Grade 8 Mathematics II_ Sem 1_1st6… · Web viewStudents use a variety of representations...

Algebra 2Scope and Sequence

Six Weeks 1: Foundations for Functions21 days: 45 minutes per day

a.5 Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to model mathematical situations to solve meaningful problems.

a.6 Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, language and communication, and reasoning (justification and proof) to make connections within and outside mathematics. Students also use multiple representations, technology, applications and modeling, and numerical fluency in problem-solving contexts.

2A.1 The student uses properties and attributes of functions and applies functions to problem situations.2A.4 The student connects algebraic and geometric representations of functions.

TEKS TAKS Obj. Instructional Scope

Possible ResourcesInstruction Assessment District

1, 10 ALGEBRA I REVIEW *SIMPLIFYING EXPRESSIONS *EVALUATIG EXPRESSIONS *ORDER OF OPERATIONS *SLOPE FROM POINTS *SIMPLIFYING SQUARE ROOTS

SIMPLIFYING EXPRESSIONS

Possible Scope and Sequence, Algebra 2 Curriculum Cluster 1: Foundations for Functions© 2005 Region 4 Education Service Center. All rights reserved. of 44




To simplify algebraic expressions,(expressions containing variables) all like terms must be combined.

Simplify 2x – (3x + 1) – 6 2x – 3x -1 -6 Use the distributive property. -x – 7 ombine like terms.

EVALUATING EXPRESSIONS

To evaluate algebraic expressions, substitute known values for the variables.

Evaluate 4x + 6y – 3w for x = 2, y = -2, and w = 5

Substitute these values for the variables





4 (2 ) + 6( - 2) – 3(5)

Perform the operations. 8 – 12 – 15 -4 -15 -19

ORDER OF OPERATIONS

When numeric expressions contain more than one operation, a rule is needed to let you know which order the operations will be performed. This rule is the Order of Operations.

Order of Operations

Step 1 Evaluate expresssions inside grouping symbolsStep 2 Evaluate all exponents.Step 3 Do all multiplication and division in order from left to rightStep 4 Do all addition and subtraction in order from left to right.

Sometimes a memory device is helpful to students. For the Order of Operations, one such device is called





PEMDAS. These letters represent the order in which the operations are to be performed. A phrase using these letters as the first letter of each word is “Please Excuse My Dear Aunt Sally.” Parentheses Exponents Multiplication Division Addition Subtraction

Ex. Evaluate this expression.

2 + 3 • 3 + 15 – 2 • 3

Multiply and divide in order from left to right.

2 + 9 + 15 – 6

Add and subtract left to right.

11 + 15 – 6 26 – 6 20

Ex. Evaluate





12 – 4 • 5 – 22

Evaluate powers

12 – 4 • 5 – 4 12 – 20 – 4 Multiply -8 – 4 Subtract -12 Subtract

Solve linear equations.

Solving equations are a very important part of geometry. Equations must be solved to fine values for lengths and measures.

Instruction:

5y + 5 = 7y – 9 Equation 5 = 2y – 9 Subtract 5y from each side 14 = 2y Add 9 to each side. 7 = y Divide each side by 2Ex.

10x – 4 = 6(x + 2) Equation

** Utilize Review sheet- Simplify/Evaluate Expressions


Textbook p. 27-30

**Utilize Review Sheet- Solving Equations




10x – 4 = 6x + 12 Use distributive property4x – 4 = 12 Subtract 6x from each side4x = 16 Add 4 to each sidex = 4 Divide each side by 4

Ex.12 = 5(- 3x + 2) – (x – 1) Equation12 = - 15x + 10 – (x – 1) Distibute 512 = - 15x + 10 – x + 1 Distribute (-)12 = - 16x + 11 Simplify1 = - 16x Subtract 11 from each side- 1 = x Divide by – 16 16

Ex.10 + 7y = 5 – y 4 3

This equation is a proportion. We solve





2A.2.A Use tools including factoring and properties of exponents to simplify expressions and to transform and solve equations.2A.4.A Identify and sketch graphs of parent functions, including linear ( f(x) = x), quadratic

proportions by cross multiplying.3 (10 + 7y ) = 4 ( 5 – y )30 + 21y = 20 – 4y Distribute30 + 25y = 20 Add 4y to each side25y = -10 Subtract 30 from each sideY = - 10 25 Divide each side by 25

Y= -2 Simplify fraction. 5

SLOPE FROM POINTS AND EQUATIONS

Key vocabulary:SlopeSlope-intercept equation

Finding Slope

The slope of a linear function is itsconstant rate of change. The slope

pp. 105 -110





(f(x)=x2), exponential, and logarithmic functions, absolute value, and square root of x.

canbe found by finding the ratio of the vertical change to the horizontal change. This is sometimes called the rise over the run. The slope of a line is the same between any two points on that line.

This is written as

slope = m = , where (x, y)

and (x1, y1) are points on the line.

Ex. Find the slope of the line that contains the points (2, -3) and (-4, 5).

m =

m = =





The slope is .

Recall from Algebra 1 that the slope-intercept form of a linear equation is y = mx + b, where m is the slope, and b is the y – intercept.

To determine slope and y – intercept from an equation, look for the values of m and b.

y = -3x + 5

In this equation, the slope is -3, and the y-intercept is 5.

When an equation is not in the form y = mx + b, solve it for y to put it in the slope intercept form.Ex. Find the slope of the line with the equation 3x – 4y = 12.

To determine this slope, solve the equation for y.





2A.2.A Use tools including factoring and properties of exponents to simplify expressions and to transform and solve equations.

3x – 4y = 12 - 4y = - 3x + 12

y =

y =

Now the equation is in the slope

intercept form. The slope is .

Simplifying Square Roots

Key Vocabulary:Radicand perfect square

For a square root to be in simplest form, the number under the radical (called the radicand), must contain no factors which are perfect squares. A perfect square is a number





2A.1A

obtained by squaring another number. 25 is a perfect square because squaring 5 gives 25.

Ex. is not in simplest form because 12 contains a factor of 4, which is a perfect square.

= = 2

Ex. Simplify .

= = 3

Dependency RelationshipsRecall from Algebra I that a function is a relationship between two quantities.





Identify the mathematical domains and ranges of functions and determine reasonable domain and range values for continuous and discrete situations.

2A.1B Collect and organize data, make and interpret scatterplots, fit the graph of a function to the data, interpret the results, and proceed to model, predict, and make decisions and critical judgments.

One quantity is input, and there is exactly one output value, or quantity, for each input value.The quantity that is output depends on the quantity that is input. Since the output value depends on the input value, the output value is the dependent variable. This makes the input value the independent variable.

Ex. The academic team at school needs to raise funds which will allow them to go out of town for competitions. They will be selling candles for this fund raiser. The candles will come in two sizes, a smaller one which will cost $5, and the larger one which will cost $8. The team hopes to raise at least $300.

Determine what the dependent and independent variables will represent.

The amount of money to be raised will depend on the number of candles sold. Therefore, the amount of money raised is the dependent variable. The number of candles sold will be the independent variable.

Domain and Range of Relations and

Textbook p. 44-45

TAKS Mathematics Preparation Grade 11 Exit (Region 4), “Input-Output Lesson,” pp. 15–32.

Algebra II Clarifying Activities www.tenet.edu/teks/math/clarifying/caalg2ks13.html

TAKS Mathematics Preparation Grade 11 Exit (Region 4), “Input-Output Lesson,” p. 33.

Algebra II Assessments (Charles A. Dana Center),“Hit the Wall,” pp. 3-10


http://www.tenet.edu/teks/math/clarifying/caalg2ks13.html






Functions

A function is graphed using a coordinate system, or coordinate plane. The coordinate system is formed by the horizontal, or x axis, and the vertical, or y axis. Usually the independent variable is graphed on the horizontal axis and the dependent variable is graphed on the vertical axis.Each input x and its output y form an order pair written in the form ( x, y), where the x value is the x – coordinate, and the y value is the y- coordinate.Any set of ordered pairs is called a relation. The domain is the set of all of the first numbers, or x values, of the ordered pairs. The range is the set of all the second numbers, or y values, of the ordered pairs.

Algebra II Clarifying Activities www.tenet.edu/teks/math/clarifying/caalg2ks13.html

Textbook p. 44-50

TAKS Mathematics Preparation Grade 11 Exit (Region 4), “Patterns and Function Relationships Lesson,” pp. 35–43.

Teacher Made Assessment








These ordered pairs are ( 1,2), ( -2,4 ), and ( 0,-3 ).

The domain is the set of x values. Written in set notation, this is {1, -2, 0}.The range is the set of y values. Written in set notation, this is {2, 4,-3}.

Interval notation is writing the sets for domain and range using the interval of the values of the numbers, and the symbols (, ), and [, ]. If we have a set of numbers written as -2< x < 6, its interval notation is written as (-2, 6). The parentheses are used because in the inequality, the -2 and 6 are not included.For the set 3 ≤ x < 8, the interval notation is written as [ 3,8), because we know the interval of numbers is from 3 to 8, and the 3 is included but the 8 is not included.

o Determine the domain and range of relations and functions from a graph, table, or symbolic representation and write in set notation or interval notation.

For example, identify the domain





and range for the function in the graph:

Determine domain and range in a problem solving context.o Investigate problem situations

and determine the domain and range.

For example: Consider a door that is 3 feet wide and can open so that it is flush to the wall. The distance, d, between the open end of the door and the doorjamb is dependent on the angle, A, at which the door is open. What is a reasonable domain and range for this situation?

www.mathbenchmarks.org


http://www.mathbenchmarks.org/




Distinguish between the domain and range of a situation and the domain and range of the function modeling the situation.

Ex. Determine a reasonable domain and range for this situation.

The freshmen science classes at school are going on a field trip to the zoo. If every student attends, there will be 80 students going. The trip is being taken on a day the school receives discounts, so the cost for each student will be $3.

The total cost for the trip depends on the number of students attending. This makes the cost for the trip the dependent variable , and the number of students attending will be the independent variable, or the domain.Therefore, a reasonable domain will be 0 to 80 students. The range will be 0 to $240.

Discrete and continuous functions.

If the domain and range of a function is a set of individual points, then the function is called a discrete function.

Algebra II Assessments (Charles A. Dana Center),“A Matter of Representation,” pp.15 -20

TAKS Mathematics Preparation Grade 11 Exit (Region 4), “Patterns and Function Relationships Lesson,” p. 44.


Textbook, p.51-56

Textbook p. 51-52




The graph of a discrete function consists of points that are not connected.A function in which you cannot name every individual point is called a continuous function. The graph of a continuous function is a continuous line or curve.

This is an example of a discrete function.





This is a continuous function.

o2A.1B Collect and organize data, make and interpret scatterplots, fit the graph of a function to the data, interpret the results, and proceed to model, predict, and make decisions and critical judgments.

Representations of Functions

Recognize and apply function notation: f(x), g(x), etc.o Find the value of a function (e.g..,

f(3 ) =).o Recognize that all functions can

be denoted in many forms, such as y =, d =, y1 =, and f(x) =. Determine which form is the most appropriate for a given situation. For example: f(3) = 10 allows students to see both the input and output values of a particular point in the functional relationship, and y = x2 + 4 can

Mathematical Models with Applications Algebra II Alignment, Teacher Handbook (Region 4,) “Data Analysis, Activities 1 – 4,” pp. 1-72–1-82.

Mathematical Models with Applications Algebra II Alignment (Region 4), “Assessment of Forensic Functions,” pp. 1-83 – 1-86.





be used when using the graphing calculator.

Illustrate a function as a table, set of ordered pairs, mapping diagram, equation, or graph. Decide which representation is the most meaningful for a given situation.For example, the function

can be represented graphically as:

In a table as:


x y−2 −14−1 −40 21 42 23 −44 −14




Or in a mapping as:

Create scatterplots using real world data. For example, represent the following data in a scatterplot:


-3

-2

-1

0

1

2

3

-28

-14

-4

2

4

x y




Determine whether relations are functions utilizing various methods such as investigating the table, definition of the function, or vertical line test.For example, the following relation is a function since each x-value maps to only one y-value.


-3

1

4

5

10

-10

2

5

12

x y




However, the following relation is NOT a function since 4 maps to both 2 and 5.

Representations of functions

Consider the function

f(x) = 2x + 1.

f(x) is the value obtained when putting in values for x. This is the function output. The output of a function is the y value. Therefore, f(x) = y.

Ex.To find the value of a function, just substitute the values for x.

When given the function,


-3

1

4

5

10

-15

-10

2

5

12

x y

Textbook p. 51-56




g(x) = x2 +3x +2find g(3).

To find this value, substitute 3 for x in the equation.

g(3) = 32 + 3(3) +2 = 9 + 9 + 2 = 20

2A.4A Identify and sketch graphs of parent functions, including linear (f(x) = x), quadratic (f(x) = x2), exponential (f(x) = ax), and logarithmic (f(x) = logax) functions, absolute value of x (f(x) = |x|), square root of x( ), and reciprocal of x (f(x) = 1/x).

2A.4BExtend parent functions with parameters such as a in f(x) = a/x and describe the effects of the parameter changes on the graph of parent functions.

2, 10

Transformations of Functions

Explore all parent functions: linear, absolute value, quadratic, square root, exponential, logarithmic and rational. use a table of values to graph

the following:

y = x, y = x2 , y = √x, etc.

Use graph paper to graph the parent equations, using a table of values.Next, graph the variations of the functions.

Pay attention to graphs that are moved horizontally or vertically, or up or

Textbook p. 59-64Graphing CalculatorsGraph paper

Mathematical Models with Applications Algebra II Alignment, Teacher Handbook (Region 4,) “Functional Transformations, Activities 1 – 3,” pp. 1-22 – 1-52.

Integrating Grade 11 TAKS Geometry Objectives Into Algebra II, Teacher Edition

Mathematical Models with Applications Algebra II Alignment, Teacher Handbook (Region 4), “Unit 1 Assessment,” pp. 1-88 – 1-93.

Algebra II Assessments (Charles A. Dana Center),“Data Dilemma,” pp. 44-42.

Algebra II Assessments (Charles A. Dana Center),“Slip Sliding Away,” pp. 43- 51. Integrating Grade 11





down.

Use symbolic representation to describe transformations on the parent function.o For example,

represents a translation of 4 units to the right and 3 units up with a vertical stretch by a factor of 2 from the parent function .

(Region 4), “Transformations and Foundations for Functions,” pp. 25 – 63.

Select from:Algebra II/Precalculus Institute (TEXTEAMS), “Ball Drop,” Introductory Activity; “Move the Monster Algebra II,” Activity 1; “Transformations on Generic Graphs,” Activity 4; and “Transformation Practice Algebra II,” Activity 5, pp. 1– 5, 31–34, 44 – 47, 48 – 53.

Textbook p. 67-73

TAKS Geometry Objectives Into Algebra II, Teacher Edition (Region 4), “Transformations and Foundations for Functions,” pp. 64–68.

www.mathbenchmarks.org

2A.1.A Identify the mathematical domains andranges of functions and determine reasonabledomain and range values for continuous and discrete situations.2A.4.B Extend parent functions with parameterssuch as a in f(x) = a/x and

TRANSFORMATIONS AND PARENT FUNCTIONS

Key vocabulary: parent functions transformations

From the lesson with the graphing calculator, you should be familiar with the graphs of parent functions.






describe the effectsof the parameter changes on the graph of theparent functions.2A.1.B Fit the graph of a function to data.2A.4.A Identify and sketch graphs of parent functions.

Functions are classified into familiesof functions. The simplest function is the parent function. It is the basic function before it has been changed in formed or is moved in any way. The movements or changes are called transformations.

We will do some graphs with tables of values to see what the transformations make happen.

The parent graph of f(x) = ǀ x ǀ has the points

X 0 1 -1f ( x ) 0 1 1

So to graph f(x) = ǀ x ǀ - 1, we will use values to determine points.

X 0 1 -1f (x) -1 0 0





This graph is just the graph of f(x) = ǀ x ǀ, moved down 1 unit.

Graph f(x ) = ǀ x – 1ǀ. We will again make a table of values for the graph.

x 0 1 -1 2f(x) 1 0 2 1

This kind of problem may require finding more points. Enough points should be found to accurately show the shape of the graph.





This is just the graph of f(x) = ǀ x ǀ, moved 1 unit to the right.

The transformations work for any of the parent functions.

f(x) = ǀ x ǀ - 3 shifts the graph of y = ǀ x ǀ down 3 units

f( x) = x2 + 3 shifts the graph of y = x2 up 3 units

f ( x ) = (x – 1 )2 shifts the graph of y = x2 one unit to the right

f(x) = -x2 + 2 turns the graph of y = x2 upside down and shifts it 2





units up

f(x) = ǀ x + 2 ǀ + 1 shifts the graph of y = ǀ x ǀ two units to the left and 1 unit up

In any functional relationship throughout the course, continue to ask students the following questions, which lead to the essence of a functions-based approach to algebra: What are the

independent/dependent variables? What is the relationship? Is the function/situation discrete or

continuous? What is an appropriate domain and

range of the function? Situation?What transformations from the parent function can be done to model the relationship between the variables?





A.7.A Analyze situations involving linear functionsand formulate linear equations or inequalities to solveproblems.A.7.C Interpret and determine the reasonableness of solutionsto linear equations and inequalities.

A.7.A Analyze situations

Solving Equations

Solve.2 ( 4x – 2 ) = 368x – 4 = 36 Use the distributive property.8x = 40 Add 4 to each side. x = 5 Divide each side by 8.

Solve.-3x – 12 = 2(x + 6) – 4- 3x – 12 = 2x + 12 – 4 Use the distributive property.-3x – 12 = 2x +8 Combine like terms.-12 = 5x +8 Add 3x to each side.- 20 = 4x Subtract 8 from each side.- 4 = x Divide each side by 4.

Solving Inequalities


p. 90-95

p. 90-95




involving linear functionsand formulate linear equations or inequalities to solveproblems.A.7.C Interpret and determine the reasonableness of solutionsto linear equations and inequalities.

Key Vocabulary: Compound Inequalities

To solve an inequality, the same steps and properties used to solveequations can be used, with one exception. The exception is whenmultiplying or dividing by a negative, reverse the inequality.

Solve and graph.

2x – 5 ≥ 52x ≥ 10 x ≥ 5

Inequalities should be written in set builder notation.

{ x | x ≥ 5 }

Read as “ the set of all x such that x is greater than or equal to 5”.





Solve and graph.

7 – 6x > 25 -6x > 18 x < -3 Divide by -6 and reverse the inequality.

{ x | x < -3}

When two or more sets are joined, the set formed is called a compound inequality. These inequalities can be formed by a conjunction





of two sets, using and, or by a disjunction of sets, using or.

Solve and graph.

-5 ≤ 2x + 1 < 9- 6 ≤ 2x < 8- 3 ≤ x < 4

{ x | -3 ≤ x < 4}

Solve and graph.

2x + 6 ≤ 10 or 3x + 2 ≥ 172x ≤ 4 3x ≥ 15 x ≤ 2 x ≥ 5





2A.10.G Use functions to model and make predictions in problem situations involving direct variation.

{ x | x ≤ 2 or x ≥ 5 }

Solving Proportions Key Vocabulary: Proportion

A proportion is an equation stating two fractions are equal.Proportions are solved by cross multiplying.

Solve.

4 (a + 3 ) = 16 Cross multiply.


p. 97-103




2A.4.A Identify and sketch

4a + 12 = 16 Use the distributive property.4a = 4 Subtract 12 from each side.a = 1 Divide each side by 4.

Solve.

4 ( x – 6 ) = 6 • 8 Cross multiply4x – 24 = 48 Use the distributive property.4x = 72 Add 24 to each side.X = 1 8 Divide each side by 4.





graphs of parent functions , including linear f(x) = x.

Graphing Linear Equations Using Tables of Values

Key Vocabulary: Linear Equations Standard Form

A linear equation is the equation of a line. Linear equations are usuallywritten in the form Ax + By = C, which is called the standard formof a linear equation.

To graph one of these equations, we will find values of x and y, which will be points on the line.

Graph y = 2x

To find x and y, we will make a table of values. We will choosevalues for x, and use the equation to find the corresponding y value.


p. 105-111




First, let x = 0.Use that value in the equation.

Y = 2xY = 2(0)Y = 0

When x = 0, y is also equal to 0.

X Y0 0

Next, we will choose x = 1.

Y = 2 ( 1 )Y = 2

X Y0 01 2





2A.4.A Identify and sketch

Although it only takes 2 points to determine a line, we will use three points to lessen the possibility of making a mistake.

Choose the next x value.We will choose 3.Using 3 in the equation gives us y = 6.

X Y0 01 23 6

We will plot these points on the coordinate plane.





graphs of parent functionsIncluding linear f(x) = x.

Graphing Linear Functions Use Slope and Intercepts

Key Vocabulary: Slope InterceptsSlope-Intercept Equation

Using slopes and intercepts is another way to graphlinear equations.

Recall from earlier algebra that slope is the ratio of vertical change to horizontal change. The x – intercept is the point where the graph crosses the x axis. The y – intercept is the point where the graph crosses the y axis.

The slope-intercept form of a linear equation is y = mx + b, where m represents the slope and b represents the y-intercept. If an equation iswritten in slope-intercept form, it can be graphed using those two points.


p. 105-110




We will graph 2x – 5y = 10 using only the slope and y-intercept.

To put the equation in slope- intercept form, we solve the equation for y.

2x – 5y = 10 -5y = - 2x + 10

y = - 2

The slope is , and the y –

intercept is -2.

Since the y – intercept is – 2, we will plot the point (0,-2). From that point, we will move 2 units up and 5 units to the right. The location of the second point is (5, 0).





We will graph y – x = 2, using x and y intercepts.

To find the x intercept,Let y = 0.

y – x = 2( 0)– x = 2- x = 2x = - 2

The x- intercept is -2.

To find the y- intercept,let x = 0.

y – x = 2y – 0 = 2y = 2





The y – intercept is 2.

Plot the points ( -2, 0) and ( 0, 2).





A.5.C Use, translate, and make connectionsamong algebraic, tabular, graphical, or verbal descriptions of linear functions.A.7.A Analyze situations involving linear functions and formulate linear equations A.7.C Interpret and determine thereasonableness of solutions to linearequations and inequalities.2A.2.A Use tools including factoringand properties of exponents to simplify

1, 2, 3, 4, 10

Exploring Graphs Activity with Graphing Calculator (Textbook Activity)

Linear FunctionSolving Linear Equations on A Graphing Calculator

EX.

Solve 5x + 6 = 3x + 7 with a graphing calculator.

Step 1. Use the y = , and let y1 be the leftside of the equation and y2 be the rightside of the equation.

Y1 = 5x + 6 Y2 = 3x + 7

p. 94 www.mathbenchmarks.org


p. 113-114





expressions and to transform and solveequations. Step 2. Go to the “Table Set” on the

calculator. Set the starting x value of thetable to 0. Set the step values to increase by .5 . Go to the “table” and you shouldsee x values, and for each x value, there isa y1 value and a y2 value.

Step 3. Scroll through the table untilyou find both y values to be the same.The corresponding x value is the solutionto the equation.

( This procedure is allowing the x valuesto change by ½. Looking at the graph ofthe equations, if the x value skips acrossthe intersection without giving a





value,the table set for the x changes should bechanged to another value, such as .25 or .33333.)


Grade 8 Mathematics II_ Sem 1_1st6… · Web viewStudents use a variety of representations...

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Transcript of Grade 8 Mathematics II_ Sem 1_1st6… · Web viewStudents use a variety of representations...