Equations Introduction

I. Intro to Equation

An equation is a mathematical sentence that says two expressions are equal. Equations always have an equal sign. An algebraic equation is one that has at least one variable. 

A linear equation is an equation that represents a straight line when graphed on a coordinate plane. The highest power of the variable in a linear equation is always 1. 

The standard form of a linear equation in one variable is:

ax + b = 0

II. Solving Equations

Sometimes when we have an algebraic expression in an equation, we would like to find a number or set of numbers that make the equation a true statement. That is, we want to find a number (or numbers) for the variable(s) which makes both sides of the equation equal. This number is called a solution to the equation and the process of finding the number is called solving (a set of numbers is called the solution set and the elements of it are called solutions).

When solving an equation for a variable, you must get the variable by itself or isolate the variable on one side. This is done by using operations such as addition, subtraction, multiplication and division in a way which maintains equality. 

Equality

Equality is a fundamental idea of algebra. It is indicated by the equal sign and means that two mathematical expressions have the same value. One way to think about equality (and equations) is as a balanced scale. As in the picture below, we can change the number of ovals on each side of the scale and it still remains balanced.

Properties of Equality

Properties of equality describe the relation between two equal quantities and that if an operation is applied on one side of the equation, then it must be applied on the other side to keep the equation balanced.

A. Solving One-Step Linear Equations by Addition or Subtraction

The addition property of equality and the subtraction property of equality state that if we add or subtract the same number on both sides of an equation, then the resulting sides of the equation will remain equal.

Khan: Representing a relationship with an equation

When you solve an equation, you find the value of the variable that makes the equation true. In order to solve the equation, you isolate the variable. Isolating the variable means rewriting an equivalent equation in which the variable is on one side of the equation and everything else is on the other side of the equation.

When the equation involves addition or subtraction, use the inverse operation to “undo” the operation in order to isolate the variable. For addition and subtraction, your goal is to change any value being added or subtracted to 0.

Example:

x - 6 = 8

x - 6 + 6 = 8 + 6

x = 14

Using the Addition Property of Equality, 6 was added to both sides of the equation to isolate the variable.

Example:

3,500 + 4,800 + x = 10,000

8,300 + x = 10,000

8,300 - 8,300 + x = 10,000 - 8,300

x = 1,700

B. Solving One-Step Linear Equations by Multiplication or Division

The multiplication property of equality states that if both the sides of an equation are multiplied by the same number, the expressions on the both sides of the equation remain equal to each other. The division property of equality tells us that if we divide both sides of an equation by the same number, the equation remains the same.

When the equation involves multiplication or division, you can “undo” these operations by using the inverse operation to isolate the variable. When the operation is multiplication or division, your goal is to change the coefficient to 1, the multiplicative identity.

Example:

3x = 24

3x/3 = 24/3

x = 8

Using the division property of equality, both sides of the equation are divided (inverse of multiplication) by 3 to isolate the variable. 

Example:

Using the multiplication property of equality, both sides of the equation are multiplied by 2 to isolate the variable. 

Example:

-7/2 = k/10

(10)-7/2 = k/10(10)

-35 = k

C. Modeling With One-Step Equations (Word Problems)

Khan: Modeling with one-step equations

Khan: Writing equations for relationships between quantities 

D. Dependent & Independent Variables

A dependent variable is a variable whose value changes based on the value of another variable. It is the effect value.  An independent variable is a variable that is being manipulated. It is the cause value. 

Example:

You are doing chores to earn your allowance. For each chore you do, you earn $3. We can write this as an equation where C = each individual chore and M = total money earned.

M = 3C

The dependent variable is M, the amount of money you earn because it is dependent on how many chores you do. The independent variable is the amount of chores you do because this is the variable you have control over. 

Practice

Khan: Equations & Inequalities Introduction

Khan: Intro to Equations

CK-12: Solving One-Step Equations

Cuemath: Properties of Equality

Splashlearn: Properties of Equality

Lumen: Beginning Algebra

Variables & Expressions

I. Intro

A variable is a symbol or (more often) a letter used to represent one or more numbers. The numbers are called the values of the variable. At an elementary level variables represent an unknown amount. 

Example:

x

A constant is a fixed numerical value. 

Example:

12

A coefficient is a number multiplied by a variable.

Example:

14c

Here, 14 is the coefficient to be multiplied by variable c.

A factor is one part of a product. Factors are the numbers and variables that are multiplied together. 

Example: 

In the term 8x, the factors are 8 and x.

A term is either a single number or variable, or the product of numbers or variables.

Example:

3x

An expression is a combination of terms that are combined by using the mathematical operations addition and subtraction. It can be thought of as a mathematical sentence. Expressions never have an equal sign. 

Example:

g + 25 - 7 

An equation is a mathematical sentence that says two expressions are equal. Equations always have an equal sign.

Example:

g + 25 - 7 = 7g

The Cross Symbol for Multiplication

In algebra, the cross symbol, X, is not used to show multiplication since the symbol can be confused as a variable

The factors of the term 5x are 5 & x

II. Evaluating Expressions

To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number. To evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations.

Example (1 variable):

Evaluate 9x - 2, when x = 5

9x - 2

= 9(5) - 2

= 45 - 2

= 43

Example (2 variables):

10 + 2p - 3r, when p = 4 & r = 5

= 10 + 2(4) - 3(5)

= 10 + 8 - 15

= 3

III. Simplify Expressions Combining Like Terms

We can simplify an expression by combining the like terms. Like terms are terms where the variables match exactly (exponents included). To do this, add the coefficients and keep the same variable.

Example:

3x + 6x = 9x

We can see why this works by writing both terms as addition problems.

Add the coefficients and keep the same variable. It doesn’t matter what x is. If you have 3 of something and add 6 more of the same thing, the result is 9 of them. For example, 3 oranges plus 6 oranges is 9 oranges.

Example: (Fraction)

Khan: Combining like terms with negative coefficients

IV. Distributive Property

The distributive property states that multiplying a number by the sum or difference of two numbers is the same as multiplying the number by each of the numbers individually and then adding or subtracting the results

Stated with variables, the distributive property of multiplication over addition states:

x(y + z) = x(y) + x(z)

We can also use the distributive property of multiplication over subtractions:

x(y - z) = x(y) - x(z)

The Distributive Property with Variables

The distributive property is useful when simplifying expressions with variables. When using variables, you may not be able to do the operation inside the parentheses first.

Example:

8(y + 4z)

= 8(y) + 8(4z)

= 8y + 32z

Example: Distributing a negative sign

5x - (3x + 2) 

=5x -3x - 2

Khan: Combining like terms with negative coefficients and distribution

 

Factoring with the distributive property

V. Writing Expressions Word Problems

Khan: Writing Expressions Word Problems

VI. Miscellaneous

Variable In Numerator

When writing a term that has a fraction and a variable in the numerator, sometimes the variable is written in the numerator, sometimes it is written on the side. Both have the same meaning. 

Dividing An Expression (Polynomial) By A Number

Dividing an expression by a number involves dividing each term individually by that number.

Example: Divide 6x² + 9x  + 12 by 3

Practice:

Khan: Variables and Expressions

MathGames: Algebra

Reference:

Varsity Tutors: Variables

Varsity Tutors: Parts of an Expression

IXL: Expressions and equations

Khan: Terms, factors, and coefficients review

Wikipedia: Variable

Wikipedia: Term

Openstax: Prealgebra 2e

IXL: The distributive property

Splashlearn: What is a variable

Prodigy: Distributive Property: 5 Effective Examples to Use in Class

Lumen: Simplify Expressions, Combine Like Terms, & Order of Operations With Real Numbers

Plane Figures Hierarchy For Basic Geometry

Plane Figures 

These are two-dimensional shapes that exist on a flat surface. They have length and width but no depth.

I. Closed Figures 

These shapes have a continuous boundary that encloses a region.

A. Polygons 

Closed figures whose boundaries are made up entirely of straight line segments with at least three sides. They are typically classified by the number of sides.

1. Triangles (3 Sides)

By Side Length

a) Isosceles Triangles

A triangle that has at least two equal sides. (Some define it as only having two equal sides, thus excluding the equilateral triangle)

b) Equilateral Triangles

A triangle in which all three sides have the same length.

c) Scalene Triangles

A triangle in which all three sides have different lengths. 

By Angles

a) Acute Triangles

A triangle that has three angles that each measure less than 90 degrees.

b) Right Triangles

A triangle that has one angle that is 90 degrees.

c) Obtuse Triangles

A triangle that has one angle that is greater than 90 degrees. 

2. Quadrilaterals (4 Sides)

a) Trapezoid

A quadrilateral with one pair of parallel sides. 

b) Kite

A kite is a quadrilateral with two distinct pair of adjacent sides of equal length. (Note: Some sources define kites broadly by not stating that the two pair are distinct. The first, restricted definition excludes  squares and rhombuses while the broad definition would include them)

c) Parallelogram

A quadrilateral in which both pairs of opposite sides are parallel and equal length. 

(1)Rectangle

i)Square

(2)Rhombus

3. Pentagons (5 Sides)

4. Hexagon (6 Sides)

5. Heptagon (7 Sides)

6. Octagon (8 Sides)

7. Nonagon (9 Sides)

8. Decagon (10 Sides)

B. Non-Polygonal Closed Figures 

If any side or part of a plane figure is curved (not straight) it is not a polygon.

1. Circles

A plane figure bounded by one curved line, and such that all straight lines drawn from it's center to the bounding line, are equal.

2. Ellipse

II. Open Figures 

These shapes do not enclose an area completely.

Math.stackexchange: Can an equilateral triangle be an isosceles triangle, too?

Cuemath: Quadrilateral

Cuemath: Triagle

Ck-12: Basic Geometry Concepts

https://www.quora.com/What-are-Polygons-and-non-polygons-What-are-their-differences

https://authenticinquirymaths.blogspot.com/2017/05/polygons-and-non-polygons.html

https://www.math.net/plane-figure

https://cryptlabs.com/polygons/

https://www.google.com/search?q=hierarchy+of+polygons&rlz=1C1CHBF_enUS876US876&oq=hierarchy+of+polygons&gs_lcrp=EgZjaHJvbWUyDwgAEEUYORiRAhiABBiKBTIICAEQABgWGB4yCAgCEAAYFhgeMg0IAxAAGIYDGIAEGIoFMg0IBBAAGIYDGIAEGIoFMg0IBRAAGIYDGIAEGIoFMgoIBhAAGKIEGIkFMgYIBxBFGDzSAQg2NTgyajBqN6gCALACAA&sourceid=chrome&ie=UTF-8#vhid=ZjLkUb2l-jrE_M&vssid=_s9T3Z-WfBoqIwbkPmLGDGA_38

Multiplication & Division Problem Structures (Word Problems)

Researchers have defined and classified multiplication and division problems in a number of different ways based on their semantic structure—that is, how the relationships are expressed in words. There are two broad categories: asymmetrical and symmetrical problems. There are three subcategories of asymmetrical problems: (1) equal grouping, (2) rate, and (3) multiplicative compare. There are two subcategories of symmetrical problems: (1) rectangular array and (2) Cartesian product.

Asymmetrical Problems

In asymmetrical situations, the quantities play different roles. For example: There are 15 cars in the parking lot and each car has 4 tires. How many tires are there in all?, 4 represents the amount in one group, 15 represents the number of groups and also acts as the multiplier. These roles are not interchangeable. If you switch the numbers (4 cars with 15 tires each) you have a different problem: the number of things in one group is 15 and the number of groups is now 4. (In an asymmetrical problem the answer is the same regardless of the role of the quantities, but that is not obvious to children.)

1. Equal Grouping

In equal grouping multiplication problems, one factor tells the number of things in a group and the other factor tells the number of equal-size groups. This second factor acts as a multiplier.

Example:

There are four basketball teams at the tournament and each team has five players. How many players
are at the tournament?, the factor 5 indicates the number of players in one group and the factor 4 indicates the number of equal groups of 5. The problem would be written as 4 x 5 = ?

Two situations result in problems being classified as an equal grouping division—either the number in each group is unknown or the number of groups is unknown. These two types of division situations are referred to as partitive division and quotitive division, respectively.

Partitive Division

Here is a partitive division problem: 

Twenty-four apples need to be placed into eight paper sacks. How many apples will you put in each sack if you want the same number in each sack?

The action involved in partitive division problems is one of dividing or partitioning a set into a predetermined number of groups. If students model this situation, 24 objects are evenly distributed into 8 different paper sacks or groups.

Quotitive Division

In quotitive division problems (sometimes referred to as repeated subtraction problems) the number of objects in each group is known, but the number of groups is unknown. 

For example: 

I have 24 apples. How many paper sacks will I be able to fill if I put 3 apples into each sack?

The action involved in quotitive division is one of subtracting out predetermined amounts. If asked to model this problem, students usually repeatedly subtract 3 objects from a group of 24 objects and then count the number of groups of 3 they removed (i.e., 8).

2. Rate

Rate problems involve a rate—a special type of ratio in which two different quantities or things are compared. Common rates are miles per gallon, wages per hour, and points per game. Rates are frequently expressed as unit rates—that is, one of the quantities in the ratio is given as a unit (e.g., price per single pound or miles per single hour). In rate problems, one number identifies the unit rate and the other tells the number of sets and acts as the multiplier. 

For example:

Concert tickets cost seven dollars each. How much will it cost for a family of four to attend the concert?

the unit rate is seven dollars per single ticket (seven to one) and the multiplier is four.

Rate problems can also be expressed as division situations. Here is a partitive division rate problem (size of one group is unknown): On the Hollingers’ trip to New York City, they drove 400 miles and used 12 gallons of gasoline. How many miles per gallon did they average? Quotitive division problems (number of equal groups is unknown) are also common in this category: Jasmine spent $108 on some new CDs. Each CD cost $18. How many did she buy?

3. Multiplicative Compare

The third type of asymmetrical problem is multiplicative compare, also called a scalar problem. Here, one number identifies the quantity in one group or set while the other number is the comparison factor.

For example:

Catherine read twelve books. Elizabeth read four times as many. How many books did Elizabeth read?, 

Here the number 12 tells us the amount in a group and the number 4 tells us how many of these groups are needed. 

In multiplicative compare division problems, either the amount in each group or the comparison factor is missing. 

The following problems are multiplicative compare division problems:

Elizabeth read 48 books during summer vacation. This is four times as many as Catherine. How many books did Catherine read during summer vacation?

Elizabeth read 48 books during summer vacation. Catherine read 12 books during summer vacation. How many times greater is the number of books Elizabeth read compared with the number of books Catherine read?

Symmetrical Problems

In symmetrical situations, on the other hand, the quantities have interchangeable roles. In symmetrical

multiplication problems it is not clear which factor is the multiplier. For example, in the problem What is the area of a room that is 10 feet by 12 feet?, either number can be the width or the length and either can be used as the multiplier.

1. Rectangular Array (Area)

The rectangular array problem, a subcategory of symmetrical problem commonly known as an area problem, is often used to introduce the idea of multiplication. Students are presented with an array (e.g., three by four) and asked to label the two sides of the array and determine the total number of square units in the array:

In rectangular array problems the role of the factors is interchangeable. For instance, when finding the area of the array above, neither the three nor the four is clearly the multiplier.

2. Cartesian Product

The other subcategory of symmetrical problems, known as Cartesian product problems, involves two sets and the pairing of elements between the sets. These problems entail a number of combinations.

For example: 

Pete’s Deli stocks four types of cold cuts and two types of cheese. How many different sandwiches consisting of one type of meat and one type of cheese are possible?

In these problems, like the rectangular array problems, neither of the two factors is clearly the multiplier.

Students tend to use tree diagrams to solve Cartesian product problems, in order to help them find all the combinations. For example, for the problem above, a student might use a diagram like this:

However, this diagram doesn’t highlight how multiplication can be used to determine the number of combinations. Teachers often have to point out the relationship between the number of objects to be combined (4 and 2) and the final number of combinations (4 ⨉ 2 = 8).

Another way to solve Cartesian product problems is to construct a rectangular array like the one below. The number of columns (4) and the number of rows (2) are the factors in the problem.

Math Matters: Understanding the math you teach

https://roomtodiscover.com/meanings-of-multiplication/

Addition & Subtraction Problem Structures (Word Problems)

A common classification scheme of addition and subtraction word problems identifies four broad categories based on the type of action or relationship in the problems: join, separate, part-part-whole, and compare. 

1. Join/Separate

Join problems and separate problems involve actions that increase or decrease a quantity, respectively. In both categories, the change occurs over time. There is an initial quantity that is changed either by adding something to it or by removing something from it, resulting in a larger or smaller final quantity. There are three types of join/separate problems: Result Unknown, Change Unknown and Initial Quantity Unknown.

Result unknown problems are generally easiest for students. Change unknown and initial quantity unknown problems are more challenging. One reason is that these problems are often presented in language that suggests one action (e.g., separating) but require using the opposite action (e.g., joining) to find the answer. For instance, the separate - initial quantity unknown problem above asks Rodney had some cookies. He ate three cookies. Now he has seven cookies left. How many cookies did Rodney have to start with? There is a separating action in the problem, but a child can solve it using addition.

- 3 = 7 becomes 7 - 3 = 

2. Part-Part-Whole

Part-part-whole problems do not use action verbs—action neither occurs nor is implied. Instead, relationships between a particular whole and its two separate parts are established. There are two common types of part-part-whole problems and a third less common one. They are: Whole Unknown, One Part Unknown and Both Parts Unknown

Both parts unknown is less common as they generally lead to multiple solution possibilities. Sometimes teachers will use these problems to test a students ability to determine all possible answers. 

Part-part-whole problems involve a comparison of “parts” (subsets) with the “whole” (set).

3. Compare

Compare problems involve a comparison of two distinct, unconnected sets. Like part-part-whole problems, compare problems do not involve action. However, they differ from part-part-whole problems in that the relationship is not between sets and subsets but between two distinct sets. There are three types of compare problems, depending on which quantity is unknown: Difference Unknown (the quantity by which the larger set exceeds the smaller set), Larger Quantity Unknown and Smaller Quantity Unknown. 

Math Matters: Understanding The Math You Teach

Math Coach Corner: Problem Structures for Addition & Subtraction

Tales From Outside The Classroom: Addition & Subtraction Word Problem Types