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Preparing a Lesson

Preparing a Lesson. Warm-up Activating Prior Knowledge. I have a box that measures 12” long, 10” deep and 8” high. Can I pack 1000 one inch cubes in the box?

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Preparing a Lesson

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  1. Preparing a Lesson

  2. Warm-upActivating Prior Knowledge • I have a box that measures 12” long, 10” deep and 8” high. Can I pack 1000 one inch cubes in the box? • I have a piece of copper that is 4 inches square, if I need a circular piece that has a radius of 2 inches, can I cut it from this piece of copper?

  3. Expected Responses to Warm-up • Volume of box – 12”x10”x8”=960 cubic inches. No, I cannot pack 1000 one inch cubes. • Area of 4 inch square = 16 square inches. • Area of circle with 2” radius = 12.56 square inches. Yes, I can cut the piece of copper into circles with a radius of 2”.

  4. The Task • My friend, Charles, wants to build a gas tank for the airplane he is building. He has a rectangular piece of aluminum that measures 3’ X 5’. He wants the tank that will hold the most fuel. Does it matter if he builds a or a fuel tank?

  5. What Standards are being addressed? • Measurement Standard for Grades 6-8 • Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision. • Develop and use formulas to determine the circumference of circles and the areas of triangles, parallelograms, trapezoids, and circles and develop strategies to find the area of more-complex shapes; • Develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders.

  6. Prerequisite Skills • Select and apply techniques and tools to accurately find length, area. • Develop and use formulas to determine the circumference of circles and the areas of rectangles and circles.

  7. Lesson Objective • Develop strategies to determine the volume of cylinders.

  8. Engage • My friend, Charles, wants to build a gas tank for the airplane he is building. He has a rectangular piece of aluminum that measures 3’ X 5’. He wants the tank that will hold the most fuel. Does it matter if he builds a or a fuel tank?

  9. Explore • Using two 4”x6” index cards, roll each into a cylinder, rolling one the short way and the other the long way. Tape them, trying not to overlap the edges. Predict whether the circumference is longer, shorter or about the same as the height. Also predict how much longer or shorter the height is. Then measure.

  10. Explore (con’t) • Suppose you filled each tube with rice or beans to compare how much each holds. Do you think each holds the same? If not, which one do you think holds more? Why?

  11. Explore (con’t) • Follow the procedure shown and record the results. Burns, Marilyn; About Teaching Mathematics, A K-8 Resource

  12. Explain • What did you find out from exploring with the two cylinders? (the volumes are different) • Why do you think this happened? (the circumference of the two cylinders are different) • What is the circumference of the tall cylinder? (3 inches)The short cylinder? (5 inches) • What have we learned previously that will help us find how much each cylinder will hold? (how to find the volume of a rectangle) • Can we develop a formula for finding the volume of a cylinder?

  13. Explain (con’t) • What do we know about the volume of a rectangle? (length X width X height) • Then what do you think we will need to know about the volume of a cylinder since length and width are not well defined? (The volume is equal to the area of the base of the cylinder times the height.)

  14. Explain (con’t) • What is the base of the cylinder? (the bottom or the circular part) • How do we find the measure of the circular part? (the area of a circle is A=(Pi)(r^2)) • For this problem we are going to use a radius of 1 inch for the short, fat cylinder and a radius .5 for the tall, thin cylinder.

  15. Explain (con’t) • Now apply the formula for volume of a cylinder to compare the volumes. • V=(Pi)(r^2)(h)

  16. Explain (con’t) Short, Fat Cylinder Radius 1” Height 3” V = (Pi)(r^2)(h) V=(3.14)(1^2)(3) V=9.42 cubic inches

  17. Explain (con’t) • Tall, thin cylinder • Radius .5” • Height 5” V = (Pi)(r^2)(h) V = (3.14)(.5^2)(5) V = 3.925 cubic inches

  18. Extend • Can you find the volume of a cylinder that has a radius greater than 1”, and a height of 3”. • Can you find the volume of a cylinder that has a radius less than 1” and a height of 3”.

  19. Extend (con’t) Fill in the chart with your findings.

  20. Extend (con’t) • What have you discovered about the volume of a cylinder? • Divergent question If a gallon of fuel requires 231 cubic inches, what dimensions would Charles have to use to make a gas tank that will hold 7.5 gallons.

  21. Student Evaluation • Check student’s papers and charts for understanding using a scoring rubric • Developing – Shows little understanding for applying the formula to the two additional problems, chart is incomplete, cannot explain what happens to the volume of a cylinder as the radius changes, no understanding of the divergent question. • Proficient- Show understanding for applying the formula to the two additional volume problems, chart is mostly complete, can adequately explain what happens to the volume of a cylinder as the radius changes, attempts with understanding to solve the divergent problem • Advanced- Students correctly solve the two additional volume problems, fill in the chart correctly with all labels, and explain fully what is happening as the radius of the circle increases or decreases. Shows understanding for the solution of the divergent question.

  22. Teacher’s Evaluation • How did the lesson go? • What misconceptions did the students have that you had not anticipated? • What enduring understandings did the students develop as a result of this lesson.

  23. Task, Discourse, Environment, Evaluation *Task – Did the task engage the students? Was the task appropriate mathematically? Did the task lead the students to discover some enduring mathematically understanding? Did the task empower the students to grow mathematically?

  24. Task, Discourse, Environment, Evaluation • Discourse • Did the questions scaffold the students to an enduring understanding? (not tell them) • Did the questions lead to student discussion? • Did the questions clear up student misconceptions?

  25. Task, Discourse, Environment, Evaluation • Environment • Was the environment open, bias free? • Did all students feel comfortable participating in the class discussion? • Did the task invite cooperation and student discussion? • Did the environment give the students a feeling of being empowered to do mathematics?

  26. Task, Discourse, Environment, Evaluation • Student Evaluation • Were the students able to evaluate their own understanding? • Did the evaluation enable students to understand and correct their misconceptions? • Did the evaluation give students an opportunity to grow mathematically?

  27. Task, Discourse, Environment, Evaluation • Teacher Evaluation • Did the teacher have an opportunity to improve the lesson? • Did the teacher change the lesson to anticipate student misconceptions and make the lesson clearer about these points? • Did the teacher develop a deeper understanding of the enduring concepts of the lesson?

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