
 11/01/13  Adjusted: 08/01/18  1 file
 Grades 8
Delivering the Mail, Assessment Variation
 Description
 Files
What we like about this task
Mathematically:
 Addresses standards: 8.F.B.4, MP.2, MP.4, and MP.6
 Requires students to have facility with all the major components of a linear function: initial value, rate of change, and how they relate to the description of a relationship (8.F.B.4)
 Builds on work with variables and equations from seventh grade (see 7.EE.B.4)
 Engages students in specifying units of measure, requiring attention to precision (MP.6)
In the classroom:
 Offers opportunity for students to share their developing thinking and understanding of interpreting functions and parts of functions
 Offers students and teachers an opportunity to see an assessmenttype task with the possibility of 3 different technologyenhanced response types
 Allows opportunities for extending the task to deepen student understanding (see Additional Thoughts below)
This task was designed to include specific features that support access for all students and align to best practice for English Language Learner (ELL) instruction. Go here to learn more about the research behind these supports. This lesson aligns to ELL best practice in the following ways:
 Provides opportunities for students to practice and refine their use of mathematical language.
 Allows for whole class, small group, and paired discussion for the purpose of practicing with mathematical concepts and language.
 Elicits evidence of student thinking both verbally and in written form.
 Includes a mathematical routine that reflects best practices to supporting ELLs in accessing mathematical concepts.
 Provides students with support in negotiating written word problems through multiple reads and/or multimodal interactions with the problem.

Making the Shifts
How does this task exemplify the instructional Shifts required by CCSSM?
Focus Belongs to the major work of eighth grade Coherence Builds on grade 7 work with expressions and equations; prepares students for more advanced modeling in high school Rigor^{} Conceptual Understanding: primary in this task
Procedural Skill and Fluency: not targeted in this task
Application: primary in this task

Task
Joshua's mail tuck travels 14 miles every day he works, and is not used at all on days he does not work. At the end of his 100th day of work the mail truck shows a mileage of 76,762.
 Fill in the blanks to express the mileage $y$ as a linear function of the number of days $x$ that Joshua has worked:
$y=$$x+$.  What are the units of the number that appears in your equation?
 What are the units of the number that appears in your equation?
 Which of the following is a correct interpretation of the number that appears in your equation? (Select all that apply.)
 In this context, which of the following is a correct interpretation of the number that appears in your equation? (Select all that apply.)
 Fill in the blanks to express the mileage $y$ as a linear function of the number of days $x$ that Joshua has worked:

Illustrative Mathematics Commentary and Solution
Commentary:
This task is part of a joint project between Student Achievement Partners and Illustrative Mathematics to develop prototype machinescorable assessment items that test a range of mathematical knowledge and skills described in the CCSSM and begin to signal the focus and coherence of the standards.
Task Purpose:
This is one of two assessment tasks illustrating the similarities and differences between the 8th grade standards in Functions and in Statistics and Probability. The first, 8.F Mail Truck, involves a situation that can be modeled exactly with a linear function. The second, 8.SP US Airports, uses a linear function to model a relationship between two quantities that show statistical variation and do not have an exact linear relationship.
In 8.F Mail Truck, each additional day of driving does correspond to exactly the same increase in the number of miles put onto the truck each day.
Mathematical Content:
This task involves constructing a linear function and interpreting its parameters in a context. Thus, this task has a medium level of complexity.
Mathematical Practices:
The task asks students to reason abstractly and quantitatively (MP.2) and directly assesses component skills related to mathematical modeling (MP.4), namely, interpreting mathematical objects in contexts.
Solution:
 $y=14x+75,362$
 miles/day
 miles
 (ii)
 (iii)
As part of an assessment, this would be a twopoint task: 1 point for parts (a)(c) and 1 point for parts (d) and (e).

Additional Thoughts
As noted in the Commentary above, this task is the first in a set of two tasks. It's interesting to view the two tasks sidebyside, as this task addresses a relationship that is linear while US Airports addresses statistical data that can be approximated to a linear relationship through the use of a model.
There are many ways this task could be extended to deepen students' understanding of functions. Teachers may want to ask students to relate this function to the equation $y=mx+b$ and discuss how the slope and $y$intercept are interpreted in the context of this task (8.F.A.3). Students could also graph the function and describe various points that lie on the graph, again in the context of the task (8.F.B). Teachers could also give students an additional function (represented in a different way) and have the students compare the properties of that function to the function represented in this task (8.F.A.2).
For more insight on the expectations for functions in grade 8, read pages 5 and 6 of the progression document, Grade 8, High School, Functions.
For more insight on the distinctions between equations and functions, read pages 2 and 3 of the progression document, High School, Algebra.
For further analysis on this task from an assessment perspective, read the Cognitive Complexity section on the Illustrative Mathematics site.