grade 8 mathematics€¦ · web viewdetermine the domain and range for linear functions in given...

Algebra IScope and Sequence

Cluster 2Introduction to Linear Functions

22 days: 45 minutes per daya5 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.a6 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.A.2 The student uses the properties and attributes of functions. A.3 The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations.A.4 The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. A.5 The student understands that linear functions can be represented in different ways and translates among their various representations.

TEKS TAKS Obj. Instructional Scope

Possible ResourcesInstruction Assessment District

A.1A Describe independent and dependent quantities in functional relationships. [TAKS (9-11) Obj. 1]

A.1B Gather and record data and use data sets to determine functional relationships between quantities. [TAKS (9-11) Obj. 1]

A.1C Describe functional relationships for given problem situations and write equations or inequalities to answer questions arising from the situations. [TAKS (9-11) Obj. 1]

2, 3, 6 Collecting and Organizing Data

Conduct an experiment that allows one to collect and organize data. Example:Ask students to drop a plastic shopping bag weighted by a paperclip attached to the handles from various heights. The students should organize their data in a table similar to the one below. This table contains sample data. Student data may vary.

TAKS Mathematics Preparation, Grade 9 (Region 4), “Paperclip Skydiving Lesson,” pp. 58-68.

Algebra I: 2000 and Beyond (TEXTEAMS), “Perimeter of Rectangles,” pp. 63-66.

Online calculator tutorial:http://www.escweb.net/math/orhttp://www.escweb.net/

TAKS Mathematics Preparation, Grade 9 (Region 4), pp. 68.

TAKS Mathematics Preparation, Grade 9 (Region 4), pp.105-106 (1, 3).

www.mathbenchmarks.org

Algebra I Assessments (Charles A. Dana Center), “Stacking Cups,” pp. 73-79.

Possible Scope and Sequence, Algebra I Curriculum Cluster 2© 2005 Region 4 Education Service Center. All rights reserved. of 30

http://www.mathbenchmarks.org/


http://www.escweb.net/math/casio/


http://www.escweb.net/math/





A.1D Represent relationships among quantities using concrete models, tables, graphs, diagrams verbal descriptions, equations, and inequalities. [TAKS (9-11) Obj. 1]

A.1E Interpret and make decisions, predictions, and critical judgments from functional relationships. [TAKS (9-11) Obj. 1]

A.2B Identify mathematical domains and ranges and determine reasonable domain and range values for given situations, both continuous and discrete. [TAKS (9-11) Obj. 2]

A.2D Collect and organize data, make and interpret scatterplots (including recognizing positive, negative, or no correlation for data approximating linear situations), and model, predict, and make decisions and critical judgments in problem situations. [TAKS (9-11) Obj. 2]

o Identify variables. Let h represent the independent variable, drop height in centimeters, and let t represent the dependent variable, the average time of the fall in seconds.

o Simplify the table.

Example:

o Determine contextual domain/range in order to make the scatterplot.

math/casio/


Drop Height

Time

50 1775 23

100 29125 41150 53175 60200 68





A.3A Use symbols to represent unknowns and variables. [TAKS (9-11) Obj. 2]

A.3B Given situations, the student looks for patterns and represents generalizations algebraically. [TAKS (9-11) Obj. 2]

A.5A Determine whether or not given situations can be represented by linear functions. [TAKS (9-11) Obj. 3]

A.5B Determine the domain and range for linear functions in given situations. [TAKS (9-11) Obj. 3]

A.5C Use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions. [TAKS (9-11) Obj. 3]

o Make a scatterplot on chart paper. Time depends on the drop height; so let drop height be the x variable and time be the y variable. Ask whether there is a positive, negative, or no correlation between the independent and dependent variables.

o Determine a trend line without using regression.Ask students to use a meter stick to “eyeball” a good trend line on their chart paper. Have them approximate a slope and y-intercept on paper.


Time vs. Drop Height




o Connect the verbal descriptions to the table, graph, and function rule (equation).

o Enter the data into a graphing calculator. Remind students to use List 1 for domain values and List 2 for range values.

o Determine an appropriate window. To determine the window, have students draw a rectangle on the chart paper that includes all the points in the domain and range. To see the axes on the graphing calculator, use -1 as minimum value for both x and y.

o Create a scatterplot. Remind students to turn on the scatterplot and to select desired settings.





o Ask them to test their rule by graphing it over their scatterplot using the equation editor. Encourage students to explore graphing functions using the equation editor. At this time they should not use the calculator to determine the linear regression. As the students try various function rules, they should begin to develop an understanding of the results on a graph of changing the rate of change (slope) and the starting point (y-intercept). In this case, the trend line is approximately y = 0.35x.

o Compare and contrast domain and range for the data and the function. The data points are discrete, and the function rule is continuous.

A.1A Describe independent and dependent quantities in functional

1, 2, 10 Linear Modeling Algebra I: 2000 and Beyond (TEXTEAMS), “Valentine’s Day Idea,”


Algebra I Assessments






relationships. [TAKS (9-11) Obj. 1]

A.1B Gather and record data and use data sets to determine functional relationships between quantities. [TAKS (9-11) Obj. 1]

A.1C Describe functional relationships for given problem situations and write equations or inequalities to answer questions arising from the situations. [TAKS (9-11) Obj. 1]

A.1D Represent relationships among quantities using concrete models, tables, graphs, diagrams verbal descriptions, equations, and inequalities. [TAKS (9-11) Obj. 1]

A.1E Interpret and make decisions, predictions, and critical judgments from functional relationships. [TAKS (9-11) Obj. 1]

A.2AIdentify and sketch the general forms of linear (y = x) and quadratic (y = x2) parent

Explore a pattern in which the beginning value is not zero.Example:The base column (beginning value) of 2 stays the same. The total number of circles changes with each step.

pp. 14-35.

Online calculator tutorial:http://www.escweb.net/math/orhttp://www.escweb.net/math/casio/

(Charles A. Dana Center), “Mosaics,” pp. 25-28.

Algebra I Assessments (Charles A. Dana Center), “CD’s for the Band,” pp. 41-44.

Algebra I Assessments (Charles A. Dana Center), “Explorer’s Glide,” pp. 45-50.

Algebra I Assessments (Charles A. Dana Center), “Hot-Air Balloon,” pp. 55-60.

Algebra I Assessments (Charles A. Dana Center), “Stretched String,” pp. 81-85.










functions. [TAKS (9-11) Obj. 2]

A.2B Identify mathematical domains and ranges and determine reasonable domain and range values for given situations, both continuous and discrete. [TAKS (9-11) Obj. 2]

A.3A use symbols to represent unknowns and variables. [TAKS (9-11) Obj. 2]

A.3B Given situations, the student looks for patterns and represents generalizations algebraically. [TAKS (9-11) Obj. 2]


A.5B Determine the domain and range for linear functions in given situations. [TAKS (9-11) Obj. 3]

A.5C Use, translate, and make

o Organize data into tables.o Identify what stays the same and

what is changing from one term to the next.

o Find rates of change from tables and graphs.

o Generalize numerical procedures from patterns in tables to find function rules.

o Create scatterplots from tables.o Use a graphing calculator:

calculator basics, enter data, set window, plot data, and enter an equation.

o Discuss reasonable domain and range values, and set an appropriate viewing window.

o Use a graphing calculator: table, tableset, graph, and trace.

o Compare and contrast graphs of proportional and non-proportional





connections among algebraic, tabular, graphical, or verbal descriptions of linear functions. [TAKS (9-11) Obj. 3]

A.7AAnalyze situations involving linear functions and formulate linear equations or inequalities to solve problems. [TAKS (9-11) Obj. 4]

A.7BInvestigate methods for solving linear equations and inequalities using concrete models, graphs, and the properties of equality,select a method, and solve the equations and inequalities. [TAKS (9-11) Obj. 4]

A.7CInterpret and determine the reasonableness of solutions to linear equations and inequalities. [TAKS (9-11) Obj. 4]

linear functions. Emphasize that the graph of a proportional relationship always includes the origin. In most real-world contexts, the graph will not include Quadrant III values.

o Compare and contrast tables of proportional and non-proportional

linear functions. is constant for

proportional relationships. is

not constant for nonproportional relationships.

Step Number of Circles

1 6 62 10 53 144 18 4.55 22 4.46 26x y=4x+2

Introduce the linear parent function

(y=x). o Compare to the graph of the

function proportional and non-proportional linear functions.

Example: Compare the graph of the





linear parent function to the graph of y=4x+2. Connect the starting point (y-intercept) with the place the graph crosses the y-axis. Compare the steepness (rate of change, slope) of the two graphs.

The foundation for functions TEKS will be integrated throughout the course. The emphasis in this section is on number concepts and the basic skills of simplifying expressions and solving equations that may be used to describe functional relationships.

A.3A use symbols to represent unknowns and variables

A.4C Connect equation notation with function notation, such as y = x + 1 and f(x) = x + 1.

2, 10Variables & Expressions

Vocabulary:Variable- a letter or symbol used to represent a value that can changeConstant- a value that does not changeNumerical expression- contains only constants & operationsAlgebraic expression- contains variables, constants, and operationsEvaluate – Solve; Find the solution

Practice writing algebraic expressions from words:

Ex. Six times a number n

Text p 6- 11





6*n= 6n

Could also be stated as:The product of 6 and n6 equal groups of n

Ex.8 less than a number x

x- 8

Could also be written as:X minus 88 subtracted from xThe difference between x and 8

Practice writing algebraic expressions from problem situations:

Ex. Mary is 5 years younger than her sister Amy, who is a years old. Write an expression to show Mary’s age.

a = Amy’s ageMary= Amy less 5 years





Mary = a – 5

Practice evaluating algebraic expressions when given values:

Ex. Given a= 5, b= 8, c= 2

Evaluate.a – c (Substitute 5 for a and 2 for c)5 – 2 = 3

ab c (Substitute 5 for a, 8 for b, & 2 for c)

(5)(8) 40 = 20 2 2

Evaluate expressions.Use standard and function notation.

Examples:Find each output for the given input.o Standard notation





A.4B Use the commutative, associative, and distributive properties to simplify algebraic expressions. [TAKS (9-11) Obj. 2]

o Function notation ; find

and

Review of Order of Operations/Properties

Vocabulary:Commutative Property – Adding numbers or multiplying numbers in any order.ie 5+3 = 3+5

Text p 40-45Text p 46-51





8 x 4 = 4 x 8

Note that the commutative property DOES NOT apply to subtraction and division

5 – 3 does NOT equal 3 – 5

Associative Property - When only adding or multiplying, you can group any of the numbers together

ie 2 +9 + 3(2 + 9) + 3 or 2 + (9+3)11 + 3 2 + 12 14 14

Ie 2 * 5 * 8 (2*5) * 8 or 2* (5*8)10 * 8 2 * 40 80 80

Distributive Property- you can multiply a number by a sum or multiply by each number in the sum and then add. The result





is the same.

ie 5 ( 2 + 7)

5 (2) + 5 (7) or 5(9)10 + 35 or 4545 = 45

Terms- parts that are added or subtractedLike Terms- terms that contain the same variables raised to the same power. Constants are also like termsie 3x and 5x are Like Termsx3 and 3x are NOT like terms

x2 and y2 are NOT like terms

Coefficient- the number multiplied by a variableie 3x- the coefficient is 3

Combining Like terms:

Ex. 3x2 + 8 – 2x + 3+ 6xx2 Terms – 3x2





A.4A Find specific function values, simplify polynomial expressions, transform and solve equations, and factor as necessary in problem situations. [TAKS (9-11) Obj. 2]

x terms - -2x, 6xconstants- 8, 3Combine all like terms to get:3x2 +4x +11

Connect a concrete model to its algebraic representation.

Simplify expressions with a concrete model such as algebra tiles.Begin with a quick review of integers (Grade 7 TEKS). Then simplify algebraic expressions of increasing difficulty, including using the application of the distributive property.

Examples:

Algebra tiles or other concrete models





A.4A TAKS Mathematics





Find specific function values, simplify polynomial expressions, transform and solve equations, and factor as necessary in problem situations. [TAKS (9-11) Obj. 2]

Simplifying Expressions

Simplify expressions without a concrete model.Example:Simplify the following expression.

Answer: x – y

6 (a-8) + 12a +14 Distribute6a – 48 + 12a + 14Combine Like Terms6a + 12a – 48 + 1418a – 34

Apply concepts to contextual problemsExample: What is the area of the shaded portion of the rectangle below in terms of x?

Answer: 84x

Preparation, Grade 9 (Region 4), “Symbolic Manipulation Lesson—Linear Functions, Simplifying Expressions,” pp. 69-77.

Text p 46- 51

Online calculator


5x

20

2x

20

8




Use a graphing calculator to: o Perform integer operations.

Commutative, associative, and distributive properties

o Perform computations with rational numbers.

o Use parentheses.o Change fractions to decimals and

vice versa.Example:

Answer: Remind students to

add any whole numbers previously subtracted before changing the decimal part to a fraction.Students should become proficient in the use of a graphing calculator as a tool, not only to explore functional relationships,

tutorial:http://www.escweb.net/math/ orhttp://www.escweb.net/math/casio/










but also as a tool to perform computations so they can more readily explore algebraic concepts and procedures.

Technology Lab 1-1:Create a Table to Evaluate ExpressionsText p 12-13

Text p 12-13


A.4B Use the commutative, associative, and distributive properties to simplify algebraic expressions. [TAKS (9-11) Obj. 2]


2, 3, 10 Solving Equations

Review how to solve one- and two-step equations using concrete models, such as algebra tiles.

Algebra Lab 2-1 : Model One-Step EquationsText p 76

Connect a concrete model to its algebraic representation.

Example: Gwen and Ruth were playing “Guess My Number.” Gwen said, “If you take 3 times my number and subtract 4, the result is 2.” How could Ruth use algebra tiles to find Gwen’s number?

TAKS Mathematics Preparation, Grade 9 (Region 4), “Symbolic Manipulation Lesson—Linear Functions, Solving Equations,” pp. 78-80

Algebra tiles or other concrete models

Hands-On Equations

Algebra Lab 2-1 : Model One-Step EquationsText p 76

Algebra I: 2000 and

TAKS Mathematics Preparation, Grade 9 (Region 4), “Symbolic Manipulation Lesson—Linear Functions,” p. 85.


Algebra I: 2000 and Beyond (TEXTEAMS), Stays the Same,” pp. 265-277.










Solve one- and two-step equations without a concrete model.Solve one step equations by adding or subtracting

The goal of solving an equation is to find the solution of the equation, or the value of the variable that makes the equation true. The ultimate goal is:Variable = ???

ie x = 3

In order to achieve this goal, we must isolate the variable by making sure all the variable terms are on one side of the equation and all of the

Beyond (TEXTEAMS), “Solving Linear Equations,” Activities 1 and 2, pp. 251-263.

Algebra I Assessments (Charles A. Dana Center), “Geothermal Energy,” pp. 51-54.

Text p 77- 82





constant terms are on the other side of the equation.

We move terms from one side of the equation to the other by “un-doing” what was done. For example if the constant number was added to the variable term, we “undo” it by subtracting from the variable term.

x + 7 = 12 x + 7 – 7 = 12 (undo addition with subtraction)

Then whatever is done to one side of the equation, must be done to the other side.

An equation has an equal sign in the middle and must balance on both sides. It is like an old-fashioned scale. If you take 7 marbles off of one side, you must take 7 marbles off of the other side, otherwise, it will tip.





x + 7 – 7 = 12 x + 7 – 7 = 12 – 7 (same thing to both sides)

7 – 7 = 0 so we are left with only x on the left side of the equation. 12 – 7 = 5 So we have found the solution to the equation, x = 5

y – 10 = 8y – 10 + 10 = 8 (undo the subtraction with addition)y – 10 + 10 = 8 + 10 (same thing to both sides)-10 +10 = 0 so we are left with only y on the left side of the equation.8 + 10 = 18So we have found the solution to the equation, y = 18





Solve one-step equations by multiplying and dividing

-3z = 21Since z is multiplied by -3, we will “undo” it by dividing by -3-3 z = 21 -3

-3 z = 21- 3 -3 Divide the other side by -3. (Must do the same thing to both sides for the equation to balance)

-3/-3= 11 * z = zTherefore we are left with only z on the left side.

Text p 84- 90





21/-3= -7So the solution to our equation is z = -7

a = 58

Since a is divided by 8, we will “undo” it by multiplying by 8.

(8)a = 5 8

(8)a = 5 (8) 8 Do the same thing to the other side by multiplying by 8.

8/8 = 11* a = aTherefore we are left with only a on the left side.5 * 8= 40So we have found the solution to the equationa = 40





Solving 2 Step Equations:

To Solve 2- step equations, reverse the order of operations. “Undo” addition/subtraction first, then “undo” multiplication/division

Ex3x + 8 = 17“undo” addition by subtracting 8 from both sides

3x + 8 – 8 = 17 – 8

3x = 9

“undo” multiplication by dividing both sides by 3

3x = 9 3 3

Text p 92-98





X = 3

Variables on both sides of the equation:Algebra Lab: Model Equations with Variables on Both sidesText p 99

Apply concepts to contextual problems.Example:One inch equals 2.54 centimeters (to the nearest hundredth). A meter is 100 centimeters. Write an equation and solve to find the number of whole inches in a meter.

Answer: 39 inches

Algebra Lab: Model Equations with Variables on Both sidesText p 99

Text p 100- 106


2, 3, 10 Solving for Y

Solve for y using a concrete model such as algebra tiles.

Connect a concrete model to its

TAKS Mathematics Preparation, Grade 9 (Region 4), “Symbolic Manipulation Lesson—Linear Functions, Simplifying Expressions,” pp. 69-

TAKS Mathematics Preparation, Grade 9 (Region 4), “Symbolic Manipulation Lesson—Linear Functions,” p. 107 (5).









algebraic representation.Example:Solve for y. 4x + 2y = 10

Key

= x = y = 1

= -x = - y = -1

Build the model.

77.

Algebra tiles or other concrete models.


4 2 10x y =

4 2 104 4x yx x

Add -4x to create zero pairs.

=




Solve for y algebraically (without a concrete model).Example:Kendra wanted to find the possible number of bicycles, x, and the number of unicycles, y, that when subtracted result in 5.She remembered that bicycles have 2 wheels, and unicycles only have 1. Kendra decided to graph the following equation in her graphing calculator to represent the word problem.

2x – y = 5Write an equation she could use.


Divide by 2 on both sides

2 4 102 2 2y x

2 5Thereforey x




Answer: y = 2x – 5 or y = -5 + 2xThen graph.

Apply concepts to contextual problems.

Example:

Which term will have 99 blocks?Possible solution::

x Process y1 2(1) – 1 12 2(2) – 1 33 2(3) – 1 54 2(4) – 1 7x 2(x) – 1 2x+1





Introduction to Linear FunctionsCurriculum-Based Assessment 2


grade 8 mathematics€¦ · web viewdetermine the domain and range for linear functions in given...

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