End of Course Exam

End of Course testing is just around the corner! And all thoughts assessment can be found in the 'Tis the Season post. This Algebra 1 bundle includes general review options of sample questions organized by the Arkansas Frameworks strands.

Happy Testing to YOU!

Solving Quadratic Equations

It's critical for students to see the need for solving quadratic equations. The bridge from graphing quadratic functions to solving quadratic equations naturally builds from the application of projectile motion. If students consider the key features within the context of projectile motion, they will quickly see the need for solving a quadratic equation to find the x-intercept(s) and the need to find the vertex of a quadratic function written in standard form. Encourage students to become fluent with all solving methods and leverage the power of which method is most effective for different scenarios.

The following unit progression stems from comparing vertex form and standard form in the context of projectile motion. Students quickly decide vertex form is their form of preference, but reality hits hard when most application settings model standard form equations. This series of activities begins with an adventure of modeling standard form equations as a square (vertex form) or rectangle (intercept form) with algebra tiles. Students discover the methods of completing the square and factoring but find more benefit in completing the square. However, they grow tired of the ugly fractions that ensue and beg for a generalized version (aka the quadratic formula) to reduce the mess. And this also expands options for graphing a quadratic function in standard form. Visit College Preparatory Mathematics for a Water Balloon Contest task (see pages 14 and 17) to assess understanding of quadratics.

HSA-REI.B.4 Resources

These activities highlight Common Core State Standard HSA-REI.B.4 included in 7th and 8th Accelerated Algebra 1.

Quadratic Functions Continued...

The previous post on Introducing Quadratic Functions started our journey with quadratic functions through solving puzzles, building the definition of parabola, and a narrative illustrated with transformed functions. The next step is to build fluency with the properties of quadratic functions represented in different ways (numerically, graphically, algebraically, and verbally). Use a table of values to determine the behavior of the parent function (aka Norm). Extend previous learning on transformations to make connections between a quadratic function graphically and algebraically in vertex form. This stations activity uses the RallyCoach structure to complete a series of transformations and construct a function model of the resulting graph. Students work with their shoulder partner and rotate through a total of 8 different stations. Each station requires one coordinate geoboard per pair and a station task card. (Note: We only used pegs because I have an aversion to rubberbands in the classroom!) The stations activity with record sheet and two practice sets are included. Be sure to emphasize intercepts and maxima/minima.

HSF-IF.C.7a Resources

If you have limited students to interpreting key features of linear and exponential functions, quadratic functions will energize the analysis routine. Intervals where the function is increasing, decreasing, positive, or negative requires detailed inspection with quadratics. For now, while students have limited solving skills, let's restrict our graphs to displays where the vertex and intercepts can be easily labeled. One key feature that may not have been addressed in prior units is symmetry; specifically, using the concept of symmetry to classify functions as even or odd. In Algebra 1, even/odd/neither classification is solely by inspection of the graph. Algebraic proof and function notation to generate evidence is reserved for Algebra 2. Another key feature included in the quadratics unit is end behavior. Although previously discussed, be sure to formalize the notation to communicate end behavior mathematically. This deck of cards can be used in a sorting activity to build the concept of even functions and odd functions or in a class building activity to practice function classification and/or end behavior.

HSF-IF.B.4 Resources

This deck of cards can be used in a class building activity to practice function notation in terms of a context that models quadratic functions. And the question set can be used in an All Write RoundRobin structure to analyze quadratic graphs. The sequence of questions prompt a great discussion of refining function models to represent a restricted domain within the context of the situation. Note: This document has been adapted from LTF to include more emphasis on key features.

HSF-IF.A.2 and HSF-IF.B.5 Resources

These team activities offer practice with average rate of change. The "Find the Function" activity reviews graphing and prompts the discussion of finding the slope of a parabola (and also the standard form for later!). Revisit the Peg Puzzle to discuss using the second differences to reveal that the generated data is quadratic. What are second differences? You're finding the slope of the curve...or the average rate of change over a specified interval...introductory calculus at its finest! A brief activity using the Showdown structure is included for students who need to refine their descriptions of average rate of change. An activity using the Simultaneous RoundTable structure is included for students to practice calculating and interpreting the average rate of change of quadratic functions.

HSF-IF.B.6 Resources

These activities highlight Common Core State Standards HSF-IF.A.2, HSF-IF.B.4, HSF-IF.B.5, HSF-IF.B.6, and HSF-IF.C.7a included in 7th and 8th Accelerated Algebra 1.

Introducing Quadratic Functions

Aside from all things trigonometry, quadratic functions are my favorite. I instantly feel old when I stop long enough to ponder the number of lessons in my classroom that have involved quadratic functions. But I'm always open to new ideas, so I hope this Genius Hour series will prompt robust discussion.

Let's start with a broad overview. Now...there are several schools of thought on building the concept of quadratic functions. You will spy an overarching theme in the quadratic functions progression below...a deep understanding of the behavior of quadratic functions which enables students to make connections within any quadratic function model.

Quadratic Functions At-A-Glance:

• Graphing Quadratic Functions in Vertex Form
• Identifying Key Characteristics from Vertex Form
• Applications in Vertex Form (includes connections to key features)
• Generating Standard Form from Vertex Form (includes operations of polynomials)
• The Pros and Cons of Vertex Form and Standard Form (includes solving by square roots)
• Rewriting Standard Form in Vertex Form and Intercept Form (includes solving by completing the square and factoring)
• Generalizing the Solving Process...Deriving the Quadratic Formula (includes finding the vertex from standard form, nature of the roots via the discriminant, and imaginary solutions)
• Applications in Standard Form (includes connections to key features)

I'm not just partial to quadratics. When I teach any function, my ongoing thread is transformations of the parent function. And since the graph provides a great visual to aid in the discovery of key features necessary for application, let's start with graphing quadratic functions. Browse the Puzzle Challenge, Geometric Discovery, Mr. Norm Parabola Narrative (or an alternate Parabolic Graphing Transformations Lab Activity), and a "Go Fish!" Quadratics Matching Game and coordinating Find the Vertex Practice Set to jumpstart your planning for building the concept of quadratic functions.

Introduction to Quadratic Functions

This introduction set highlights Common Core State Standards HSF-IF.B.4 and HSF-IF.C.7a included in 7th and 8th Accelerated Algebra 1.

Arithmetic and Geometric Sequences

I remember the details like it was yesterday...three years ago...on a dusty bus filled with excited tweens...traveling four hours to a state math competition...and I filled the Saturday hours with studying the Algebra 1 Common Core State Standards. Yep. It's no secret. I'm a math nerd. But clearly, it's never too late to learn something new. Confession: I had never made the connection between sequences and functions. And Monday's algebra class couldn't come soon enough!

I remember learning sequences via formulas involving a, n, d, or r. It was like a giant puzzle with my efforts to find all the necessary values to substitute into the formula. You nailed it...process! Unfortunately, I have taught plenty of sequence lessons in precalculus in this same manner. I love that CCSS place the connection at the beginning of the progression when students first learn linear and exponential functions. The vast majority of my students view a table of values and naturally spy the sequence, rather than spying the relationship between both variables (function!). So it only seems natural to use the sequence to build the function. Enjoy the beauty of building this concept!

Arithmetic and Geometric Sequences Resources

This concept begins in MATH-8 with constructing linear functions and continues in Algebra 1 when students extend previous learning to compare linear and exponential functions.

These activities highlight Common Core State Standards HSF-BF.A.1a, HSF-BF.A.2, HSF-IF.A.3, HSF-LE.A.1, HSF-LE.A.2, and HSF-LE.A.3 included in 7th and 8th Accelerated Algebra 1.