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Title Suggested Time Frame Algebra I Unit 8 4th & 5th Six Weeks Suggested Duration – 22 days Guiding Questions Quadratic Equations and Modeling Big Ideas/Enduring Understandings Factoring a quadratic equation can be used to solve real-world problems. Quadratic Equations can be used to solve real-world problems. How can you use factoring a quadratic equation to solve real-world problems? How can you use quadratic equations to solve real-world problems? Vertical Alignment Expectations TEA Vertical Alignment Grades 5-8, Algebra 1 Sample Assessment Question Coming Soon……………………………… The resources included here provide teaching examples and/or meaningful learning experiences to address the District Curriculum. In order to address the TEKS to the proper depth and complexity, teachers are encouraged to use resources to the degree that they are congruent with the TEKS and research-based best practices. Teaching using only the suggested resources does not guarantee student mastery of all standards. Teachers must use professional judgment to select among these and/or other resources to teach the district curriculum. Some resources are protected by copyright. A username and password is required to view the copyrighted material. A portion of the District Specificity and Examples are a product of the Austin Area Math Supervisors TEKS Clarifying Documents available on the Region XI Math website. Algebra I Unit 8 Updated November 19, 2015 Page 1 of 13 Algebra I Unit 8 Ongoing TEKS Math Processing Skills A.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to: (A) apply mathematics to problems arising in everyday life, society, and the workplace; (B) use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution; (C) select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems; • Focus is on application • Students should assess which tool to apply rather than trying only one or all (E) create and use representations to organize, record, and communicate mathematical ideas; • Students should evaluate the effectiveness of representations to ensure they are communicating mathematical ideas clearly Students are expected to use appropriate mathematical vocabulary and phrasing when communicating ideas (F) analyze mathematical relationships to connect and communicate mathematical ideas; and • Students are expected to form conjectures based on patterns or sets of examples and non-examples (G) display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication • Precise mathematical language is expected. (D) communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate; Algebra I Unit 8 Updated November 19, 2015 • Page 2 of 13 Algebra I Unit 8 Knowledge and Skills with Student Expectations District Specificity/ Examples A.4 Linear functions, equations, and inequalities. The student applies the mathematical process standards to formulate statistical relationships and evaluate their reasonableness based on real-world data. The student is expected to: (C) write, with and without technology, linear functions that provide a reasonable fit to data to estimate solutions and make predictions from realworld problems. • • • • • Linear function Association/correlation Linear regression Correlation coefficient Strength Resources listed and categorized to indicate suggested uses. Any additional resources must be aligned with the TEKS. HMH Algebra I Unit 8 A.4(C) Collect data and model with a linear equation with and without technology. • • • • Algebra I Unit 8 Updated November 19, 2015 Vocabulary Suggested Resources Create a scatter plot from two quantitative variables. Describe the form, strength and direction of the relationship. Categorize data as linear or not. Use algebraic methods and technology to fit a linear function to the data. Use the function to predict values. Explain the meaning of the slope and y-‐intercept in context. Page 3 of 13 Algebra I Unit 8 Example: 1. If you have a keen ear and some crickets, can the cricket chirps help you predict the temperature? What does 20 cricket chirps tell you? 2. The model is used to draw conclusions: The line estimates that, on average, each added chirp predicts an increase of about 3.29 degrees Fahrenheit. What does this represent? A.6 Quadratic functions and equations. The student applies the mathematical process standards when using properties of quadratic functions to write and represent in multiple Algebra I Unit 8 Updated November 19, 2015 • • • • • • • Domain Range X-values Y-values Function Inequality Greater than (>) Page 4 of 13 ways, with and without technology, quadratic equation. The student is expected to: (A) determine the domain and range of quadratic functions and represent the domain and range using inequalities. • A.6(A) • • Determine domain and range from a graph, table, equation, or verbal situation. Represent in the form of an inequality. Misconceptions The student may confuse x and y values. The student may confuse which inequality symbol to use ( < or >, ≤ or ≥, etc.) The student may have trouble interpreting the independent and dependent variables in a real-world situation. The student may have difficulty determining how a problem situation can limit the domain or the range. • • • • • • • Algebra I Unit 8 Greater than or equal to (≥) Less than (<) Less than or equal to (≤) Open circle Closed circle Between Strictly between Between, inclusive Example: Algebra I Unit 8 Updated November 19, 2015 Page 5 of 13 Algebra I Unit 8 Solution: Domain: All real numbers or all real solutions Range: y ≥ 6.2 or {y| y ≥ 6.2} A.7 Quadratic functions and equations. The student applies the mathematical process standards when using graphs of quadraic functions and their realted transformations to represent in multiple ways and determine, with and without A.7(B) technology, the solutions to equations. Linear factors are the zeros of the quadratic function, such as The student is (x+3)(x+2), where ‐3 and ‐2 are your x‐intercepts (or zeros). expected to: Example: (B) describe the relationship between y=(x+4)(x-5) the linear factors of Set both linear factors equal to zero. quadratic expressions x+4=0 and the zeros of their x=-4 x-5=0 associated quadratic x=5 functions. Algebra I Unit 8 Updated November 19, 2015 • • • • • Quadratic function x-intercept(s) Zero(s) Solutions(s) Factors Page 6 of 13 A.8 Quadratic functions and equations. The student applies the mathematical process standards to solve, with and without technology, quadratic equations and evaluate the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on real-world data. The student is expected to: (A) solve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formula. Algebra I Unit 8 Updated November 19, 2015 • Quadratic equations • Solutions • Real solutions • Factor • Square root • Coefficients • Quadratic function • Regression Algebra I Unit 8 A.8(A) Students need to be familiar with all forms of solving quadratic equations, and be able to recognize when one form is more appropriate over another. Misconceptions • When factoring, the student may confuse whether pairs of numbers have a common product or common sum. • The student may make sign errors when determining solutions from factors. Page 7 of 13 • (B) write, using technology, quadratic functions that provide a resonable fit to data to estimate solutions and make predictions for real-world problems. • Algebra I Unit 8 When evaluating the quadratic formula, the student may make arithmetic mistakes involving integers, squares, square roots, fractions, and/or the order of operations. Students may have difficulty remembering how to complete the square due to lack of understanding of the process of factoring perfect square trinomials. Example 1 Find the solutions to the following quadratic equation: x2 + 5x + 6 = 0 Solution: x2 x2 + 5x + 6 = 0 + 5x + 6 = (x + 3)(x + 2) 0 = (x + 3)(x + 2) 0 = x + 3 or 0 = x + 2 x = −3 or x = −2 Example 2 Find the solutions to the following quadratic equation: (x − 4)2 = 5 Solution: Algebra I Unit 8 Updated November 19, 2015 (x − 4)2 = 5 �(x − 4)2 = 5 Page 8 of 13 Algebra I Unit 8 x – 4 = √5 x = ±√5 x = 4 + √5 and x = 4 - √5 Example 3 Find the solutions to the following quadratic equation: Solution: +7 +7 +9 +9 x2 + 6x − 7 = 0 x2 + 6x − 7 = 0 x2 + 6x = 7 x2 + 6x + 9 = 16 (x + 3)2 = 16 �(x + 3)2 = √16 x + 3 = ±4 x = −3 ± 4 x = −7 and x = 1 Example 4 Find the solutions to x! + 3x − 4 = 0 using the quadratic formula. Solution below: Algebra I Unit 8 Updated November 19, 2015 Page 9 of 13 Algebra I Unit 8 So, our solutions are x=-4 and x=1. A.8(B) Students need to be familiar with how to use a graphing calculator to write quadratic functions. Example: • Create a scatter plot from two quantitative variables. • Describe the form, strength and direction of the relationship. • Categorize data as quadratic or not. Use algebraic methods and technology to fit a quadratic function to the data. Use the function to predict values. • Explain the meaning of the x-‐intercepts in concept. Algebra I Unit 8 Updated November 19, 2015 Page 10 of 13 A.10 Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to rewrite in equivalent forms and perform operations on polynomial expressions. The student is expected to: (E) factor, if possible, trinomials with real factors in the form ax2 + bx + c, including perfect square trinomials of degree two. A.10(E) Factor Misconceptions The student may make errors in distribution or combining like terms when checking the factors of a trinomial (especially with minus signs and/or negative numbers). The student may think you can “distribute” the exponent to both terms in a binomial that is being squared. For example, students may incorrectly rewrite (2x+3)2 as (2x)2 + (3)2 = 4x2 + 9. Or, students may incorrectly factor 16y2 – 25 as (4y – 5)2. The students may “factor” polynomials that are prime. The student may factor a polynomial, but not its simplest form. For example, students may factor 8x2 + 4x -12 as (4x + 6)(2x – 2) rather than 4(2x + 3)(x – 1). • Factor • Polynomial • Trinomial • Terms • Leading coefficient • Product • Sum • Binomial • Difference • Square (or, perfect square) Algebra I Unit 8 Example 1 (perfect square trinomial): Factor x2 + 6x + 9 Solution: (x + 3)(x + 3) = (x + 3)2 Example 2: (F) decide if a binomial can be written as the Solution: diffence of two squares and, if possible, use the Algebra I Unit 8 Updated November 19, 2015 Factor x2+ 2x − 15 (x − 3)(x + 5) Page 11 of 13 structure of a difference of two squares to rewrite the binomial. Algebra I Unit 8 A.10(F) To factor a difference of squares, students must be able to recognize if a term is a square or not. Example 1: x2 − 64 Solution: (x − 8)(x + 8) Example 2: 9y2 − 121 Solution: (3y − 11x)(3y + 11x) Example 3: Solution: x4 − 81 (x2 − 9)( x2 + 9) (x − 3)(x + 3)( x2 + 9) If students have a hard time understanding why this pattern works each time when factoring a difference of squares, have Algebra I Unit 8 Updated November 19, 2015 Page 12 of 13 them FOIL out the factored solution. They will end up with the original binomial. Algebra I Unit 8 Updated November 19, 2015 Algebra I Unit 8 Page 13 of 13