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Algebra 1A and 1B (Instructional Sequence 1 & 2 are Algebra 1A; 3 & 4 are Algebra 1B) Lesson Plan Instr COS Date Course of Study Objective Suggested Lesson Resources/Activities (description, Seq # Implemented number, etc.) Explain how the definition of the meaning of rational A1133 exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals 3 1 in terms of rational exponents. [N-RN1] Example: We define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. Rewrite expressions involving radicals and rational exponents A1133 3 2 using the properties of exponents. [N-RN2] Explain why the sum or product of two rational numbers is A1121, A1125, A1131 rational; that the sum of a rational number and an irrational 1 3 number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. [N- RN3] Use units as a way to understand problems and to guide the (graphs) A1316, A1615, A1617; (area/surface solution of multistep problems; choose and interpret units area) GE711, GE721, GE723, GE725, GE727, 1,2, 4 consistently in formulas; choose and interpret the scale and GE1011, GE1013, GE1015, GE1017, GE1031, 3,4 the origin in graphs and data displays. [N-Q1] ge1428 (WS); (volume) GE1021, GE1023, GE1025, GE1027, GE1033 Define appropriate quantities for the purpose of descriptive 1,2,4 5 modeling. [N-Q2] Choose a level of accuracy appropriate to limitations on 1 6 measurement when reporting quantities. [N-Q3] Interpret expressions that represent a quantity in terms of its 1 7 context.* [A-SSE1] a. Interpret parts of an expression such as terms, factors, and A1211, A1231, a11612 (WS) coefficients. [A-SSE1a] 1 7a Example: Interpret P(1+r)n as the product of P and a factor not depending on P. b. Interpret complicated expressions by viewing one or more of A11431, a11612 (WS) their parts as a single entity. [A-SSE1b] 1 7b Example: Interpret P(1+r)n as the product of P and a factor not depending on P. 2010 Alabama Course of Study Mathematics 1 Algebra 1A and 1B (Instructional Sequence 1 & 2 are Algebra 1A; 3 & 4 are Algebra 1B) Use the structure of an expression to identify ways to rewrite Same as A-SSE1a and A-SSE1b it. [A-SSE2] 4 8 Example: See x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2). Choose and produce an equivalent form of an expression to 4 9 reveal and explain properties of the quantity represented by the expression.* [A-SSE3] a. Factor a quadratic expression to reveal the zeros of the A11011, A11021, A11031, a11628 function it defines. [A-SSE3a] 4 9a Example: The expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. b. Complete the square in a quadratic expression to reveal the A11151, A11153 4 9b maximum or minimum value of the function it defines. [A- SSE3b] c. Determine a quadratic equation when given its graph or 4 9c roots. d. Use the properties of exponents to transform expressions A1911, A1913, A1915, a2270 4 9d for exponential functions. [A-SSE3c] Understand that polynomials form a system analogous to the A1921, A1923, A1931, a2180, a2190, a11627 integers; namely, they are closed under the operations of (WS) 4 10 addition, subtraction, and multiplication; add, subtract, and multiply polynomials. [A-APR1] Create equations and inequalities in one variable, and use (solving equations) A1311, A1313, A1321, A1331; them to solve problems. Include equations arising from linear (writing equations) A1341, a11614 (WS); and quadratic functions, and simple rational and exponential (ratios/proportions) A1411, a11615 (WS); 1,2 11 functions. [A-CED1] (inequalities) A1511, A1521, A1523, A1531, A1541, A1551, a11618 (WS); (exponential) A11211, A11213 Create equations in two or more variables to represent (arithmetic) A1661; (graphing) A1751, A1761; relationships between quantities; graph equations on (functions) A1851, A11111, A11321, A11421, 1,2 12 coordinate axes with labels and scales. [A-CED2] a2060, a2170, a2320, a2360, a2370, a2380, a11632 (WS), a2810 (WS), a2850 (WS), a2870 (WS), a2890 (WS) Represent constraints by equations or inequalities, and by (solving systems of equations) A1341, A1811, 1,2,3 13 systems of equations and/or inequalities and interpret A1821, A1831, A1841, a11614 (WS), a11624 2010 Alabama Course of Study Mathematics 2 Algebra 1A and 1B (Instructional Sequence 1 & 2 are Algebra 1A; 3 & 4 are Algebra 1B) solutions as viable or non-viable options in a modeling (WS); (solving systems of inequalities) A1843, context. [A-CED3] a11625 (WS); A11411, a2310, a2490, a2500, Example: Represent inequalities describing nutritional and cost a2900 (WS) constraints on combinations of different foods. Rearrange formulas to highlight a quantity of interest, using A2040 the same reasoning as in solving equations. [A-CED4] 1 14 Example: Rearrange Ohm’s law V = IR to highlight resistance R. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, 1 15 starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. [A-REI1] Solve linear equations and inequalities in one variable, (solving equations) A1311, A1313, A1321, A1331; including equations with coefficients represented by letters. (solving inequalities) A1511, A1521, A1523, 1,2 16 [A-REI3] A1531, A1541, A1551, a11618 (WS); (literal equations) a2040 4 17 Solve quadratic equations in one variable. [A-REI4] a. Use the method of completing the square to transform any A11151, A11153, A11161, A11163, a11630 (WS) quadratic equation in x into an equation of the form (x - p)2 = q 4 17a that has the same solutions. Derive the quadratic formula from this form. [A-REI4a] b. Solve quadratic equations by inspection (e.g., for x2 = 49), A11021, A11031, A11111, A11121, A11131, taking square roots, completing the square and the quadratic A11141, A11151, A11153, A11161, A11163, formula, and factoring as appropriate to the initial form of the a11628 (WS), a11629 (WS), a11630 (WS) 4 17b equation. Recognize when the quadratic formula gives complex solutions, and write them as a ± bi for real numbers a and b. [A-REI4b] Prove that, given a system of two equations in two variables, A1831, a11624 (WS) replacing one equation by the sum of that equation and a 3 18 multiple of the other produces a system with the same solutions. [A-REI5] Solve systems of linear equations exactly and approximately A1811, A1821, A1831, A2480, A11624 (WS) 3,4 19 (e.g., with graphs), focusing on pairs of linear equations in two variables. [A-REI6] 4 20 Solve a simple system consisting of a linear equation and a A2030, A2480, A2490 2010 Alabama Course of Study Mathematics 3 Algebra 1A and 1B (Instructional Sequence 1 & 2 are Algebra 1A; 3 & 4 are Algebra 1B) quadratic equation in two variables algebraically and graphically. [A-REI7] Example: Find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. Understand that the graph of an equation in two variables is A1751, A1761, A1851, A11111, A11321, A11421, the set of all its solutions plotted in the coordinate plane, A2060, A2170, A2320, A2360, A2370, A2380, 1 21 often forming a curve (which could be a line). [A-REI10] A11632 (WS), A2810 (WS), A2820 (WS), A2870 (WS), A2880 (WS), A2890 (WS) Explain why the x-coordinates of the points where the graphs A1821, A1831, A2480, A2490, A11624 (WS) of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, 1,4 22 make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.* [A-REI11] Graph the solutions to a linear inequality in two variables as a A2500, A2510, A2900 (WS) half-plane (excluding the boundary in the case of a strict 3,4 23 inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. [A-REI12] Understand that a function from one set (called the domain) A1631, A2010, A2020 to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function 1 24 and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). [F-IF1] Use function notation, evaluate functions for inputs in their A2010 1,2 25 domains, and interpret statements that use function notation in terms of a context. [F-IF2] Recognize that sequences are functions, sometimes defined A1661, A11231, A2590, A2600 recursively, whose domain is a subset of the integers. [F-IF3] 2,3 26 Example: The Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥1. For a function that models a relationship between two A1751, A1761 1,2,4 27 quantities, interpret key features of graphs and tables in A11111, A11321 terms of the quantities, and sketch graphs showing key A11421, tr080, tr090, tr100, tr130, tr140 2010 Alabama Course of Study Mathematics 4 Algebra 1A and 1B (Instructional Sequence 1 & 2 are Algebra 1A; 3 & 4 are Algebra 1B) 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; end behavior; and periodicity.* [F-IF4] Relate the domain of a function to its graph and, where A2020 applicable, to the quantitative relationship it describes.* [F- A2080 IF5] A2090 1,2,4 28 Example: If the function h(n) gives the number of person-hours A2320 it takes to assemble n engines in a factory, then the positive A2360 integers would be an appropriate domain for the function. Calculate and interpret the average rate of change of a A1711 function (presented symbolically or as a table) over a 1,2 29 specified interval. Estimate the rate of change from a graph.* [F-IF6] Graph functions expressed symbolically and show key 1,2 30 features of the graph, by hand in simple cases and using 3,4 technology for more complicated cases.* [F-IF7] 1,2 a. Graph linear and quadratic functions, and show intercepts, A1641, A1751,A1761 30a 3,4 maxima, and minima. [F-IF7a] A11111 b. Graph square root, cube root, and piecewise-defined A11321 1,2 30b functions, including step functions and absolute value 3,4 functions. [F-IF7b] c. Graph exponential and logarithmic functions, showing a2370, a2380, tr080, tr090, tr100, tr130, tr140 1,2 30c intercepts and end behavior, and trigonometric functions, 3,4 showing period, midline, and amplitude. [F-IF7e] Write a function defined by an expression in different but 1,2 31 equivalent forms to reveal and explain different properties of 3,4 the function. [F-IF8] a. Use the process of factoring and completing the square in a Aa11011, A11021, A11031, A11151, A11153 1,2 quadratic function to show zeros, extreme values, and 31a 3,4 symmetry of the graph, and interpret these in terms of a context. [F-IF8a] b. Use the properties of exponents to interpret expressions for A1911, A1913, A1915, A11211, A11213 1,2 31b exponential functions. [F-IF8b] 3,4 Example: Identify percent rate of change in functions such as y 2010 Alabama Course of Study Mathematics 5 Algebra 1A and 1B (Instructional Sequence 1 & 2 are Algebra 1A; 3 & 4 are Algebra 1B) = (1.02)t, y = (0.97)t, y = (1.01)12t, and y = (1.2)t/10, and classify them as representing exponential growth and decay. Compare properties of two functions each represented in a A1761, A11111, A11321, A11421, tr080, tr090, different way (algebraically, graphically, numerically in tables, tr100, tr130, tr140 or by verbal descriptions). [F-IF9] 2,3,4 32 Example: Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. 1,2 Write a function that describes a relationship between two 33 3,4 quantities.* [F-BF1] 1,2 a. Determine an explicit expression, a recursive process, or A1651, A1771, a2070 33a 3,4 steps for calculation from a context. [F-BF1a] b. Combine standard function types using arithmetic A1921, A1923, A1931, A11462, A1661 operations. [F-BF1b] 1,2 33b Example: Build a function that models the temperature of a 3,4 cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Write arithmetic and geometric sequences both recursively A1661, A11231, a2590, aqa2600 2,3 34 and with an explicit formula, use them to model situations, and translate between the two forms.* [F-BF2] Identify the effect on the graph of replacing f(x) by f(x) + k, k a2100, a2110, tr110, tr120 f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. 2,3,4 35 Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. [F-BF3] 1 36 Find inverse functions. [F-BF4] a. Solve an equation of the form f(x) = c for a simple function f a2130 that has an inverse, and write an expression for the inverse. [F- 1 36a BF4a] Example: f(x) =2x3 or f(x) = (x+1)/(x-1) for x ≠ 1. Distinguish between situations that can be modeled with 3,4 37 linear functions and with exponential functions. [F-LE1] a. Prove that linear functions grow by equal differences over A1711, A1715, A11211, A11213 3,4 37a equal intervals, and that exponential functions grow by equal 2010 Alabama Course of Study Mathematics 6 Algebra 1A and 1B (Instructional Sequence 1 & 2 are Algebra 1A; 3 & 4 are Algebra 1B) factors over equal intervals. [F-LE1a] b. Recognize situations in which one quantity changes at a A1711, A1715, A1716 3,4 37b constant rate per unit interval relative to another. [F-LE1b] c. Recognize situations in which a quantity grows or decays by A11211, A11213 3,4 37c a constant percent rate per unit interval relative to another. [F- LE1c] Construct linear and exponential functions, including A1341, A1661, A1771, A11231 arithmetic and geometric sequences, given a graph, a 1,2,3 38 description of a relationship, or two input-output pairs (include reading these from a table). [F-LE2] Observe, using graphs and tables, that a quantity increasing A1761, A11111, A11211, A11213 exponentially eventually exceeds a quantity increasing 4 39 linearly, quadratically, or (more generally) as a polynomial function. [F-LE3] Interpret the parameters in a linear or exponential function in A1761, A11211, A11213, A11221 2,3 40 terms of a context. [F-LE5] Represent data with plots on the real number line (dot plots, A1613, A1615 4 41 histograms, and box plots). [S-ID1] Use statistics appropriate to the shape of the data a2420, a2430 distribution to compare center (median, mean) and spread 4 42 (interquartile range, standard deviation) of two or more different data sets. [S-ID2] Interpret differences in shape, center, and spread in the a2420, a2430 4 43 context of the data sets, accounting for possible effects of extreme data points (outliers). [S-ID3] Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context 4 44 of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. [S-ID5] Represent data on two quantitative variables on a scatter 2 45 plot, and describe how the variables are related. [S-ID6] a. Fit a function to the data; use functions fitted to data to A1791, a2440, a2450 solve problems in the context of the data. Use given functions 2 45a or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. [S-ID6a] 2010 Alabama Course of Study Mathematics 7 Algebra 1A and 1B (Instructional Sequence 1 & 2 are Algebra 1A; 3 & 4 are Algebra 1B) b. Informally assess the fit of a function by plotting and a2440, a2450 2 45b analyzing residuals. [S-ID6b] c. Fit a linear function for a scatter plot that suggests a linear A1791 2 45c association. [S-ID6c] Interpret the slope (rate of change) and the intercept A1791 1,2 46 (constant term) of a linear model in the context of the data. [S-ID7] Compute (using technology) and interpret the correlation 2 47 coefficient of a linear fit. [S-ID8] 2 48 Distinguish between correlation and causation. [S-ID9] Describe events as subsets of a sample space (the set of outcomes), using characteristics (or categories) of the 4 49 outcomes, or as unions, intersections, or complements of other events (―or,‖ ―and,‖ ―not‖). [S-CP1] Understand that two events A and B are independent if the probability of A and B occurring together is the product of 4 50 their probabilities, and use this characterization to determine if they are independent. [S-CP2] 2010 Alabama Course of Study Mathematics 8