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From an Activity in a Textbook to an Open-Ended Problem: Developing Students’ Mathematical Thinking and Communication

From an Activity in a Textbook to an Open-Ended Problem: Developing Students’ Mathematical Thinking and Communication. Soledad A. Ulep University of the Philippines National Institute for Science and Mathematics Education Development. For Questions 1 and 2. Teachers’ Reliance on Textbooks.

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From an Activity in a Textbook to an Open-Ended Problem: Developing Students’ Mathematical Thinking and Communication

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  1. From an Activity in a Textbook to an Open-Ended Problem: Developing Students’ Mathematical Thinking and Communication Soledad A. Ulep University of the Philippines National Institute for Science and Mathematics Education Development

  2. For Questions 1 and 2

  3. Teachers’ Reliance on Textbooks • Grade 10 mathematics teachers of Sta. Lucia High School use the list of learning competencies , textbooks and teacher’s manual provided by the DepED, and other reference materials in preparing their lessons. • In our lesson study group, our goal is to develop students’ mathematical thinking through problem solving. • So we teach mathematics through problem solving.

  4. An Activity from a Textbook to Introduce Polynomial Function

  5. An Open-ended Problem to Introduce Polynomial Function Christmas is fast approaching. Lucy wants to give a personalized gift to her friends. She plans to make a box with an open top where she can store the gifts. Before making the actual box, she wants to try it first by using a plain sheet of grid paper measuring 10 cm by 16 cm. If you were Lucy, what possible boxes can you make? Among the boxes that you made, which do you prefer and why?

  6. Pointers to follow: • Construct an open-top box of different sizes using sheets of grid paper, pair of scissors, and tape. • Use one sheet of grid paper for each box. • Do not remove any part of the sheet of grid paper. You can cut the paper but not cut off any part. • Avoid any folds on the top part of the box. • Do not waste the sheets of grid paper so that you can make many boxes.

  7. One Method Making Boxes with Open Top

  8. Changing Quantities • The length of a side of a square that is cut at each corner of a sheet of grid paper is equal to the height of the box formed. • The quantities that change as the height of a box changes are: 1) length and width of a box 2) perimeter of the different faces of a box 3) area of the different faces of a box 4) volume of a box 5) surface area of a box

  9. Mathematical Relationships • dimensions of a box height = x length of a box, l(x) = 16 – 2x width of a box, w(x) = 10 -2x • perimeter P(x) of the different faces of a box P1(x) = 32 – 2x P2 (x) = 52 – 8x P3(x) = 20 - 2x

  10. area A(x) of the different faces of a box A1(x) = 16x – 2x2 A2(x) = 160 - 52x + 4x2 A3(x) = 10x - 2x2 • volume V(x) of a box V(x) = 4x3 - 52x2 + 160x • surface area of a box S(x)= -4x2 + 160

  11. Mathematical Thinking and Communication • Polynomial function is introduced to relate the identity, linear, quadratic, and cubic functions based on what the students already know about polynomials and functions. • Devising own ways of solving an open-ended problem, connecting previous and new learning, and representing mathematical relationships promoted students’ mathematical thinking and communication.

  12. For Questions 3 and 4

  13. Teachers’ Use of Other Educational Tools • Grade 10 teachers of Sta. Lucia High School do not use overhead projectors and rarely use computers to teach mathematics. • They mainly use blackboards.

  14. Visual Representations of Mathematical Relationships: Using Geogebra Geogebra is a free downloadable dynamic geometry software that can show the graph, equation, and specified points on the graph of a mathematical relationship.

  15. Dimensions of a box height

  16. Perimeter of different faces height

  17. Area of different faces height

  18. Volume height

  19. Surface area height

  20. Further Explorations using Geogebra 1. Perimeter of different nets height 2. How will the graphs look like if the context of the problem is disregarded?

  21. More Mathematical Explorations and Thinking Using Geogebra makes it easy to make further mathematical explorations. These lead to more mathematical connections and further problem posing.

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