1: Translating Expressions

Size: px

Start display at page:

Download "1: Translating Expressions"

Josephine Hamilton
6 years ago
Views:

1 Algebra I

2 1: Translating Expressions Translate between words and math symbols, such as: The sum of twice a number and 3 Answer: 2n + 3 Also translate math symbols into verbal language. 5(x + 3) Answer: The product of 5 and the sum of a number and 3 Include appropriate vs. inappropriate ways to write symbols for multiplication and for division (Don t use x as multiplication; use a fraction bar for division) Include special operations: squared, cubed, square root, cube root, absolute value Use real world examples: Lisa has three gift cards and two concert tickets in her wallet, but she gave away four business cards. Answer: 3gc + 2ct 4bc Have them circle the operation word and underline the values, drawing arrows to connect them (similar to diagramming a sentence in English). Vocabulary to use: Operation Constant Coefficient Variable Expression Equation Verbal Algebraic Numerical 5 blocks 8/14-18 Content from earlier grades: 7.13a SOL A.1: The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

3 2: Evaluating Expressions & Order of Operations (a) Order of operations of a numerical expression using only a fourfunction calculator; (b) TI Calculator Introduction; (c) Using order of operations given a replacement set Be able to use PEMDAS (or GEMDAS). Know what operation should happen at each step of a problem. Grouping symbols to bring up for graphing calculator use: o (), {}, [], radicals, fraction bar Include absolute value, square root, and cube root expressions. 5 blocks 8/21-25 Emphasize parenthesis, esp fraction bars, radical signs, and (-3) 2 vs -3 2 [STO ] button Evaluate Content from earlier grades: 7.13b; 8.1a, b; 8.4 SOL A.1: The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

4 3: Simplifying Expressions & Properties & Solving Equations (a) Combining Like Terms (b) Commutative and associative properties (c) Distributive Property (d) Distribute then combine like terms (e) Field Properties (commutative, associative, distributive, additive and multiplicative identity, additive and multiplicative inverse) Simplify vs evaluate vs solve 5 blocks 8/28 9/1 Students need to be able to identify which property is used in each step in simplifying an expression. 5x + y = y + 5x ß Commutative Simplify expressions with powers, multiple variables, etc. 5x 3y - 7x + 2y + 4: The sign is attached to the term after it, so it s a negative 3y and a positive 2y. We always add when combining. -5(5x 7) = -25x + 35 a negative 5 times a negative 7 is a positive 35 Content SOL A.4: from earlier grades: 7.16 a, b, c, d, e; The student will solve multistep linear equations in two variables, including: (b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets.

5 3: Simplifying Expressions & Properties & Solving Equations continued a. Properties of Equality Reflexive, transitive, symmetric, and substitution axioms of equality, emphasize vocabulary: addition, subtraction, multiplication, and division properties of equality b. Solving Equations c. Infinitely Many Solutions/No Solution d. Verify solutions by graphing the expression on each side separately and finding the x-coordinate of the point of intersection. e. Equation Word Problems translating equations f. Proportion Equations g. Literal Equations formulas Solve Solve modeling & pictorial equations (online tools, manipulatives) SOL A.4: The student will solve multistep linear and quadratic equations in two variables, including: (a) solving Content literal from equations earlier grades: (formulas) 7.14a, for b; a given 8.15a, variable; c (b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets; (d) solving multistep linear equations algebraically and graphically; (f) solving real-world problems involving equations. Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.

6 4: Simplifying Radicals 4 blocks 9/5 9/8 a. Prime Factorization b. Simplifying radicals Square root of monomial expressions Cube root of whole numbers only Factor Prime Composite Radical expression Radicand (argument) Index Square root Cube root Content from earlier grades: 7.1d; 8.5a, b SOL A.3: The student will express the square roots and cube roots of whole numbers and the square root of a monomial algebraic expression in simplest radical form.

7 5: Laws of Exponents Simplify Monomials a. Rule 1: multiplying monomials; incl. zero as an exponent b. Rule 2: dividing monomials c. Rule 3: power to a power d. Rule 4: negative exponents Scientific Notation Express numbers in scientific notation, and perform operations, using the laws of exponents. 9 blocks 9/11-21 By operation Addition (combining like terms): Add the coefficient; leave the exponents Multiplication: Multiply the coefficient, add the exponents Division: Divide the coefficient, subtract the exponents Power: Raise the coefficient, multiply the exponents coefficient variable base exponent; power scientific notation standard notation Content from earlier grades: 7.1b, c; 8.1a SOL A.2: The student will perform operation on polynomials, including: (a) applying the laws of exponents to perform operations on expressions

8 6: Polynomials operations a. adding (combining like terms) & subtracting (distributive property) emphasize again that neg * neg = pos b. multiplying a polynomial by a monomial (distributive property, & refer to Rule 1 of Exponents) & multiplying polynomials (including multiplying binomials and multiplying a trinomial by a binomial) FOIL double distribute four square long multiplication c. special products perfect squares difference of two squares d. dividing a polynomial by a monomial (compare to distributive property) Use real world situations modeled by polynomials. Operations Coefficients Exponents add/sub add/sub none multiply multiply add divide divide subtract exponentiation raise to power multiply by power 10 Days 9/25 10/6 Review and Benchmark 10/11 10/14 Content from earlier grades: SOL A.2: The student will perform operations on polynomials, including: (b) adding, subtracting, multiplying, and dividing polynomials

9 7: Factoring Polynomials 10 blocks 10/16-27 a. Factoring gcf s 1) GCF of a set of integers 2) GCF of a set of monomials; 3) GCF of polynomials (undistributing) b. Factoring binomials/trinomials 1) Factoring by grouping x 2 + b 1 x + b 2 x + c 2) Factoring x 2 + bx + c 3) Factoring ax 2 + bx + c Prime polynomials 4) Factoring special products difference of two-squares perfect squares NLVM and Gizmos Factor Prime Composite GCF Polynomial, Trinomial, binomial c. Graph the polynomials & the factors, and recognize the relationship between the factors & the x-intercepts d. dividing a polynomial by a binomial (by factoring and canceling common factors) SOL A. 2: The student will perform operation on polynomials, including: (c) factoring completely first- and second-degree binomials and trinomials in one or two variables. Graphing calculators will be used as a tool for factoring and for confirming algebraic factorizations.

10 8: Solving Quadratic Equations (A) Solve by factoring (B) Solve by quadratic formula (C) Solve by graphing (relate to zero, x-intercepts, roots, etc) (D) Real world quadratic equation problems Algeblocks - NLVM Gizmo 14.5 blocks 10/30 11/17 prime composite factor product linear vs quadratic quadratic term vs linear term vs constant term Content from earlier grades: SOL A.4: The student will solve multistep linear and quadratic equations in two variables, including: (f) solving real-world problems involving equations. Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.

11 9: Relations & Functions a. Relations: Ø Expressed as ordered pairs, tables, mapping, graphs Ø Identify Domain and Range (use set builder notation) Ø As a linear equation b. Functions: Ø Function or not? Ø Use Functional Notation Ø Function machines Ø Zeros (algebraically and graphically) Ø x and y intercepts (algebraically and graphically) Be able to go back and forth between relations in their various forms (table, mapping, ordered pairs, graph, equation in two variables) Understand that equations show relations as well. Solutions to these equations are ordered pairs of the form (x, y) where x and y satisfies the equation. Determine whether relations are functions or not. Function Machines 1 Gizmo Input x-coordinate domain independent variable abscissa output y-coordinate range dependent variable ordinate Zero 2 blocks 11/20-21 Content from earlier grades: 7.12; 8.14, 8.17 SOL A.7: The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including a) determining whether a relation is a function; b) domain and range; c) zeros of a function; d) x- and y-intercepts; e) finding the values of a function for elements in its domain; and f) making connections between and among multiple representations of functions including concrete, verbal, numeric, graphic, and algebraic.

12 10: Slope & Writing Equations of Lines 13a Find the slope of a line given the graph of a line Identify graphs of lines with positive, negative, zero, and undefined slopes. Find the slope of a line given the equation of a line Find the slope of a line given two points on the line. Graph lines given point and slope. 13b Parallel lines have the same slope and perpendicular lines have the negative reciprocal slope. Start with counting slope (based on graphing unit). Proceed to definition of and practice of delta. Then learn slope as (change in y)/(change in x), leading into the slope formula. Then use slope as the rate of change (especially in word problems). parallel perpendicular rate of change 10 blocks 11/27 12/8 Content SOL A.6 from earlier grades: The student will graph linear equations and linear inequalities in two variables, including a) determining the slope of a line when given an equation of the line, the graph of the line, or two points on the line. Slope will be described as rate of change and will be positive, negative, zero, or undefined;

13 10: Slope & Writing Equations of Lines continued (a) Given the graph of a line. (b) Given a slope and y-intercept (c) Given a point and a slope (point-slope formula or y=mx+b) (d) Given 2 points (e) vertical & horizontal line equations Relating part c of the left hand side to the pointslope formula seems to stick a little better with students than using y = mx + b. Slope intercept form Point-slope form REVIEW and MIDTERM 12/12-21 Content from earlier grades: SOL A.6: The student will graph linear equations and linear inequalities in two variables, including: (b) writing the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line.

14 11: Solving Inequalities Solve the inequality; give solution in set-builder notation Graph the solution Verify the solutions on the graphing calculator The justification step is one that has been featured prominently in the change of SOLs, so that s an area we need to focus on. Perhaps have them solve inequality first and then next to each step put the justification. Include real-world problems involving inequalities. Dimodbann? (Did I Multiply Or Divide By A Negative Number?) solution to an equation being a value vs. a solution to an inequality being a range of values equation vs. inequality set builder notation 2 blocks 1/3-5 Content from earlier grades: 7.15a, b; 8.15b, c SOL A.5: The student will solve multistep linear inequalities in two variables, including (a) solving multistep linear inequalities algebraically and graphically; (b) justifying steps used in solving inequalities, using axioms of inequality and properties of order that are valid for the set of real numbers and its subsets; (c) solving real-world problems involving inequalities

15 12: Graphing Linear Equations & Inequalities 8 blocks 11/27 12/8 a) Graphing lines in y=mx+b form from a table of values from the slope and y-intercept b) Horizontal and Vertical lines. c) Standard form X- and Y- Intercept graphing putting into y=mx+b form d) Graphing Linear Inequalities Content from earlier grades: 8.16 SOL A.6: The student will graph linear equations and linear inequalities in two variables.

16 13: Systems of Linear Equations & Inequalities Solve by: graphing substitution elimination one solution no solution infinitely many solutions strict inequality 9 blocks 1/8-19 Use set builder notation for the answers. Include problems with no solution and with infinitely many solutions. SOL A.4 The student will solve multistep linear and quadratic equations in two variables, including e) solving systems of two linear equations in two variables algebraically and graphically f) solving real-world problems involving systems of equations. SOL A.5 The student will solve multistep linear inequalities in two variables, including d) solving systems of inequalities

17 14: Relations & Functions a. Relations: Ø Expressed as ordered pairs, tables, mapping, graphs Ø Identify Domain and Range (use set builder notation) Ø As a linear equation b. Functions: Ø Function or not? Ø Use Functional Notation Ø Function machines Ø Zeros (algebraically and graphically) Ø x and y intercepts (algebraically and graphically) Be able to go back and forth between relations in their various forms (table, mapping, ordered pairs, graph, equation in two variables) Understand that equations show relations as well. Solutions to these equations are ordered pairs of the form (x, y) where x and y satisfies the equation. Determine whether relations are functions or not. Function Machines 1 Gizmo Input x-coordinate domain independent variable abscissa output y-coordinate range dependent variable ordinate Zero 2 blocks 11/21-22 Content from earlier grades: 7.12; 8.14, 8.17 SOL A.7: The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including a) determining whether a relation is a function; b) domain and range; c) zeros of a function; d) x- and y-intercepts; e) finding the values of a function for elements in its domain; and f) making connections between and among multiple representations of functions including concrete, verbal, numeric, graphic, and algebraic.

18 15: Direct and Inverse Variation Direct proportion: constant of proportionality is the RATIO between the dependent & independent variables Inverse proportion: constant of proportionality is the PRODUCT between the dependent & independent variables Emphasize that in a direct variation, an input of zero gives an output of zero, especially in the word problem situations. The graph always goes through the origin. Constant of proportionality dependent variable independent variable 5 blocks 2/12-16 Review and Benchmark 2/27 3/3 Content from earlier grades: SOL A. 8: The student, given a situation in a real-world context, will analyze a relation to determine whether a direct or inverse variation exists, and represent a direct variation algebraically and graphically and an inverse variation algebraically.

19 16: Transformations 5 blocks 1/22-26 Starting with parent function y=x, describe transformations by changing the slope or the y-intercept. changes in slope reflections, dilations, or both changes in y-intercept - translations dilation reflection translation parent function Content from earlier grades: SOL A. 6: The student will graph linear equations and linear inequalities in two variables

20 17: Scatterplots and Curves of Best fit Drawing lines/curves of best fit. Using calculator to find the equation of the curve of best fit (STAT edit) Using equation of curve of best fit to make predictions Sample size randomness bias 5 blocks 3/5-9 Design experiments, evaluate reasonableness of a mathematical model Content from earlier grades: 8.13a, b SOL A.11: The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve real-world problems, using mathematical models. Mathematical models will include linear and quadratic functions.

21 18: Statistics Begin with mean, median, mode, and box and whisker plots (really only have about 1 block to devote to his). Move into measures of dispersion. univariate measures of center measures of dispersion o variance o standard deviation o mean absolute deviation o outliers o z-scores 10 block 2/26 3/2 3/12-16 Review for SOL 4/17 5/5 Content from earlier grades: 5.16a, d; 6.15a, b SOL A.9: The student, given a set of data, will interpret variation in real-world contexts and calculate and interpret mean absolute deviation, standard deviation, and z-scores. SOL A.10: The student will compare and contrast multiple univariate data sets, using box-and-whisker plots.

Equations and Inequalities

Equations and Inequalities Algebra I SOL Expanded Test Blueprint Summary Table Blue Hyperlinks link to Understanding the Standards and Essential Knowledge, Skills, and Processes Reporting Category Algebra I Standards of Learning