## Phase II: The Real Numbers

My 9th grade Regular Algebra I classes will be the focus of the year-long ImagineIT project. I've taught this class for five years and am thrilled to approach it from STEM viewpoints. A total of 54 students populate the two sections of algebra; one section is a co-taught class, which means there will be a group of diverse learners to consider. (At the time of this writing, I don't know how many diverse learners or ELL students I have.) The class meets daily for 51 minutes, except for Fridays, when we meet for 44 minutes. I find that of all of the students that I teach, freshmen are most malleable; my best chance of getting buy-in with new ideas lies with this class.

I will attempt successfully to connect each of the topics I teach to the Real Number Line (RNL). The motivation for this project comes from frustration with student understanding of numbers - their number sense - which makes for a challenge with basic computation and when we dive into more abstract topics in algebra.

We typically cover the following topics in algebra:

(1) Properties/Order of Operations

(2) Operations with Rational Numbers

(3) Solving Equations

(4) Graphing Equations (including Functions & Domain and Range)

(5) Slope and Slope-Intercept Form

(6) Inequalities

(7) Systems of Equations

(8) Polynomials

(9) Factoring Quadratic Expressions

(10) Quadratic Equations

As often as appropriate, I plan to connect each of these topics to the RNL. This running theme will become familiar - and, I hope, reassuring- to students from the first week of school. Furthermore, an emphasis on the RNL will deepen student understanding of numbers and lay the groundwork for abstract thinking. Some topics, such as Operations with Rational Numbers, Graphing Equations, Identifying Domain and Range, and Inequalities, connect to the RNL more easily than others; Polynomials, on the other hand, may be difficult to connect, but I look forward to thinking creatively about how best to do this. Some of my favorite activities (click on the Crossing the River and Two Balloons buttons) feature the RNL as it relates to graphing lines. The most interesting discussion each year digs into the ideas of 0, undefined and indeterminate: what they mean, how they appear in equations, and why they matter. Students debate the similarities and differences with ideas in an attempt to make sense and use of them.

I will attempt successfully to connect each of the topics I teach to the Real Number Line (RNL). The motivation for this project comes from frustration with student understanding of numbers - their number sense - which makes for a challenge with basic computation and when we dive into more abstract topics in algebra.

We typically cover the following topics in algebra:

(1) Properties/Order of Operations

(2) Operations with Rational Numbers

(3) Solving Equations

(4) Graphing Equations (including Functions & Domain and Range)

(5) Slope and Slope-Intercept Form

(6) Inequalities

(7) Systems of Equations

(8) Polynomials

(9) Factoring Quadratic Expressions

(10) Quadratic Equations

As often as appropriate, I plan to connect each of these topics to the RNL. This running theme will become familiar - and, I hope, reassuring- to students from the first week of school. Furthermore, an emphasis on the RNL will deepen student understanding of numbers and lay the groundwork for abstract thinking. Some topics, such as Operations with Rational Numbers, Graphing Equations, Identifying Domain and Range, and Inequalities, connect to the RNL more easily than others; Polynomials, on the other hand, may be difficult to connect, but I look forward to thinking creatively about how best to do this. Some of my favorite activities (click on the Crossing the River and Two Balloons buttons) feature the RNL as it relates to graphing lines. The most interesting discussion each year digs into the ideas of 0, undefined and indeterminate: what they mean, how they appear in equations, and why they matter. Students debate the similarities and differences with ideas in an attempt to make sense and use of them.

Each of these activities lends itself to some sort of theatre that I have not tried in the past but may implement this year (perhaps the 'A' in STEAM). For instance, students can act out Crossing the River before using manipulatives to solve the problem. With Two Balloons, they can continue the graphs on chart paper to ensure that they understand - from a graphical and algebraic perspective - the intersection of the two lines. A gallery walk with 'curators' will help to solidify their understanding. Finally, students can have a more formalized debate on the differences among 0, undefined and indeterminate. A jury can 'decide' whether in fact these ideas are actually different. The jury's decision will inform whether in fact students grasp these important differences. These performances of understanding will supplement, or in some cases replace, traditional quizzes and tests.

I look to make more allowances for students to wrestle with the above ideas, and to tie all topics back to the RNL, in order to deepen their understanding of pre-algebra and algebra. Creating time and space for students to explore more deeply ideas that matter - that is, some of those listed above - means less lecture from me, but, I hope, an enriching experience for all of us. I aim to act as the facilitator or math coach, to push kids to gain a better understanding of these topics through experiment and discussion. I plan to use websites like Desmos, a graphing site introduced to me this summer, to flesh out ideas, such as slope and systems of equations, that are challenging to the kids. As we don't have a reliable computer lab and we don't lend students tablets, I may have to resort to allowing students to use their smartphones. My hesitation with this is that students can become quickly distracted with their phones and monitoring their attentiveness could be difficult. Perhaps this risk is worth the reward of using technology to make sense of tough material.