# Into Math Grade 6 Module 10 Lesson 1 Answer Key Represent Equations in Tables and Graphs

We included HMH Into Math Grade 6 Answer Key PDF Module 10 Lesson 1 Represent Equations in Tables and Graphs to make students experts in learning maths.

## HMH Into Math Grade 6 Module 10 Lesson 1 Answer Key Represent Equations in Tables and Graphs

I Can represent a situation using a table, equation, or graph.

A baby gorilla had a mass of 2,620 grams at birth, 2,960 grams after 10 days, and 3,300 grams after 20 days. What is the average number of grams that the baby gorilla grew each day during the first 10 days and during the next 10 days? Show your work.

Given,
A baby gorilla had a mass of 2,620 grams at birth, 2,960 grams after 10 days, and 3,300 grams after 20 days.
(2960 – 2620)/10 = 340/10 = 34 grams per day
(3300 – 2960)/(20-10) = 340/10 = 34 grams per day
Thus the average growth for the baby gorilla is 34 grams per day for both the first 10 days and the next 10 days.

Turn and Talk If the baby gorilla continued to grow at the same rate for the first month after birth, what would be the baby gorilla’s mass 26 days after birth? Explain how you found your answer.
We can multiply the average growth rate per day 34 by 6.
Then we can add that amount to 3300 grams to get a total mass of 3504 grams for 26 days after the birth of the baby gorilla.

Build Understanding

Many real-world situations involve two variable quantities in which one quantity depends on the other. The quantity that depends on the other quantity is the dependent variable, and the quantity that it depends on is the independent variable. In general, the dependent variable is typically recorded on the y-axis.

Question 1.
The Jackson family is driving to visit family members who live out of state. They record the following information about their trip.

A. Which of the two quantities is the independent quantity, and which is the dependent quantity? Explain.
Let us consider hours = x
Miles = y
y = 65x
If x = 2 then y = 65(2) = 130
x = 3 then y = 65(3) = 195
x = 4 then y = 65(4) =260
x = 5 then y = 65(5) =325
All the x values are the independent quantity and all the y values are the dependent quantities.

B. How far does the car travel each hour? How do you know?
The car traveled per 2 hours = 130
The car travelled per each hour = 130/2 = 65
The car traveled per hour = 65.

C. Suppose that d represents the distance traveled by the car in miles and that t represents the time in hours the car has been traveling. How can you model the relationship between the distance traveled, in miles, by the car and the time, in hours, using an equation? Use d for distance and t for time in hours.

D. A student represented this relationship as 65t = d. Is this student correct? Why or why not?
A student represented this relationship as 65t = d is the correct answer.
If t = 2 then d = 65(2) = 130
t = 3 then d = 65(3) = 195
t = 4 then d = 65(4) =260
t = 5 then d = 65(5) =325

Turn and Talk Describe how the table and the equation are related.
You can construct a table using the equation by changing the independent and dependent variables.

Step It Out

When one variable depends on a second variable, the relationship between the variables can be represented in a table or by an equation. An equation can be used to express the dependent variable in terms of the independent variable.

Question 2.
Pablo walks dogs after school. His earnings e, in dollars, are represented by the equation e = 24h, where h is the number of hours he works.

A. Complete the table.

Given that the equation is e = 24h
h = number of hours he works.
If h = 1 then e = 24(1) = 24
h = 2 then e = 24(2) = 48
h = 3 then e = 24(3) = 72
h = 4 then e = 24(4) = 96

B. To determine values to the table, multiply each value of the independent variable, , by to find each corresponding value of the dependent variable, .
To determine values to the table, multiply each value of the independent variable h by 24 to find each corresponding value of the dependent variable e.

Question 3.
Each egg carton holds one dozen eggs. The equation n = 12c shows the number of eggs n in 1 carton.
A. Use the equation to complete the table.

Given that the equation is n = 12c
n = number of eggs in one carton.
If c = 1 then n = 12(1) = 12
c = 2 then n = 12(2) = 24.
c = 5 then n = 12(5) = 60.
c = 8 then n = 12(8) = 96

B. To find the missing values in the table, sometimes I multiplied a value of the independent variable, , by to get each corresponding value of the dependent variable, . Other times I divided a value of the dependent variable by to get the value of the independent variable, c.
To find the missing values in the table, sometimes I multiplied a value of the independent variable c by 12 to get each corresponding value of the dependent variable n.
Other times I divided a value of the dependent variable by 12 to get the value of the independent variable c.

Turn and Talk Do the equations and the tables describe the same relationships? Explain how you know.
Yes, the equations and the tables show the same relationships.
In the equations, the independent and dependent variables are varied the then represent the table.

Question 4.
The distance in meters that Kaycia hikes in x minutes is represented by the equation y = 75x.
A. Complete the table.

Given that,
The equation is y = 75x
The distance that Kaycia hikes = x
If x = 5 then y = 75(5) = 375
x = 15 then y = 75(15) = 1125
If y = 1500 then x = 1500/75 =20
y = 2250 then x = 2250/75 = 30

B. How did you find the value of y when x = 5?
Given that the equation is y = 75x
Substitute the 5 in the x place.
Then y = 75(5) = 375.

C. How did you find the value of x when y = 1,500?
Given that the equation is y = 75x
Substitute the 1500 in the y place.
1500 = 75x
x = 1500/75
x = 20

D. Write the ordered pairs from the table.

The ordered pairs are (5, 375), (15, 1250), (20, 1500), (30, 2250).

E. Graph the ordered pairs on the coordinate grid.

F. Your points should follow a straight line pattern. Because Kaycia is hiking between the times represented by points, you can draw a line through the points. Draw this line on the graph.
Answer: Yes, the points follow the straight-line pattern in the graph. Because the Kaycia hikes between the time and meters.

Turn and Talk Why might graphing a set of paired values be better than displaying them in a table?
Answer: Graphing a set of paired values is better than displaying them in a table Because in graphing the data is shown in a visual nature and it shows the overall shape of the data.

Check Understanding

Question 1.
The equation p = $$\frac{2}{5}$$m relates the number of book pages p Ellen reads in m minutes. What is the dependent variable? What is the independent variable?
Complete the table.

Explanation:
The equation p = $$\frac{2}{5}$$m relates the number of book pages p Ellen reads in m minutes
Let us find the values of p and m from the information given in the above table.
m = 10
p = $$\frac{2}{5}$$m
p = $$\frac{2}{5}$$ . 10
p = 2 × 2 = 4
p = 4
p = 8
p = $$\frac{2}{5}$$m
8 = $$\frac{2}{5}$$m
m = (8 × 5)/2
m = 20
p = 12
p = $$\frac{2}{5}$$m
12 = $$\frac{2}{5}$$m
m = (12 × 5)/2
m = 30
m = 60
p = $$\frac{2}{5}$$m
p = $$\frac{2}{5}$$ . 60
p = 2 × 12 = 24
p = 24
m is the dependent variable and p is the independent variable.

Question 2.
A store sells a box of breakfast cereal for $3. Write an equation to relate the number of boxes b to total sales T. Answer: Given, A store sells a box of breakfast cereal for$3.
b = number of boxes
T = total sales
Number of boxes = total amount from the sale of cereal/cost per box
b = T/3
T = 3b

Question 3.
The total cost C, in dollars, to dry clean a certain number of shirts s is given by the equation C = 3.25s.
A. What is the dependent variable?
What is the independent variable?
C is the dependent variable and s is the independent variable.
Because the value of C depends on the value of s.

B. Complete the table to relate the total cost to the number of shirts dry cleaned.

The equation is C = 3.25s
i. s = 3
C = 3.25 × 3 = 9.75
ii. C = $16.25 16.25 = 3s s = 16.25/3 s = 5.41 iii. C =$26
26 = 3s
s = 26/3
s = 8.6
iv. C = 3.25 × 10
C = 32.5

C. Plot the points on a graph.

D. Reason Should you draw a straight line through the points on the graph, or should you leave them as separate points? Explain.

Question 4.
Model with Mathematics Dennis is floating down the river on a raft. The table shows the distance he travels over time. Write an equation that represents the number of kilometers k he travels in h hours.

Given,
Table of values of hours traveled and distance covered.
For (0.5, 0.75), (2, 3) = (3 – 0.75)/(2 – 0.5)
= 2.25/1.5 km/hr
= 1.5 km/hr
For (2, 3), (3, 4.5) = (4.5 – 3)/(3 – 2)
= 1.5/1 = 1.5 km/hr
So, here speed = slope = 1.5 km/hr
Distance = speed × time
k = 1.5h
where,
k = distance in km
h = time in hours
Hence, k = 1.5h

Question 5.
Gary has a sister Tina. Their ages are related by the equation G = T + 4.5 where G stands for Gary’s age and T stands for Tina’s age.

A. Complete the table to show Gary’s and Tina’s ages over time.
G = T + 4.5
G = 7
7 = T + 4.5
T = 7 – 4.5
T = 2.5
G = 10
G = T + 4.5
10 = T + 4.5
10 – 4.5 = T
T = 5.5
T = 7
G = 7 + 4.5
G = 11.5
T = 10
G = 10 + 4.5
G = 14.5

B. Write the ordered pairs from the table.

The ordered pairs from the table are (2.5, 7) (5.5, 10), (7, 11.5), (10, 14.5)

C. Graph the ordered pairs on the coordinate grid.

D. Reason Should you draw a straight line through the points on the graph, or should you leave them as separate points? Explain.
Answer: There is no need to draw a straight line through the points on the graph.

Question 6.
The equation y = 1.19x, gives the costy, in dollars, of x boxes of pasta. Complete the table to represent the relationship.

The equation will be y = 1.19x
y = 1.19 × 2 = 2.38
y = 1.19 × 3 = 3.57
y = 1.19 × 5 = 5.95
y = 1.19 × 8 = 9.52

Model with Mathematics For Problems 7-8, write an equation to represent each table of values.

Question 7.

228/4 = 57
342/6 = 57
By this we can say that the equation would be
d = 57t

Question 8.

3.8 – 0 = 3.8
4.2 – 0.4 = 3.8
5.1 – 1.3 = 3.8
7.3 – 3.5 = 3.8
x – y = 3.8

I’m in a Learning Mindset!

How does understanding of independent and dependent variables impact my ability to complete a table and graph?
The independent variable belongs on the x-axis (horizontal line) of the graph and the dependent variable belongs on the y-axis (vertical line). The dependent variable is the column variable and the Independent Variable is the row variable.

Lesson 10.1 More Practice/Homework

Question 1.
The price of green beans at a supermarket is shown in the table. Write an equation for the total cost C, in dollars, of p pounds of green beans.

The price of green beans at a supermarket is shown in the table.
The equation is
C = 3p
where C is the cost and p is the pounds of green beans.

Question 2.
STEM Joules and calories are two different units of energy. The equation j = 0.239c relates the measures, where c stands for calories and j stands for joules. Complete the table.

Joules and calories are two different units of energy.
j = 0.239c
j = 0.239 × 3 = 0.717
j = 0.239 × 7 = 1.673
j = 0.239 × 10 = 2.39
j = 0.239 × 50 = 11.95

Question 3.
Math on the Spot A car wash attendant counted the number of cars washed and the total amount of money earned. The company charged the same price for each car washed and earned $165 for 15 cars,$231 for 21 cars, and $275 for 25 cars. Write an equation for the relationship between the number of cars washed, n, and the number of dollars earned, m. Answer: The information given is shown in the table. We can get that one more car washed$11 more earned.
So, the equation is a linear function
m = an + b
(15, 165) and (21, 231) are solutions to the equation
165 = 15a + b
a = 11  and b = 0
So, the equation is m = 11n

Question 4.
Data for the fastest long-distance train in the United States as of 2018 are shown, where t is the time in minutes, and d is the distance in miles.

A. What is the dependent variable? What is the independent variable?
d = 2.5t
d is the dependent variable and t is the independent variable.

B. Complete the table.

By substituting the known values in the given equation we can complete the table.
d = 2.5t
t = 3
d = 2.5 × 3 = 7.5
d = 15
15 = 2.5 × t
t = 15/2.5
t = 6
t = 10
d = 2.5 × 10 = 25
d = 28
28 = 2.5 × t
t = 28/2.5
t = 11.2

Model with Mathematics For Problems 5-6, write an equation representing each table of values.

Question 5.

y = x + 8
Explanation:
x = 17
y = 17 + 8 = 25
x = 14
y = 14 + 8 = 22
x = 29
y = 29 + 8 = 37
Thus the equation will be y = x + 8.

Question 6.

g = 15d
where g is the dependent variable and d is the independent variable.
Explanation:
135/9 = 15
180/12 = 150
By this we can say that the equation would be g = 15d

Test Prep

Question 7.
The perimeter p of a regular hexagon is found using the equation p = 6s. The length of each side is s. Complete the table to represent the relationship.

The perimeter p of a regular hexagon is found using the equation p = 6s.
substitute the values given in the above equation to complete the table.
s = 5
p = 6s
p = 6 × 5
p = 30
s = 12
p = 6 × 12
p = 72
p = 21
21 = 6s
s = 21/6
s = 7/2 = 3.5
p = 54
54 = 6s
s = 54/6
s = 9

Question 8.
Does the graph match the values in the table?

The graph does / does not match the values in the table.
Answer: Yes the graph shown in the above matches the data given in the table.

Question 9.
The distance m in miles, traveled by an airplane in h hours of flying is shown in the table. What equation represents the situation?

m = 540h
m is the dependent variable and h is the independent variable.
m = 810
h = 1.5
810/1.5 = 540
1350/1.5 = 540
So, the constant will be 540 and the dependent variable will be h.
So, the equation will be m = 540h

Question 10.
Write an equation to represent the values in the table.

n + 0.5 = p
Explanation:
3 – 2.5 = 0.5
3.7 – 3.2 = 0.5
5 – 4.5 = 0.5
5.9 – 5.4 = 0.5
So, by this we can estimate the equation to represent the values given in the above table.
So, the equation is p = n + 0.5
where p is the dependent variable and n is the independent variable.

Spiral Review

Question 11.
Write an inequality that represents the sentence, “y is less than 3.” Then graph the inequality.

Sally pays $50.00 for a meal that costs$38.13. She tells the server to keep the difference as a tip. How much is the tip?
Sally pays $50.00 for a meal that costs$38.13. She tells the server to keep the difference as a tip.
50 – 38.13 = $11.87 Thus the tip is$11.87