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HS Algebra Math Question Stem Bank: Common Core State Standards Standards for Mathematical Practice Seeing Structure in Expressions Interpret the structure of expressions. CCSS.MATH.CONTENT.HSA.SSE.A.1 Interpret expressions that represent a quantity in terms of its context.* CCSS.MATH.CONTENT.HSA.SSE.A.1.A Interpret parts of an expression, such as terms, factors, and coefficients. CCSS.MATH.CONTENT.HSA.SSE.A.1.B Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P. CCSS.MATH.CONTENT.HSA.SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 ­ y4as (x2)2 ­ (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 ­ y2)(x2 + y2). Write expressions in equivalent forms to solve problems. CCSS.MATH.CONTENT.HSA.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.* CCSS.MATH.CONTENT.HSA.SSE.B.3.A Factor a quadratic expression to reveal the zeros of the function it defines. CCSS.MATH.CONTENT.HSA.SSE.B.3.B Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. CCSS.MATH.CONTENT.HSA.SSE.B.3.C Use the properties of exponents to transform expressions for exponential functions. For 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%. CCSS.MATH.CONTENT.HSA.SSE.B.4 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. For example, calculate mortgage payments.* HS Algebra Math Question Stem Bank: Common Core State Standards Standards for Mathematical Practice Seeing Structure in Expressions Standard HSA. SSE.A.1 Basic: Conceptual ­ “Understand” Standard: Procedural ­ “Doing” Identify the different parts of the The product of ­5, x, and y is represented by expression and explain their meaning the expression ­5xy. If the value of x is within the context of the problem… negative, what must be said about the value of Ex: SSE.A.1.A y in order for the product to remain negative? How many terms does the expression 3x3 − 4x2 − 5x + 2 have? Or what is the coefficients of the expression? or what is the constant of the expression? Expanded: Application Given the expression 20(4x)+500 represents the cost in dollars of the materials and labor needed to build a square fence with side length x feet around a playground. Interpret the constants and coefficients of the expression in context. Ex: SSE.A.1.B Without expanding the expression, explain how 4(xy − 1)2 + 3(xy − 1) + 1
can be viewed as having the structure of a quadratic equation. HSA. SSE.A.2 Rewrite this expression in different equivalent forms (factoring, combining like terms) Expand the expression 2(x − 1)2 − 4 to show that is a quadratic expression of the form ax2 + bx + c . HSA. SSE.B.3 Given a quadratic function, explain the meaning of zeros in the function. That is if f (x) = (x − c)(x − a) Then
f (a) = 0 f(c) = 0 Given a quadratic equation, explain the meaning of zeros graphically. Solve 8m2 + 20m = 12 for m by factoring The length of a rectangle is 3 m greater than its width. the area of the rectangle is 54 meters squared. Find the length and the width. The expression 3m2 − 15m is the income gathered by promoters of a rock concert based on the ticket price, m. For what value(s) of m would the promoters break even? HSA.SSE.B
.4 Develop the formula for the sum of a finite geometric series when the ratio is not 1. Calculate the height of a tree after n years given the initial height of 10 feet HS Algebra Math Question Stem Bank: Common Core State Standards Standards for Mathematical Practice and the growth rate of .34 feet per year... Calculate the mortgage payment with the following parameters... Arithmetic with Polynomials & Rational Expressions Perform arithmetic operations on polynomials. CCSS.MATH.CONTENT.HSA.APR.A.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. Understand the relationship between zeros and factors of polynomials. CCSS.MATH.CONTENT.HSA.APR.B.2 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). CCSS.MATH.CONTENT.HSA.APR.B.3 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. Use polynomial identities to solve problems. CCSS.MATH.CONTENT.HSA.APR.C.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 ­ y2)2 + (2xy)2 can be used to generate Pythagorean triples. CCSS.MATH.CONTENT.HSA.APR.C.5 (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of xand y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.1 Rewrite rational expressions. CCSS.MATH.CONTENT.HSA.APR.D.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) +r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. CCSS.MATH.CONTENT.HSA.APR.D.7 (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. HS Algebra Math Question Stem Bank: Common Core State Standards Standards for Mathematical Practice Arithmetic with Polynomials & Rational Expressions Standard Basic: Conceptual ­ “Understand” Standard: Procedural ­ “Doing” Expanded: Application HSA. APR.A.1 Understand the definition of a polynomial. Understand the concepts of combining like terms and closure Multiply (x+3)(x+2) HSA. APR.B.2 Understand and apply the Remainder Theorem. Understand that a is a root of a polynomial function if and only if x­a is a factor of the function. HSA. APR.B.3 Find zeros of a polynomial when the polynomial is factors. Use zeros of a function to sketch a graph of the function. The graphs below represent functions defined by polynomials. For which function are the zeros of the polynomials 2 and ­3? HSA. APR.C.4 Understand that polynomial identities include, but are not limited to, the product of the sum and difference of two terms, the difference of two squares, and the sum and difference of two cubes, the square of a binomial, etc. Illustrate how polynomial identities are used to determine numerical relationships such as 252 = (20 + 5)2 = 202 + 2 * 20 * 5 + 52 HS Algebra Math Question Stem Bank: Common Core State Standards Standards for Mathematical Practice HSA. APR.C.5 For small values of n use Pascal’s Triangle to determine the coefficients of the binomial expansion. Use the Binomial Theorem to find the nth term in the expansion of a binomial to a positive power HSA. APR.D.6 Rewrite rational expressions. HSA. APR.D.7 Understand that rational expressions are closed under addition, subtraction, multiplication and division (by a nonzero expressions) Simplify rational expressions by adding, subtracting, multiplying or dividing. HS Algebra Math Question Stem Bank: Common Core State Standards Standards for Mathematical Practice Creating Equations Create equations that describe numbers or relationships. CCSS.MATH.CONTENT.HSA.CED.A.1 Create equations and inequalities in one variable and use them to solve problems.Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CCSS.MATH.CONTENT.HSA.CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CCSS.MATH.CONTENT.HSA.CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. CCSS.MATH.CONTENT.HSA.CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R. Creating Equations Standard HSA. CED.A.1 Basic: Conceptual ­ “Understand” Standard: Procedural ­ “Doing” Expanded: Application A family’s cell phone plan costs $80 per month for 1500 minutes and 35 cents over the limit. This month the family paid $145.35. By how much did they exceed their limit? The length of the shortest side of a triangle is 8 inches. The lengths of the other two sides are represented by consecutive odd integers. Which equation could be used to find the HS Algebra Math Question Stem Bank: Common Core State Standards Standards for Mathematical Practice other sides of the triangle? HSA. CED.A.2 The school bookstore rents graphing calculators for $5 per month. It also collects a non­refundable fee of $10.00 for the school year. Write a rule for the total cost (c) of renting a calculator as a function of them number of months, m. HSA. CED.A.3 Given the inequality y< 2x­6 and y< 4x+5, the point (2, 3) is: a. a solution to both inequalities b. a solution to y<2x­6 only c. a solution to y< 4x+5 only d. not a solution to either. A student purchases a box of candy. The label says one serving of 5 candies contains 100 calories. On the axes below, graph the function C where C(x) represents the number of calories in x candies. HSA. CED.A.4 In the equation 3x − 2y = 24 the first step to isolating y would likely be: The formula for finding the Area of a triangle is A = 1/2bh , where b is the length of the base and h is the height of the triangle. Solve this formula for h. Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. CCSS.MATH.CONTENT.HSA.REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. CCSS.MATH.CONTENT.HSA.REI.A.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Solve equations and inequalities in one variable. HS Algebra Math Question Stem Bank: Common Core State Standards Standards for Mathematical Practice CCSS.MATH.CONTENT.HSA.REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. CCSS.MATH.CONTENT.HSA.REI.B.4 Solve quadratic equations in one variable. CCSS.MATH.CONTENT.HSA.REI.B.4.A Use the method of completing the square to transform any quadratic equation in xinto an equation of the form (x ­ p)2 = q that has the same solutions. Derive the quadratic formula from this form. CCSS.MATH.CONTENT.HSA.REI.B.4.B Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Solve systems of equations. CCSS.MATH.CONTENT.HSA.REI.C.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. CCSS.MATH.CONTENT.HSA.REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. CCSS.MATH.CONTENT.HSA.REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = ­3x and the circle x2 + y2 = 3. CCSS.MATH.CONTENT.HSA.REI.C.8 (+) Represent a system of linear equations as a single matrix equation in a vector variable. CCSS.MATH.CONTENT.HSA.REI.C.9 (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater). Represent and solve equations and inequalities graphically. CCSS.MATH.CONTENT.HSA.REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). CCSS.MATH.CONTENT.HSA.REI.D.11 Explain why the x­coordinates 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); find the solutions approximately, e.g., using technology to graph the functions, 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.* HS Algebra Math Question Stem Bank: Common Core State Standards Standards for Mathematical Practice CCSS.MATH.CONTENT.HSA.REI.D.12 Graph the solutions to a linear inequality in two variables as a half­plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half­planes. Reasoning with Equations & Inequalities Standard Basic: Conceptual ­ “Understand” Standard: Procedural ­ “Doing” Expanded: Application HSA. REI.A.1 In the equation 3x­4y=12 the first step in solving the equation would be... Which ordered pair is not in the solution set of y >− 1/2 x + 5 and y ≥ 3x − 2 ... HSA. REI.A.2 Solve: √x − 4 = 7 Solve: √x + 5 = 11 HSA. REI.B.3 Solve: 6x­12<2x­10 HSA. REI.B.4 If the quadratic formula is used to find the roots of A student is given the graph of the function y = 1/2 x2 − 4 . Find the zeros the equation x2 − 6x − 19 = 0 , the correct roots in simplest radical form. are... HSA. REI. C.5 When using elimination to solve the Solve a system of linear equation using following system of equation, the elimination first step you would most likely do is... HSA. REI.C.6 A business is looking to hire a painter. There are two local companies to use. Company A charges a flat rate of $100 per hour. Company B charges $50 per hour plus paint and equipment fee of $1500. Write a system of equations 3
Solve the system of equation using the graphs of the linear equations to find the intersections. HS Algebra Math Question Stem Bank: Common Core State Standards Standards for Mathematical Practice representing how much each company charges…. Determine the number of hours worked at which the cost of each company is the same... HSA. REI.C.7 Solve a system containing a linear equation and a quadratic equation in two variables. HSA. REI.C.8 Write a system of linear equations a single matrix equation. HSA. REI.C.9 HSA. REI.D.10 Understand that all solutions to an equation in two variable are contained on the graph of that equation. HSA. REI.D.11 Find the intersection point of the two equations by creating a chart or graph. HSA. REI.D.12 Graph the solution set to a system of linear inequalities in two variables as the intersection of their corresponding half­planes.