etched with a pair of perpendicular diameters, as shown in the drawing at the right. Using the string provided, students work in small groups to transfer (theta, x) and (theta, y) data from the circle to a Cartesian system, creating the graphs. Extensions of the activity help students understand the concepts of periodicity and radian measure. Instructions for the activities are included with the set. Model #113 (Unit Circle Set) $83.00 + $10.00 Shipping.

Unit Circle Clock
Use the most-watched object in your classroom to teach the Unit Circle. The clock is 14" in diameter with a black plastic case and has normal hands with clockwise rotation. Complementary radian angles are color-coded for easy identification. One AA battery is included--it's classroom ready. Model #114 (Unit Circle Clock) $29.00 + $6.00 Shipping.	Model #114 [Photo]

Fraction Clock
This is a wall clock that indicates time normally and as fractional parts of an hour. It has a black plastic case and is 14" in diameter. One AA battery is included--it's classroom ready. Model #120 (Fraction Clock) $29.00 + $6.00 Shipping.	Model #120 [Photo]

Monochord

No Musical Background is Necessary Use the 1-stringed instrument to show the application of an exponential function to music. The instrument has a sliding fret which may be positioned anywhere along the 60 cm scale. Creating a tone is similar to that using a guitar--position the fret, press the string against the fret, and pluck the stirng. Calculate the string lengths and produce the notes of the Pythagorean or Chromatic scales. Applications in Algebra, PreCalculus, Math Modeling, and Physics. Lesson plans and classroom activities covering 1-2 weeks of instruction are included. The instrument is pre-strung, but will require tuning. A spare string, along with stringing and tuning instructions, is included with the instrument. Model #115 (Monochord) $60.00 + $6.00 Shipping. Model #116 (Class Set of 6 Monochords) $330.00 + $15.00 Shipping.

Model #115

If you have any questions, or if you need a model to help present a concept, please give me a call, write, or send and e-mail. If a custom model can be developed so that it has a wide appeal, I can usually give you a much better price than if it is a one-time project.

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