Prasanna

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson 6.2 Answer Key Describing Functions.

Texas Go Math Grade 8 Lesson 6.2 Answer Key Describing Functions

Essential Question
What are some characteristics that you can use to describe functions?

Texas Go Math Grade 8 Lesson 6.2 Explore Activity Answer Key

Investigating a Constant Rate of Change
The U.S. Department of Agriculture defines heavy rain as rain that falls at a rate of 1.5 centimeters per hour.
A. The table shows the total amount of rain that falls in various amounts of time during a heavy rain. Complete the table.

B. Plot the ordered pairs from the table on the coordinate plane at the right.
C. How much rain falls in 3.5 hours? ___
D. Plot the point corresponding to 3.5 hours of heavy rain.

E. What do you notice about all of the points you plotted?
F. Is the total amount of rain that falls a function of the number of hours that rain has been falling? Why or why not?

Reflect

Question 1.
Suppose you continued to plot points for times between those in the table, such as 1.2 hours or 4.5 hours. What can you say about the locations of these points?
Answer:
These points will lie on the straight line of the graph of the data.

Your Turn

Go Math Grade 8 Lesson 6.2 Answer Key Question 2.
State whether the relationship between x and y in y = 0.5x is proportional or nonproportional. Then graph the function.

Answer:
y = 0.5x Given

Proportional relationship Compare the equation with the general linear equation y = mx + b. Since b = 0, the relationship is proportional

Choose several values for the input x. Substitute these values of x in the equation to find the output y.

Graph the ordered pairs. Then draw a line through the points to represent the solution.

Your Turn

Question 3.
A soda machine makes \(\frac{2}{3}\) gallons of soda every minute. The total amount y that the machine makes in x minutes is given by the equation y = \(\frac{2}{3}\)x. Use the table and graph to determine whether the relationship between x and y is linear and, if so, if it is proportional.


Answer:
First, we compare the equation with the general linear equation y = mx + b. y = \(\frac{2}{3}\)x is in the form y = mx + b, with m = \(\frac{2}{3}\) and b = 0. Therefore the equation is a linear equation. Since b = 0, the relationship is proportional.
Then we choose several values for the input x. We substitute these values of x in the equation to find the output y.

We graph the ordered pairs and we draw a line through the points to represent the solutions of the function.

Texas Go Math Grade 8 Lesson 6.2 Guided Practice Answer Key

Plot the ordered pairs from the table. Then graph the function represented by the ordered pairs and tell whether the function is linear or nonlinear. Tell whether the function is proportional. (Examples 1 and 2)

Question 1.
y = 5 – 2x

Answer:
y = 5 – 2x

Complete the table

Graph the ordered pairs. Then draw a line through the points to represent the solution.
A linear relationship Graph of a Linear function is a straight-line

Lesson 6.2 Representing Functions Answer Key Question 2.
y = 2 – x²

Answer:
y = 2 – x² Given

Complete the table

Graph the ordered pairs. Then draw a line through the points to represent the solution.
Non linear relationship Graph of a Linear function is a straight line

Explain whether each equation is a linear equation. (Example 2)

Question 3.
y = x² – 1
Answer:
y = x² – 1
Insert several vaLues for X:
x = 2
y = 2² – 1 (Substitute the given value of x for x)
= 4 – 1 (Simplify)
= 3 (Subtract)
x = 3
y = 3² – 1 (Substitute the given value of x for x)
= 9 – 1 (Simplify)
= 8 (Subtract)
x = 4
y = 4² – 1 (Substitute the given value of x for x)
= 16 – 1 (Simplify)
= 15 (Subtract)
The rate of change is not constant, so the equation is non-linear.
Non-linear

Question 4.
y = 1 – x
Answer:
y = 1 – x Given
The equation is in the form of a linear equation, hence is a linear equation.
Compare the equation with the general linear equation y = mx + b.

Essential Question Check-In

Question 5.
Explain how you can use a table of values, an equation, and a graph to determine whether a function represents a proportional relationship.
Answer:
From a table, determine the ratio \(\frac{y}{x}\). If it is constant the relationship is proportional.

From a graph, note if the graph passes through the origin. The graph of the proportional relationship must pass through the origin (0, 0).

From an equation, compare with the general linear form of the equation, y = mx + b. If b = 0, the relationship is proportional.

Texas Go Math Grade 8 Lesson 6.2 Independent Practice Answer Key

Lesson 6.2 Answer Key 8th Grade Go Math Question 6.
State whether the relationship between x and y in y = 4x – 5 is proportional or nonproportional. Then graph the function.

Answer:
First, we compare the equation with the general linear equation y = mx + b. y = 4x – 5 is in the form y = mx + b, with m = 4 and b = -5. Therefore, the equation is a linear equation. Since b ≠ 0, the relationship is non-proportional.
Then, we choose several values for the input x. We substitute these values of x in the equation to find the output y.

We graph the ordered pairs and we draw a line through the points to represent the solutions of the function.

Go Math Lesson 6.2 Describing Function Relationships Answer Key Question 7.
The Fortaleza telescope in Brazil is a radio telescope. Its shape can be approximated with the equation y = 0.01 3x². Is the relationship between x and y linear? Is it proportional? Explain.
Answer:
The linear function has the form y = mx + b, where m and b are real numbers. Every equation in the form of y = mx + b is a linear equation. Equations that cannot be written in this form is not linear equations and not linear functions.

Given the equation y = 0.013x², we can definitely say that the relationship of x and y is not linear because the equation is not in the form of y = mx + b. As we can see, in the given equation there is variable x that is being raised to the 2 power or simply x² which is not noticeable in the form of linear equation y = mx + b. So, it is not linear.

Since the given is not a linear equation, we cannot identify having a proportional or nonproportional relationship, then the equation is not proportional

Question 8.
Kiley spent $20 on rides and snacks at the state fair. If x is the amount she spent on rides, and y is the amount she spent on snacks, the total amount she spent can be represented by the equation x + y = 20. Is the relationship between x and y linear? Is it proportional? Explain.
Answer:
Owen
x + y = 20 Given
y = -x + 20 Rewriting the equation
It is linear Compare the equation with the general linear equation y = mx + b.
It is not proportional Since b ≠ 0, the relationship is not proportionaL

Question 9.
Represent Real-World Problems The drill team is buying new uniforms. The table shows y, the total cost in dollars, and x, the number of uniforms purchased.

a. Use the data to draw a graph. Is the relationship between x and y linear? Explain.
Answer:

Draw the graph
x and y are linear Graph of a linear relationship is a straight line.

b. Use your graph to predict the cost of purchasing 12 uniforms.
Answer:
The cost of 12 uniforms is $720

Go Math 8th Grade Answer Key Pdf Lesson 6.2 Reteach Answer Key Question 10.
Marta, a whale calf in an aquarium, is fed a special milk formula. Her handler uses a graph to track the number of gallons of formula y the calf drinks in x hours. Is the relationship between x and y linear? Is it proportional? Explain.

Answer:
The relationship is linear The data lies on a straight line

The relationship is proportional. The graph passes through the origin

Question 11.
Critique Reasoning A student claims that the equation y = 7 is not a linear equation because it does not have the form y = mx + b. Do you agree or disagree? Why?
Answer:
Disagree The equation can be written in the form y = mx + b where m is 0. The graph of the solutions is a horizontal line.

Question 12.
Make a Prediction Let x represent the number of hours you read a book and y represent the total number of pages you have read. You have already read 70 pages and can read 30 pages per hour. Write an equation relating x hours and y pages you read. Then predict the total number of pages you will have read after another 3 hours.
Answer:
In the problem, the number of pages you have read already which is 70 pages and you can read 30 pages per hour, let us determine the equation relating x hours and y pages you read. Also, let us predict the total number of pages you will have read after another 3 hours.

Representation:
Let x be the number of hours you read a book
y be the number of pages you have read
m be the number of pages per x hours
b be the number of pages you have read already

To determine the equation, let us apply the slope-intercept form y = mx + b. By using the representation, we can
formulate the equation given m = 30 and b = 70, that is y = 30x + 70

Now, Let us predict the total number of pages you will have read after another 3 hours. Using the equation y =
30x + 70, substitute the 3 as the value of x.
y = 30x + 70
y = 30(3) + 70
y = 90 + 70
y = 160

Thus, the number of pages you will have read after another 3 hours is 160 pages.
y = 30x + 70; 160 pages

Texas Go Math Grade 8 Lesson 6.2 H.O.T. Focus On Higher Order Thinking Answer Key

Lesson 6.2 Describing Functions Answer Key Go Math 8th Grade Question 13.
Draw Conclusions Rebecca draws a graph of a real-world relationship that turns out to be a set of unconnected points. Can the relationship be linear? Can it be proportional? Explain your reasoning.
Answer:
The relationship is linear if all, the points lie on the same line.

If the relationship is linear and passes through the origin, it is proportional.

Question 14.
Communicate Mathematical Ideas Write a real-world problem involving a proportional relationship. Explain how you know the relationship is proportional.
Answer:
The amount of money earned at a car wash is a proportional relationship. When there is 0 cars washed, $0 is earned. The amount of money earned increases by the unit cost of car wash.

Go Math 8th Grade Lesson 6.2 Answer Key Describing Functions Question 15.
Justify Reasoning Show that the equation y + 3 = 3(2x + 1) is linear and that it represents a proportional relationship between x and y.
Answer:
y + 3 = 3(2x + 1) Given
y + 3 = 6x + 3 Simplify using distributive property
y + 3 – 3 = 6x + 3 – 3 Subtract 3 from each side
y = 6x
Compared with the general linear equation y = mx + b
Proportional Since b = 0, it is a proportional relationship

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson 4.1 Answer Key Representing Linear Nonproportional Relationships.

Texas Go Math Grade 8 Lesson 4.1 Answer Key Representing Linear Nonproportional Relationships

Essential Question
How can you use tables, graphs, and equations to represent linear nonproportional situations?

Your Turn

Question 1.
Francisco makes $12 per hour doing part-time work on Saturdays. He spends $4 on transportation to and from work. The equation y = 12x – 4 gives his earnings y, after transportation costs, for working x hours. Make
a table of values for this situation.

Answer:
Given equation y = 12x – 4.
Choose several values for x z and substitute in the equation.
Make a table.

How did we get values for y:
For x = 1 : y = 12 ∙ 1 – 4 = 8
For x = 2 : y = 12 ∙ 2 – 4 = 20
For x = 3: y = 12 ∙ 3 – 4 = 32
For x = 4 : y = 12 ∙ 4 – 4 = 44

Choose several values for x and substitute in the equation. Make a table.

Texas Go Math Grade 8 Lesson 4.1 Explore Activity Answer Key 

Examining Linear Relationships
Recall that a proportional relationship is a relationship between two quantities in which the ratio of one quantity to the other quantity is constant. The graph of a proportional relationship is a line through the origin. Relationships can have a constant rate of change but not be proportional.

The entrance fee for Mountain World theme park is $20. Visitors purchase additional $2 tickets for rides, games, and food. The equation y = 2x +20 gives the total cost, y, to visit the park, including purchasing x tickets.

Step 1
Complete these exercises to review skills you will need for this chapter.

Step 2
Plot the ordered pairs from the table. Describe the shape of the graph.
Step 3
Find the rate of change between each point and the next. Is the rate constant?
Step 4
Calculate \(\frac{y}{x}\) for the values in the table. Explain why the relationship between number of tickets and total cost is not proportional.

Reflect

Lesson 4.1 Go Math Grade 8 Answer Key Pdf Question 2.
Analyze Relationships Would it make sense to add more points to the graph from x = 0 to x = 10? Would it make sense to connect the points with a line? Explain.
Answer:
The given equation is y = 2x + 20, which shows the proportion of the total cost and the number of additional
tickets.

To get the values for y, include the given values of x in the given equation.

After entering the points on the graph, we get a straight line, which does not pass through the origin.

To find the rate of change between each point and the next one, we need to get the slope of the line. The
standard form of the given equation is y = mx + b, so we are calculating the m (slope).
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Look at the dots from the graph:

The rate is constant because the ratio between each and the next point is the same.

In the previous step, we have calculated the rate of change. The rate is constant, but the linear relationship is not proportional because the line does not pass through the origin.

Take a look at the standard form of the equation again: y = mx + b. when the b is not equal to 0, b ≠ 0, the linear
function won’t pass the origin on the graph.

Your Turn

Question 3.
Make a table and graph the solutions of the equation y = -2x + 1.


Answer:
y = -2x + 1.

Choose several values for x and substitute in the equation to find y. Make a table.

Plot the ordered pairs from the table. Then draw a line connecting the points to represent all the possible solutions.

Texas Go Math Grade 8 Lesson 4.1 Guided Practice Answer Key 

Make a table of values for each equation. (Example 1)

Question 1.
y = 2x + 5

Answer:
For the given equation y = 2x + 5, defaulted are values for x. To find the values for y, include the given values of x in the given equation.

Go Math Grade 8 Lesson 4.1 Answer Key Question 2.
y = \(\frac{3}{8}\)x – 5

Answer:
y = \(\frac{3}{8}\)x – 5 Given

Choose several values for x and substitute in the equation to find y. Make a table.

Explain why each relationship is not proportional. (Explore Activity)

Question 3.

First calculate \(\frac{y}{x}\) for the values in the table.
Answer:
\(\frac{3}{0}\) = undefined Find \(\frac{y}{x}\)
\(\frac{7}{2}\) = 3.5
\(\frac{11}{4}\) = 2.75
\(\frac{15}{6}\) = 2.5
\(\frac{19}{8}\) = 2.375
The ratio is not constant, hence relationship is not proportional.

Question 4.

Answer:
The graph is a straight line but does not pass through the origin.

Complete the table for the equation. Then use the table to graph the equation. (Example 2)

Question 5.
y = x – 1


Answer:
y = x – 1 Given
Choose several values for x and substitute in the equation to find y. Make a table.

Plot the ordered pairs from the table. Then draw a line connecting the points to represent all the possible solutions.

Essential Question Check-In

Question 6.
How can you choose values for x when making a table of values representing a real-world situation?
Answer:
When choosing values for x in a real-world situation, you choose positive values with an appropriate interval to represent the array of data.

Texas Go Math Grade 8 Lesson 4.1 Independent Practice Answer Key 

State whether the graph of each linear relationship is a solid line or a set of unconnected points. Explain your reasoning.

Question 7.
The relationship between the number of $4 lunches you buy with a $100 school lunch card and the money remaining on the card
Answer:
Each time you buy lunch the amount on the card will decrease by 4. This means that this function will be y = 100 – 4x, where x is the number of lunches and y remaining money.
For example:
When we buy one lunch (x = 1), we have 96 left.
That way we can only buy 25 lunches for 100.
The number x must be an integer because we cannot buy a fractional part of a lunch. So we have a set of unconnected points:{( 1, 96), (2, 92), …, (25, 0)}, not a solid line.

Set of unconnected points

Texas Go Math Lesson 4.1 Answer Key Grade 8 Question 8.
The relationship between time and the distance remaining on a 3-mile walk for someone walking at a steady
rate of 2 miles per hour
Answer:
The person who is going to walk starts from the starting point 0. Thus, the first point on the graph is (0, 0), the starting point. The second thing we know is that a person has crossed 2 miles in an hour, so that’s another point on the graph (1, 2).
Those data are based on the rate of steady walking.
Since the rate of steady walking is the same all the time, it is constant, and time can be measured in fraction, we can read the value for x on 3 miles (y) from the graph. → x = 1.5 = \(\frac{3}{2}\)

The state of this graph of the linear relationship is a solid line.

Question 9.
Analyze Relationships Simone paid $12 for an initial year’s subscription to a magazine. The renewal rate is $8 per year. This situation can be represented by the equation y = 8x + 12, where x represents the number of years the subscription is renewed and y represents the total cost.
a. Make a table of values for this situation.

Answer:
y = 8x + 12
to make a table, we need to take some values for x, then include them into the equation y = 8x + 12 to get the values for y.

b. Draw a graph to represent the situation.
Answer:
enter to the graph obtained values of x and y

c) Explain why this relationship is not proportional.
Answer:
When the value of b is not equal to 0, b ≠ 0. the linear function does not pass through the origin. If the line does not pass the origin, the linear function is NON-proportional.
In our case b = 12 so it is not proportional.

d. Does it make sense to connect the points on the graph with a solid line? Explain.
Answer:
If we were to connect the obtained points on the graph, we would get a straight line, but that makes no sense.
The subscription works in a way that it is not possible to pay any amount other than $ 8 per year and it is not
possible to pay the subscription amount for just half a year or a few months. Any other point on the graph would
not be an actual indicator of the total cost.

Lesson 4.1 Representing Linear Non-Proportional Relationships Answer Key Question 10.
Analyze Relationships A proportional relationship is a linear relationship because the rate of change is constant (and equal to the constant of proportionality). What is required of a proportional relationship that is not required of a general linear relationship?
Answer:
All proportional relationships are linear relationships, but all linear relationships are not proportional relationships. So, the only difference between those is b from the equation y = mx + b. The linear function is proportional if the b = 0 and passes through the Origin. If b ≠ 0, then it is non-proportional, but it is still linear.

Question 11.
Communicate Mathematical Ideas Explain how you can identify a linear nonproportional relationship from a table, a graph, and an equation.
Answer:
In a table, the ratios \(\frac{y}{x}\) will not be equal.
A graph will not pass through the origin.
An equation will be in the form y = mx + b where b ≠ 0

Texas Go Math Grade 8 Lesson 4.1 H.O.T. Focus On Higher Order Thinking Answer Key 

Question 12.
Critique Reasoning George observes that for every increase of 1 in the value of x, there is an increase of 60 in the corresponding value of y. He claims that the relationship represented by the table is proportional. Critique George’s reasoning.

Answer:
\(\frac{90}{1}\) = 90 Find \(\frac{y}{x}\)
\(\frac{150}{2}\) = 75
\(\frac{210}{3}\) = 70
\(\frac{270}{4}\) = 67.5
\(\frac{330}{5}\) = 66
The ratio is not constant, hence the relationship cannot be proportional.

Go Math Middle School Grade 8 Answer Key Pdf Lesson 4.1 Reteach Answer Key Question 13.
Make a Conjecture Two parallel lines are graphed on a coordinate plane. How many of the lines could represent proportional relationships? Explain.
Answer:
The linear function can be proportional just if b = 0, and passes through the origin. If there are two lines on the
graph, just one can pass through the origin. Thus, if one line passes through the origin, the other one won’t, so just the ONE line can represent the proportional relationship.

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson Texas Go Math Grade 8 Lesson 6.1 Answer Key Identifying and Representing Functions.

Texas Go Math Grade 8 Lesson 6.1 Answer Key Identifying and Representing Functions

Essential Question
How can you identify and represent functions?

Texas Go Math Grade 8 Lesson 6.1 Explore Activity Answer Key 

Explore Activity
Understanding Relationships
Carlos needs to buy some new pencils from the school supply store at his school. Carlos asks his classmates if they know how much pencils cost. Angela says she bought 2 pencils for $0.50. Paige bought 3 pencils for $0.75, and Spencer bought 4 pencils for $1.00.

Carlos thinks about the rule for the price of a pencil as a machine. When he puts the number of pencils he wants to buy into the machine, the machine applies a rule and tells him the total cost of that number of pencils.

A. Use the prices in the problem to fill in total cost in rows i – iii of the table.

B. Describe any patterns you see. Use your pattern to determine the cost of 1 pencil.

C. Use the pattern you identified to write the rule applied by the machine. Write the rule as an algebraic expression and fill in rule column row iv of the table.

D. Carlos wants to buy 12 pencils. Use your rule to fill in row v of the table to show how much Carlos will pay for 12 pencils.

Reflect

Question 1.
How did you decide what operation to use in your rule?
Answer:
The rule would be to multiply the number of pencils to the unit rate to find the total cost.

Identifying and Representing Functions Lesson 6.1 Answer Key Question 2.
What If? Carlos decides to buy erasers in a package. There are 6 pencil-top erasers in 2 packages of erasers.
Answer:
To write the rule as an algebraic expression, we will denote the number of pencil-top erasers with variables y and x will represent the package of erasers.

a. Write a rule in words for the number of packages Carlos needs to buy to get x erasers. Then write the rule as an algebraic expression.
Answer:
Knowing that Carlos got six pencil-top erasers in two packages, we can conclude that one package results in
three erasers, so we can write it down as:
x = \(\frac{y}{3}\)
Therefore, we can claim that for every bought package, Carlos gets three pencil-top erasers.

b. How many packages does Carlos need to buy to get 18 erasers?
Answer:
To calculate how many packages Carlos has to buy to get 18 erasers, we will use the given formula as following:
x = \(\frac{18}{3}[/latex
x = 6

Reflect

Question 3.
Is it possible for a function to have more than one input value but only one output value? Provide an illustration to support your answer.

Answer:
We know that a function assigns exactly one output to each input. Therefore, it is possible for a function to have more than one input value but only one output value. Below, you can find an illustration of this case.

Your Turn

Determine whether each relationship is a function. Explain.

Question 4.

Answer:
Function A function assigns exactly one output to each input.

Identifying and Representing Functions Answer Key Question 5.

Answer:
Not function A function assigns exactly one output to each input.

Reflect

Question 6.
What is always true about the numbers in the first column of a table that represents a function? Why must this be true?
Answer:
To determine the answer here, we will review the definition of a function.

In the first column of a table that represents a function, there should be no repetition of values. By the definition, a table of values represents a function only if each input value has one output value.
Therefore, two values in one column would mean that the table does not represent a function.

Your Turn

Determine whether each relationship is a function. Explain.

Question 7.

Answer:
Function There is no repetition in the first column. Each input value is paired with only one output value.

Question 8.

Answer:
Not a function There is a repetition in the first column which is 8. Hence, 8 has two outputs 21 and 34.

Question 9.
Many real-world relationships are functions. For example, the amount of money made at a car wash is a function of the number of cars washed. Give another example of a real-world function.
Answer:
Profit at the coffee shop can be viewed as a function, considering the fact that it depends on the other variable which is the amount of sold products.

Therefore, we can say that the profit is a function of the number of sold products

Your Turn

Lesson 6.1 Identifying and Representing Functions Answer Key Question 10.
The graph shows the relationship between the heights and weights of the members of Team Members of a basketball team. Is the relationship represented by the graph a function? Explain.

Answer:
Not a function Input values are paired with more than one output values (70, 165) and (70, 174).

Texas Go Math Grade 8 Lesson 6.1 Guided Practice Answer Key

Complete each table. In the row with x as the input, write a rule as an algebraic expression for the output. Then complete the last row of the table using the rule. (Explore Activity)

Question 1.

Answer:
Unit cost of ticket = [latex]\frac{40}{2}\) = 20

Total Cost = 20x
where $x$ is the number of tickets

Question 2.

Answer:
Number of pages per minute = \(\frac{1}{2}\) = 0.5
Total Cost = 0.5x
where $x$ is the number of minutes

Question 3.

Answer:
Unit cost of Muffins = \(\frac{2.25}{1}\) = 2.25
Total Cost = 2.25x
where $x$ is the number of muffins

Determine whether each relationship is a function. (Examples 1 and 2)

Go Math Lesson 6.1 Identifying and Representing Functions Answer Key Question 4.

Answer:
Function Each input is assigned to exactly one output

Question 5.

Answer:
To determine the answer here, we will review the definition of a function.

If a certain table of values represents a function, there should be no repetition of values in the first column because in a function each input value is paired with one output value.

Therefore, the given relationship does not represent a function, because value 4 repeats two times in the first column and results in two output values.

Question 6.
The graph shows the relationship between the weights of 5 packages and the shipping charge for each package. Is the relationship represented by the graph a function? Explain.

Answer:
Explanation A:
Because inputs are paired with output value and there is no repeated value, the relationship is a function.

Explanation B:
To determine the answer, we will write values from the graph in one table. First column represents the weight and the second one represents the shipping cost. Table is shown below.

Considering the fact that there is no repetition of input values and every input value is paired with one output value, we can claim that given relationship is a function.

Essential Question Check-In

Question 7.
What are four different ways of representing functions? How can you tell if a relationship is a function?
Answer:
A function is defined as the relationship in which the input value is paired with exactly one output value. In representing functions, we have four different ways. These are using Mapping Diagrams, Tables, Graphs, and Equations.

A mapping diagram represents a function if each input value is paired with only one output value.

Relationships of functions between input values and output values can also be represented using tables. The values in the first column are the input values. The values in the second column are the output values.

Graphs can be used to displays relationships between two sets of numbers. Each point on a graph represents an ordered pair

A function can be represented by an equation with two variables. One variable is the domain and the other is the
range.

By the description, the four ways of representing a function are through Mapping Diagrams, Tables, Graphs, and
Equations Also, we can say that a relationship is a function if the input value is paired with exactly one output value.

Texas Go Math Grade 8 Lesson 6.1 Independent Practice Answer Key

Determine whether each relationship represented by the ordered pairs is a function. Explain.

Question 8.
(2, 2), (3, 1), (5, 7), (8, 0), (9, 1)
Answer:
A function is defined as a relation such that the input value is paired with exactly one output value.

To determine if the given ordered pairs (2, 2), (3, 1), (5, 7), (8, 0), (9, 1) is a function, we can write it in table form, wherein x-coordinate is represented as input and y-coordinate is represented as output.
|input (x) | Output(y)|
|——|——|
|2 |2|
|3|1|
|5 | 7|
|8| 0|
| 0|1|

As we can notice in the table, one output value is paired with two input values these are (3, 1) and (0, 1), however
each input value is paired only with one output value, so still, the relationship is a function.

A function; Each input value is paired only with one output value.

Lesson 6.1 Answer Key 8th Grade Representing Functions Answer Key Question 9.
(0, 4), (5, 1), (2, 8), (6, 3), (5, 9)
Answer:
A function is defined as a relation such that the input value is paired with exactly one output value.

To determine if the given ordered pairs (0,1), (5, 1), (2,8), (6,3), (5,9) is a function, we can write it in table form, wherein the x-coordinate is represented as input and the y-coordinate is represented as output
|Input (x) | Output (y)l
|–|–|
|0|4|
|5|1|
|2 | 8|
|6| 3|
|5| 9|

As we can notice in the table, one input value “5” is paired with two output values “1 and 9”, (5, 1) and (5, 9), so the relationship is not a function.
Not a function; The input value 5 is paired with more than one output value 1 and 9.

Question 10.
Draw Conclusions Joaquin receives $0.40 per pound for 1 to 99 pounds of aluminum cans he recycles. He receives $0.50 per pound if he recycles more than 100 pounds. Is the amount of money Joaquin receives a function of the weight of the cans he recycles? Explain your reasoning.
Answer:
Yes The amount of money increases with the weight of the cans. No weight will result in the same amount of money earned.

Question 11.
A biologist tracked the growth of a strain of bacteria, as shown in the graph.

a. Explain why the relationship represented by the graph is a function.
Answer:
The relationship is a function as each input has been assigned exactly one output. There is only one number of bacteria for each number of hours.

b. What is It? Suppose there was the same number of bacteria for two consecutive hours. Would the graph still represent a function? Explain.
Answer:
Yes. If the number of bacteria for two consecutive hours is the same, one input will still be paired with one output hence the relationship is still a function.

Question 12.
Multiple Representations Give an example of a function in everyday life and represent it as a graph, a table, and a set of ordered pairs. Describe how you know it is a function.


Answer:
The cost of a bouquet of flowers and the number of flowers in the bouquet is a function. Unit cost of flowers = $0.85 and x is the number of flowers. Hence. C = 0.85x


Ordered Pairs
(2, 1.7), (4, 3.4), (6, 5.1), (8, 6.8), (10, 8.5)
Each value of input is paired with exactly one output.

The graph shows the relationship between the weights of six wedges of cheese and the price of each wedge.

Lesson 6.1 Identifying and Representing Functions Go Math Question 13.
Is the relationship represented by the graph a function? Justify your reasoning. Use the words ‘input” and output” in your explanation, and connect them to the context represented by the graph.

Answer:
Yes, the relationship represented by the graph is a function. Each input (weight) in the graph is paired with exactly one output (price).

Yes, the relationship represented by the graph is a function.

Question 14.
Analyze Relationships Suppose the weights and prices of additional wedges of cheese were plotted on the graph. Is that likely to change your answer to question 13? Explain your reasoning.

Answer:
Explanation A:
No, since the weight of the cheese is directly proportional to the cost of wedges of cheese.

Explanation B:
If weights and prices of additional wedges of cheese were plotted on the graph, it would still result in a function

The relationship would still result in a function because the price of wedges and weight are proportional values, meaning that for every input (weight) there would be only one output (price).

Texas Go Math Grade 8 Lesson 6.1 H.O.T Focus On Higher Order Thinking Answer Key

Question 15.
Justify Reasoning A mapping diagram represents a relationship that contains three different input values and four different output values. Is the relationship a function? Explain your reasoning.
Answer:
No. Since there are three inputs and four outputs, one of the inputs will have more than one output, hence the relationship cannot be a function.

Question 16.
Communicate Mathematical Ideas An onion farmer is hiring workers to help harvest the onions. He knows that the number of days it will take to harvest the onions is a function of the number of workers he hires. Explain the use of the word “function” in this context.
Answer:
In the problem, An onion farmer is hiring workers to help harvest the onions. He knows that the number of days it will take to harvest the onions is a function of the number of workers he hires. The concept of the function is being applied in the context.

The word function in the context of the problem tells that the input or the independent variable x is the number of workers the onion farmers hire on the farm, while the output or the dependent variable y is the number of days it will take to harvest the onions.

The function in the problem is used to describe that the input will greatly affect the output since the more workers hired will result in fewer days of harvest As x increases, y decreases or vice versa. In relation to function, each worker has a unique element or contribution to the output

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson 4.5 Answer Key Solving Systems of Linear Equations by Graphing.

Texas Go Math Grade 8 Lesson 4.5 Answer Key Solving Systems of Linear Equations by Graphing

Explore Activity
Investigating Systems of Equations

You have learned several ways to graph a linear equation in slope-intercept form. For example, you can use the slope and y-intercept or you can find two points that satisfy the equation and connect them with a line.

A. Graph the pair of equations together:

B. Explain how to tell whether (2, -1) is a solution of the equation y = 3x – 2 without using the graph.
Answer:

C. Explain how to tell whether (2, -1) is a solution of the equation y = -2x + 3 without using the graph.
Answer:

D. Explain how to use the graph to tell whether the ordered pair (2, -1) is a solution of either equation.
Answer:

E. Find the point of intersection of the two lines. Check by substitution to determine if it is a solution to both equations.

Reflect

Question 1.
A system of linear equations has infinitely many solutions. Does that mean any ordered pair in the coordinate plane is a solution?
Answer:
When a system of linear equations has infinitely many solutions, it means that graphing those equations will give us the same line. Therefore, the solutions of this system would be all the ordered pairs that lie on this line. Having infinitely many solutions does not mean that any ordered pair ¡n the coordinate plane is a solution.

Go Math Grade 8 Answer Key Pdf Lesson 4.5 Question 2.
Can you show algebraically that both equations in part B represent the same line? If so, explain how.
Answer:
y = 3x – 3
Distribute 3 to the terms within the parentheses.
y = 3(x – 1) = 3x – 3
Both equations in part B represent the same line.

y = 3(x – 1) = 3x – 3

Your Turn

Solve each system by graphing. Check by substitution.

Question 3.

Answer:
y = -x + 2
y = -4x – 1

Graph each of the equations.
Solution: (-1, 3) The solution of the linear systems is the intersection of the two equations

3 = -(-1) + 2 Check by substituting in the original equations
3 = 1 + 2
3 = 3
y = -4x – 1
3 = -4(-1) – 1
3 = 4 – 1
3 = 3

Question 4.

Answer:
Start by graphing each equation.
Find the point of intersection of the two lines.
It appears to be (1, 3). Check by substitution to determine if it is a solution to both equations.
y = -2x + 5
3 = -2(1) + 5
3 = -2 + 5
3 = 3
y = 3x
3 = 3(1)
3 = 3
The solution of the system is (1, 3).

The solution of the system (1, 3)

Reflect

Question 5.
Conjecture Why do you think the graph is limited to the first quadrant?
Answer:
The solution is restricted to the first quadrant as the number of hot dogs and the number of drinks cannot be negative.

Your Turn

Texas Go Math Grade 8 Lesson 4.5 Answer Key Question 6.
During school vacation, Marquis wants to go bowling and play laser tag. He wants to play 6 total games but needs to figure out how many of each he can play if he spends exactly $20. Each game of bowling is $2 and each game of laser tag is $4.

a. Let x represent the number of games Marquis bowls and let y represent the number of games of laser tag Marquis plays. Write a system of equations that describes the situation. Then write the equations in slope-intercept form.
Answer:
x + y = 6 Let x be the number of bowling games and y be the number of laser tag games
2x + 4y = 20
x + y – x = -x + 6 Rewrite in slope-intercept form
y = -x + 6
2x + 4y – 2x = -2x + 20
4y = -2x + 20
y = –\(\frac{2}{4}\)x + \(\frac{20}{4}\)
y = –\(\frac{1}{2}\)x + 5

b. Graph the solutions of both equations.
Answer:

Graph the equations on the same coordinate plane
y = -x + 6
y = –\(\frac{1}{2}\)x + 5

c. How many games of bowling and how many games of laser tag will Marquis play?
Answer:
Marquis should play 2 games of bowling and 4 games of laser tag
Solution

Texas Go Math Grade 8 Lesson 4.5 Guided Practice Answer Key

Question 1.

Answer:
y = 3x – 4
y = x + 2
We are given a system of equations

Graph the equations on the same coordinate plane
(3, 5) is the solution of the system of equations.
The solution of the linear system of equations is the intersection point of the two equations.
y = 3x – 4 Check by substitution to determine if it is a solution to both equations.
5 = 3.3 – 4
5 = 9 – 4
5 = 5
True
y = 3 + 2
5 = 5
True
(3, 5) is the solution of the system of linear equations.

Go Math 8th Grade Answer Key Lesson 4.5 Question 2.

Answer:
x – 3y = 2 Given
-3x + 9y = -6

x – 3y – x = -x + 2 Rewrite in slope-intercept form
-3y = -x + 2
y = \(\frac{1}{3}\)x – \(\frac{2}{3}\)
-3x + 9y + 3x = 3x – 6
9y = 3x – 6
y = \(\frac{3}{9}\)x – \(\frac{6}{9}\)
y = \(\frac{1}{3}\)x – \(\frac{2}{3}\)

Graph the equations on the same coordinate plane

Solution: Infinitely many solutions
The solution of the linear system of equations is the intersection of the two equations

Question 3.
Mrs. Morales wrote a test with 15 questions covering spelling and vocabulary. Spelling questions (x) are worth 5 points and vocabulary questions (y) are worth 10 points. The maximum number of points possible on the test is 100. (Example 2)
a. Write an equation in slope-intercept form to represent the number of questions on the test.
Answer:
x + y = 15
The sum of two types of questions is 15
x + y – x = -x + 15
Rewrite in slope-intercept form
y = -x + 15

b. Write an equation in slope-intercept form to represent the total number of points on the test.
Answer:
5x + 10y = 100
The total number of points on the test is 100
5x + 10y – 5x = -5x + 100 Rewrite in slope-intercept form
10y = -5x + 100
y = –\(\frac{5}{10}\)x + \(\frac{100}{10}\)
y = –\(\frac{1}{2}\)x + 10

c. Graph the solutions of both equations.
Answer:

Graph the equations on the same coordinate plane

d. Use your graph to tell how many of each question type are on the test.
Answer:
Solution: (10, 5)
There should be 10 spelling questions and 5 intersections of the two equations.
The solution of the linear system of equations is the vocabulary questions.

Essential Question Check-In

Question 4.
When you graph a system of linear equations, why does the intersection of the two lines represent the solution of the system?
Answer:
To solve a system of linear equations means to find the solutions that satisfy all the equations of that system. When we graph a system of linear equations, the intersection point lies on the line of each equation, which means that satisfies all, the equations. Therefore, it is considered to be the solution for that system.

Texas Go Math Grade 8 Lesson 4.5 Independent Practice Answer Key

Question 5.
Vocabulary A _________ is a set of equations that have the same variables.
Answer:
A “system of equations” is a set of equations that have the same variables.

Go Math Answer Key Grade 8 System of Linear Equations Question 6.
Eight friends started a business. They will wear either a baseball cap or a shirt imprinted with their logo while working. They want to spend exactly $36 on the shirts and caps. The shirts cost $6 each and the caps cost $3 each.
a. Write a system of equations to describe the situation. Let x represent the number of shirts and let y represent the number of caps.
_________
Answer:
x – the number of shirts;
y – the number of caps;
(The sum of all shirts and caps.)x + y = 8
(The total cost of all shirts and caps is 36 dollars.)6x + 3y = 36

b. Graph the system to find the solution. Verify the solution. What does it represent?

Answer:
Write the equations in slope-intercept form. Then graph.
x + y = 8
y = 8 – x
y = -x + 8 and (Divide both sides by 3.)
6x + 3y = 36
2x + y = 12
y = -2x + 12
Graph the equations y = -x + 8 and y = -2x + 12.
Use the graph to identify the solution of the system of equations.
Check your answer by substituting the ordered pair into both equations.
Apparent solution: (4, 4)
x + y = 8
4 + 4 = 8
8 = 8
and
6(4) + 3(4) = 36
24 + 12 = 36
36 = 36
The point (4, 4) is a solution of both equations.
Eight friends should order 4 shirts and 4 caps.

The point (4, 4) is a solution of both equations
Eight friends should order 4 shirts and 4 caps

Question 7.
Multistep The table shows the cost for bowling at two bowling alleys.

a. Write a system of equations, with one equation describing the cost to bowl at Bowl-o-Rama and the other
describing the cost to bowl at Bowling Pinz. For each equation, let x represent the number of games played and let y represent the total cost.
Answer:
x – the number of games played;
y – the total cost;
(The total cost to bowl at Bowl-o-Rama)y = 2 + 2.5x
(The total cost to bowl at Bowling Pinz)y = 4 + 2x

b. Graph the system to find the solution. Verify the solution. What does it represent?

Answer:
Graph the equations y = 2 + 2.5x and y = 4 + 2x.
Use the graph to identify the solution of the system of equations.
Check your answer by substituting the ordered pair into both equations.
Apparent solution: (4, 12)
y = 2 + 2.5x
12 = 2 + 2.5(4)
12 = 2 + 10
12 = 12 and
y = 4 + 2x
12 = 4 + 2(4)
12 = 4 + 8
12 = 12
Point (4, 12) is a solution or both equations.
The total cost for four games is 12 dollars.

The point (4, 12) is a solution of both equations.
The total cost for four games is 12 dollars.

Graphing Linear Equations Vocabulary Answer Key Question 8.
Multi-Step Jeremy runs 7 miles per week and increases his distance by 1 mile each week. Tony runs 3 miles per week and increases his distance by 2 miles each week. In how many weeks will Jeremy and Tony be running the same distance? What will that distance be?
Answer:
Jeremy’s distance after x weeks:
y = x + 7
Tony’s distance after x weeks:
y = 2x + 3
To find after how many weeks will Jeremy and Tony be running the same distance and what will the distance be, we can solve the system of those two linear equations:
y = x + 7
y = 2x + 3
We graph each equation on the same coordinate plane and we find the intersection point of those two lines, which is the solution of the system.

The solution of the system of linear equation is (4, 11), which means that after 4 weeks Jeremy and Tony will be running the same distance and that distance would be 11 miles.

After 4 weeks Jeremy and Tony will be running the same distance and that distance would be 11 miles.

Question 9.
Critical Thinking Write a real-world situation that could be represented by the system of equations shown below.

Answer:
Explanation A:
The entry fee for the first gym is $10 and for every hour that you spend there, you pay an extra $. If we denote with x the number of hours that somebody spends at the gym and with y the total cost, we have:
y = 4x + 10
On the other hand, the entry fee of the second gym is $15 and for every hour that you spend there, you pay an extra $3. If we denote with x the number of hours that somebody spends at the gym and with y the total cost, we
have:
y = 3x + 15
To find after how many hours the costs of the gyms will be the same and what will that cost be, we can solve the system of linear equations:
y = 4x + 10
y = 3x + 15

Let y be the total cost of skating and x be the rental cost of the stakes per hour. The constant represents the entry fee to the skating arena. In the first equation, $4 per hour is the rental cost of stakes and $10 is the entry fee. In the second equation, $3 per hr is the rental cost of stakes and $15 is the entry fee.

Texas Go Math Grade 8 Lesson 4.5 H.O.T. Focus On Higher Order Thinking Answer Key

Texas Go Math Grade 8 Answer Key Lesson 4.5 Question 10.
Multistep The table shows two options provided by a high-speed Internet provider.

a. In how many months will the total cost of both options be the same? What will that cost be?
Answer:
x – the number of months;
y – the total cost of one option after x months;
(The total cost for Option 1) after x months y = 50 – 30x
(The total cost for Option 2) after x months y = 40x
Check-in how many months the total cost of both options will be the same.
(Subtract 30x from both sides.)40x = 50 + 30x
(Divide both sides by 10.)10x = 50
(The total cost of both options will be the same in 5 months.)x = 5
(For x = 5)y = 40x
y = 40(5)
(The total cost will be 200 dollars.)y = 200

b. If you plan to cancel your Internet service after 9 months, which is the cheaper option? Explain.
Answer:
y = 50 + 30(9)
y = 50 + 270
(The total cost for Option 1) after 9 months y = 320 dollars
y = 40x
y = 40(9)
(The total cost for Option 2) after 9 months y = 360 dollars
After 9 months Option 1 is cheaper than Option 2

Texas Go Math Grade 8 Pdf Linear Equations Vocabulary Question 11.
Draw Conclusions How many solutions does the system formed by x – y = 3 and ay – ax + 3a = 0 have for a nonzero number a? Explain.
Answer:
x – y = Given
ay – ax + 3a = 0
ay – ax + 3a – 3a = 0 Rearranging the left side of the 2nd equation by subtracting 3a from both sides.
ay – ax = -3a
a(y – x) = 3a Divide both sides by -a
y – x = 3
x – y = 3
Since both equations are the same, the system of linear equations has infinitely many solutions.

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Module 1 Quiz Answer Key.

Texas Go Math Grade 8 Module 1 Quiz Answer Key

Texas Go Math Grade 8 Module 1 Ready to Go On? Answer Key

1.1 Rational and Irrational Numbers

Write each fraction as a decimal.

Question 1.
\(\frac{7}{20}\) ____
Answer:
To write \(\frac{7}{20}\) as a decimal. we divide the numerator by the denominator until the remainder is zero or until the digits in the quotient to repeat. We add as many zeros after the decimal point in the dividend as needed.

\(\frac{7}{20}\) = 0.35

Texas Go Math Squared Grade 8 Module 1 Question 2.
\(\frac{14}{11}\) ____
Answer:
Let x = \(1 . \overline{27} .\) Since \(1 . \overline{27}\) has two repeating digits multiplied on both sides of the equation with 100;
x = \(1 . \overline{27} .\)
100x = 100 × \(1 . \overline{27} .\)
100x = \(127 . \overline{27}\) Subtract x = \(1 . \overline{27}\) from both sides
99x = 126
x = \(\frac{14}{11}\) = 1\(\frac{3}{11}\)
x = 1\(\frac{3}{11}\)

Question 3.
1\(\frac{7}{8}\) ____
Answer:
First, we convert the mixed number to an improper fraction:
1\(\frac{7}{8}\) = 1 + \(\frac{7}{8}\)
= \(\frac{8}{8}\) + \(\frac{7}{8}\)
= \(\frac{15}{8}\)
To write \(\frac{15}{8}\) as a decimal, we divide the numerator by the denominator until the remainder is zero or until the digits in the quotient begin to repeat.
We add as many zeros after the decimal point in the dividend as needed.

1\(\frac{7}{8}\) = 1.875

Find the two square roots of each number.

Question 4.
81 ____
Answer:
x² = 81
\(\sqrt{x^{2}}\) = \(\sqrt{81}\) Take square root from both sides of the equation
x = ± 9

Question 5.
1600 ____
Answer:
x³ = 343
\(\sqrt[3]{x^{3}}\) = \(\sqrt[3]{343}\) Take square root from both sides of the equation
x = 7

Grade 8 Mathematics Session 1 Answer Key Question 6.
\(\frac{1}{100}\) ____
Answer:
x² = \(\frac{1}{100}\) Take square root from both sides of the equation
\(\sqrt{x^{2}}\) = \(\sqrt{\frac{1}{100}}\) Solve for x
x = ±\(\frac{1}{10}\)

Question 7.
A square patio has an area of 200 square feet. How long is each side of the patio to the nearest 0.05? ____
Answer:
The area of a square is found by multiplying the side of the square by itself
Therefore, to find the side of the square, we have to take the square root of the area
Let’s denote with A the area of the patio and with s each side of the square. We have:
A = 200
A = s . s
s = \(\sqrt{A}\) = \(\sqrt{200}\)
Following the steps as in “Explore Activity 1”on page 9, we can make an estimation for the irrationaL number:
196 < 200 < 225
\(\sqrt{196}\) < \(\sqrt{200}\) < \(\sqrt{225}\)
14 < \(\sqrt{200}\) < 15
We see that 200 is much closer to 196 than to 225, therefore the square root of it should be between 14 and 14.5 To make a better estimation, we pick some numbers between 14 and 14.5 and calculate their squares:
14.1² = 198.81
14.2² = 201.64
14.1 < \(\sqrt{200}\) < 14.2
\(\sqrt{200}\) ≈ 14.15
We see that 200 is much closer to 14.1 than to 14.2, therefore the square root of it should be between 14.1 and 14.15.
If we round to the nearest 0.05, we have:
s = 14.15

Each side of the patio is 14.15 feet long.

1.2 Sets of Real Numbers

Write all names that apply to each number.

Question 8.
\(\frac{121}{\sqrt{121}}\) _____
Answer:
Using \(\sqrt{121}\) = 11 we can see that \(\frac{121}{\sqrt{121}}\) = \(\frac{121}{11}\) = 11
Therefore it is whole, integer, rational, real

Whole, integer, rational, real

Module 1 Test Answers Texas Go Math Grade 8 Question 9.
\(\frac{\pi}{2}\) _____
Answer:
π is an irrational. number. When multiplying any rational number (in this case \(\frac{1}{2}\)) with an irrational number we get an irrational number.
Therefore it is an irrational, real number.

Irrational, real.

Question 10.
Tell whether the statement “All integers are rational numbers” is true or false. Explain your choice.
Answer:
All integers are rational numbers” is true, because every integer can be expressed as a fraction with a denominator equal to 1. The set of integers is a subset of rational numbers.

True

1.3 Ordering Real Numbers

Compare. Write <>, or =.

Question 11.

Answer:
Given
\(\sqrt{8}\) + 3 ? 8 + \(\sqrt{8}\) Given

4 < 8 < 9 Approximate \(\sqrt{8}\) Approximate \(\sqrt{8}\)
\(\sqrt{4}\) < \(\sqrt{8}\) < \(\sqrt{9}\)
2 < \(\sqrt{8}\) < 3

1 < 3 < 4
\(\sqrt{1}\) < \(\sqrt{3}\) < \(\sqrt{4}\)
1 < \(\sqrt{3}\) < 2

\(\sqrt{8}\) + 3 is between 5 and 6. Use the estimations to simplify the expressions
8 + \(\sqrt{3}\) is between 9 and 10.

\(\sqrt{8}\) + 3 < 8 + \(\sqrt{3}\) Compare

\(\sqrt{8}\) + 3 < 8 + \(\sqrt{3}\)

Grade 8 Math Module 1 Answer Key Question 12.

Answer:
Approximate the value of \(\sqrt{5}\) by comparing the number 5 to the closest perfect squares:
4 < 5 < 9
\(\sqrt{4}\) < \(\sqrt{5}\) < \(\sqrt{9}\)
2 < \(\sqrt{5}\) < 3
So, \(\sqrt{5}\) lies between 2 and 3.

In the same way, approximate the value of \(\sqrt{11}\) by comparing the number 11 to the closest perfect squares:
9 < 11 < 16
\(\sqrt{9}\) < \(\sqrt{11}\) < \(\sqrt{16}\)
3 < \(\sqrt{11}\) < 4 So, \(\sqrt{11}\) lies between 3 and 4. \(\sqrt{5}\) + 11 lies between 13 and 14, while \(\sqrt{11}\) + 5 lies between 8 and 9. Therefore: \(\sqrt{5}\) + 11 > \(\sqrt{11}\) + 5

Order the numbers from least to greatest.

Question 13.

Answer:
Comparing 99 to the closest perfect squares gives us a estimation of \(\sqrt{99}\):
81 < 99 < 100
\(\sqrt{81}\) < \(\sqrt{99}\) < \(\sqrt{10}\)
9 < \(\sqrt{99}\) < 10
A good approximation for \(\sqrt{99}\) is 9.95 because 9.95² ≈ 99
Second, π ≈ 3.14 ⇒ π² ≈ (3.14)² ≈ 9.86
Also, \(9 . \overline{8}\) ≈ 9.88
So because of: 9.86 < 9.88 < 9.95 the order is π², \(9 . \overline{8}\), \(\sqrt{99}\)

Question 14.

Answer:
\(\sqrt{\frac{1}{25}}\), \(\frac{1}{4}\), \(0 . \overline{2}\) Given
\(\sqrt{\frac{1}{25}}\) = \(\frac{1}{5}\) = 0.2 Evaluate
\(\frac{1}{4}\) = 0.25 Evaluate
\(0 . \overline{2}\) = 0.222 ≈ 0.22 Underlined number is the repeating number after decimal

Graph on the number line.
\(\sqrt{\frac{1}{25}}\), \(0 . \overline{2}\), \(\frac{1}{4}\) From least to greatest.

Essential Question

Question 15.
How are real numbers used to describe real-world situations?
Answer:
In real-world situations, we use real numbers to count or make measurements. They can be seen as a convention for us to quantify things around us, for example, the distance, the temperature, the height, etc.

Texas Go Math Grade 8 Module 1 Mixed Review Texas Test Prep Answer Key

Texas Test Prep

Selected Response

Question 1.
The square root of a number is 9. What is the other square root?
(A) -9
(B) -3
(C) 3
(D) 81
Answer:
(A) -9

Explanation:
We know that every positive number has two square roots, one positive and one negative. We are given the
principal square root (9), so the other square root would be its negative (-9).
To prove that, we square both numbers and compare the results:
9 . 9 = 81
(-9) . (-9) = 81

Question 2.
A square acre of land is 4840 square yards. Between which two integers is the length of one side?
(A) between 24 and 25 yards
(B) between 69 and 70 yards
(C) between 242 and 243 yards
(D) between 695 and 696 yards
Answer:
(B) between 69 and 70 yards

Explanation:
The area of a square is found by multiplying the side of the square by itself.
Therefore, to find the side of the square, we have to take the square root of the area.
Let’s denote with À the area of the Land and with s each side of the square. We have:
À = 4840
A = s . s
s = \(\sqrt{A}\) = \(\sqrt{4840}\)
Following the steps as in “Explore Activity 1” on page 9, we can make an estimation for the irrational number:

Each side of the land is between 69 and 70 yards.

Real Numbers Module Quiz Answer Key Question 3.
Which of the following is an integer but not a whole number?
(A) -9.6
(B) -4
(C) 0
(D) 3.7
Answer:
(B) -4

Explanation:
-4 is an integer but not a whole number Whole numbers are not negative

Question 4.
Which statement is false?
(A) No integers are irrational numbers.
(B) All whole numbers are integers.
(C) No real numbers are irrational numbers.
(D) All integers greater than 0 are whole numbers.
Answer:
(C) No real numbers are irrational numbers.

Explanation:
Option C is false Rational and irrational numbers are real numbers.

Question 5.
Which set of numbers best describes the displayed weights on a digital scale that shows each weight to the nearest half pound?
(A) whole numbers
(B) rational numbers
(C) real numbers
(D) integers
Answer:
(B) rational numbers

Explanation:
Rational number The scale weigh nearest \(\frac{1}{2}\) to pound

Question 6.
Which of the following is not true?

Answer:
(C)

Explanation:

A.
π ≈ 3.14 ⇒ π² ≈ (3.14)² ≈ 9.86
2π + 4 ≈ 10.28
It follows that π² < 2π + 4 so the given statement is true B. π ≈ 3.14 ⇒ 3π ≈ 3 × 3.14 = 9.42 Number 9 is given It follows that 3π > 9 so the given statement is true

C.
\(\sqrt{27}\) ≈ 5.2 because 5.2² = 27.04 ⇒ \(\sqrt{27}\) + 3 ≈ 8.2
\(\frac{17}{2}\) = 8\(\frac{1}{2}\) = 8.5
It follows that \(\sqrt{27}\) + 3 < \(\frac{17}{2}\), so the given statement is not true

D.
\(\sqrt{24}\) ≈ 4.9 because 4.9² = 24.01 ⇒ 5 – \(\sqrt{24}\) ≈ 0.101
Number 1 is given.
It follows that 5 – \(\sqrt{24}\) < 1 so the given statement is true

Option (C) is True

Question 7.
Which number is between \(\sqrt{21}\) and \(\frac{3 \pi}{2}\)?
(A) \(\frac{14}{3}\)
(B) 2\(\sqrt{6}\)
(C) 5
(D) π + 1
Answer:
(A) \(\frac{14}{3}\)

Explanation:
\(\sqrt{21}\) and \(\frac{3 \pi}{2}\) Given
\(\sqrt{21}\) = 4.58 Evaluate
\(\frac{3 \pi}{2}\) = \(\frac{3 * 3.14}{2}\) = 4.71

\(\frac{14}{3}\) = 4.67 Option A

2\(\sqrt{6}\) = 4.90 Option B

5 Option C

π + 1 = 3.14 + 1 = 4.14 Option D

Question 8.
What number is shown on the graph?

(A) π + 3
(B) \(\sqrt{20}\) + 2
(C) \(\sqrt{14}\) + 2.5
(D) \(6 . \overline{14}\)
Answer:
(B) \(\sqrt{20}\) + 2

Explanation:
Approximate every option and see which fits the point showed on the graph.
We can see that the given point on the graph lies between 6.4 and 6.5 but closer to 6.5.
π + 3 ≈ 3.14 + 3 = 6.14 This option obviously isn’t our answer because it isn’t between 6.4 and 6.5.
Next
\(\sqrt{4}\) + 2.5 = 2 + 2.5 = 4.5 This option also obviously isn’t our answer because it isn’t between 6.4 and 6.5
Next
\(\sqrt{20}\) + 2 ≈ 4.47 + 2 = 6.47 We can see that this fits out graph since it is between 6.4 and 6.5
Finally,
\(6 . \overline{14}\) ≈ 6.14 This option also obviously isn’t our answer because it isn’t between 6.4 and 6.5
So the final answer is that only the option C is shown on the graph.

Texas Go Math Grade 8 Module 1 Test Answers Question 9.
Which list of numbers is in order from least to greatest?
(A) 3.34, \(\frac{10}{3}\), π, \(\frac{11}{4}\)
(B) \(\frac{10}{3}\), 3.3, \(\frac{11}{4}\), π
(C) π, \(\frac{10}{3}\), \(\frac{11}{4}\), 3.3
(D) \(\frac{11}{4}\), π, 3.3, \(\frac{10}{3}\)
Answer:
(D) \(\frac{11}{4}\), π, 3.3, \(\frac{10}{3}\)

Explanation:
• To write \(\frac{10}{3}\) as a decimal, we divide the numerator by the denominator until the remainder is zero or until the digits in the quotient begin to repeat.
We add as nìanv zeros after the decimal point in the dividend as needed.

when a decimal lias one or more digits that repeat indefinitely, we write the decimal with a bar over the repeating digit(s). In our case, 3 repeats indefinitely.

• To write \(\frac{11}{4}\) as a decimal, we divide the numerator by the denominator until the remainder is zero or until the digits in the quotient begin to repeat.
We add as many zeros after the decimal point in the dividend as

Plot numbers on number line:

Gridded Response

Question 10.
What is the decimal equivalent of the fraction \(\frac{28}{25}\)?

Answer:
a. To find the length of an edge (x) substitute V = 1728 and solve for X by taking cube roots from both sides of
the equation:
1728 = x³
x = 12
So the length of the edge is 12 inches.

b. Since cubes are made of squares, what we really need to calculate is the area of a square that has a length of
the side x = 12
Use formula for area of a square A = x²:
A = x² = 12² = 144 inches.
So, the area of one side of the cube is 144 square inches.

c. Like we said in b., a cube is made out of squares, precisely, it is made out of 6 squares.
To find the surface area of the cube we need to add all the areas from the squares that make the cube, so:
S = 6 × r = 6 × 114 = 864 inches squared
So, the surface area of the cube is 864 square inches.

d. To get the desired surface area in square feet divide the surface area in square inches by 144 (since 1 ft = 12 in
⇒ 1 ft² = 144 in²)
S = \(\frac{864}{144}\) = 6 square feet.

a. x = 12 inches
b. A = 144 square inches
c. S = 864 square inches
d. S = 6 square feet

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Module 4 Answer Key Nonproportional Relationships.

Texas Go Math Grade 8 Module 4 Answer Key Nonproportional Relationships

Essential Question
How can you use non proportional relationships to solve real-world problems?

Texas Go Math Grade 8 Module 4 Are You Ready? Answer Key

Complete these exercises to review the skills you will need for this chapter.

Find each difference.

Question 1.
3 – (-5) _______
Answer:
3 – (-5) Given
3 + 5 To subtract an integer, add its opposite.
|3| + |-5| Since the signs are the same, so find the sum of the absolute values
3 + 5 = 8
8 Use the sign of the number with the greater absolute value.

Grade 8 Module 4 Answer Key Question 2.
-4 – 5 _______
Answer:
-4 – 5 Given
-4 + (-5) The signs are the same, so find the sum of the absolute values.
|-4| + |-5|
4 + 5 = 9
-9 Use the sign of the number with the greater absolute value

Question 3.
6 – 10 _____
Answer:
6 – 10 Given
6 + (-10) The signs are different so find the difference of the absolute values.
(-10) + (6)
|-10| – |6|
10 – 6 = 4
-4 Use the sign of the number with the greater absolute value.

Question 4.
-5 – (-3) ________
Answer:
-5 – (-3) Given
-5 + 3 To subtract an integer, add its opposite.
|-5| – |3| Since the signs are different, so find the difference of the absolute values
5 – 3 = 2
-2 Use the sign of the number with the greater absolute value

Question 5.
8 – (-8) ________
Answer:
8 – (-8) Given
8 + 8 To subtract an integer, add its opposite.
|8| + |8| Since the signs are different, so find the difference of the absolute values
8 + 8 = 16
16 Use the sign of the number with the greater absolute value

Module 4 Non-Proportional Relationships Answer Key Question 6.
9 – 5 = _____
Answer:
9 – 5 Given
9 + (-5) To subtract an integer, add its opposite.
|9| – |5| Since the signs are different, so find the difference of the absolute values
9 – 5 = 4
4 Use the sign of the number with the greater absolute value

Question 7.
-3 – 9 ______
Answer:
-3 – 9 Given
-3 + (-9) The signs are the same, so find the sum of the absolute values.
|-3| + |-9|
3 + 9 = 12
-12 Use the sign of the number with the greater absolute value
-12

Question 8.
0 – (-6) _____
Answer:
0 – (-6) Given
0 + 6 To subtract an integer, add its opposite.
|0| + |6| Since the signs are different, so find the difference in the absolute values
0 + 6 = 6
6 Use the sign of the number with the greater absolute value
6

Module 4 Grade 8 Math Answer Key Question 9.
12 – (-9) _____
Answer:
12 – (-9) Given
12 + 9 To subtract an integer, add its opposite.
|12| + |9| Since the signs are same, so find the difference of the absolute values
12 + 9 = 21
21 Use the sign of the number with the greater absolute value
21

Question 10.
-6 – (-4) _______
Answer:
-6 – (-4) Given
-6 + 4 To subtract an integer, add its opposite.
|-6| – |4| Since the signs are different, so find the difference of the absolute values
6 – 4 = 2
-2 Use the sign of the number with the greater absolute value

Question 11.
-7 – 10 _______
Answer:
-7 – 10 Given
-7 + (-10) To subtract an integer, add its opposite.
|-7| + |-10| Since the signs are the same, so find the difference between the absolute values
7 + 10 = 17
-17 Use the sign of the number with the greater absolute value
-17

Module 4 Non-Proportional Relationships Grade 8 Answer Key Question 12.
5 – 14 = ___
Answer:
5 – 14 Given
5 + (-14) The signs are different, so find the difference of the absolute values
(-14) + (5)
|-14| – |5|
14 – 5 = 11
-11 Use the sign of the number with the greater absolute value
-11

Graph each point on the coordinate grid.

Question 13.
B (0, 5)
Answer:
To graph a point at (0, 5) start at the origin.
Then move 5 points up.
Graph point B(0, 5).

To graph a point at (0, 5) start at the origin.
Then move 5 points up.

Grade 8 Math Module 4 Answer Key Question 14.
C (8, 0)
Answer:

Question 15.
D (5, 7)
Answer:

Question 16.
E(2, 3)
Answer:

Texas Go Math Grade 8 Module 4 Reading Start-Up Answer Key
Visualize Vocabulary
Use the ✓ words to complete the diagram. You can put more than one word in each box.

Understand Vocabulary

Complete the sentences using the preview words.

Question 1.
Any ordered pair that satisfies all the equations in a system is a _____
Answer:
Any ordered pair that satisfies all the equations in a system is a Solution

Texas Go Math Grade 8 Module 4 Answer Key Question 2.
A _________________________ is an equation whose solutions form a straight line on a coordinate plane.
Answer:
A ‘linear equation” is an equation whose solutions form a straight line on a coordinate plane

Question 3.
A ________________________ is a set of two or more equations that contain two or more variables.
Answer:
A System of Equations is a set of two or more equations that contain two or more variables.

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson 4.4 Answer Key Proportional and Nonproportional Situations.

Texas Go Math Grade 8 Lesson 4.4 Answer Key Proportional and Nonproportional Situations

Essential Question
How can you distinguish between proportional and nonproportional situations?

Your Turn

Determine if each of the following graphs represent a proportional or nonproportional relationship.

Question 1.

Answer:
The starting point of the graph is (0, 2) which means that it does not pass through the origin. Some functions will have a proportional relationship if the line passes through the origin, but it does not. So, this graph represents the nonproportional relationship.

8th Grade Independent Practice Answer Key Lesson 4.4 Question 2.

Answer:
Proportional relationship It does pass through the origin

Reflect

Question 3.
Communicate Mathematical Ideas In a proportional relationship the ratio \(\frac{y}{x}\) is constant. Show that this ratio is not constant for the equation y = a – 14.
Answer:
y = a – 14 Given
y = 16 – 14 = 2 Find a few values of y by substituting different values of a
y = 17 – 14 = 3
y = 18 – 14 = 4
\(\frac{2}{16}\) = 0.125 Find the ratio \(\frac{y}{x}\), where x represents a
\(\frac{3}{17}\) = 0.176
\(\frac{4}{18}\) = 0.222
The ratio is not constant.

Go Math 8th Grade Lesson 4.4 Proportional or Non-Proportional Question 4.
What If? Suppose another equation represents Keith’s age in months y given his age in years a. Is this relationship proportional? Explain.
Answer:
y = 12a Since there are 12 months in a year

y = 12(1) = 12 Find a few values of y by substituting different values of a
y = 12(2) = 24
y = 12(3) = 36
\(\frac{12}{1}\) = 12 Find the ratio \(\frac{y}{x}\), where x represents a
\(\frac{24}{2}\) = 12
\(\frac{36}{3}\) = 12
Since the ratio is constant, the relationship is proportional.

Your Turn

Determine if each of the following equations represents a proportional or nonproportional relationship.

Question 5.
d = 65t
Answer:
d = 65t Comparing with the linear equation
y = mx + b
Proportional relationship Since y-intercept b = 0

Question 6.
p = 0.1s + 2000
Answer:
P = 0.1s + 2000 Comparing with linear relationship y = mx + b
This is a Non-Proportional relationship Since y-intercept b is not equal to 0
This is a Non-Proportional Relationship

Go Math Grade 8 Lesson 4.4 Answer Key Question 7.
n = 450 – 3p
Answer:
n = 450 – 3p Comparing with the linear equation y = mx + b
Non-proportional relationship Since y-intercept b = 450

Question 8.
36 = 12d
Answer:
36 = 12d Comparing with the linear equation y = mx + b
Non-proportional relationship The graph will not pass through the origin. It is a horizontal line.

Your Turn

Determine if the linear relationship represented by each table is a proportional or nonproportional relationship.

Question 9.

Answer:
To know if some linear relationship is proportional or nonproportional, we need to use ratio \(\frac{y}{x}\), which needs to be constant for proportionality.
From the given table, we have values for x and y, so just put them into the calculation.
\(\frac{y}{x}\) = \(\frac{30}{2}\) = 15
= \(\frac{90}{8}\) = 11.25
= \(\frac{150}{14}\) = 10.71
Thus, the obtained values are not constant so this Linear relationship is not proportional.

Lesson 4.4 Proportional and Non-Proportional Relationships Question 10.

Answer:
\(\frac{1}{5}\) = 0.2 Find the ratio \(\frac{y}{x}\)
\(\frac{8}{40}\) = 0.2
\(\frac{13}{65}\) = 0.2
The ratio is constant, hence the relationship is proportional.

Your Turn

Question 11.
Compare and contrast the following two situations.

Answer:
c = 20h Test-Prep Center A

The cost of attending Test-Prep Center A is proportional. The hourly rate is $20 which is less than Test-Prep Center B. Comparing with the linear equation y = mx + b where b = O
C = 25h – 100 Test-Prep Center B where h is the number of hours and 1oo is a coupon

The cost of attending Test-Prep Center B is non-proportional. The hourly rate is $25. It offers a coupon for an initial credit but its hourly rate is higher. Center B will cost more in the long run. Comparing with the Linear equation y = mx + b

Texas Go Math Grade 8 Lesson 4.4 Guided Practice Answer Key

Determine if each relationship is a proportional or nonproportional situation. Explain your reasoning. (Example 1, Example 2, Example 4)

Question 1.

Look at the origin.
Answer:
From the graph, we can see that the line is passing through the origin. When some linear function is passing
through the origin its relationship is proportional, and the y-intercept (‘b”) is equal to 0.

Question 2.

Answer:
From the graph, we can see that the line is not passing through the origin. When some linear function is not passing through the origin its relationship is nonproportional, and the y-intercept (“b”) is b ≠ 0.

Lesson 4 Skills Practice Proportional and Non-Proportional Relationships Answer Key Question 3.
q = 2p + \(\frac{1}{2}\)
Compare the equation with y = mx + b
Answer:
q = 2p + \(\frac{1}{2}\)
The equation is in the form y = mx + b, with p being used as the variable instead of x and q instead of y. The value of in is 2, and the value of b is \(\frac{1}{2}\). Since b is not 0, the relationship presented through the above equation is non-proportional.

Non-proportional

Question 4.
v = \(\frac{1}{10}\)u
Answer:
To determine if the relationship of function is proportional or nonproportional, let us compare the equation in
slope-intercept form. After comparing the equations, determine the y-intercept b, if b = 0, then the relationship is
proportional, if b ≠ 0, then it is a non-proportional relationship.

Given the equation v = \(\frac{1}{10}\)u, it is a pattern of y = mx + b, where variable y is being used as y and u as r. The value of m is \(\frac{1}{10}\), and the value of b = 0.

Since the value of b is equal to 0, then the relationship between u and v is proportional.
See the explanation.

The tables represent linear relationships. Determine if each relationship is a proportional or nonproportional situation. (Example 3, Example 4)

Question 5.

Find the quotient of y and x.
Answer:
To know if some linear relationship is proportional or nonproportional, we need to use ratio \(\frac{y}{x}\), which needs to be constant for proportionality.

From the given table, we have values for x and y, so just put them into my calculation
\begin(aligned}
\dfrac{y}(x}&=\dfrac{12}{3)=4\
&=\dfrac{36}{91}=4\
&=\dfrac{84}{21}=4
\end{atigned}
Thus, the obtained values are constant so this linear relationship is proportional.

Go Math 8th Grade Lesson 4.4 Answer Key Question 6.

Answer:
\(\frac{4}{22}\) = \(\frac{2}{11}\) Find the ratio \(\frac{y}{x}\)
\(\frac{8}{46}\) = \(\frac{4}{23}\)
\(\frac{10}{58}\) = \(\frac{5}{29}\)
Since the ratio is not constant, the relationship is non-proportional.

Question 7.
The values in the table represent the number of households that watched three TV shows and the ratings of the shows. The relationship is linear. Describe the relationship in another way. (Example 4)

Answer:
The number of households that watched TV shows represents the æ and the TV show rating represents y. We know that the relationship is Linear, but we need to find out if the linear relationship is proportional or nonproportional.
To know if some linear relationship is proportional or nonproportional, we need to use ratio \(\frac{y}{x}\), which needs to be constant for proportionality.
Given values of x and y, put into the calculation

Thus, the obtained values are constant, so this linear relationship is proportional.

Essential Question Check-In

Question 8.
How are using graphs, equations, and tables similar when distinguishing between proportional and nonproportional linear relationships?
Answer:
The ratio between y to x is constant when the relationship is proportional. Graphs, tables, and equations all can be used to find the ratio. The ratio is not constant when the relationship is non-proportional.

Texas Go Math Grade 8 Lesson 4.4 Independent Practice Answer Key

Proportional and Non-Proportional Lesson 4.4 Answer Key Question 9.
The graph shows the weight of a cross-country team’s beverage cooler based on how much sports drink it contains.

a. Is the relationship proportional or nonproportional? Explain.
Answer:
Non-proportional
The graph does not pass through the origin. The graph of a proportional relationship must pass through the origin

b. Identify and interpret the slope and the y-intercept.
Answer:
Slope = \(\frac{12-10}{4-0}\) = 0.5 Finding the slope using any two given points by Slope(m) = (y₂ – y₁) ÷ (x₂ – x₁) where (x₂, y₂) = (4, 12)
(x₁, y₁) = (0, 10)
The slope shows that each cup of sports drink weighs $ 0.5 lb.
y-intercept = 10 From the graph when x = 0
the y-intercept is the weight of the empty cooler, which is 10 lbs

For 10-11, tell if the relationship between a rider’s height above the first floor and the time since the rider stepped on the elevator or escalator is proportional or nonproportional. Explain.

Question 10.
The elevator paused for 10 seconds after you stepped on before beginning to rise at a constant rate of 8 feet per second.
Answer:
The rider’s height above the first floor represents the y-axis and the time since the rider stepped on the elevator or escalator represents the x-axis.
From the moment the rider steps on the elevator, the distance traveled at a certain time is counted. Only after 10 seconds, the elevator did start moving. For the first 10 seconds, the elevator did not move at all there wasn’t any shift in that time.
If we tried to show this with a graph, the first point would be (10, 0), which means that the graph would start to rise constantly only with the starting point 10 on the x-axis. If this is the starting point and not (0, 0), i.e. the origin, this linear relationship is not proportional.

Go Math Lesson 4.4 8th Grade Practice Answer Key Question 11.
Your height, h, in feet above the first floor on the escalator is given by h = 0.75t, where t is the time in seconds.
Answer:
The given equation h = 0.75t is the form of the equation y = mx + b, where h is the value instead of y, and t is the value instead of x. In the given equation there is no y-intercept (b), thus our b is equal to 0, b = 0. Some linear relationships are proportional if the y-intercept is equal to 0, so the given linear relationship is proportional.

Question 12.
Analyze Relationships Compare and contrast the two graphs.

Answer:
Graph A represents a linear relationship while Graph B represents a exponential relationship. They both pass through the origin and the value of y increases with an increase in x.

Question 13.
Represent Real-World Problems Describe a real-world situation where the relationship is linear and nonproportional.
Answer:
The dentist receives a monthly fee of 20 for each patient and an additional 5 for each patient’s arrival. The
equation is y = 5x + 20, where y represents total monthly income and x represents the number of patients admitted. The 20 fee represents the y – intercept(b). This relationship is linear but it is not proportional because the value of y-intercept is bigger than 0. i.e.b\not=0$.

Texas Go Math Grade 8 Lesson 4.4 H.O.T. Focus On Higher Order Thinking Answer Key

Question 14.
Mathematical Reasoning Suppose you know the slope of a linear relationship and one of the points that its graph passes through. How can you determine if the relationship is proportional or nonproportional?
Answer:
Use the graph and the given point to determine the second point. Connect the two points by a straight line. If the
graph passes through the origin, the relationship is proportional and if the graph does not pass through the origin, the relationship is non-proportional.

Texas Go Math Grade 8 Pdf Practice and Homework Lesson 4.4 Question 15.
Multiple Representations An entrant at a science fair has included information about temperature conversion in various forms, as shown. The variables F, C, and K represent temperatures in degrees Fahrenheit, degrees Celsius, and Kelvin, respectively.

a. Is the relationship between kelvins and degrees Celsius proportional? Justify your answer in two different ways.
Answer:
1st – Equation B, K = C + 273.15, includes the values of temperatures in kelvins and degrees Celsius. This
equation is the form of the equation y = mx + b. The value of y-intercept (b) in equation B is bigger than 0, and
when b ≠ 0 the linear relationship is not proportional.

2nd – To know if some linear relationship is proportional or nonproportional, we need to use ratio \(\frac{y}{x}\), which needs to be constant for proportionality. From the given table plot the values of x (degrees Celsius) and y (kelvin) into the calculation.
y = \(\frac{281.15}{8}\) = 35.14
= \(\frac{288.15}{15}\) = 19.21
= \(\frac{309.15}{36}\) = 8.59
From the obtained values we see that they are not constant which means that the given equation is not proportional

b. Is the relationship between degrees Celsius and degrees Fahrenheit proportional? Why or why not?
Answer:
Equation A can also be compared to equation y = mx + b. So when the value of y-intercept (b) in the equation is bigger than 0, when b ≠ 0, the linear relationship is not proportional. To conclude, the relationship between degrees Celsius and degrees Fahrenheit is not proportional.

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson 2.2 Answer Key Scientific Notation with Negative Powers of 10.

Texas Go Math Grade 8 Lesson 2.2 Answer Key Scientific Notation with Negative Powers of 10

Essential Question
How can you use scientific notation to express very small quantities?

Texas Go Math Grade 8 Lesson 2.2 Explore Activity Answer Key 

You can use what you know about writing very large numbers in scientific notation to write very small numbers in scientific notation.

A typical human hair has a diameter of 0.000025 meter. Write this number in scientific notation.

A. Notice how the decimal point moves in the list below. Complete the list.

B. Move the decimal point in 0.000025 to the right as many places as necessary to find a number that is greater than or equal to 1 and less than 10. What number did you find? ____
C. Divide 0.000025 by your answer to B. _____
Write your answer as a power of 10. ____
D. Combine your answers to B and c to represent 0.000025 in scientific notation. ____

Reflect

Question 1.
When you move the decimal point, how can you know whether you are increasing or decreasing the number?
Answer:
When the decimal point moves to the right, the number is increasing and if it moves to the left, the is decreasing.

Question 2.
Explain how the two steps of moving the decimal and multiplying by a power of 10 leave the value of the original number unchanged.
Answer:
The power of 10 represents the number of times the decimal is moved, hence multiplying will leave the value of the original number unchanged.

Reflect

Question 3.
Critical Thinking When you write a number that is less than 1 in scientific notation, how does the power of 10 differ from when you write a number greater than 1 in scientific notation?
Answer:
When we write a number that is less than 1 in scientific notation, the power of 10 is negative, whereas when we write a number that is greater than 1 in scientific notation, the power of 10 is positive

Your Turn
Write each number in scientific notation.

Question 4.
0.0000829
Answer:
0.0000829 Given
8.29 Place the decimal point
5 Count the number of places the decimal point is moved.
8.29 × 10^-5 Multiply 8.29 times a power of 10.
Since 0.0000829 is less than 1, the decimal point moves to the right and the exponent on 10 is negative.
8.29 × 10^-5

Scientific Notation with Positive Powers of 10 Practice and Problem Solving Question 5.
0.000000302
Answer:
0.000000302 Given
3.02 Place the decimal point
7 Count the number of places the decimal point is moved.
3.02 × 10^-7 Multiply 3.02 times a power of 10. Since the original number is less than 1, the decimal point moves to the right and the exponent on 10 is negative.
3.02 × 10^-7

Question 6.
A typical red blood cell in human blood has a diameter of approximately 0.000007 meters. Write this diameter in scientific notation. ________________________
Answer:
0.000007 Given
7.0 Place the decimal point
6 Count the number of places the decimal point is moved.
7.0 × 10^-6 Multiply 7.0 times a power of 10. Since the original number is less than 1, the decimal point moves to the right and the exponent on 10 is negative.
7.0 × 10^-6

Reflect

Question 7.
Justify Reasoning Explain whether 0.9 × 10^-5 is written in scientific notation. If not, write the number correctly in scientific notation.
Answer:
In order for a number to be written in scientific notation they must be of the format:
c × 10ⁿ (1)
where c is a decimal number greater or equal to 1 and less than 10 and n is an integer
Since in the given number, 0.9 < 1 this number is not written in scientific notation.
A correct way to write that number in scientific notation would be to move the decimal point one place to the
right because of that the exponent would also decrease by one:
9× 10^-6 (2)

Question 8.
Which number is larger, 2 × 10^-3 or 3 × 10^-2? Explain.
Answer:
2 × 10^-3 Given
and
3 × 10^-2
3 × 10^-2 is larger when comparing exponents, -2 > -3

Your Turn

Write each number in standard notation.

Question 9.
1.045 × 10^-6
Answer:
1.045 × 10^-6 Given
6 Use the exponent of the power of 10 to see how many places to move the decimal point.
places
0.000001045 Place the decimal point Since you are going to write a number less than 1.045, move the decimal to the left. Add placeholder zeros if necessary.
The number
1.045 × 10^-6 in standard notation is 0.000001045

Go Math Grade 8 Lesson 2.2 Answer Key Question 10.
9.9 × 10^-5
Answer:
9.9 × 10^-5 Given
5 Use the exponent of the power of 10 to see how many places to move the decimal point.
places
0.000099 Place the decimal point Since you are going to write a number less than 9.9, move the decimal to the left. Add placeholder zeros if necessary.
The number
9.9 × 10^-5 in standard notation is 0.000099

Question 11.
Jeremy measured the length of an ant as 1 × 10^-2 meters. Write this length in standard notation.
Answer:
1 × 10^-2 Given
2 Use the exponent of the power of 10 to see how many places to move the decimal point.
places
0.01 Place the decimal point Since you are going to write a number less than 1, move the decimal to the left. Add placeholder zeros if necessary.
The number
1 × 10^-2 in standard notation is 0.01 meter
0.01

Texas Go Math Grade 8 Lesson 2.2 Guided Practice Answer Key 

Write each number in scientific notation. (Explore Activity and Example 1)

Question 1.
0.000487
Hint: Move the decimal right 4 places.
Answer:
0.000487 Given
4.87 Place the decimal point
4 Count the number of places the decimal point is moved.
4.87 × 10^-4 Multiply 4.87 times a power of 10.
Since 0.000487 is less than 1, the decimal point moves to the right and the exponent on 10 is negative.
4.87 × 10^-4

Question 2.
0.000028
Hint: Move the decimal right 5 places.
Answer:
Move the decimal point as many places as necessary to find a number that is greater that or equal to 1 and less
than 10:
2.8 Place the decimal point (1)
5 places Count the number of places you moved the decimal point (2)
2.8 × 10^-5 Multiply the number from step (1) with 10^-5 (3)
Note: in step (3) you multiplied 2.8 with 10^-5 because you moved the decimal point 5 places to the right
2.8 × 10^-5

Go Math Grade 8 Lesson 2.2 Question 3.
0.000059
Answer:
0.000059 Given
5.9 Place the decimal point
5 Count the number of places the decimal point is moved.
5.9 × 10^-5 Multiply 5.9 times a power of 10. Since 0.000059 is less than 1, the decimal point moves to the right and the exponent on 10 is negative
5.9 × 10^-5

Question 4.
0.0417
Answer:
0.0417 Given
4.17 Place the decimal point
2 Count the number of places the decimal point is moved.
4.17 × 10^-2 Multiply 4.17 times a power of 10. Since 0.0417 is less than 1, the decimal point moves to the right and the exponent on 10 is negative
4.17 × 10^-2

Question 5.
Picoplankton can be as small as 0.00002 centimeter.
Answer:
0.00002 Given
2.0 Place the decimal point
5 Count the number of places the decimal point is moved.
2 × 10^-5 Multiply 2 times a power of 10. Since 0.00002 is less than 1, the decimal point moves to the right and the exponent on 10 is negative
2 × 10^-5 centimeter

Question 6.
The average mass of a grain of sand on a beach is about 0.000015 gram.
Answer:
0.000015 gram Given
1.5 Place the decimal point
5 Count the number of places the decimal point is moved.
1.5 × 10^-5 Multiply 1.5 times a power of 10. Since 0.000015 is less than 1, the decimal point moves to the right and the exponent on 10 is negative
1.5 × 10^-5 gram

Write each number in standard notation. (Example 2)

Question 7.
2 × 10^-5
Hint: Move the decimal left 5 places.
Answer:
2 × 10^-5 Given
5 Use the exponent of the power of 10 to see how many places to move the decimal point.
places
0.00002 Place the decimal point. Since you are going to write a number less than 2, move the decimal to the left. Add placeholder zeros if necessary.
The number
2 × 10^-5 in standard notation is 0.00002

Question 8.
3.582 × 10^-6
Hint: Move the decimal left 6 places.
Answer:
3.582 × 10^-6 Given
6 Use the exponent of the power of 10 to see how many places to move the decimal point.
places
0.000003582 Place the decimal point. Since you are going to write a number less than 3.582, move the decimal to the left. Add placeholder zeros if necessary.
The number
3.582 × 10^-6 in standard notation is 0.000003582

Question 9.
8.3 × 10^-4
Answer:
8.3 × 10^-4 Given
4 Use the exponent of the power of 10 to see how many places to move the decimal point.
places
0.00083 Place the decimal point. Since you are going to write a number less than 8.3, move the decimal to the left. Add placeholder zeros if necessary.
The number
8.3 × 10^-4 in standard notation is 0.00083

Go Math Grade 8 Lesson 2.2 Answer Key Question 10.
2.97 × 10^-2
Answer:
2.97 × 10^-2 Given
2 Use the exponent of the power of 10 to see how many places to move the decimal point.
places
0.0279 Place the decimal point. Since you are going to write a number less than 2.97, move the decimal to the left. Add placeholder zeros if necessary.
The number
2.7 × 10^-2 in standard notation is 0.0297

Question 11.
9.06 × 10^-5
Answer:
9.06 × 10^-5 Given
5 Use the exponent of the power of 10 to see how many places to move the decimal point.
places
0.0000906 Place the decimal point. Since you are going to write a number less than 9.06, move the decimal to the left. Add placeholder zeros if necessary.
The number
9.06 × 10^-5 in standard notation is 0.0000906

Question 12.
4 × 10^-5
Answer:
(1) First, use the exponent of the power of 10 to see how many places to move the decimal point. Since we have 10^-5 we have to move the decimal point 5 places.
(2) Place the decimal point, since you are going to write a number less than 4, move the decimal point to the left Add placeholder zeros if necessary:
4 × 10^-5 = 0.00004.
0.00004

Question 13.
The average length of a dust mite is approximately 0.0001 meter. Write this number in scientific notation. (Example 1)
Answer:
Move the decimal point as many places as necessary to find a number that is greater than or equal to 1 and less
than 10:
1 Place the decimal point (1)
4 places Count the number of places you moved the decimal point (2)
1 × 10^-4 Multiply the number from step (1) with 10^-4 (3)
Note: in step (3) you multiplied 1 with 10^-4 because you moved the decimal point 5 places to the right because 0.0001 is smaller than 1.
1 × 10^-4 meter.

Question 14.
The mass of a proton is about 1.7 × 10^-24 gram. Write this number in standard notation. (Example 2)
Answer:
1.7 × 10^-24 gram Given
24 Use the exponent of the power of 10 to see how many places to move the decimal point
places

0.0000000000000000000000017 Place the decimal point Since you are going to write a number less than 1.7, move the decimal to the left Add placeholder zeros if necessary.
The number
1.7 × 10^-24 in
standard notation is
0.0000000000000000000000017 gram

Essential Question Check-In

Question 15.
Describe how to write 0.0000672 in scientific notation.
Answer:
(1) Place the decimal point such that the new number is larger or equal to 1 but less than 10.
0.0000672 ⇒ 6.72
(2) Count the number of places you moved the decimal. point: 5 places.
(3) Multiply 6.72 by 10^-5 (because you moved the decimal 5 places to the right the exponent is negative)
6.72 × 10^-5

Texas Go Math Grade 8 Lesson 2.2 Independent Practice Answer Key 

Use the table for problems 16-21. Write the diameter of the fibers in scientific notation.

Question 16.
Alpaca
__________
Answer:
0.00277 Fiber diameter of Alpaca
2.77 Place the decimal point
3 Count me number 0f places the decimal point is moved.
2.77 × 10^-3 Multiply 2.77 times a power of 10. Since 0.00277 is less than 1, the decimal point moves to the right and the exponent on 10 is negative.
2.77 × 10^-3

Question 17.
Angora rabbit
_________
Answer:
0.0013 Fiber diameter of Angora rabbit
1.3 Place the decimal point
3 Count me number 0f places the decimal point is moved.
1.3 × 10^-3 Multiply 1.3 times a power of 10. Since 0.0013 is less than 1, the decimal point moves to the right and the exponent on 10 is negative.
1.3 × 10^-3

Go Math Grade 8 Lesson 2.2 Homework Answers Question 18.
Llama
_____
Answer:
0.0035 Fiber diameter of Llama
3.5 Place the decimal point
3 Count me number 0f places the decimal point is moved.
3.5 × 10^-3 Multiply 3.5 times a power of 10. Since 0.0035 is less than 1, the decimal point moves to the right and the exponent on 10 is negative.
3.5 × 10^-3

Question 19.
Angora goat
______
Answer:
0.0045 Fiber diameter of Angora goat
4.5 Place the decimal point
3 Count me number 0f places the decimal point is moved.
4.5 × 10^-3 Multiply 4.5 times a power of 10. Since 0.0045 is less than 1, the decimal point moves to the right and the exponent on 10 is negative.
4.5 × 10^-3

Question 20.
Orb web spider
Answer:
0.015 Fiber diameter of Orb web spider
1.5 Place the decimal point
2 Count me number 0f places the decimal point is moved.
1.5 × 10^-2 Multiply 1.5 times a power of 10. Since 0.0045 is less than 1, the decimal point moves to the right and the exponent on 10 is negative.
1.5 × 10^-2

Question 21.
Vicuna
Answer:
0.0008 Fiber diameter of Vicuna
8.0 Place the decimal point
4 Count me number 0f places the decimal point is moved.
8.0 × 10^-4 Multiply 8.0 times a power of 10. Since 0.0008 is less than 1, the decimal point moves to the right and the exponent on 10 is negative.
8.0 × 10^-4

Question 22.
Make a Conjecture Which measurement would be least likely to be written in scientific notation: the thickness of a dog hair, the radius of a period on this page, the ounces in a cup of milk? Explain your reasoning.
Answer:
Both the thickness of a dog hair and the radius of a period on a page are very small, numbers (Lengths), something that we tend to write in scientific notation (because it is easier to work with when written in scientific notation). On the other hand, it is likely that the number of ounces in a cup of milk is a number that we can easily write and work with, so we don’t need to write it in scientific notation, so it is reasonable to suspect that it is the least likely to be written in scientific notation.
The number of ounces in a cup of milk

Question 23.
Multiple Representations Convert the length 7 centimeters to meters. Compare the numerical values when both numbers are written in scientific notation.
Answer:
There are 100 centimeters in 1 meter, or 1 centimeter = 0.01 meters
Because of this 7 centimeters = 0.07 meters
Write 0.07 in scientific notation by moving the decimal point two places to the right:
0.07 = 7 × 10^-2 meter
Since 7 centimeters is a number already greater or equal to 1 and less than 10 to get a scientific notation we just
multiply it with 10⁰ = 1
7 = 7 × 10⁰ centimeters
Finally we can see that when in scientific notation the first factor is the same but the exponents of the second factor differ by 2.

In scientific notation the first factor is the same but the exponents of the second factor differ by 2.

Question 24.
Draw Conclusions A graphing calculator displays 1.89 × 10^-12 as 1.89E12. How do you think it would display 1.89 × 10^-12? What does the E stand for?
Answer:
We are told that 1.89. 10¹² will be displayed as 1.89e¹² in a scientific calculator.
We can conclude that here e stands for exponent of 10 and 1.89 . 10^-12 will be displayed as 1.89e -12

Question 25.
Communicate Mathematical Ideas When a number is written in scientific notation, how can you tell right away whether or not it is greater than or equal to 1 ?
Answer:
In a scientific notation, if the exponent of 10 is negative, the number is smaller than 1. Otherwise, it is greater or equal to 1.

Lesson 2.2 Independent Practice Go Math Grade 8 Question 26.
The volume of a drop of a certain liquid is 0.000047 liter. Write the volume of the drop of liquid in scientific notation.
Answer:
0.000047 liter Given
4.7 Place the decimal point
5 Count the number of places the decimal point is moved.
4.7 × 10^-5 Multiply 4.7 times a power of 10. Since 0.000047 is less than 1, the decimal point moves to the right and the exponent on 10 is negative.
4.7 × 10^-5

Question 27.
Justify Reasoning If you were asked to express the weight in ounces of a ladybug in scientific notation, would the exponent of the 10 be positive or negative? Justify your response.
Answer:
Since the weight of a ladybug in ounces ¡s a number smaller than 1 we would need to move the decimal point to the right. Therefore the exponent of the 10 would be a negative number.

Negative, since the weight of a ladybug is smaller than 1.

Physical Science The table shows the length of the radii of several very small or very large items. Complete the table.

Question 28.

Answer:
(1) Move the decimal point to the left and remove extra zeros:
1,740,000 ⇒ 1.74
(2) Count the number of places you moved the decimal. point: 6 places
(3) Multiply 1.74 by 10⁶ (because we moved the decimal point 6 places to the left 1.74 × 10⁶
1.74 × 10⁶ meters

Question 29.

Answer:

Question 30.

Answer:

Question 31.

Answer:

Question 32.

Answer:

Question 33.

Answer:

Question 34.
List the items in the table in order from the smallest to largest.
Answer:

Compare the powers of 10 and when they are same, compare the first factor. Based on this rule, the items are arranged from smallest to largest.

Texas Go Math Grade 8 Lesson 2.2 H.O.T. Focus On Higher Order Thinking Answer Key

Question 35.
Analyze Relationships Write the following diameters from least to greatest. 1.5 × 10^-2m 1.2 × 10²m 5.85 × 10^-3 m 2.3 × 10^-2 m 9.6 × 10^-1 m
Answer:
Compare the powers of 10 (the smaller the exponent the smaller the number), if some are the same then compare the first factors:
5.85 × 10^-3 < 1.5 × 10^-2 < 2.3 × 10^-2 < 9.6 × 10^-1 < 1.2 × 10²

5.85 × 10^-3, 1.5 × 10^-2, 2.3 × 10^-2, 9.6 × 10^-1, 1.2 × 10²

Question 36.
Critique Reasoning Jerod’s friend Al had the following homework problem:
Express 5.6 × 10^-7 in standard form.
Al wrote 56,000,000. How can Jerod explain Al’s error and how to correct it?
Answer:
5.6 × 10^-7 Given
Error is that the decimal is moved in the moved in the right direction and not the left direction.
Since the exponent of 10 is -7, the decimal should be moved to the left.

7 Use the exponent of the power of 10 to see how many places to move the decimal point
places
0.00000056 Place the decimal point Since you are going to write a number less than 5.6, move the decimal to the left. Add placeholder zeros if necessary.

Question 37.
Make a Conjecture Two numbers are written in scientific notation. The number with a positive exponent is divided by the number with a negative exponent. Describe the result. Explain your answer.
Answer:
Since numbers in scientific notation always have a factor that is a power of 10, dividing two numbers in scientific
notation will again be a power of 10. Using the rule:
\(\frac{a^{m}}{a^{n}}\) = a^m-n
we know that if n is a negative number, the final result will be a positive number since subtracting a negative
number gives a positive number.
So finally, the result will be like this:
Let n be a positive number so the exponent in the denominator is -n, a negative number.
\(\frac{a \times 10^{m}}{b \times 10^{-n}}\) = \(\frac{a}{b}\) × 10^m+n

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Unit 1 Study Guide Review Answer Key.

Texas Go Math Grade 8 Unit 1 Study Guide Review Answer Key

Texas Go Math Grade 8 Unit 1 Exercises Answer Key

Find the two square roots of each number. If the number is not a perfect square, approximate the values to the nearest 0.05. (Lesson 1.1)

Module 1 Real Numbers

Question 1.
16 ___________
Answer:
\(\sqrt{16}\) = 4, since 4 . 4 = 16
–\(\sqrt{16}\) = -4, since (-4) . (-4) = 16
Two square roots of 16 are +4 and -4.

Grade 8 Mathematics Unit 1 Answer Key Question 2.
\(\frac{4}{25}\) _______________
Answer:

Two square roots of \(\frac{4}{25}\) are +\(\frac{2}{5}\) and –\(\frac{2}{5}\)

Question 3.
225 _________
Answer:
\(\sqrt{225}\) = 15, since 15 . 15 = 225
–\(\sqrt{225}\) = -15, since (-15) . (-15) = 225
Two square roots of 225 are +15 and -15.

Question 4.
\(\frac{1}{49}\) _______________
Answer:

Two square roots of \(\frac{1}{49}\) are +\(\frac{1}{7}\) and –\(\frac{1}{7}\)

Question 5.
\(\sqrt{10}\) _______
Answer:
Since 10 is not a perfect square, approximate 10 with two closest perfect squares:
9 < 10 < 1.6
Take square root to find in between what numbers lies \(\sqrt{10}\):
\(\sqrt{9}\) < \(\sqrt{10}\) < \(\sqrt{16}\)
Solve:
3 < \(\sqrt{10}\) < 4
We now know that \(\sqrt{10}\) lies between 3 and 4 but since 10 is closer to 9, \(\sqrt{10}\) is closer to 3. To find a better approximation pick a few number between 3 and 4 but closer to 3 and square them to see where they lie relating to 10:
3.10² = 9.61 < 10
3.15² ≈ 9.92 < 10
3.2² = 10.24 > 10
We see that 3.15² is closer to 10 than 3.2² so we can conclude that the best approximation to the nearest 0.05 is 3.15

Unit 1 Study Guide Answer Key Texas Go Math Grade 8 Question 6.
\(\sqrt{18}\) _______
Answer:
Since 18 is not a perfect square, approximate 18 with the two closest perfect squares:
16 < 18 < 25
Take square root to find in between what numbers lie \(\sqrt{18}\):
\(\sqrt{16}\) < \(\sqrt{18}\) < \(\sqrt{25}\)
Solve:
4 < \(\sqrt{18}\) < 5
We now know that \(\sqrt{18}\) lies between 4 and 5 but since 18 is closer to 16, \(\sqrt{18}\) is closer to 4.
To find a better approximation pick a few numbers between 4 and 5 but closer to 4 and square them to see where they lie relating to 18:
4.15² ≈ 17.22 < 18
4.2² = 17.64 < 18 4.25 ≈ 18.06 > 18
We see that 4.25² is closer to 18 than 4.2² so we can conclude that the best approximation to the nearest 0.05 is 4.25

Write all names that apply to each number. (Lesson 1.2)

Question 7.
\(\frac{2}{3}\) _______________
Answer:
Since \(\frac{2}{3}\) isn’t an integer or a whole number but is expressed as a ratio of two integers we can conclude that it is a rational, real number.

Rational, real.

Question 8.
–\(\sqrt{100}\) _______
Answer:
–\(\sqrt{100}\)
-10
Rational, integer, real

Grade 8 Math Unit 1 Assessment Answer Key Question 9.
\(\frac{15}{5}\) _______________
Answer:
Since \(\frac{15}{5}\) is a ratio of two integers it is a rational real number. But, since
\(\frac{15}{5}\) = 3
it is also an integer and a whole number.

Whole, integer, rational, real

Question 10.
\(\sqrt{21}\) _______
Answer:
\(\sqrt{21}\)
Irrational, real

Compare. Write <, >, or . (Lesson 1.3)

Question 11.

Answer:
\(\sqrt{7}\) + 5 ? 7 + \(\sqrt{5}\) Given
\(\sqrt{7}\) is between 2 and 3 Estimate the value of \(\sqrt{7}\)
\(\sqrt{5}\) is between 2 and 3 Estimate the value of \(\sqrt{5}\)
\(\sqrt{7}\) + 5 is between 7 and 8 Use approximations to simply the expressions
7 + \(\sqrt{5}\) is between 9 and 10
\(\sqrt{7}\) + 5 < 7+ \(\sqrt{5}\) Compare

Question 12.

Answer:
6 + \(\sqrt{8}\) ? \(\sqrt{6}\) + 8 Given
\(\sqrt{8}\) is between 2 and 3 Estimate the value of \(\sqrt{8}\)
\(\sqrt{6}\) is between 2 and 3 Estimate the value of \(\sqrt{6}\)
6 + \(\sqrt{8}\) is between 8 and 9 Use approximations to simply the expressions
\(\sqrt{6}\) + 8 is between 10 and 11
6 + \(\sqrt{8}\) < \(\sqrt{6}\) + 8 Compare

Question 13.

Answer:
\(\sqrt{4}\) – 2 ? 4 – \(\sqrt{2}\) Given
\(\sqrt{4}\) = 2 Estimate the value of \(\sqrt{4}\)
\(\sqrt{2}\) is between 1 and 2 Estimate the value of \(\sqrt{2}\)
\(\sqrt{4}\) – 2 = 2 – 2 = 0 Use approximations to simply the expressions
4 – \(\sqrt{2}\) is between 3 and 2
\(\sqrt{4}\) – 2 < 4 – \(\sqrt{2}\) Compare

Order the numbers from least to greatest. (Lesson 1.3)

Question 14.
\(\sqrt{81}\), \(\frac{72}{7}\), 8.9
Answer:
First, express au. numbers in (simplified) decimal form so we could easily compare them.
\(\sqrt{81}\) = 9 Simplify (81 is a perfect square)
\(\frac{72}{7}\) = 10.29 Using long division divide 72 by 7
8.9 Already in the simplest form
Now we can compare:
8.9 < 9 < 10.26 (1)
From (1) it now follows that:
From least to greatest: 8.9 < \(\sqrt{81}\) < \(\frac{72}{7}\)

Algebra 1 Unit 1 Study Guide Answer Key Question 15.
\(\sqrt{7}\), 2.55, \(\frac{7}{3}\)
Answer:
\(\sqrt{7}\) lies between 2 and 3 Estimate the value of \(\sqrt{7}\)
Since $7$ is approximately midway
between $4$ and $9$, hence
2.65² = 6.50
2.6² = 6.76
2.65² = 7.02
Since $2.65^{2} = 7.02$ $\sqrt{7}
\approx 2.65$

2.55 Given

\(\frac{7}{3}\) = 2.33 Divide 7 by 3

Graph on the number line

\(\frac{7}{3}\), 2.55, \(\sqrt{7}\)

Module 2 Scientific Notation

Essential Question
How can you use scientific notation to solve real-world problems?
Answer:
Scientific notation is used to write very large or very small numbers using fewer digits. See how scientists use this notation to describe astronomical distances, such as the distance between planets, or microscopic distances, such as the length of a blood cell

Exercises
Write each number in scientific notation. (Lessons 2.1, 2.2)

Question 1.
3000 _________
Answer:
3000 in the scientific notation is 3 × 10³
Here move the decimal 3 times to the left then it is 3. But the three are less than 10.
Moved the decimal to the left is the exponent and the exponent is positive.

Question 2.
0.000015 _____
Answer:
0.000015 in the scientific notation is 1.5 × 10^-5.
Here move the decimal 5 times to the left then it is 1.5. But the three are less than 10.
Moved the decimal to the left is the exponent and the exponent is negative.

Question 3.
25,500,000 _____
Answer:
25,500,000 Given
2.55 Move the decimal point 7 places to the left. Remove the extra zeros.
10, 000, 000 Divide the original number by the result from above.
10⁷ Write the answer as power of 10.
2.55 × 10⁷ Write the product of the resuLts from step 1 and 2
2.55 × 10⁷

Question 4.
0.00734 _________
Answer:
0.00734 Given
7.34 Place the decimal point
3 Count the number of places the decimal point is moved.
7.34 × 10^-3 Multiply 7.34 times a power of 10. Since 0.00734 is less than 1, the decimal point moves to the right and the exponent on 10 is negative.
7.34 × 10^-3

Write each number in standard notation. (Lessons 2.1, 2.2)

Question 5.
5.23 × 10⁴ __________________________
Answer:
5.23 × 10⁴ Given
4 Use the exponent of the power of 10 to determine the number paces to move the decimal point
places
52,300 Place the decimal point Since you are going to write a number greater than 5.23, move the decimal point to the right. Add placeholder zeros if necessary.
52,300

Review and Preview Answer Key Grade 8 Question 6.
1.05 × 10⁶ __________________________
Answer:
Given that 1.05 × 10⁶
10⁶ = 1000000
1.05 × 1000000 = 1050000.

Question 7.
4.7 × 10^-1 __________________________
Answer:
Given that 4.7 × 10^-1.
10^-1 = 0.1
4.7 × 0.1 = 0.47.

Question 8.
1.33 × 10^-5 __________________________
Answer:
1.33 × 10^-5 Given
5 Use the exponent of the power of 10 to determine the number paces to move the decimal point
places
0.0000133 Place the decimal point Since you are going to write a number greater than 1.33, move the decimal point to the right. Add placeholder zeros if necessary.
0.0000133

Use the information in the table to write each weight in scientific notation. (Lessons 2.1, 2.2)

Question 9.
Ant _________
Answer:
Given that the weight of the ant = 0.000000661.
0.000000661 in the scientific notation is 6.61 × 10^-7.
Here move the decimal 7 times to the left then it is 6.61. But the three are less than 10.
Moved the decimal to the left is the exponent and the exponent is negative.

Question 10.
butterfly _________
Answer:
Given that the weight of the butterfly = 0.00000625.
0.00000625 in the scientific notation is 6.25 × 10^-6.
Here move the decimal 6 times to the left then it is 6.25. But the three are less than 10.
Moved the decimal to the left is the exponent and the exponent is negative.

Question 11.
elephant _________
Answer:
Given that the weight of the elephant = 9900
9900 in the scientific notation is 9.9 × 10³
Here move the decimal 3 times to the left then it is 9.9.But the three are less than 10.
Moved the decimal to the left is the exponent and the exponent is positive.

Texas Go Math Grade 8 Unit 1 Performance Tasks Answer Key

Question 1.
Careers In Math Astronomer An astronomer is studying Proxima Centauri, which is the closest star to our Sun. Proxima Centauri is 39,900,000,000,000,000 meters away.
a. Write this distance in scientific notation.
Answer:
39,900,000,000,000,000 Distance in meters
3.99 Move the decimal point 16 places to the left. Remove the extra zeros
10, 000, 000, 000, 000, 000 Divide the original number by the result from Step 1. Write the answer as the power of 10.
10¹⁶
3.99 × 10¹⁶ Write the product of the results from the above

b. Light travels at a speed of 3.0 × 10⁸ m/s (meters per second). How can you use this information to calculate the time in seconds it takes for light from Proxima Centauri to reach Earth? How many seconds does it take? Write your answer in scientific notation.
Answer:
speed = 3.0 × 10⁸ speed in meters per second

\(\frac{3.99}{3}\) = 1.33 Find the quotient by dividing the decimals and using the laws of exponents
\(\frac{10^{16}}{10^{8}}\) = 10^16-8 = 10⁸
1.33 × 10⁸ Combine the answers to write the distance in scientific notation
It takes $1.33 \times 10^{8}$ seconds
for the light to reach Earth

c. Knowing that 1 year = 3.1536 × 10⁷ seconds, how many years does it take for light to travel from Proxima Centauri to Earth? Write your answer in standard notation. Round your answer to two decimal places.
Answer:
1.33 × 10⁸ seconds From part b
1 yr = 3.1536 × 10⁷

Find the quotient by dividing the decimals and using the Laws of exponents
0.4217 × 10¹ combine the answers to write the distance in scientific notation
4.22 Write in standard notation
It takes $4.22$ years for the light to reach Earth

Unit 1 Review Answer Key Math Grade 8 Question 2.
Cory is making a poster of common geometric shapes. He draws a square with a side length of 4³ cm, an equilateral triangle with a height of \(\sqrt{200}\) cm, a circle with a circumference of 8π cm, a rectangle with length \(\frac{122}{5}\) cm, and a parallelogram with base 3.14 cm.

a. Which of these numbers are irrational?
Answer:
\(\sqrt{200}\) cm and 8π cm are irrational Irrational numbers are real numbers that cannot be expressed in the form \(\frac{a}{b}\)

b. Write the numbers in this problem in order from least to greatest. Approximate π as 3.14.
Answer:
4³ = 64 Evaluate the power
\(\sqrt{200}\) Estimate the value of \(\sqrt{200}\)
lies between $14$ and $15$ Since
$200$ is closer to $196 = 14^{2}$,
hence
14.1² = 198.81
14.15² = 200.22
14.2² = 201.64
Since $14.15”{2} = 200.22$
$\sqrt{200} \approx 14.15$
8π = 8 * 3.14 = 25.12 π = 3.14
\(\frac{122}{5}\) = 24.4 Divide 122 by 5
3.14 Given
Graph on the number line
3.14, \(\sqrt{200}\), \(\frac{122}{5}\), 8π, 4³ From least to greatest

c. Explain why 3.14 is rational, but π is not.
Answer:
3.14 can be expressed in the form \(\frac{a}{b}\) but π cannot be expressed in fraction form. It gives a value which is neither a terminating or repeating decimal.

3.14 is rational but π is not

Texas Go Math Grade 8 Unit 1 Mixed Review Texas Test Prep Answer Key

Selected Response

Question 1.
A square on a large calendar has an area of 4220 square millimeters. Between which two integers is the length of one side of the square?
(A) between 20 and 21 millimeters
(B) between 64 and 65 millimeters
(C) between 204 and 205 millimeters
(D) between 649 and 650 millimeters
Answer:
(B) between 64 and 65 millimeters

Explanation:
The area of a square is found by multiplying the side of the square by itself. Therefore, to find the side of the
square, we have to take the square root of the area.
Let’s denote with A the area of a square of the calendar and with s each side of the square. We have:
A = 4220
A = s • s
s = \(\sqrt{A}\) = \(\sqrt{4220}\)
Following the steps as in “Explore activity 1” on page 9, we can make an estimation for the irrational number:

Therefore, the side of the square is between 64 and 65 millimeters.

8th Grade Math Unit 1 Study Guide Answer Key Question 2.
Which of the following numbers is rational but not an integer?
(A) -9
(B) -4.3
(C) 0
(D) 3
Answer:
(B) -4.3

Explanation:
-4.3 is not an integer
Integers are whole numbers that can be positive, negative or zero.

Question 3.
Which statement is false?
(A) No integers are irrational numbers.
(B) All whole numbers are integers.
(C) All rational numbers are real numbers.
(D) All integers are whole numbers.
Answer:
(D) All integers are whole numbers.

Explanation:
All integers are whole numbers is false
Whole numbers are non-negative while integers are negative, positive or zeros.

Question 4.
Which set best describes the numbers displayed on a telephone keypad?
(A) whole numbers
(B) rational numbers
(C) real numbers
(D) integers
Answer:
The rational numbers are displayed on the telephone keypad.
Option B is the correct answer.

Question 5.
In 2011, the population of Laos was about 6.586 × 10⁶ people. What is this number written in standard notation?
(A) 6,586 people
(B) 658,600 people
(C) 6,586,000 people
(D) 65,860,000 people
Answer:
(C) 6,586,000 people

Explanation:
6.586 × 10⁶ Population in 2011

6 Use exponent of the power of 10 to determine the number places to move the decimal point
places

6, 586, 000
Place the decimal point Since you are going to write a number greater than 6.586, move decimal point to the right Add placeholder zeros if necessary.
There are 6, 586, 000 people in $2011$

Question 6.
Which of the following is not true?

Answer:
(D) 6 – \(\sqrt{35}\) < 0 Explanation: a. Since both 16 and 4 are perfect squares we can evaluate them easily:
and the statement is TRUE.

b. We know that π ≈ 3.14 which means that 4π ≈ 4 × 3.14 = 12.56 Obviously, 12.56 > 12
so,
4π > 12
and the statement is TRUE

c. Lets first evaluate \(\sqrt{18}\) and \(\frac{15}{2}\)
Since 18 is not a perfect square find two closest squares that 1 lies in between:
16 < 18 < 25
\(\sqrt{16}\) < \(\sqrt{18}\) < \(\sqrt{25}\)
4 < \(\sqrt{18}\) < 5
Since 18 is closer to 16 a good approximation for \(\sqrt{18}\) is 4 but we are certain that \(\sqrt{18}\) is not larger or equal to 5
To express \(\frac{15}{2}\) as a fraction use long division to divide 15 by 2
\(\frac{15}{2}\) = 7.5
Now we can see that
\(\sqrt{18}\) + 2 < 5 + 2 = 7
\(\frac{15}{2}\) = 7.5
We can conclude that \(\sqrt{18}\) + 2 < \(\frac{15}{2}\)
and the statement is TRUE.

d. Since
6² = 36
36 > 35
by taking square roots from both sides we can see that
\(\sqrt{35}\) < \(\sqrt{36}\) = 6 but now 6 – \(\sqrt{35}\) > 0
It now follows that the statement is NOT TRUE.

Question 7.
Which number is between \(\sqrt{50}\) and \(\frac{5 \pi}{2}\)?
(A) \(\frac{22}{3}\)
(B) 2\(\sqrt{8}\)
(C) 6
(D) π + 3
Answer:
(A) \(\frac{22}{3}\)

Explanation:
\(\sqrt{50}\) Given
and \(\frac{5 \pi}{2}\)
\(\sqrt{50}\) lies between 7 and 8 Evaluate the value of \(\sqrt{50}\)
7.0² = 49.00
7.05² = 49.70
7.15² = 51.12
Since $7.05^{2} = 49.7$, $\sqrt{50}
\approx 7.05$

\(\frac{5 * 3.14}{2}\) = 7.85 Evaluate \(\frac{5 \pi}{2}\)
The number should be between 7.05 and 7.85
\(\frac{22}{3}\) = 7.33

8th Grade Math Unit 1 Study Guide Answer Key Question 8.
What number is indicated on the number line?

(A) π + 4
(B) \(\frac{152}{20}\)
(C) \(\sqrt{14}\) + 4
(D) \(7 . \overline{8}\)
Answer:
Given that the number \(\sqrt{14}\) + 4 is indicated on the number line.
\(\sqrt{14}\) + 4 = 7.7416.
Option C is the correct answer.

Question 9.
Which of the following is the number 5.03 × 10^-5 written in standard form?
(A) 503,000
(B) 50,300,000
(C) 0.00503
(D) 0.0000503
Answer:
The standard form of the number 5.03 × 10^-5
5.03 × 0.00001 = 0.0000503
Here move the decimal 5 times to the left then it is 5.03. But the three are less than 10.
Moved the decimal to the left is the exponent and the exponent is negative.
Option D is the correct answer.

Question 10.
In a recent year, about 20,700,000 passengers traveled by train in the United States. What is this number written in scientific notation?
(A) 2.07 × 10¹ passengers
(B) 2.07 × 10⁴ passengers
(C) 2.07 × 10⁷ passengers
(D) 2.07 × 10⁸ passengers
Answer:
(C) 2.07 × 10⁷ passengers

Explanation:
20,700,000 Given
2.07 Move the decimal point 7 places to the left Remove the extra zeros
10,000,000 Divide the original number by the result from above.
10⁷ Write the answer as the power of 10.
Write the product of the results above
2.07 × 10⁷
There are $2.07 \times 10^{7}$ passengers that travel by train

Unit 1 End of Unit Assessment Answer Key Grade 8 Question 11.
A quarter weighs about 0.025 pounds. What is this weight written in scientific notation?
(A) 2.5 × 10^-2 pound
(B) 2.5 × 10¹ pound
(C) 2.5 × 10^-1 pound
(D) 2.5 × 10² pound
Answer:
(A) 2.5 × 10^-2 pound

Explanation:
We need to express the number 0.025 in scientific notation (a × 10ⁿ, where a is a number greater or equal to 1 but less than 10.
In that spirit, we see we need to move the decimal point 2 places (to the right) so the power of 10 needs to be 2
(or -2). Since the given number is a number smaller than 1 the exponent in the power of 10 is going to be negative (-2).
0.025 = 2.5 × 10^-2

Question 12.
Which of the following is the number 3.0205 × 10^-3 written in standard notation?
(A) 0.00030205
(B) 0.0030205
(C) 3.0205
(D) 3020.5
Answer:
The standard form of the number 3.0205 × 10 power of -3 = 0.00030205.
Option A is the correct answer.

Question 13.
A human fingernail has a thickness of about 4.2 × 10^-4 meters. What is this width written in standard notation?
(A) 0.0000042 meter
(B) 0.000042 meter
(C) 0.00042 meter
(D) 0.0042 meter
Answer:
(C) 0.00042 meter

Explanation:
a We are given: 4.2 × 10^-4 meter Since the power of 10 is a negative 4 we need to move the decimal place 4
places to the left to get a number written in scientific notation.
4.2 × 10^-4 meter = 0.00042 meter Move decimal point and add placeholder zeros

b. Since 1 meter = 0.001 millimeter or 1 millimeter = 10³ meter we can calculate how many millimeters is 4.2 × 10^-4 meter
4.2 × 10^-4 meter = (4.2 × 10^-4) × 10³ millimeter = 0.42 millimeter
We can conclude that their measurements agree.

c. Since the thickness of a human fingernail is a very small number it is more appropriate to express it in millimeters than in meters (because a millimeter is 1000 times smaller than a meter).

Gridded Response

Question 14.
The square root of a number is -18. What is the other square root?

Answer:
A square root always has 2 numbers, one is positive and other is negative.
So, one number is -18 and the other number is 18

Hot Tip!
Underline keywords given in the test question so you know for certain what the question is asking.

Question 15.
Jerome is writing a number in scientific notation. The number is greater than one million and less than ten million. What will be the exponent in the number Jerome writes?

Answer:
Given that Jerome is writing the number in the scientific notation.
The number is greater than one million and less than ten million.
The exponent in the number is
1 million = 10⁶
10 million = 10⁷
The number is > 10⁶ < 10⁷
The exponent number is 6.

Grade 8 Math Unit 1 Performance Assessment Task 1 Answer Key Question 16.
Write the number 3.3855 × 10² in standard notation.

Answer:
3.3855 × 10² in the standard form is 3.3855 × 100 = 338.55

Texas Go Math Grade 8 Unit 1 Vocabulary Preview Answer Key

Use the puzzle to preview key vocabulary from this unit. Unscramble the circled letters to answer the riddle at the bottom of the page.

Question 1.
TCREEFP
SEAQUR

Answer:
PERFECT
SQUARE

Question 2.
NOLRATAI
RUNMEB

Answer:
RATIONAL
NUMBER

Question 3.
PERTIANEG
MALCEDI

Answer:
PERTAINGE
DECIMAL

Question 4.
LAER
SEBMNUR

Answer:
REAL
NUMBERS

Question 5.
NIISICFTCE
OITANTON

Answer:
SCIENTIFIC
NOTATION

1. Has integers as its square roots. (Lesson 1-1)
2. Any number that can be written as a ratio of two integers. (Lesson 1-1)
3. A decimal in which one or more digits repeat infinitely. (Lesson 1-1)
4. The set of rational and irrational numbers. (Lesson 1-2)
5. A method of writing very large or very small numbers by using powers of 10. (Lesson 2-1)

Q: What keeps a square from moving?
A: ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___!
Answer:
SQUARE ROOTS

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson 3.3 Answer Key Interpreting the Unit Rate as Slope.

Texas Go Math Grade 8 Lesson 3.3 Answer Key Interpreting the Unit Rate as Slope

How do you interpret the unit rate as slope?

Texas Go Math Grade 8 Lesson 3.3 Explore Activity Answer Key 

Relating the Unit Rate to Slope
A rate is a comparison of two quantities that have different units, such as miles and hours. A unit rate is a rate in which the second quantity in the comparison is one unit.

A storm is raging on Misty Mountain. The graph shows the constant rate of change of the snow level on the mountain.

A. Find the slope of the graph using the points (1, 2) and (5, 10). Remember that the slope is the constant rate of change.

B. Find the unit rate of snowfall in inches per hour. Explain your method.

C. Compare the slope of the graph and the unit rate of change in the snow level. What do you notice?

D. Which point on the graph tells you the slope of the graph and the unit rate of change in the snow level? Explain how you found the point.

Reflect

Question 1.
What If? Without referring to the graph, how do you know that the point (1, \(\frac{4}{3}\)) is on the graph?
Answer:
Lets find the slope:
Slope = \(\frac{8-4}{6-3}\) = \(\frac{4}{3}\)
This holds for every two points on the graph, so it has to hold for some of the given points and the point we want to check. Substitute the point (6, 8) for (1, \(\frac{4}{3}\)) and see if it still holds.
\(\frac{\frac{4}{3}-4}{1-3}\) = \(\frac{4}{3}\) = Slope
for x = Time and y = Amount
Therefore, the point (1, \(\frac{4}{3}\)) is on the line.
By checking if the equation for the slope (which we calculated) holds for the given point we see that it does, so it is on the line.

Your Turn

Interpreting the Unit Rate as Slope Go Math 8th Grade Pdf Question 2.
Tomas rides his bike at a steady rate of 2 miles every 10 minutes. Graph the situation. Find the unit rate of this proportional relationship.

Answer:
Make a table
Graph the solution
slope = \(\frac{\text { Rise }}{\text { Run }}\) = \(\frac{4-2}{20-10}\) = \(\frac{2}{10}\) = \(\frac{1}{5}\) Find the slope
The unit rate and slope of a graph of the ride is \(\frac{1}{5}\) mi per min

Reflect

Question 3.
Describe the relationships among the slope of the graph of Well A’s rate, the equation representing Well A’s rate, and the constant of proportionality.
Answer:
The slope of the graph of Well, A’s rate, the equation representing Well A’s rate, and the constant of proportionality are the same. They all represent the amount of oil pumped from Well A in an hour.

Your Turn

Question 4.
The equation y = 375x represents the relationship between x, the time that a plane flies in hours, and y, the distance the plane flies in miles for Plane A. The table represents the relationship for Plane B. Find the slope of the graph for each plane and the plane’s rate of speed. Determine which plane is flying at a faster rate of speed.

Answer:
Notice that for every value of x and y for Plane A the given equation has to hold. So
y = 375x
therefore
k = \(\frac{y}{x}\) = 375 is the slope of the graph for Plane A.
Using the table we can find the slope of the graph for Plane B:
\(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) = \(\frac{850-425}{2-1}\) = 425
Therefore, the slope of the graph of Plane B is greater than the slope of the graph of Plane A. We can conclude that Plane B is faster because it flies more miles per hour (faster rate of speed).
Plane B is flying at a faster rate of speed.

Texas Go Math Grade 8 Lesson 3.3 Guided Practice Answer Key 

Give the slope of the graph and the unit rate. (Explore Activity and Example 1)

Question 1.
Jorge: 5 miles every 6 hours

Answer:
we are given that Jorge travels 5 miles every 6 hours. ‘Therefore, we construct the table and draw a graph:

slope = \(\frac{\text { rise }}{\text { run }}\) = \(\frac{5}{6}\)
The unit of the distance travelled and the slope of the graph of the relationship is equal, \(\frac{5}{6}\) miles per hour.
\(\frac{5}{6}\) miles per hour.

Unit Rate as Slope Grade 8 Math Answer Key Pdf Question 2.
Akiko

Answer:
slope = \(\frac{\text { Rise }}{\text { Run }}\) = \(\frac{10-5}{8-4}\) = \(\frac{5}{4}\)
The unit rate is \(\frac{5}{4}\) miles per hour.
Slope: \(\frac{5}{4}\), unit rate: \(\frac{5}{4}\) miles per hour

Question 3.
The equation y = 0.5x represents the distance Henry hikes in miles over time in hours. The graph represents the rate that Clark hikes. Determine which hiker is faster. Explain. (Example 2)

Answer:
y = 0.5x Equation representing Henry’s rate
Use the equation to make a table
Slope = Unit rate = \(\frac{2-1}{4-2}\) = \(\frac{1}{2}\) miles per hour
Use the table to find the slope of the graph of Plane A
Slope = Unit rate = \(\frac{18-6}{12-4}\) = \(\frac{12}{8}\) miles per hour
Use the graph to find the slope of the graph of Clark
Clark is faster Compare the units
\(\frac{3}{2}\) > \(\frac{1}{2}\)

Write an equation relating the variables in each table. (Example 2)

Question 4.

Answer:
Find the slope and with it the unit rate:
Slope = \(\frac{30-15}{2-1}\) = 15 = Unit rate
Now we know that for every x and y the following equation holds:
\(\frac{y}{x}\) = 15
therefore
y = 15x

Practice and Homework Lesson 3.3 Answer Key 8th Grade Question 5.

Answer:
Slope = Unit rate = \(\frac{12-6}{32-16}\) = \(\frac{6}{16}\) = \(\frac{3}{8}\) Find the slope using the table
y = \(\frac{3}{8}\)x Writing the equation

Essential Question Check-In

Question 6.
Describe methods you can use to show a proportional relationship between two variables, x and y. For each method, explain how you can find the unit rate and the slope.
Answer:
Explanation A:
We can represent a proportional relationship between two variables x and y using a table. For each point we find
the ratio of y over x. If this ratio is constant over each point, then the relationship is proportional. To find the unit rate and slope, we have:
Slope = Unit Rate = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

We can represent a proportional relationship between two variables x and y using an equation of the form y = kx, where k is a number called the constant of proportionality. The slope and unit rate are equal to the constant of proportionality, however we can construct a table from the given equation and proceed as described above.

We can represent a proportional relationship between two variables x and y using a graph. The graph will be a line that passes through the origin (0, 0). To find the unit rate and slope, we have:
Slope = Unit Rate = \(\frac{\text { rise }}{\text { run }}\)

Explanation B:
You can use a table to find the ratio of each point. If the ratio is constant, the relationship is proportional. You
can find the unit rate and slope by:
Slope(m) = UnitRate = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

You can use a graph to find the ratio of the rise for equal intervals of run. You can find the unit rate and slope by:
Slope(m) = UnitRate = \(\frac{\text { rise }}{\text { run }}\)

Texas Go Math Grade 8 Lesson 3.3 Independent Practice Answer Key 

Interpreting Unit Rate as Slope 8th Grade Go Math Answer Key Question 7.
A Canadian goose migrated at a steady rate of 3 miles every 4 minutes.
a. Fill in the table to describe the relationship.

Answer:

Top row: 12, 16; Bottom row: 3, 6, 15

b. Graph the relationship.

Answer:

Click for the graph.

c. Find the slope of the graph and describe what it means in the context of this problem.
Answer:
The slope of the graph is
\(\frac{6-3}{8-4}\) = \(\frac{3}{4}\)
in this case, this means that a Canadian goose migrates 3 miles every 4 minutes or \(\frac{3}{4}\) miles every minute.
The goose migrates \(\frac{3}{4}\) miles every minute.

Question 8.
Vocabulary A unit rate is a rate in which the in the comparison is one unit.
Answer:
A unit is a rate in which the second quantity in the comparison is one unit.

Question 9.
The table and the graph represent the rate at which two machines are bottling milk in gallons per second.

a. Determine the slope and unit rate of each machine
Answer:
(1) Using the given table, calculate the slope and unit rate of the first machine.
Slope = Unit rate = \(\frac{1.2-0.6}{2-1}\) = \(\frac{0.6}{1}\) = 0.6
The unit rate for the first machine is therefore 0.6 gallons per second.

(2) Using the given graph, find points to substitute into the formula for the slope.
Two points on the line for the second machine are: (8, 6) and (16, 12).
Now calculate the slope and unit rate of the second machine using the points you just found.
Slope = Unit rate = \(\frac{12-6}{16-8}\) = \(\frac{6}{8}\) = 0.75
The unit rate for the second machine is therefore 0.75 gallons per second
The slope and unit rate of the first machine is 0.6 (gallons per second), and of the second 0.75 (gallons per
second).

b. Determine which machine is working at a faster rate.
Answer:
Comparing the unit rates of both machines we can see that the second machine bottles more milk per second than the first machine. Therefore it is working at a faster rate.
The second machine is faster.

Texas Go Math Grade 8 Lesson 3.3 Answer Key Question 10.
Cycling The equation y = \(\frac{1}{9}\)x represents the distance y, in kilometers, that Patrick traveled in x minutes while training for the cycling portion of a triathlon. The table shows the distance y Jennifer traveled in x minutes in her training. Who has the faster training rate?

Answer:
From y = \(\frac{1}{9}\)x we can make a table for Patrick’s unit rate in kilometers per minute.

Now,
\(\frac{2-1}{18-9}\) = \(\frac{1}{9}\)
Therefore, Patrick’s unit rate is \(\frac{1}{9}\) kilometers per minute.

Lets find the slope and the unit rate of jennifer
Slope = \(\frac{8-5}{64-40}\) = \(\frac{3}{24}\) = \(\frac{1}{8}\)
Therefore, Jennifer’s unit rate is \(\frac{1}{8}\) kilometers per minute.
We can conclude that Patrick has a faster training rate.

Question 11.
Analyze Relationships There is a proportional relationship between minutes and cost in dollars. The graph passes through the point (1, 4.75). What is the slope of the graph? What is the unit rate? Explain.
Answer:
If we know that there is a proportional relationship between minutes and dollars per minute, then the graph is a
straight line. Also since that line passes through (1, 4.75) we know that for 1 minute it costs 4.75 dollars to print.
Therefore the slope is 4.75 and the unit rate is 4.75 dollars per minute.

The slope is 4.75 and the unit rate is 4.75 dollars per minute.

Question 12.
Draw Conclusions Two cars start at the same time and travel at different constant rates. The graph of the distance in miles given the time in hours for Car A passes through the point (0.5, 27.5), and the graph for Car B passes through the point (4, 240). Which car is traveling faster? Explain.
Answer:
Unit Rate = \(\frac{27.5}{0.5}\) = 55 Find the unit rate for Car A.
Car $A$ travels $55$ miles in $1$ hr

Unit Rate = \(\frac{240}{4}\) = 60 Find the unit rate for Car B.
Car $B$ travels $60$ miles in $1$ hr
Car B is faster Compare the unit rates
60 > 55

Lesson 3.3 Interpreting the Unit Rate as Slope Reteach Answer Key Question 13.
Critical Thinking The table shows the rate at which water is being pumped into a swimming pool.

Use the unit rate and the amount of water pumped after 12 minutes to find how much water will have been pumped into the pool after 13\(\frac{1}{2}\) minutes. Explain your reasoning.
Answer:
Unit rate = \(\frac{216-36}{12-2}\) = \(\frac{180}{10}\) = 18 Find the unit rate
18 gal of water is pumped into the swimming pool every minute.

Additional water = 18 * 1\(\frac{1}{2}\) = 27 After 1\(\frac{1}{2}\) min after 12 min

Total = 216 + 27 = 243gal The total water after 13\(\frac{1}{2}\) min