What makes a quality math curriculum? Our school district has been fortunate enough to be selected to pilot the Desmos Curriculum, based on the work from Illustrative Mathematics and Open Up Resources. In preparation for the upcoming school year, I have spent some time going through the lessons. When it comes to curriculum, NCTM (2014) states that “An excellent mathematics program includes a curriculum that develops important mathematics along coherent learning progressions” (Principles to Action, p. 70). Coherent learning progressions value connecting content in meaningful ways across grade levels, units, and individual lessons. The linking of mathematical ideas is one of the many aspects of the Desmos Curriculum that stands out.
Connecting Mathematical Ideas
In middle school mathematics, developing an understanding of linear relationships is a focal point for student learning. In particular, connecting the ideas of similar triangles, proportionality, unit rate, and slope serve as a foundation for all things linear (Boaler, Munson, Williams, 2020). Water Slide, a previous activity that Desmos has redone, is one of my favorite lessons that I have worked through thus far. The experience weaves similar triangles, proportional relationships, and unit rates to build a conceptual understanding of slope.
The lesson begins with students altering triangles to create a “smooth water slide.” What I love about the start of this lesson is the visual relationship between the triangles and slide void of any mathematical language—a welcoming entry point to making sense of slope. The premise is simple, adjust the triangles to make the slide work. Students are successful when they can create triangles the satisfy the conditions of making a smooth slide. Feedback is visual and engaging and gives purpose to creating similar triangles.
Bumpy Versus Smooth Slide
The lesson continues by layering numerical values to illustrate the proportional relationships among triangles and the shared unit rate. Students enter values and see how they impact their slide, only when they create triangles with the same unit rate will their slide work. Desmos offers little direct instruction for students but instead guides them to discover the relationships among the triangles and numbers as mathematicians.
Exploring Proportional Relationships
The learning progression’s final piece has students practice computing slope to foster procedural fluency from the conceptual understanding the lesson developed in the initial screens. Even at this point in the lesson, Desmos defines slope only as steepness. It does not try to bog students down with formulas or complex academic language. Students build their understanding first and create a personal working definition that supports their mathematical identity. They can then apply this definition to the practice problems that follow the water slide experience. As students reflect on the learning experience and apply their new knowledge, they can see how all the mathematical ideas used in this lesson: similar triangles, proportional relationships, unit rate, and slope connect as they continue to explore the story of linear relationships. A well-constructed set of mathematical connections, such as those in this lesson, is an indication of a quality curriculum.
Practicing Calculating Slope
On a final note, I lead a group of teachers through this lesson as it is essential for teachers to experience lessons as students. At one point in the lesson, students can create a slide with a slope they choose. It was interesting that all the teachers tried to build the most harrowing slide possible. I don’t know what this necessarily says about our math teachers, maybe they have been around middle schoolers too long. However, if the lesson was able to spark the curiosity of teachers who already know the story of slope, our students will be captivated.
Teacher Created Slides
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA :NCTM, National Council of Teachers of Mathematics.
Boaler, J., Munson, J., & Williams, C. (2020). Mindset mathematics: Visualizing and investing big ideas, grade 8. Hoboken, NJ: Jossey-Bass.