options: Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.
problem_activity: Activity
In this activity, students are asked to interpret in context the slope and vertical intercept of a linear model given a scatter plot and the equation for a linear model that fits the data well. The linear model is also used to interpolate and extrapolate information about the data in context.
Monitor for students who:
estimate values based on the graph
use the equation for the linear model to find the values
Launch
Arrange students in groups of 2.
Student Facing
The scatter plot shows the sale price of several food items,
\(y\)
, and the cost of the ingredients used to produce those items,
\(x\)
, as well as a line that models the data. The line is also represented by the equation
\(y = 3.48x + 0.76\)
.
[IMG]
What is the predicted sale price of an item that has ingredients that cost \$1.50? Explain or show your reasoning.
What is the predicted ingredient cost of an item that has a sale price of \$7? Explain or show your reasoning.
What is the slope of the linear model? What does that mean in this situation?
What is the
\(y\)
-intercept of the linear model? What does this mean in this situation? Does this make sense?
Student Response
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Anticipated Misconceptions
Although the grid in the picture does not look like squares, each line in both the horizontal and vertical directions are 0.5 apart. Tell students who are confused to look at the values listed on the axes and identify the values attached to a few of the grid lines.
Activity Synthesis
The purpose of this discussion is for students to describe where to find the slope and vertical intercept from a scatter plot and interpret the values in terms of the context.
Select previously identified students to share in the order listed in the narrative. Ask students:
“Are the values obtained from looking at the graph close to the values calculated using the equation?” (Yes, the values are close.)
“Which method will you use to predict values based on the model?” (Maybe a mixture of both methods. Using the equation produces a more precise value from the model, but it is also important to look at the graph to get a sense of how well the model fits the data where I am predicting.)
“Why do you think the intercept for the model is not
\((0,0)\)
?” (Companies may still try to charge money for items that cost nothing to produce. They need to pay for the salaries of their workers, research and development, as well as other things that require the company to make money more than just the cost of making the item. It may also be the case that it does not make sense to interpret the
\(y\)
-intercept, since we have no evidence to believe that the same linear relationship will hold there.)
Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct.
Before students share their interpretations of the slope, present an ambiguous explanation. For example: “The slope means that as sales increase, the cost of ingredients increases.” Ask students to identify inaccuracies, critique the reasoning, and write a correct explanation. As students discuss with a partner, listen for students who identify and clarify the ambiguous language in the statement. Invite students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to describe what happens to the sale price when the cost of ingredients increases by
$
1. This helps students evaluate, and improve on, the written mathematical arguments of others, as they clarify the meaning of slope in context.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Action and Expression: Internalize Executive Functions.
Provide students with a two-column graphic organizer. Label one column “graph” and another column “equation.” Ask students to transfer their answers into the corresponding column on their organizer. When the whole group shares, they can continue to use the organizer capturing the answer if anyone used the alternative method for each question.
Supports accessibility for: Language; Organization


correct_option_index: 10