HRCE Program Department

Pre-calculus 11

Note: Outcomes T01 and T02 from the Pre-calculus 12 Curriculum have been moved to the Pre-calculus 11 Curriculum as T03 and T04.

Pre-calculus 11 Curriculum Document
Pre-calculus 12 Curriculum Document - Please reference pg 35 - 47 for additional details around outcomes T03 and T04 (formerly T01 and T02 in Pre-calculus 12).
Pre-calculus 11 Outcomes At-a-Glance
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.

T01 Students will be expected to demonstrate an understanding of angles in standard position (0° to 360°).

T01.01 Sketch an angle in standard position, given the measure of the angle.
T01.02 Determine the reference angle for an angle in standard position.
T01.03 Explain, using examples, how to determine the angles from 0° to 360° that have the same reference angle as a given angle.
T01.04 Illustrate, using examples, that any angle from 90° to 360° is the reflection in the x-axis and/or the y-axis of its reference angle.
T01.05 Determine the quadrant in which a given angle in standard position terminates.
T01.06 Draw an angle in standard position given any point P (x, y) on the terminal arm of the angle.
T01.07 Illustrate, using examples, that the points P (x, y), P (−x, y), P (−x, −y), and P (x, −y) are points on the terminal sides of angles in standard position that have the same reference angle.

T02 Students will be expected to solve problems, using the three primary trigonometric ratios for angles from 0° to 360° in standard position.

T02.01 Determine, using the Pythagorean theorem or the distance formula, the distance from the origin to a point P (x, y) on the terminal arm of an angle.
T02.02 Determine the value of sinθ , cosθ , or tanθ , given any point P (x, y) on the terminal arm of angle T .
T02.03 Determine, without the use of technology, the value of sinθ , cosθ , or tanθ , given any point P (x, y) on the terminal arm of angle θ , where θ = 0°, 90°, 180°, 270°, or 360°.
T02.04 Determine the sign of a given trigonometric ratio for a given angle, without the use of technology, and explain.
T02.05 Solve, for all values of θ , an equation of the form sinθ = a or cosθ = a, where −1 ≤ a ≤ 1, and an equation of the form tanθ = a, where a is a real number.
T02.06 Determine the exact value of the sine, cosine, or tangent of a given angle with a reference angle of 30°, 45°, or 60°.
T02.07 Describe patterns in and among the values of the sine, cosine, and tangent ratios for angles from 0° to 360°.
T02.08 Sketch a diagram to represent a problem.
T02.09 Solve a contextual problem, using trigonometric ratios.

T03 Students will be expected to demonstrate an understanding of angles in standard position, expressed in degrees and radians. [CN, ME, R, V]

T03.01 Sketch, in standard position, an angle (positive or negative) when the measure is given in degrees.
T03.02 Describe the relationship among different systems of angle measurement, with emphasis on radians and degrees.
T03.03 Sketch, in standard position, an angle with a measure of one radian.
T03.04 Sketch, in standard position, an angle with a measure expressed in the form k radians, where k Q .
T03.05 Express the measure of an angle in radians (exact value or decimal approximation), given its measure in degrees.
T03.06 Express the measure of an angle in degrees, given its measure in radians (exact value or decimal approximation).
T03.07 Determine the measures, in degrees or radians, of all angles in a given domain that are coterminal with a given angle in standard position.
T03.08 Determine the general form of the measures, in degrees or radians, of all angles that are coterminal with a given angle in standard position.
T03.09 Explain the relationship between the radian measure of an angle in standard position and the length of the arc cut on a circle of radius r, and solve problems based upon that relationship.

T04 Students will be expected to develop and apply the equation of the unit circle. [CN, R, V]

T04.01 Derive the equation of the unit circle from the Pythagorean theorem.
T04.02 Describe the six trigonometric ratios, using a point P (x, y) that is the intersection of the terminal arm of an angle and the unit circle.
T04.03 Generalize the equation of a circle with centre (0, 0) and radius r.

Additional Resources and Activities for T01 and T02 (angles in standard position and trig ratios):

Rock, Paper, Triggers - Each person secretly picks a trig function (Sine, Cosine or Tangent) for themselves, and an angle to send to the other person. Then, once ready, both reveal and each person thinks about… TheirFunction(AngleSentToThem). Whoever’s value is higher wins. No need for exact values, just figure out which one is bigger (and DNE automatically loses). This is really good for number sense (no calculators), for thinking about what values of the different functions are possible, and where those values are on the unit circle.

Unit Circle Birthdays - Ask students to think of the year as a circle (with 365 days instead of 360 degrees). Have students determine where on this circle year their birthday would be (a Julian date calendar may help with this). If there birthday was on the 246th day of the year then their birthday would be at about 246 degrees. Ask students to then estimate this spot as a radian in decimals and to the nearest 12th of pi.

Additional Resources and Activities for T03 and T04 (radians and the unit circle *These outcomes were previously in Precal12):

Visualizing Radians in GeoGebra - An interactive applet to demonstrate the distance of a radian around the circumference of a circle.
Measuring Angles in Radians - A short worksheet for students to measure angles in radians based.
Radians Polygraph Desmos Activity - 16 radian measurements from -2π to 2π in increments of π/4.
Pizza Slices Desmos Activity - Have students guess how many slices of a pizza make a whole and explore the idea behind degree measurement and radian measurement.
Which One Doesn’t Belong: Sine Functions (number 46) - 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 powers and exponents.
Radians and Degrees War - Students mentally compare angle measurements in radians and degrees using the format of the classic card game of War and a special set of cards.

A Prelude to Unit Circle Trigonometry - Sam describes an activity to prepare students to understand the unit circle. Instead of starting with a circle, he primes students with thinking about x and y coordinates for other shapes. In this case he used a square, a rotated square and a triangle. Some very interesting graphs result.

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