Algebra: Patterns and Equations

Master of Arts in Child Study and Education

Math Fundamentals for Elementary Teachers

Introduction

We encourage teachers to complete the modules and practice problems at their own pace. While the modules are intended to build upon one other, they can also stand alone, depending on your individual needs. The order of the modules as laid out below provides a reasonably logical progression of skills and concepts.

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1. Whole Numbers and Operations
2. Rational Numbers (Fractions)
3. Algebra (Patterns and Equations)
4. Integers and Operations
5. Rational Numbers (Fractions, Decimals, Percents)
6. Algebra (Linear Relationships)
7. Geometry
8. Measurement

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Note that the corresponding problem set is linked at the end of each module.

Learning Objectives

• Patterns
• Representing numerical patterns and relationships using pictures, tables, and graphs.
• Describing numerical patterns and relationships in words
• Algebraic Expressions and Equations
• Understanding the concept of equivalence
• Identifying the parts of an equation
• Defining and recognizing variables in algebraic expressions
• Modeling real world situations using algebraic expressions
• Simplifying, Balancing, and Solving
• Simplifying algebraic expressions by collecting like terms
• Balancing equations using the concept of equivalence
• Solving algebraic equations by isolating the variable of interest

What is Algebra?

• Algebra is generalized arithmetic.
• The word “algebra” is a Latin form of the Arabic word al-jabr, meaning “reunion of broken parts.”
• We see the significance of this etymology in the concept of balancing equations, which plays a major role in algebra.
• There are two conceptual foundations of algebraic reasoning:
• Patterns and functions (understanding consistent relationships)
• Equivalence and equations (understanding balance and “sameness”)

Introduction to Patterns: Video

Representing Patterns

• Patterns exist all around us! Mathematics is one way to interpret and make sense of patterns in the world.
• We can represent patterns in multiple ways.
• We will be focusing on the following representations:
• Pictures
• Tables
• Graphs

Representing Patterns: Pictures

Consider the following picture:

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This picture shows a sequence of 5 different figures. What patterns do you see?

Representing Patterns: Pictures

• Each figure represents a term of the pattern.
• The first figure (term 1) has a length of 1 unit and a width of 1 unit. The second figure (term 2) has a length of 2 and a width of 2.
• This pattern continues for each figure. As the term number increases by 1, the vertical and horizontal dimensions increase by 1 unit.

Representing Patterns: Pictures

• We can also find a pattern by counting the red squares in each figure.
• The first figure has 1 red square. The second figure has 3 red squares. The third figure has 5 red squares.
• We can describe this pattern! As the term number increases by 1, the number of red squares increases by 2.

Representing Patterns: Pictures

• We have found a relationship between consecutive terms of the pattern (as the term number increases by 1, the number of red squares increases by 2). This is sometimes called a recursive relationship in which each term is considered in relationship to the term that precedes it. Thinking recursively, it would take a long time to find the 72nd figure in the pattern!
• Can we also find a relationship between the term number and the number of red squares? This is called a functional relationship, in which each term has the same relationship to its term number.

Representing Patterns: Pictures

• The 2nd figure (term 2) has 2 horizontal red squares and 1 vertical red square.
• The 5th figure (term 5) has 5 horizontal red squares and 4 vertical red squares.
• Extending this pattern, term n would have n horizontal red squares and n - 1 vertical red squares.

Representing Patterns: Pictures

• Term 1 has 1 + 0 = 1 red square. Term 2 has 2 + 1 = 3 red squares.
• Term 5 has 5 + 4 = 9 red squares.
• Term n would have n + (n - 1) = 2n - 1 red squares! Thus, Term 72 (or the 72nd figure) would have 143 red squares.

Representing Patterns: Tables

We can represent the same geometric pattern in a numerical table.

Let n be the term number, and let r_n be the associated number of red squares.

Representing Patterns: Tables

Our earlier relationship between term number and number of red squares still holds when the pattern is written as a table!

Representing Patterns Algebraically

In this pattern, for any value of n, there is a general relationship between n and r_n.

This relationship helps us summarize the pattern into one neat equation:

r_n = 2n - 1

We can extend our table and picture to higher values of n using this relationship.

Patterns are important foundations of algebraic reasoning. We started with the first five terms of a pattern, and we found a general formula to describe any term of the pattern.

Representing Patterns: Graphs

We can also represent the same pattern with a graph. Each point on the graph is represented by (n, r_n), where n (term number) is on the x-axis (horizontal axis) and r_n (number of red squares) is on the y-axis (vertical axis).

We can see for every unit increase in n there is a corresponding increase of 2 units in r_n.

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y

x

Representing Patterns: Try it Yourself!

Consider the growing pattern below. Using the strategies from the previous slides:

1. Describe the pattern in words.
2. Find the number of orange squares in the nth figure.
1. Hint: Look at the the lighter and darker orange squares.
2. Find the number of orange squares in the 203rd figure.
3. Represent the pattern using a table.
4. Represent the pattern using a graph.

Representing Patterns: Solution

1. As the term number increases by 1, the number of orange squares in the figure increases by 3. (There are many different ways to describe this pattern, this is one potential solution!)
2. a. # of light orange squares + # of dark orange squares = (2n - 1) + (n - 1)�b. 607

Representing Patterns: Solution

3.

4.

What is an Equation?

• An equation conveys that two quantities are of equal value.
• For instance, the equation x + 2 = 6 tells us that “the value of the left side (x + 2) is equal to the value of the right side (6)”
• Possible parts of an equation are shown in the example below:

Parts of an Equation

• Variable: A symbol (e.g., “x”) representing any value that makes an equation correct
• Constant: A fixed number in the equation
• Coefficient: Number used to multiply a variable
• Operator: Symbol that shows an operation (such as +, ×)

Exercise: Identify the parts of the following equations: 6x + 13 = 19; x - 9 = 2; 3x = 23.

How many green blocks are needed to balance the second scale?

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You have just solved a two-variable equation!

Terms and Expressions

• A term can be either a constant or a variable, or numbers and variables multiplied together.
• Terms are separated by + or - signs.
• For example, the equation 9x + 2y + 3 = 5 has 4 terms (9x, 2y, 3, and 5).
• An expression is a group of terms.
• Using this vocabulary, we can begin to discuss algebraic expressions.

Writing Algebraic Expressions and Equations

• To write an algebraic expression or set up an equation, you need to identify which operators (+, -, x) connect the different terms in the problem .
• When multiplying and dividing variables by whole numbers, you do not typically use x or ÷ in the resulting expression (4 × n → 4n; n ÷ 3 → n/3).

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Try it Yourself!

Write an algebraic expression to represent each of the following situations:

1. Jane is 5 cm taller than her younger sister who is x cm tall. How tall is Jane?
2. Mike has three times more marbles than Ryan who has m marbles. How many marbles does Mike have?
3. Kurtis shared p slices of pizza evenly among himself and his 4 friends. How many slices does each person get?

Answers: x + 5, 3m, p/5

Algebraic Expressions and Equations: Example

A math tutor charges $40 per hour and spends $4 per day on public transportation. Write an algebraic expression to represent her daily earnings.

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Solution:

Let x represent the number of hours the tutor works in one day.

The tutor’s net daily earnings (amount paid for tutoring - transportation costs) can be represented by the following expression: 40x - 4.

(Note that x is a variable since its value varies with the number of hours worked in a day. The number 4 is a constant since transportation costs are the same each day.)

Algebraic Expressions and Equations: Try it Yourself!

Write an algebraic expression to represent each of the following situations:

1. A start-up company with x employees has $2000 to distribute to its employees as a holiday bonus. How much money will each employee receive if the bonus is distributed evenly?
2. Lisa gave 13 cookies and Mark gave y cookies to the Smith sisters to share evenly among themselves. If there are 3 Smith sisters, how many cookies did each girl receive?

Algebraic Expressions and Equations: Try it Yourself!

Write an algebraic expression to represent each of the following situations:

• A start-up company with x employees has $2000 to distribute to its employees as a holiday bonus. How much money will each employee receive if the bonus is distributed evenly? Each employee will receive 2000/x dollars.
• Lisa gave 13 cookies and Mark gave y cookies to the Smith sisters to share evenly among themselves. If there are 3 Smith sisters, how many cookies did each girl receive? The girls received (13+y)/3 cookies each.

Simplifying Algebraic Expressions

• Simplifying algebraic expressions involves writing an expression in its most compact form without changing its value.
• This involves combining like terms and constants.
• Like terms are terms with identical variables (such as x or y). Identical variables with exponents (such as the 2 in x²⁾ can only be combined when the exponents are also the same.

For example:

• 12x and 3x are like terms.
• 6x² and 2x² are like terms.
• 2x and 4y are not like terms.
• 7x and 9x² are not like terms.

Simplifying Algebraic Expressions

We simplify an algebraic expression as follows:

• Combine the constants (e.g., 2 and 7)
• Combine like terms using addition or subtraction
• Remove brackets by multiplying factors

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Example: Simplify the expression 3x + 7 - 2x + 10.

3x + 7 - 2x + 10 = 3x - 2x + 7 + 10

= x + 17

Like terms collected

Like terms and constants combined

Distributive Property

• The distributive property states that an expression in the form of A(B + C) can be solved as A(B + C) = AB + AC
• This distributive property similarly applies to subtraction: A(B – C) = AB – AC
• This means that A is distributed across B and C.

Applying the Distributive Property

• Some expressions take the form a(b+c). The brackets indicate that the expression within the brackets (b+c) is multiplied by the preceding term (a).
• When all terms are constants, we can solve using an agreed-upon order of operations.

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• If we apply the distributive property to the above example we see that it yields the same answer.

Simplifying Expressions: Try it Yourself!

Simplify the following expressions. It is conventional to place variables in alphabetical order and constants after all variables.

1. 10 + 20x - 12
2. 5x² + 2x² - 3y
3. 2(2 + x) + 3(x + y)
4. 5m + 12(2 + 1) - m
5. 6n - 8n + 2n³

Simplifying Expressions: Solutions

Simplify the following expressions. It is conventional to place variables in alphabetical order and constants after all variables.

• 10 + 20x - 12 = 20x -2
• 5x² + 2x² - 3y = 7x² - 3y
• 2(2 + x) + 3(x + y) = 5x + 3y + 4
• 5m + 12(2 + 1) - m = 4m + 36
• 6n - 8n + 2n³ = -2n + 2n³

Revisiting Balancing

To balance a scale, we need equal weights on each side.

But what if we do not know the value of one amount? Can we still balance the equation?

Consider the drawing on the right.

• If we remove 2 kilograms from the left side, what do we need to do to the left side to ensure the scales remain balanced?
• What does that tell us about the value of x?

2

x

3

2

Balancing Equations

Balancing Equations

To maintain equality, whatever you do to one side of an equation you must do to the other side.

Take the equation x + y = 7.

• If we subtract 3 from the RHS (right-hand side), we must subtract 3 from the LHS (left-hand side)
• x + y - 3 = 7 - 3

This procedure applies to other operations as well.

Exercises:

• Say we have x + y = y + z. What would the RHS be if we subtract y from the LHS?
• Say we have 3x = 12. What would the LHS be if we divide by 3 on the RHS?
• Say we have a - 3 = 0. What would the RHS be if we add 3 to the LHS?

Solving Equations by Isolating the Variable

Keeping equations balanced allows us to solve for a given variable. For example, if we want to know how much profit p a company makes, we subtract their operating costs c from their revenue r. This can be represented as p = r - c.

• Say we know the profit p and the operating costs c. What is the revenue r?

To answer this question, we need to isolate r. We rearrange the equation to leave r by itself on one side of the equation, and move everything else to the other side. This involves re-balancing the equation at each step.

• Since p = r - c, then we can add c to the RHS to isolate r. To maintain the balance, we have to add c to the LHS as well. Thus p + c = r.

To make our solution clearer, we can rewrite this as r = p + c.

Balancing Equations: Try it Yourself!

Balance the following equations:

• Suppose x + 4 = y + 4. What would the RHS be if we subtract 4 from the LHS?
• Suppose x - 7 = 3. What would the RHS be if we add 7 to the LHS?

Solve for the variable in the following equalities:

• Solve for x in 5x = 15y
• Solve for y in 5x = 15y
• Solve for a in a/b + c = 0

Algebra: Patterns and Equations (Problem Set)

Sources for Images

Slide 20, 24: https://www.mathsisfun.com/algebra/definitions.html

Slide 33: https://miro.medium.com/max/742/1*X_t_I5VOdpaMxi-yKsImMQ.png

Slide 34: https://www.mathsisfun.com/algebra/introduction.html

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