Pre-calculus 11 Pacing Guide - This pacing guide replaces the previous yearly plan. It has been updated to reflect removed outcomes and provide flexibility for responsive instruction.
Pre-calculus 11 Desmos Activity Collection - A collection of online student Desmos activities organized by unit.
EAL Support - Desmos offers a free suite of math software tools, including the Desmos Graphing Calculator and Scientific Calculator, as well as free digital classroom activities. Click on the globe in the tool bar to access the site in other languages.
RF03 Students will be expected to analyze quadratic functions of the form y = a(x – p)² + q and determine the vertex, domain and range, direction of opening, axis of symmetry, x-intercept, and y-intercept.
(This outcome will focus on quadratic functions written in the vertex form, f(x) = a(x - h)² + k . Note that some of the performance indicators for this outcome are in italics. This is because they were addressed in Mathematics 11, and it is the intent of this course to extend and deepen student understanding of these performance indicators.)
RF03.01 Explain why a function given in the form y = a(x – p)² + q is a quadratic function.
RF03.02 Compare the graphs of a set of functions of the form y = ax² to the graph of y = x² , and generalize, using inductive reasoning, a rule about the effect of a.
RF03.03 Compare the graphs of a set of functions of the form y = x² + q to the graph of y = x², and generalize, using inductive reasoning, a rule about the effect of q.
RF03.04 Compare the graphs of a set of functions of the form y = (x – p)² to the graph of y = x² , and generalize, using inductive reasoning, a rule about the effect of p.
RF03.05 Determine the coordinates of the vertex for a quadratic function of the form, y = a(x - p)² + q and verify with or without technology.
RF03.06 Generalize, using inductive reasoning, a rule for determining the coordinates of the vertex for quadratic functions of the form y = a(x - p)² + q.
RF03.07 Sketch the graph of y = a(x – p)² + q, using transformations, and identify the vertex, domain and range, direction of opening, axis of symmetry, and x- and y-intercepts.
RF03.08 Explain, using examples, how the values of a and q may be used to determine whether a quadratic function has zero, one, or two x-intercepts.
RF03.09 Write a quadratic function in the form y = a(x – p)² + q for a given graph or a set of characteristics of a graph.
RF04 Students will be expected to analyze quadratic functions of the form y = ax² + bx + c to identify characteristics of the corresponding graph, including vertex, domain and range, direction of opening, axis of symmetry, x-intercept and y-intercept, and to solve problems.
RF04.01 Explain the reasoning for the process of completing the square as shown in a given example.
RF04.02 Write a quadratic function given in the form y = ax² + bx + c as a quadratic function in the form y = a(x – p)² + q by completing the square.
RF04.03 Identify, explain, and correct errors in an example of completing the square.
RF04.04 Determine the characteristics of a quadratic function given in the form y = ax² + bx + c, and explain the strategy used.
RF04.05 Sketch the graph of a quadratic function given in the form y = ax² + bx + c.
RF04.06 Verify, with or without technology, that a quadratic function in the form y = ax² + bx + c represents the same function as a given quadratic function in the form y = a(x – p)² + q.
RF04.07 Write a quadratic function that models a given situation, and explain any assumptions made.
RF04.08 Solve a problem, with or without technology, by analyzing a quadratic function.
RF05 Students will be expected to solve problems that involve quadratic equations.
RF05.01 Explain, using examples, the relationship among the roots of a quadratic equation, the zeros of the corresponding quadratic function, and the x-intercepts of the graph of the quadratic function.
RF05.02 Derive the quadratic formula, using deductive reasoning.
RF05.03 Solve a quadratic equation of the form ax² + bx + c = 0 by using strategies such as
determining square roots
factoring
completing the square applying the quadratic formula
graphing its corresponding function
RF05.04 Select a method for solving a quadratic equation, justify the choice, and verify the solution.
RF05.05 Explain, using examples, how the discriminant may be used to determine whether a quadratic equation has two, one, or no real roots, and relate the number of zeros to the graph of the corresponding quadratic function.
RF05.06 Identify and correct errors in a solution to a quadratic equation.
RF05.07 Solve a problem by
analyzing a quadratic equation
determining and analyzing a quadratic equation
RF01 Students will be expected to factor polynomial expressions of the following form where a, b, and c are rational numbers.
ax² + bx + c, a ≠ 0
a² x² – b² y², a z 0, b ≠ 0
a[f(x)]² + b[f(x)] + c, a ≠ 0
a² [f(x)]² – b² [g(y)]² , a ≠ 0, b ≠ 0
RF01.01 Factor a given polynomial expression that requires the identification of common factors.
RF01.02 Determine whether a given binomial is a factor for a given polynomial expression, and explain why or why not.
RF01.03 Factor a given polynomial expression of the form
ax² + bx + c, a ≠ 0
a² x² – b² y² , a ≠ 0, b ≠ 0
RF01.04 Factor a given polynomial expression that has a quadratic pattern, including
a[f(x)]² + b[f(x)] + c, a ≠ 0
a² [f(x)]² – b² [g(y)]² , a ≠ 0, b ≠ 0
Additional Resources and Activities for RF03 and RF04 (analyze quadratic functions, completing the square ):
Desmos Polygraph: Parabolas - This is a great online activity for a class to try out. One student selects a parabola. Another student tries to guess which parabola was chosen by asking yes/no questions. Student practice mathematical communication in this activity.
Parabola Menu Task - Nat uses a problem structure called a menu. Students are asked to satisfy ten specifications (in whatever combinations they desire) by using as few functions as possible. Students have to determine which specifications go well together and which don't. The complexity increases as students attempt to use fewer and fewer functions. Jay Chow has turned this menu task into an online Desmos Activity.
Which One Doesn’t Belong : Quadratics - The WODB number routine encourages mathematical thinking, reasoning and promotes discourse in the classroom that includes all students. This Google slides file includes a selection of WODB images focused on quadratics (from Which One Doesn't Belong) .
Desmos Activity: Completing the Square Introduction - An introduction to completing the square. Starting with virtual algebra tiles (using a Polypad screen) and moving to symbolic form.
Completing the Square in Mathigon Polypad - Polypad is a virtual manipulative that you can use to demonstrate completing the square. You can even split algebra tiles to work with fractions (you can't do that with physical tiles!). Here is a Desmos activity using this feature.
Completing the Square using the "Box" Method - Using the "box" method for teaching multiplication of binomials can be extended to completing the square and may reach more visual learners.
Completing the Square Fill in the Blanks and Harder Fill in the Blanks (a ≠ 1) - Students fill in a table with columns for different features of each expression to practice completing the square. From Dr. Austin Maths. Solutions for Fill in the Blanks and Solutions for Harder Fill in the Blanks.
Forming Quadratics Lesson from the Mathematics Assessment Project - This lesson unit is intended to help you assess how well students are able to understand what the different algebraic forms of a quadratic function reveal about the properties of its graphical representation. In particular, the lesson will help you identify and help students who have the following difficulties: Understanding how the factored form of the function can identify a graph’s roots; Understanding how the completed square form of the function can identify a graph’s maximum or minimum point; Understanding how the standard form of the function can identify a graph’s intercept.
Additional Resources and Activities for RF05 (solve quadratic equations):
Super Mario Quadratics Desmos Activity - In this activity, students will construct various quadratics to collect coins and stars in a series of Super Mario levels. Students begin by modifying quadratics in Vertex form and Factored form before constructing their own quadratics to move through progressively more difficult levels.
Solving Quadratic Equations Problems - Several geometry problems using quadratic equations from Dr. Austin Maths. Solutions.
Completing the Squared - Analyzing worked examples - These google slides contain worked examples for students to analyze and generalize from. The final slide shows a series of questions with fading out solution steps to move from fully guided to problem solving. You might also solve these equations using Graspable Math. Here is an example of Graspable Math in action.
M&M Catapult Project - Students use experimental data from firing M&M's from a small catapult on the floor to calculate the equation of a quadratic. They then place the catapult on a desk and have to use the equation to predict where the M&M will land.
Two Squares are Equal - This classroom task is meant to elicit a variety of different methods of solving a quadratic equation. Some are straightforward; some are simple but clever; some use tools (using a graphing calculator). Some solution methods will work on an arbitrary quadratic equation, while others may have difficulty or fail if the quadratic equation is not given in a particular form, or if the solutions are not rational numbers.
Additional Resources and Activities for RF01 (factor polynomial expressions ):
Activities from Math 10 AN05 could be used for students to review factoring or for teachers to assess prior knowledge.
Factoring Quadratics Tarsia Puzzle - Students cut out and then put together 16 pieces of a puzzle to form a triangle. Each side of the puzzle should match a trinomial to its factored form. These are fairly simple trinomials to factor and a good review of Math 10. From Mrs. Dekker. Tarsia files can be created with the free Tarsia Formulator program.
Venn Diagram Activity for Quadratics (and linear functions) and teachers notes - To be able to fill in an eight-region diagram shows a really deep understanding of the distinctions that a good student has to be able to make, while the weaker student finds being faced with a picture such as this much less threatening than a page of exercises. From Rich Starting Points for A Level Maths.
Factoring Quadratics from Open Middle - Fill in the empty boxes (a, b and c in ax^2 +bx +c=0) with whole numbers 0 through 9, using each number at most once, so that the solutions are integers. How many different possibilities are there? How many give you 2 integer solutions, how many give you 1 integer solution, how many give you non-integer solutions (like rationals), and how many give you no solutions?
Unit 2 Cumulative Review
Quadratic Equations Circuit Training - Students complete a sequence of 12 quadratics problems on a self-checking worksheet. The answer to each problem leads to the next problem until student have completed the circuit and found their way back to question #1.