# Lesson 6

Actual Data vs. Predicted Data

These materials, when encountered before Algebra 1, Unit 3, Lesson 6 support success in that lesson.

## 6.1: Which One Doesn’t Belong: Data Representations (5 minutes)

### Warm-up

This warm-up prompts students to compare four images. It gives students a reason to use language precisely (MP6) and gives the opportunity to hear how they use terminology and talk about characteristics of the items in comparison to one another. To allow all students to access the activity, each item has one obvious reason it does not belong. Encourage students to move past the obvious reasons and find reasons based on mathematical properties.

### Launch

Arrange students in groups of 2–4. Display the images for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together find at least one reason each item doesn’t belong.

### Student Facing

Which one doesn’t belong?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct. During the discussion, ask students to explain the meaning of any terminology they use, such as rate of change, slope, or linear function. Also, press students on unsubstantiated claims.

## 6.2: Predicting Sales (15 minutes)

### Activity

The purpose of this activity is for students to understand the difference between actual data and information predicted by the linear model. This prepares students to understand that a line of best fit, while representative of the data, is different from the actual data and to start thinking about what differences occur between the two. This is useful for when students learn about residuals in the associated Algebra 1 lesson.

### Student Facing

price per eyeglasses (dollars) | 8 | 9 | 10 | 15 | 16 | 17 | 20 | 22 | 26 | 28 |
---|---|---|---|---|---|---|---|---|---|---|

number of sales | 850 | 800 | 900 | 789 | 703 | 725 | 658 | 640 | 614 | 540 |

price per eyeglasses (dollars) | 30 | 34 | 37 | 40 | 42 | 48 | 50 | 55 | 57 | 60 |
---|---|---|---|---|---|---|---|---|---|---|

number of sales | 520 | 425 | 380 | 370 | 370 | 305 | 175 | 136 | 75 | 25 |

- How many sales does the model estimate will be made when the eyeglasses are \$50 each? Explain or show your reasoning.
- How many sales were actually made when the eyeglasses were \$50 each?
- How many times did the model estimate fewer sales than what were actually made? List the coordinates.
- How many times were the predicted number of sales and actual number of sales equivalent? List the coordinates.
- Find a point for which the model predicted there would be at least 25 more sales than were actually made?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of this discussion is for students to understand that the actual data can differ from a linear model even when the model is a good one. Discuss how students used the graph and table to answer the questions. Here are sample questions to promote class discussion.

- “Is the table more helpful in answering questions about the actual data or the values estimated by the linear model?” (the actual data)
- “Which is more helpful in answering questions about the values estimated by the linear model: the graph, the equation, or the table?” (The graph that displays the linear model and the equation that represents the linear model. The graph is helpful in finding how many times the model and the actual sales were equivalent. The equation is helpful in finding when there was a difference of at least 25 between actual sales and the amount of sales predicted using the linear model.)
- “When asked about estimates, how do you use the graph to answer the question?” (The line on the graph shows every estimate made using the linear model \(y=1,\!000-16x\). Finding a point on the line that corresponds to specific information gives additional information about estimated values.)
- “When asked about what was actually sold, how do you use the graph to to answer the question?” (The plotted points show how many sales were actually made when the eyeglasses were \(x\) dollars.)
- “How can you tell when the linear model and the actual data are equivalent?” (The plotted points are directly on the line when the two are equivalent.)
- “How can you tell when the linear model predicts a sales amount that is more than the actual sales amount?” (If the linear model predicts sales that are more than the actual amount, then the line will be above the plotted points.)

## 6.3: Predictions (20 minutes)

### Activity

The purpose of this activity is for students to practice interpreting a linear model and understanding the difference between the actual data and the linear model predictions. Students can use the equation and graph to complete the table, which allows them to synthesize the graph, equation, and table to understand that they represent the same information. Then, students use the graph, equation, and completed table to interpret the data and predictions.

### Student Facing

Priya’s family keeps track of the number of miles on each trip they take over the summer and the amount spent on gas for the trip. The model, represented by \(y=50 + 0.15x\), is graphed with a scatter plot.

Use the graph and equation to complete the table. Then, use the graph, equation, and table to answer the questions.

distance (miles) | amount spent on gas (dollars) | estimated amount spent on gas (dollars) |
---|---|---|

50 | 60 | |

70 | 65 | |

100 | 75 | |

60 | 67 | |

110 | 60 | |

140 | 65 | |

80 | 68 | |

150 | 80 | |

160 | 76 |

- When Priya’s family drove 85 miles, they spent $68 on gas. How much did they expect to spend based on the linear model?
- How far had the family gone when they spent $80 on gas?
- How far does the model estimate the family should have driven when they spent $80 on gas?
- Are there any instances for which the model’s estimated amount spent on gas is equivalent to the actual amount spent on gas?
- Circle one option.
- In general, the model predicts the family will spend more on gas than they actually spend.
- In general, the model predicts the family will spend less on gas than they actually spend.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of this discussion is for students to interpret data and predictions using a linear model that is represented in three ways: using an equation, a graph, and a table. Discuss how students used the representations to answer the questions. Here are sample questions to promote class discussion:

- “How can the linear equation be used to complete the table with estimated amounts spent on gas?” (To use the equation to complete the table, substitute the given value for \(x\) and solve for the estimated \(y\).)
- “How can you use the graph to complete the estimated amount spent on gas?” (To use the graph to complete the table, find the point on the line associated with the given \(x \) to find the estimated \(y\)-value. Be sure to use the coordinates from the line, not the plotted points on the scatter plot.)