Rigorous Curriculum Design
Rigorous Curriculum Design
Unit Planning Organizer
|Subject(s) |High School Mathematics |
|Grade/Course |Math I Standards |
|Unit of Study |Unit 6: Quadratic Functions |
|Unit Type(s) |❑Topical ❑X Skills-based ❑ Thematic |
|Pacing |12 days for Semester Block & A-Day/B-Day; 25 days for Middle School |
|Unit Abstract |
|An important nonlinear function category is quadratics. Understanding characteristics of quadratic functions and connections between various |
|representations are developed in this unit. In the table form of a quadratic function, the change in the rate of change distinguishes it from |
|a linear relationship. In particular, looking at the second rates of change or differences is where a constant value occurs. The symmetry of |
|the function values can be found in the table. The graphical form shows common characteristics of quadratic functions including maximum or |
|minimum values, symmetric shapes (parabolas), location of the y-intercept, and the ability to determine roots of the function. This unit |
|explores the polynomial form [f (x) = ax2 + bx + c] and factored form |
|[f (x) = a (x -p ) (x - q)] of quadratic functions and the impact of changing the parameters a, b, and c. Connections should be made between |
|each explicit form and its graph and table. Real-world situations that can be modeled by quadratic functions include projectile motion, |
|television dish antennas, revenue and profit models in business, and the shape of suspension bridge cables. Students learn to distinguish |
|relationships between variables that are functions from those that are not. They use f(x) notation to represent functions and identify domain|
|and range of functions. |
|Common Core Essential State Standards |
|Conceptual Category: Functions |
| |
|Domain: 1) The Real Number System (N-RN) |
|2) Seeing Structure in Expressions (A-SSE) |
|3) Arithmetic with Polynomials & Rational Expressions (A-APR) |
|4) Creating Equations (A-CED) |
|5) Interpreting Functions (F-IF) |
|6) Building Functions (F-BF) |
|7) Linear, Quadratic & Exponential Models (F-LE) |
| |
|Clusters: 1) Reason quantitatively and use units to solve problems. |
|2) Interpret the structure of expressions. |
|3) Perform arithmetic operations on polynomials. |
|4) Create equations that describe numbers or relationships. |
|5) Understand the concept of a function and use function notation. |
|Interpret functions that arise in applications in terms of the context. |
|Analyze functions using different representations. |
|6) Build a function that models a relationship between two quantities. |
|7) Construct and compare linear and exponential models and exponential |
|models and solve problems. |
| |
| |
|Standards: 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.2 DEFINE appropriate quantities for the purpose of descriptive |
|modeling. |
| |
|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 [pic] as |
|the product of P and a factor not depending on P. |
| |
|Note: At this level, limit to linear expressions, exponential expressions with |
|integer exponents and quadratic expressions. |
| |
| |
|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.3 CHOOSE and PRODUCE an equivalent form of an expression to |
|REVEAL and EXPLAIN properties of the quantity represented by |
|the expression. |
| |
|a. FACTOR a quadratic expression to REVEALl the zeros of the |
|function it defines. |
| |
|Note: At this level, the limit is quadratic expressions of the form |
|[pic]. |
| |
| |
| |
|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. |
| |
|Note: At this level, limit to addition and subtraction of quadratics |
|and multiplication of linear expressions. |
| |
| |
|A-CED.2 CREATE equations in two or more variables to represent |
|relationships between quantities; GRAPH equations on coordinate |
|axes with labels and scales. |
| |
|Note: At this level, focus on linear, exponential and quadratic. |
|Limit to situations that involve evaluating exponential functions for |
|integer inputs. |
| |
|F-IF.1 UNDERSTAND that a function from one set (called the domain) to |
|another set (called the range) ASSIGNS to each element of the |
|domain exactly one element of the range. If f is a function and x is an |
|element of its domain, then f(x) DENOTES the output of f |
|corresponding to the input x. The graph of f is the graph of the |
|equation y = f(x). |
| |
|F-IF.2 USE function notation, EVALUATE functions for inputs in their |
|domains, and INTERPRET statements that use function notation in |
|terms of a context. |
| |
| |
|F-IF.4 For a function that MODELS a relationship between two quantities, |
|INTERPRET key features of graphs and tables in terms of the |
|quantities and SKETCH graphs showing key features given a verbal |
|description of the relationship. Key features include: intercepts; |
|intervals where the function is increasing, decreasing, positive, or |
|negative; relative maximums and minimums; symmetries; end |
|behavior; and periodicity. |
| |
|Note: At this level, focus on linear, exponential and quadratic functions; |
|no end behavior or periodicity. |
| |
| |
|F-IF.7 GRAPH functions expressed symbolically and show key features of |
|the graph, by hand in simple cases and using technology for more |
|complicated cases. |
| |
|a. GRAPH linear and quadratic functions and SHOW intercepts, |
|maxima, and minima. |
| |
| |
|F-IF.8 WRITE a function defined by an expression in different but equivalent |
|forms to REVEAL and EXPLAIN different properties of the function. |
| |
|a. USE the process of factoring and completing the square in a |
|quadratic function to show zeros, extreme values, and symmetry of |
|the graph, and INTERPRET these in terms of a context. |
| |
|Note: At this level, only factoring expressions of the form [pic] is |
|expected. Completing the square is not addressed at this level. |
| |
| |
|F-IF.9 COMPARE properties of two functions each represented in a |
|different way (algebraically, graphically, numerically in tables, or by |
|verbal descriptions). |
| |
|Note: At this level, focus on linear, exponential and quadratic functions. |
| |
|F-BF.1 WRITE a function that describes a relationship between two |
|quantities. |
| |
|b. COMBINE standard function types using arithmetic operations. For |
|example, BUILD a function that models the temperature of a cooling |
|body by adding a constant function to a decaying exponential, and |
|RELATE these functions to the model. |
| |
|Note: At this level, limit to addition or subtraction of constant to linear, |
|exponential or quadratic functions or addition of linear functions to |
|linear or quadratic functions. |
| |
| |
| |
|F-LE.3 OBSERVE using graphs and tables that a quantity increasing |
|exponentially eventually EXCEEDS a quantity increasing linearly, |
|quadratically, or (more generally) as a polynomial function. |
| |
|Note: At this level, limit to linear, exponential, and quadratic functions; |
|general polynomial functions are not addressed. |
| |
| |
| |
|Standards for Mathematical Practice |
|1. Make sense of problems and persevere in solving them. |
|2. Reason abstractly and quantitatively. |
|3. Construct viable arguments and critique the reasoning of others. |
| |
|4. Model with mathematics. |
|5. Use appropriate tools strategically. |
|6. Attend to precision. |
|7. Look for and make use of structure. |
|8. Look for and express regularity in repeated reasoning. |
| |
|“UNPACKED STANDARDS” |
| |
|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. |
| |
|Ex. When finding the area of a circle using the formula [pic], which unit of measure would be appropriate for the radius? |
|square feet |
|inches |
|cubic yards |
|pounds |
| |
| |
|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. |
| |
| |
|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. |
| |
|Ex. What quantities could you use to describe the best city in North Carolina? |
|Ex. What quantities could you use to describe how good a basketball player is? |
| |
| |
|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. At this level, limit to linear expressions, exponential expressions with integer exponents, and |
|quadratic expressions. |
| |
|Ex. 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. |
| |
|A-SSE.1b Students group together parts of an expression to reveal underlying structure. For example, consider the expression |
|[pic]that represents income from a concert where p is the price per ticket. The equivalent factored form, [pic], 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, limit to linear expressions, exponential expressions with integer exponents, and quadratic expressions. |
| |
|Ex. Without expanding, explain how the expression [pic] can be viewed as having the structure of a quadratic expression. |
| |
| |
|A.SSE.2 Students rewrite algebraic expressions by combining like terms or factoring to reveal equivalent forms of the same |
|expression. |
|Ex. Expand the expression [pic] to show that it is a quadratic expression of the form [pic]. |
| |
| |
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|A-SSE.3a Students factor quadratic expressions and find the zeros of the quadratic function they represent. Zeroes are the |
|x-values that yield a y-value of 0. Students should also explain the meaning of the zeros as they relate to the problem. |
|For example, if the expression x2 – 4x + 3 represents the path of a ball that is thrown from one person to another, then the |
|expression (x – 1)(x – 3) represents its equivalent factored form. The zeros of the function, (x – 1)(x – 3) = y would be x |
|= 1 and x = 3, because an x-value of 1 or 3 would cause the value of the function to equal 0. This also indicates the ball |
|was thrown after 1 second of holding the ball, and caught by the other person 2 seconds later. At this level, limit to |
|quadratic expressions of the form ax2 + bx + c. |
| |
|Ex. The expression [pic] 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? |
| |
| |
|A-APR.1 The Closure Property means that when adding, subtracting or multiplying polynomials, the sum, difference, or product |
|is also a polynomial. Polynomials are not closed under division because in some cases the result is a rational expression |
|rather than a polynomial. At this level, limit to addition and subtraction of quadratics and multiplication of linear |
|expressions. |
| |
|A-APR.1 Add, subtract, and multiply polynomials. At this level, limit to addition and subtraction of quadratics and |
|multiplication of linear expressions. |
|Ex. If the radius of a circle is [pic] kilometers, what would the area of the circle be? |
| |
|Ex. Explain why[pic] does not equal [pic]. |
| |
| |
|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, focus on linear, exponential and quadratic equations.|
|Limit to situations that involve evaluating exponential functions for integer inputs. |
| |
|Ex. In a woman’s professional tennis tournament, the money a player wins depends on her finishing place in the standings. |
|The first-place finisher wins half of $1,500,000 in total prize money. The second-place finisher wins half of what is left; |
|then the third-place finisher wins half of that, and so on. |
|Write a rule to calculate the actual prize money in dollars won by the player finishing in nth place, for any positive |
|integer n. |
|Graph the relationship that exists between the first 10 finishers and the prize money in dollars. |
|What pattern do you notice in the graph? What type of relationship exists between the two variables? |
| |
| |
|F-IF.1 The domain of a function is the set of all x-values, which you control and therefore is called the independent |
|variable. The range of a function is the set of all y- values and is dependent on a particular x-value, thus called the |
|dependent variable. Students should experience a variety of types of situations modeled by functions. Detailed analysis of |
|any particular class of functions should not occur at this level. Students will apply these concepts throughout their future|
|mathematics courses. |
|Ex. When is an equation a function? Explain the notation that defines a function. |
|Ex. Describe the domain and range of a function and compare the concept of domain and range as it relates to a function. |
| |
|F-IF.2 Using function notation, evaluate functions and explain values based on the context in which they are in. |
|Ex. Evaluate f(2) for the function [pic]. |
| |
| |
|Ex. The function [pic] describes the height h in feet of a tennis ball x seconds after it is shot straight up into the air |
|from a pitching machine. Evaluate [pic] and interpret the meaning of the point in the context of the problem. |
| |
| |
|F-IF.4 When given a table or graph of a function that models a real-life situation, explain the meaning of the |
|characteristics of the graph in the context of the problem. The characteristics described should include rate of change, |
|intercepts, maximums/minimums, symmetries, and intervals of increase and/or decrease. At this level, focus on linear, |
|exponential, and quadratic functions; no end behavior or periodicity. |
| |
|Ex. Below is a table that represents the relationship between daily profit, P for an amusement park and the number of paying|
|visitors in thousands, n. |
| |
|n |
|P |
| |
|0 |
|0 |
| |
|1 |
|5 |
| |
|2 |
|8 |
| |
|3 |
|9 |
| |
|4 |
|8 |
| |
|5 |
|5 |
| |
|6 |
|0 |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|What are the x-intercepts and y-intercepts and explain them in the context of the problem. |
|Identify any maximums or minimums and explain their meaning in the context of the problem. |
|Determine if the graph is symmetrical and identify which shape this pattern of change develops. |
|Describe the intervals of increase and decrease and explain them in the context of the problem. |
| |
|Ex. A rocket is launched from 180 feet above the ground at time t = 0. The function that models this situation is given by |
|h(t) = – 16t2 + 96t + 180, where t is measured in seconds and h is height above the ground measured in feet. |
| |
|a. What is the practical domain for t in this context? Why? |
|b. What is the height of the rocket two seconds after it was launched? |
|c. What is the maximum value of the function and what does it mean in context? |
|d. When is the rocket 100 feet above the ground? |
|e. When is the rocket 250 feet above the ground? |
|f. Why are there two answers to part e but only one practical answer for part d? |
|g. What are the intercepts of this function? What do they mean in the context of this problem? |
|h. What are the intervals of increase and decrease on the practical domain? What do they mean in the context of the |
|problem? |
| |
| |
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|F-IF.7a Students should graph functions given by an equation and show characteristics such as but not limited to intercepts,|
|maximums, minimums, and intervals of increase or decrease. Students may use calculators or a CAS for more difficult cases. |
|Ex. Graph [pic], identifying it’s intercepts and maximum or minimum. |
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|F-IF.8a Students should take a function and manipulate it in a different form so that they can show and explain special |
|properties of the function such as; zeros, extreme values, and symmetries. |
| |
|Students should factor and complete the square to find special properties and interpret them in the context of the problem. |
|Keep in mind when completing the square, the coefficient on the x2 variable must always be one and what you add in to the |
|problem, you must also subtract from the problem. In other words, we are adding zero to the problem in order to manipulate |
|it and get it in the form we want. At this level, only factoring expressions of the form ax2 + bx + c, is expected. |
|Completing the square is not addressed. |
| |
|Ex. Suppose you have a rectangular flower bed whose area is 24ft2. The shortest side is (x-4)ft and the longest side is |
|(2x)ft. Find the length of the shortest side. |
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|F-IF.9 Students should compare the properties of two functions represented by verbal descriptions, tables, graphs, and |
|equations. For example, compare the growth of two linear functions, two exponential functions, or one of each. At this |
|level, limit to linear, exponential, and quadratic functions. |
| |
| |
|Ex. Compare the functions represented below. Which has the lowest minimum? |
| |
|a. f(x) = 3x2 +13x +4 b. [pic] |
| |
| |
|F-BF.1b Students should take standard function types such as constant, linear and exponential functions and add, subtract, |
|multiply and divide them. Also explain how the function is effected and how it relates to the model. At this level, limit |
|to addition or subtraction of a constant function to linear, exponential, or quadratic functions or addition of linear |
|functions to linear or quadratic functions. |
| |
|F-LE.3 When students compare graphs of various functions, such as linear, exponential, quadratic, and polynomial they should |
|see that any values that increase exponentially eventually increases or grows at a faster rate than values that increase |
|linearly, quadratically, or any polynomial function. At this level, limit to linear, exponential, and quadratic functions; |
|general polynomial functions are not addressed. |
| |
| |
|“Unpacked” Concepts |“Unwrapped” Skills |COGNITION |
|(students need to know) |(students need to be able to do) |DOK |
| |N-Q.1 | |
| | | |
|Numbers can be interpreted as quantities with |I can label units through multiple steps of a |1 |
|appropriate units, scales, and levels of accuracy|problem. | |
|to effectively model and make sense of real world| | |
|problems. |I can choose appropriate units for real world |1 |
| |problems involving formulas. | |
| | | |
| |I can use and interpret units when solving formulas.| |
| | |2 |
| | | |
| |I can choose an appropriate scale and origin for |1 |
| |graphs and data displays. | |
| | |2 |
| |I can interpret the scale and origin for graphs and | |
| |data displays. | |
| | | |
| |N-Q.2 | |
| | |3 |
| |I can identify the variables or quantities of | |
| |significance from the data provided. | |
| | | |
| |I can identify or choose the appropriate unit of |3 |
| |measure for each variable or quantity. | |
| | | |
| | | |
| |A-SSE.1a, b | |
| | | |
|Expressions can be written in multiple ways using|I can define expression, term, factor, and |1 |
|the rules of algebra; each version of the |coefficient. | |
|expression tells something about the problem it | | |
|represents. |I can interpret the real-world meaning of the terms,|2 |
| |factors, and coefficients of an expression in terms | |
| |of their units. | |
| | | |
| |I can group the parts of an expression differently | |
| |in order to better interpret their meaning. |3 |
| | | |
| |I can define expression, term, factor, and |1 |
| |coefficient. | |
| | |2 |
| |I can interpret the real-world meaning of the terms,| |
| |factors, and coefficients of an expression in terms | |
| |of their units. |3 |
| | | |
| |I can group the parts of an expression differently | |
| |in order to better interpret their meaning. | |
| | | |
| |A-SSE.2 |3 |
| |I can look for and identify clues in the structure | |
| |of expressions (e.g., like terms, common factors, | |
| |difference of squares, perfect squares) in order to |2 |
| |rewrite it another way. | |
| | |3 |
| |I can explain why equivalent expressions are | |
| |equivalent. |3 |
| | | |
| |I can apply models for factoring and multiplying | |
| |polynomials to rewrite expressions. | |
| | | |
| |A-SSE.3a | |
| | | |
| |I can factor a quadratic expression (ax2+bx+c) to | |
| |find the zeros of the function it represents. | |
| |A-APR.1 | |
|Algebraic expressions, such as polynomials and | | |
|rational expressions, symbolize numerical |I can apply the definition of an integer to explain |2 |
|relationships and can be manipulated in much the |why adding, subtracting, or multiplying two integers| |
|same way as numbers. |always produces an integer. |2 |
| | | |
| |I can apply the definition of polynomial to explain | |
| |why adding, subtracting, or multiplying two |3 |
| |polynomials always produces a polynomial. | |
| | |3 |
| |I can add and subtract polynomials. | |
| | | |
| |I can multiply polynomials. | |
| | | |
| | | |
| |A-CED.2 | |
| | |1 |
|Relationships between numbers can be represented |I can identify the variables and quantities | |
|by equations, inequalities, and systems. |represented in |2 |
| |real world problems. | |
| | |3 |
| |I can determine the best model for the real-world | |
| |problem (e.g. linear, quadratic). |3 |
| | | |
| |I can write the equation that best models the |3 |
| |problem. | |
| | | |
| |I can set up coordinate axes using an appropriate | |
| |scale | |
| |and label the axes. | |
| | | |
| |I can graph equations on coordinate axes with | |
| |appropriate labels and scales. | |
| |F-IF.1 | |
|Equations, verbal descriptions, graphs, and | |1 |
|tables provide insight into the relationship |I can define relation, domain, and range. | |
|between quantities. | |2 |
| |I can define a function as a relation in which each | |
| |input (domain) has exactly one output (range). | |
| | |2 |
| |I can determine if a graph, table or set of ordered | |
| |pairs represents a function. |2 |
| | | |
| |I can determine if states rules (both numeric and | |
| |non-numeric) produce ordered pairs that represent a | |
| |function. |1 |
| | | |
| |F-IF.2 | |
| | | |
| |I can convert a table, graph, set of ordered pairs, |2 |
| |or description into function notation by identifying| |
| |the rule used to turn inputs into outputs and |3 |
| |writing the rule. | |
| | |3 |
| |I can identify the numbers that are not in the | |
| |domain of a function. |2 |
| | | |
| |I can choose inputs that make sense based on a |3 |
| |problem situation. | |
| | | |
| |I can analyze the input and output values of a | |
| |function based on a problem situation. |3 |
| |F-IF.4 | |
| |I can locate the information that explains what each|3 |
| |quantity represents. | |
| |I can interpret the meaning of an ordered pair |1 |
| |(e.g., the ordered pair (9,90) could mean that a | |
| |person earned $90 after working 9 hours). |2 |
| |I can determine if negative inputs make sense in the| |
| |problem situation. | |
| |I can determine if negative outputs make sense in |2 |
| |the problem situations. | |
| |I can identify the y-intercept. | |
| | |2 |
| |I can use the definition of function to explain why | |
| |there can only be one y-intercept |2 |
| |I can use the problem situation to explain what the | |
| |y-intercept means. | |
| |I can identify the x-intercept(s). |3 |
| |I can use the definition of function to explain why | |
| |some functions have more than one x-intercept. | |
| |I can use the problem situation to explain what an |3 |
| |x-intercept means. | |
| |I can use the problem situation to explain where and| |
| |why the function is increasing or decreasing. |3 |
| |I can use the problem situation to explain why the | |
| |function has symmetry. |3 |
| |F-IF.7a | |
| |I can explain that the minimum or maximum of a | |
| |quadratic is called the vertex. |3 |
| |I can identify whether the vertex of a quadratic | |
| |will be a minimum or a maximum by looking at the |2 |
| |equation. | |
| |I can find the y-intercept of a quadratic by |1 |
| |substituting 0 for x and evaluating. | |
| |I can estimate the vertex and x-intercepts of a |2 |
| |quadratic by evaluating different values of x. | |
| |I can graph a quadratic using evaluated points. | |
| |I can use technology to graph a quadratic and to | |
| |find precise values for the x-intercept(s) and the | |
| |maximum or minimum. |1 |
| | | |
| |F-IF.8a | |
| |I can explain that there are three forms of | |
| |quadratic functions: standard form, vertex form, and|1 |
| |factored form. | |
| | |2 |
| |I can explain that standard form is [pic]. | |
| |I can explain that factored form is [pic], where x1 | |
| |and x2 are x intercepts of the function. |2 |
| |II can find the x-intercepts of a quadratic written | |
| |in factored form. |2 |
| |I can use the x-intercepts of a quadratic to find | |
| |the axis of symmetry . |2 |
| |I can use the axis of symmetry of a quadratic to | |
| |find the vertex of a parabola. |2 |
| |I can convert a standard for quadratic to factored | |
| |form by factoring. | |
| | | |
| |F-IF.9 | |
| |I can compare properties of two functions when |3 |
| |represented in different ways (algebraically, | |
| |graphically, numerically in tables, or by verbal | |
| |descriptions) | |
| |F-LE.3 | |
|Lines, exponential functions, and parabolas each |I can use graphs or tables to compare the output |2 |
|describe a specific pattern of change. |values of linear, quadratic, polynomial, and | |
| |exponential functions. | |
| | |2 |
| |I can estimate the intervals for which the output of| |
| |one function is greater than the output of another | |
| |function when given a graph or table. | |
| | |2 |
| |I can use technology to find the point at which the | |
| |graphs of two functions intersect. | |
| | |2 |
| |I can use the points of intersection to precisely | |
| |list the intervals for which the output of one | |
| |function is greater than the output of another | |
| |function. |2 |
| | | |
| |I can use graphs or tables to compare the rate of | |
| |change of linear, quadratic, polynomial and |2 |
| |exponential functions. | |
| | | |
| |I can explain why exponential functions eventually | |
| |have greater output values than linear, quadratic , | |
| |or polynomial functions by comparing simple | |
| |functions of each type. | |
|Essential Questions |Corresponding Big Ideas |
|In what ways can the choice of units, quantities, and levels of |Interpret numbers as quantities with appropriate units, scales, and |
|accuracy impact a solution? |levels of accuracy to effectively model and make |
| |sense of real world problems. |
|Why do we structure expressions in different ways? |Expressions can be written in multiple ways using the rules of |
| |algebra; each version of the expression tells something about the |
| |problem it represents. |
|How can the properties of the real number system be useful when |Algebraic expressions, such as polynomials and rational expressions, |
|working with polynomials and rational expressions? |symbolize numerical relationships and can be manipulated in much the |
| |same way as numbers. |
|How can I use algebra to describe the relationship between sets of |Relationships between numbers can be represented by equations, |
|numbers? |inequalities, and systems. |
|How can the relationship between quantities best be represented? |Equations, verbal descriptions, graphs, and tables provide insight |
| |into the relationship between quantities. |
|When does a function best model a situation? |Lines, exponential functions, and parabolas each describe a specific |
| |pattern of change. |
|Vocabulary |
|Units, scale, origin, expression, term, factor, coefficient, equivalent, polynomial, closure property, integers, linear, quadratic, coordinate|
|axes, labels, x-intercept, y-intercept, increase, decrease, maximum, minimum, symmetry, function, domain, range |
|Language Objectives |
|Key Vocabulary |
|N-Q.1 N-Q.2 |SWBAT define and give examples of vocabulary (above) specific to the standards. |
|A-SSE.1 a,b A-SSE.2 | |
|A-SSE.3 a A-APR.1 | |
|A-CED.2 | |
|F-IF.1 | |
|F-IF.2 | |
|F-IF.4 F-IF.7a | |
|F-IF.8a F-IF.9 | |
|F-BF.1b F-LE.3 | |
|Language Function |
|N-Q.1,2 |SWBAT use given units and the context of a problem as a way to determine if the solution to a multi-step |
| |problem is reasonable (e.g. length problems dictate different units than problems dealing with a measure |
| |such as slope) |
| | |
| |SWBAT interpret units or scales used in formulas or represented in graphs. |
|A-SSE.1 |SWBAT interpret parts of an expression, such as terms, factors, and coefficients in terms of the context. |
|F-IF.1, 2 |SWBAT write algebraic rules as functions and interpret the meaning of expressions involving function |
| |notation. |
|Language Skills |
| F-IF.1, 2 |SWBAT to understand the meaning of domain and range and to understand the relationship between those sets |
| |and input and output values, respectively. |
|F-IF.9 |SWBAT use a variety of function representations (algebraically, graphically, numerically in tables, or by |
| |verbal descriptions) to compare and contrast properties of two functions. |
|F-LE.3 |SWBAT compare tables and graphs of linear and exponential functions to observe that a quantity increasing |
| |exponentially exceeds all others to solve mathematical and real-world problems. |
|Language Structures |
|N-CED.2 |SWBAT justify which quantities in a mathematical problem or real-world situation are dependent and |
| |independent of one another and which operations represent those relationships. |
|F-IF.4 |SWBAT sketch graphs showing key features of a function that models a relationship between two quantities |
| |from a given verbal description of the relationship. |
|Lesson Tasks |
|N-Q.3 |SWBAT choose and justify a level of accuracy and/or precision appropriate to limitations on measurement when|
| |reporting quantities. |
|F-IF.4 |SWBAT sketch graphs showing key features of a function that models a relationship between two quantities |
| |from a given verbal description of the relationship. |
|Language Learning Strategies |
|F-BF.1b |SWBAT, given a real-world situation or mathematical problem, |
| |build standard functions to represent relevant relationships/ quantities; determine which arithmetic |
| |operation should be performed to build the appropriate combined function; and |
| |relate the combined function to the context of the problem. |
|Information and Technology Standards |
|HS.TT.1.1 Use appropriate technology tools and other resources to access information. |
|HS.TT.1.2 Use appropriate technology tools and other resources to organize information. |
|Instructional Resources and Materials |
|Physical |Technology-Based |
|Core Plus Contemporary |CPMP-Tools Software |
|Mathematics in Context (2nd | |
|Edition) – Unit 7 |NCTM Illuminations([pic] ) |
|Course 1, Unit 7, |Egg Launch Contest: Students will represent quadratic functions as a table, with a graph, and with an equation.|
| |They will compare data and move between representations. |
| |[pic] |
| |Hanging Chains: Both ends of a small chain will be attached to a board with a grid on it to (roughly) form a |
| |parabola. Students will choose three points along the curve and use them to identify an equation. Repeating the|
| |process, students will discover how the equation changes when the chain is shifted. |
| |[pic] |
| |Texas Instruments ([pic] ) |
| |Applications of Parabolas(TI-84+): In this activity, students will look for both number patterns and visual |
| |shapes that go along with quadratic relationships. Two applications are introduced after some basic patterns in|
| |the first two problems. |
| |[pic] |
| |Exploring the Vertex Form of the Quadratic Function(TI-84+): Students explore the vertex form of the parabola |
| |and discover how the vertex, direction, and width of the parabola can be determined by studying the parameters.|
| |They predict the location of the vertex of a parabola expressed in vertex form. |
| |[pic] |
| |Pass the Basketball – Linear and Quadratic Activities: Many teachers have probably seen a linear version of |
| |this activity. Students determine the time it takes for different numbers of students to pass a ball from one |
| |student to the next. If the students pass the ball at a relatively constant rate, the data collected and |
| |graphed (time versus number of students) can be modeled by a linear function. The activity can be modified to |
| |collect data that is logically modeled by a quadratic function. Questions are provided for each version of the |
| |activity. A basketball and a stopwatch are needed for both activities. |
| |[pic]
| |pdf |
| |Kitchen Parabolas: Students use kitchen bowls to determine the equation of a quadratic function that closely |
| |matches a set of points. |
| |[pic] |
| |Quadratic Functions: Quadratic Functions are explored through two lessons in this unit. The first lesson |
| |requires students to explore quadratic functions by examining the family of functions described by y = a (x - |
| |h)2 + k. In the second, students explore quadratic functions by using a motion detector known as a Calculator |
| |Based Ranger (CBR) to examine the heights of the different bounces of a ball. Students will represent each |
| |bounce with a quadratic function of the form y = a (x - h)2 + k. |
| |[pic] |
| |Toothpicks and Transformations: The lesson begins with a review of transformations of quadratic functions, |
| |vertical and horizontal shifts, and stretches and shrinks. First, students match the symbolic form of the |
| |function to the appropriate graph, then given the graphs, students analyze the various transformations and |
| |determine the equation for the functions. This review is followed by an activity where students explore a |
| |mathematical pattern that emerges as they build a geometric design with toothpicks. Students examine the |
| |recursive nature of the relationship. An explicit model for the relation is developed, and a third model is |
| |developed by examining the scatterplot and determining the equation from the transformations. Finally, the |
| |class uses the graphing calculators to develop another model and to verify that all of the models;factored |
| |form, vertex form, and general form;are equivalent. |
| |[pic] |
| |GeoGebra ([pic] ) |
| | Quadratic Fun 1: This geogebra applet allows the user to explore the relationship between the value of a in |
| |f(x)=a(x−h)2+k on the shape vertex of a parabola. Also the relationship between and the axis of symmetry and |
| |the vertex of the parabola is explored. |
| |[pic] |
| |Quadratic Fun 2: This applet explores how knowing the vertex and an additional point on the parabola can help |
| |generate the entire parabola. In addition, using the previous information, the student is asked to calculate a |
| |in the equation f(x)=a(x−h)2+k. |
| |[pic] |
| |Vertical Motion Interactvity: The motion of a mortar shell shot directly up from the top of a cliff is used to |
| |simulate free fall motion. Included are some very good questions or students to consider about the meaning of |
| |points along the path of the object. The good questions and worksheet provide scenarios to consider and pose |
| |questions for students to explore. |
| |[pic] |
| |Professional Resources |
| |NCTM () |
| |Focus in High School Mathematics: Reasoning and Sense Making: This publication elevates reasoning and sense |
| |making to a primary focus of secondary mathematics teaching. It shifts the teachers’ role from acting as the |
| |main source of information to fostering students’ reasoning to make sense of the mathematics. |
| |[pic] |
| |Focus in High School Mathematics: Reasoning and Sense Making in Algebra: Reasoning about and making sense of |
| |algebra are essential to students' future success. This book examines the five key elements (meaningful use of |
| |symbols, mindful manipulation, reasoned solving, connecting algebra with geometry, and linking expressions and |
| |functions: identified in Focus in High School mathematics: Reasoning and Sense Making in more detail and |
| |elaborates on the associated reasoning habits. |
| |[pic] |
| |Articles from National Council of Teachers of Mathematics () |
| |Articles available as free downloads to NCTM members, or for a fee to non-members |
| |Eraslan, A. and Aspinwall, L. (2007). Connecting Research to Teaching: Quadratic Functions:Students’ Graphic |
| |and Analytic Representations. |
| |Mathematics Teacher, 101(3), 223. Retrieved February 18, 2011 from |
| |[pic] |
| | |
| |Math Assessment Project |
| | |
| | |
| | |
| |Assessment Tasks |
| |1) Patchwork: Build a Function |
| | |
| |2) Functions |
| | |
| | |
| | |
| |Activities for Students: Using Graphs to Introduce Functions: Hands-on, open-ended activities that encourage |
| |problem solving, reasoning, communication, and mathematical connections. |
| | |
| |Domain and Range - Graphically!: This demo is designed to help students use graphical representations of |
| |functions to determine the domain and range. |
| |[pic] |
| |Professional Resources |
| |Articles from National Council of Teachers of Mathematics () |
| |Articles available as free downloads to NCTM members, or for a fee to non-members. |
| |Hartter, B. (2009). A Function or Not a Function? That is the Question. Mathematics Teacher, 103(3), 200. |
| |Retrieved on March 7, 2012 from |
| | |
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