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Algebra 1 View details

1. Using Inductive Reasoning to Create Equations

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Chapter 1

1.

Using Inductive Reasoning to Create Equations

Inductive reasoning serves as a practical method for identifying patterns in numerical and visual data. This method is particularly useful in algebra and geometry. The lesson delves into the importance of variables, which are symbols used to represent unknown quantities. It also covers linear expressions and equations, which are foundational elements in algebra. These concepts are not just theoretical; they have real-world applications. For instance, understanding how to form equations can help in financial planning or in engineering problems. The lesson also emphasizes that equations often contain one or more unknown values called variables, which can be solved to make the equation true.

Lesson Settings & Tools

	14 Theory slides
	9 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Using Inductive Reasoning to Create Equations

Slide of 14

Mathematics can be used to model real-life situations. This is helpful to understand those situations and find solutions to problems that might arise in such scenarios. This lesson will discuss the concepts of variable, linear expression, equation, and linear equation.

Catch-Up and Review

Here is a recommended reading before getting started with this lesson.

Base Units

Challenge

Organizing Tables in a Restaurant

A group of people at a restaurant is waiting for a table for a family dinner. Waiters began adding tables one by one until the entire group was seated as requested.

People sitting around tables at a restaurant

Is there an expression to represent the number of people in terms of the number of tables?

Discussion

Variable

Sometimes a quantity is unknown or its value may change. If this is the case, the best way to represent this quantity is using a variable.

A variable is a symbol used to represent an unknown quantity. Often, variables represent fixed but unknown numbers. Variables are usually denoted with letters such as

x .

x + 1 = 8

In this case,

x

is a variable in an equation. Solving this equation for

x

will identify the value of

x

that makes the equation true.

x + 1 = 8 \Leftrightarrow x = 7

Alternatively, a variable can be used to represent a quantity that changes.

Izabella ’ s income varies depending on the number of hours she works . She receives a fixed salary of $ 100 per week plus $ 4 per hour .

In this case, Izabella's weekly income could be different every week, depending on the number of hours she works. Therefore, the use of a variable is appropriate. Let

x

be the hours that Izabella works in a week. Her income can be written by adding the fixed

$ 100

to the

$ 4

per hour.

Izabella ’ s Weekly Income 4 x + 100

Discussion

Linear Expression

There are many real-life situations that can be modeled by combining numbers and variables in algebraic expressions. A special case of algebraic expressions are linear expressions.

Concept

Linear Expression

A linear term is an algebraic expression that includes a coefficient multiplied by a variable with an exponent of one. A linear expression is an expression that includes at least one linear term and any constant terms. No other type of terms may be included. The most common form of a linear expression is given below.

The linear expression ax+b, where a is the coefficient, x is the variable, ax is the linear term, and b is the constant term.

In this expression, $a$ and $b$ are real numbers, with $a \neq = 0 .$ To completely understand the definition of a linear expression, some important concepts will be be broken down. Consider the example linear expression $x - 5 y + 2 .$

$x - 5 y + 2$
Concept	Explanation	Example
Term	Parts of an expression separated by a $+$ or $-$ sign.	$x,$ $- 5 y,$ $2$
Coefficient	A constant that multiplies a variable. If a coefficient is $1,$ it does not need to be written due to the Identity Property of Multiplication.	$1,$ $- 5$
Linear Term	A term that contains exactly one variable whose exponent is $1 .$	$x,$ $- 5 y$
Constant Term	A term that contains no variables. It consists only of a number with its corresponding sign.	$2$

The following table shows some examples of linear and non-linear expressions.

Linear Expressions	Non-linear Expressions
$3 x$	$5$
$- 5 y + 1$	$2 x y - 3$
$3 x - \frac{1}{2} y + 2$	$\frac{1}{x} - 2$
$π x + 6 y$	$5 x^{2} + x - 1$

It is important to keep in mind that before classifying an expression, it must be written in simplest form.

3 x + 4 - 2 x + 9 - x = 13

The expression

3 x + 4 - 2 x + 9 - x

looks like a linear expression, but the result after simplifying is

13,

which is a constant and therefore not a linear expression.

Example

Classifying the Parts of a Linear Expression

Ramsha, Tiffaniqua, Zosia, and some of their friends are spending the summer at math camp. One of their first lessons is about identifying parts of linear expressions. They are given the following table to fill in based on the linear expressions in the top row.

	$6 x - 1$	$2 + x$	$- 2 x + 3 - 9 y$
Term(s)
$x -$ term
$y -$ term
Linear Term(s)
Coefficient(s)
Constant Term

Help them fill in the table.

Answer

	$6 x - 1$	$2 + x$	$- 2 x + 3 - 9 y$
Term(s)	$6 x$ and $- 1$	$2$ and $x$	$- 2 x,$ $3,$ and $- 9 y$
$x -$ term	$6 x$	$x$	$- 2 x$
$y -$ term	-	-	$- 9 y$
Linear Term(s)	$6 x$	$x$	$- 2 x$ and $- 9 y$
Coefficient(s)	$6$	$1$	$- 2$ and $- 9$
Constant Term	$- 1$	$2$	$3$

Hint

When there is no number in front of a variable, the coefficient is $1 .$ The $x -$ and $y -$ terms are the terms that contain the variables $x$ and $y,$ respectively.

Solution

Start by finding the terms of each expression. Recall that the terms are the parts separated by addition or subtraction signs.

	$6 x - 1$	$2 + x$	$- 2 x + 3 - 9 y$
Term(s)	$6 x$ and $- 1$	$2$ and $x$	$- 2 x,$ $3,$ and $- 9 y$
$x -$ term
$y -$ term
Linear Term(s)
Coefficient(s)
Constant Term

The $x -$ term is the term that contains the variable $x .$

	$6 x - 1$	$2 + x$	$- 2 x + 3 - 9 y$
Term(s)	$6 x$ and $- 1$	$2$ and $x$	$- 2 x,$ $3,$ and $- 9 y$
$x -$ term	$6 x$	$x$	$- 2 x$
$y -$ term
Linear Term(s)
Coefficient(s)
Constant Term

Similarly, the $y -$ term is the term that contains the variable $y .$ In this case, the first two expressions do not have a $y -$ term.

	$6 x - 1$	$2 + x$	$- 2 x + 3 - 9 y$
Term(s)	$6 x$ and $- 1$	$2$ and $x$	$- 2 x,$ $3,$ and $- 9 y$
$x -$ term	$6 x$	$x$	$- 2 x$
$y -$ term	-	-	$- 9 y$
Linear Term(s)
Coefficient(s)
Constant Term

The linear terms are the terms that contain only one variable, raised to the power of $1 .$

	$6 x - 1$	$2 + x$	$- 2 x + 3 - 9 y$
Term(s)	$6 x$ and $- 1$	$2$ and $x$	$- 2 x,$ $3,$ and $- 9 y$
$x -$ term	$6 x$	$x$	$- 2 x$
$y -$ term	-	-	$- 9 y$
Linear Term(s)	$6 x$	$x$	$- 2 x$ and $- 9 y$
Coefficient(s)
Constant Term

Next, the coefficients are the numbers that multiply a variable. When there is no number in front of a variable, the coefficient is $1 .$

	$6 x - 1$	$2 + x$	$- 2 x + 3 - 9 y$
Term(s)	$6 x$ and $- 1$	$2$ and $x$	$- 2 x,$ $3,$ and $- 9 y$
$x -$ term	$6 x$	$x$	$- 2 x$
$y -$ term	-	-	$- 9 y$
Linear Term(s)	$6 x$	$x$	$- 2 x$ and $- 9 y$
Coefficient(s)	$6$	$1$	$- 2$ and $- 9$
Constant Term

Finally, the constant term is any term without a variable.

	$6 x - 1$	$2 + x$	$- 2 x + 3 - 9 y$
Term(s)	$6 x$ and $- 1$	$2$ and $x$	$- 2 x,$ $3,$ and $- 9 y$
$x -$ term	$6 x$	$x$	$- 2 x$
$y -$ term	-	-	$- 9 y$
Linear Term(s)	$6 x$	$x$	$- 2 x$ and $- 9 y$
Coefficient(s)	$6$	$1$	$- 2$ and $- 9$
Constant Term	$- 1$	$2$	$3$

Pop Quiz

Identifying the Parts of a Linear Expression

Select the required parts of the given linear expressions.

Example

Modeling Distance Traveled Using Linear Expressions

Ramsha is on the second day of a hiking excursion at math camp. On the first day, she hiked from base camp to the first station.

Distance from Ramsha to the Base camp and 1st station in between

Her current distance from base camp, in miles, is given by the following linear expression.

Distance From the Base Camp 1.5 x + 10

Here,

x

is the number of hours Ramsha walks on the second day.

a What does the coefficient

1.5

represent in this context?

b What does the linear term

1.5 x

represent in this context?

c What does the constant term

10

represent in this context?

Answer

a The speed at which Ramsha walks.

b The distance Ramsha walks on the second day.

c The distance from base camp to the first station.

Hint

a The expression

1.5 x + 10

represents distance, in miles. Therefore, the units of

1.5 x

are miles. Also, the units of the variable

x

are hours. This means that the units of the coefficient must be miles per hour.

b The expression represents distance. Therefore, each term in the expression also represents distance.

c The constant term represents a fixed distance.

Solution

a Start by representing Ramsha's current position from the base camp in the diagram.

Note that $1.5 x + 10$ represents a distance and the units of each term are miles. However, the units of $x$ are hours. This gives a clue of the units of the coefficient $1.5 .$

In order to get rid of hours and end up with miles, the units of the coefficient must be miles per hour. Therefore, the coefficient represents speed, or more precisely, the speed at which Ramsha walks.

b Since

x

represents the number of hours Ramsha has walked on the second day, then

1.5 x

represents the distance Ramsha has walked on the second day, starting at the first station.

c Finally, since

1.5 x + 10

represents the distance from base camp and

1.5 x

the distance walked on the second day, then the constant term

10

represents the distance Ramsha walked on the first day — that is, the distance from base camp to the first station.

Discussion

Inductive Reasoning

In the previous example, an expression modeling a real-life scenario was provided. However, most of the time such expressions are not provided and must instead be created. The use of inductive reasoning can help with this process.

Inductive reasoning is the process of finding patterns in specific observations and writing a conclusion or conjecture. Since the conjecture is based on observations, it might be false. For example, suppose an observer notices that all the birds around them are white. The observer might inductively reason that all birds in the world are white, which is not true.

Man watching four white doves, then makes a conjecture. Look and behold, he sees a red bird!

Inductive reasoning is a practical method used in geometry and algebra for recognizing visual and numerical patterns. For instance, use the first three figures in the following diagram to guess the shape and the number of cubes in the fourth figure.

Four figures following a pattern.

After observing the first three figures, it is reasonable to conclude that the number of cubes increases by

3

from one figure to the next — one on the top, one on the front, and one on the right. Since the figure starts with

1

cube and each step adds

3

cubes, an expression can be written to model the number of cubes in each figure.

1 + 3 n

Here,

n

represents the number of steps after Figure

1 .

For example, Figure

2

is one step after Figure

1,

so

n = 1 .

This expression allows the calculation of the number of cubes in any figure of the pattern.

	$n$	Number of Cubes
Figure $1$	$0$	$1 + 3 \cdot 0 = 1$
Figure $2$	$1$	$1 + 3 \cdot 1 = 4$
Figure $3$	$2$	$1 + 3 \cdot 2 = 7$
Figure $4$	$3$	$1 + 3 \cdot 3 = 10$
Figure $121$	$120$	$1 + 3 \cdot 120 = 361$

Example

Expressing the Cost of an Amusement Park Trip

At the end of the math camp, the students all visit an amusement park. Tiffaniqua did so well at camp that she won a coupon that allows her entire friend group to go on rides for $x$ dollars per ride. By the end of the day, the group has gone on $11$ rides and each person has eaten one hot dog.

The price of a hot dog is

d

dollars and the group is formed by

k

people, including Tiffaniqua.

Write an expression that represents the total amount of money the group paid.

They want to split the total cost evenly. Write an expression representing the amount each person will pay.

Hint

The total amount of money paid is the sum of the amount paid for the $11$ rides and the amount spent in hot dogs. The cost per person is the total cost divided by the number of people.

Solution

Start by writing what each letter represents.

x d k \to cost of a ride \to cost of a hot dog \to number of people

Thanks to the coupon, the entire group pays

x

dollars per ride. Since they went on

11

rides, they paid

11 x

dollars.

Cost of the rides : 11 x

The cost of the hot dogs must be added to the previous amount. The price of a hot dog is

d

dollars. Since there are

k

people, the total cost is

k d

dollars.

Cost of the hot dogs : k d

Finally, the expression representing the total amount the group paid is the sum of the amount paid in rides and the amount paid in hot dogs.

Total cost : 11 x + k d

If the group wants to divide the cost evenly, then the total cost must be divided by the total number of people. In this case, the expression above must be divide by

k .

Cost per person : \frac{1 1 x + k d}{k}

Discussion

Equations

In the previous two examples, each situation was represented by an algebraic expression. There are some cases where writing an expression is not enough to model a situation. In such cases, an equation might be required.

Concept

Equation

An equation is a type of mathematical relation that indicates that two quantities are equal. Equations often contain one or more unknowns values called variables. Some examples are shown below.

equations with one and two variables

Solving an equation with a variable means to find the value or values of each variable that make the equation true.

Discussion

Linear Equation

Equations always contain an equals sign, while expressions do not. In fact, an equation can be seen like a statement that connects two expressions with an equals sign.

Translating an equation to words.

When learning about equations, a good first step is to start with linear equations, which involve only one variable.

Concept

Linear Equation

A linear equation is an equation with at least one linear term and any number of constants. No other types of terms may be included. Linear equations in one variable have the following form, where $a$ and $b$ are real numbers and $a \neq = 0 .$

a x + b = 0 or a x = b

The graph of a linear equation is a line.

Example

Money Saving

Zosia wants to go shopping after camp, but she realizes that she spent too much money at the amusement park. She decides to start saving money and go shopping when she has

$ 25

saved up. She puts

$ 15

in her piggy bank and plans to add

$ 0.50

every day.

Piggy Bank Animation

Let

t

be the number of days Zosia has been saving money. Write an equation that models the moment when her savings reaches

$ 25 .

Hint

Zosia's savings can be modeled by a linear expression of the form $a t + b,$ where $a$ and $b$ are real numbers.

Solution

First, write an expression modeling Zosia's savings. To do this, note that the amount of money in the piggy bank increases by a constant amount every day. Therefore, it can be modeled by a linear expression.

a t + b

The coefficient

a

is the rate of change and the constant

b

is the initial amount. Since Zosia starts with

$ 15

and saves

$ 0.50

every day, it can be said that

a = 0.5

and

b = 15 .

0.5 t + 15

The expression above represents the money in the piggy bank after

t

days. Finally, the expression above has to be equated to something else.

0.5 t + 15 = ?

To figure out what comes on the right-hand side, translate the equation above into words.

The money in the piggy bank after t days is equal to \dots

Since the aim is to model the moment in which Zosia reaches

$ 25,

the right-hand side of the equation is

25 .

$0.5 t + 15 = 25$

Example

Counting the Candy From the Store

Now that she has money saved up, Zosia is finally going to the mall. At the candy store, she buys

3

bags containing the same number of candies. The store also gives her

2

free candy samples. When she arrives home, she opens the bags and counts a total of

17

candies.

Candy Bags Animation

Let

x

be the number of candies in each bag. Write an equation that models the total amount of candies Zosia got from the store.

Hint

What is the coefficient for the $x -$ term in this situation?

Solution

First, write an expression that models the number of candies from the store. To do so, note that the total number of candies depends on the number of candies in each bag, the number of bags, and the number of extra candies. A linear expression can be used.

a x + b

x

represents the number of candies in each bag. The coefficient

a

is the number of bags, and the constant

b

is the number of free candies. Then,

a = 3

and

b = 2 .

3 x + 2

The expression above represents the total number of candies after opening the

3

bags with

x

candies each. Since an equation is required, the expression above has to be equated to something else.

3 x + 2 = ?

Zosia counted

17

total candies. Therefore, the right-hand side of the equation is

17 .

3 x + 2 = 17

Closure

Writing Algebraic Expressions to Model Real-Life Situations

This lesson discussed linear expressions and linear equations, as well as how to use inductive reasoning to create them. Countless real-life situations can be modeled by algebraic expressions and equations. Consider again the challenge presented at the beginning of the lesson.

People sitting around tables at a restaurant

As the waiters add more tables for the group of people at the restaurant for the family dinner, they begin to notice a pattern. Help the waiters by writing an algebraic expression that represents the number of people they can seat in terms of the number of tables

x .

Hint

Notice how the number of people sat increases with each table added.

Solution

Begin by making sense of the situation. At least $1$ table is needed for the family dinner. In this case, $4$ people can be seated.

For each subsequent table, $two$ more people are seated.

A linear expression can be used to model this situation.

a x + b

Here,

a = 2

because adding a table will add two more seated people. It can be noted that there will be always

2

people

at

the

ends

of the big table.

Therefore, it can be said that

b = 2 .

With this, the expression is complete.

Number of people = 2 x + 2

Using Inductive Reasoning to Create Equations

Exercises

Kriz has $$ 1000$ saved. They need a total of $$ 2000$ to buy a new gaming laptop.

Gaming laptop on sale for 2000 dollars

In order to reach this amount, they plan to save

$ 50

per week from their part-time job. Let

w

be the number of weeks that Kriz has saved. Write an equation that can be used to find after how many weeks Kriz will reach their goal.

To start writing an equation, we need to put the equality sign in the middle. ... = ... Kriz saves $50 each week, so we can find their new savings by multiplying 50 by the number of weeks passed since they started saving money. Let's put this on the left-hand side of the equation! 50w = ... This money will be used for their new laptop. Since Kriz started with $1000, we can find the total amount of money saved by adding these amounts. 1000+50w = ... Since Kriz aims to buy a $2000 gaming laptop, we will finish our equation by setting this expression equal to 2000. 1000+50w=2000

LaShay is making cookies for her cousins. She bakes the cookies in small batches of $5 .$

Five cookies on a baking sheet

After baking all the batches, she eats two cookies. Write an equation that can be used to find the total number of cookies

T

that LaShay is going to give to her cousins. Write the answer in terms of the number of batches

b

she made.

To start writing an equation, we need to put the equality sign in the middle. ... = ... LaShay bakes 5 cookies in each batch, so we multiply 5 by the number of batches b she made. Let's put this on the right-hand side of the equation! ... = 5b These cookies will be given to LaShay's cousins, but not before she eats two of them. This means that we will subtract 2 from the total amount of cookies LaShay baked. ... = 5b-2 Finally, we write the total amount of cookies to be given to her cousins on the left-hand side of the equation. T=5b-2

Magdalena wants to play $2$ games of golf this weekend.

Magdalena thinking of golf

A packet of practice golf balls contains

4

balls. Magdalena wants

20

balls for each game of golf she will play. Write an equation that can be used to find the number of packets

p

of golf balls that Magdalena should buy.

To start writing an equation, we need to put the equality sign in the middle. ... = ... We know from the exercise that a single pack of golf balls contains 4 balls. This means that if Magdalena buys p packs, she will have 4 p golf balls. Let's write that on the right-hand side of the equality. ... = 4 p This value should be equal to the number of golf balls Magdalena wants for each game, 20, multiplied by the number of games she will play, 2. 20* 2=4p This can be simplified by performing the multiplication on the left-hand side of the equation. 40=4p

Using Inductive Reasoning to Create Equations

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