Grade 11 | Mathematics

Action steps

1. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities.

2. Sketch graphs showing the following key features: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

3. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

4. Fit a function to the data; use functions fitted to data to solve problems in the context of the data.

5. Informally assess the fit of a linear function by plotting and analyzing residuals.

6. Fit a linear function for a scatter plot that suggests a linear association.

7. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Action steps

1. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

2. Know there is a complex number i such that i squared=-1, and every complex number has the form a + bi with a and b real.

3. Use the relation i squared =-1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

4. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p) squared = q that has the same solutions.

5. Solve quadratic equations by inspection (e.g., for x squared = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation.

6. Recognize when the quadratic formula gives complex solutions and write them as a + bi for real numbers a and b.

7. Solve quadratic equations with real coefficients that have complex solutions.

8. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

9. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

Action steps

1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

2. Write a function that describes a relationship between two quantities.

3. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

4. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

5. Sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; and end behavior.

6. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative).

7. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

8. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

9. Explain why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x).

10. Find the solutions of the equations y=f(x) and y=g(x) approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations.

Action steps

1. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

2. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

3. Rewrite expressions involving radicals and rational exponents using the properties of exponents.

4. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

5. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

6. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

Action steps

1. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

2. Write geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

3. Construct exponential functions, including geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

4. Interpret the parameters in an exponential function in terms of a context.

5. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.

6. Graph exponential and logarithmic functions, showing intercepts and end behavior.

7. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Action steps

1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

3. Graph trigonometric functions, showing period, midline, and amplitude.

4. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

5. Prove the Pythagorean identity sin2(θ ) + cos2(θ ) = 1 and use it to find sin(θ ), cos(θ ), or tan(θ ), given sin(θ ), cos(θ ), or tan(θ ), and the quadrant of the angle.

Action steps

1. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages.

2. Recognize that there are data sets for which such a procedure is not appropriate.

3. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

4. Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

5. Recognize the purposes of and differences among sample surveys; explain how randomization relates to each.

6. Decide if a specified model is consistent with results from a given data- generating process, e.g., using simulation.

7. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

8. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

9. Evaluate reports based on data.