Rational Numbers (Eighth Grade Math)

TEKS Math Background: Rational Numbers

At Grade 8, students compare and order rational numbers in various forms including integers, percents, and positive and negative fractions and decimals. 8.1. A. Analysis of TAKS results shows that students struggle with this student expectation because….

Often, students find equivalent forms and/or compare and order rational numbers, including positive and negative integers, fractions and decimals. Special emphasis gets placed on this process because….

New at Grade 8 is expressing numbers in scientific notation, including negative exponents *8.1.d) To prepare for scientific notation, students review and extend their Grade 7 exponent work. They also write numbers in scientific notation.

2.1. Factors
TEKS 8.1.B
• A number that is divisible by n is also divisible by each factor of n. For example, any number divisible by 12 is also divisible by 2, 3, 4 and 6.
• The fundamental theorem of arithmetic states that every integer greater than 1 can be expressed as a product of prime fators in one and only 1 way, except for the order of factors.
• The number 1 is neither prime nor composite. The number 2 is the only even prime number.


A number is divisible by a second number is the number can be divided by he second number with a remainder of 0. A prime number is a whole number greater than 1 with exactly two factors, 1 and the number itself. A composite number is a whole number greater than 1 with more than two factors.

A composite number written as a product of rpime numbers is the prime factorization of the number. The greatest common factor (GCF) of two or mre number is the largest number that is a factor of all the numbers.

Example: 84 = 2 x 2 x 3 x 7
308 = 2 x 2 x 7 x 11
The GCF of 84 and 308 is 2 x 2 x 7, or 28.

2.2 Equivalent Forms of Rational Numbers

• Dividing the numerator and denominator of a rational number by common factors relies on two properties: a/a = 1 & a x 1 = a.
• Terminating decimals and repeating decimals are rational numbers. Non-terminating decimals that do not repeat, such as pi Π and√3 are irrational numbers.

A rational number is a number that can be written in the form a/b, where a is an integer and b is any nonzero integer. Two integers a and b are relatively prime if 1 is their only common factor. A fraction a/b is in simplest form when a and b are relatively prime. A terminating decimal is a decimal that stops. A decimal that repeats the same digit or group of digits forever is a repeating decimal.

Example 1/3 = 0.33333 = 0.3

5/12 = 0.41666…= 0.416

2.3 & 2.4 Comparing, Ordering, Adding and Subtracting Rational Numbers

TEKS 8.1. A 8.2.A. and 8.2.B Obj. 3
• You can always find a common denominator for two fractions by multiplying the two denominators, but this may not be the least common denominator.
• When two rational numbers have a common positive denominator, the greater numberator identifies the greater rational number.
• We add or subtract rational numbers with unlike denominators by changng to a simple calculation with like denominators.

Example: Evaluate 5(n+4) - 2 for n = 6.
5(6+4) – 2 = 5 (10) – 2 = 50 – 2 = 48.

1.2 Integers and Absolute Value
TEKs *.2A. Obj. 1 (814(a) Obj. 6

• Zero is the only integer that is its own opposite. Zero is neither positive nor negative. The absolute value of zero is zero
• A number and its opposite have the same absolute value.

Numbers that are the same distance from zero non a number line bu tin opposite directions are opposites, where 0 is its own opposite. 4 and -4 are opposites. Integers are the set of whole numbers and their opposites. A number’s distance from zero on the number line is its absolute value. The absolute value of -4, written as |-4|, is 4, and |4| is also 4.

1.3. Adding and Subtracting Integers

• You can rewrite subtracting a number as adding the opposite of that number.
• The opposite of the opposite of a number is the number itself.

Two numbers whose sum is 0 are additive inverses.
1.3 Adding Integers

Same sign: the sum of two positive integers is positive.
The sum of two negative integers is positive.

Different signs. Finds the absolute value of each. Subtract the lesser absolute value from the greater. The sum has the sign of the integer with the greater absolute value. The sum of opposites is zero.

1.3 Subtracting Integers

To subtract an integer, add its opposite.

a - b = c is equivalent to a = b + c
a + (-b) = (-B) + b + c = c, so a – b = a + (-b).

• Inverse operations undo each other. Addition and subtraction are inverse operations. Multiplication and division oare inverse operations.
• Dividing by a number is the same as multiplying by the reciprocal of that number.
1.4 Multiplying or Dividing Two Integers

The product or quotient of two integers with the same sign is positive.
The product or quotient of two integers with different signs is negative.

1.5 Properties of Numbers
• The Distributive Property combines two operations: 1) multiplication, 2) either addition or subtraction.
• The Distributive Property can be written and used in several different forms.
Commutative Property of Addition and Multiplication
a + b = b + a a x b = b x a
Associative Properties of Addition & Multiplication
(a + b) + c = a + (b + c) (a x b) x c = a x (b x c)
Identity Properties of Addition and Multiplication
a + 0 = 0 + a = a a x 1 = 1 x a = a
Distributive Property
a(b + c) = ab + ac (b + c)a = ba + ca
a(b - c) = ab - ac (b - c)a = ba - ca



1.6 & 1.7 Solving Equations by Adding, Subtracting, Multiplying and Dividing
MU
• Addition and subtraction are inverse operations that undo each other. Multiplication and division are also inverse operations that undo each other.
• When you solve an equation. Any operation that you do to one side, you must also do to the other.


A mathematical sentence with an equal sign is an equation. A solution to an equation is any value that makes the equation true. To find a solution of an equation, isolate the variable by using inverse operations (operations that undo each other) and the properties of equality.


Addition and Subtraction Properties of Equality
If you add the same number to each side of an equation, the two sides remain equal.
Arithmetic Algebra
10 = 5(2), so 10 + 3 = 5(2) + 3 If a = b, then a + c = b + c
If you subtract the same number from each side of an equation, the two sides remain equal.
Arithmetic Algebra
10 = 5(2), so 10 – 3 = 5(2) – 3 If a = b then a – c = b - c

Graphs & Functions (Algebra 1)

TEKS Math Background: Algebra 1 Chapter 5. Graphs and Functions

In Algebra 1, students describe functional relationships for given problem situations and write equations to answer questions arising form the situations (A.1. (C). Analysis of TAKS results shows that students struggle with this new student expectation.
[Camille, on blog put images of each of the problems the state isolated as representative of the TEKS.]

Students write function rules using tables and apply this skill to write function rules for given situations. We should place special emphasis on mastery of this skill, because it focuses on (A.1) (c).
• Also new for Algebra 1 is analyzing data and representing situations involving inverse variation A.11.B. Students use data given in tables and graphs to determine inverse variation… A.11.B.

Overview of Big Ideas behind Graphs * Functions in Algebra 1:

In this chapter, students further their knowledge of functions by building on the exploration of patterns in Chapter 1. Students move from the specific case of equations in one variable, found in Chapter 3, to the study of functions in two variables. Functional relationship relates the value of one variable, such as y, to the value of another variable, such as x. Functional relationships can be represented visually by graphs.

Students begin the chapter by graphing situations and interpreting the situations represented by a graph. They then use the vocabulary and mathematical notation for functions, and model functions with rules, tables and graphs. Students examine two particular functions called a direct variation and an inverse variation. Students will use what they learn about inverse variation in this lesson when they study rational functions in Chapter 12. Finally, they explore the special patterns in arithmetic sequence.

Relating Graph to Events
TEKS A.1.A A.!D. Obj. 1 A.2.C Obj. 2

The lesson will help students see how they can use mathematics to make a general representation of a specific relationship that occurs in their daily lives. Through graphing, students explore the idea of change, such as changes in speed, altitude, etc. (Fast and Furious). In addition to change, student investigates relationships between quantities by using graphs. Students will continue their work on graphs in their lessons on ….

Relations and Functions
TEKS A.1.D Obj. 1 a.4.A Obj. 2 A.4.C.

Building on their understanding of functions discussed in Chapter 1, students
Further their knowledge of functions by determining whether a relation is a function. Students should learn to correctly read f(x), as “F of x.” This will help them distinguish function notation from the more familiar notation used to show the multiplication of two variables, such as ab or a(b). Explain that the context usually makes the difference clear, and certain letters, such as f, g, h and k, are most commonly used for function notation.

Function Rules, Tables and Graphs: TEKS. A.1.A Obj 1. A.2. B Obj 2. A.5 c a.6 E

Help students see that the rule f(x) = x + 3 represents an infinite number of points, each of which can be written as an ordered pair in the form (x, x + 3). Similarly, the straight line that is the graph of y = x + 3 includes all these points, since the line extends forever. However, a table represents only a sample of points from this solution set—enough points to determine and check the straight line that contains all the point sin the solution set. Using these three views of a function, students can show whether a given function models discrete or continuous data.

Make sure students understand that graphs represent the relationship between two quantities. Although in this lesson the quantities are abstract variables in equations. Remind students (initially) that they learned that graphs can present the relationship between two real quantities, such as speed and time. Help students think of real world situations that could be represented by the graph of a quadratic equation, such as the flight of a projectile.

Writing a function Rule (TEKS A.1C Obj. 1 A.2B A.3B Obj 2
When students write a rule for a table of values, they must make sure that the apparent pattern does indeed hold true for each pair of points listed in the table. While two points are enough to determine a line, at least a third point should be used as a check to find and correct errors in calculation.

The fact that two points determine a line may not at first be obvious to students until they try to find a second and different line that includes the same two points.

Direct Variation

TEKs a.1.e Obj. 1

Variation is a statement of how one quantity changes relative to the way another quantity changes. A variation may be expressed as an equation that states a constant relationship between the values of two variables.

In direct variation, as the absolute vale of one variable increases, the abs. value of the second variable also increases proportionally. Thus for example in y = - 2x, as x increases from 10 to 20, the absolute value of y increases from 20 to 40. This means that ratio of y to x remains equal to the constant k, or y/x = k. Another way to express this direct relationship is with the proportion y1/x1 = y2/x2. So, in this example, -20/10 = -40/20.


Tell students that there are other kinds of variation, such as inverse variation, which will be discussed in the next lesson.
Inverse Variation TEKS a.1D Obj 2. A.11 B

Students extend their knowledge of variation studied in prior lessons to the study of inverse variation. In an inverse variation, as the value of one variable increases, the value of the other variable decreases so that the product of the two variables remains constant. Another way to express this inverse relationship is with an equation in the form of xy = k.

Make sure that students understand the difference between a direct variation and an inverse variation. They should recognize how the graphs of each reflect their respective definitions.


Describing Number Patterns

TEKS A.1.D Obj 1. a.3B, A.4.A Obj 2

An inductive argument reasons from observations of particular instances, (e.g. this crow is black, that crow is black) to a generality (all crows are black). Not all conclusions reached in this way are correct, but this kind of reasoning is very often used in daily life, and is often effective.

Students can use inductive reasoning to identify patterns, or sequences in sets of numbers. An arithmetic sequence is formed by adding a specific number of to each term after the first. Another type of sequence, the geometric sequence (see changed TEKS), will be introduced later. Deductive reasoning, as a next stage, using formal logic, will confirm or discredit their conclusions.

Solving Linear Equations and Their Graphs

Chapter 6, Algebra 1 Solving Linear Equations and their Graphs

I. Rate of Change and Slope
Focus on the TEKS you must put in the actual objects from the released tests.

In algebra 1, students develop the concept of slope as rate of change and determine slopes from graphs, tables and algebraic representations A.6.A. Analysis of TAKS Results shows that students struggle with this expectation.
In their first lessons, students solve for the rate of change using numbers and graphs. In Lessons 6-2, and 605, students find slopes using various slope forms and explore the parallel and perpendicular lines in Lesson 6. Place special emphasis on these lessons because they all focus on A.6.A.

Also, emphasize interpretation of the meaning of slope and intercepts. A.6.B. Students determine and analyze slope in different situations in Lessons 6-1, 6-3 and 6-5 (a.6.B).

Overview of Big Ideas Affecting Slope and Linear Equations

In the previous chapter, students learned the connection between a function and its graph. In this chapter, students focus on the connection between a linear equation and its graph. This topic is introduced first by a discussion of rate of change that leads to the definition of the slope of a line. The idea of slope is then expanded to include the SIF of a linear equation, and the use of this form to graph an equation. Using these skills, students write linear functions and interpret graphs to solve real world problems. Students then learn to write a linear equation in standard form, followed by graphing and writing linear equations when given various kinds of information about a line.

The characteristics of the slopes and graphs of both parallel and perpendicular lines are discussed. Students apply the preceding skills as they write the equation of a trend line and a line of best fit. Graphing a linear equation is extended to graphing and translating absolute value equations.
I. Rate of Change and Slope A.6.A A.6.B Obj. 3
The steepness of a road going up a hill and the pitch of a roof can both be expressed as the ratio of the rise over the run, or the slope. Students may at first confuse the order of the ration, since they are accustomed to putting x first when writing the coordinates of an ordered pair. This may lead them to place the difference in x values, or run, incorrectly in the numerator. Emphasize that the rise is the vertical change and the run is the horizontal change.


II. Slope Intercept Form A.2.A, Obj. 2, a.6.a. a.6.D, a.6.E, A.6.G Obj 3 a.4 C.

The lesson begins by showing students that all linear functions are related. That is linear functions are a family of functions where f(x) = x is the linear parent function.
Remind students that in Chapter 5, they learned that a function in the form of y = kx is a direct variation. Have student compare y = kx to y = mx + b . Lead students to understand that the slop of a linear function through the origin is a constant of variation.

Students may find it helpful to express slope as change in y/change in x, where x delta means change in as the change in y divided by the change in x.

III. Applying Linear Functions

A.1.E Obj. 1, a.2.C Obj. 2; a.6.B a.6F, Obj 3. a.5.B
Building on the concepts and skills discussed in the previous two lessons, in this section students use linear functions to model real world situations. They analyze application problems, write linear functions and create graphs to represent these functions. Point out to students that importance of choosing a reasonable domain to represent the original situation.

IV. Standard Form
Tell students that a linear equation in standard form is useful in making quick graphs. Suggest that an efficient way of graphing an equation in this form is to create a table of values for the equation by substituting 0 for x and finding the corresponding y value. Then substituting 0 for y and finding the corresponding x value. If more points are needed, it may be convenient to evaluate the equation using 1 for x and then for y.
Explain to students why finding a third point is helpful.

V. Point Slope From and Writing Linear Equations
A.5.A, A.6.A. a.6.B a.6.D Obj 3
Point out to students that using the point slope form of a linear equation is a generalization of the more specific slope-intercept form, in which the point is specifically designated to be the y-intercept. Students can use the point slope form to write equations that model sets of data.




VI. Parallel and Perpendicular Lines
A.6.A, A.6.D Obj 3.

Two lines in the same plane that have the same slope must either be parallel or coincide. In the latter case, the two equations are equivalent, as for example, x + 1 = y and 3 x = 3y – 3. (Student may not realize that this is just another case of equivalent equations having exactly the same solution set.) The family of lines with a given slope, m = 5, for example, consists of infinitely many lines, all which are parallel.
Discuss with the students the slopes of a family of lines with one common point. Graph examples of such lines. By finding the slope of each line, show students that for two perpendicular lines that the slope of one line is the negative reciprocal of the other.


VII. Scatter Plots and Equations of Lines TEKs a.1.B Obj 1 A.2.C a.2.D obj 2.

Real-world data, when graphed, rarely falls exactly along a line. Data are classified as linear if the points approximate a line on a graph. When this is the case, there are techniques to approximate an equation for a line of best fit. The graphing calculator makes this tasks easy (input TI-84 activities), by correlating equations of “best fit” to actual entered data.

Because real-world data many not be linear, the range (the difference between the value of the greatest point and the least) is called a measure of scatter. Measures of scatter indicate whether the measurements in a distribution are bunched together or scattered apart. In statistics, other measures of scatter include deviation, variance, and standard deviation.


VIII. Graphing Absolute Value Equations

TEKS Prepares for TEKS 2.A.4.A, 2.A.4.B
Translation is one of three commonly used transformations: translation or slide, rotation or turn, and reflection or flip. These three transformations are called isometries or rigid motions, because the size and the shape of the figure remain unchanged.

Dilation, such as enlarging a figure using copying machines, is another transformation but it is not rigid motion because distances within the figure are changed. Students learn that graphs of various absolute value equations can be thought of as translations of the equation y = |x|. Remind students that they studied absolute value equations in Chapter 4.

Systems of Linear Equations

Chapter 7 Algebra 1, Resources - Systems of Equations and Inequalities
TEKS Focus
In Algebra 1, students analyze situations and formulate systems of linear equations in two unknowns to solve problems. A.8.A. TAKS Analysis again reveals that students struggle with this expectation.

In Lessons 7-1 through 7-3 students solve problems involving systems of linear equations using graphing, substitution and elimination of one variable. In Lesson 7-4, students apply the appropriate method to solve real world problems. Place special emphasis on these lessons because they focus on A.8.A.

New for Algebra 1, students investigate methods for solving linear inequalities using graphs A.7.B, Student us graphs to solve systems of linear inequalities in Lesson 7-5. a.7.B

Big Ideas Overview for Systems of Equations and Inequalities

This chapter examines systems of two linear equations and system of two linear inequalities. Students first graph two equations on a single coordinate system and identify the point of intersection to find the solution. When the solution to the system is not apparent graphically, algebraic methods such as substitution and elimination enable students to find the coordinates of the point of intersection. Students will learn that some systems may be solved easily by graphing, while other systems may need to be solved algebraically or with a graphing calculator.


7.1 Solving Systems by Graphing A.8.A. A.8.B, A.8.C Objective 4

Two or more linear equations together (with the same variables) are called a system of linear equations. The solution to a system of linear equations is any ordered pair that satisfies all equations in the system.

The graph of a system of linear equations can take three distinct forms: the lines intersect so there is one solution. The lines are parallel so there are no solutions, or their lines are the same so there are infinitely many solutions.

Students often find an infinite number of solutions perplexing. Remind them that the solution to a linear equation such as y = 2x + 1 is the infinite set of ordered pairs (x, y). that makes the equation true. The graph of this solution set forms a line that never ends. So two lines that lie on top of one another intersect at each and every point.

Consider reviewing with students the y-intercept form of a linear equation, y = mx + b, and how to graph a line. For example, find the slope m and y-intercept b, plot the point (0.b) use the slope as a ratio of rise over run to plot one or more points, and connect the points with a straight line. Students can also use the slopes and y-intercepts to identify the number of solutions in a system: different slopes mean the lines intersect at one point, same slope but different y-intercepts mean the lines are parallel and have no intersections, and same slope and same y-intercept means the lines have infinitely many solutions.

7.2 Solving Systems Using Substitution
A.8.A, A.8.B, Objective 4

The substitution method involves replacing one variable with an equivalent expression containing only the other variable, solving for the remaining variable, and then substituted the value for the other variable into either original equation, and finally solving for the value of the original variable.

Some students have difficulty deciding which equation and variable to being solving for. Have students identify al l variables with coefficients of 1 0r -1 and choose one of these to solve for...

You may also want to present that systems of equations with no solution result in false equations, such as 7 = -2, while systems with infinitely many solutions will result in a universally true equation, such as 3 + 2 = 5. In both special cases, the variable is eliminated from the equation.



7.3 Solving Systems Using Elimination TEKS A.8(A). A.8.B, A.8.C Obj. 4
As with the substitution method, the goal of the elimination method is to obtain a one-variable equation, to solve for the value of the variable, and to substitute this value into either original equation to find the value of the other variable. The primary difference between the methods involves how the one-variable equation is obtained.

Guide students in identifying which variable is easiest to eliminate. When coefficients are not the same or not opposites, multiplication is needed. Remind students to multiply all terms on both sides of the equation, since they are applying the Multiplication Property of Equality.

Many students also find subtraction of polynomials difficult. Remind students that subtracting a negative is the same as adding a positive. (-- ) Consider showing students how they can take the opposite of the entire polynomial and add. Point out that this is simply another application of the Multiplication Property of Equality using -1 as the multiplier.

7.4 Application of Linear Systems
A.8.A, A.8.B, A.8.C. Obj 4.

In theory, any system of linear equations can be solved by graphing and then reading the coordinates of the point of intersection form the graph. However, unless the x-value and y-values of the solution are integers, accurately determining the value of the coordinates may be difficult. The TRACE function of a graphing calculator can be used for an approximate solution.

Similarly, any system of linear equations can be solved by substitution, but the calculations may be quite difficult and lengthy when neither equation has a variable with a coefficient of 1 or -1, such as rewriting 7x + 3y = 13 -3/7y + 13/7.

As with the other methods, it is theoretically possible to solve any system of two linear equations by first multiplying to make the coefficients of one of the variables into additive inverses, then using the elimination method. However, the calculations again may be too cumbersome to make this the best approach.

Finally, there are some systems that can be reasonably solved by any one of the methods taught. One student might decide to use substitution while another might elect to multiply an equation and use elimination.


7.5 Linear Inequalities
TEKS a.1.C, a.1.D, Obj 1, A.7.B Obj. 4

Many students benefit from a comparison of linear inequalities in one variable with linear inequalities in two variables, as shown initially in these lessons. Emphasize that One-variable solutions are graphed on a coordinate plane.

A linear inequality in two variables is different from a linear equation in two essential ways. First, the solution set is a region defined by a boundary line rather than a point or line. Second, multiplying both sides of an inequality by a number less than zero reverses the direction of the inequality symbol.

In general, when an inequality is written so that y is isolated on the left, the < and < symbols mean that the solution region is below the boundary line and the > and > symbols mean the solution is above the boundary line. Use a dashed line for < and >, and a solid line for < and > symbols. The < and > symbols indicate that the boundary line is included in the solution region while < and > symbols indicate the boundary line is not included.


7.6 Systems of Linear Inequalities

Two or more linear inequalities together are called a system of linear inequalities. When picturing the solution region for a system of linear inequalities, it is visually powerful to use different colors to help clarify exactly which points make up the solution set. Tit is also important to emphasize whether or not points on the various boundary lines of the regions are included in the solution set by using either solid or dashed lines.

Because it easy for students to forget to reverse the inequality sign when multiplying a negative, it is particularly important to test points in the solutions region. It is also possible to graph a set of linear inequalities on the graphing calculator, using the shaded area to confirm test points.

Best Practices Math Books

Integers (8.1)

The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. A formal way of stating this: the integers are the only integral domain whose positive elements are well-ordered, and which has order preserved under addition. Like the natural numbers, the integers form a countably infinite set. The set of all integers is usually denoted in mathematics by a boldface Z (or blackboard bold, ), which stands for Zahlen (German for "numbers").