Pre-Calculus 12 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 12 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 calculator in other languages.
RF11 - To find the roots of a polynomial, Pre-Calculus 12 (McAskill et al. 2012) uses the integral zero theorem to relate the factors of a polynomial and the constant term of the polynomial. The integral zero theorem is a special case of the rational root theorem where the polynomial has a leading coefficient of 1. Pre-Calculus 12 (McAskill et al. 2012) does not mention the rational root theorem. The resource will need to be supplemented in this area.
RF12 - Students should also be able to identify when the function is positive, f (x) > 0, and when it is negative, f (x) > 0.
Note that outcome RF08 (Students will be expected to solve problems that involve quadratic inequalities in one variable) has been removed from Pre-calculus 11 Curriculum, so students will not have prior knowledge to access when completing these problems. Please see pages 205 - 213 in the Pre-calculus 11 Curriculum Document for additional detail).
RF12 - Students are not expected to use transformations to graph polynomial functions.
RF11 Students will be expected to demonstrate an understanding of factoring polynomials of degree greater than 2 (limited to polynomials of degree ≤ 5 with integral coefficients). [C, CN, ME]
RF11.01 Explain how long division of a polynomial expression by a binomial expression of the form x - a, a ∈ Z is related to synthetic division.
RF11.02 Divide a polynomial expression by a binomial expression of the form x - a, a ∈ Z, using long division or synthetic division.
RF11.03 Explain the relationship between the linear factors of a polynomial expression and the zeros of the corresponding polynomial function.
RF11.04 Explain the relationship between the remainder when a polynomial expression is divided by x - a, a ∈ Z, and the value of the polynomial expression at x = a (remainder theorem).
RF11.05 Explain and apply the factor theorem to express a polynomial expression as a product of factors.
RF12 Students will be expected to graph and analyze polynomial functions (limited to polynomial functions of degree ≤ 5 ). [C, CN, T, V]
RF12.01 Identify the polynomial functions in a set of functions, and explain the reasoning.
RF12.02 Explain the role of the constant term and leading coefficient in the equation of a polynomial function with respect to the graph of the function.
RF12.03 Generalize rules for graphing polynomial functions of odd or even degree.
RF12.04 Explain the relationship among the zeros of a polynomial function, the roots of the corresponding polynomial equation, and the x-intercepts of the graph of the polynomial function.
RF12.05 Explain how the multiplicity of a zero of a polynomial function affects the graph.
RF12.06 Sketch, with or without technology, the graph of a polynomial function.
RF12.07 Solve a problem by modelling a given situation with a polynomial function and analyzing the graph of the function.
Additional Resources and Activities for RF11 (factoring polynomials - synthetic division, factor theorem, rational root theorem):
Polynomials WODB - A "Which One Doesn't Belong?" discussion prompt to review the characteristics of polynomials: degree, roots, reflections, intercepts, etc. Many more WODB prompts can be found at WODB.ca.
Polynomial Expressions Menu Task - This activity can be used as a review of polynomial terminology. A Math Menu is a collection of constraints that appear as an unordered list generally about 6-10 constraints long. Each menu prescribes a type of mathematical object, in this case a polynomial expression, that needs to be designed to satisfy these constraints. A variety of menus task are available on Nat Banting's Menu Math page.
Algebraic Division - Four methods for polynomial division. Created by Jo Morgan.
Dividing Polynomials Using the Box Method - How to divide polynomials using the "box method" . Created by Sarah Carter. This video from YouTube demonstrates this method: Polynomial Division with the Box Method.
Polynomial Factoring Row Game - Student A finds the roots for the polynomials in the left hand column, while Student B finds the roots for the polynomials in the right hand column. The sum of the roots in each row should be the same! If not, work together to find your mistake.
Factoring Polynomials with Mystery Coefficients - Students work to complete a table using the given information. Some coefficients are missing and students must solve for them first before factoring. Activity from Starting Points Maths.
Factor Theorem Fill in the Blanks - Students work to factor cubic polynomials by filling in the blanks in a table using the factor Theorem. Activity from Dr. Austin Maths.
Additional Resources and Activities for RF12 (graph and analyze polynomial functions):
Representing Polynomials Graphically - In this task, students work collaboratively in pairs or threes, matching functions to their graphs and creating new examples. From Mathematics Assessment Project
Graph Me If You Can activity - Students work in groups. One student has a graph attached to their back and has to ask questions of the other students about its features in order to draw it. Students take turns being the sketch artist. A great communication activity. Inspired by an activity by Julia Anker.
Graphing a Polynomial question - An open question from Marian Small's NCTM presentation, "You graph a function and there are at least three roots that are equally spaced. What might the function be?"
Sketching Polynomials Desmos Activity - Student use the key features (x-int, y-int, leading coefficient, degree) to graph polynomials up to degree 5. Based on a Desmos activity by Martin Smith.
Polynomial FUNctions – Multiplicity Roots Video - Julie talks about how students came to understand multiplicty of roots using songs and shares a short video. Created by Julie Reulbach.
Match My Polynomial Desmos Activity - In this activity, students work through a series of "match my graph" challenges for polynomial functions designed to build understanding of roots, end behavior, multiplicity, and vertical dilation.
Polynomial Functions Menu Task - A Math Menu is a collection of constraints that appear as an unordered list generally about 6-10 constraints long. Each menu prescribes a type of mathematical object, in this case a polynomial function, that needs to be designed to satisfy these constraints. A variety of menus task are available on Nat Banting's Menu Math page.
Des-man by Desmos - Des-man is an opportunity for students to use creativity to draw fun faces with math, and learn about domain and range in the process.
Unit 3 Cumulative Review
Roots and Factors of Polynomials Quizizz formative assessment - questions for students to check their understanding of factoring and graphing polynomial functions.