Problem Solving

Paper Folding Challenge

Goal of the lesson.

• To find the point at which two triangles of equal areas can be formed with the folding of a paper. (Middle School 8 th grade)

What to do?

• Every student folds and the class observes others and discusses similarity and differences (Tell/show them how to fold the paper) • Launch the idea of what they see and how can they tie it into mathematics • Put students into groups

Content Discussions Student Driven • Angle measures (vertical, complimentary, supplementary, right, obtuse, acute) • Perpendicular Iines, parallel lines, transversals, hypotenuse, legs • Polygons, perimeter, area & formulas, scale factor, ratios, similar triangles, (scalene, obtuse, & right triangles)

Questions to ask individual groups?

• What, if any patterns do you see?

• Any relationships to geometry?

• What would happen to the polygons if you folded it differently? Why?

• What are you going to do to find the same area of the triangles?

• Can you make connections to vocabulary terms?

More questions • How do you calculate area of a triangle?

• Do the areas of the triangles increase/decrease the same as the point is moved?

• What happens to the sides of the triangle as the point is moved?

Monitor Students • Look for discussion of vocabulary terms • Getting organized with data definitions • Ask what do they see? How do they know?

• Label everything?

What students will attempt • Massively fold forever • Slide the point along the bottom • Measuring everything • Tick marks • Cut out shapes • Text previous classes for answers

Expected student responses • Similar triangles • Just some triangles • Trapezoid, pentagon, quadrilaterals • Sliding the folding point • Changes in size, perimeter, area.

• There is no spoon

Student Difficulties • Diverse levels of understanding or relating to geometry • Wanting to know exactly what to do from the teacher only • Not used to working with others in groups

Students end discussion • Describe what they did • Why did they do it that way • Have students comment on other strategies • Would they use different strategies and why • Why are some strategies easier/better than others

McAllen e-PCMI • Meiling Dang • Steve Ferguson • Raul Hinojosa • Louis Shoe • Armando Soto • Dai Tolkov