# Study Guide

## Field 004: Mathematics

### Sample Constructed-Response Item 1

#### Competency 0007 Pedagogical 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 Learning Standards for Mathematics:1

Interpreting Functions (AI-F.IF)

Graph linear, quadratic, and exponential functions and show key features.

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 Constructed-Response Item 1

Students will plot graphs of linear, quadratic, and exponential 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 linear, quadratic, and exponential graphs. 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 equals 2 x plus 1, y equals x squared plus 4 x plus 4, and y equals 3 times 2 sup x  and lead a discussion asking students what they see as similarities/differences in the three graphs: one linear, one quadratic, and one exponential. 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, quadratic, and exponential 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), between all the quadratic graphs (some have a maximum, some a minimum for a vertex, number of x- and y-intercepts the graph has), and between all the exponential graphs (some increasing, some decreasing, where they "level out," and number of x- and y-intercepts).

Following discussion, each student would receive a worksheet with a combination of 5 linear, quadratic, and exponential 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, quadratic, or exponential based on shape. Students would record their observations about the degree of the equations (linear are first degree; quadratic are second degree; and exponential do not have degrees because they are raised to a variable power instead of a constant power). The next task on the worksheet would include a different set of equations: linear, quadratic, and exponential. 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 curve will be formed? What are the similarities and differences among linear, quadratic, and exponential 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, quadratic, and exponential 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.

### Sample Constructed-Response Item 2

#### Competency 0007 Pedagogical 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 Learning Standards for Mathematics:2

Conditional Probability and the Rules of Probability (AII-S.CP)

4. Interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and calculate conditional probabilities.

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

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

### Sample Strong Response to Constructed-Response Item 2

Learning Objectives:

- Students will create a two-way table.

- Using the two-way table and the formula The probability of A, given that B occurred, is the probability of A and B happening together divided by the probability of A occurring. students will determine the probabilities of events.

Conceptual Understanding, Skills, Prerequisite Knowledge:

Students should already possess prerequisite knowledge and skills of conditional probability and principles of probability such as all probabilities are greater than or equal to zero or less than or equal to 1. Students should also have conceptual knowledge of terminology associated with probability such as event, ratios, outcomes, sample space and symbolism.

Instructional Strategy: Collaborative Learning

Prior to placing students in collaborative groups, the teacher will model how to use a two-way table to determine conditional probability of given events. Students would observe how the teacher generates a two-way table. For example, the teacher's table may show genres of music (jazz, country, rock, rap) in columns across the top and age groupings (11 to 20, 21 to 30, 31 to 40, 41 to 50) in rows down the side of table with predetermined data that completes the table. The teacher would guide a whole class discussion to ensure students understand how to use the teacher-generated table to determine the conditional probability of events. Discussion would result in students understanding the probability for musical preference if a person falls between the ages of 11 to 20 versus the ages of 41 to 50, etc.

Next, the teacher would assign students to collaborative groups of four and would distribute a blank template of a 5 by 5 table to each group. The template would be void of any information with the exception of the word "Totals" as the label in the bottom row and the last heading across the columns. It would be explained to students that each student in the group would brainstorm an idea for categories to be displayed on the table and the group would then collectively decide which student's ideas would be used to represent the group's first two-way table. While students are brainstorming and discussing ideas to use on the table, the teacher would check in with each group and review categories selected by the group. Once the teacher has approved a set of categories for the groups' tables, students would insert theoretical data for those categories into the individual cells of the table so the two-way table is completed accurately. Each student in the group would then independently determine a conditional probability of a different event  the probability of A, given that B occurred  represented by the values in their group's two-way table.

Building a Viable Argument: Peer to Peer Evaluation & Discussion

Students within each group would exchange their probability statement with another student in the group. Students will then determine the conditional probability of their peer's event. The two students would compare probabilities and justify their respective values to their partner to build a viable argument. To extend discussion, peers could also be asked to review principles of probability and explain events that have a probability of 0 or 1.

Method for Assessing Students' Progress: Checklist

The teacher was able to assess students' knowledge throughout instruction by guiding class discussion and conducting frequent checks for understanding while students created two-way tables in collaborative groups and determined probability of events. During these checks for understanding, the teacher would annotate and monitor student understanding of the learning objectives via a checklist. This checklist would serve to document and inform the teacher of how well students progressed toward the learning objectives and whether future instruction would be required to ensure that instructional goals are met.

### Performance Characteristics for a Constructed-Response Item

The following characteristics guide the scoring of the response to a constructed-response item.

 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 a Constructed-Response Item

A score will be assigned to the response to a 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.
U The 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.
B No response.

#### Acknowledgments

1From the New York State Education Department. New York State Next Generation Mathematics Learning Standards. Internet. Available from http://www.nysed.gov/curriculum-instruction/new-york-state-next-generation-mathematics-learning-standards; accessed 2/20/2019.

2Ibid.