8th Grade Math I Curriculum Guide Caldwell County Schools · 2016. 9. 10. · 3 8th Grade Math I...

1 8th Grade Math I Curriculum Guide – Caldwell County Schools - December 2015

8th Grade Math I Curriculum Guide Caldwell County Schools

Revised December 2015

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Domain Cluster Standard

N.Q Quantities A.CED Creating Equations

Reason quantitatively and use units to solve problems. (N.Q.1-3) Create equations that describe numbers or relationships. (A.CED.1-4)

N.Q.1★ N.Q.2★ N.Q.3★ A.CED.3★

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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.1★ This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential.

This standard is included throughout Math I, II and III. Units are a way for students to understand and make sense of problems. Use units as a way to understand problems and to guide the solution of multi-step problems

x Students use the units of a problem to identify what the problem is asking. They recognize the information units provide about the quantities in context and use units as a tool to help solve multi-step problems. Students analyze units to determine which operations to use when solving a problem. For example, 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.)

Another example, the length of a spring increases 2 cm for every 4 oz. of weight attached. Determine how much the spring will increase if 10 oz. are attached: (

) ( ) .

This can be extended into a multi-step problem when asked for the length of a 6 cm spring after 10 oz. are attached: (

) ( ) .

Choose and interpret units consistently in formulas

x Students choose the units that accurately describe what is being measured. Students understand the familiar measurements such as length (unit), area (unit squares) and volume (unit cubes). They use the structure of formulas and the context to interpret units less familiar.


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N.Q.2★ Define appropriate quantities for the purpose of descriptive modeling.

N.Q.3★ Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

For example, if

then the unit for density is

.

Choose and interpret the scale and the origin in graphs and data displays

x When given a graph or data display, students read and interpret the scale and origin. When creating a graph or data display, students choose a scale that is appropriate for viewing the features of a graph or data display. Students understand that using larger values for the tick marks on the scale effectively “zooms out” from the graph and choosing smaller values “zooms in.” Students also 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. They are also aware that apparent intercepts may not correspond to the actual x- or y-intercepts of the graph of a function.


This standard is included in Math I, II and III. Throughout all three courses, students define the appropriate variables to describe the model and situation represented.

Example(s): Explain how the units cm, cm2, and cm3 are related and how they are different. Describe situations where each would be an appropriate unit of measure.


This standard is included in Math I, II and III throughout all three courses. Students understand the tool used determines the level of accuracy that can be reported for a measurement.

Example(s): x When using a ruler, one can legitimately report accuracy to the nearest division. If a ruler has centimeter divisions,

then when measuring the length of a pencil the reported length must be to the nearest centimeter, or x In situations where units constant a whole value, as the case with people. An answer of 1.5 people would reflect a

level of accuracy to the nearest whole based on the fact that the limitation is based on the context. Students use the measurements provided within a problem to determine the level of accuracy.

Example: If lengths of a rectangle are given to the nearest tenth of a centimeter then calculated measurements should be reported to no more than the nearest tenth.

Students recognize the effect of rounding calculations throughout the process of solving problems and complete calculations with the highest degree of accuracy possible, reserving rounding until reporting the final quantity.


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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★ This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential. This standard is included in Math I, II and III. Throughout all three courses, students: x Recognize when a constraint can be modeled with an equation, inequality or system of equations/inequalities. These

constraints may be stated directly or be implied through the context of the given situation. x Create, select, and use graphical, tabular and/or algebraic representations to solve the problem.

In Math I, focus on linear equations and inequalities. Linear programming is not expected; students should approach optimization problems using a problem-solving process such as using a table. Represent constraints by equations or inequalities:

Example: The relation between quantity of chicken and quantity of steak if chicken costs $1.29/lb and steak costs $3.49/lb, and you have $100 to spend on a dinner party where chicken and steak will be served.

a. Write a constraint b. Justify your reasoning for writing the constraint as either an equation or an inequality c. Determine two solutions d. Graph the equation or inequality and identify your solutions

Represent constraints by a system of equations or inequalities:

Example: A club is selling hats and jackets as a fundraiser. Their budget is $1500 and they want to order at least 250 items. They must buy at least as many hats as they buy jackets. Each hat costs $5 and each jacket costs $8.

a. Write a system of inequalities to represent the situation. b. Graph the inequalities.

Interpret solutions as viable or nonviable options in a modeling context.

a. If the club buys 150 hats and 100 jackets, will the conditions be satisfied? b. What is the maximum number of jackets they can buy and still meet the conditions?


1st Nine Weeks

Solving Equations and Inequalities

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Domain Cluster Standard 8.EE Expressions and Equations A.REI Reasoning with Equations and Inequalities A.CED Creating Equations A.SSE Seeing Structure in Expressions

Analyze and solve linear equations and pairs of simultaneous linear equations. - Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: intersecting, parallel lines, coefficient, distributive property, like terms, substitution, system of linear equations Understanding solving equations as a process of reasoning and explaining the reasoning. (A.REI.1-2) Solve Equations and Inequalities in one variable. (A.REI.3-4) Create equations that describe numbers or relationships. (A.CED.1-4) Interpret the structure of expressions. (A.SSE.1-2)

8.EE.7a, b † A.REI.1 A.REI.3 A.CED.4★ A.SSE.1a, b ★

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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 This standard is included in Math I, II and III. Students should focus on solving all applicable equation types presented in Math I and be able to extend and apply their reasoning to other types of equations in future courses. 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. x When solving equations, students will use the properties of equality to justify and explain each step obtained from the

previous step, assuming the original equation has a solution, and develop an argument that justifies their method. x For quadratic functions, justifications for steps to show zeroes align with F.IF.8 and should include rewriting the equation

into an equivalent form using factoring and using the multiplicative property of zero to write the equations that allow the zeroes to be shown.

Example: Assuming an equation has a solution; construct a convincing argument that justifies each step in the solution process. Justifications may include the associative, commutative, and division and identity properties, combining like terms, etc.


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( )

Method 1: ( ) ( )

( )

( )

Method 2:

Construct a viable argument to justify a solution method. x Properties of operations can be used to change expressions on either side of the equation to equivalent expressions. In

addition, adding the same term to both sides of an equation or multiplying both sides by a non-zero constant produces an equation with the same solutions.

Challenge students to justify each step of solving an equation. Transforming to is possible because 5 = 5, so adding the same quantity to both sides of an equation makes the resulting equation true as well. Each step of solving an equation can be defended, much like providing evidence for steps of a geometric proof.

A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

A.REI.3 Solve:

x

x

x Solve for x.

x

x

x Solve ( ) ( ) for m.


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8.EE.7 Solve linear equations in one variable. a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

8.EE.7 Students solve one-variable equations including those with the variables being on both sides of the equals sign. Students recognize that the solution to the equation is the value(s) of the variable, which make a true equality when substituted back into the equation. Equations shall include rational numbers, distributive property and combining like terms. Example 1: Equations have one solution when the variables do not cancel out. For example, 10x – 23 = 29 – 3x can be solved to x = 4. This means that when the value of x is 4, both sides will be equal. If each side of the equation were treated as a linear equation and graphed, the solution of the equation represents the coordinates of the point where the two lines would intersect. In this example, the ordered pair would be (4, 17). 10 • 4 – 23 = 29 – 3 • 4 40 – 23 = 29 – 12 17 = 17

Example 2: Equations having no solution have variables that will cancel out and constants that are not equal. This means that there is not a value that can be substituted for x that will make the sides equal. –x + 7 – 6x = 19 – 7x Combine like terms –7x + 7 = 19 – 7x Add 7x to each side 7 ≠ 19

This solution means that no matter what value is substituted for x the final result will never be equal to each other.

If each side of the equation were treated as a linear equation and graphed, the lines would be parallel.

Example 3: An equation with infinitely many solutions occurs when both sides of the equation are the same. Any value of x will produce a valid equation. For example the following equation, when simplified will give the same values on both sides.

– (36a – 6) = (4 – 24a) –18a + 3 = 3 – 18a If each side of the equation were treated as a linear equation and graphed, the graph would be the same line. Students write equations from verbal descriptions and solve.

Example 4: Two more than a certain number is 15 less than twice the number. Find the number. Solution: n + 2 = 2n – 15 17 = n

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A.CED.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.

A.CED.4★ This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential.

This standard is included in Math I, II and III. Throughout all three courses, students solve multi-variable formulas or literal equations for a specific variable. In Math I, limit to formulas that are linear in the variable of interest, or to formulas involving squared or cubed variables.

Example: Solve for radius r.

Explicitly connect this to the process of solving equations using inverse operations.

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.

A.SSE.1a.★ This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential.

This standard is included in Math I, II and III. Throughout all three courses, students interpret expressions that represent quantities.

In Math I, the focus is on linear expressions, exponential expressions with integer exponents and quadratic expressions. Throughout all three courses, students:

x Explain the difference between an expression and an equation x Use appropriate vocabulary for the parts that make up the whole expression x Identify the different parts of the expression and explain their meaning within the context of a problem x Decompose expressions and make sense of the multiple factors and terms by explaining the meaning of the individual

parts

Note: Students should understand the vocabulary for the parts that make up the whole expression, be able to identify those parts, and interpret their meaning in terms of a context. Interpret parts of an expression, such as terms, factors, and coefficients. x Students recognize that the linear expression has two terms, m is a coefficient, and b is a constant.

x Students extend beyond simplifying an expression and address interpretation of the components in an algebraic

expression. For example, a student recognizes that in the expression , “2” is the coefficient, “2” and “x” are factors, and “1” is a constant, as well as “2x” and “1” being terms of the binomial expression. Also, a student recognizes that in the expression ( ) , 4 is the coefficient, 3 is the factor, and x is the exponent. Development and proper use of mathematical language is an important building block for future content. Using real-world context examples, the nature of algebraic expressions can be explored.


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A.SSE.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.

Example: The height (in feet) of a balloon filled with helium can be expressed by where s is the number of seconds since the balloon was released. Identify and interpret the terms and coefficients of the expression. Example: A company uses two different sized trucks to deliver sand. The first truck can transport cubic yards, and the second cubic yards. The first truck makes S trips to a job site, while the second makes 𝑇 trips. What do the following expressions represent in practical terms? (https://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/531) and (http://www.mcphersonmath.com/2015/04/pta-series-1-establish-goals-to-focus.html)

a. 𝑇 b. c. 𝑇 d.

A.SSE.1b.★ Interpret complicated expressions by viewing one or more of their parts as a single entity. x Students view mx in the expression as a single quantity.

Example: 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. Example: A rectangle has a length that is 2 units longer than the width. If the width is increased by 4 units and the length increased by 3 units, write two equivalent expressions for the area of the rectangle. Solution: The area of the rectangle is ( )( ) . Students should recognize ( ) as the length of the modified rectangle and ( ) as the width. Students can also interpret as the sum of the three areas (a square with side length x, a rectangle with side lengths 9 and x, and another rectangle with area 20 that have the same total area as the modified rectangle. Example: Given that income from a concert is the price of a ticket times each person in attendance, consider the equation that represents income from a concert where p is the price per ticket. What expression could represent the number of people in attendance? Solution: 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. Students recognize ( ) as a single quantity for the number of people in attendance. Example: The expression ( ) is the amount of money in an investment account with interest compounded annually for n years. Determine the initial investment and the annual interest rate. Note: The factor of 1.055 can be rewritten as (1 + 0.055), revealing the growth rate of 5.5% per year.

Connection to N.Q.1

Another example, the length of a spring increases 2 cm for every 4 oz. of weight attached. Determine how much the spring will increase if 10 oz. are attached:(

) ( ) . This can be extended into a multi-step problem when asked for the

length of a 6 cm spring after 10 oz. are attached: (

) ( ) .

https://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/531


http://www.mcphersonmath.com/2015/04/pta-series-1-establish-goals-to-focus.html

http://www.mcphersonmath.com/2015/04/pta-series-1-establish-goals-to-focus.html


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8.EE Expressions and Equations

8.F Functions

A.REI Reasoning with Equations and Inequalities A.CED Creating Equations F.IF Interpreting Functions F.LE Linear, Quadratic, and Exponential Models F.BF Building Functions

Understand the connections between proportional relationships, lines, and linear equations. - Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: unit rate, proportional relationships, slope, vertical, horizontal, similar triangles, y-intercept.

Define, evaluate, and compare functions. - Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: functions, y-value, x-value, vertical line test, input, output, rate of change, linear function, non-linear function

Use functions to model relationships between quantities. - Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: linear relationship, rate of change, slope, initial value, y-intercept

Represent and solve equations and inequalities graphically. (A.REI.10-12) Create equations that describe numbers or relationships. (A.CED.1-4) Understand the concept of a function and use function notation. (F.IF.1-3) Interpret functions that arise in applications in terms of the context. (F.IF.4-6) Analyze functions using different representations. (F.IF. 7-9) Interpret expressions for functions in terms of the situation they model. (F.LE.4-6) Build a function that models a relationship between two quantities. (F.BF.1-2)

Build new functions from existing functions. (F.BF.3)

8.EE.5 † 8.EE.6 †

8.F.1 † 8.F.2 † 8.F.3* †

8.F.4* † 8.F.5 †

A.REI.10* A.CED.1★ * A.CED.2★ *

F.IF.1 F.IF.2* F.IF4★ * F.IF.5★ F.IF.6★

F.IF.7a★ * F.IF.9* F.LE.5★ * F.BF.1a★ F.BF.1b★ F.BF.3


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d Common Core

Standard Unpacking What does this standard mean that a student will know and be able to do?

8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

8.EE.5 Students build on their work with unit rates from 6th grade and proportional relationships in 7th grade to compare graphs, tables and equations of proportional relationships. Students identify the unit rate (or slope) in graphs, tables and equations to compare two proportional relationships represented in different ways. Example 1: Compare the scenarios to determine which represents a greater speed. Explain your choice including a written description of each scenario. Be sure to include the unit rates in your explanation.

Scenario 1: Scenario 2: y = 55x x is time in hours y is distance in miles Solution: Scenario 1 has the greater speed since the unit rate is 60 miles per hour. The graph shows this rate since 60 is the distance traveled in one hour. Scenario 2 has a unit rate of 55 miles per hour shown as the coefficient in the equation. Given an equation of a proportional relationship, students draw a graph of the relationship. Students recognize that the unit rate is the coefficient of x and that this value is also the slope of the line.

8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

8.EE.6 Triangles are similar when there is a constant rate of proportionality between them. Using a graph, students construct triangles between two points on a line and compare the sides to understand that the slope (ratio of rise to run) is the same between any two points on a line. Example 1: The triangle between A and B has a vertical height of 2 and a horizontal length of 3. The triangle between B and C has a vertical height of 4 and a horizontal length of 6. The simplified ratio of the vertical height to the horizontal length of both triangles is 2 to 3, which

also represents a slope of for the line, indicating that the triangles are similar. Given an equation in slope-intercept form, students graph the line represented. ��

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Students write equations in the form y = mx for lines going through the origin, recognizing that m represents the slope of the line.

Example 2: Write an equation to represent the graph to the right. Solution: y = – x Students write equations in the form y = mx + b for lines not passing through the origin, recognizing that m represents the slope and b represents the y-intercept. Solution: y = x – 2

8.F.3* Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

8.F.3* Students understand that linear functions have a constant rate of change between any two points. Students use equations, graphs and tables to categorize functions as linear or non-linear. Example 1: Determine if the functions listed below are linear or non-linear. Explain your reasoning.

1. y = –2x2 + 3 2. y = 0.25 + 0.5(x – 2) 3. A = πr2

4. 5. X Y 1 12 2 7 3 4 4 3 5 4 6 7

Solution:

1. Non-linear 2. Linear 3. Non-linear 4. Non-linear; there is not a constant rate of change 5. Non-linear; the graph curves indicating the rate of change is not constant.

(3, 0)

(6, 2)

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8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

8.F.4 Students identify the rate of change (slope) and initial value (y-intercept) from tables, graphs, equations or verbal descriptions to write a function (linear equation). Students understand that the equation represents the relationship between the x-value and the y-value; what math operations are performed with the x-value to give the y-value. Slopes could be undefined slopes or zero slopes. Tables: Students recognize that in a table the y-intercept is the y-value when x is equal to 0. The slope can be determined by finding the ratio between the change in two y-values and the change between the two corresponding x-values. Example 1: Write an equation that models the linear relationship in the table below.

x y –2 8 0 2 1 –1

Solution: The y-intercept in the table below would be (0, 2). The distance between 8 and –1 is 9 in a negative direction Æ –9;

the distance between –2 and 1 is 3 in a positive direction. The slope is the ratio of rise to run or or = –3. The equation

would be y = –3x + 2.

Graphs: Using graphs, students identify the y-intercept as the point where the line crosses the y-axis and the slope as the

runrise .

Example 2: Write an equation that models the linear relationship in the graph below. Solution: The y-intercept is 4. The slope is ¼ , found by moving up 1 and right 4 going from (0, 4) to (4, 5). The linear equation would be y = ¼ x + 4. Equations: In a linear equation the coefficient of x is the slope and the constant is the y-intercept. Students need to be given the equations in formats other than y = mx + b, such as y = ax + b (format from graphing calculator), y = b + mx (often the format from contextual situations), etc.

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Point and Slope: Students write equations to model lines that pass through a given point with the given slope. Example 2: A line has a zero slope and passes through the point (–5, 4). What is the equation of the line? Solution: y = 4 Example 3: Write an equation for the line that has a slope of ½ and passes though the point (–2, 5) Solution: y = ½ x + 6 Students could multiply the slope ½ by the x-coordinate –2 to get –1. Six (6) would need to be added to get to 5, which gives the linear equation. Students also write equations given two ordered pairs. Note that point-slope form is not an expectation at this level. Students use the slope and y-intercepts to write a linear function in the form y = mx +b. Contextual Situations: In contextual situations, the y-intercept is generally the starting value or the value in the situation when the independent variable is 0. The slope is the rate of change that occurs in the problem. Rates of change can often occur over years. In these situations it is helpful for the years to be “converted” to 0, 1, 2, etc. For example, the years of 1960, 1970, and 1980 could be represented as 0 (for 1960), 10 (for 1970) and 20 (for 1980). Example 4: The company charges $45 a day for the car as well as charging a one-time $25 fee for the car’s navigation system (GPS). Write an expression for the cost in dollars, c, as a function of the number of days, d, the car was rented. Solution: C = 45d + 25 Students interpret the rate of change and the y-intercept in the context of the problem. In Example 4, the rate of change is 45 (the cost of renting the car) and that initial cost (the first day charge) also includes paying for the navigation system. Classroom discussion about one-time fees vs. recurrent fees will help students model contextual situations.

F.IF.6★ Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

F.IF.6★ This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential. Students calculate the average rate of change of a function given a graph, table, and/or equation.

x The average rate of change of a function ( ) over an interval [a, b] is

( ) ( )

.


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Example: What is the average rate at which this bicycle rider traveled from four to ten minutes of her ride?

Example: The plug is pulled in a small hot tub. The table gives the volume of water in the tub from the moment the plug is pulled, until it is empty. What is the average rate of change between:

x 60 seconds and 100 seconds? x 0 seconds and 120 seconds? x 70 seconds and 110 seconds?

A–REI.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).

A.REI.10* This standard is included in Math I, II and III. In Math I, students focus on linear and exponential equations and are able to adapt and apply that learning to other types of equations in future courses. Students can explain and verify that every point (cx, y) on the graph of an equation represents all values for x and y that make the equation true.

Example: Which of the following points are on the graph of the equation How many points are on this graph? Explain. a. (4, 0) b. (0, 10) c. (–1, 7.5) d. (2.3, 5)

Example: Verify that (–1, 60) is a solution to the equation (

) . Explain what this means for the graph of the

function.


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★ * This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential. This standard is included in Math I, II and III. Throughout all three courses, students recognize when a problem can be modeled with an equation or inequality and are able to write the equation or inequality. Students create, select, and use graphical, tabular and/or algebraic representations to solve the problem. In Math I, focus on linear and exponential contextual situations that students can use to create equations and inequalities in one variable and use them to solve problems.

Example: The Tindell household contains three people of different generations. The total of the ages of the three family members is 85.

a. Find reasonable ages for the three Tindells. b. Find another reasonable set of ages for them. c. In solving this problem, one student wrote ( ) ( )

1. What does C represent in this equation? 2. What do you think the student had in mind when using the numbers 20 and 56? 3. What set of ages do you think the student came up with?

Example: Mary and Jeff both have jobs at a baseball park selling bags of peanuts. They get paid $12 per game and $1.75 for each bag of peanuts they sell. Create equations, that when solved, would answer the following questions:

a. How many bags of peanuts does Jeff need to sell to earn $54? b. How much will Mary earn if she sells 70 bags of peanuts at a game? c. How many bags of peanuts does Jeff need to sell to earn at least $68?

Example: Phil purchases a used truck for $11,500. The value of the truck is expected to decrease by 20% each year. When will the truck first be worth less than $1,000? Example: A scientist has 100 grams of a radioactive substance. Half of it decays every hour. How long until 25 grams remain? Be prepared to share any equations, inequalities, and/or representations used to solve the problem.

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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★ * This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential. This standard is included in Math I, II and III. Throughout all three courses, students: x Create equations in two variables x Graph equations on coordinate axes with labels and scales clearly labeling the axes defining what the values on the axes

represent and the unit of measure. Students also select intervals for the scale that are appropriate for the context and display adequate information about the relationship.

x Students interpret the context and choose appropriate minimum and maximum values for a graph.


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In Math I, focus on linear, exponential and quadratic contextual situations for students to create equations in two variables. Limit exponential situations to only ones involving integer input values. Linear equations can be written in a multitude of ways; and are commonly used forms (given that x and y are the two variables). Students should be flexible in using multiple forms and recognizing from the context, which is appropriate to use in creating the equation.

Example: The FFA had a fundraiser by selling hot dogs for $1.50 and drinks for $2.00. Their total sales were $400. a. Write an equation to calculate the total of $400 based on the hot dog and drink sales. b. Graph the relationship between hot dog sales and drink sales.

Example: A spring with an initial length of 25 cm will compress 0.5 cm for each pound applied.

a. Write an equation to model the relationship between the amount of weight applied and the length of the spring. b. Graph the relationship between pounds and length. c. What does the graph reveal about limitation on weight?

8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1 1Function notation is not required in Grade 8.

8.F.1 Students understand rules that take x as input and gives y as output is a function. Functions occur when there is exactly one y-value is associated with any x-value. Using y to represent the output we can represent this function with the equations y = x2 + 5x + 4. Students are not expected to use the function notation f(x) at this level.

Students identify functions from equations, graphs, and tables/ordered pairs.

Graphs Students recognize graphs such as the one below is a function using the vertical line test, showing that each x-value has only one y-value;

whereas, graphs such as the following are not functions since there are 2 y-values for multiple x-value.


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Tables or Ordered Pairs Students read tables or look at a set of ordered pairs to determine functions and identify equations where there is only one output (y-value) for each input (x-value).

Functions Not A Function

x y 0 3 1 9 2 27

x y 16 4 16 –4 25 5 25 –5

{(0, 2), (1, 3), (2, 5), (3, 6)} Equations Students recognize equations such as y = x or y = x2 + 3x + 4 as functions; whereas, equations such as x2 + y2 = 25 are not functions.

8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

8.F.2 Students compare two functions from different representations.

Example 1: Compare the following functions to determine which has the greater rate of change.

Function 1: y = 2x + 4

x y –1 –6 0 –3 2 3

Function 2: Solution: The rate of change for function 1 is 2; the rate of change for function 2 is 3. Function 2 has the greater rate of change.

Example 2: Compare the two linear functions listed below and determine which has a negative slope.

x y 0 20 1 16.50 2 13.00 3 9.50

Function 1: Gift Card Samantha starts with $20 on a gift card for the bookstore. She spends $3.50 per week to buy a magazine. Let y be the amount remaining as a function of the number of weeks, x. Function 2: Calculator rental The school bookstore rents graphing calculators for $5 per month. It also collects a non-refundable fee of $10.00 for the school year. Write the rule for the total cost © of renting a calculator as a function of the number of months (m). c = 10 + 5m

Solution: Function 1 is an example of a function whose graph has a negative slope. Both functions have a positive starting amount; however, in function 1, the amount decreases 3.50 each week, while in function 2, the amount increases 5.00 each month.


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NOTE: Functions could be expressed in standard form. However, the intent is not to change from standard form to slope-intercept form but to use the standard form to generate ordered pairs. Substituting a zero (0) for x and y will generate two ordered pairs. From these ordered pairs, the slope could be determined. Example 3: 2x + 3y = 6 Let x = 0: 2(0) + 3y = 6 Let y = 0: 2x + 3(0) = 6 3y = 6 2x = 6 3y = 6 2x = 6 3 3 2 2 y = 2 x = 3 Ordered pair: (0, 2) Ordered pair: (3, 0) Using (0, 2) and (3, 0) students could find the slope and make comparisons with another function.

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.1 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. While F.IF.1 is not repeated in any other course, students will apply these concepts throughout their future mathematics courses.

In Math I, draw examples from linear, quadratic, and exponential functions. Students use the definition of a function to determine whether a relationship is a function given a table, graph or words.

Example: Determine which of the following tables represent a function and explain why.

Table A Table B x y r T 0 1 0 0 1 2 1 2 2 2 1 3 3 4 4 5

Given the function ( ), students explain that input values are guaranteed to produce unique output values and use the function rule to generate a table or graph. They identify x as an element of the domain, the input, and ( ) as an element in the range, the output. The domain of a function given by an algebraic expression, unless otherwise specified, is the largest possible domain.

Example: A pack of pencils cost $0.75. If n number of packs are purchased then the total purchase price is represented by the function ( ) .

a. Explain why t is a function. b. What is a reasonable domain and range for the function t?

Students recognize that the graph of the function, f, is the graph of the equation y = f (x) and that (x, f (x)) is a point on the graph of f.


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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.2* This standard is included in Math I, II and III. In Math I, students should focus on linear and exponential functions and be able to extend and apply their reasoning to other types of functions in future courses. Use function notation, evaluate functions for inputs in their domains

Example: Evaluate ( ) for the function ( ) ( ) . Example: Evaluate ( ) for the function ( ) ( )

Interpret statements that use function notation in terms of a context

You placed a ham in the oven and, after 45 minutes, you take it out. Let f be the function that assigns to each minute after you placed the yam in the oven, its temperature in degrees Fahrenheit. Write a sentence for each of the following to explain what it means in everyday language. a. ( ) b. ( ) ( ) c. ( ) ( ) d. ( ) ( )

Example: The rule ( ) ( ) represents the amount of a drug in milligrams ( ), which remains in the bloodstream after x hours. Evaluate and interpret each of the following: a. ( ) b. ( ) ( ). What is the value of k? c. ( )

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.

F.IF.4 ★ * This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential. This standard is included in Math I, II and III. Throughout all three courses, students interpret the key features of a variety of different functions. There are two categories of functions to consider. The first are the “classical” functions usually referred to by families. In Math I, students focus on linear, exponential, and quadratic functions. (Connect to F.IF.7) There are also functions that are not included into families. For example, plots over time represent functions as do some scatterplots. These are often functions that “tell a story” hence the portion of the standard that has students sketching graphs given a verbal description. Students should have experience with a wide variety of these types of functions and be flexible in thinking about functions and key features using tables and graphs. When given a table or graph of a function that models a real-life situation, explain the meaning of the characteristics of the table or graph in the context of the problem.


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The following are examples of “classical” family of functions:

# of words Cost to place ad ($) 50 0 60 0.50 70 1 80 1.50 90 2

100 2.5

Key features of a linear function are slope and intercepts.

Example: The local newspaper charges for advertisements in their community section. A customer has called to ask about the charges. The newspaper gives the first 50 words for free and then charges a fee per word. Use the table at the right to describe how the newspaper charges for the ads. Include all important information.

The following are examples of functions that “tell a story”:

There are a number of videos on this site http://graphingstories.com Some are aligned to Math I while others are more appropriate for Math 2 or 3. The following are suggested videos for Math I: x Water Volume x Weight x Bum Height Off Ground x Air Pressure x Height of Stack Example: Marla was at the zoo with her mom. When they stopped to view the lions, Marla ran away from the lion exhibit, stopped, and walked slowly towards the lion exhibit until she was halfway, stood still for a minute then walked away with her mom. Sketch a graph of Marla’s distance from the lions’ exhibit over the period of time when she arrived until she left.

Example: Describe a situation, in detail, that could be represented by the graph at the right.

F.IF.5★ Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n

F- IF.5★ This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential.

This standard is included in Math I, II and III. Throughout all three courses, students relate the domain of a function to its graph and, where applicable, the quantitative relationship it describes. Focus on linear, exponential, and quadratic function types in this course.

http://graphingstories.com/


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engines in a factory, then the positive integers would be an appropriate domain for the function.

Example: An all-inclusive resort in Los Cabos, Mexico provides everything for their customers during their stay including food, lodging, and transportation. Use the graph at the right to describe the domain of the total cost function.

Example: Jennifer purchased a cell phone and the plan she decided upon charged her $50 for the phone and $0.10 for each minute she is on the phone. (The wireless carrier rounds up to the half minute.) She has budgeted $100 for her phone bill. What would be the appropriate domain for the cost as a function of the total minutes she used the phone? Describe what the point (10, 51) represents in the problem.

Example: Maggie tosses a coin off of a bridge into a stream below. The distance the coin is above the water is modeled by the equation where x represents time in seconds. What is a reasonable domain for the function?

Example: Oakland Coliseum, home of the Oakland Raiders, is capable of seating 63,026 fans. For each game, the amount of money that the Raiders’ organization brings in as revenue is a function of the number of people, n, in attendance. If each ticket costs $30, find the domain of this function.

Sample Response: Let r represent the revenue that the Raider's organization makes, so that ( ). Since n represents a number of people, it must be a nonnegative whole number. Therefore, since 63,026 is the maximum number of people who can attend a game, we can describe the domain of f as follows: Domain = {n: and n is an integer}. (Students should be able to understand and express using this notation.)

The deceptively simple task above asks students to find the domain of a function from a given context. The function is linear and if simply looked at from a formulaic point of view, students might find the formula for the line and say that the domain and range are all real numbers. However, in the context of this problem, this answer does not make sense, as the context requires that all input and output values are non-negative integers, and imposes additional restrictions.

F.LE.5★ * Interpret the parameters in a linear or exponential function in terms of a context.

F.LE.5 ★ * This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential. Use real-world situations to help students understand how the parameters of linear and exponential functions depend on the context.

Example: A plumber who charges $50 for a house call and $85 per hour can be expressed as the function . If the rate were raised to $90 per hour, how would the function change?

Example: Lauren keeps records of the distances she travels in a taxi and what it costs:

Distance d in miles Fare f in dollars 3 8.25 5 12.75

11 26.25


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a. If you graph the ordered pairs ( ) from the table, they lie on a line. How can this be determined without graphing them?

b. Show that the linear function in part a. has equation . c. What do the 2.25 and the 1.5 in the equation represent in terms of taxi rides.

Example: A function of the form ( ) ( ) is used to model the amount of money in a savings account that earns 8% interest, compounded annually, where n is the number of years since the initial deposit. a. What is the value of r? Interpret what r means in terms of the savings account? b. What is the meaning of the constant P in terms of the savings account? Explain your reasoning. c. Will n or ( ) ever take on the value 0? Why or why not? Example: The equation ( ) models the rising population of a city with 8,000 residents when the annual growth rate is 4%. a. What would be the effect on the equation if the city’s population were 12,000 instead of 8,000? b. What would happen to the population over 25 years if the growth rate were 6% instead of 4%?

F.BF.1★ Write a function that describes a relationship between two quantities. a.★Determine an explicit expression, a recursive process, or steps for calculation from a context.

F.BF.1★ This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential.

This standard is included in Math I, II and III. Throughout all three courses, students write functions that describe relationships. Students will

x Analyze a given problem to determine the function expressed by identifying patterns in the function’s rate of change.

x Specify intervals of increase, decrease, constancy, and relate them to the function’s description in words or graphically

F.BF.1a★ Focus on linear and exponential, however other non-linear and non-exponential patterns (quadratic) should be included as a point of comparison to linear and exponential function types. Students are not expected to create a formal recursive process; however explicit forms are expected for linear, exponential and quadratic functions.

Example: The height of a stack of cups is a function of the number of cups in the stack. If a 7.5” cup with a 1.5” lip is stacked vertically, determine a function that would provide you with the height based on any number of cups. Hint: Start with height of one cup and create a table, list, graph or description that describes the pattern of the stack as an additional cup is added.


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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.

Age Value 1 $1575 2 $1200 3 $900 4 $650 5 $500 6 $400 7 $300

Example: The price of a new computer decreases with age. Examine the table by analyzing the outputs.

a. Informally describe a recursive relationship. Note: Determine the average rate at which the computer is decreaseing over time.

b. Analyze the input and the output pairs to determine an explicit function that represents the value of the computer when the age is known.

Example: Sidewalk Patterns from http://map.mathshell.org/tasks.php?unit=HA06&collection=9&redir=1

From context, write an explicit expression or describe the calculations needed to model a function between two quantities.

Example: A carpet company uses the template to the right to design rugs. Write an expression for the area of the shaded region as it relates to the variable length x.

Determine an explicit expression, a recursive process, or steps for calculation from a context.

Example: A movie theater has to pay 8% in monthly taxes on its profit. The profit from tickets and the concession stand is $7.75 per person. The theater has expenses of $15,000 per month. Build a function to represent the amount the theater must pay in taxes each month as a function of the number of people attending the movies. Example: A single bacterium is placed in a test tube and splits in two after one minute. After two minutes, the resulting two bacteria split in two, creating four bacteria. This process continues. a. How many bacteria are in the test tube after 5 minutes? 15 minutes? b. Write a recursive rule to find the number of bacteria in the test tube from one minute to the next. c. Convert this rule into explicit form. How many bacteria are in the test tube after one hour?

F.BF.1b★ Combine standard function types using arithmetic operations. At this level, a constant may be combined to linear, quadratic or exponential functions and a linear function may be combined to a linear or quadratic function.

Example: A ball is thrown straight up into the air starting at 2 meters with an initial upward velocity of 20 meters per second. Write a function rule that relates the height (in meters) of the ball and the time (in seconds). Note that gravity has an effect on the height with the distance fallen represented by 4.9t2 where t is time in seconds.

Example: A retail store has two options for discounting items to go on clearance.

x Option 1: Decrease the price of the item by 15% each week. x Option 2: Decrease the price of the item by $5 each week.

If the cost of an item is $45, write a function rule for the difference in price between the two options.

Example: Blake has a monthly car payment of $225. He has estimated an average cost of $0.32 per mile for gas and maintenance. He plans to budget for the car payment the minimal he needs with an additional 3% of his total budget for incidentals that may occur. Build a function that gives the amount Blake needs to budget as a function of the number of miles driven.

http://map.mathshell.org/tasks.php?unit=HA06&collection=9&redir=1


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F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

F.BF.3 This standard is included in Math I, II and III. The standard addresses specific functions and specific effects throughout each course. In Math I, focus on vertical and horizontal translations of linear and exponential functions.

Example: Describe how ( ) compares to ( ) represented in the graph.

Example: Given the graph of ( ) whose x-intercept is 3, find the value of k if ( ) resulted in the graph having an x-intercept of .

Example: Describe how the graph of ( ) compares to ( ) if k positive. If k is negative.

Example: The graph of ( ) is shown at right. The graph of ( ) ( ) . Find the value of k.

8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

8.F.5 Given a verbal description of a situation, students sketch a graph to model that situation. Given a graph of a situation, students provide a verbal description of the situation. Example 1: The graph below shows a John’s trip to school. He walks to his Sam’s house and, together, they ride a bus to school. The bus stops once before arriving at school. Describe how each part A – E of the graph relates to the story. Solution:

A John is walking to Sam’s house at a constant rate. B John gets to Sam’s house and is waiting for the bus. C John and Sam are riding the bus to school. The bus is moving at a constant rate, faster than John’s walking rate. D The bus stops. E The bus resumes at the same rate as in part C.

Example 2: Describe the graph of the function between x = 2 and x = 5? Solution: The graph is non-linear and decreasing.


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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.7★ * This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential. In Math I, students graph linear, exponential and quadratic functions expressed symbolically. (Connect to F.IF.4) Students recognize how the form of the linear function provides information about key features such as slope and y-intercept. Also, the x-intercept can be revealed by determining the input value x such that ( ) .

Example: Describe the key features of the graph ( )

and use the key features to create a sketch of the function.

F.IF.9* Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

F.IF.9* This standard is included in Math I, II and III. Throughout all three courses, students compare properties of two functions. The representations of the functions should vary: table, graph, algebraically, or verbal description. In Math I, students focus on linear, exponential, and quadratic.

t ( ) 0.2 38.6 0.4 32.2 0.6 25.8 0.8 19.4 1.0 13

Example: David and Fred each throw a baseball into the air. The velocity of David’s ball is given by ( ) where v is in feet per second and t is in seconds. The velocity of Fred’s ball is given in the table. What is the difference in the initial velocity between the two throws?

Example: Examine the two functions represented below. Compare the x-intercepts and find the difference between the minimum values.

( ) x –7 –6 –5 –4 –3 –2 –1

( ) 4 1.5 0 –0.5 0 1.5 4

Example: Joe is trying to decide which job would allow him to earn the most money after a few years. x His first job offer agrees to pay him $50 per week. If he does a good job, they will give him a 2% raise each year. x His other job offer agrees to pay him $40 per week and if he does a good job, they will give him a 3% raise each

year. Which job would you suggest Joe take? Justify your reasoning.


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F.IF Interpreting Functions F.BF Building Functions

Understand the concept of a function and use function notation. (F.IF.1-3) Build a function that models a relationship between two quantities. (F.BF.1-2)

F.IF.3* F.BF.2★ *

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F.IF.3* Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

F.IF.3* A sequence can be described as a function, with the input numbers consisting of a subset of the integers, and the output numbers being the terms of the sequence. The most common subset for the domain of a sequence is the Natural numbers {1, 2, 3, …}; however, there are instances where it is necessary to include {0} or possibly negative integers. Whereas, some sequences can be expressed explicitly, there are those that are a function of the previous terms. In which case, it is necessary to provide the first few terms to establish the relationship.

Example: The sequence 1, 1, 2, 3, 5, 8, 13, 21,… has an initial term of 1 and subsequent terms. The first 7 terms are displayed. Since each term is unique to its position in the sequence then, by definition, the sequence is a function. Explore the sequence and evaluate each of the following:

a. ( ) b. ( ) c. ( )

Example: A theater has 60 seats in the first row, 68 seats in the second row, 76 seats in the third row, and so on in the same increasing pattern.

a. If the theater has 20 rows of seats, how many seats are in the twentieth row? b. Explain why the sequence is considered a function. c. What is the domain of the sequence? Explain what the domain represents in context.

Connect to arithmetic and geometric sequences (F.BF.2). Emphasize that arithmetic and geometric sequences are examples of linear and exponential functions, respectively.

F.BF.2★ * Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

F.BF.2★ * This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential. This standard is included in Math I and 3.


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In Math I, students use informal recursive notation (such as 𝑇 is intended). NOW – NEXT equations are the first step in the process of formalizing a sequence using function notation. The NOW – NEXT representation allows students to explore and understand the concept of a recursive function before being introduced to symbolic notation in Math 3 such as or ( ) ( ) . (Connect to F.IF.3)

In an arithmetic sequence, each term is obtained from the previous term by adding the same number each time. This number is called the common difference. In a geometric sequence, each term is obtained from the previous term by multiplying by a constant amount, called the common ratio.

Example: Given the following sequences, write a recursive rule: a. 2, 4, 6, 8, 10, 12, 14, … b. 1000, 500, 250, 125, 62.5, …

NOW – NEXT equations are informal recursive equations that show how to calculate the value of the next term in a sequence from the value of the current term. The arithmetic sequence represented by 𝑇 is the recursive form of a linear function. The common difference c corresponds to the slope m in the explicit form of a linear function, . The initial value of the sequence d may or may not correspond to the y-intercept, b. This varies based on the given information.

Example: If the sequence is 37, 32, 27, 22, 17, … it can be written as 37, 37–5, 37–5–5, 37–5-5–5, 37–5-5–5–5, etc., so that students recognize an initial value of 37 and a –5 as the common difference.

The geometric sequence represented by 𝑇 is the recursive form of an exponential function. The common ratio b corresponds to the base b in the explicit form of an exponential function, y = abx. The initial value may or may not correspond to the y-intercept, a. This varies based on the given information.

Example: If the sequence is 3, 6, 12, 24,… it can be written as 3,3(2), 3(2)(2), 3(2)(2)(2), etc., so that students recognize an initial value of 3 and 2 is being used as the common ratio.

Note: Write out terms in a table in an expanded form to help students see what is happening. Translate between the two forms

Example: A concert hall has 58 seats in Row 1, 62 seats in Row 2, 66 seats in Row 3, and so on. The concert hall has 34 rows of seats. a. Write a recursive formula to find the number of seats in each row. How many seats are in row 5? b. Write the explicit formula to determine which row has 94 seats?


2nd Nine Weeks

Systems of Equations and Inequalities

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8.EE Expressions and Equations A.REI Reasoning with Equations and Inequalities

Understand the connections between proportional relationships, lines, and linear equations. - Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: unit rate, proportional relationships, slope, vertical, horizontal, similar triangles, y-intercept. Solve Systems of Equations (REI.5-6) Represent and solve equations and inequalities graphically. (A.REI.10-12)

8.EE.8a,b,c A.REI.5 A.REI.6 A.REI.11★ * A.REI-12

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8.EE.8 Analyze and solve pairs of simultaneous linear equations. a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

8.EE.8 Systems of linear equations can also have one solution, infinitely many solutions or no solutions. Students will discover these cases as they graph systems of linear equations and solve them algebraically.

Students graph a system of two linear equations, recognizing that the ordered pair for the point of intersection is the x-value that will generate the given y-value for both equations. Students recognize that graphed lines with one point of intersection (different slopes) will have one solution, parallel lines (same slope, different y-intercepts) have no solutions, and lines that are the same (same slope, same y-intercept) will have infinitely many solutions.

By making connections between algebraic and graphical solutions and the context of the system of linear equations, students are able to make sense of their solutions. Students need opportunities to work with equations and context that include whole number and/or decimals/fractions. Students define variables and create a system of linear equations in two variables.

Example 1: 1. Plant A and Plant B are on different watering schedules. This affects their rate of growth. Compare the growth of the two

plants to determine when their heights will be the same.

Plant A Plant B W H W H 0 4 (0, 4) 0 2 (0, 2) 1 6 (1, 6) 1 6 (1, 6) 2 8 (2, 8) 2 10 (2, 10) 3 10 (3, 10) 3 14 (3, 14)

Solution: Let W = number of weeks Let H = height of the plant after W weeks


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c. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

2. Based on the coordinates from the table, graph lines to represent each plant. Solution: 3. Write an equation that represents the growth rate of Plant A and Plant B.

Solution: Plant A H = 2W + 4 Plant B H = 4W + 2 4. At which week will the plants have the same height?

Solution: 2W + 4 = 4W + 2 Set height of Plant A equal to height of Plant B 2W – 2W + 4 = 4W – 2W+ 2 Solve for W 4 = 2W + 2 4 – 2 = 2W + 2 – 2 2 = 2W 2 2 1 = W

After one week, the height of Plant A and Plant B are both 6 inches. Check: 2(1) + 4 = 4(1) + 2 2 + 4 = 4 + 2 6 = 6 Given two equations in slope-intercept form (Example 1) or one equation in standard form and one equation in slope-intercept form, students use substitution to solve the system. Example 2: Solve: Victor is half as old as Maria. The sum of their ages is 54. How old is Victor? Solution: Let v = Victor’s age v + m = 54 Let m = Maria’s age v = ½ m ½ m + m = 54 Substitute ½m for v in the first equation 1½ m = 54 m = 36 If Maria is 36, then substitute 36 into v + m = 54 to find Victor’s age of 18. Note: Students are not expected to change linear equations written in standard form to slope-intercept form or solve systems using elimination.

Plant B

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For many real world contexts, equations may be written in standard form. Students are not expected to change the standard form to slope-intercept form. However, students may generate ordered pairs recognizing that the values of the ordered pairs would be solutions for the equation. For example, in the equation above, students could make a list of the possible ages of Victor and Maria that would add to 54. The graph of these ordered pairs would be a line with all the possible ages for Victor and Maria.

Victor Maria 20 34 10 44 50 4 29 25

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A.REI.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.

A.REI.5 The focus of this standard is to provide mathematics justification for the addition (elimination) method of solving systems of equations ultimately transforming a given system of two equations into a simpler equivalent system that has the same solutions as the original system. Build on student experiences in graphing and solving systems of linear equations from middle school to focus on justification of the methods used. Include cases where the two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution); connect to GPE.5, which requires students to prove the slope criteria for parallel lines.

Example(s): Use the system { to explore what happens graphically with different combinations of the

linear equations. a. Graph the original system of linear equations. Describe the solution of the system and how it relates to the

solutions of each individual equation. (Note: connect to A.REI.10)

b. Add the two linear equations together and graph the resulting equation. Describe the solutions to the new equation and how they relate to the system’s solution.

c. Explore what happens with other combinations such as: i. Multiply the first equation by 2 and add to the second equation

ii. Multiply the second equation by –2 and add to the first equation iii. Multiply the second equation by –1 and add to the first equation iv. Multiply the first equation by –1 and add to the second equation

d. Are there any combinations that are more informative than others?

e. Make a conjecture about the solution to a system and any combination of the equations.

Example: Given that the sum of two numbers is 10 and their difference is 4, what are the numbers? Explain how your answer can be deduced from the fact that the two numbers, x and y, satisfy the equations and .


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A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

A.REI.6 The system solution methods can include but are not limited to graphical, elimination/linear combination, substitution, and modeling. Systems can be written algebraically or can be represented in context. Students may use graphing calculators, programs, or applets to model and find approximate solutions for systems of linear equations.

Example: José had 4 times as many trading cards as Philippe. After José gave away 50 cards to his little brother and Philippe gave 5 cards to his friend for his birthday, they each had an equal amount of cards. Write a system to describe the situation and solve the system. Example: A restaurant serves a vegetarian and a chicken lunch special each day. Each vegetarian special is the same price. Each chicken special is the same price. However, the price of the vegetarian special is different from the price of the chicken special.

x On Thursday, the restaurant collected $467 selling 21 vegetarian specials and 40 chicken specials. x On Friday, the restaurant collected $484 selling 28 vegetarian specials and 36 chicken specials.

What is the cost of each lunch special?

Example: Solve the system of equations: and . Use a second method to check your answer.

A.REI.11★ * Explain why the x-coordinates 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.

A.REI.11★ * This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential. This standard is included in Math I, II and III. Throughout all three courses, students use graphs, tables and algebraic methods (including substitution and elimination for systems) to solve equations and systems of equations. Students understand that numerical solution methods (data in a table used to approximate an algebraic function) and graphical solution methods may produce approximate solutions, and algebraic solution methods produce precise solutions that can be represented graphically or numerically. Students may use graphing calculators or programs to generate tables of values, graphs, or solve a variety of functions. In Math I, focus on linear and exponential functions.

Example: The functions ( ) and ( ) give the lengths of two different springs in centimeters, as mass is added in grams, m, to each separately.

a. Graph each equation on the same set of axes. b. What mass makes the springs the same length? c. What is the length at that mass? d. Write a sentence comparing the two springs.

Example: Solve the following equations by graphing. Give your answer to the nearest tenth.

a. ( ) b.


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A.REI.12* Graph the solutions to a linear inequality in two variables as a half- plane (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 half-planes.

A.REI.12* In Math I, students graph linear inequalities and systems of linear inequalities.

Graph the solutions to a linear inequality in two variables as a half-plane, excluding the boundary for non-inclusive inequalities.

Example: Graph the following inequalities:

Students understand that the solutions to a system of inequalities in two-variables are the points that lie in the intersection of the corresponding half-planes.

Example: Compare the solution to a system of equations to the solutions of a system of inequalities. Example: Describe the solution set of a system of inequalities.

Graph the solution set to a system of linear inequalities in two variables as the intersection of their corresponding half-planes.

Example: Graph the solution set for the following system of inequalities:

Example: Graph the system of linear inequalities below and determine if (3, 2) is a solution to the system.

Connect to A.CED.2 and A.CED.3

Rational and Irrational

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8.NS The Number System

Know that there are numbers that are not rational, and approximate them by rational numbers.

- Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: Real Numbers, Irrational numbers, Rational numbers, Integers, Whole numbers, Natural numbers, radical, radicand, square roots, perfect squares, cube roots, terminating decimals, repeating decimals, truncate

8.NS.1 † 8.NS.2 †


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8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

8.NS.1 Students understand that Real numbers are either rational or irrational. They distinguish between rational and irrational numbers, recognizing that any number that can be expressed as a fraction is a rational number. The diagram below illustrates the relationship between the subgroups of the real number system Students recognize that the decimal equivalent of a fraction will either terminate or repeat. Fractions that terminate will have denominators containing only prime factors of 2 and/or 5. This understanding builds on work in 7th grade when students used long division to distinguish between repeating and terminating decimals.

Students convert repeating decimals into their fraction equivalent using patterns or algebraic reasoning.

One method to find the fraction equivalent to a repeating decimal is shown below.

Example 1: Change 0. to a fraction. x Let x = 0.444444…..

x Multiply both sides so that the repeating digits will be in front of the decimal. In this example, one digit repeats so both sides are multiplied by 10, giving 10x = 4.4444444….

x Subtract the original equation from the new equation.

10x = 4.4444444…. – x = 0.444444….. 9x = 4 x Solve the equation to determine the equivalent fraction. 9x = 4 9 9

x = Additionally, students can investigate repeating patterns that occur when fractions have denominators of 9, 99, or 11.

Example 2: is equivalent to 0. is equivalent to 0. , etc.

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8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

8.NS.2 Students locate rational and irrational numbers on the number line. Students compare and order rational and irrational numbers. Students also recognize that square roots may be negative and written as – Example 1: Compare and Solution: Statements for the comparison could include:

and are between the whole numbers 1 and 2 is between 1.7 and 1.8

is less than Additionally, students understand that the value of a square root can be approximated between integers and that non-perfect square roots are irrational. Example 2: Find an approximation of x Determine the perfect squares is between, which would be 25 and 36. x The square roots of 25 and 36 are 5 and 6 respectively, so we know that is between 5 and 6. x Since 28 is closer to 25, an estimate of the square root would be closer to 5. One method to get an estimate is to divide 3

(the distance between 25 and 28) by 11 (the distance between the perfect squares of 25 and 36) to get 0.27. The estimate of would be 5.27 (the actual is 5.29).

Radicals and Exponents

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Domain Cluster Standard 8.EE Expressions and Equations N.RN The Real Number System

Analyze and solve linear equations and pairs of simultaneous linear equations. - Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: intersecting, parallel lines, coefficient, distributive property, like terms, substitution, system of linear equations Extend the properties of exponents to rational exponents. (N.RN.1-2)

8.EE.1 8.EE.2 8.EE.3 8.EE.4 N.RN.1 N.RN.2

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8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.

8.EE.1 In 6th grade, students wrote and evaluated simple numerical expressions with whole number exponents (ie. 53 = 5 • 5 • 5 = 125). Integer (positive and negative) exponents are further developed to generate equivalent numerical expressions when multiplying, dividing or raising a power to a power. Using numerical bases and the laws of exponents, students generate equivalent expressions.

Students understand: x Bases must be the same before exponents can be added, subtracted or multiplied. (Example 1) x Exponents are subtracted when like bases are being divided (Example 2) x A number raised to the zero (0) power is equal to one. (Example 3) x Negative exponents occur when there are more factors in the denominator. These exponents can be expressed as

a positive if left in the denominator. (Example 4) x Exponents are added when like bases are being multiplied (Example 5) x Exponents are multiplied when an exponents is raised to an exponent (Example 6) x Several properties may be used to simplify an expression (Example 7)

Example 1: = Example 2: = = = = Example 3: 60 = 1 Students understand this relationship from examples such as . This expression could be simplified as = 1. Using

the laws of exponents this expression could also be written as 62–2 = 60. Combining these gives 60 = 1.

8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

8.EE.2 Students recognize perfect squares and cubes, understanding that non-perfect squares and non-perfect cubes are irrational. Students recognize that squaring a number and taking the square root √ of a number are inverse operations; likewise, cubing

a number and taking the cube root are inverse operations.

Example 1: 42 = 16 and = ±4

NOTE: (–4)2 = 16 while –42 = –16 since the negative is not being squared. This difference is often problematic for students, especially with calculator use.

Example 2: and NOTE: There is no negative cube root since multiplying 3 negatives would give a negative.

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This understanding is used to solve equations containing square or cube numbers. Rational numbers would have perfect squares or perfect cubes for the numerator and denominator. In the standard, the value of p for square root and cube root equations must be positive. Example 3: Solve: x2 = 25

Solution: = ± x = ±5 NOTE: There are two solutions because 5 • 5 and –5 • –5 will both equal 25.

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8.EE.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.

8.EE.3 Students use scientific notation to express very large or very small numbers. Students compare and interpret scientific notation quantities in the context of the situation, recognizing that if the exponent increases by one, the value increases 10 times. Likewise, if the exponent decreases by one, the value decreases 10 times. Students solve problems using addition, subtraction or multiplication, expressing the answer in scientific notation. Example 1: Write 75,000,000,000 in scientific notation. Solution: 7.5 x 1010

Example 2: Write 0.0000429 in scientific notation. Solution: 4.29 x 10–5

Example 3: Express 2.45 x 105 in standard form. Solution: 245,000

Example 4: How much larger is 6 x 105 compared to 2 x 103 Solution: 300 times larger since 6 is 3 times larger than 2 and 105 is 100 times larger than 103. Example 5: Which is the larger value: 2 x 106 or 9 x 105? Solution: 2 x 106 because the exponent is larger

8.EE.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose

8.EE.4 Students understand scientific notation as generated on various calculators or other technology. Students enter scientific notation using E or EE (scientific notation), * (multiplication), and ^ (exponent) symbols. Example 1: 2.45E+23 is 2.45 x 1023 and 3.5E–4 is 3.5 x 10–4 (NOTE: There are other notations for scientific notation depending on the calculator being used.)

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units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

Students add and subtract with scientific notation. Example 2: In July 2010 there were approximately 500 million Facebook users. In July 2011 there were approximately 750 million Facebook users. How many more users were there in 2011. Write your answer in scientific notation. Solution: Subtract the two numbers: 750,000,000 – 500,000,000 = 250,000,000 Æ 2.5 x 108 Students use laws of exponents to multiply or divide numbers written in scientific notation, writing the product or quotient in proper scientific notation. Example 3: (6.45 x 1011)(3.2 x 104) = (6.45 x 3.2)(1011 x 104) Rearrange factors = 20.64 x 1015 Add exponents when multiplying powers of 10 = 2.064 x 1016 Write in scientific notation Example 4: 3.45 x 105 3.45 Subtract exponents when dividing powers of 10 6.7 x 10-2 6.7 = 0.515 x 107 Write in scientific notation = 5.15 x 106 Example 5: (0.0025)(5.2 x 104) = (2.5 x 10–3)(5.2 x 105) Write factors in scientific notation = (2.5 x 5.2)(10–3 x 105) Rearrange factors = 13 x 10 2 Add exponents when multiplying powers of 10 = 1.3 x 103 Write in scientific notation

Example 6: The speed of light is 3 x 10 8 meters/second. If the sun is 1.5x 1011 meters from earth, how many seconds does it take light to reach the earth? Express your answer in scientific notation. Solution: 5 x 102 (light)(x) = sun, where x is the time in seconds (3 x 10 8 )x = 1.5 x 1011 1.5 x 1011 3 x 10 8

Students understand the magnitude of the number being expressed in scientific notation and choose an appropriate corresponding unit. Example 7: 3 x 108 is equivalent to 300 million, which represents a large quantity. Therefore, this value will affect the unit chosen.

x 105 – (–2)


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N.RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define

⁄ to be the cube root of

5 because we want ( ⁄ )

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to hold, so (

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must equal 5.

N.RN.1 The meaning of an exponent relates the frequency with which a number is used as a factor. So indicates the product where 5 is a factor 3 times. Extend this meaning to a rational exponent, then

⁄ indicates one of three equal factors whose product is 125.

Students recognize that a fractional exponent can be expressed as a radical or a root. For example, an exponent of a ⁄ is equivalent to a cube root; an exponent of ⁄ is equivalent to a fourth root.

Students extend the use of the power rule, ( ) from whole number exponents i.e., ( ) to rational exponents.

They compare examples, such as ( ⁄ )

⁄ to (√ ) to establish a connection between radicals and

rational exponents: ⁄ √ and, in general,

⁄ = √ .

Example: Determine the value of x a.

b. ( ) Example: A biology student was studying bacterial growth. The population of bacteria doubled every hour as indicated in the following table:

# of hours of observation 0 1 2 3 4 Number of bacteria cells (thousands) 4 8 16 32 64

How could the student predict the number of bacteria every half hour? every 20 minutes?

Solution: If every hour the number of bacteria cells is being multiplied by a factor of 2 then on the half hour the number of cells is increasing by a factor of

⁄ . For every 20 minutes, the number of cells is increasing by a factor of

⁄ .

N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

N.RN.2 The standard is included in Math I and Math II.

Example: Simplify √

Example: Simplify ( ) ⁄

Example: Simplify

Example: Simplify √

Solution: √ √ √ √ √

Note: Students should be able to simplify square roots in Math I. This is a foundational skill for simplifying the solutions generated by using the quadratic formula in Math II.


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Domain Cluster Standard A.APR Arithmetic with Polynomials and Rational Expressions A.SSE Seeing Structure in Expressions F.IF Interpreting Functions A.CED Creating Equations

Perform arithmetic operations on polynomials. (A.APR.1) Interpret the structure of expressions. (A.SSE.1-2) Write equations in equivalent forms to solve problems. (A.SSE.3) Interpret functions that arise in applications in terms of the context. (F.IF.4-6) Analyze functions using different representations. (F.IF.7-9) Create equations that describe numbers or relationships. (A.CED.1-4)

A.APR.1 A.SSE.2 A.SSE.3★ F.IF.4★ * F.IF.7a★ * F.IF.8a F.IF.9* A.CED.2★ *

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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 This standard is included in Math I, II and III. Throughout all three courses, students operate with polynomials. In Math I, focus on adding and subtracting polynomials (like terms) and multiplication of linear expressions. The primary strategy for this cluster is to make connections between arithmetic of integers and arithmetic of polynomials. In order to understand this standard, students need to work toward both understanding and fluency with polynomial arithmetic. Furthermore, to talk about their work, students will need to use correct vocabulary, such as integer, monomial, polynomial, factor, and term.

Example: Write at least two equivalent expressions for the area of the circle with a radius of kilometers. Example: Simplify each of the following: a. ( ) ( ) b. ( ) ( )

Example: A town council plans to build a public parking lot. The outline below represents the proposed shape of the parking lot.


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a. Write an expression for the area, in square feet, of this proposed parking lot. Explain the reasoning you used to find the expression.

b. The town council has plans to double the area of the parking lot in a

few years. They plan to increase the length of the base of the parking lot by p yards, as shown in the diagram below.

Write an expression in terms of x to represent the value of p, in feet. Explain the reasoning you used to find the value of p.

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 This standard is included in Math I, II and III.

Rewrite algebraic expressions in different equivalent forms such as factoring or combining like terms. Use factoring techniques such as common factors, grouping, the difference of two squares, or a combination of methods to factor quadratics completely. Students should extract the greatest common factor (whether a constant, a variable, or a combination of each). If the remaining expression is a factorable quadratic, students should factor the expression further. Students should be prepared to factor quadratics in which the coefficient of the quadratic term is an integer that may or may not be the GCF of the expression. Students rewrite algebraic expressions by combining like terms or factoring to reveal equivalent forms of the same expression. Connect to multiplying linear factors in A.APR.1 and interpreting factors in A.SSE.1b.

Example: Rewrite the expression ( ) into an equivalent quadratic expression of the form . Example: Rewrite the following expressions as the product of at least two factors and as the sum or difference of at least two totals. a. b. c.


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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 reveal the zeros of the function it defines.

A.SSE.3a ★ This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential.

Students factor quadratic expressions and find the zeroes of the quadratic function they represent. Students should be prepared to factor quadratics in which the coefficient of the quadratic term is an integer that may or may not be the GCF of the expression. Students explain the meaning of the zeroes as they relate to the problem. Connect to A.SSE.2

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. a. Rewrite the expression to reveal the linear factors. b. Identify the zeroes and intercepts of the expression and interpret what they mean in regards to the context. c. How long is the ball in the air?

Note: The standard A.REI.4 is in Math 2 (not Math I) and includes solving quadratic equations in one variable. For this standard in Math I, focus on quadratic expressions and how the factors can reveal the zeroes.

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.7★ * This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential.

In Math I, students graph linear, exponential and quadratic functions expressed symbolically. (Connect to F.IF.4) Students recognize how the form of the quadratic function provides information about key features such as intercepts and maximum/minimum values.

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. Note: Students can identify the max/min point by using the symmetry of the graph. If the x-intercepts are at and then the minimum point is when . Using the function, students can use as the input and calculate the output to get the maximum point.

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

F.IF.8 F.IF.8a is included in Math I, II and III. Students use the property of exponents to rewrite exponential expressions in different yet equivalent forms. Students use the form to explain the special characteristics of the function. In Math I, focus on factoring as a process to show zeroes, extreme values, and symmetry of the graph. Students should also interpret these features in terms of a context. At this level, completing the square is not expected. Students should be prepared


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to factor quadratics in which the coefficient of the quadratic term is an integer that may or may not be the GCF of the expression. Students must use the factors to reveal and explain properties of the function, interpreting them in context. Factoring just to factor does not fully address this standard.

Example: The quadratic expression represents the height of a diver jumping into a pool off a platform. Use the process of factoring to determine key properties of the expression and interpret them in the context of the problem.

Connect to F.IF.7a when students are graphing the functions to show key features and F.IF.4 when they are interpreting the key features given a graph or a table.


F.IF.4★ * This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential. This standard is included in Math I, II and III. Throughout all three courses, students interpret the key features of a variety of different functions. There are two categories of functions to consider. The first are the “classical” functions usually referred to by families.

In Math I, students focus on linear, exponential, and quadratic functions. (Connect to F.IF.7) There are also functions that are not included into families. For example, plots over time represent functions as do some scatterplots. These are often functions that “tell a story” hence the portion of the standard that has students sketching graphs given a verbal description. Students should have experience with a wide variety of these types of functions and be flexible in thinking about functions and key features using tables and graphs. When given a table or graph of a function that models a real-life situation, explain the meaning of the characteristics of the table or graph in the context of the problem. The following are examples of “classical” family of functions:

# paying visitors (thousands)

Daily profit (thousands of $)

0 0 1 5 2 8 3 9 4 8 5 5 6 0

Key features of a quadratic function are intervals of increase/decrease, positive/negative, maximum/minimum, symmetry, and intercepts.

Example: The table represents the relationship between daily profit for an amusement park and the number of paying visitors. a. What are the x-intercepts and y-intercepts and explain them in the

context of the problem. b. Identify any maximums or minimums and explain their meaning in

the context of the problem. c. Determine if the graph is symmetrical and identify which shape

this pattern of change develops. d. Describe the intervals of increase and decrease and explain them in

the context of the problem.


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Example: The graph represents the height (in feet) of a rocket as a function of the time (in seconds) since it was launched. Use the graph to answer the following: 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?

The following are examples of functions that “tell a story”: There are a number of videos on this site http://graphingstories.com Some are aligned to Math I while others are more appropriate for Math 2 or 3. The following are suggested videos for Math I: x Water Volume x Weight x Bum Height Off Ground x Air Pressure x Height of Stack Example: Marla was at the zoo with her mom. When they stopped to view the lions, Marla ran away from the lion exhibit, stopped, and walked slowly towards the lion exhibit until she was halfway, stood still for a minute then walked away with her mom. Sketch a graph of Marla’s distance from the lions’ exhibit over the period of time when she arrived until she left.




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d F.IF.9* Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

F.IF.9* This standard is included in Math I, II and III. Throughout all three courses, students compare properties of two functions. The representations of the functions should vary: table, graph, algebraically, or verbal description.

In Math I, students focus on linear, exponential, and quadratic.

t ( ) 0.2 38.6 0.4 32.2 0.6 25.8 0.8 19.4 1.0 13



( ) x –7 –6 –5 –4 –3 –2 –1

( ) 4 1.5 0 –0.5 0 1.5 4


year. Which job would you suggest Joe take? Justify your reasoning.


A.CED.2★ * This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential. This standard is included in Math I, II and III. Throughout all three courses, students: x Create equations in two variables x Graph equations on coordinate axes with labels and scales clearly labeling the axes defining what the values on the axes


x Students interpret the context and choose appropriate minimum and maximum values for a graph. In Math I, focus on linear, exponential and quadratic contextual situations for students to create equations in two variables. Limit exponential situations to only ones involving integer input values.


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3rd Nine Weeks

Quadratic equations can be written in a multitude of ways; and ( )( ) are commonly used forms (given that x and y are the two variables). Students should be flexible in using multiple forms and recognizing from the context, which is appropriate to use in creating the equation.

Example: The cheerleaders are launching t-shirts into the stands at a football game. They are launching the shirts from a height of 3 feet off the ground at an initial velocity of 36 feet per second. What function rule shows a t-shirt’s height h related to the time t? (Use 16t² for the effect of gravity on the height of the t-shirt.) Example: The local park is designing a new rectangular sandlot. The sandlot is to be twice as long as the original square sandlot and 3 feet less than its current width. What must be true of the original square lot to justify that the new rectangular lot has more area? Note: Students can construct the equations for each area and graph each equation. Students should select scales for the length of the original square and the area of the lots suitable for the context.

Pythagorean Theorem, Distance, and Midpoint

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8.G Geometry G.GPE Expressing Geometric Properties with Equations

Understand and apply the Pythagorean Theorem. (8.G.6-8) - Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: right triangle, hypotenuse, legs, Pythagorean Theorem, Pythagorean triple Use Coordinates to Prove Simple Geometric Theorems Algebraically (G.GPE.4-7)

8.G.6 8.G.7 8.G.8 G.GPE.6 G.GPE.7★

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8.G.6 Explain a proof of the Pythagorean Theorem and its converse.

8.G.6 Using models, students explain the Pythagorean Theorem, understanding that the sum of the squares of the legs is equal to the square of the hypotenuse in a right triangle. Students also understand that given three side lengths with this relationship forms a right triangle. Example 1: The distance from Jonestown to Maryville is 180 miles, the distance from Maryville to Elm City is 300 miles, and the distance from Elm City to Jonestown is 240 miles. Do the three towns form a right triangle? Why or why not?


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Solution: If these three towns form a right triangle, then 300 would be the hypotenuse since it is the greatest distance. 1802 + 2402 = 3002

32400 + 57600 = 90000 90000 = 90000 These three towns form a right triangle.

8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Example 1: The Irrational Club wants to build a tree house. They have a 9-foot ladder that must be propped diagonally against the tree. If the base of the ladder is 5 feet from the bottom of the tree, how high will the tree house be off the ground?

Solution: a2 + 52 = 92 a2 + 25 = 81 a2 = 56

= a = or ~7.5 Example 2: Find the length of d in the figure to the right if a = 8 in., b = 3 in. and c = 4in. Solution: First find the distance of the hypotenuse of the triangle formed with legs a and b. 82 + 32 = c2 642 + 92= c2 73 = c2

= in. = c

The is the length of the base of a triangle with c as the other leg and d is the hypotenuse. To find the length of d:

2 + 42 = d2 73 + 16 = d2

89 = d2

= in. = d

Based on this work, students could then find the volume or surface area.

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8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

8.G.8 One application of the Pythagorean Theorem is finding the distance between two points on the coordinate plane. Students build on work from 6th grade (finding vertical and horizontal distances on the coordinate plane) to determine the lengths of the legs of the right triangle drawn connecting the points. Students understand that the line segment between the two points is the length of the hypotenuse. NOTE: The use of the distance formula is not an expectation. Example 1: Solution: Find the length of . 1. Form a right triangle so that the given line segment is the hypotenuse. 2. Use Pythagorean Theorem to find the distance (length) between the two points. Example 2:

Find the distance between (–2, 4) and (–5, –6).

Solution: The distance between –2 and –5 is the horizontal length; the distance between 4 and –6 is the vertical distance. Horizontal length: 3 Vertical length: 10 102 + 32 = c2 100 + 9 = c2 109 = c2

= = c

Students find area and perimeter of two-dimensional figures on the coordinate plane, finding the distance between each segment of the figure. (Limit one diagonal line, such as a right trapezoid or parallelogram)

G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

G.GPE.6 This standard is in Math I and II. In Math I, students find the midpoint of a line segment. The midpoint partitions the ratio into 1:1 and thus from either direction the point is the same. Given two points on a line, find the point that divides the segment into an equal number of parts. The midpoint is always halfway between the two endpoints. The x-coordinate of the midpoint will be the mean of the x-coordinates of the endpoints and the y-coordinate will be the mean of the y-coordinates of the endpoints.

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Example: If you are given the midpoint of a segment and one endpoint. Find the other endpoint. a. midpoint: ( ) endpoint: ( ) b. midpoint: ( ) endpoint: ( )

Example: Jennifer and Jane are best friends. They placed a map of their town on a coordinate grid and found the point at which each of their house lies. If Jennifer’s house lies at (9, 7) and Jane’s house is at (15, 9) and they wanted to meet in the middle, what are the coordinates of the place they should meet?

G.GPE.7★ Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

G.GPE.7★ This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential. This standard provides practice with the distance formula and its connection with the Pythagorean Theorem. Use the coordinates of the vertices of a polygon graphed in the coordinate plane and use the distance formula to compute the perimeter and to find lengths necessary to compute the area.

Example: Calculate the area of triangle ABC with altitude CD, given A (–4, –2), B(8, 7), C(1, 8) and D(4, 4). Example: Find the perimeter and area of a rectangle with vertices at C (–1, 1), D(3, 4), E(6, 0), F (2, –3). Round your answer to the nearest hundredth.

Exponentials

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8.F Functions F.LE Linear, Quadratic, and Exponential Models F.IF Interpreting Functions

Define, evaluate, and compare functions. (8.F.1-3) - Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: functions, y-value, x-value, vertical line test, input, output, rate of change, linear function, non-linear function Construct and compare linear, quadratic, and exponential models and solve problems. (F.LE.1-3) Interpret expressions for functions in terms of the situation they model. (F.LE.4-6) Understand the concept of a function and use function notation. (F.IF.1-3)

Interpret functions that arise in applications in terms of the context. (F.IF.4-6)

8.F.3* (nonlinear examples) F.LE.1a,b,c ★ F.LE.2★ F.LE.3★ F.LE.5★ * F.IF.2*

F.IF.4★ *


A.CED Creating Equations A.REI Reasoning with Equations and Inequalities

Analyze functions using different representations. (F.IF. 7-9) Create equations that describe numbers or relationships. (A.CED.1-4) Represent and solve equations and inequalities graphically. (A.REI.10-12)

F.IF.7e★ F.IF.8b F.IF.9* A.CED.1★ * A.CED.2★ * A.REI.10* A.REI.11★ *

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F.LE.1★ Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

F.LE.1★ This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential. Students distinguish between a constant rate of change and a constant percent rate of change.

Example: Town A adds 10 people per year to its population, and town B grows by 10% each year. In 2006, each town has 145 residents. For each town, determine whether the population growth is linear or exponential. Explain. Example: Sketch and analyze the graphs of the following two situations. What information can you conclude about the types of growth each type of interest has? x Lee borrows $9,000 from his mother to buy a car. His mom charges him 5% interest a year, but she does not

compound the interest. x Lee borrows $9,000 from a bank to buy a car. The bank charges 5% interest compounded annually.

Students recognize situations where one quantity changes at a constant rate per unit interval relative to another.

Example: A streaming movie service has three monthly plans to rent movies online. Graph the equation of each plan and analyze the change as the number of rentals increase. When is it beneficial to enroll in each of the plans?

Basic Plan: $3 per movie rental Watchers Plan: $7 fee + $2 per movie with the first two movies included with the fee Home Theater Plan: $12 fee + $1 per movie with the first four movies included with the fee

Note: The intent of this example is not to formally introduce piecewise functions, but to provide a realistic context to compare different rates.

Students recognize situations where one quantity changes as another changes by a constant percent rate. x When working with symbolic form of the relationship, if the equation can be rewritten in the form ( ) ,

then the relationship is exponential and the constant percent rate per unit interval is r. x When working with a table or graph, either write the corresponding equation and see if it is exponential or locate at

least two pairs of points and calculate the percent rate of change for each set of points. If these percent rates are the same, the function is exponential. If the percent rates are not all the same, the function is not exponential.


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Example: A couple wants to buy a house in five years. They need to save a down payment of $8,000. They deposit $1,000 in a bank account earning 3.25% interest, compounded quarterly. How long will they need to save in order to meet their goal?

Example: Carbon 14 is a common form of carbon which decays exponentially over time. The half-life of Carbon 14, that is the amount of time it takes for half of any amount of Carbon 14 to decay, is approximately 5730 years. Suppose we have a plant fossil and that the plant, at the time it died, contained 10 micrograms of Carbon 14 (one microgram is equal to one millionth of a gram). a. Using this information, make a table to calculate how much Carbon 14 remains in the fossilized plant after n number

of half-lives. b. How much carbon remains in the fossilized plant after 2865 years? Explain how you know. c. When is there one microgram of Carbon 14 remaining in the fossil?

F.LE.2★ Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

F.LE.2★ This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential.

In Math I, students construct linear and exponential functions.

Students use graphs, a verbal description, or two-points (which can be obtained from a table) to construct linear or exponential functions.

Example: Albuquerque boasts one of the longest aerial trams in the world. The tram transports people up to Sandia Peak. The table shows the elevation of the tram at various times during a particular ride.

Minutes into the ride 2 5 9 14 Elevation in feet 7069 7834 8854 10,129

a. Write an equation for a function that models the relationship between the elevation of the tram and the number of minutes into the ride.

b. What was the elevation of the tram at the beginning of the ride? c. If the ride took 15 minutes, what was the elevation of the tram at the end of the ride?

Example: After a record setting winter storm, there are 10 inches of snow on the ground! Now that the sun is finally out, the snow is melting. At 7 am there were 10 inches and at 12 pm there were 6 inches of snow.

a. Construct a linear function rule to model the amount of snow. b. Construct an exponential function rule to model the amount of snow. c. Which model best describes the amount of snow? Provide reasoning for your choice.

Note: In order to write the exponential function as the amount of snow for every hour, connect to F.IF.8b. Students could start with ( ) where x is the number of 5 hour periods then rewrite it to be ( )(

) ( ) ( ) where x

is the number of hours since 7 am. Students need to be able to construct exponential functions from a table or two points that may or may not include 0 as an input value. Also Connect to F.LE.1, F.BF.1, and F.BF.2


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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.

F.LE.3★ This standard is included in Math I and Math III.

In Math I, focus comparisons between linear, exponential, and quadratic growth.

Students know that the value of the exponential function eventually is greater than the other function types.

Example: Kevin and Joseph each decide to invest $100. Kevin decides to invest in an account that will earn $5 every month. Joseph decided to invest in an account that will earn 3% interest every month.

a. Whose account will have more money in it after two years? b. After how many months will the accounts have the same amount of money in them? c. Describe what happens as the money is left in the accounts for longer periods of time.

Example: Compare the values of the functions ( ) , ( ) , and ( ) for .

F.LE.5★ * Interpret the parameters in a linear or exponential function in terms of a context.

F.LE.5★ * This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential. Use real-world situations to help students understand how the parameters of linear and exponential functions depend on the context.

Example: A plumber who charges $50 for a house call and $85 per hour can be expressed as the function . If the rate were raised to $90 per hour, how would the function change?

Example: Lauren keeps records of the distances she travels in a taxi and what it costs:

Distance d in miles Fare f in dollars 3 8.25 5 12.75

11 26.25

a. If you graph the ordered pairs ( ) from the table, they lie on a line. How can this be determined without graphing them?

b. Show that the linear function in part a. has equation . c. What do the 2.25 and the 1.5 in the equation represent in terms of taxi rides.

Example: A function of the form ( ) ( ) is used to model the amount of money in a savings account that earns 8% interest, compounded annually, where n is the number of years since the initial deposit.

a. What is the value of r? Interpret what r means in terms of the savings account? b. What is the meaning of the constant P in terms of the savings account? Explain your reasoning. c. Will n or ( ) ever take on the value 0? Why or why not?

Example: The equation ( ) models the rising population of a city with 8,000 residents when the annual growth rate is 4%.

a. What would be the effect on the equation if the city’s population were 12,000 instead of 8,000? b. What would happen to the population over 25 years if the growth rate were 6% instead of 4%?


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F.IF.4★ * This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential. This standard is included in Math I, II and III. Throughout all three courses, students interpret the key features of a variety of different functions. There are two categories of functions to consider. The first are the “classical” functions usually referred to by families. In Math I, students focus on linear, exponential, and quadratic functions. (Connect to F.IF.7) There are also functions that are not included into families. For example, plots over time represent functions as do some scatterplots. These are often functions that “tell a story” hence the portion of the standard that has students sketching graphs given a verbal description. Students should have experience with a wide variety of these types of functions and be flexible in thinking about functions and key features using tables and graphs. When given a table or graph of a function that models a real-life situation, explain the meaning of the characteristics of the table or graph in the context of the problem. The following are examples of “classical” family of functions: Key features of an exponential function include y-intercept and increasing/decreasing.

Example: Jack planted a mysterious bean just outside his kitchen window. Jack kept a table (shown below) of the plant’s growth. He measured the height at 8:00 am each day.

Day 0 1 2 3 4 Height (cm) 2.56 6.4 16 40 100

a. What was the initial height of Jack’s plant? b. How is the height changing each day? c. If this pattern continues, how tall should Jack’s plant be after 8 days?

The following are examples of functions that “tell a story”:

There are a number of videos on this site http://graphingstories.com Some are aligned to Math I while others are more appropriate for Math 2 or 3. The following are suggested videos for Math I: x Water Volume x Weight x Bum Height Off Ground x Air Pressure x Height of Stack

Example: Marla was at the zoo with her mom. When they stopped to view the lions, Marla ran away from the lion exhibit, stopped, and walked slowly towards the lion exhibit until she was halfway, stood still for a minute, then walked away with her mom. Sketch a graph of Marla’s distance from the lions’ exhibit over the period of time when she arrived until she left.



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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. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

F.IF.7★ * This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential.

F.IF.7e is included in Math I, II and III. In Math I, students graph linear, exponential and quadratic functions expressed symbolically. (Connect to F.IF.4) Students recognize how the form of the exponential function provides information about key features such as initial values and rates of growth or decay.

Example: The function ( ) ( ) models the amount of aspirin left in the bloodstream after x hours. Graph the function showing the key features of the graph. Interpret the key features in context of the problem.

F.IF.8* Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y=(1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.

F.IF.8* Students use the property of exponents to rewrite exponential expressions in different yet equivalent forms. Students use the form to explain the special characteristics of the function. Connect to F.IF.7a when students are graphing the functions to show key features and F.IF.4 when they are interpreting the key features given a graph or a table. Students can determine if an exponential function models growth or decay. Students can also identify and interpret the growth or decay factor. Students can rewrite an expression in the form ( ) as ( ) . They can identify as the growth or decay factor. Students recognize that when the factor is greater than 1, the function models growth and when the factor is between 0 and 1 the function models decay.


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Example: The projected population of Delroysville is given by the function ( ) ( ) where t is the number of years since 2010. You have been selected by the city council to help them plan for future growth. Explain what the function ( ) ( ) means to the city council members. Example: Suppose a single bacterium lands on one of your teeth and starts reproducing by a factor of 2 every hour. If nothing is done to stop the growth of the bacteria, write a function for the number of bacteria as a function of the number of days. Example: The expression ( ) represents the amount of a drug in milligrams that remains in the bloodstream after x hours. a. Describe how the amount of drug in milligrams changes over time. b. What would the expression ( ) represent? c. What new or different information is revealed by the changed expression?

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★ * This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential.

This standard is included in Math I, II and III. Throughout all three courses, students recognize when a problem can be modeled with an equation or inequality and are able to write the equation or inequality. Students create, select, and use graphical, tabular and/or algebraic representations to solve the problem.

In Math I, focus on linear and exponential contextual situations that students can use to create equations and inequalities in one variable and use them to solve problems.

Example: The Tindell household contains three people of different generations. The total of the ages of the three family members is 85.

a. Find reasonable ages for the three Tindells. b. Find another reasonable set of ages for them. c. In solving this problem, one student wrote ( ) ( )

1. What does C represent in this equation? 2. What do you think the student had in mind when using the numbers 20 and 56? 3. What set of ages do you think the student came up with?

Example: Mary and Jeff both have jobs at a baseball park selling bags of peanuts. They get paid $12 per game and $1.75 for each bag of peanuts they sell. Create equations, that when solved, would answer the following questions:

a. How many bags of peanuts does Jeff need to sell to earn $54? b. How much will Mary earn if she sells 70 bags of peanuts at a game? c. How many bags of peanuts does Jeff need to sell to earn at least $68?

Example: Phil purchases a used truck for $11,500. The value of the truck is expected to decrease by 20% each year. When will the truck first be worth less than $1,000? Example: A scientist has 100 grams of a radioactive substance. Half of it decays every hour. How long until 25 grams remain? Be prepared to share any equations, inequalities, and/or representations used to solve the problem.


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A.REI.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).

A.REI.10* This standard is included in Math I, II and III.

In Math I, students focus on linear and exponential equations and are able to adapt and apply that learning to other types of equations in future courses.

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.

Example: Which of the following points are on the graph of the equation How many points are on this graph? Explain. a. (4, 0) b. (0, 10) c. (-1, 7.5) d. (2.3, 5)

Example: Verify that (–1, 60) is a solution to the equation ( ) . Explain what this means for the graph of the

function.

A.REI.11★ * Explain why the x-coordinates 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.

A.REI.11★ * This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential.

This standard is included in Math I, II and III. Throughout all three courses, students use graphs, tables and algebraic methods (including substitution and elimination for systems) to solve equations and systems of equations. Students understand that numerical solution methods (data in a table used to approximate an algebraic function) and graphical solution methods may produce approximate solutions, and algebraic solution methods produce precise solutions that can be represented graphically or numerically. Students may use graphing calculators or programs to generate tables of values, graphs, or solve a variety of functions. In Math I, focus on linear and exponential functions.

Example: The functions ( ) and ( ) give the lengths of two different springs in centimeters, as mass is added in grams, m, to each separately.

a. Graph each equation on the same set of axes. b. What mass makes the springs the same length? c. What is the length at that mass? d. Write a sentence comparing the two springs.

Example: Solve the following equations by graphing. Give your answer to the nearest tenth.

a. ( ) b.


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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.2* This standard is included in Math I, II and III. In Math I, students should focus on linear and exponential functions and be able to extend and apply their reasoning to other types of functions in future courses.

Use function notation, evaluate functions for inputs in their domains Example: Evaluate ( ) for the function ( ) ( ) . Example: Evaluate ( ) for the function ( ) ( )

Interpret statements that use function notation in terms of a context

You placed a ham in the oven and, after 45 minutes, you take it out. Let f be the function that assigns to each minute after you placed the yam in the oven, its temperature in degrees Fahrenheit. Write a sentence for each of the following to explain what it means in everyday language. a. ( ) b. ( ) ( ) c. ( ) ( ) d. ( ) ( )

Example: The rule ( ) ( ) represents the amount of a drug in milligrams ( ), which remains in the bloodstream after x hours. Evaluate and interpret each of the following: a. ( ) b. ( ) ( ). What is the value of k? c. ( )

F.IF.9* Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

F.IF.9* This standard is included in Math I, II and III. Throughout all three courses, students compare properties of two functions. The representations of the functions should vary: table, graph, algebraically, or verbal description.

In Math I, students focus on linear, exponential, and quadratic. t ( ) 0.2 38.6 0.4 32.2 0.6 25.8 0.8 19.4 1.0 13



( ) x –7 –6 –5 –4 –3 –2 –1

( ) 4 1.5 0 -0.5 0 1.5 4


year.

Which job would you suggest Joe take? Justify your reasoning.


8.F.3* (nonlinear examples) Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

8.F.3* Students understand that linear functions have a constant rate of change between any two points. Students use equations, graphs and tables to categorize functions as linear or non-linear. Example 1: Determine if the functions listed below are linear or non-linear. Explain your reasoning.

1. y = –2x2 + 3 2. y = 0.25 + 0.5(x – 2) 3. A = πr2

4. 5.

X Y 1 12 2 7 3 4 4 3 5 4 6 7

Solution: 1. Non-linear 2. Linear 3. Non-linear 4. Non-linear; there is not a constant rate of change 5. Non-linear; the graph curves indicating the rate of change is not constant.

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A.CED.2★ * This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential.

This standard is included in Math I, II and III. Throughout all three courses, students: x Create equations in two variables x Graph equations on coordinate axes with labels and scales clearly labeling the axes defining what the values on the axes


x Students interpret the context and choose appropriate minimum and maximum values for a graph.

In Math I, focus on linear, exponential and quadratic contextual situations for students to create equations in two variables. Limit exponential situations to only ones involving integer input values.

Exponential equations can be written in a couple of ways; and ( ) are the most common forms (given that x and y are variables). Students should be flexible in using all forms and recognizing from the context which is appropriate to use in creating the equation.

Example: 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.


a. Write a rule to calculate the actual prize money in dollars won by the player finishing in nth place, for any positive integer n.

b. Graph the relationship between the first 10 finishers and the prize money in dollars. c. What pattern is indicated in the graph? What type of relationship exists between the two variables?

Geometric Sequences

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F.BF Building Functions F.IF Interpreting Functions

Build a function that models a relationship between two quantities. (F.BF.1-2) Understand the concept of a function and use function notation. (F.IF.1-3)

F.BF.2★ * F.IF.3*

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F.BF.2★ * Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

F.BF.2★ * This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential. This standard is included in Math I and 3. In Math I, students use informal recursive notation (such as 𝑇 is intended). NOW – NEXT equations are the first step in the process of formalizing a sequence using function notation. The NOW – NEXT representation allows students to explore and understand the concept of a recursive function before being introduced to symbolic notation in Math 3 such as or ( ) ( ) . (Connect to F.IF.3) In an arithmetic sequence, each term is obtained from the previous term by adding the same number each time. This number is called the common difference. In a geometric sequence, each term is obtained from the previous term by multiplying by a constant amount, called the common ratio.

Example: Given the following sequences, write a recursive rule: a. 2, 4, 6, 8, 10, 12, 14, … b. 1000, 500, 250, 125, 62.5, …

NOW – NEXT equations are informal recursive equations that show how to calculate the value of the next term in a sequence from the value of the current term. The arithmetic sequence represented by 𝑇 is the recursive form of a linear function. The common difference c corresponds to the slope m in the explicit form of a linear function, . The initial value of the sequence d may or may not correspond to the y-intercept, b. This varies based on the given information.


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Example: If the sequence is 37, 32, 27, 22, 17, … it can be written as 37, 37–5, 37–5–5, 37–5–5–5, 37–5–5–5–5, etc., so that students recognize an initial value of 37 and a –5 as the common difference.

The geometric sequence represented by 𝑇 is the recursive form of an exponential function. The common ratio b corresponds to the base b in the explicit form of an exponential function, y = abx. The initial value may or may not correspond to the y-intercept, a. This varies based on the given information.

Example: If the sequence is 3, 6, 12, 24,… it can be written as 3,3(2), 3(2)(2), 3(2)(2)(2), etc., so that students recognize an initial value of 3 and 2 is being used as the common ratio.

Note: Write out terms in a table in an expanded form to help students see what is happening. Translate between the two forms

Example: A concert hall has 58 seats in Row 1, 62 seats in Row 2, 66 seats in Row 3, and so on. The concert hall has 34 rows of seats. a. Write a recursive formula to find the number of seats in each row. How many seats are in row 5? b. Write the explicit formula to determine which row has 94 seats?

F.IF.3* Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

F.IF.3* A sequence can be described as a function, with the input numbers consisting of a subset of the integers, and the output numbers being the terms of the sequence. The most common subset for the domain of a sequence is the Natural numbers {1, 2, 3, …}; however, there are instances where it is necessary to include {0} or possibly negative integers. Whereas, some sequences can be expressed explicitly, there are those that are a function of the previous terms. In which case, it is necessary to provide the first few terms to establish the relationship.

Example: The sequence 1, 1, 2, 3, 5, 8, 13, 21,… has an initial term of 1 and subsequent terms. The first 7 terms are displayed. Since each term is unique to its position in the sequence then, by definition, the sequence is a function. Explore the sequence and evaluate each of the following: a. ( ) b. ( ) c. ( )

Example: A theater has 60 seats in the first row, 68 seats in the second row, 76 seats in the third row, and so on in the same increasing pattern. a. If the theater has 20 rows of seats, how many seats are in the twentieth row? b. Explain why the sequence is considered a function. c. What is the domain of the sequence? Explain what the domain represents in context.

Connect to arithmetic and geometric sequences (F.BF.2). Emphasize that arithmetic and geometric sequences are examples of linear and exponential functions, respectively.


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8.SP Statistics and Probability S.ID Interpreting Categorical and Quantitative Data

Investigate patterns of association in bivariate data. - Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: bivariate data, scatter plot, linear model, clustering, linear association, non-linear association, outliers, positive association, negative association, categorical data, two-way table, relative frequency Summarize, Represent, and Interpret Data on Two Categorical and Quantitative Variables. (S.ID.5-6) Interpret Linear Models (S.ID.7-9)

8.SP.1 8.SP.2 8.SP.3 8.SP.4 S.ID.5★ S.ID.6a, b, c ★ * S.ID.7★ S.ID.8★ S.ID.9

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8.SP.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

8.SP.1 Bivariate data refers to two-variable data, one to be graphed on the x-axis and the other on the y-axis. Students represent numerical data on a scatter plot, to examine relationships between variables. They analyze scatter plots to determine if the relationship is linear (positive, negative association or no association) or non-linear. Students can use tools such as those at the National Center for Educational Statistics to create a graph or generate data sets. (http://nces.ed.gov/nceskids/createagraph/default.aspx) Data can be expressed in years. In these situations it is helpful for the years to be “converted” to 0, 1, 2, etc. For example, the years of 1960, 1970, and 1980 could be represented as 0 (for 1960), 10 (for 1970) and 20 (for 1980).

Example 1: Data for 10 students’ Math and Science scores are provided in the chart. Describe the association between the Math and Science scores.

Student 1 2 3 4 5 6 7 8 9 10 Math 64 50 85 34 56 24 72 63 42 93 Science 68 70 83 33 60 27 74 63 40 96

Solution: This data has a positive association. Example 2: Data for 10 students’ Math scores and the distance they live from school are provided in the table below. Describe the association between the Math scores and the distance they live from school.


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Student 1 2 3 4 5 6 7 8 9 10 Math 64 50 85 34 56 24 72 63 42 93 Distance from School (miles)

0.5 1.8 1 2.3 3.4 0.2 2.5 1.6 0.8 2.5

Solution: There is no association between the math score and the distance a student lives from school. Example 3: Data from a local fast food restaurant is provided showing the number of staff members and the average time for filling an order are provided in the table below. Describe the association between the number of staff and the average time for filling an order.

Number of Staff 3 4 5 6 7 8 Average time to fill order (seconds) 56 24 72 63 42 93

Solution: There is a positive association. Example 4: The chart below lists the life expectancy in years for people in the United States every five years from 1970 to 2005. What would you expect the life expectancy of a person in the United States to be in 2010, 2015, and 2020 based upon this data? Explain how you determined your values.

Date 1970 1975 1980 1985 1990 1995 2000 2005 Life Expectancy (in years) 70.8 72.6 73.7 74.7 75.4 75.8 76.8 77.4

Solution: There is a positive association. Students recognize that points may be away from the other points (outliers) and have an effect on the linear model. NOTE: Use of the formula to identify outliers is not expected at this level.

Students recognize that not all data will have a linear association. Some associations will be non-linear as in the example below:


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8.SP.2 Know that straight lines are widely used to model relationships and between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, informally assess the model fit by judging the closeness of the data points to the line.

8.SP.2 Students understand that a straight line can represent a scatter plot with linear association. The most appropriate linear model is the line that comes closest to most data points. The use of linear regression is not expected. If there is a linear relationship, students draw a linear model. Given a linear model, students write an equation.

8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

8.SP.3 Linear models can be represented with a linear equation. Students interpret the slope and y-intercept of the line in the context of the problem.

Example 1: 1. Given data from students’ math scores and absences, make a scatterplot.

2. Draw a linear model paying attention to the closeness of the data points on either side of the line.

3. From the linear model, determine an approximate linear equation that models the given data (about y = – x + 95)

4. Students should recognize that 95 represents the y-intercept and – represents the slope of the line. In the context of

the problem, the y-intercept represents the math score a student with 0 absences could expect. The slope indicates that the math scores decreased 25 points for every 3 absences.

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5. Students can use this linear model to solve problems. For example, through substitution, they can use the equation to determine that a student with 4 absences should expect to receive a math score of about 62. They can then compare this value to their line.

8.SP.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

8.SP.4 Students understand that a two-way table provides a way to organize data between two categorical variables. Data for both categories needs to be collected from each subject. Students calculate the relative frequencies to describe associations. Example 1: Twenty-five students were surveyed and asked if they received an allowance and if they did chores. The table below summarizes their responses.

Receive Allowance

No Allowance

Do Chores 15 5 Do Not Do Chores 3 2

Of the students who do chores, what percent do not receive an allowance? Solution: 5 of the 20 students who do chores do not receive an allowance, which is 25%

S.ID.5 ★ Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

S.ID.5 ★ This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential.

When students are proficient with analyzing two-way frequency tables, build upon their understanding to develop the vocabulary. x The table entries are the joint frequencies of how many subjects displayed the respective cross-classified values. x Row totals and column totals constitute the marginal frequencies. x Dividing joint or marginal frequencies by the total number of subjects define relative frequencies, respectively.


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Conditional relative frequencies are determined by focusing on a specific row or column of the table. They are particularly useful in determining any associations between the two variables. Students are flexible in identifying and interpreting the information from a two-way frequency table. They complete calculations to determine frequencies and use those frequencies to describe and compare.

Example: At the NC Zoo 23 interns were asked their preference of where they would like to work. There were three choices: African Region, Aviary, or North American Region. There were 13 who preferred the African Region, 5 of them were male. There were 6 who preferred the Aviary, 2 males and 4 females. A total of 4 preferred the North American Region and only 1 of them was female.

a. Use the information on the NC Zoo Internship to create a two way frequency table. b. Use the two-way frequency table from the NC Zoo Internship, calculate:

i. the percentage of males who prefer the African Region ii. the percentage of females who prefer the African Region

c. How does the percentage of males who prefer the African Region compare to the percentage of females who prefer the African Region?

d. 15% of the paid employees are male and work in the Aviary. How does that compare to the interns who are male and prefer to work in the Aviary? Explain how you made your comparison.

S.ID.6★ Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models. b. Informally assess the fit of a function by plotting and analyzing residuals. c. Fit a linear function for a scatter plot that suggests a linear association.

S.ID.6★ This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential. In Math I, focus on linear and exponential. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models.

Example: In an experiment, 300 pennies were shaken in a cup and poured onto a table. Any penny ‘heads up’ was removed. The remaining pennies were returned to the cup and the process was repeated. The results of the experiment are shown below.

# of Rolls 0 1 2 3 4 5 # of Pennies 300 164 100 46 20 8

Write a function rule suggested by the context and determine how well it fits the data. Note: The rule suggested by the context is ( ) since the probability of the penny remaining is 50%.


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Example: Which of the following equations best models the (babysitting time, money earned) data?

Informally assess the fit of a function by plotting and analyzing residuals.

Year

(0=1990) Tuition

Rate 0 6546 1 6996 2 6996 3 7350 4 7500 5 7978 6 8377 7 8710 8 9110 9 9411

10 9800

A residual is the difference between the actual y-value and the predicted y-value ( ̂), which is a measure of the error in prediction. (Note: ̂ is the symbol for the predicted y-value for a given x-value.) A residual is represented on the graph of the data by the vertical distance between a data point and the graph of the function. A residual plot is a graph that shows the residuals on the vertical axis and the independent variable on the horizontal axis. If the points in a residual plot are randomly dispersed around the horizontal axis, a linear regression model is appropriate for the data; otherwise, a non-linear model is more appropriate. Example: The table to the left displays the annual tuition rates of a state college in the U.S. between 1990 and 2000, inclusively. The linear function ( ) has been suggested as a good fit for the data. Use a residual plot to determine the goodness of fit of the function for the data provided in the table.

Miles driven

(In thousands) Price

(In dollars) 22 17,998 29 16,450 35 14,998 39 13,998 45 14,599 49 14,988 55 13,599 56 14,599 69 11,998 70 14,450 86 10,998

Fit a linear function for a scatter plot that suggests a linear association. Example: The data below give number of miles driven and advertised price for 11 used models of a particular car from 2002 to 2006.

a. Use your calculator to make a scatter plot of the data. b. Use your calculator to find the correlation coefficient for the data above.

Describe what the correlation means in regards to the data. (Connect to S.ID.8) c. Use your calculator to find an appropriate linear function to model the

relationship between miles driven and price for these cars. d. How do you know that this is the best-fit model?


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S.ID.7★ Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

S.ID.7★ This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential.

Example: Data was collected of the weight of a male white laboratory rat for the first 25 weeks after its birth. A scatterplot of the rat’s weight (in grams) and the time since birth (in weeks) shows a fairly strong, positive linear relationship. The linear regression equation W=100+40t (where W = weight in grams and t = number of weeks since birth) models the data fairly well.

a. What is the slope of the linear regression equation? Explain what it means in context. b. What is the y-intercept of the linear regression equation? Explain what it means in context.

S.ID.8★ Compute (using technology) and interpret the correlation coefficient of a linear fit.

S.ID.8★ This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential. The correlation coefficient, r, is a measure of the strength and direction of a linear relationship between two quantities in a set of data. The magnitude (absolute value) of r indicates how closely the data points fit a linear pattern. If , the points all fall on a line. The closer | | is to 1, the stronger the correlation. The closer | | is to zero, the weaker the correlation. The sign of r indicates the direction of the relationship – positive or negative.

Note: Students often misinterpret the correlation coefficient to mean that a given linear model is a good or bad fit for the data especially since it is often provided when completing a linear regression on the calculator. The correlation coefficient, r, measures the strength and the direction of a linear relationship between two variables. It can be calculated without a linear model. It is independent of the linear model and thus does not describe the fit of the model. Students should use residuals to assess the fit of the function.

Example: The correlation coefficient of a given data set is 0.97. List three specific things this tells you about the data.

S.ID.9★ Distinguish between correlation and causation.


Some data leads observers to believe that there is a cause and effect relationship when a strong relationship is observed. Students should be careful not to assume that correlation implies causation. The determination that one thing causes another requires a controlled randomized experiment.

Example: A study found a strong, positive correlation between the number of cars owned and the length of one’s life. Larry concludes that owning more cars means you will live longer. Does this seem reasonable? Explain your answer. Example: Choose two variables that could be correlated because one is the cause of the other; defend and justify selection of variables.

Explore the website http://tylervigen.com/ for correlations for discussion regarding causation and association.

http://tylervigen.com/


4th Nine Weeks

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8.G Geometry G.CO Congruence

G.GPE Expressing Geometric Properties with Equations

G.GMD Geometric Measurement and Dimension

Understand congruence and similarity using physical models, transparencies, or geometry software. (8.G.1-5) - Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: translations, rotations, reflections, line of reflection, center of rotation, clockwise, counterclockwise, parallel lines, congruence, ≅, reading A’ as “A prime”, similarity, dilations, pre-image, image, rigid transformations, exterior angles, interior angles, alternate interior angles, angle-angle criterion, deductive reasoning, vertical angles, adjacent, supplementary, complementary, corresponding, scale factor, transversal, parallel Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres. (8.G.9) - Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: cones, cylinders, spheres, radius, volume, height, Pi Experiment with transformation in the plane. (G.CO.1)

Use Coordinates to Prove Simple Geometric Theorems Algebraically (G.GPE.4-7)

Explain Volume Formulas and Use Them to Solve Problems (G.GMD.1-3)

8.G.1a, b, c † 8.G.2 † 8.G.3 † 8.G.4 † 8.G.5 † 8.G.9 † G.CO.1

G.GPE.4 G.GPE.5

G.GMD.1 G.GMD.3★

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8.G.1 Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines.

8.G.1 Students use compasses, protractors and rulers or technology to explore figures created from translations, reflections and rotations. Characteristics of figures, such as lengths of line segments, angle measures and parallel lines, are explored before the transformation (pre-image) and after the transformation (image). Students understand that these transformations produce images of exactly the same size and shape as the pre-image and are known as rigid transformations.


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8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

8.G.2 This standard is the students’ introduction to congruency. Congruent figures have the same shape and size. Translations, reflections and rotations are examples of rigid transformations. A rigid transformation is one in which the pre-image and the image both have exactly the same size and shape since the measures of the corresponding angles and corresponding line segments remain equal (are congruent). Students examine two figures to determine congruency by identifying the rigid transformation(s) that produced the figures. Students recognize the symbol for congruency (≅) and write statements of congruency. Example 1: Is Figure A congruent to Figure A’? Explain how you know. Solution: These figures are congruent since A’ was produced by translating each vertex of Figure A 3 to the right and 1 down. Example 2: Describe the sequence of transformations that results in the transformation of Figure A to Figure A’. Solution: Figure A’ was produced by a 90º clockwise rotation around the origin.

8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

8.G.3 Students identify resulting coordinates from translations, reflections, and rotations (90º, 180º and 270º both clockwise and counterclockwise), recognizing the relationship between the coordinates and the transformation.

Translations Translations move the object so that every point of the object moves in the same direction as well as the same distance. In a translation, the translated object is congruent to its pre-image. Triangle ABC has been translated 7 units to the right and 3 units up. To get from A (1, 5) to A’ (8, 8), move A 7 units to the right (from x = 1 to x = 8) and 3 units up (from y = 5 to y = 8). Points B and C also move in the same direction (7 units to the right and 3 units up), resulting in the same changes to each coordinate.


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Reflections A reflection is the “flipping” of an object over a line, known as the “line of reflection”. In the 8th grade, the line of reflection will be the x-axis and the y-axis. Students recognize that when an object is reflected across the y-axis, the reflected x-coordinate is the opposite of the pre-image x-coordinate (see figure below).

Likewise, a reflection across the x-axis would change a pre-image coordinate (3, -8) to the image coordinate of (3, 8) -- note that the reflected y-coordinate is opposite of the pre-image y-coordinate. Rotations A rotation is a transformation performed by “spinning” the figure around a fixed point known as the center of rotation. The figure may be rotated clockwise or counterclockwise up to 360º (at 8th grade, rotations will be around the origin and a multiple of 90º). In a rotation, the rotated object is congruent to its pre-image.

Consider when triangle DEF is 180˚ clockwise about the origin. The coordinate of triangle DEF are D(2,5), E(2,1), and F(8,1). When rotated 180˚ about the origin, the new coordinates are D’(-2,-5), E’(-2,-1) and F’(-8,-1). In this case, each coordinate is the opposite of its pre-image (see figure below).

Dilations A dilation is a non-rigid transformation that moves each point along a ray which starts from a fixed center, and multiplies distances from this center by a common scale factor. Dilations enlarge (scale factors greater than one) or reduce (scale factors less than one) the size of a figure by the scale factor. In 8th grade, dilations will be from the origin. The dilated figure is similar to its pre-image. The coordinates of A are (2, 6); A’ (1, 3). The coordinates of B are (6, 4) and B’ are (3, 2). The coordinates of C are (4, 0) and C’ are (2, 0). Each of the image coordinates is ½ the value of the pre-image coordinates indicating a scale factor of ½.


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The scale factor would also be evident in the length of the line segments using the ratio: image length pre-image length Students recognize the relationship between the coordinates of the pre-image, the image and the scale factor for a dilation from the origin. Using the coordinates, students are able to identify the scale factor (image/pre-image).

Students identify the transformation based on given coordinates. For example, the pre-image coordinates of a triangle are A(4, 5), B(3, 7), and C(5, 7). The image coordinates are A(-4, 5), B(-3, 7), and C(-5, 7). What transformation occurred?

8.G.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two- dimensional figures, describe a sequence that exhibits the similarity between them.

8.G.4 Similar figures and similarity are first introduced in the 8th grade. Students understand similar figures have congruent angles and sides that are proportional. Similar figures are produced from dilations. Students describe the sequence that would produce similar figures, including the scale factors. Students understand that a scale factor greater than one will produce an enlargement in the figure, while a scale factor less than one will produce a reduction in size.

Example1: Is Figure A similar to Figure A’? Explain how you know. Solution: Dilated with a scale factor of ½ then reflected across the x-axis, making Figures A and A’ similar. Students need to be able to identify that triangles are similar or congruent based on given information. Example 2: Describe the sequence of transformations that results in the transformation of Figure A to Figure A’. Solution: 90° clockwise rotation, translate 4 right and 2 up, dilation of ½. In this case, the scale factor of the dilation can be found by using the horizontal distances on the triangle (image = 2 units; pre-image = 4 units)


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8.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

8.G.5 Students use exploration and deductive reasoning to determine relationships that exist between the following: a) angle sums and exterior angle sums of triangles, b) angles created when parallel lines are cut by a transversal, and c) the angle-angle criterion for similarity of triangle. Students construct various triangles and find the measures of the interior and exterior angles. Students make conjectures about the relationship between the measure of an exterior angle and the other two angles of a triangle. (The measure of an exterior angle of a triangle is equal to the sum of the measures of the other two interior angles) and the sum of the exterior angles (360°). Using these relationships, students use deductive reasoning to find the measure of missing angles. Students construct parallel lines and a transversal to examine the relationships between the created angles. Students recognize vertical angles, adjacent angles and supplementary angles from 7th grade and build on these relationships to identify other pairs of congruent angles. Using these relationships, students use deductive reasoning to find the measure of missing angles. Example 1: You are building a bench for a picnic table. The top of the bench will be parallel to the ground. If m⦟1 = 148°, find m⦟2 and m⦟3. Explain your answer. Solution: Angle 1 and angle 2 are alternate interior angles, giving angle 2 a measure of 148°. Angle 2 and angle 3 are supplementary. Angle 3 will have a measure of 32° so the m⦟2 + m⦟3 = 180° Example 2: Show that m⦟3 + m⦟4 + m⦟5 = 180° if line l and m are parallel lines and t1 and t2 are transversals. Solution: ⦟1 + ⦟2 + ⦟3 = 180° ⦟5 ≅⦟1 Corresponding angles are congruent; therefore, ⦟1 can be substituted for ⦟5. ⦟4 ≅ ⦟2 Alternate interior angles are congruent; therefore, ⦟4 can be substituted for ⦟2. Therefore ⦟3 + ⦟4 + ⦟5 = 180° Students can informally conclude that the sum of the angles in a triangle is 180° (the angle-sum theorem) by applying their understanding of lines and alternate interior angles.

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Example 3: In the figure below Line X is parallel to Line . Prove that the sum of the angles of a triangle is 180°.

Solution: Angle a is 35° because it alternates with the angle inside the triangle that measures 35°. Angle c is 80° because it alternates with the angle inside the triangle that measures 80°. Because lines have a measure of 180°, and angles a + b + c form a straight line, then angle b must be 65° Æ 180 – (35 + 80) = 65. Therefore, the sum of the angles of the triangle is 35° + 65° + 80°. Example 4: What is the measure of angle 5 if the measure of angle 2 is 45° and the measure of angle 3 is 60°? Solution: Angles 2 and 4 are alternate interior angles; therefore, the measure of angle 4 is also 45°. The measure of angles 3, 4 and 5 must add to 180°. If angles 3 and 4 add to 105°, the measure of angle 5 must equal 75°. Students construct various triangles having line segments of different lengths but with two corresponding congruent angles. Comparing ratios of sides will produce a constant scale factor, meaning the triangles are similar. Students solve problems with similar triangles.

G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

G.CO.1 This standard is in Math I and III. Students recognize the importance of having precise definitions and use the vocabulary to accurately describe figures and relationships among figures. Students define angles, circles, perpendicular lines, parallel lines, and line segments precisely using the undefined terms.

Example: Draw an example of each of the following and justify how it meets the definition of the term. x Angle x Circle x Perpendicular lines x Parallel lines x Line segment

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G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

G.GPE.4 Students determine the slope of a line and the length of a line segment using a variety of methods including counting intervals and using formulas. Students have prior experience with the slope formula. Students need to develop the distance formula as an application of the Pythagorean Theorem.

Use the concepts of slope and distance to prove that a figure in the coordinate system is a special geometric shape. Students recognize the length of the line segment is the same as the distance between the end points.

Example: The coordinates for the vertices of quadrilateral MNPQ are ( ), ( ), ( ), and ( ). a. Classify quadrilateral MNPQ. b. Identify the properties used to determine your classification c. Use slope and length to provide supporting evidence of the properties

Example: If quadrilateral ABCD is a rectangle, where ( ), ( ), ( ) and ( ) is unknown. a. Find the coordinates of the fourth vertex. b. Verify that ABCD is a rectangle providing evidence related to the sides and angles.

Conics is not the focus at this level, therefore students do not need to know about circles on the coordinate plane. Connect to G.GPE.5 to understand how slope can be used to determine parallel and perpendicular lines.

G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

G.GPE.5 Use the formula for the slope of a line to determine whether two lines are parallel or perpendicular. Two lines are parallel if they have the same slope and two lines are perpendicular if their slopes are opposite reciprocals of each other. In other words, the product of the slopes of lines that are perpendicular is ( ).

Example: Suppose a line k in a coordinate plane has slope

. a. What is the slope of a line parallel to k? Why must this be the case? b. What is the slope of a line perpendicular to k? Why does this seem reasonable?

Find the equations of lines that are parallel or perpendicular given certain criteria.

Example: Two points ( ), ( ) determines a line, ⃡ . a. What is the equation of the line AB? b. What is the equation of the line perpendicular to ⃡ . passing through the point ( )?

Connect to F.LE.2.

8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

8.G.9 Students build on understandings of circles and volume from 7th grade to find the volume of cylinders, finding the area of the base πr2 and multiplying by the number of layers (the height). find the area of the base and multiply by the number of layers

V = πr2h


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Students understand that the volume of a cylinder is 3 times the volume of a cone having the same base area and height or that the volume of a cone is the volume of a cylinder having the same base area and height.

A sphere can be enclosed with a cylinder, which has the same radius and height of the sphere (Note: the height of the cylinder

is twice the radius of the sphere). If the sphere is flattened, it will fill of the cylinder. Based on this model, students understand that the volume of a sphere is the volume of a cylinder with the same radius and height. The height of the cylinder is the same as the diameter of the sphere or 2r. Using this information, the formula for the volume of the sphere can be derived in the following way: V = πr2h cylinder volume formula V = πr2h multiply by since the volume of a sphere is the cylinder’s volume

V = πr22r substitute 2r for height since 2r is the height of the sphere

V = πr3 simplify Students find the volume of cylinders, cones and spheres to solve real world and mathematical problems. Answers could also be given in terms of Pi. Example 1: James wanted to plant pansies in his new planter. He wondered how much potting soil he should buy to fill it. Use the measurements in the diagram below to determine the planter’s volume. Solution: V = πr2h V = 3.14 (40)2(100) V = 502,400 cm3 The answer could also be given in terms of π: V = 160,000 π

V = πr2h or 𝑉 𝜋𝑟

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Example 2: How much yogurt is needed to fill the cone to the right? Express your answers in terms of Pi. Solution: V = πr2h V = π (32)(5) V = π (45) V = 15 π cm3

Example 3: Approximately, how much air would be needed to fill a soccer ball with a radius of 14 cm? Solution: V = πr3 V = (3.14)(143) V = 11.5 cm3

“Know the formula” does not mean memorization of the formula. To “know” means to have an understanding of why the formula works and how the formula relates to the measure (volume) and the figure. This understanding should be for all students. Note: At this level composite shapes will not be used and only volume will be calculated.

G.GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

G.GMD.1 Circumference of a circle:

Students begin with the measure of the diameter of the circle or the radius of the circle. They can use string or pipe cleaner to represent the measurement. Next, students measure the distance around the circle using the measure of the diameter. They discover that there are 3 diameters around the circumference with a small gap remaining. Through discussion, students conjecture that the circumference is the length of the diameter times. Therefore, the circumference can be written as . When measuring the circle using the radius, students discover there are 6 radii around the circumference with a small gap remaining. Students conjecture that the circumference is the length of the radius 2 times. Therefore, the circumference of the circle can also be expressed using .

Area of a circle:

Students may use dissection arguments for the area of a circle. Dissect portions of the circle like pieces of a pie and arrange the pieces into a figure resembling a parallelogram as indicated below. Reason that the base is half of the circumference and the height is the radius. Students use the formula for the area of a parallelogram to derive the area of the circle.

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Volume of a cylinder:

Students develop the formula for the volume of a cylinder based on the area of a circle stacked over and over again until the cylinder has the given height. Therefore the formula for the volume of a cylinder is . This approach is similar to Cavalieri’s principle. In Cavalieri’s principle, the cross-sections of the cylinder are circles of equal area, which stack to a specific height.

Volume of a pyramid or cone:

For pyramids and cones, the factor will need some explanation. An informal demonstration can be done using a

volume relationship set of plastic shapes that permit one to pour liquid or sand from one shape into another. Another way to do this for pyramids is with Geoblocks. The set includes three pyramids with equal bases and altitudes that will stack to form a cube. An algebraic approach involves the formula for the sum of squares ( ). After the coefficient

has been justified for the formula of the volume of the pyramid (

), one can argue that it

must also apply to the formula of the volume of the cone by considering a cone to be a pyramid that has a base with infinitely many sides.

Informal limit arguments are not the intent at this level.

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G.GMD.3★ Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

G.GMD.3★ This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential.

Example: The Southern African Large Telescope (SALT) is housed in a cylindrical building with a domed roof in the shape of a hemisphere. The height of the building wall is 17 m and the diameter is 26 m. To program the ventilation system for heat, air conditioning, and dehumidifying, the engineers need the amount of air in the building. What is the volume of air in the building?

Formulas for pyramids, cones, and spheres will be given.

http://mathworld.wolfram.com/Circle.html



Statistics U

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S.ID Interpreting Categorical and Quantitative Data

Summarize, Represent, and Interpret Data on a Single Count or Measurement Variable. (S.ID.1-3)

S.ID.1★ S.ID.2★ S.ID.3★

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S.ID.1★ Represent data with plots on the real number line (dot plots, histograms, and box plots).

S.ID.1★ This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential. A statistical process is a problem-solving process consisting of four steps:

1. Formulating a statistical question that anticipates variability and can be answered by data. 2. Designing and implementing a plan that collects appropriate data. 3. Analyzing the data by graphical and/or numerical methods. 4. Interpreting the analysis in the context of the original question.

The four-step statistical process was introduced in Grade 6, with the recognition of statistical questions. In middle school students describe center and spread in a data distribution. In Math I, students need to become proficient in the first step of generating meaningful questions and choose a summary statistic appropriate to the characteristic of the data distribution, such as the shape of the distribution or the existence of extreme data points. Students will graph numerical data on a real number line using dot plots, histograms, and box plots.

x Analyze the strengths and weakness inherent in each type of plot by comparing different plots of the same data. x Describe and give a simple interpretation of a graphical representation of data.

Students should use appropriate tools strategically. The use of a graphing calculator to represent the data should be utilized. Emphasis must be on analyzing the data display to make decisions, not on making the display itself.

Example: The following data set shows the number of songs downloaded in one week by each student in Mrs. Jones class: 10, 20, 12, 14, 12, 27, 88, 2, 7, 30, 16, 16, 32, 25, 15, 4, 0, 15, 6. Choose and create a plot to represent the data


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S.ID.2★ Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.


Given two sets of data or two graphs, students x Identify the similarities and differences in shape, center and spread x Compare data sets and summarize the similarities and difference between the shape, and measures of center and

spreads of the data sets x Use the correct measure of center and spread to describe a distribution that is symmetric or skewed x Identify outliers and their effects on data sets

The mean and standard deviation are most commonly used to describe sets of data. However, if the distribution is extremely skewed and/or has outliers, it is best to use the median and the interquartile range to describe the distribution since these measures are not sensitive to outliers.

Day Tip Amount Sunday $50 Monday $45 Wednesday $48 Friday $125 Saturday $85

Example: You are planning to take on a part time job as a waiter at a local restaurant. During your interview, the boss told you that their best waitress, Jenni, made an average of $70 a night in tips last week. However, when you asked Jenni about this, she said that she made an average of only $50 per night last week. She provides you with a copy of her nightly tip amounts from last week. Calculate the mean and the median tip amount.

a. Which value is Jenni’s boss using to describe the average tip? Why do you think he chose this value?

b. Which value is Jenni using? Why do you think she chose this value? c. Which value best describes the typical amount of tips per night? Explain why.

Example: Delia wanted to find the best type of fertilizer for her tomato plants. She purchased three types of fertilizer and used each on a set of seedlings. After 10 days, she measured the heights (in cm) of each set of seedlings. The data she collected is shown below. Construct box plots to analyze the data. Write a brief description comparing the three types of fertilizer. Which fertilizer do you recommend that Delia use? Explain your answer.

Fertilizer A 7.1 6.3 1.0 5.0 4.5 5.2 3.2 4.6 2.4 5.5 3.8 1.5 6.2 6.9 2.6

Fertilizer B 11.0 9.2 5.6 8.4 7.2 12.1

10.5 14.0 15.3 6.3 8.7 11.3

17.0 13.5 14.2

Fertilizer C 10.5 11.8 15.5 14.7 11.0 10.8 13.9 12.7 9.9 10.3 10.1 15.8 9.5 13.2 9.7

S.ID.3★ Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

S.ID.3★ This is a modeling standard which means students choose and use appropriate mathematics to analyze situations. Thus, contextual situations that require students to determine the correct mathematical model and use the model to solve problems are essential. Students understand and use the context of the data to explain why its distribution takes on a particular shape (e.g. Why is the data skewed? are there outliers?)


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= Modeling Standards Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★).

† = Compacted Standards for 7th Grade

Grouping of Standards: Standards are grouped into units, but within units, standards are not sequenced for instruction.

NOTE: This document is based on North Carolina DPI’s Unpacking Documents as of December 2015. As NCDPI staff is continually revising and updating these tools, it is important to also refer to the actual Unpacking Documents at http://www.ncdpi.wikispaces.net/ for the most current information.

The September 2015 adjustments to the standards notes are identified in purple to distinguish the instructional implications of the adjustments.

References This document is based on North Carolina DPI’s Unpacking Documents at http://maccss.ncdpi.wikispaces.net/Unpacking+Documents. These documents include examples, illustrations and references from the following websites:

Arizona Department of Education http://www.azed.gov/standards-practices/mathematics-standards-2/

Graphing Stories: http://graphingstories.com

Spurious Correlations: http://tylervigen.com/

Wolfram Mathworld: http://mathworld.wolfram.com/Circle.html

Illustrative Mathematics: https://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/531

Examples: Why does the shape of the distribution of incomes for professional athletes tend to be skewed to the right? Why does the shape of the distribution of test scores on a really easy test tend to be skewed to the left? Why does the shape of the distribution of heights of the students at your school tend to be symmetrical?

Students understand that the higher the value of a measure of variability, the more spread out the data set is. Measures of variability are range (100% of data), standard deviation (68-95-99.7% of data), and interquartile range (50% of data).

Example: On last week’s math test, Mrs. Smith’s class had an average of 83 points with a standard deviation of 8 points. Mr. Tucker’s class had an average of 78 points with a standard deviation of 4 points. Which class was more consistent with their test scores? How do you know?

Students explain the effect of any outliers on the shape, center, and spread of the data sets.

Example: The heights of Washington High School’s basketball players are: 5 ft. 9in, 5 ft. 4in, 5 ft. 7 in, 5ft. 6 in, 5 ft. 5 in, 5 ft. 3 in, and 5 ft. 7 in. A student transfers to Washington High and joins the basketball team. Her height is 6 ft. 10in. a. What is the mean height of the team before the new player transfers in? What is the median height? b. What is the mean height after the new player transfers? What is the median height? c. What affect does her height have on the team’s height distribution and stats (center and spread)? d. How many players are taller than the new mean team height?

e. Which measure of center most accurately describes the team’s average height? Explain.

Standards with * will be taught in multiple units – Linear with linear, quadratic with quadratic, etc.

http://www.ncdpi.wikispaces.net/

http://maccss.ncdpi.wikispaces.net/Unpacking+Documents

http://www.azed.gov/standards-practices/mathematics-standards-2/


http://tylervigen.com/



8th Grade Math I Curriculum Guide Caldwell County Schools · 2016. 9. 10. · 3 8th Grade Math I...

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Transcript of 8th Grade Math I Curriculum Guide Caldwell County Schools · 2016. 9. 10. · 3 8th Grade Math I...