# Study Guide

## Field 004: Mathematics

### Sample Constructed-Response Item

##### Competency 0007Pedagogical Content Knowledge

Use the information below to complete the task that follows.

As a mathematics teacher, you are preparing to teach a lesson to address an aspect of the following standard from the New York State P to 12 Common Core Learning Standards:

Interpreting Functions (F–IF)
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases, and using technology for more complicated cases.

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

Prepare a response of approximately 400 to 600 words in which you:

• state the student learning objectives for this lesson;
• describe the conceptual understanding, skills, and prerequisite knowledge that students need in order to understand the content described by this learning standard;
• describe an effective instructional strategy that promotes student understanding of the content described by the learning standard;
• describe a method for helping students build a viable argument related to the learning standard; and
• describe a method for assessing students' progress toward the goal of understanding the content described by the learning standard.

### Sample Strong Response to the Constructed-Response Assignment

Students will plot graphs of linear and quadratic functions by hand and by using a graphing calculator.

Students will compare and analyze graphs to identify key features (degree of function, intercepts, maxima, minima, increasing, decreasing, etc.).

To understand the new content that will be addressed in this lesson, students will need prerequisite knowledge that a graph is the set of solutions for a given equation, as well as the general shape of a linear and quadratic graph. They must know how to determine the degree of a polynomial. Students will require the skills to plot points on the coordinate plane and connect the points to form a graph and skills in using a graphing calculator to graph an equation in y = format. Conceptually, students will need to understand the relationship between the function and the graph.

An instructional strategy to achieve the learning objectives would be guided instruction. I will begin by showing the graphs, y = 2x + 1 and y = x squared + 4x + 4, and lead a discussion asking students what they see as similarities/differences in the two graphs: one linear, one quadratic. Terms such as x-intercept, y-intercept, vertex, etc. will be defined and students would identify these on the sample graphs. Throughout discussion, students will view additional graphs of linear and quadratic functions, identifying the type of function each graph displays and locating intercepts for each and the vertex for the parabolas. I would check for understanding as students explained how they determined which type of function each graph was, and how they located the key features. Students would demonstrate knowledge of similarities/differences between all the linear graphs (some increasing, some decreasing, number of x- and y-intercepts) and between all the quadratic graphs (some have a maximum, some a minimum for a vertex, number of x- and y-intercepts the graph has).

Following discussion, each student would receive a worksheet with a combination of 5 linear and quadratic equations. First, they will insert each function into the graphing calculator and sketch the appropriate graph next to the given equations. Students will identify whether the graph is linear or quadratic, based on shape. Students would record their observations about the degree of the equations (linear are first degree; quadratic are second degree). The next task on the worksheet would include a different set of equations, both linear and quadratic. Students would predict the type of graph each equation will produce, complete a data table by evaluating the equation for given x values, and plot the graph. Students will use the graph to identify its intercepts and maximum or minimum point.

To help students build a viable argument, students would be assigned a partner to compare work, confirm understanding and discuss questions like: What features in an equation reveal whether a line or a parabola will be formed? What are the similarities and differences between linear and quadratic graphs?

Assessment would be ongoing throughout the lesson. I would check for understanding at the beginning, during discussion, to ensure students are ready for the independent work. I would also circulate among students during the independent work and paired discussion to assist students who were struggling or offer a challenging extension for students who finished before others and showed mastery of the learning objectives. At the close of class, I would engage students in a whole group discussion to summarize what was learned about key features of linear and quadratic graphs and different methods of graphing them. The worksheet would be turned in at the end of class so I could evaluate student work and use it to inform future instruction.

### Performance Characteristics for Constructed-Response Item

The following characteristics guide the scoring of responses to the constructed-response assignment.

Completeness The degree to which the response addresses all parts of the assignment The degree to which the response demonstrates the relevant knowledge and skills accurately and effectively The degree to which the response provides appropriate examples and details that demonstrate sound reasoning

### Score Scale for Constructed-Response Item

A score will be assigned to the response to the constructed-response item according to the following score scale.

Score Point Score Point Description
4 The "4" response reflects a thorough command of the relevant knowledge and skills:
• The response thoroughly addresses all parts of the assignment.
• The response demonstrates the relevant knowledge and skills with thorough accuracy and effectiveness.
• The response is well supported by relevant examples and details and thoroughly demonstrates sound reasoning.
3 The "3" response reflects a general command of the relevant knowledge and skills:
• The response generally addresses all parts of the assignment.
• The response demonstrates the relevant knowledge and skills with general accuracy and effectiveness.
• The response is generally supported by some examples and/or details and generally demonstrates sound reasoning.
2 The "2" response reflects a partial command of the relevant knowledge and skills:
• The response addresses all parts of the assignment, but most only partially; or some parts are not addressed at all.
• The response demonstrates the relevant knowledge and skills with partial accuracy and effectiveness.
• The response is partially supported by some examples and/or details or demonstrates flawed reasoning.
1 The "1" response reflects little or no command of the relevant knowledge and skills:
• The response minimally addresses the assignment.
• The response demonstrates the relevant knowledge and skills with minimum accuracy and effectiveness.
• The response is minimally supported or demonstrates significantly flawed reasoning.
UThe response is unscorable because it is unrelated to the assigned topic or off-task, unreadable, written in a language other than English or contains an insufficient amount of original work to score.
BNo response.