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 Module 2 Quiz Answer Key.

Texas Go Math Grade 8 Module 2 Quiz Answer Key

Texas Go Math Grade 8 Module 2 Ready to Go On? Answer Key

2.1 Scientific Notation with Positive Powers of 10

Write each number in scientific notation.

Question 1.
2,000 ______________
Answer:
2,000 Given
2.0 Move the decimal point 3 places to the left. Remove the extra zeros.
1000 Divide the original number by the result from above.
10³ Write the answer as the power of 10.
2.0 × 10³ Write the product of the results from the above.
2.0 × 10³

Module 2 Form A Module Test Answer Key Question 2.
91,007,500 ____________
Answer:
91,007,500 Given
91,007,500 Move the decimal point 7 places to the left. Remove the extra zeros.
10000000 Divide the original number by the result from above.
10⁷ Write the answer as the power of 10.
9.10075 × 10⁷ Write the product of the results from the above.
9.10075 × 10⁷

Question 3.
On average, the Moon’s distance from Earth is about 384,400 km. What is this distance in scientific notation? __________________
Answer:
Given that,
The average distance of the moon from the earth is 384,400 km.
The scientific notation of the distance 384400 is 3.844 x 10⁵.
Here move the decimal 5 times to the left then it is 3.844. But it is less than 10.
Moved the decimal to the left is the exponent and the exponent is positive.

Write each number in standard notation.

Question 4.
1.0395 × 10⁹ _______________
Answer:
1.0395 x 10⁹ in the standard form is
10⁹ = 1,000,000,000
1.0395 x 1,000,000,000 = 1,039,500,000.

Question 5.
4 × 10² _______________
Answer:
4 x 10² in the standard form is
10² = 100
4 x 100 = 400

Go Math Grade 8 Module 2 Answer Key Pdf Question 6.
The population of Indonesia was about 2.48216 × 10⁸ people in 2011. What is this number in standard notation? _______________
Answer:
Given that the population of Indonesia is 2.48216 x 10⁸
2.48216 x 10⁸ in the standard form is
10⁸ = 100000000
2.48216 x 100000000 = 248,216,000

2.2 Scientific Notation with Negative Powers of 10

Write each number in scientific notation.

Question 7.
0.02 _______________
Answer:
0.02 in the scientific notation is 2 x 10^-2
Here move the decimal 2 times to the left then it is 2. But two is less than 10.
Moved the decimal to the left is the exponent and the exponent is negative.

Question 8.
0.000701 ____
Answer:
0.000701 Given
7.01 Place the decimal point
4 Count the number of places the decimal point is moved
7.01 × 10⁴ Multiply 7.01 times a power of 10. Since 0.000701 is less than 1, the decimal point moves to the right and the exponent on 10 is negative.

Write each number in standard notation.

Question 9.
8.9 × 10^-5 ______________
Answer:
8.9 × 10^-5 Given
5 Use the exponent of the power of 10 to see how many places to move the decimal point.
places
0.000089 Place the decimal point Since you are going to write a number less than 8.9, move the decimal to the left. Add placeholder zeros if necessary.
0.000089

Module 2 Exponents and Scientific Notation Module Quiz Question 10.
4.41 × 10^-2 ______________
Answer:
4.41 × 10^-2 Given
2 Use the exponent of the power of 10 to see how many places to move the decimal point.
places
0.0441 Place the decimal point Since you are going to write a number less than 4.41, move the decimal to the left. Add placeholder zeros if necessary.
0.0441

Complete the table.

Question 11.

Answer:
Given that,
The distance structure of the standard notation = 0.000009m.
0.000009m in the scientific notation is
The distance of the structure in the scientific notation is 9 x 10^-6
Here move the decimal 6 times to the left then it is 9. But nine is less than 10.
Moved the decimal to the left is the exponent and the exponent is negative.

Question 12.

Answer:
Given that,
The distance of the structure in the scientific notation is 9.5 x 10^-9
9.5 x 10^-9
10^-9 = 0.000000009
9.5 x 0.000000009 = 0.0000000855
The distance structure of the standard notation = 0.0000000855

Question 13.

Answer:
Given that,
The distance structure of the standard notation = 0.000009m.
0.000009m in the scientific notation is
The distance of the structure in the scientific notation is 9 x 10^-6
Here move the decimal 6 times to the left then it is 9. But the nine is less than 10.
Moved the decimal to the left is the exponent and the exponent is negative.

Essential Question

Question 14.
How is scientific notation used in the real world?
Answer:
Scientific notation is used to write very small and very large numbers using fewer digits.
It is used to describe astronomical distances such as the distance between the cities and used in the microscopic distance means the length of blood cells.

Texas Go Math Grade 8 Module 2 Mixed Review Texas Test Prep Answer Key

Selected Response

Question 1.
Which of the following is the number 90 written in scientific notation?
(A) 90 × 10²
(B) 9 × 10²
(C) 90 × 10¹
(D) 9 × 10¹
Answer:
The number 90 in the scientific notation is 9 × 10¹
Here move the decimal 1 times to the left then it is 9. But the nine is less than 10.
Moved the decimal to the left is the exponent and the exponent is positive.
Option D is the correct answer.

Grade 8 Math Module 2 Answer Key Question 2.
About 786,700,000 passengers traveled by plane in the United States in 2010. What is this number written in scientific notation?
(A) 7,867 × 10⁵ passengers
(B) 7.867 × 10² passengers
(C) 7.867 × 10⁸ passengers
(D) 7.867 × 10⁹ passengers
Answer:
(C) 7.867 × 10⁸ passengers

Explanation:
786,700,000 Given
7.867 Move the decimal point 8 places to the left. Remove the extra zeros.
100000000 Divide the original number by the result from Step 1.
10⁸ Write the answer as power of 10.
7.867 × 10⁸ Write the product of the results from above
There are 7.867 × 10⁸ passengers

Question 3.
In 2011, the population of Mali was about 1.584 × 10⁷ people. What is this number written in standard notation?
(A) 1.584 people
(B) 1,584 people
(C) 15,840,000 people
(D) 158,400,000 people
Answer:
(C) 15,840,000 people

Explanation:
1.584 × 10⁷ Given
7 Use the exponent of the power of 10 to determine the number of places to move the decimal point places

15,840,000 Place the decimal point. Since you are going to write a number greater than 1.584, move the decimal to the right. Add placeholder zeros if necessary.
There are
15,840,000 people

Question 4.
The square root of a number is between 7 and 8. Which could be the number?
(A) 72
(B) 83
(C) 51
(D) 66
Answer:
(C) 51

Explanation:
7 and 8 Given

49 and Find the squares of the number
64
Since the square root lies between
$7$ and $8$, the square of the number
lies between $49$ and $64$, hence it
could be $51$

Module 2 Exponents and Scientific Notation Answer Key Question 5.
Pilar is writing a number in scientific notation. The number is greater than ten million and less than one hundred million. Which exponent will Pilar use?
(A) 10
(B) 7
(C) 6
(D) 2
Answer:
The pilar is writing a number that number is in the scientific notation.
The number is greater than 1 million and less than 100 million.
1 million = 1000000 = 10⁶
100 million = 100000000 = 10⁸
Greater than 10⁶ and less than 10⁸
The exponent is 7,
Option B is the correct answer.

Question 6.
Place the numbers in order from least to greatest.
0.24, 4 × 10^-2, 0.042, 2 × 10^-4, 0.004
(A) 2 × 10^-4, 4 × 10^-2, 0.004, 0.042, 0.24
(B) 0.004, 2 × 10^-4, 0.042, 4 × 10^-2, 0.24
(C) 0.004, 2 × 10^-4, 4 × 10^-2, 0.042, 0.24
(D) 2 × 10^-4, 0.004, 4 × 10^-2, 0.042, 0.24
Answer:
(D) 2 × 10^-4, 0.004, 4 × 10^-2, 0.042, 0.24

Explanation:
0.24, 4 × 10^-2, 0.042, 2 × 10^-4, 0.004
2.4 × 10^-1 Express the numbers in scientific notation
4 × 10^-2
4.2 × 10^-2
2 × 10^-4
4 × 10^-3

2 × 10^-4 Compare the power of 10 and if they are same
4 × 10^-3 compare the first factor
4 × 10^-2 Placed in the order from least to greatest
4.2 × 10^-2
2.4 × 10^-1
2 × 10^-4, 0.004, 4 × 10^-2, 0.042, 0.24 placed in the order from least to greatest

Question 7.
Which of the following is the number 1.0085 × 10^-4 written in standard notation?
(A) 10,085
(B) 1.0085
(C) 0.00010085
(D) 0.000010085
Answer:
1.0085 x 10^-4 in the standard notation is
10^-4 = 0.0001
1.0085 x 0.0001 = 0.00010085
Multiplying 1.0085 with 0.0001 then you get 0.00010085.
Option C is the correct answer.

Module 2 Exponents and Scientific Notation Answer Key Question 8.
A human hair has a width of about 6.5 × 10^-5 meters. What is this width written in standard notation?
(A) 0.00000065 meter
(B) 0.0000065 meter
(C) 0.000065 meter
(D) 0.00065 meter
Answer:
(C) 0.000065 meter

Explanation:
6.5 × 10^-5 Given
5 Use the exponent of the power of 10 to see how many places to move the decimal point.
places
0.000065 Place the decimal point Since you are going to write a number less than 6.5, move the decimal to the left. Add placeholder zeros if necessary.
The width in standard notation is $0.000065$ meter

Gridded Response

Question 9.
Write 2.38 × 10^-1 in standard form.

Answer:
2.38 x 10^-1 in the standard form is
10^-1 = 0.1
2.38 x 0.1 = 0.238
Multiplying the 2.38 with 0.1 you get 0.238.

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 4.3 Answer Key Graphing Linear Nonproportional Relationships using Slope and y-Intercept.

Texas Go Math Grade 8 Lesson 4.3 Answer Key Graphing Linear Nonproportional Relationships using Slope and y-Intercept

Essential Question
How can you graph a line using the slope and y-intercept?

Reflect

Question 1.
Draw Conclusions How can you use the slope of a line to predict the way the line will be slanted? Explain.
Answer:
If the slope is positive, the graph will be slanted upwards and if the slope is negative, the graph will be slanted downwards.

Your Turn

Graph each equation.

Question 2.
y = \(\frac{1}{2}\)x + 1

Answer:
From the given equation y = \(\frac{1}{2}\)x + 1, we can see that y-intercept (‘b”) is 1. The first point is (0, b) → (0, 1).

To find the second point we can use the slope (“m”), thus m = \(\frac{1}{2}\). Since the slope is positive, we are counting 1 unit up, and 2 units to the right, and our next point is (2, 2). Plot it in the graph:

Texas Go Math Grade 8 Representing Linear Non-Proportional Relationships Answer Key Question 3.
y = -3x + 4

Answer:
y = -3x + 4
The y-intercept is b = 4
Plot the point that contains the y-intercept:
(0, 4)
The slope is m = \(\frac{-3}{1}\)
Use the slope to find a second point.
From
(0, 4)
count $-3$ unit down and $1$ unit right. The new point is
(1, 1)
Draw a line through the points.

Example 2.
Ken has a weekly goal for the number of calories he will burn by taking brisk walks.
The equation y = -300x + 2400 represents the number of calories y Ken has left to burn after x hours of walking.
A. Graph the equation y = -300x + 2400.
Step 1
Write the slope as a fraction.
m = \(\frac{-300}{1}\) = \(\frac{-600}{2}\) = \(\frac{-900}{3}\)
Step 2
Plot the point for the y-intercept:
(0, 2400).
Step 3
Use the slope to locate a second point.

From (0, 2400), count down 900 and right 3.
The new point is (3, 1500).
Step 4
Draw a line through the two points.

B. After how many hours of walking will Ken have 600 calories left to burn? After how many hours will he reach his weekly goal?
Step 1
Locate 600 calories on the y-axis. Read across and down to the x-axis.
Ken will have 600 calories left to burn after 6 hours.

Step 2
Ken will reach his weekly goal when the number of calories left to burn is 0. Because every point on the x-axis has a y-value of 0, find the point where the line crosses the x-axis.
Ken will reach his goal after 8 hours of brisk walking.

Your Turn

What If? Ken decides to modify his exercise plans from Example 2 by slowing the speed at which he walks. The equation for the modified plan is y = -200x + 2400.

Question 4.
Graph the equation.
Answer:
y = -200x + 2400.
Write the slope as a fraction
\(\frac{-200}{1}\) = \(\frac{-400}{2}\)
The y-intercept is b = 2400
Plot the point that contains the y-intercept:
(0, 2400)
The slope is m = \(\frac{-400}{2}\)
Use the slope to find a second point
From
(0, 2400)
count $400$ unit down and $2$ unit right. The new point is
(2, 2000)
Draw a line through the points

Grade 8 Math Answer Key Proportional Relationships Answer Key Question 5.
How does the graph of the new equation compare with the graph in Example 2?
Answer:
The y-intercept of the second graph ¡s the same as the y-intercept of the first graph: (0, 2400).
The slope (m) of the second graph (-200) is greater than the slope of the first graph (-300), therefore we can see
that Ken will reach his weekly goal earlier in the first graph compared with the second one.
The x-intercept of the first graph is (8, 0), while the x-intercept of the second graph is (12, 0).

Question 6.
Will Ken have to exercise more or less to meet his goal? Explain.
Answer:
He will have to exercise more to meet his goal. He burns fewer calories per hour or the calories left to burn will decrease more slowly per hour.

Question 7.
Suppose Ken decides that instead of walking, he will jog, and jogging burns 600 calories per hour. How do you think this would change the graph?
Answer:
The slope of the new graph will become -600, but the y-intercept remains the same. The graph would be steeper and the line would intersect the x-axis when x = 4 hrs

Math Talk
Mathematical processes
What do the slope and the y-intercept of the line represent in this situation?

Texas Go Math Grade 8 Lesson 4.3 Guided Practice Answer Key

Graph each equation using the slope and the y-intercept. (Example 1)

Question 1.
y = \(\frac{1}{2}\)x – 3
slope = ____ y-intercept = ___

Answer:
y = \(\frac{1}{2}\)x – 3
The y-intercept is b = – 3
Plot the point that contains the y-intercept:
(0, -3)
The slope is m = \(\frac{1}{2}\)
Use the slope to find a second point
From
(0, -3)
count $1$ unit up and $2$ unit right. The new point is
(2, -2)

Draw a line through the points

Go Math 8th Grade Lesson 4.3 Writing Linear Equations Answer Key Question 2.
y = -3x + 2
slope = ____ y-intercept = ___

Answer:
y = -3x + 2
The y intercept is b = 2
Plot the point that contains the y-intercept:
(0, 2)
The slope is m = \(\frac{-3}{1}\)
Use the slope to find a second point
From
(0, 2)
count $3$ unit down and $1$ unit right. The new point is
(1, -1)
Draw a line through the points

Question 3.
A friend gives you two baseball cards for your birthday. Afterward, you begin collecting them. You buy the same number of cards once each week. The equation y = 4x + 2 describes the number of cards, y, you have after x weeks. (Example 2)
a. Find and interpret the slope and the intercept of the line that represents this situation. Graph the equation y = 4x + 2.
Answer:
The standard form of the given equation y = 4x + 2 is y = mx + b, so we already have values for the slope and
y-intercept (“b”). Thus, m = 4 and b = 2. The first point to plot in the graph is (0, b) → (0, 2).

Using the slope we can find the second point but we have to know that m = 4 is the same as m = \(\frac{4}{1}\) because we need to move the first point up/down for x units (number in the numerator) and several units right (number in the denominator). So, from the first point, we are counting 4 units up and 1 unit right and get the second point (16). Plot it in the graph.

b. Discuss which points on the line do not make sense in this situation. Then plot three more points on the line that do make sense.
Answer:
To draw the points that will, be in the parameters of the function, I will include certain values for x in the given equation (x indicates the number of weeks that have passed). The numbers I choose must be integers because the values obtained for x y must also be integers. The y-axis indicates the number of cards collected and in reality, can not be counted as fractions.

So, for the x will take 2 (two weeks…), 3, and 5.
Thus:
y = 4x + 2
y = 4 . 2 + 2
y = 8 + 2
y = 10
(2, 10)
y = 4 . 3 + 2
y = 12 + 2
y = 14
(3, 14)
y = 4 . 5 + 2
y = 20 + 2
y = 22
(5, 22)

Any point chosen in fractions between integers on the x-axis makes no sense because, in reality, it does not correspond to the parameters.

Essential Question Check-In

Question 4.
Why might someone choose to use the y-intercept and the slope to graph a line?
Answer:
When the relationship is given ¡n the form y = mx b, the y-intercept (b) and the slope (m) are easily accessible and easily calculable. Therefore, it is a good practice to use them to graph the line.

Texas Go Math Grade 8 Lesson 4.3 Independent Practice Answer Key

Question 5.
Science A spring stretches in relation to the weight hanging from it according to the equation y = 0.75x + 0.25 where x is the weight in pounds and y is the length of the spring in inches.
a. Graph the equation.
Answer:
y = 0.75x + 0.25
Slope (m) = 0.75
and
y-intercept = 0.25

Plot the point that contains the y-intercept:
(0, 0.25)
The slope is m = \(\frac{0.75}{1}\)
Use the slope to find a second point.
From
(0, 0.25)
count $0.75$ unit up and $1$ unit right. The new point is
(1, 1)
Draw a line through the points

b. Interpret the slope and the y-intercept of the line.
Answer:
The slope represents the increase in the length of spring in inches for each increase of pounds of weight. y-intercept represents the un-stretched length of the spring when there is no weight attached.

c. How long will the spring be if a 2-pound weight is hung on it? Will the length double if you double the weight? Explain
Answer:
When there is a 2-pound weight hung, the length of the spring would be 1.75 inches.
No, when there is a 4-pound weight hung, the length of the spring would be 3.25 inches and not 3.5 inches.

Look for a Pattern Identify the coordinates of four points on the line with each given slope and y-intercept.

Go Math 8th Grade Lesson 4.3 Homework Answers Question 6.
slope = 5, y-intercept = -1
Answer:
Slope = 5
y-intercept = -1

Plot the point that contains the y-intercept:
(0, -1)

The slope is m = \(\frac{5}{1}\)
Use the slope to find a second point
From
(0, -1)
count $5$ unit up and $1$ unit right. The new point is
(1, 4)
Follow the same procedure to find the remaining three points.
(2, 9)
(3, 14)

Question 7.
slope = -1, y-intercept = 8
Answer:
To determine the coordinates of four points on the line given the slope and y-intercept, Let us use the slope-
intercept form y = mx + b, where n is the slope and b is the y-intercept that gives a point of (0, b)

Given the yinterceptt8, b = 8, we can have the first point (0, 8)

Using the given slope = -1, m = -1 which is the same as m = \(\frac{-1}{1}\) from point (0, 8), move 1 unit down since the numerator of the slope is -1, and then move 1 unit to the right since the denominator is 1, it gives us the second point (1, 7).
Using the same slope, from point (1, 7), move 1 unit down, and then 1 unit right, it gives us the third point (2, 6)
Getting the next point from the point (2, 6), moving 1 unit down and 1 unit right it gives us the point (3, 5).
(0, 8), (1, 7), (2, 6), (3, 5)

Question 8.
slope = 0.2, y-intercept = 0.3
Answer:
Slope = 0.2
y-intercept = 0.3
Plot the point that contains the y-intercept:
(0, 0.3)
The slope is m = \(\frac{0.2}{1}\)
Use the slope to find a second point
From
(0, 0.3)
count $0.2$ unit up and $1$ unit right. The new point is
(1, 0.5)
Follow the same procedure to find the remaining three points.
(2, 0.07)
(3, 0.9)

Question 9.
slope = 1.5, y-intercept = -3
Answer:
To determine the coordinates of four points on the Line given the slope and y-intercept, let us use the slope-
intercept form y = mx + b, where m is the slope and b is the y-intercept that gives a point of (0, b)
Given the y-intercept = -3, b = -3, we can have the first point (0, -3).
Using the given slope = 1.5,m = 1.5 which is the same as m = \(\frac{1.5}{1}\) from point (0, -3), move 1.5 unit up since the numerator of the slope is 1.5, and then move 1 unit to the right since the denominator is 1, it gives us the second point (1, -1.5).
Using the same slope, from the point (1, -1.5), move 1.5 unit up, and then 1 unit right it gives the third point (2, 0)
Getting the next point, from the point (2, 0), moving 1.5 units up and 1 unit right it gives the point (3, 1.5).
(0, -3), (1, -1.5), (2, 0), (3, 1.5)

Go Math 8th Grade Pdf Lesson 4.3 Practice Answer Key Question 10.
slope = –\(\frac{1}{2}\) y-intercept = 4
Answer:
To determine the coordinates of four points on the line given the slope and y-intercept, let us use the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept that gives a point of (0, b)
Given the y-intercept = 4, b = 4, we can have the first point (0. 4).
Using the given slope = –\(\frac{1}{2}\), m = –\(\frac{1}{2}\) which is the same as m = –\(\frac{1}{2}\), from point (0, 4), move 1 unit down since the numerator of the slope is -1, and then move 2 units to the right since the denominator is 2, it gives us the second point (2, 3).
Using the same slope, from the point (2, 3), move 1 unit down, and then 2 units right, it gives the third point (4, 2)
Getting the next point from the point (4, 2), move 1 unit down and 2 units right, it gives the point (6, 1)
(0, 4), (2, 3), (4, 2), (6, 1)

Question 11.
slope = \(\frac{2}{3}\), y-intercept = -5
Answer:
To determine the coordinates of four points on the line given the slope and y-intercept, let us use the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept that gives a point of (0, b)
Given the y-intercept = -5, b = -5, we can have the first point (0, -5)
Using the given slope = \(\frac{2}{3}\)m = \(\frac{2}{3}\), from point (0, -5), move 2 units up since the numerator of the slope is 2, and then move 3 units to the right since the denominator is 3, it gives us the second point (3, -3)
Using the same slope, from the point (3, -3), move 2 units up, and then 3 units right, it gives the third point (6, -1)
Getting the next point from the point (6, -1), moving 2 units down and 3 units right it gives the point (9, 1).
(0, -5), (3, -3), (6, -1), (9, 1)

Question 12.
A music school charges a registration fee in addition to a fee per lesson. Music lessons last 0.5 hour. The equation y = 40x + 30 represents the total cost y of x lessons. Find and interpret the slope and y-intercept of the line that represents this situation. Then find four points on the line.
Answer:
y = 40x + 30
Slope = 40
y – intercept = 30
Slope represents the fee of the classes per lesson and y intercept represents the registration fee.
Plot the point that contains the y-intercept:
(0, 30)
The slope is m = \(\frac{40}{1}\)
Use the slope to find a second point
From
(0, 30)
count $40$ unit up and $1$ unit right. The new point is (1, 70)
Follow the same procedure to find the remaining three points
(2, 110)
(3, 150)

Math Grade 8 Answer Key Pdf Slope Intercept Form Answer Key Question 13.
A public pool charges a membership fee and a fee for each visit. The equation y = 3x + 50 represents the cost y for x visits.
a. After locating the y-intercept on the coordinate plane shown, can you move up three gridlines and right one gridline to find a second point? Explain.
Answer:
y = 3x + 50
Yes
Since the horizontal and vertical gridlines each represent 25 units, hence moving up 3 gridlines and right 1 gridline represent a slope of \(\frac{75}{25}\) or
3

b. Graph the equation y = 3x + 50. Then interpret the slope and y-intercept.
Answer:
Slope = 3
y-intercept = 50
Plot the point that contains the y-intercept:
(0, 50)

The slope is m = \(\frac{3}{1}\)
Use the slope to find a second point.
From
(0, 50)
count $3$ unit up and $1$ unit right. The new point is
(1, 53)
Draw a line through the points

The slope represents the fee per visit and y-intercept represents the membership fee.

c. How many visits to the pool can a member get for $200?
Answer:
You would get 50 visits for $200.

Texas Go Math Grade 8 Lesson 4.3 H.O.T. Higher Order Thinking Answer Key

Question 14.
Explain the Error A student says that the slope of the line for the equation y = 20 – 15x is 20 and the y-intercept is 15. Find and correct the error.
Answer:
y = 20 – 15x
The slope is -15 as it represents the change in y per unit change in x. The y-intercept is 20 when x = 0

Question 15.
Critical Thinking Suppose you know the slope of a linear relationship and a point that its graph passes through. Can you graph the line even if the point provided does not represent the y-intercept? Explain.
Answer:
Yes. You can plot the given point and use the slope to find a second point. Connect the points by drawing a line.

Go Math Grade 8 Answer Key Pdf Lesson 4.3 Practice Algebra Answers Question 16.
Make a Conjecture Graph the lines y = 3x, y = 3x – 3, and y = 3x + 3. What do you notice about the lines? Make a conjecture based on your observation.

Answer:
Following the steps of “Example 1” on page 101, we graph the lines:
y = 3x
y = 3x – 3
y = 3x + 3

We notice that the lines are parallel to each other: the slopes of the lines are equal but the y-intersection point differs.

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 3.2 Answer Key Rate of Change and Slope.

Texas Go Math Grade 8 Lesson 3.2 Answer Key Rate of Change and Slope

Essential Question
How do you find a rate of change or a slope?

Your Turn

Question 1.
The table shows the approximate height of a football after it is kicked. Tell whether the rates of change are constant or variable. Find the rates of change:
The rates of change are

Answer:
Input variable: Time (s) Identify the input and output variables
Output variable: Height (ft)

Texas Go Math Grade 8 Lesson 3.2 Explore Activity Answer Key 

Using Graphs to Find Rates of Change
You can also use a graph to find rates of change.

The graph shows the distance Nathan bicycled over time. What is Nathan’s rate of change?

A. Find the rate of change from 1 hour to 2 hours.

B. Find the rate of change from 1 hour to 4 hours.

C. Find the rate of change from 2 hour to 4 hours.

D. Recall that the graph of a proportional relationship is a line through the origin. Explain whether the relationship between Nathan’s time and distance is a proportional relationship.

Reflect

Rate of Change and Slope Answer Key Go Math Grade 8 Question 2.
Make a Conjecture Does a proportional relationship have a constant rate of change?
Answer:
Yes. The following equation holds:
\(\frac{y}{x}\) = k
Where k is the constant rate of change
Yes

Question 3.
Does it matter what interval you use when you find the rate of change of a proportional relationship? Explain.
Answer:
No. It does not matter what interval you use as long the order of points remains the same. When y is subtracted from y₂, then x₁ should be subtracted from x₂.

Your Turn

Question 4.
The graph shows the rate at which water is leaking from a tank. The slope of the line gives the leaking rate in gallons per minute.

Rise = ___________________________
Run = _________________________
Rate of leaking ___ gallon(s) per minute
Answer:
The two given points are
(4, 3) and (8, 6)
Let x₁ = 4, x₂ = 8 and y₁ = 3, Y₂ = 6 We can now calculate the rise, run and slope:
rise = y₂ – y₁ = 6 – 3 =3
run = x₂ – x₁ = 8 – 4 = 4
slope = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) = \(\frac{3}{4}\)
Rise = 3, Run = 4 and slope = \(\frac{3}{4}\)

Texas Go Math Grade 8 Lesson 3.2 Guided Practice Answer Key 

Tell whether the rates of change are constant or variable. (example 1)

Question 1.
building measurements ____

Answer:
(1) First identify the input and output variables
Input variable: Feet, output variable: Yards.
(2) Now find the rates of change.

The rates of change are constant: 3 feet per yard.
Constant rate of change

Question 2.
computers sold ____

Answer:
Input variable: Week Identify the input and output variables
Output variable: Number sold

The rates of change are variable.

Question 3.
distance an object falls ____

Answer:
Input variable: Week Identify the input and output variables
Output variable: Number sold

The rates of change are variable.
Input variable: Time Identify the input and output variables
Output variable: Distance

The rates of change are variable.

Go Math Answer Key Grade 8 Lesson 3.2 Answer Key Question 4.
cost of sweaters ____

Answer:
(1) First identify the input and output variables.
Input variable: Number, output variable: Cost.

(2) Now find the rates of change.

The rates of change are constant: 19 dollars per sweater.
The rates of change are constant.

Erica walks to her friend Philip’s house. The graph shows Erica’s distance from home over time. (Explore Activity)

Question 5.
Find the rate of change from 1 minute to 2 minutes.


Answer:
Find the rate of change from 1 minute to 2 minutes.
= 200 ft per min

Question 6.
Find the rate of change from 1 minute to 4 minutes. ________
Answer:

Therefore the rate of change from 1 minute to 4 minutes is 200 feet per minute.
200 feet per minute.

Find the slope of each line. (Example 2)

Question 7.

Slope = ____
Answer:
Rise is the difference in values represented by the axis
Rise = 4 – 0 = 4

Run = -2 – 0 = -2 Run is the difference in values represented by the x-axis

Slope = \(\frac{4}{-2}\) = -2

Go Math 8th Grade Slope and Rate of Change Answer Key Question 8.

Slope = ____
Answer:
Rise = 3 – 0 = 3 Rise is the difference in values represented by y axis
Run = 2 – 0 = 2 Run is the difference in values represented by x axis
Slope = \(\frac{3}{2}\)

Essential Question Check-In

Question 9.
If you know two points on a line, how can you find the rate of change of the variables being graphed?
Answer:
If the two given points are (x₁, y₁) and (x₂, y₂) we can find the slope
slope = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
since the slope of a line is the ratio of the change in y-values for a segment of the graph to the corresponding change in x-values.

You can calculate the slope (See inside for more)

Texas Go Math Grade 8 Lesson 3.2 Independent Practice Answer Key 

Question 10.
Rectangle EFGH is graphed on a coordinate plane with vertices at E(-3, 5), F(6, 2), G(4, -4), and H(-5, -1).

a. Find the slopes of each side.
Answer:
We are given the vertices of the rectangle EFGH:
E(-3, 5) F(6, 2) G(4, -4) H(-5, -1)

a) In general, the slope of a line is the ratio of the change in y-values for a segment of the graph to the corresponding change in x-values.
Slope = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Slope of EF:

b. What do you notice about the slopes of opposite sides?
Answer:
The slopes of opposite sides are equal
slope_EF = slope_GH = – \(\frac{1}{3}\)
Slope_FG = slope_HE = 3

c. What do you notice about the slopes of adjacent sides?
Answer:
The slopes of adjacent sides are negative reciprocals of each other (3 and –\(\frac{1}{3}\))

Go Math Book Grade 8 Answer Key Lesson 3.2 Answer Key Question 11.
A bicyclist started riding at 8:00 a.m The diagram below shows the distance the bicyclist had traveled at different times. What was the bicyclist’s average rate of speed in miles per hour?

Answer:
We are given that the bicyclist rides the first 4.5 miles for 18 minutes and the other 7.5 miles for 30 minutes. To find the rate of speed in miles/hour. First, we need to convert minutes into hours (divide by 60 since 1 hour has 60 minutes).
• Convert 18 minutes into hours:

• Convert 30 minutes into hours:

The rate of change for the first 0.3 hours:
\(\frac{4.5}{0.3}\) = \(\frac{45}{3}\) = 15
The rate of change for the next 0.5 hours:
\(\frac{7.5}{0.5}\) = \(\frac{75}{5}\) = 15
In conclusion, the bicyclist’s average rate of speed is 15 miles per hour.

Question 12.
Multistep A line passes through (6, 3), (8,4), and (n, -2). Find the value of n.
Answer:
Using the first two points:
(x₁, y₁) = (6, 3) and (x₂, y₂) = (8, 4) find the slope of the line.
Slope = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) = \(\frac{4-3}{8-6}\) = \(\frac{1}{2}\)
Now use any of the first two points ant the last point to find the value of n
by using the definition of a slope of the line:
Slope = \(\frac{1}{2}\) = \(\frac{4-(-2)}{8-n}\) = \(\frac{6}{8-n}\)
\(\frac{6}{12}\) = \(\frac{6}{8-n}\)
Multiply the numerator and denominator of the left side by 6
12 = 8 – n
n = -4

Go Math Grade 8 Answers 3.2 Rate of Change and Slope Question 13.
A large container holds 5 gallons of water. It begins leaking at a constant rate. After 10 minutes, the container has 3 gallons of water left.
a. At what rate is the water leaking?
Answer:
At 0 minutes, there are 5 gallons of water. At 10 minutes, there are 3 gallons of water. Given

Rate of = \(\frac{5-3}{0-10}\) = \(\frac{2}{-10}\) = –\(\frac{1}{5}\) = -0.2
The rate of water leakage is $1$ gallon every $5$ minute or $0.2$ gallon per min

b. After how many minutes will the container be empty?
Answer:
No of minutes = = 25
\(\frac{5}{0.2}\) = 25 The number of minutes is determined by dividing the total volume of container by rate of water leakage.
It will take $25$ minutes for the container to be empty.

Question 14.
Critique Reasoning Billy found the slope of the line through the points (2, 5) and (-2, -5) using the equation \(\frac{2-(-2)}{5-(-5)}\) = \(\frac{2}{5}\). What mistake did he make?
Answer:
By definition, the slope is the change in y-values (rise) for a segment of the graph to the corresponding change in
Slope = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Meanwhile, Billy wrote it as \(\frac{x_{-} 2-x_{-} 1}{y_{2}-y_{1}}\)
So the real slope is \(\frac{5}{2}\)
Billy got the reciprocal of the slope.

Lesson 3.2 Rate of Change and Slope Reteach Answer Key Question 15.
Multiple Representations Graph parallelogram ABCD on a coordinate plane with vertices at A(3, 4), 8(6, 1), C(0, -2), and D(-3, 1).

Answer:
We are given the vertices of the parallelogram ABCD:
Ä(3, 4)
B(6, 1)
C(0, -2)
D(-3, 1)

a. Find the slope of each side.
Answer:
In general, the slope of a line is the ratio of the change in y-values for a segment of the graph to the corresponding change in x-values.
slope = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

b. What do you notice about the slopes?
Answer:
The slopes of opposite sides are equal.
slope_AB = slope_CD = – 1
slope_BC = Slope_DA = \(\frac{1}{2}\)

c. Draw another parallelogram on the coordinate plane. Do the slopes have the same characteristics?
Answer:
If we calculate the slopes of the sides of the parallelogram shown below in the same way as calculated above,
we will see that they have the same characteristics.

Texas Go Math Grade 8 Lesson 3.2 H.O.T. Focus On Higher Order Thinking Answer Key 

Question 16.
Communicate Mathematical Ideas Ben and Phoebe are finding the slope of a line. Ben chose two points on the line and used them to find the slope. Phoebe used two different points to find the slope. Did they get the same answer? Explain.
Answer:
Yes, they did. The slope of a line can be calculated using every two points on the line, since it is the same lane it has a unique (singular) slope

Yes, the slope is the same for every two points.

Lesson 3.2 Rate of Change and Slope Answer Key Question 17.
Analyze Relationships Two lines pass through the origin. The lines have slopes that are opposites. Compare and contrast the lines.
Answer:
Since the slopes of the lines are opposites (if one line has a slope k the other has -k) it means they are equally
steep, but since one has a positive slope it is slanted upwards from left to right and the other with the negative slope is slanted downwards.

One line is upward facing and the other is downward, but both are equally steep.

Question 18.
Reason Abstractly What is the slope of the x-axis? Explain.
Answer:
The slope of a line is the ratio of the change in y-values for a segment of the graph to the corresponding change in x-values x-axis is a horizontal line that never changes its y-position, so the numerator of the ratio is zero
Therefore, the slope of the x-axis is 0.

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 3 Quiz Answer Key.

Texas Go Math Grade 8 Module 3 Quiz Answer Key

Texas Go Math Grade 8 Module 3 Ready to Go On? Answer Key

3.1 Representing Proportional Relationships

Question 1.
Find the constant of proportionality for the table of values.

Answer:
\(\frac{3}{2}\) = 1.5
\(\frac{4.5}{3}\) = \(\frac{3}{2}\) = 1.5
\(\frac{6}{4}\) = \(\frac{3}{2}\) = 1.5
\(\frac{7.5}{5}\) = \(\frac{3}{2}\) = 1.5
Therefore the constant of proportionality is 1.5
The constant of proportionality is 1.5.

Grade 8 Math Module Answer Key Quiz Answers Question 2.
Phil is riding his bike. He rides 25 miles in 2 hours, 37.5 miles in 3 hours, and 50 miles in 4 hours. Find the constant of proportionality and write an equation to describe the situation.
Answer:
Let’s first find the tabLe of the given situation:

The constant of proportionality is 12.5
An equation to describe the situation is y = 12.5x where y is the distance and æ the time.
\(\frac{25}{2}\) = 12.5
\(\frac{37.5}{3}\) = \(\frac{25}{2}\) = 12.5
\(\frac{50}{4}\) = \(\frac{25}{2}\) = 12.5
The constant of proportionality is 12.5.
An equation to describe the situation is y = 12.5x where y is the distance and x the time

Constant of proportionality is 12.5, and the equation y = 12.5x

3.2 Rate of Change and Slope

Find the slope of each tine.

Question 3.

Answer:
We choose two points on the line and we find the change in y-values and the change in x-values as we move from
one point to the other.
rise = +3, since we move up
run = +1, since we move right
Therefore,
slope = \(\frac{+3}{+1}\)

slope = 3

Grade 8 Math Module 3 Quiz Answer Key Question 4.

Answer:
Using the graph find 2 points that are on the given line, for example: (0, 0) and (1, -5). Now find the slope
Slope = \(\frac{-5-0}{1-0}\) = -5

The slope is —5.

3.3 Interpreting the Unit Rate as Slope

Question 5.
What is the slope of the data in the table?

Answer:
The formula for the slope m= y2 – y1/x2 x1.
Here
x1 = 13.5
x2 = 18
y1 = 3
y2 = 4
m = 4 – 3/18 -13.5
m = 1/ 4.5.
m = 0.22.
And
x1 = 18
x2 = 22.5
y1 = 4
y2 = 5
m = 5 – 4/22.5 – 18
m = 1/ 4.5.
m = 0.22.
And
x1 = 22.5
x2 = 27
y1 = 5
y2 = 6
m = 6 – 5/27 -22.5
m = 1/ 4.5.
m = 0.22.
The slope of the data is 0.22

3.4 Direct Variation

Question 6.
A wheelchair ramp rises 2.25 feet for every 25 feet of horizontal distance it covers. What is the slope of the ramp? ________
Answer:
Given that,
Wheelchair ramp rises = 2.25 feet per every 25 feet of horizontal distance.
Slope of the ramp = m = 2.25/25 = 0.09
The slope of the ramp = 0.09

Essential Question

Grade 8 Math Module 3 Answer Key Question 7.
What is the relationship among direct variation, lines, rates of change, and slope?
Answer:
The relationship among direct variation, lines, rates of change, and the slope is the variables of the x-axis and the y-axis is the proportional relational ship that has the slope it means the direct variation is determined by calculating the rate of change in the line.

Texas Go Math Grade 8 Module 3 Mixed Review Texas Test Prep Answer Key

Selected Response

Question 1.
Lupe’s heart beats 288 times in 4 minutes. Which equation represents the number of times her heart beats per minute?
(A) y = \(\frac{1}{288}\)x
(B) y = \(\frac{1}{72}\)x
(C) y = 72x
(D) y = 288x
Answer:
Given that,
Lupe’s heartbeat = 288 times per 4 minutes.
Let x be the number of minutes.
Heartbeat per minute equal to = 288x/4
= 72x
Option C is the correct answer.

Question 2.
Prasert earns $9 an hour. Which table represents this proportional relationship?

Answer:
(A)

Explanation:
y = 9x Relationship representing earning at $9 per hr. where x is the number of hour

y = 9(2) = 18 When x = 2, 3, 4, 6, 8
y = 9(3) = 27
y = 9(4) = 36
y = 9(6) = 54
y = 9(8) = 72
Table $A$ represents the situation

Texas Go Math Grade 8 Module 3 Test Answers Question 3.
A factory produces widgets at a constant rate. After 4 hours, 3,120 widgets have been produced. At what rate are the widgets being produced?
(A) 630 widgets per hour
(B) 708 widgets per hour
(C) 780 widgets per hour
(D) 1,365 widgets per hour
Answer:
(C) 780 widgets per hour

Explanation:
Let’s denote with x the change ¡n time (hours) and with y the change in the number of widgets produced. We
know that x = 4 and y = 3120. To find the rate of change at which the widgets are being produced, we have:
rate = \(\frac{y}{x}\)
= \(\frac{3120}{4}\)
= 780
In conclusion, 780 widgets are produced every hour.

Question 4.
A full lake begins dropping at a constant rate. After 4 weeks it has dropped 3 feet. What is the unit rate of change in the lake’s level compared to its full level?
(A) 0.75 feet per week
(B) 1.33 feet perweek
(C) -0.75 feet per week
(D) -1.33 feet per week
Answer:
(A) 0.75 feet per week

Explanation:
\(\frac{3}{4}\) = 0.75 Find the unit rate by
The lake leaks at $0.75$ feet per week

Question 5.
What is the slope of the line below?

(A) -2
(B) – \(\frac{1}{2}\)
(C) \(\frac{1}{2}\)
(D) 2
Answer:
(B) – \(\frac{1}{2}\)

Explanation:
Using the graph find two points on the line. For example: (3, 0) and (-3, 3). Now using those two points calculate
the slope:
Slope = \(\frac{0-3}{3-(-3)}\) = \(\frac{-3}{6}\) = –\(\frac{1}{2}\)

Grade 8 Module 3 Math Quiz Answer Key Question 6.
Jim earns $41.25 in 5 hours. Susan earns $30.00 in 4 hours. Pierre’s hourly rate is less than Jim’s, but more than Susan’s. What is his hourly rate?
(A) $6.50
(B) $7.75
(C) $7.35
(D) $8.25
Answer:
(B) $7.75

Explanation:
\(\frac{41.25}{5}\) = 8.25 Unit rate is given by
Jim’s unit rate is $\$8.25$per hr$
$\dfrac{30}{4} = 7.5$ $Susan’s unit
rate is$\$7.5$ per hr

Pierre’s hourly rate = $7.75 per hour Pierre’s hourly rate is less than Jim’s but more than Susan’s
7.5 < x < 8.25

Gridded Response

Question 7.
Joelle kept track of her daily reading in the table shown below. What is her fastest daily reading rate in pages per minute?

Answer:
Given that,
Monday timing time = 20 minutes
Monday number of pages = 44
Number of minutes per minute = 44/20 = 2.2
Tuesday timing time = 40 minutes
Tuesday number of pages = 70
Number of minutes per minute = 70/40 = 1.75
Wednesday timing time = 25 minutes
Wednesday number of pages = 56
Number of minutes per minute = 56/25 = 2.24
The fastest daily reading rate is 2.24 pages per minute.

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 1.3 Answer Key Ordering Real Numbers.

Texas Go Math Grade 8 Lesson 1.3 Answer Key Ordering Real Numbers

Essential Question
How do you order a set of real numbers?

Reflect

Question 1.
If 7 + \(\sqrt{5}\) is equal to \(\sqrt{5}\) plus a number, what do you know about the number? Why?
Answer:
We are told that 7 + \(\sqrt{5}\) is equal to\(\sqrt{5}\) plus a number. Lets denote with x the unknown number. We have:
7 + \(\sqrt{5}\) = \(\sqrt{5}\) + x
If we subtract \(\sqrt{5}\) in both sides we get:
7 + \(\sqrt{5}\) – \(\sqrt{5}\) = \(\sqrt{5}\) + x – \(\sqrt{5}\)
x = 7

The unknown number is 7 (whole number, integer, rational, real number).

Go Math Grade 8 Algebra Lesson 1.3 Question 2.
What are the closest two integers that \(\sqrt{300}\) is between?
Answer:
\(\sqrt{300}\) Given
289 < 300 < 324 Find the perfect squares of the number that 300 is between

17 < \(\sqrt{300}\) < 18 Take square roots
The close two integers are $17$ and $18$

Your Turn

Compare. Write <, >, or .

Question 3.
\(\sqrt{2}\) + 4 2 + \(\sqrt{4}\)
Answer:
First approximate \(\sqrt{2}\) by comparing 2 to the closest perfect squares and then take square root from both sides of the equation.
1 < 2 < 4
\(\sqrt{1}\) < \(\sqrt{2}\) < \(\sqrt{4}\) \(\sqrt{2}\) is between 1 and 2. Next since 4 is a perfect square, we know that \(\sqrt{4}\) = 2. \(\sqrt{2}\) + 4 is between 5 and 6 2 + \(\sqrt{4}\) is equal to 4. So, \(\sqrt{2}\) + 4 > 2 + \(\sqrt{4}\)
\(\sqrt{2}\) + 4 > 2 + \(\sqrt{2}\)

Question 4.
\(\sqrt{12}\) + 6 12 + \(\sqrt{6}\)
Answer:
\(\sqrt{12}\) + 6 ? 12 + \(\sqrt{6}\) Given
\(\sqrt{12}\) is between 3 and 4 Approximate \(\sqrt{12}\)

\(\sqrt{6}\) is between 2 and 3 Approximate \(\sqrt{6}\)

\(\sqrt{12}\) + 6 is between 9 and 10 Use approximations to simply the expressions
12 + \(\sqrt{6}\) is between 14 and 15
\(\sqrt{12}\) + 6 < 12 + \(\sqrt{6}\) Compare

Your Turn

Order the numbers from least to greatest. Then graph them on the number line.

Go Math Grade 8 Lesson 1.3 Independent Practice Answer Key Question 5.
\(\sqrt{5}\), 2.5, \(\sqrt{3}\) ____

Answer:
\(\sqrt{5}\), 2.5, \(\sqrt{3}\) Given
4 < 5 < 9 Approximate \(\sqrt{5}\). Find the perfect squares that 5 is between. Take square roots.
2 < \(\sqrt{5}\) < 3
\(\sqrt{5}\) is between 2 and 3
2.15² = 4.62 Find better estimates for \(\sqrt{5}\) so it can be compared to 2.5.
2.2² = 4.84 5 is closer to 4,; try numbers between 2 and 2.5
2.25² = 5.06
Since $2.25^{2}$ is closer to $5$, an approximate value is $2.25$
1 < 3 < 4 Approximate \(\sqrt{3}\). Find the perfect squares that 3 is between Take square roots.
1 < \(\sqrt{3}\) < 2
\(\sqrt{3}\) is between 1 and 2

1.6² = 2.56 Find better estimates for \(\sqrt{3}\).
1.7² = 2.89 3 is closer to 4, hence try numbers between 1.5 and 2
1.8² = 3.24
Since $1.7^{2}$ is closer to $3$, an approximate value is $1.7$
Graph on a Number line
\(\sqrt{3}\), \(\sqrt{5}\), 2.5 From the least to greatest

Math Talk
Mathematical Processes
If real numbers a, b, and c are in order from least to greatest, what is the order of their opposites from least to greatest? Explain.

Question 6.
π², 10, \(\sqrt{75}\) ____

Answer:
π², 10, \(\sqrt{75}\) Given
64 < 75 < 81 Approximate \(\sqrt{75}\) Find the perfect squares that 75 is between. Take square roots.
8 < \(\sqrt{75}\) < 9
\(\sqrt{75}\) is between 8 and 9

8.6² = 73.96 Find better estimates for \(\sqrt{75}\)
8.65² = 74.82 75 is closer to 81, hence try numbers between 8.5 and 9
8.78.6² = 75.69
Since $8.65^{2}$ is closer to $75$, an approximate value is $8.65$
π² = (3.14)² = 9.86 Find π²
Graph on number line
\(\sqrt{75}\), π, 10

Your Turn

Question 7.
Four people have found the distance in miles across a crater using different methods. Their results are given below.
Jonathan: \(\frac{10}{3}\) Elaine: \(3 . \overline{45}\), José: 3\(\frac{1}{2}\), Lashonda: \(\sqrt{10}\)
Order the distances from greatest to least.
Answer:
\(\frac{10}{3}\) = \(3 . \overline{3}\) ≈ 3.33 Express the given numbers as decimals.
\(3 . \overline{45}\) = 3.454545 ≈ 3.45
3\(\frac{1}{2}\) = 3.5
\(\sqrt{10}\) is between 3.1 and 3.2. Since 3.2² = 10.24, an approximate value is 3.2.
Graph on number line
3\(\frac{1}{2}\) , \(3 . \overline{45}\), \(\frac{10}{3}\), \(\sqrt{10}\) From greatest to least

Texas Go Math Grade 8 Lesson 1.3 Guided Practice Answer Key 

Compare. Write <, >, or =. (Example 1)

Question 1.
\(\sqrt{3}\) + 2 \(\sqrt{3}\) + 3
Answer:
\(\sqrt{3}\) + 2 ? \(\sqrt{3}\) + 3
\(\sqrt{3}\) is between 1 and 2 Approximate \(\sqrt{3}\)
\(\sqrt{3}\) + 2 is between 3 and 4 Use approximations to simply the expressions
\(\sqrt{3}\) + 3 is between 4 and 5
\(\sqrt{3}\) + 2 < \(\sqrt{3}\) + 3 Compare
\(\sqrt{3}\) + 2 < \(\sqrt{3}\) + 3

Go Math 8th Grade Compare and Order Real Numbers Answer Key Question 2.
\(\sqrt{11}\) + 15 \(\sqrt{8}\) + 15.
Answer:
First approximate \(\sqrt{11}\) by comparing 11 to the closest perfect squares and then take square root from both sides of the equation
9 < 11 < 16
\(\sqrt{9}\) < \(\sqrt{11}\) < \(\sqrt{16}\)
3 < \(\sqrt{11}\) < 4
\(\sqrt{11}\) is between 3 and 4.
Next, approximate\(\sqrt{8}\) by comparing 8 to the closest perfect squares and then take the square root from both sides of the equation.
4 < 8 < 9
\(\sqrt{4}\) < \(\sqrt{8}\) < \(\sqrt{9}\)
2 < \(\sqrt{8}\) < 3 \(\sqrt{8}\) is between 2 and 3.
\(\sqrt{11}\) + 15 is then in between 18 and 19. \(\sqrt{8}\) + 15 is between 17 and 18
Therefore, \(\sqrt{11}\) + 15 > \(\sqrt{8}\) + 15

\(\sqrt{11}\) + 15 > \(\sqrt{8}\) + 15

Practice and Homework Lesson 1.3 8th Grade Question 3.
\(\sqrt{6}\) + 5 6 + \(\sqrt{5}\)
Answer:
\(\sqrt{6}\) + 5 ? 6 + \(\sqrt{5}\) Given

4 < 6 < 9 Approximate \(\sqrt{6}\)
\(\sqrt{4}\) < \(\sqrt{6}\) < \(\sqrt{9}\)
2 < \(\sqrt{6}\) < 3

4 < 5 < 9 Approximate \(\sqrt{5}\)
\(\sqrt{4}\) < \(\sqrt{5}\) < \(\sqrt{9}\)
2 < \(\sqrt{5}\) < 3
\(\sqrt{6}\) + 5 is between 7 and 8 Use approximately to simplify the expressions
6 + \(\sqrt{5}\) is between 8 and 9
\(\sqrt{6}\) + 5 < 6 + \(\sqrt{5}\) Compare

Question 4.
\(\sqrt{9}\) + 3 9 + \(\sqrt{3}\)
Answer:
\(\sqrt{9}\) + 3 ? 9 + \(\sqrt{3}\)
\(\sqrt{9}\) = 3 Approximate \(\sqrt{9}\)
\(\sqrt{3}\) is between 1 and 2 Approximate \(\sqrt{3}\)
\(\sqrt{9}\) + 3 = 3 + 3 = 5 use approximations to simplify the expressions
9 + \(\sqrt{3}\) is between 10 and 11
\(\sqrt{9}\) + 3 < 9 + \(\sqrt{3}\) Compare
\(\sqrt{9}\) + 3 < 9 + \(\sqrt{3}\)

Question 5.
\(\sqrt{17}\) – 3 -2 + \(\sqrt{5}\)
Answer:
\(\sqrt{17}\) – 3 ? -2 + \(\sqrt{5}\) Given
\(\sqrt{17}\) is between 4 and 5 Approximate \(\sqrt{17}\)
\(\sqrt{5}\) is between 2 and 3 Approximate \(\sqrt{5}\)

\(\sqrt{17}\) – 3 is between 1 and 2 Use approximations to simplify the expressions
-2 + \(\sqrt{5}\) is between 0 and 1
\(\sqrt{17}\) – 3 > -2 + \(\sqrt{5}\) Compare

Compare and Order Numbers Lesson 1.3 Question 6.
10 – \(\sqrt{8}\) 12 – \(\sqrt{2}\)
Answer:
10 – \(\sqrt{8}\) ? 12 – \(\sqrt{2}\) Given
\(\sqrt{2}\) is between 1 and 2 Approximate \(\sqrt{8}\)
\(\sqrt{2}\) is between 1 and 2 Approximate \(\sqrt{2}\)

10 – \(\sqrt{8}\) is between 8 and 7 Use approximations to simplify the expressions
12 – \(\sqrt{2}\) is between 11 and 10

10 – \(\sqrt{8}\) < 12 – \(\sqrt{2}\) Compare

Question 7.
\(\sqrt{7}\) + 2 \(\sqrt{10}\) – 1.
Answer:
\(\sqrt{7}\) + 2 ? \(\sqrt{10}\) – 1. Given
\(\sqrt{7}\) is between 2 and 3 Approximate \(\sqrt{8}\)
\(\sqrt{10}\) is between 3 and 4 Approximate \(\sqrt{10}\)

\(\sqrt{7}\) + 2 is between 4 and 5 Use approximations to simplify the expressions
\(\sqrt{10}\) – 1 is between 2 and 3

\(\sqrt{7}\) + 2 > \(\sqrt{10}\) – 1 Compare

Question 8.
\(\sqrt{17}\) + 3 3 + \(\sqrt{11}\)
Answer:
\(\sqrt{17}\) + 3 ? 3 + \(\sqrt{11}\) Given
\(\sqrt{17}\) is between 4 and 5 Approximate \(\sqrt{17}\)
\(\sqrt{11}\) is between 3 and 4 Approximate \(\sqrt{11}\)

\(\sqrt{17}\) + 3 is between 7 and 8 Use approximations to simplify the expressions
3 + \(\sqrt{11}\) is between 6 and 7

\(\sqrt{7}\) + 3 > 3 + \(\sqrt{11}\) Compare

Practice and Homework Lesson 1.3 Answer Key 8th Grade Question 9.
Order \(\sqrt{3}\), 2π, and 1.5 from least to greatest. Then graph them on the number line. (Example 2)
\(\sqrt{3}\) is between ___ and ___, so \(\sqrt{3}\) ≈ ____.
π ≈ 3.14, so 2π ≈ ____.

From least to greatest, the numbers are ____ , _____
______
Answer:
\(\sqrt{3}\) is between 1.7 and 1.8 since 1.7² = 2.89, a good approximation for \(\sqrt{3}\) is 1.7.
1.5 is in good form so we don’t need to change it.
π ≈ 3.14 ⇒ 2π ≈ 6.28

Graphing:
A = 1.7, B = 1.5 and C = 6.28.

From least to greatest, the numbers are 1.5, \(\sqrt{3}\), 2π

From least to greatest, the numbers are 1.5, \(\sqrt{3}\), 2π. Click for the graph.

Question 10.
Four people have found the perimeter of a forest using different methods. Their results are given in the table. Order their calculations from greatest to least. (Example 3)

Answer:
\(\sqrt{3}\) – 2 Estimate the value of \(\sqrt{3}\)
\(\sqrt{3}\) is between 4 and 5
Since, $17$ is closer to $16$, the estimated value is $4.1$
1 + \(\frac{\pi}{2}\) Where π = 3.14
1 + \(\frac{3.14}{2}\) = 2.57

\(\frac{12}{5}\) = 2.4 Evaluate the fraction
2.5 Given
Graph on number line
\(\sqrt{17}\) – 2, 1 + \(\frac{\pi}{2}\), 2.5, \(\frac{12}{5}\) From greatest to least

Essential Question Check-in

Question 11.
Explain how to order a set of real numbers.
Answer:
To order real numbers first express them in the decimal form. If all numbers can’t be calculated to a finite decimal,
approximate them to a chosen accuracy (number of digits behind the decimal point, where more digits equal higher accuracy). Then you can easily place them on a number line.

Express numbers in decimal form and approximate.

Texas Go Math Grade 8 Lesson 1.3 Independent Practice Answer Key 

Order the numbers from least to greatest.

Question 12.
\(\sqrt{7}\), 2, \(\frac{\sqrt{8}}{2}\)
Answer:
\(\sqrt{7}\), 2, \(\frac{\sqrt{8}}{2}\) Given
\(\sqrt{7}\) Estimate the value of \(\sqrt{7}\)
\(\sqrt{7}\) is between 2 and 3
Since, $7$ is closer to $9$, $2.65^{2} = 7.02$, hence the estimated value is $2.65$

2 Given

\(\frac{\sqrt{8}}{2}\) Estimate the value of \(\sqrt{8}\) and simplify
\(\sqrt{8}\) is between 2 and 3
Since, $8$ is closer to $9$, $2.85^{2} = 8.12$, hence the estimated value is $2.85$
\(\frac{2.85}{2}\) = 1.43
Graph on a number line
\(\frac{\sqrt{8}}{2}\), 2, \(\sqrt{7}\) From least to greatest

Go Math Grade 8 Answer Key Algebra Lesson 1.3 Question 13.
\(\sqrt{10}\), π, 3.5
Answer:
\(\sqrt{10}\) is between 3.1 and 3.2. In some cases, this approximation is precise enough, but this is not the case. Since we have a given number π ≈ 3.14 that ¡s also between 3.1 and 3.2 we must be more precise. So, an even better approximation is 3.15 because 3.15² ≈ 9.92.
π ≈ 3.14

We can now order the numbers from [east to greatest easily:
Because 3.14 < 3.15 < 3.5 it follows that π < \(\sqrt{10}\) < 3.5
Numbers from least to greatest π < \(\sqrt{10}\) < 3.5

Question 14.
\(\sqrt{220}\), -10, \(\sqrt{100}\), 11.5
Answer:
\(\sqrt{220}\), -10, \(\sqrt{100}\), 11.5 Given
196 < 220 < 225 Approximate \(\sqrt{220}\)
\(\sqrt{196}\) < \(\sqrt{220}\) < \(\sqrt{225}\)
14 < \(\sqrt{220}\) < 15
\(\sqrt{220}\) ≈ 14.5
\(\sqrt{100}\) = 10 Calculate \(\sqrt{100}\)
Plot on a number line
-10, \(\sqrt{100}\), 11.5, \(\sqrt{220}\) Order from least to greatest
-10, \(\sqrt{100}\), 11.5, \(\sqrt{220}\)

Question 15.
\(\sqrt{8}\), -3.75, 3, \(\frac{9}{4}\)
Answer:
\(\sqrt{8}\), -3.75, 3, \(\frac{9}{4}\) Given
\(\sqrt{8}\) Estimate the value of \(\sqrt{8}\)
\(\sqrt{8}\) is between 2 and 3
Since, $8$ is closer to $9$, $2.85^{2} = 8.12$, hence the estimated value is $2.85$
-3.75 Given
3
\(\frac{9}{4}\) = 2.25 Evaluate the fraction
Graph on the number line
-3.75, \(\frac{9}{4}\), \(\sqrt{8}\), 3 From least to greatest

Ordering Real Numbers 8th Grade Question 16.
Your sister is considering two different shapes for her garden. One is a square with side lengths of 3.5 meters, and the other is a circle with a diameter of 4 meters.
a. Find the area of the square. _______
Answer:
To find the area of a square use the formula: A = x² where x is the side of the square and the area is represented by A.
So, A = 35² = 12.25

To find the area of a square use the formula: B = r²π where r is the radius of the circle and the area is represented by B.
So, B = 2²π = 12.57 (Diameter d is twice the radius r, so r = 2.)

The garden shaped like a circle would give her more space to plant because it has a bigger area.
A = 3.5² = 12.25

b. Find the area of the circle. ____
Answer:
B = 2²π = 12.57

c. Compare your answers from parts a and b. Which garden would give your sister the most space to plant?
Answer:
The circle-shaped garden has more space to plant.

Question 17.
Winnie measured the length of her father’s ranch four times and got four different distances. Her measurements are shown in the table.

a. To estimate the actual length, Winnie first approximated each distance to the nearest hundredth. Then she averaged the four numbers. Using a calculator, find Winnie’s estimate.
Answer:
\(\sqrt{60}\) ≈ 7.75 Evaluate the lengths to nearest hundredth (using calculator)
\(\frac{58}{8}\) ≈ 7.25
\(7 . \overline{3}\) ≈ 7.33
7\(\frac{3}{5}\) ≈ 7.60
Average = $\dfrac{7.75 + 7.25 + 7.33
+ 7.60}{4} = 7.4815$

b. Winnie’s father estimated the distance across his ranch to be \(\sqrt{56}\) km. How does this distance compare to Winnie’s estimate?
Answer:
\(\sqrt{56}\) ≈ 7.4833 Using claculator
They are nearly identical

Give an example of each type of number.

Question 18.
a real number between \(\sqrt{13}\) and \(\sqrt{14}\) _____
Answer:
Since, \(\sqrt{13}\) ≈ 3.61 < 3.7 < \(\sqrt{14}\) ≈ 3.74
3.7 is a good example of a real. number that is in between \(\sqrt{13}\) and \(\sqrt{14}\)

3.7

Question 19.
an irrational number between 5 and 7 _________
Answer:
Square the given numbers to find their perfect squares and then a square root of any number in between those two is an example of an irrational number between 5 and 7:
5² = 25
7² = 49
So, one example is \(\sqrt{31}\)
\(\sqrt{31}\)

Question 20.
A teacher asks his students to write the numbers shown from least to greatest. Paul thinks the numbers are already in order. Sandra thinks the order should be reversed. Who is right?

Answer:
Approximate every given number in decimal form:
Use dosest perfect squares to approximate
100 < 115 < 121
\(\sqrt{100}\) < \(\sqrt{115}\) < \(\sqrt{121}\)
10 < \(\sqrt{115}\) < 11
A good approximation for \(\sqrt{115}\) is 10.7 since 10.7² = 114.5
\(\frac{115}{11}\) ≈ 10.45
Finally, that gives us the approximated numbers that the teacher gave them in the task: 10.7, 10.45, 10.5624
We can conclude that neither is right because they are not in any order

Neither are right The numbers aren’t in any order.

Question 21.
Math History There is a famous irrational number called Euler’s number, often symbolized with an e. Like π, it never seems to end. The first few digits of e are 2.7182818284.

a. Between which two square roots of integers could you find this number?
Answer:
2.718281828
The square of e lies between 7 and 8
2.72² = 7.3984
Hence, it lies between $\sqrt{7} =
2.65$ and $\sqrt{8} = 2.82$

b. Between which two square roots of integers can you find π?
Answer:
The square of π lies between 9 and 10
3.142
3.14² = 9.8596
Hence, it lies between $\sqrt{9} = 3$
and $\sqrt{1ø} = 3.16$

Texas Go Math Grade 8 Lesson 1.3 H.O.T. Focus On Higher Order Thinking Answer Key 

Question 22.
Analyze Relationships There are several approximations used for π, including 3.14 and \(\frac{22}{7}\).π is approximately 3.14159265358979…

a. Label π and the two approximations on the number line.

Answer:
a) We plot on number line π, 3.14 and \(\frac{22}{7}\):

b. Which of the two approximations is a better estimate for π? Explain.
Answer:
As we can see from the number line, \(\frac{22}{7}\) is closer to π, so we can conclude that \(\frac{22}{7}\) is a better estimation for π.

c. Find a whole number x in \(\frac{x}{113}\) so that the ratio is a better estimate for π than the two given approximations. ______
Answer:
\(\frac{355}{113}\) is a better estimation for π, because
\(\frac{355}{113}\) = 3.14159292035 ≈ 3.14159265358979 = π

Go Math 8th Grade Lesson 1.3 Compare and Order Real Numbers Question 23.
Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line, what is the fewest number of distinct points that need to be graphed? Explain.
Answer:
The fewest number of distinct points that need to be graphed is 2. Because there are both rational and irrational
numbers in the set and every rational number is not irrational and every irrational is not rational they can’t all be the same.
Therefore it could be that there are some numbers that repeat but there are at least two different numbers.

Two. Click for proof.

Question 24.
Critique Reasoning Jill says that \(12 . \overline{6}\) is less than 12.63. Explain her error.
Answer:
The line over the digit 6 means that it is repeating forever. So,
\(12 . \overline{6}\) = 12.666666….
If we round that number to two decimal Places we have \(12 . \overline{6}\) ≈ 12.67.
Obviously, 12.67 > 12.63

12.67 > 12.63. Click for proof.

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 3.4 Answer Key Direct Variation.

Texas Go Math Grade 8 Lesson 3.4 Answer Key Direct Variation

Essential Question
How can you solve problems involving direct variation?

Your Turn

Question 1.
The table shows the widths and lengths of United States flags. Determine whether the data set shows direct variation. If so, write an equation that describes the relationship.

Answer:

Reflect

Go Math Lesson 3.4 8th Grade Direct Variation Question 2.
Does the equation y = \(\frac{3}{2}\)x – 5 show a direct variation? Why or why not?
Answer:

Your Turn

Question 3.
One brand of motorcycle uses an oil-to-gasoline ratio as shown in the graph. The amount of oil that should be added varies directly with the amount of gasoline. Write a direct variation equation that describes the relationship. Use your equation to determine the amount of oil that should be added to 6.5 gallons of gasoline.

Answer:

Texas Go Math Grade 8 Lesson 3.4 Guided Practice Answer Key 

Determine whether the data sets show direct variation. If so, write an equation that describes the relationship. (Example 1)

Question 1.

Answer:

Question 2.

Answer:

Question 3.

Answer:

Go Math Grade 8 Lesson 3.4 Answer Key Question 4.

Answer:

Question 5.
The number of cups in a measure varies directly as the number of tablespoons. Write a direct variation equation that describes the relationship. Use your equation to determine the number of cups in 56 tablespoons. (Example 2)

Answer:

Question 6.
The number of calories varies directly with the number of grams of protein. Write a direct variation equation that describes the relationship. Use your equation to determine the number of calories from 25 grams of protein. (Example 2)

Answer:

Essential Question Check-In

Question 7.
How can you solve problems involving direct variation?
Answer:

Texas Go Math Grade 8 Lesson 3.4 Independent Practice Answer Key 

Question 8.
Vocabulary A _______ is a relationship that can written be as y = kx, where k is a ____.
Answer:

Direct Variation 8th Grade Math Lesson 3.4 Answer Key Question 9.
Which equation does NOT represent a direct variation?
(A) y = \(\frac{1}{4}\)
(B) y = -4x
(C) y = 5x + 1
(D) y = 6x
Answer:

Question 10.
Environment Mischa bought an energy-efficient washing machine. The amount of water she saves per wash load compared to her old washer is shown in the table.

a. Determine whether the relationship is a direct variation. If so, write an equation that describes the relationship.
Answer:

b. How much water will she save when washing 8 loads?
Answer:

Question 11.
Sandy wants to build a square garden. Complete the table for the different side lengths.

a. Does the perimeter of a square vary directly with the side length? If so, write an equation that describes the relationship. Explain your answer.
Answer:

b. Does the area of a square vary directly with the side length? Why or why not?
Answer:

c. Sandy decides to build her garden with a side length of 3 feet. The border she buys for the perimeter costs $1.99 for a 1.5-foot piece. The soil she buys covers an area of 3 square feet and costs $4.99 a bag. How much does Sandy spend on border and soil for her garden? Explain.
Answer:

Direct Variation Worksheet Answer Key Question 12.
The three-toed sloth is an extremely slow animal. Use the graph to write a direct variation equation for the distance y a sloth will travel in x minutes. How long will it take the sloth to travel 24 feet?

Answer:

Texas Go Math Grade 8 Lesson 3.4 H.O.T. Focus On Higher Order Thinking Answer Key

Question 13.
Critique Reasoning Martin is told that a graph includes the points (2, 5) and (4, 10). He says that this is the graph of the direct variation y = 2.5x. Do you agree? Explain.
Answer:

Texas Go Math Grade 8 Pdf Download Lesson 3.4 Question 14.
Make a Conjecture If you can write a direct variation equation that relates y to x,y = kx, then you can write a direct variation equation that relates x to y, x = k₁y. Make a conjecture about how the constants of variation are related. Use the table to help you decide.

Answer:

Question 15.
Analyze Relationships One graph of a direct variation equation goes through point A in Quadrant I which is not at the origin. A second graph of a different direct variation equation goes through a point that is one unit to the right of A. Which direct variation equation has a greater constant of variation? Explain.
Answer:

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 3 Answer Key Proportional Relationships.

Texas Go Math Grade 8 Module 3 Answer Key Proportional Relationships

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

Texas Go Math Grade 8 Module 3 Are You Ready? Answer Key

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

Write each fraction as decimal.

Question 1.
\(\frac{3}{8}\)
Answer:
To express \(\frac{3}{8}\) as a decimal, we write the fraction as a division problem.

Therefore, \(\frac{3}{8}\) = 0.375

Go Math Grade 8 Module 3 Proportional Relationships Module Answer Key Question 2.
\(\frac{0.3}{0.4}\)
Answer:
First, multiply the numerator and the denominator by 10 so that the denominator is a whole number.
\(\frac{0.3 \times 10}{0.4 \times 10}\) = \(\frac{3}{4}\)
Now write the fraction as a division problem, place a decimal point in the quotient, and divide as whole numbers:

0.75

Question 3.
\(\frac{0.13}{0.2}\)
Answer:
First multiply the numerator and the denominator by 10 so that the denominator is a whole number.
\(\frac{0.13 \times 10}{0.2 \times 10}\) = \(\frac{1.3}{2}\)
Now write the fraction as a division problem. place a decimal point in the quotient and divide as whole numbers:

0.65

Question 4.
\(\frac{0.39}{0.75}\)
Answer:
First, multiply the numerator and the denominator by 100 so that the denominator is a whole number.
\(\frac{0.39 \times 100}{0.75 \times 100}\) = \(\frac{39}{75}\)
Now write the fraction as a division problem, place a decimal point in the quotient and divide as whole numbers:

0.52

Question 5.
\(\frac{4}{5}\)
Answer:
Write the fraction as a division problem, place a decimal point in the quotient and divide as whole numbers

\(\frac{4}{5}\) = 0.8

Question 6.
\(\frac{0.1}{2}\)
Answer:
First, we multiply the numerator and the denominator by a power of 10 so that we get whole numbers.
\(\frac{0.1}{2}\) = \(\frac{0.1 \cdot 10}{2 \cdot 10}\) = \(\frac{1}{20}\)
To express \(\frac{1}{20}\) as a decimal, we write the fraction as a division problem.

Therefore,
\(\frac{0.1}{2}\) = 0.05

Grade 8 Math Module 3 Answer Key Question 7.
\(\frac{3.5}{14}\)
Answer:
First, we multiply the numerator and the denominator by a power of 10 so that we get whole numbers.
\(\frac{3.5}{14}\) = \(\frac{3.5 \cdot 10}{14 \cdot 10}\) = \(\frac{35}{140}\)
To express \(\frac{35}{140}\) as a decimal, we write the fraction as a division problem.

Therefore, \(\frac{3.5}{14}\) = 0.25

Question 8.
\(\frac{7}{14}\)
Answer:
To express \(\frac{7}{14}\) as a decimal, we write the fraction as a division problem.

\(\frac{7}{14}\) = 0.5

Go Math Grade 8 Module 3 Answer Key Question 9.
\(\frac{0.3}{10}\)
Answer:
First, we multiply the numerator and the denominator by a power of 10 so that we get whole numbers.
\(\frac{0.3}{10}\) = \(\frac{0.3 \cdot 10}{10 \cdot 10}\) = \(\frac{3}{100}\)
To express \(\frac{3}{100}\) as a decimal, we write the fraction as a division problem.

\(\frac{0.3}{10}\) = 0.03

Solve each proportion for x.

Question 10.
\(\frac{20}{18}\) = \(\frac{10}{x}\) ______
Answer:
\(\frac{20}{18}\) = \(\frac{10}{x}\) Given
\(\frac{20 \div 2}{18 \div 2}\) = \(\frac{10}{x}\) Divide 20 ÷ 2 = 10, so divide the numerator and denominator by 2
\(\frac{10}{9}\) = \(\frac{10}{x}\)
x = 9 compare

Grade 8 Module 3 Proportional Relationships Question 11.
\(\frac{x}{12}\) = \(\frac{30}{72}\) ______
Answer:
\(\frac{x}{12}\) = \(\frac{30}{72}\) Given
\(\frac{x}{12}\) = \(\frac{30 \div 6}{72 \div 6}\) Divide 72 ÷ 6 = 12, so divide the numerator and denominator by 6.
\(\frac{x}{12}\) = \(\frac{5}{12}\)
x = 5 compare

Question 12.
\(\frac{x}{4}\) = \(\frac{4}{16}\) ______
Answer:
\(\frac{x}{4}\) = \(\frac{4}{16}\) Given
\(\frac{x}{4}\) = \(\frac{4 \div 4}{16 \div 4}\) Divide 16 ÷ 4 = 4, so divide the numerator and denominator by 4.
\(\frac{x}{4}\) = \(\frac{1}{4}\)
x = 1 compare

Question 13.
\(\frac{11}{x}\) = \(\frac{132}{120}\) ______
Answer:
\(\frac{11}{x}\) = \(\frac{132}{120}\) Given
\(\frac{11}{x}\) = \(\frac{132 \div 12}{120 \div 12}\) Divide 132 ÷ 12 = 11, so divide the numerator and denominator by 12.
\(\frac{11}{x}\) = \(\frac{11}{10}\)
x = 10 compare

Question 14.
\(\frac{36}{48}\) = \(\frac{x}{4}\) ______
Answer:
\(\frac{36}{48}\) = \(\frac{x}{4}\) Given
\(\frac{36 \div 12}{48 \div 12}\) = \(\frac{x}{4}\) Divide 48 ÷ 12 = 4, so divide the numerator and denominator by 12.
\(\frac{3}{4}\) = \(\frac{x}{4}\)
x = 3 compare

Go Math Grade 4 Module 3 Answer Key Question 15.
\(\frac{x}{9}\) = \(\frac{21}{27}\) ______
Answer:
\(\frac{x}{9}\) = \(\frac{21}{27}\) Given
\(\frac{x}{9}\) = \(\frac{21 \div 3}{27 \div 3}\) Divide 27 ÷ 3 = 9, so divide the numerator and denominator by 12.
\(\frac{x}{9}\) = \(\frac{7}{9}\)
x = 7 Compare
x = 7

Question 16.
\(\frac{24}{16}\) = \(\frac{x}{2}\) ______
Answer:
\(\frac{24}{16}\) = \(\frac{x}{2}\) Given
\(\frac{24 \div 8}{16 \div 8}\) = \(\frac{x}{2}\) Divide 16 ÷ 8 = 2, so divide the numerator and denominator by 8.
\(\frac{3}{2}\) = \(\frac{x}{2}\)
x = 3 Compare
x = 3

Question 17.
\(\frac{30}{15}\) = \(\frac{6}{x}\) ______
Answer:
\(\frac{30}{15}\) = \(\frac{6}{x}\) Given
\(\frac{30 \div 5}{15 \div 5}\) = \(\frac{6}{x}\) Divide 30 ÷ 5 = 6, so divide the numerator and denominator by 5.
\(\frac{6}{3}\) = \(\frac{6}{x}\)
x = 3 Compare
x = 3

Module 3 Answer Key Grade 8 Answer Key Question 18.
\(\frac{3}{x}\) = \(\frac{18}{36}\) ______
Answer:
\(\frac{3}{x}\) = \(\frac{18}{36}\)
\(\frac{3}{x}\) = \(\frac{18 \div 6}{36 \div 6}\) Divide 18 ÷ 6 = 3, so divide the numerator and denominator by 6.
\(\frac{3}{x}\) = \(\frac{3}{6}\)
x = 6

Texas Go Math Grade 8 Module 3 Reading Start-Up Answer Key

Visualize Vocabulary
Use the ✓ words to complete the diagram.

Understand Vocabulary

Match the term on the left to the definition on the right.

Answer:
1. (B) A unit rate is B. A rate in which the second quantity in the comparison is one unit.
2. (A) Constant of proportionality is A. A constant ratio of two variables is related proportionally.
3. (C) A proportional relationship is C. A relationship between two quantities in which the ratio of one quantity to the other quantity is constant.

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 1.2 Answer Key Sets of Real Numbers.

Texas Go Math Grade 8 Lesson 1.2 Answer Key Sets of Real Numbers

Essential Question
How can you describe relationships between sets of real numbers?

Your Turn

Write all names that apply to each number.

Question 1.
A baseball pitcher has pitched 12\(\frac{2}{3}\) innings.

Answer:
12\(\frac{2}{3}\) = \(\frac{38}{3}\) is a fraction, so it is rational and real.

Rational, real

Question 2.
The length of the side of a square that has an area of 10 square yards. __________________
Answer:
10 is a whole number, integer, real and a rational number
Length of the side of the given square is \(\sqrt{10}\) and it is an irrational (5 is a whole number that is not a perfect square), real number.

10 – integer, whole, real, rational
\(\sqrt{10}\) – irrational, real

Your Turn

Tell whether the given statement is true or false. Explain your choice.

Question 3.
All rational numbers are integers.
Answer:
All rational numbers are integers

False. Every integer is a rational number but every rational number is not an integer Ration numbers such as \(\frac{3}{8}\) and –\(\frac{7}{2}\) are not integers.

Texas Go Math Grade 8 Pdf Lesson 1.2 Real Numbers Answer Key Question 4.
Some irrational numbers are integers.
Answer:
False, every integer is a rational number (Integers are a subset of rational numbers), but no rational number ¡s an
irrational number and no irrational number is a rational number. So no irrational number is an integer.

False. Click for explanation

Your Turn

Identify the set of numbers that best describes the situation. Explain your choice.

Question 5.
the amount of water in a glass as it evaporates
Answer:
The set of real, numbers best describes the situation since the number of water in a glass can be any number greater or even to 0.

Real numbers.

Question 6.
the number of seconds remaining when a song is playing, displayed as a negative number
Answer:
Set of integers best describes the situation since seconds can’t be a fraction or an irrational number.

Integers.

Texas Go Math Grade 8 Lesson 1.2 Guided Practice Answer Key 

Write all names that apply to each number. (Example 1)

Question 1.
\(\frac{7}{8}\)
Answer:
\(\frac{7}{8}\)

Rational Number, Real

Question 2.
\(\sqrt{36}\)
Answer:
\(\sqrt{36}\) = 6
Natural Number, Rational, whole number, integer, real

Question 3.
\(\sqrt{24}\)
Answer:
\(\sqrt{24}\) is an irrational, real number

Irrational, real

Question 4.
0.75
Answer:
Since 0.75 can be expressed as \(\frac{3}{4}\) it is a rational, real number but it is not an integer.

Rational, real

Question 5.
0
Answer:
Rational, Whole number, Integer, real

Go Math Book Grade 8 Answer Key Lesson 1.2 Answer Key Question 6.
–\(\sqrt{100}\)
Answer:
–\(\sqrt{100}\) = -10
Rational, integer, real

Question 7.
\(5 . \overline{45}\)
Answer:
Rational Numbers, real number

Question 8.
\(-\frac{18}{6}\)
Answer:
\(-\frac{18}{6}\) = -3
Rational, Integer, real

Tell whether the given statement is true or false. Explain your choice. (Example 2)

Question 9.
All whole numbers are rational numbers.
Answer:
True

Explanation:
The statement is true. Whole numbers are a subset of integers, which are a subset of rational numbers. It follows
that any whole number is also a rational number.
Also notice that you can write every whole number as a rational one, by expressing it as a fraction with “1” for it
denominator.

Go Math Grade 8 Lesson 1.2 Answer Key Question 10.
No irrational numbers are whole numbers.
Answer:
True

Explanation:
No irrational numbers are whole numbers

True Whole numbers are ration numbers.

Identify the set of numbers that best describes each situation. Explain your choice. (Example 3)

Question 11.
the change ¡n the value of an account when given to the nearest dollar
Answer:
Integers.
The change can be a whole dollar amount and can be positive, negative or zero

Question 12.
the markings on a standard ruler

Answer:
Rational
The ruler is marked every \(\frac{1}{16}\)th inch.

Essential Question Check-In

Question 13.
What are some ways to describe the relationships between sets of numbers?
Answer:
There are two ways that we have been using until now to describe the relationships between sets of numbers:

• Using as scheme or a diagram like the one on page 15.
• Verbal description, for example, ‘All irrational numbers are real numbers.”

Texas Go Math Grade 8 Lesson 1.2 Independent Practice Answer Key

Write all names that apply to each number. Then place the numbers in the correct location on the Venn diagram.

Question 14.
\(\sqrt{9}\) _____
Answer:
Since \(\sqrt{9}\) = 3, it is a whole number, integer, rational and real number

It should be placed inside the area for whole numbers.

Whole, integer, rational and real number. Place inside the area for whole numbers.

Lesson 1.2 Go Math 8th Grade Question 15.
257 _____
Answer:
257
whole, integer, rational, real

Question 16.
\(\sqrt{50}\) _____
Answer:
\(\sqrt{50}\)
irrational, real

Question 17.
8\(\frac{1}{2}\) _____
Answer:
8\(\frac{1}{2}\) = \(\frac{17}{2}\) is a fraction so it is a rational number and a real number, but it is not an integer nor whole.

It should be placed inside the area for rational numbers but outside the area for integers.

Rational, real. Place inside area for rational numbers and outside of integers.

Question 18.
16.6 ______
Answer:
The decimal number 16.6 means “166 tenths” so it can be expressed as a fraction: \(\frac{166}{10}\). Therefore it is a rational, real number.

It should be placed in the area for rational numbers but outside of the area for integers.

Rational, real. Place in the area for rational numbers but outside of the area for integers.

Question 19.
\(\sqrt{16}\) _____
Answer:
\(\sqrt{16}\) = 4
Rational, whole, integers, real

Identify the set of numbers that best describes each situation. Explain your choice.

Go Math 8 Grade Answer Key Lesson 1.2 Answer Key Question 20.
the height of an airplane as it descends to an airport runway
Answer:
Set of real numbers best describes the given situation because the height of the airplane is a real number larger
than or equal to 0. The plane doesn’t jump from one height to the other, for example from 100 to 99, rather it descends gradually hitting every height on its way down.

Real numbers.

Question 21.
the score with respect to par of several golfers: 2, -3, 5, 0, -1
Answer:
Integers. The scores are counting numbers, their opposites and zero.

Question 22.
Critique Reasoning Ronald states that the number \(\frac{1}{11}\) is not rational because, when converted into a decimal, it does not terminate. Nathaniel says it is rational because it is a fraction. Which boy is correct? Explain.
Answer:
Nathaniel is correct, if a number can be expressed as a ratio in the form \(\frac{a}{b}\) where a and b are integers and b is not 0, it is a rational number.
Obviously in this case a = 1 and b = 11, so both are integers and b ≠ 0

Nathaniel is correct.

Question 23.
Critique Reasoning The circumference of a circular region is shown. What type of number best describes the diameter of the circle? Explain your answer.

Answer:
The formula for the circumference of a circular region is C = 2rπ where r is the radius of the circle.
C = 2rπ
π = 2rπ Substitute C = π
1 = 2r Divide both sides of the equation with π

Since the diameter of the circle is twice the length of its radius: d = 2r we get d = 1. Therefore, the type of
number that best describes the diameter of the circle is a whole number

Whole number

Question 24.
Critical Thinking A number is not an integer. What type of number can it be?
Answer:
Since it is not an integer it can not be a whole number because every whole number is also an integer. It can be
any other type of number so: rational or irrational real number

Rational or irrational real number.

Simple Solutions Math Grade 8 Answer Key Pdf Lesson 1 Question 25.
A grocery store has a shelf with half-gallon containers of milk. What type of number best represents the total number of gallons?
Answer:
\(\frac{1}{2}\) gallons
Rational number

Texas Go Math Grade 8 Lesson 1.2 H.O.T. Focus On Higher Order Thinking Answer Key

Question 26.
Explain the Error Katie said, “Negative numbers are integers.” What was her error?
Answer:
Her error is that not alt negative numbers are integers, for example, the number –\(\frac{1}{2}\) is obviously a negative number but it is not an integer.

Not all negative numbers are Integers.

Question 27.
Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational? Explain.
Answer:
Yes, in some cases you can use a calculator to tell if the number is rational or not
For example, if we wonder if the number 0.5 is rational we can just type 0.5 in the calculator and it will give us a
result \(\frac{1}{2}\)”. Since we have a representation of the number 0.5 as a ratio of two integers we can conclude that it is in fact a rational number

This method doesn’t always work since calculators show a limited number of digits and sometimes that is not
enough to tell it there is a repeating decimal or not.

Yes, but not always.

Practice and Homework Lesson 1.2 Answer Key 8th Grade Question 28.
Draw Conclusions The decimal \(0 . \overline{3}\) represents \(\frac{1}{3}\). What type of number best describes \(0 . \overline{9}\), which is 3 ∙ \(0 . \overline{3}\)? Explain.
Answer:
Let
x = \(0 . \overline{9}\)
10x = 10 × \(0 . \overline{9}\). Since \(0 . \overline{9}\) has one repeating decimal, multiply both sides with 10
10x = \(9 . \overline{9}\). Subtract x = \(0 . \overline{9}\) from both sides
9x = 9
x = 1
Since x = 1 the type of number that best describes \(0 . \overline{9}\) is a whole number.

A whole number.

Question 29.
Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form. Why is this?
Answer:
We say that a number is irrational if it not rational, so if it can’t be expressed as a ratio \(\frac{a}{b}\) where a and b are integers and b is not 0.

The problem why irrational numbers can’t be precisely represented is that unlike some rational numbers that
have digits repeating forever, for example \(\frac{1}{3}\) = \(0 . \overline{3}\), so the digit 3 goes on forever, irrational numbers don’t have a ‘rule” that we can observe. This means that to know what digit is on any place behind the decimal point we have to calculate it (where in the previous example we know that if we express the number \(\frac{1}{3}\) as a decimal we can say with certainty exactly what digit is in what place, even if in this case it is very simple because all digits are 3)
For example \(\sqrt{2}\) ≈ 1.414213562…

There is no pattern/rule for the repeating decimals. Click for an explanation.

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 2 Answer Key Scientific Notation.

Texas Go Math Grade 8 Module 2 Answer Key Scientific Notation

Essential Question
How can you use scientific notation to solve real-world problems?

Texas Go Math Grade 8 Module 2 Are You Ready? Answer Key

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

Write each exponential expression as a simplified number.

Question 1.
10² _______
Answer:
10² Given
10 * 10 = 100.00 Evaluate

Texas Go Math Grade 8 Module 2 Answer Key Question 2.
10³ _______
Answer:
10³ Given
10 * 10 * 10 = 1,000.00 Evaluate

Question 3.
10⁵ _______
Answer:
10⁵ Given
10 * 10 * 10 * 10 * 10 = 100,000 Evaluate

Question 4.
10⁷ _______
Answer:
10⁷ Given
10 * 10 * 10 * 10 * 10 * 10 * 10 = 10,000,000.00 Evaluate

Find each product or quotient.

Question 5.
45.3 × 10³
Answer:
45.3 * 10³ Identify the number of zeros in the power of 10
45.3 * 1,000 When multiplying, move the decimal point to the right the same number of places as the number of zeros
45,300

Question 6.
7.08 ÷ 10²
Answer:
7.08 ÷ 10² Identify the number of zeros in the power of 10
7.08 ÷ 100 When dividing, move the decimal point to the left the same number of places as the number of zeros.
0.0708

Texas Go Math 8th Grade Module 2 Answer Key Question 7.
0.00235 × 10⁶
Answer:
0.00235 × 10⁶ Identify the number of zeros in the power of 10

0.00235 × 1,000, 000 When multiplying, move the decimal point to the right the same number of places as the number of zeros
2,350

Question 8.
3,600 ÷ 10⁴
Answer:
3600 ÷ 10⁴ Identify the number of zeros in the power of 10

3600 ÷ 10000 When dividing, move the decimal point to the left the same number of places as the number of zeros.
0.36

Question 9.
0.5 × 10²
Answer:
0.5 × 10² Identify the number of zeros in the power of 10
0.5 × 100 When multiplying, move the decimal point to the right the same number of places as the number of zeros
50

Question 10.
67.7 ÷ 10⁵
Answer:
67.7 ÷ 10⁵ Identify the number of zeros in the power of 10
67.7 ÷ 100000 When dividing, move the decimal point to the left the same number of places as the number of zeros.
0.000677

Texas Go Math Grade 8 Module 2 Question 11.
0.0057 × 10⁴
Answer:
0.0057 × 10⁴
Identify the number of zeros in the power of 10
0.0057 × 10,000 When multiplying, move the decimal point to the right the same number of places as the number of zeros
57

Question 12.
195 ÷ 10⁶
Answer:
195 ÷ 10⁶ Identify the number of zeros in the power of 10
195 ÷ 1000000 When dividing, move the decimal point to the left the same number of places as the number of zeros.
0.000195

Texas Go Math Grade 8 Module 2 Reading Start-Up Answer Key

Visualize Vocabulary

Use the ✓ words to complete the Venn diagram. You can put more than one word in each section of the diagram.

Understand Vocabulary

Complete the sentences using the preview words.

Question 1.
A number produced by raising a base to an exponent is a _______
Answer:
A number produced by raising a base to an exponent is a power.

Texas Go Math Grade 8 Module 2 Pdf Question 2.
____ is a method of writing very large or very small numbers by using powers of 10.
Answer:
Scientific notation is a method of writing very large or very small numbers by using powers of 10.

Question 3.
A ____ is any number that can be expressed as a ratio of two integers.
Answer:
A rational number is any number that can be expressed as a ratio of two integers.

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 2.1 Answer Key Scientific Notation with Positive Powers of 10.

Texas Go Math Grade 8 Lesson 2.1 Answer Key Scientific Notation with Positive Powers of 10

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

Using Scientific Notation
Scientific notation is a method of expressing very large and very small numbers as a product of a number greater than or equal to 1 and less than 10, and a power of 10.

The weights of various sea creatures are shown in the table. Write the weight of the blue whale in scientific notation.

A. Move the decimal point in 250,000 to the left 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? ____

B. Divide 250,000 by your answer to A . Write your answer as a power of 10.

C. Combine your answers to
A and B to represent 250,000.
Repeat steps A through C towrite the weight of the whale shark in scientific notation.

Reflect

Question 1.
How many places to the left did you move the decimal point to write 41,200 in scientific notation? ___________________________________
Answer:
We need to move the decimal point 4 places to the left to write 42,200 in scientific notation.
41,200 = 41,200.0 = 4.1200 × 10⁴ = 4.12 × 10⁴

Four places to the left.

Go Math Grade 8 Lesson 2.1 Answer Key Question 2.
What is the exponent on 10 when you write 41,200 in scientific notation?
Answer:
To write 41,200 in scientific notation, we move the decimal point 4 places to the right and we multiply by 10⁴.
41,200 = 4.12 • 10⁴

4 places to the left

Your Turn

Write each number in scientific notation.

Question 3.
6,400
Answer:
6,400 Given
6.400 Move the decimal point 3 places to the left.
6.4 Remove extra zeros
1000 Divide the original number by the result from above
10³ Write the answer as power of 10.
6 × 10³ Write the product of the results from above
6 × 10³

Question 4.
570,000,000,000
Answer:
570,000,000,000 Given
5.70000000000 Move the decimal point $. Remove extra zeros.
100, 000, 000, 000 Divide the original number by the result from above.
10¹¹ Write the answer as the power of 10.
5.7 × 10¹¹ Write the product of the results from above
5.7 × 10¹¹

Question 5.
A light-year is the distance that light travels in a year and is equivalent to 9,461,000,000,000 km. Write this distance in scientific notation.
Answer:
9,461,000, 000,000 Given
9,461000000000 Move the decimal point 11 places to the left.
9.461 Remove extra zeros.

1, 000, 000, 000, 000 Divide the original number by the result from above.
10¹² Write the answer as the power of 10.
9.461 × 10¹² km Write the product of the results from above
9.461 × 10¹² km

Lesson 2.1 Extra Practice Powers and Exponents Answer Key Question 6.
3.5 × 10⁶ means that the decimal should be moved 6 decimals. Placeholder zeros are added as necessary. Moving one decimal gives 35 and the remaining five decimals are represented by placeholder zeros.

Question 7.
To express a number in scientific notation we follow some steps:

1. We move the decimal point until we get a number that is greater than or equal to 1 and less than 10.
2. We divide the original number with the one we got from Step 1.
3. We write the product of the results from Step I and Step 2.
   In our case. the given number is already greater than 1 and less than 10. So it can be written as:
   5.3 = 5.3 10⁰
   The exponent on 10 is 0.

The exponent on 10 is 0

Question 8.
7.034 × 10⁹ Given

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

7,034 000000 Place the decimal point Since you are going to write a number greater than 7.035, move the decimal point to the right. Add placeholder zeros if necessary. The number
7.034 × 10⁹ in standard form is 7, 034,000,000

Question 9.
2.36 × 10⁵ Given
5 places Use the exponent of the power of 10 to determine the number of places to move the decimal point

236,000 Place the decimal point Since you are going to write a number greater than 2.36, move the decimal point to the right Add placeholder zeros if necessary.
The number 2.36 × 10⁵ in standard form is 236,000

Lesson 2.1 Extra Practice Scientific Notation Answer Key Question 10.
5 × 10⁶ Given
6 places Use the exponent of the power of 10 to determine the number of places to move the decimal point
5, 000, 000 Place the decimal point Since you are going to write a number greater than 5, move the decimal point to the right. Add placeholder zeros if necessary.
The number 5 × 10⁶ grams in standard form is 5, 000, 000 grams

Texas Go Math Grade 8 Lesson 2.1 Guided Practice Answer Key 

Question 1.
58,927 Given

5.8927 Move the decimal point 4 places to the Left.

10,000 Divide the original number by the result from above.
10⁴ Write the answer as the power of 10.
5.8927 × 10⁴ Write the product of the results from above
5.8927 × 10⁴

Question 2.
1, 304, 000,000 Given
1.304000000 Move me decimal point 9 places to the left.
1.304 Remove extra zeros.

1,000,000 Divide the original number by the result from above.
10⁹ Write the answer as power of 10.
1.304 × 10⁹ Write the product of the results from above
1.304 × 10⁹

Question 3.
6,730,000 Given

6.730000 Move the decimal point 6 places to the left.
6.73 Remove extra zeros.
1,000,000 Divide the original number by the result from above.
10⁶ Write the answer as power of 10.
6.73 × 10⁶ Write the product of the results from above
6.73 × 10⁶

Question 4.
13 300 Given (1)
1.3300 Move the decimal point L places to the left (2)
1.33 Remove extra zeroes (3)
10000 = 10⁴ Divide the original number by 1.33 (4)
1.33 × 10⁴ Multiply numbers from above

1.33 × 10⁴

Question 5.
97, 700, 000, 000, 000, 000, 000, 000 Given

9.7700000000000000000000 Move the decimal point 22 places to the left
9.77 Remove extra zeros.
10,000, 000,000,000, 000,000,000 Divide the original number by the result from above.
10²² Write the answer as the power of 10.
9.77 × 10²² Write the product of the results from above

Scientific Notation with Positive Powers of 10 Answer Key Question 6.
384,000 Given
3.84000 Move the decimal point 5 places to the left
3.84 Remove extra zeros.
100,000 Divide the original number by the result from above.
10⁵ Write the answer as power of 10.

3.84 × 10⁵ Write the product of the results from above
3.84 × 10⁵

Question 7.
4 × 10⁵ Given
5 places Use exponent of the power of 10 to determine the number places to move the decimal point
400,000 Place the decimal point Since you are going to write a number greater than 4, move decimal point to the right. Add placeholder zeros if necessary.
The number 4 × 10⁵ in standard form is 400, 000

Question 8.
1.8499 × 10⁹ Given

9 places use exponent of the power of 10 to determine the number places to move the decimal point
1,849,900,000 Place the decimal point Since you are going to write a number greater than 1.8499, move decimal point to the right. Add placeholder zeros if necessary.
The number 1.8499 × 10⁹ in standard form is 1, 849, 900, 000

Question 9.
6.41 × 10³ Given
3 places use exponent of the power of 10 to determine the number places to move the decimal point
6,410 Place the decimal point Since you are going to write a number greater than 6.41, move decimal point to the right. Add placeholder zeros if necessary.
The number 6.41 × 10³ in standard form is 6, 410

Question 10.
8.456 × 10⁷ Given
7 places use exponent of the power of 10 to determine the number places to move the decimal point
84, 560, 000 Place the decimal point Since you are going to write a number greater than 8.456, move decimal point to the right. Add placeholder zeros if necessary.
The number
8.456 × 10⁷ in standard form is 84, 560, 000

Question 11.
8 × 10⁵ Given
5 places use exponent of the power of 10 to determine the number places to move the decimal point
800,000 Place the decimal point Since you are going to write a number greater than 8, move decimal point to the right. Add placeholder zeros if necessary.
The number
8 × 10⁵ in standard form is 800, 000

Question 12.
9 × 10¹⁰ Given
10 places use exponent of the power of 10 to determine the number places to move the decimal point
90, 000, 000, 000 Place the decimal point Since you are going to write a number greater than 8, move decimal point to the right. Add placeholder zeros if necessary.
The number 9 × 10¹⁰ in standard form is 90, 000, 000, 000

Question 13.
5.4 × 10⁴ Given
4 places use exponent of the power of 10 to determine the number places to move the decimal point
54, 000 Place the decimal point Since you are going to write a number greater than 5.4, move decimal point to the right. Add placeholder zeros if necessary.
The time in standard form is 54, 000 seconds

Go Math Grade 8 Lesson 2.1 Answer Key Question 14.
7.6 × 10⁶ Given
6 places use exponent of the power of 10 to determine the number places to move the decimal point
7, 600, 000 Place the decimal point Since you are going to write a number greater than 7.6, move decimal point to the right. Add placeholder zeros if necessary.
The time in standard form is 7, 600, 000 cans

Question 15.
3,482,000.000 Given (1)
3.482000000 Move the decimal point 9 places to the left. (2)
3.482 Remove extra zeroes (3)
1000000000 = 10⁹ Divide the original number by 3.482 (4)
3.482 × 10⁹ Multiply numbers from steps (3) and (4) (5)

Rewrite the number as a decimal that has a whole between 1 and 10 (not including 10) multiplied by 10⁹ where 9 is the number of places you moved the decimal. Click for details!

Texas Go Math Grade 8 Lesson 2.1 Independent Practice Answer Key 

Paleontology Use the table for problems 16-21. Write the estimated weight of each dinosaur in scientific notation.

Question 16.
Apatosaurus
Answer:

6.6 × 10⁴

Question 17.
Argentinosaurus _____
Answer:
The weight of Argentinosaurus is 220, 000 pounds. Write that number in a scientific notation.

2.2 × 10⁵

Question 18.
Brachiosaurus _____
Answer:
The weight of Brachiosaurus is 100, 000 pounds. Write that number in a scientific notation.

1 × 10⁵

Go Math Grade 8 Lesson 2.1 Answer Key Question 19.
Camarasaurus ____
Answer:
The weight of Camarasaurus is 40, 000 pounds. Write that number in a scientific notation.

4 × 10⁴

Question 20.
Ceriosauriscus ________________________
Answer:
Estimated weight of Cetiosauriscus is 19,850 pounds. To express this number in scientific notation:
Move the decimal point 4 places to the left, so we get a number that is greater than or equal to 1 and less
than 10. We remove the extra zeros.
1.9800 = 1.98
Divide the original number by the result from above. Write the answer as a power of 10.
\(\frac{19,800}{1.98}\) = 10,000 = 10⁴
Write the product of the results from above.
19,800 = 1.98 • 10⁴

Question 21.
Diplodocus _____________________
Answer:
The weight of Diplodocus is 50,000 pounds. Write that number in a scientific notation.

5 × 10⁴

Question 22.
A single little brown bat can eat up to 1000 mosquitoes in a single hour. Express in scientific notation how many
mosquitoes a little brown bat might eat in 10.5 hours.
Answer:
Since a Little brown bat can eat up to 1, 000 mosquitoes in an hour it can eat 10.5 times more in 10.5 hours, SO:
10.5 × 1,000 = 10,500
Write 10, 500 in a scientific notation.

1.05 × 10⁴

Question 23.
Multistep Samuel can type nearly 40 words per minute. Use this information to find the number of hours it would take him to type 2.6 × 10⁵ words.
Answer:
To find the member of hours N, we need to divide the total number of words by typing speed (words per minute). We have:
N = \(\frac{2.6 \cdot 10^{5}}{40}\)
To simplify, we write 40 in a scientific form as 4 • 10¹:

To write \(\frac{2.6}{4}\) 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 needed.

Therefore,
N = 0.65 . 10⁴
N = 0.65 . 10³ minutes

To convert from minutes to hours, we divide the result by 60:
N = \(\frac{6.5 \cdot 10^{3}}{60}\)
To simplify, we write 60 in a scientific form as 6 . 10¹:

To write \(\frac{6.5}{6}\) 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.

108.3 hours = 108 hours 20 minutes

Lesson 2.1 Integer Exponents Answer Key Pdf Question 24.
Entomology A tropical species of mite named Archegozetes longisetosus is the record holder for the strongest insect in the world. It can lift up to 1.182 × 10³ times its own weight.

a. If you were as strong as this insect, explain how you could find how many pounds you could lift.
Answer:
Number of pounds you can lift by multiplying 1.182 × 10³ by your weight.
Since you are as strong as the ant which can lift up to 1.182 × 10³ its own weight.

b. Complete the calculation to find how much you could lift, in pounds, if you were as strong as an Archegozetes longisetosus mite. Express your answer in both scientific notation and standard notation.
Answer:
Number of pounds = 100 * 1.182 × 10³ Let weight = 100 pounds
10² * 1.182 × 10³, Simplify
1.182 × 10⁵ In scientific notation. Add the exponent of 10
118200 In standard notation. Move the decimal 5 places to the right. Add placeholder zeros if necessary.

Question 25.
During a discussion in science class, Sharon learns that at birth an elephant weighs around 230 pounds. In four herds of elephants tracked by conservationists, about 20 calves were born during the summer. In scientific notation, express approximately how much the calves weighed altogether.
Answer:
Total weight of calves = 230 * 20 = 4600 where each elephant calves weight 230 pounds and 20 calves are born

4.600 Move the decimal point 4 places to the left
Remove extra zeros.

1, 000 Divide the original number by the result from above
10³ Write the answer as the power of 10.
4.6 × 10³ Write the product of the results from above
The total weight is $4.6 \times
10^{3}$ pounds
4.6 × 10³ pounds

Question 26.
Classifying Numbers Which of the following numbers are written in scientific notation?
0.641 × 10³ 9.999 × 10⁴
2 × 10¹ 4.38 × 5¹⁰
Answer:
In order for a number to be written in scientific notation it must be of the format
c × 10³
where c is a decimal number greater or equal to 1 and less than 10 and n is an integer, let’s name that rule: Rule (1).
Using this rule we have to check which numbers are written in scientific notation:
0.641 × 10³ Not in scientific notation, correct would be 641 × 10²
9.999 × 10⁴ This is in scientific notation according to Rule (1)
2 × 10¹ This is in scientific notation according to Rule (1).
4.38 × 5¹⁰ Not in scientific notation because it is not in the correct form according to Rule (1)

9.999 × 10⁴ and 2 × 10¹ are written in scientific notation.

Question 27.
Explain the Error Polly’s parents’ car weighs about 3500 pounds. Samantha, Esther, and Polly each wrote the weight of the car in scientific notation. Polly wrote 35.0 × 10², Samantha wrote 0.35 × 10⁴, and Esther wrote 3.5 × 10⁴.

a. Which of these girls, if any, is correct?
Answer:
3500 pounds Given

None of the girls is correct Correct scientific notation: 3.5 × 10³

b. Explain the mistakes of those who got the question wrong.
Answer:
Potty did not express the number such first part is greater than or equal to 1 and less than 10
Scientific notation is a way of expressing numbers as a product of numbers greater than or equal to 1 and
less than 10 and 10
Samantha did not express the number such first part is greater than or equal to $1$ and less than $10$ Esther did not express the exponent of $10$ correctly

Question 28.
Justify Reasoning If you were a biologist counting very large numbers of cells as part of your research, give several reasons why you might prefer to record your cell counts in scientific notation instead of standard notation.
Answer:
Reason 1
It is a standard way of writing results in a scientific environment accepted and used by scientists all over the world.

Reason 2
It is easier to compare large numbers since you only have to compare the exponents on the power 10ⁿ or if they are the same number between 1 and 10 that are multiplied by 10’

Reason 3
It is much easier to write very large (or very small) numbers and take less space, for example:
15, 000, 000. 000, 000, 000, 000 = 1.5 × 10¹⁹
1) It is a notation accepted around the world and used in science.
2) It is easy to compare numbers.
3) It is easier to write large numbers.

Texas Go Math Grade 8 Lesson 2.1 H.O.T. Focus On Higher Order Thinking Answer Key 

Question 29.
Draw Conclusions Which measurement would be least likely to be written in scientific notation: number of stars in a galaxy, number of grains of sand on a beach, speed of a car, or population of a country? Explain your reasoning.
Answer:
Scientific notation is used to express measurements that are extremely large or extremely small. The number of stars in a galaxy and the number of grains of sand on a beach are extremely large, so we use scientific notation for those. Comparing the speed of a car and the population of a country, it is clear that the speed of a car is a smaller number. Therefore, the speed of a car is less likely to be written in scientific notation.

The speed of a car is less likely to be written in scientific notation.

Lesson 2.1 Scientific Notation with Positive Powers of 10 Answer Key Question 30.
Analyze Relationships Compare the two numbers to find which is greater. Explain how you can compare them without writing them in standard notation first.
4.5 × 10⁶ 2.1 × 10⁸
Answer:
We can easily compare them by just comparing the exponents of the power 10ⁿ. In this case, we have 10⁶ and 10⁸, since 10⁸ > 10⁶ every number greater or equal to 1 and less than 10 multiplied by 10⁶ is smaller than any number greater or equal to 1 and less than 10 multiplied by 10⁸
We can conclude: 4.5 × 10⁶ < 2.1 × 10⁸

Comparing the exponents we have: 4.5 × 10⁶ < 2.1 × 10⁸

Question 31.
Communicate Mathematical Ideas To determine whether a number is written in scientific notation, what test can you apply to the first factor, and what test can you apply to the second factor?
Answer:
Scientific notation is in the form of c × 10ⁿ so c is the first factor and 10ⁿ is the second factor.
To the first factor, we can apply the test: if it is a decimal number greater than or equal to 1 but less than 10 it can
be a first factor in a scientific notation.
To the second factor, we can apply the test: if it’s a power of 10 it can be a second factor in a scientific notation.

The first factor has to be greater or equal to 1 and less than 10.
The second factor must be a power of 10.