Getting Started |
| Misconception/Error The student is unable to correctly determine the reading rates from either the table or the graph of the functions. |
| Examples of Student Work at this Level The student:
- Compares the total number of pages read for some portion of the domain or for a specific value of the domain.
- Divides the page number by the hours for a particular ordered pair of values to determine each person’s reading rate.
- Graphs Jordan’s data and concludes that his reading rate is greater because each graphed point is above Alyssa’s line.
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| Questions Eliciting Thinking Did Jordon and Alyssa start reading on the same page? Does this make a difference?
What do the coordinates of the points on the graph tell you?
How can you determine how many pages each read in one hour? |
| Instructional Implications Review linear functions and the various ways that they can be described (with equations, tables, graphs, and verbal descriptions). Focus on the rate of change and initial value in a linear function, and relate these components of the equation to the slope and y-intercept of the graph. Review how to calculate a value of one variable given a value of the other. Provide additional examples of linear functions that model the relationship between real-world quantities and ask the student to identify and compare properties of functions represented in different ways.
Make explicit the difference between the constant of proportionality (y divided by x) in a proportional relationship and the rate of change (change in y divided by the change in x) in other linear relationships. Remind the student that the graph of a proportional relationship always passes through the origin. Assist the student in identifying the initial value of each linear function and to explain its meaning in the context of the problem. |
Moving Forward |
| Misconception/Error The student is able to correctly determine the reading rate of only one of the two representations. |
| Examples of Student Work at this Level The student:
- Correctly calculates Jordon’s reading rate but compares it to the y-intercept of Alyssa’s graph.
- Correctly calculates Alyssa’s rate but is unable to correctly calculate Jordan’s rate.
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| Questions Eliciting Thinking What is the x-coordinate of the y-intercept? What does it represent? What does the y-intercept actually tell you?
How did you calculate each person’s reading rate? |
| Instructional Implications Review the important properties of linear functions (e.g., rate of change and initial value) and how to identify and interpret them. Provide sample graphs, tables, equations, and verbal descriptions of functions, and ask the student to identify the x-intercept, y-intercept, rate of change, an x-value when a y-value is given, and a y-value when an x-value is given. Ask the student to describe in general terms the significance of each of these properties of a function (e.g., slope is the increase in a y-value when an x-value increases by one, y-intercept is the y-value when the x-value is zero).
Provide the student with a linear function represented by a table. Have the student represent the same function with a graph and an equation. Help the student to identify the rates of change in all three representations. Then provide the student with two different functions (e.g., one represented by a graph and the other represented by an equation). Challenge the student to determine and compare the rates of change of the functions.
Consider implementing MFAS task Competing Functions (8.F.1.2) for further assessment. |
Almost There |
| Misconception/Error The student is unable to correctly interpret the reading rates in the context of the problem. |
| Examples of Student Work at this Level The student correctly identifies the numerical value of each reading rate. However, the student:
- Is unable to assign a unit or explain what these values mean in the context of the problem.
- Incorrectly describes the unit of each rate.
- Does not assign a unit to the rates and draws an incorrect conclusion.
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| Questions Eliciting Thinking What do these values, 45 and 50, actually mean?
What are the units of measure for the rates you found? |
| Instructional Implications Guide the student to identify the unit of each rate (i.e., pages per hour) and ask the student to explain what the rates mean in the context of the problem (e.g., Jordan is reading at a rate of 45 pages per hour while Alyssa is reading at a rate of 50 pages per hour).
Provide additional opportunities to interpret and explain the rate of change, initial value, and selected solutions of linear functions in the context of word problems. |
Got It |
| Misconception/Error The student provides complete and correct responses to all components of the task. |
| Examples of Student Work at this Level The student identifies Jordan’s reading rate as 45 pages per hour and Alyssa’s reading rate as 50 pages per hour. The student concludes that Alyssa is reading faster than Jordan.
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| Questions Eliciting Thinking How did you know to use the rate of change formula instead of dividing y by x and finding a constant rate?
How did you know to use the slope of the line?
If Alyssa reads faster, why is Jordan on a higher page number after two hours of reading? |
| Instructional Implications Encourage the student to explore the properties of the functions further. Have the student graph the data from the table and compare the y-intercepts of both graphs. Have the student explain the meaning of the y-intercepts in context. |
