Factoring and expanding linear expressions with rational coefficients (7.EE.1) Understand that rewriting expressions into equivalent can reveal other relationships between quantities (7.EE.2) Generally, all SMPs can be applied in every standard. The following SMPs can be highlighted for this standard. 4 – Model with mathematics Connections Disciplinary Literacy Interpreting the factors in context (NC.M1.A-SSE.1b) Understanding the relationship between factors, solutions, and zeros (NC.M1.A-APR.3) Solving quadratic equations (NC.M1.A-REI.4) Rewriting quadratic functions into different forms to show key features of the function (NC.M1.F-IF.8a) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in all oral and written communication. Students should be able to compare and contrast the zeros of a function and the solutions of a function. New Vocabulary: quadratic expression Mastering the Standard Comprehending the Standard Assessing for Understanding Students factor a quadratic in the form where is an integer in order to reveal the zeroes of the quadratic function. Students use the linear factors of a quadratic function to explain the meaning of the zeros of quadratic functions and the solutions to quadratic equations in a real world problem. Students should understand that the reasoning behind rewriting quadratic expressions into factored form is to reveal different key features of a quadratic function, namely the zeros/x-intercepts. Example: The expression represents the height of a coconut thrown from a person in a tree to a basket on the ground where x is the number of seconds. Rewrite the expression to reveal the linear factors. Identify the zeroes and intercepts of the expression and interpret what they mean in regards to the context. How long is the ball in the air? Example: Part A: Three equivalent equations for are shown. Select the form that reveals the zeros of without changing the form of the equation. Part B: Select all values of for which . (from the Smarter Balanced Assessment Consortium) Students should understand that the reasoning behind rewriting quadratic expressions into factored form is to reveal the solutions to quadratic equations. Example: A vacant rectangular lot is being turned into a community vegetable garden with a uniform path around it. The area of the lot is represented by where is the width of the path in meters. Find the width of the path surrounding the garden. Instructional Resources Tasks Additional Resources Back to:
Algebra – Arithmetic with Polynomial Expressions NC.M1.A-APR.1 Perform arithmetic operations on polynomials. Build an understanding that operations with polynomials are comparable to operations with integers by adding and subtracting quadratic expressions and by adding, subtracting, and multiplying linear expressions. Concepts and Skills The Standards for Mathematical Practices Pre-requisite Connections Add, subtract, factor and expand linear expressions (7.EE.1) Understand that rewriting expressions into equivalent can reveal other relationships between quantities (7.EE.2) Generally, all SMPs can be applied in every standard. The following SMPs can be highlighted for this standard. 7 – Look for an make use of structure Connections Disciplinary Literacy Rewrite expressions using the properties of exponents (NC.M1.N-RN.2) Understanding the process of elimination (NC.M1.A-REI.5) Rewrite a quadratic function to reveal key features (NC.M1.F-IF.8a) Building functions to a relationship (NC.M1.F-BF.1b) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in all oral and written communication. Students should be able to compare operations with polynomials to operations with integers. New Vocabulary: polynomial, quadratic expression Mastering the Standard Comprehending the Standard Assessing for Understanding Students connect knowledge of integer operations to polynomial operations. At the Math 1 level, students are only responsible for the following operations: adding and subtracting quadratic expressions adding, subtracting, and multiplying linear expressions Students should be able to rewrite polynomial expressions using the properties of operations. Example: Write at least two equivalent expressions for the area of the circle with a radius of kilometers. Example: Simplify each of the following: Example: The area of a trapezoid is found using the formula , where is the area, is the height, and and are the lengths of the bases. What is the area of the above trapezoid? (NCDPI Math I released EOC #33)
NC.M1.F-IF.2, NC.M1.F-IF.5) Mastering the Standard Comprehending the Standard Assessing for Understanding Students understand that the graph of an equation is the set of all ordered pairs that make that equation a true statement. This standard contains no limitation and so applies to all function types, including those functions that a student cannot yet algebraically manipulate. Students can explain and verify that every point (x, y) on the graph of an equation represents all values for x and y that make the equation true. Students should be able to assess if a point is a solution to an equation. Example: Consider three points in the plane, Find the equation of the line through and . Use your equation in (a) to show that is on the same line as and . () Example: of the following points are on the graph of the equation Which of the following points are of the graph of the equation? How do you know? (4, 0) (0, 10) (-1, 7.5) (2.3, 5) Example: Verify that is a solution to the equation . Explain what this means for the graph of the function. Example: Without graphing, determine if the ordered pair is on the graph of . Explain. Example: The graph below shows the height of a hot air balloon as a function of time. Use the graph to answer the following: What is the height of the hot air balloon 10 minutes after it has left the ground? Approximately, when will the hot air balloon reach a height of 600 feet? Explain what the on this graph represents. Instructional Resources Tasks Additional Resources (Illustrative Mathematics) Back to: Algebra – Reasoning with Equations and Inequalities NC.M1.A-REI.11 Represent and solve equations and inequalities graphically Build an understanding of why the x-coordinates of the points where the graphs of two linear, exponential, or quadratic equations and intersect are the solutions of the equation and approximate solutions using a graphing technology or successive approximations with a table of values. Concepts and Skills The Standards for Mathematical Practices
Students should be aware of the key functions typically associated with each function type. Linear functions – domain & range, rate of change, intercepts, increasing/decreasing Quadratic functions – domain and range, x-intercepts (zeros), intervals of increasing and decreasing, intervals of positive and negative values, maximums and minimums, and end behavior Exponential functions – domain and range, rate of change, increasing or decreasing (growth and decay), intervals of positive and negative values, end behavior At the Math 1 students should not be exposed to finding the line of symmetry of a quadratic function using the formula , unless it is developed conceptually. This concept this concept can be developed with a study of the quadratic formula. If the students need to find the line of symmetry (not a requirement of Math 1), they can find the midpoint of the zeros of the function. Example: Without using the graphing capabilities of a calculator, sketch the graph of and identify the x-intercepts, y-intercept, and the maximum or minimum point. Example: The function models the amount of aspirin left in the bloodstream hours. Graph the function showing the key features of the graph. Interpret the key features in context of the problem. Students should be able to identify key feature of linear, quadratic and exponential functions from the graphical representation. Example: Which of the following is the function graphed below? (NCDPI Math 1 released EOC #4 modified) Example: Which of the following could be the function of a real variable x whose graph is shown below? Explain. *This task could be modified for a Math 1 classroom to not use vertex form. () Instructional Resources Tasks Additional Resources (Illustrative Mathematics) Back to: Functions – Interpreting Functions NC.M1.F-IF.8a Analyze functions using different representations.