The Algebra I session was focused on presenting the rationale behind the lessons of Module 4, “Polynomial and Quadratic Expressions, Equations and Functions.” Participants discussed many scaffolding ideas, which reconfirmed the power of the Algebra I standards and how they lay the foundational pieces for the mathematics studied in Algebra II and Calculus (exploratory exercise 2 from lesson 8).
The first hour of the session was dedicated to the mathematician in us all, as it provided an adult experience/perspective that dealt with the foundation of quadratics. This foundation was explored in three pieces:
- How do sequences and their leading diagonals coupled with basic polynomial building blocks help generate quadratic equations?
- Gravity is why we study quadratics. If acceleration is a constant, the formula for defining height is a quadratic.
- Can any U-shaped graph be represented by a quadratic? Answer: No.
Participants engaged in an excellent activity that is not in the modules, but should be part of every mathematics classroom. Educators taped the ends of a chain together so that the lowest part of the chain was at the origin. They then identified points that the chain went through. The task was to create a quadratic equation to fit the curve. Looks can be deceiving, and not all of the chain curves ended up being quadratic. This was a challenging, thought-provoking activity for all, and a question not usually tackled until Algebra II.
For most of the morning and part of the afternoon, participants focused on Topic A, which covers the following:
- Reverse multiplication (factoring) and the connection to the geometric models (rectangles);
- The zero product property;
- Graphing a quadratic equation;
- The importance symmetry; and
- Relating quadratics and their graphs to a real-world context.
The lessons for Topic A are built off the foundation created for multiplying polynomials using the distributive property from grades 6-8. The theme throughout was to get students to stop learning passively and become engaged in the life of a quadratic. Let students determine on their own what it means to factor and how to go about doing it. Lessons 8-10 allow students to explore the symmetry that exists in the graph of a quadratic function and how this symmetry proves helpful in determining the vertex.
How does a quadratic equation written in factored form help us with the graph? Example 2 from lesson 9 was discussed. Providing the function for determining the motion of the ball, which we always do, really isn’t good enough. We need to be asking the students where the formula came from. This question pulls out language that students need to identify with such as y-intercept, maximum height, roots (x-intercepts), domain, and other words that feel natural in the context of the questioning that is going on in the classroom. How is factoring helpful here with respect to this problem? What does it enable you to find? There was never any mention of using the formula for the axis of symmetry; do the students really need it?
Overall, the lessons are fully packed mathematically. Participants raised questions about how to develop students’ fluency with factoring when moving at a quick pace through discovery and not using rules or formulas. The rapid white board exchange was presented as one method of developing fluency with factoring. The appearance of dual intensity, how the balance of procedure and fluency are both crucial pieces when working with quadratics, was seen throughout this session.