Of all the courses we've developed, Algebra 1 is the one we've spent the most time planning, debating, and brainstorming. Why? Because it sets the stage for all of high school math. Both in terms of content and how a student views themselves as a math learner. For many people, Algebra 1 was the course that caused them to develop a distaste for math. Something that used to be concrete became abstract and elusive. We want to change that! We designed the first three units of our Algebra 1 curriculum to give students opportunities for sense making and time to develop reasoning skills.
We start our year in Algebra 1 by identifying, describing, representing, and generalizing patterns. The goal is to give students multiple opportunities to describe relationships between variables and express how change is occurring. This process is made concrete through the use of various visual pattern tasks as students first ““see”” the change and then describe the change with words, colors, tables, or algebraic expressions. Students will also work on noticing structure that allows them to generalize a rule for a pattern. This involves communicating about ideas at a level where the focus is no longer on a particular instance but rather on patterns and relationships between particular instances that allow for the development of a generalized case. Students will begin to use generalizing language such as always, every time, the pattern is, the rule is, any number, for all numbers, etc. An emphasis is placed on the Standards for Mathematical Practice, specifically Practices 7 and 8, where students are asked to look for and make use of structure and use repeated reasoning.
There are two types of sequences we look at in particular: arithmetic sequences and geometric sequences. This is to provide the foundation for our upcoming units on linear relationships (Unit 2) and exponential functions (Unit 8). While we do not expect students to write explicit formulas for any given arithmetic or geometric sequence, we want to emphasize patterns of growth that rely on repeated addition and repeated multiplication. We introduce the vocabulary words of term, common difference, and common ratio.
Big ideas: Identifying, describing, and representing patterns in multiple representations; problem-solving
Our second unit is all about linear relationships. While students have some experience with lines and linear equations from middle school, our goal is to deepen students’ understanding of linear growth and specifically describing situations that model linear growth. Students will interpret and create graphical, verbal, numerical, and algebraic representations of linear relationships and continually make connections between those representations.
We begin the unit with a look at proportional relationships, primarily to introduce tools and visual models that will help develop students’ reasoning skills that do not rely on memorizing rote procedures. In the second lesson, we graph proportional relationships as a way to review the coordinate plane and basic graphing skills. We want students to understand that ordered pairs do not just denote distinct data points but that they also represent solutions to an equation whose graph can be described as a collection of points.
Throughout the rest of the unit we work to build strong conceptual understanding around the rate of change and the values of a linear relationship, and link these to graphical features such as the slope of a line and the x- and y-intercepts. Most of all, we want students to be flexible with their reasoning, being able to use any features of the linear graph or equation to determine any other feature.
Big idea: Creating and connecting algebraic, tabular, graphical, and verbal representations of linear relationships.
This unit builds upon Unit 2 with a focus on solving linear equations and inequalities–one of the most important skills students need to master in an Algebra 1 course. Instead of dividing these lessons by ““type”” of equation (one-step, two-step, multi-step), we want to introduce students to a variety of strategies that can be used to solve equations, built upon strong conceptual understanding of what an equation is and what a solution to an equation represents. We will use visual models like double number lines and bar models as well as mental models like working backwards, balancing, or isolating the impact in order to build fluency and sense making around solving equations. Finally, we specifically choose contexts that encourage students to use intuition and logic to solve for the unknown quantity. The goal is not to teach students a procedure but to help students collect a variety of flexible strategies that can be applied to many different problems. Our goal in providing many different strategies is not just that each student will be able to choose one strategy that works for them, but that students would begin to make decisions about which strategy works best in any given scenario, based on what information is presented in the problem.
In the second half of the unit we turn to representing situations with inequalities and solving inequalities. In general, our approach to solving inequalities will be to find the boundary point (i.e. solving the related equation) and then reasoning about the direction of the inequality. This can be done by testing values on either side of the solution to the related equation, using the context of the problem to make sense of the relationship between the quantities, or analyzing the structure of the inequality more abstractly using relational reasoning and visual models like a number line.
Big ideas: Understanding equivalence, relational reasoning
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