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Ancient Egyptians and Their Neighbors, An Activity Guide, by Marian Broida; ISBN 1-55652- 360-2
An Ancient Greek Temple, by John Malam; ISBN 1904194680
Cubes, Cones, Cylinders, & Spheres, by Tana Hoban; ISBN 0-688-15326-7
How To Draw What You See, by Rudy de Reyna; ISBN 0-8230-2375-3
Background For Teachers:
The volume of the cylinder is the amount of space inside the cylinder. To find the volume, students will first need to know the area of the circular base. The formula for the area of the circle is the same as that used in surface area or π r2. They next must know the height of the cylinder. To find the volume, they must multiply the area of the circle by the height of the cylinder. The formula is V = Bh where B = the area of the circular base or π r2 and h = the height of the cylinder. Thus, the final formula is V = π r2h. Volume is measured in cubic units.
It is helpful to consider the net of the cylinder to see how surface
area is determined. When a cylinder is taken apart and looked at as
a net, there are two congruent circles and a rectangle. The surface
area includes the sum of the areas of each circle (the bases) and the
rectangle (the lateral surface). Thus, students must have previously
learned how to find the areas of circles and rectangles. The formula for
the area of a circle is π r2. The formula for the area of a rectangle is bh.
When looking at the net, students will need to see that the dimensions of the rectangle are equal to the circumference of the circle and the
height of the cylinder. The formula is developed as follows: S = 2πr2 +
2πrh where S is the surface area, r is the radius of the base, and h is the
height of the cylinder. Surface area is measured in square units.
Intended Learning Outcomes:
2. Become effective problem solvers by selecting appropriate methods, employing a variety of strategies, and exploring alternative approaches to solve problems.
3. Reason logically, using inductive and deductive strategies and justify conclusions.
Hold up a cylinder. Ask students to identify the geometric solid. Review the parts of a cylinder with the students, focusing on the two circular faces called bases and the curved surface. Note that the two circular bases are parallel and congruent. Point out the circumference of the circular bases and the height of the cylinder.
Tell students we are going to play a game with some cylinders. Each student will need a copy of the handout Which Is Larger? on which to mark their answers. The teacher will need to have a collection of ten cylindrical objects hidden from student view in a tub. Have each cylinder numbered one through ten. One at a time, hold up a cylinder and have students predict through their powers of observation which is larger, the height of the cylinder or the circumference of the base. After students have recorded their estimate for all ten objects, it is time to check their answers. One at a time, measure the height and circumference of each cylinder in front of the group. A possible discovery is that the circumference is often larger than the height. It is a common misconception that the reverse is true.
(NOTE: The activities outlined in Instructional Procedures are intended to be taught sequentially. They will take several lessons/ days to complete with students.)
Pierce, Rebecca L., & Adams, Cheryll M. Tiered lessons. Gifted Child Today, Spring 2004, Vol. 27, Number 2, p58-65.
Based on tenets of differentiated instruction supported by the NCTM, the authors of this article define tiered lessons and outline eight steps to designing them. The three main ways to differentiate a lesson are guided by student’s readiness, interest, or learning profile. Grouping for differentiated instruction is designed to be flexible from one lesson to the next.
Pugalee, David K. Writing, mathematics, and metacognition: looking for connections through students’ work in mathematical problem solving. School Science and Mathematics, May 2001, Vol. 101, Number 5, p236-245.
This study looked for evidence of a metacognitive framework based on students’ writing about mathematical problem-solving processes. Students’ writing was analyzed from the introduction of a topic through the execution of problems on the same topic. The findings proved that a metacognitive framework is established through the process of writing. Furthermore, the author emphasizes the importance of writing as an integral part of mathematics curriculum.
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