In this book, we focus on a set of big ideas that extend across the eighth-grade curriculum, bringing in a greater focus on geometrical thinking. Geometry has been a neglected part of the eighth-grade curriculum for some time.
Ginsberg, Cooke, Leinwand, Noell, and Pollock (2005) investigated US students’ geometrical experiences, looking at the international tests TIMSS and PISA, and found that US students spend 50% less time on geometry than students in other countries. Not surprisingly given this lack of attention, students’ achievement in these areas was also significantly lower than students in other countries (Driscoll, DiMatteo, Nikula, & Egan, 2007).
Many teachers and students associate geometry with rules, remembering their high school years reproducing two-column proofs. This is the unfortunate outcome of a misguided approach to mathematics, when important ideas are lost as mathematical thinking is reduced to a set of rules. What is more critical to geometry is reasoning and adaptability. In this big idea, we introduce the ideas of congruence and similarity. Rather than just learning definitions for these, students look at cases and consider deeply the question, How do we know if two shapes are congruent or similar? Definitions play a part, but the most important act is reasoning; students should be encouraged to consider such questions as, What do we know now about this shape? What else do we need to know? Can I move or adapt my shape to give me more information? Can I convince someone else that my shapes are similar or congruent? What would I use to convince them? A great starting discussion for this sequence of lessons would be the question, What does it mean to be the same? Transformational geometry, congruence, and similarity are key ideas. We have chosen to focus our attention on triangles, the building blocks of geometric shapes and the coordinate plane, an important visual space for algebra.
In the Visualize activity, students are asked to consider the question, How do we know when two figures are the same? We ask students to study triangles where their vertices are provided. As students plot the points and connect the vertices with segments, they are asked to determine which triangles are congruent. We have created triangles that are congruent but may not appear so because they have been rotated and flipped. Others combine to make a triangle. Students can hone their detective skills by investigating each set. Students explore the key ideas visually.
In the Play activity, students are asked to transform shapes, rotating and reflecting them. We think that students will enjoy working out how one shape turns into another, developing patterns that explain the transformations. This is the work of computer animation, which has been important to the cartoon filmmaking industry for many years. Students will be given the opportunity to create their own puzzle transformations, which they can share with each other.
The Investigate activity provides students the experience of continuous transformations that are repeated over and over again. Students will be invited to design their own shape and think about what happens when they repeat the same transformation on the shape. In doing so, they will become pattern creators, which we hope they will find exciting. The work will help them understand what happens when transformations happen continuously, and the patterns that can result.
