lesson 8 homework 6.2

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Eureka Math Grade 6 Module 6 Lesson 8 Answer Key

Engage ny eureka math grade 6 module 6 lesson 8 answer key, eureka math grade 6 module 6 lesson 8 example answer key.

Example 1: Comparing Two Data Distributions

Robert’s family is planning to move to either New York City or San Francisco. Robert has a cousin in San Francisco and asked her how she likes living in a climate as warm as San Francisco. She replied that it doesn’t get very warm in San Francisco. He was surprised by her answer. Because temperature was one of the criteria he was going to use to form his opinion about where to move, he decided to investigate the temperature distributions for New York City and San Francisco. The table below gives average temperatures (in degrees Fahrenheit) for each month for the two cities.

Use the data in the table provided in Example 1 to answer the following:

Exercise 1. Calculate the mean of the monthly average temperatures for each city. Answer: The mean of the monthly temperatures for New York City is 63 degrees. The mean of the monthly temperatures for San Francisco is 64 degrees.

Exercise 2. Recall that Robert is trying to decide where he wants to move. What is your advice to him based on comparing the means of the monthly temperatures of the two cities? Answer: Since the means are almost the some, it looks like Robert could move to either city. Even though the question asks students to focus on the means, they might make a recommendation that takes variability into account.

For example, they might note that even though the means for the two cities are about the same, there are some much lower and much higher monthly temperatures for New York City and use this as a basis to suggest that Robert move to San Francisco.

Example 2: Understanding Variability

Maybe Robert should look at how spread out the New York City monthly temperature data are from the mean of the New York City monthly temperatures and how spread out the San Francisco monthly temperature data are from the mean of the San Francisco monthly temperatures. To compare the variability of monthly temperatures between the two cities, it may be helpful to look at dot plots. The dot plots of the monthly temperature distributions for New York City and San Francisco follow.

Exercises 3 – 7:

Use the dot plots above to answer the following:

Exercise 3. Mark the location of the mean on each distribution with the balancing A symbol. How do the two distributions compare based on their means? Answer: Place ∆ at 63 for New York City and at 64 for Son Francisco. The means are about the same.

Exercise 4. Describe the variability of the New York City monthly temperatures from the New York City mean. Answer: The temperatures are spread out around the mean. The temperatures range from a low of around 39 °F to a high of 85 °F.

Exercise 5. Describe the variability of the San Francisco monthly temperatures from the San Francisco mean. Answer: The temperatures aæ clustered around the mean. The temperatures range from a low of 57 °F to a high of 70 °F.

Exercise 6. Compare the variability in the two distributions. Is the variability about the same, or is it different? If different, which monthly temperature distribution has more variability? Explain. Answer: The variability is different. The variability in New York City is much greater than the variability in San Francisco.

Exercise 7. If Robert prefers to choose the city where the temperatures vary the least from month to month, which city should he choose? Explain. Answer: He should choose San Francisco because the temperatures vary the least, from a low of 57 °F to a high of 70 °F. New York City has temperatures with more variability, from a low of 39°F to o high of 85°F.

Example 3: Considering the Mean and Variability in a Data Distribution

The mean is used to describe a typical value for the entire data distribution. Sabina asks Robert which city he thinks has the better climate. How do you think Robert responds? Answer: He responds that they both have about the same mean but that the mean is a better measure or a more precise measure of a typical monthly temperature for San Francisco than it is for New York City.

Sabina is confused and asks him to explain what he means by this statement. How could Robert explain what he means? Answer: The temperatures in New York City in the winter months are in the 40’s and in the summer months are in the 80’s. The mean of 63 isn’t very close to those temperatures. Therefore, the mean is not a good indicator of a typical monthly temperature. The mean is a much better indicator of a typical monthly temperature in SanFrancisco because the variability of the temperatures there is much smaller.

Exercises 8 – 14:

Consider the following two distributions of times it takes six students to get to school in the morning and to go home from school in the afternoon.

Exercise 9. What is the mean time to get from home to school in the morning for these six students? Answer: The mean is 14 minutes.

Exercise 10. What is the mean time to get from school to home in the afternoon for these six students? Answer: The mean is 14 minutes.

Exercise 11. For which distribution does the mean give a more accurate indicator of a typical time? Explain your answer. Answer: The morning mean is a more accurate indicator. The spread in the afternoon data is far greater than the spread in the morning data.

Distributions can be ordered according to how much the data values vary around their means. Consider the following data on the number of green jelly beans in seven bags of jelly beans from each of five different candy manufacturers (AllGood, Best, Delight, Sweet, and Vum). The mean in each distribution is 42 green jelly beans.

Exercise 12. Draw a dot plot of the distribution of the number of green jelly beans for each of the five candy makers. Mark the location of the mean on each distribution with the balancing A symbol.

Answer: The dot plots should each hove a balancing A symbol located at 42.

Exercise 13. Order the candy manufacturers from the one you think has the least variability to the one with the most variability. Explain your reasoning for choosing the order. Answer: Note: Do not be critical; answers and explanations may vary. One possible answer: In order from least to greatest: All Good, Sweet, Vum, Delight, Best. The data points are all close to the mean for all good, which indicates it has the least variability, followed by Sweet and Yum. The data points are spread farther from the mean for Delight and Best, which indicates they have the greatest variability.

Exercise 14. For which company would the mean be considered a better indicator of a typical value (based on least variability)? Answer: The mean for All Good would be the best indicator of a typical value for the distribution.

Eureka Math Grade 6 Module 6 Lesson 8 Problem Set Answer Key

Question 1. The number of pockets in the clothes worn by seven students to school yesterday was 4, 1, 3, 4, 2, 2, 5. Today, those seven students each had three pockets in their clothes.

b. For each distribution, find the mean number of pockets worn by the seven students. Show the means on the dot plots by using the balancing symbol. Answer: The mean of both dot plots is 3.

c. For which distribution is the mean number of pockets a better indicator of what is typical? Explain. Answer: There is certainly variability in the data for yesterday’s distribution, whereas today’s distribution has none. The mean of 3 pockets is a better indicator (more precise) for today’s distribution.

Question 2. The number of minutes (rounded to the nearest minute) it took to run a certain route was recorded for each of five students. The resulting data were 9, 10, 11, 14, and 16 minutes. The number of minutes (rounded to the nearest minute) it took the five students to run a different route was also recorded, resulting in the following data: 6, 8, 12, 15, and 19 minutes.

b. Do the distributions have the same mean? What is the mean of each dot plot? Answer: Yes, Both distributions have the same mean, 12 minutes.

c. In which distribution is the mean a better indicator of the typical amount of time taken to run the route? Explain. Answer: Looking at the dot plots, the times for the second route are more varied than those for the first route. So, the mean for the first route is a better indicator (more precise) of a typical value.

Question 3. The following table shows the prices per gallon of gasoline (in cents) at five stations across town as recorded on Monday, Wednesday, and Friday of a certain week.

a. The mean price per day for the five stations is the same for each of the three days. Without doing any calculations and simply looking at Friday’s prices, what must the mean price be? Answer: Friday’s prices ore centered at 360 cents. The sum of the distances from 360 for values above 360 is equal to the sum of the distances from 360 for values below 360, so the mean is 360 cents.

b. For which daily distribution is the mean a better indicator of the typical price per gallon for the five stations? Explain. Answer: From the dot plots, the mean for Monday is the best indicator of o typical price because there is the least variability in the Monday prices.

Eureka Math Grade 6 Module 6 Lesson 8 Exit Ticket Answer Key

Question 1. Consider the following statement: Two sets of data with the same mean will also have the same variability. Do you agree or disagree with this statement? Explain. Answer: Answers will vary, but students should disagree with this statement. There are many examples in this lesson that could be used as the basis for an explanation.

b. For which distribution, if either, would the mean be considered a better indicator of a typical value? Explain your answer. Answer: Variability in the distribution for girls is less than ¡n the distribution for boys, so the mean of 3 goals for the girls is a better indicator of a typical value.

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Extra Practice: Math 6 - Lesson 6.2.1 (Part 2)

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Curriculum  /  Math  /  6th Grade  /  Unit 4: Rational Numbers  /  Lesson 6

Rational Numbers

Lesson 6 of 13

Criteria for Success

Tips for teachers, anchor problems, problem set, target task, additional practice.

Order integers and rational numbers. Explain reasoning behind order using a number line.

Common Core Standards

Core standards.

The core standards covered in this lesson

The Number System

6.NS.C.6.C — Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

6.NS.C.7.A — Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret -3 > -7 as a statement that -3 is located to the right of -7 on a number line oriented from left to right.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Know that if a number $$a$$  is to the right of number $$b$$ , then $$a$$  will always be bigger than $$b$$ ; if a number  $$b$$  is to the left of number $$a$$ , then $$b$$  will always be smaller than $$a$$ .
• Understand that opposites of numbers have opposite orders of the original numbers; if a positive number $$a$$  is less than a positive number $$b$$ , then the opposites of $$a$$  and $$b$$  have the opposite order, or $$-a$$  is greater than $$-b$$  (e.g., $$3$$  is less than $$5$$ , but  $$-3$$  is greater than $$-5$$ ).
• Order rational numbers from least to greatest or greatest to least.

Suggestions for teachers to help them teach this lesson

• Lesson 6 and Lesson 7 are related; In Lesson 6, students determine the order for rational numbers when given a set of numbers and explain their thinking using a number line. In Lesson 7, students extend this understanding to compare rational numbers and interpret their order as it relates to real-world situations.
• Encourage students to use or draw a number line when one is not provided in the problem (see lesson notes for Lesson 1). This will continue to be a great tool for students to work precisely and accurately and avoiding common misconceptions.

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Consider the set of numbers $$6$$ , $${4 \frac{1}{2}}$$ , $$2$$ , and $$5$$ , and answer the questions that follow.

a.   Graph the numbers on the number line and list the numbers in order from least to greatest. 

b.   Write the opposites of each number and graph them on the number line. 

c.   Order the opposites from least to greatest. 

d.   Is −5 greater than −2? Explain using your number line. 

Guiding Questions

Order the rational numbers below from least to greatest.  

$${5, -4, -\frac{1}{3}, \frac{10}{3}, 3, 0, -4\frac{1}{4}, -4\frac{3}{4}}$$  

What strategies did you use to determine the correct order?

A student orders three rational numbers from least to greatest as: $${{{{{{{{{-5}}}}}}}}}$$ , $${{{{{{{{{-5}}}}}}}}} \frac {1}{3}$$ ,  $$6$$

Are the numbers ordered correctly? Choose the best statement below.

A set of suggested resources or problem types that teachers can turn into a problem set

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Tram knows that $$1\frac{1}{6}$$ is less than $$1\frac{2}{3}$$ . He wonders if this means that $$-1\frac{1}{6}$$ is also less than $$-1\frac{2}{3}$$ .  Help Tram determine and understand the correct order of $$-1\frac{1}{6}$$ and $$-1\frac{2}{3}$$ , from least to greatest. 

Student Response

An example response to the Target Task at the level of detail expected of the students.

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

• Multiple-choice questions where students select the correct order of a set of numbers
• Multiple-choice questions where students select true statements about a set of numbers, similar to Anchor Problem 3
• EngageNY Mathematics Grade 6 Mathematics > Module 3 > Topic B > Lesson 8 — Problem Set
• Open Up Resources Grade 6 Unit 7 Practice Problems — Lesson 4, Problems 1-3
• EngageNY Mathematics Grade 6 Mathematics > Module 3 > Topic B > Lesson 7 — Problem Set

Topic A: Understanding Positive and Negative Rational Numbers

Extend the number line to include negative numbers. Define integers.

6.NS.C.6 6.NS.C.6.C

Use positive and negative numbers to represent real-world contexts, including money and temperature.

Use positive and negative numbers to represent real-world contexts, including elevation.

Define opposites and label opposites on a number line. Recognize that zero is its own opposite.

6.NS.C.6.A 6.NS.C.6.B

Find and position integers and rational numbers on the number line.

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Topic B: Order and Absolute Value

6.NS.C.6.C 6.NS.C.7.A

Compare and interpret the order of rational numbers for real-word contexts.

Write and interpret inequalities to compare rational numbers in real-world and mathematical problems.

6.NS.C.7.A 6.NS.C.7.B

Define absolute value as the distance from zero on a number line. 

Model magnitude and distance in real-life situations using order and absolute value.

6.NS.C.7.C 6.NS.C.7.D

Topic C: Rational Numbers in the Coordinate Plane

Use ordered pairs to name locations on a coordinate plane. Understand the structure of the coordinate plane.

6.NS.C.6.B 6.NS.C.6.C

Reflect points across axes and determine the impact of reflections on the signs of ordered pairs.

Calculate vertical and horizontal distances on a coordinate plane using absolute value in real-world and mathematical problems.

6.NS.C.7.C 6.NS.C.8

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Home > CCA > Chapter Ch6 > Lesson 6.1.2

Lesson 6.1.1, lesson 6.1.2, lesson 6.1.3, lesson 6.1.4, lesson 6.2.1, lesson 6.2.2, lesson 6.2.3, lesson 6.2.4, lesson 6.2.5.

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Texas Go Math Grade 8 Lesson 6.2 Answer Key Describing Functions

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

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

Texas Go Math Grade 8 Lesson 6.2 Explore Activity Answer Key

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

Texas Go Math Grade 8 Lesson 6.2 Guided Practice Answer Key

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

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

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

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

Essential Question Check-In

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

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

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

Texas Go Math Grade 8 Lesson 6.2 Independent Practice Answer Key

Go Math Lesson 6.2 Describing Function Relationships Answer Key Question 7. The Fortaleza telescope in Brazil is a radio telescope. Its shape can be approximated with the equation y = 0.01 3x 2 . Is the relationship between x and y linear? Is it proportional? Explain. Answer: The linear function has the form y = mx + b, where m and b are real numbers. Every equation in the form of y = mx + b is a linear equation. Equations that cannot be written in this form is not linear equations and not linear functions.

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

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

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

The relationship is proportional. The graph passes through the origin

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

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

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

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

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

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

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

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

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

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

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

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