# The Real Number System N-RN

Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/ThemeUnpackings)

The Real Number System N-RN
Common Core Cluster
Extend the properties of exponents to rational exponents.
Common Core Standard / Unpacking
What does this standard mean that a student will know and be able to do?
N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. / N-RN.2 Students should be able to use the properties of exponents to rewrite expressions involving rational exponents as expressions using radicals.
Ex. The expression can be written as ( )2 or as .
Write these expressions in radical form. How would you confirm that these forms are equivalent?
Considering that , which form would be easier to simplify without a calculator? Why?
Ex. When calculating , Kyle entered 9^3/2 into his calculator. Karen entered it as. They came out with
different results. Which was right and how could the other student modify their input?
Quantities N-Q
Common Core Cluster
Reason quantitatively and use units to solve problems.
Common Core Standard / Unpacking
What does this standard mean that a student will know and be able to do?
N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. / N-Q.1Use units as a tool to help solve multi-step problems. For example, students should use the units assigned to quantities in a problem to help identify which variable they correspond to in a formula. Students should also analyze units to determine which operations to use when solving a problem. Given the speed in mph and time traveled in hours, what is the distance traveled? From looking at the units, we can determine that we must multiply mph times hours to get an answer expressed in miles: . Note that knowledge of the distance formula is not required to determine the need to multiply in this case.)
N-Q.1 Based on the type of quantities represented by variables in a formula,choose the appropriate units to express the variables and interpret the meaning of the units in the context of the relationships that the formula describes.
N-Q.1 When given a graph or data display, read and interpret the scale and origin. When creating a graph or data display, choose a scale that is appropriate for viewing the features of a graph or data display. Understand that using larger values for the tick marks on the scale effectively “zooms out” from the graph and choosing smaller values “zooms in.” Understand that the viewing window does not necessarily show the x- or y-axis, but the apparent axes are parallel to the x- and y-axes. Hence, the intersection of the apparent axes in the viewing window may not be the origin. Also be aware that apparent intercepts may not correspond to the actual x- or y-intercepts of the graph of a function.
Ex. A group of students organized a local concert to raise awareness for The American Diabetes Foundation.
They have several expenses for promoting and operating the concert and will be making money through selling
tickets. Their profit can be modeled by the formula P = x(4,000 – 250x) – 7500. Graph the profit model using an
appropriate scale and explain your reasoning.
N-Q.2 Define appropriate quantities for the purpose of descriptive modeling. / N-Q.2 Define the appropriate quantities to describe the characteristics of interest for a population. For example, if you want to describe how dangerous the roads are, you may choose to report the number of accidents per year on a particular stretch of interstate. Generally speaking, it would not be appropriate to report the number of exits on that stretch of interstate to describe the level of danger.
N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. / N-Q.3 Understand that the tool used determines the level of accuracy that can be reported for a measurement. For
example, when using a ruler, you can only legitimately report accuracy to the nearest division. Further, if I use a
ruler that has centimeter divisions to measure the length of my pencil, I can only report its length to the nearest
centimeter. Also, when using a calculator, measures can be given to the nearest desired decimal place of a
particular unit.
Ex. What is the accuracy of a ruler with 16 divisions per inch?
Ex. What would an appropriate level of accuracy be when studying the number of Facebook users?
Ex. Vivian, John’s mother is a chemist, and she brought home a very delicate and responsive measuring
instrument. Her children enjoyed learning how to use the device by measuring the weight of pennies one at a time.
Here is a list from lightest to heaviest (weights are in milligrams).
2480 2484 2487 2491 2493 2495 2496 2498 2501 2503 2506 2507 2511 2515 2516
1. Given the information above, what do you think is the best estimate of the weight of a penny in units?
Explain your reasoning.
2. Vivian and John’s Aunt, Maria, said she had a penny that was counterfeit. They decided to weigh the penny
and discovered the penny weighed 2541 milligrams.
John said that because the penny weighed more than all of their other measurements, it must be counterfeit.
Vivian does not believe it is counterfeit based only on one weight measurement; she believes if they weigh the
penny again it might be closer to the weight of the other pennies. Vivian also had a thought that if they measured
the weights of more pennies, that Aunt Maria’s penny might not seem so strange. Why might this be so?
Ex. What would an appropriate level of accuracy be when measuring the length of a beach, from sand to water?
Explain your reasoning.
Seeing Structure in Expressions A-SSE
Common Core Cluster
Interpret the structure of expressions.
Common Core Standard / Unpacking
What does this standard mean that a student will know and be able to do?
A-SSE.1 Interpret expressions that represent a quantity in terms of its context.
a.Interpret parts of an expression, such as terms, factors, and coefficients.
b.Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret as the product of P and a factor not depending on P. / A-SSE.1a. Students manipulate the terms, factors, and coefficients in difficult expressions to explain the meaning of the individual parts of the expression. Use them to make sense of the multiple factors and terms of the expression. For example, consider the expression 10,000(1.055)5. This expression can be viewed as the product of 10,000 and 1.055 raised to the 5th power. 10,000 could represent the initial amount of money I have invested in an account. The exponent tells me that I have invested this amount of money for 5 years. The base of 1.055 can be rewritten as (1 + 0.055), revealing the growth rate of 5.5% per year.
Ex. The expression −4.9t2 + 17t + 0.6 describes the height in meters of a basketball t seconds after it has been
thrown vertically in the air. Interpret the terms and coefficients of the expression in the context of this situation.
A-SSE.1b Students group together parts of an expression to reveal underlying structure.For example, consider the expression that represents income from a concert where p is the price per ticket. The equivalent factored form, , shows that the income can be interpreted as the price times the number of people in attendance based on the price charged. At this level, include polynomial expressions.
Ex. What information related to symmetry is revealed by rewriting the quadratic formula as x = ?
A-SSE.2 Use the structure of an expression to identify ways to rewrite it. For 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). / A.SSE.2 Students rewrite algebraic expressions by combining like terms or factoring to reveal equivalent forms of the same expression.
Ex. The expression 4000p2 − 250p represents the income at a concert, where p is the price per ticket. Rewrite
this expression in another form to reveal the expression that represents the number of people in attendance based
on the price charged.
Seeing Structure in Expressions A-SSE
Common Core Cluster
Write expressions in equivalent forms to solve problems.
Common Core Standard / Unpacking
What does this standard mean that a student will know and be able to do?
A-SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
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 ≈
(1.012)12t to reveal the
approximate equivalent monthly interest rate if the annual rate is 15%. / A-SSE.3c Use properties of exponents to write an equivalent form of an exponential function to reveal and explain specific information about the rate of growth or decay.
Ex. The equation y = 14000(0.8)x represents the value of an automobile x years after purchase. Find the yearly and the monthly rate of depreciation of the car.
Arithmetic with Polynomials and Rational Expressions A-APR
Common Core Cluster
Perform arithmetic operations on polynomials.
Common Core Standard / Unpacking
What does this standard mean that a student will know and be able to do?
A-APR.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. / A-APR.1 The Closure Property means that when adding, subtracting or multiplying polynomials, the sum, difference, or product is also a polynomial.
A-APR.1 Add, subtract, and multiply polynomials. At this level, add and subtract any polynomial and extend multiplication to as many as three linear expressions.
Common Core Cluster
Understand the relationship between zeros and factors of polynomials.
Common Core Standard / Unpacking
What does this standard mean that a student will know and be able to do?
A-APR.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. / A-APR.3 Find the zeros of a polynomial when the polynomial is factored. Then use the zeros to sketch the graph. At this level, limit to quadratic expressions.
Ex. Given the function y = 2x2 + 6x – 3, list the zeros of the function and sketch the graph.
Ex. Sketch the graph of the function f(x) = (x + 5)2. What is the multiplicity of the zeros of this function? How
does the multiplicity relate to the graph of the function?
Creating Equations A-CED
Common Core Cluster
Create equations that describe numbers or relationships.
Common Core Standard / Unpacking
What does this standard mean that a student will know and be able to do?
A-CED.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. / A-CED.1 From contextual situations, write equations and inequalities in one variable and use them to solve problems. Include linear and exponential functions. At this level, extend to quadratic and inverse variation (the simplest rational) functions and use common logs to solve exponential equations.
(Level II)
Ex. A pool can be filled by pipe A in 3 hours and by pipe B in 5 hours. When the pool is full, it can be drained by
pipe C in 4 hours. If the pool is initially empty and all three pipes are open, write an equation to find how long it
takes to fill the pool. What do you think the student had in mind when using the numbers 20 and 56?
A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. / A-CED.2 Given a contextual situation, write equations in two variables that represent the relationship that exists between the quantities. Also graph the equation with appropriate labels and scales. Make sure students are exposed to a variety of equations arising from the functions they have studied. At this level, extend to simple trigonometric equations.
Ex. The intensity of light radiating from a point source varies inversely as the square of the distance from the
source. Write an equation to model the relationship between these quantities given a fixed energy output.
A-CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. / A-CED.3 Use constraints which are situations that are restricted to develop equations and inequalities and systems of equations or inequalities. Describe the solutions in context and explain why any particular one would be the optimal solution. Extend to linear-quadratic and linear-inverse variation (simplest rational) systems of equations.
A-CED.3 When given a problem situation involving limits or restrictions, represent the situation symbolically using an equation or inequality. Interpret the solution(s) in the context of the problem. When given a real world situation involving multiple restrictions, develop a system of equations and/or inequalities that models the situation. In the case of linear programming, use the Objective Equation and the Corner Principle to determine the solution to the problem.
Ex. Imagine that you are a production manager at a calculator company. Your company makes two types of
calculators, a scientific calculator and a graphing calculator.
a. Each model uses the same plastic case and the same circuits. However, the graphing calculator requires
20 circuits and the scientific calculator requires only 10. The company has 240 plastic cases and 3200
circuits in stock. Graph the system of inequalities that represents these constraints.
b. The profit on a scientific calculator is \$8.00, while the profit on a graphing calculator is \$16.00. Write
an equation that describes the company’s profit from calculator sales.
c. How many of each type of calculator should the company produce to maximize profit using the stock on
hand?
A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrangeOhm’s law V = IR to highlight resistance R. / A-CED.4 Solve multi-variable formulas or literal equations, for a specific variable. Explicitly connect this to the process of solving equations using inverse operations. At this level, extend to compound variation relationships.
(Level II)
Ex. If , solve for T2.
Reasoning with Equations and Inequalities A-REI
Common Core Cluster
Understand solving equations as a process of reasoning and explain the reasoning.
Common Core Standard / Unpacking
What does this standard mean that a student will know and be able to do?
A-REI.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. / A-REI.1 Relate the concept of equality to the concrete representation of the balance of two equal quantities. Properties of equality are ways of transforming equations while still maintaining equality/balance. Assuming an equation has a solution, construct a convincing argument that justifies each step in the solution process with mathematical properties.
A-REI.2 Solve simple rational and
radical equations in one variable, and give examples showing how
extraneous solutions may arise. / A-REI.2 Solve simple rational and radical equations in one variable and provide examples of how extraneous
solutions arise. Add context.
Ex. Solvefor x.
Ex. Mary solvedfor x and got x = -2, and x = 1. Are all the values she found solutions to the equation?
Why or why not?
Ex. Solve for x.
Reasoning with Equations and Inequalities A-REI
Common Core Cluster
Solve equations and equalities in one variable.
Common Core Standard / Unpacking
What does this standard mean that a student will know and be able to do?
A-REI.4 Solve quadratic equations in one variable.
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. / A-REI.4bSolve quadratic equations in one variable by simple inspection, taking the square root, factoring, and
completing the square. Add context or analysis. At this level, limit solving quadratic equations by inspection, taking square roots, quadratic formula, and factoring when lead coefficient is one. Writing complex solutions is not expected; however, recognizing when the formula gives complex solutions is expected.
Ex. Find the solution to the following quadratic equations:
a. x2 – 7x – 18 = 0
b. x2 = 81
c. x2 – 10x + 5 = 0
A-REI.4b Use the quadratic formula to solve any quadratic equation, recognizing the formula always produces
solutions. Write the solutions in the form a ± bi, where a and b are real numbers.
Students should understand that the solutions are always complex numbers of the form a ± bi . Real solutions are
produced when b = 0, and pure imaginary solutions are found when a = 0. The value of the discriminant,
, determines how many and what type of solutions the quadratic equation has.