Part Two: Mathematics
Preparing for the Multi-Subject CST

Welcome! This module will begin to prepare you for the Math portion of the Multi-Subject Content Specialty Test.

A few things to note:

• This test is divided into Competencies, Sections, and Performance Indicators
• Competencies: There are 5 Competencies on this exam. The first three are about mathematical ability and the second two are about mathematical instruction.
• 1. Numbers and Operations [5%]
• 2. Ratios, Proportions, and Number Systems [30%]
• 3. Algebra, Measurement, Geometry, and Data [35%]
• 4. Instruction [10%]
• 5. Constructed Response - Analysis, Synthesis, and Application [20%]
• Sections: Each Competency is divided into sections
• Performance Indicators: Do not get overwhelmed by the number of indicators on this exam, as there are many overlaps.

The pages that follow will show groups of performance indicators, give you some samples problems and offer some advice on how to learn this content if you are having difficulty. At the end of this module, there is a list of resources that you can use for more practice in any area. Let's get started!

# Performance Indicators

Section 1.1: Operations and Algebraic Thinking

1.1a Applies operations and relationships between operations

1.1b Analyzes properties of factors and multiples

1.2f Applies number theory concepts (primes, divisibility, factors, LCM, GCF)

1.2g Extends number operations to fractions and performs operations on fractions

1.1c Applies strategies for writing and interpreting numerical expressions

1.1d Generates and analyzes patterns and relationships and identifies apparent features of patterns that are not explicit in the rule used to generate them

1.1e Applies and extends principles of arithmetic and the order of operations to algebraic expressions, equations, and inequalities

1.1f Uses properties of operations to generate equivalent expressions

1.1g Analyzes and solves linear equations and inequalities and pairs of simultaneous linear equations

1.1h Solves mathematical and real-world problems using numerical and algebraic expressions and equations

Section 1.2: Number and Operations--Base Ten and Fractions

1.2a Demonstrates knowledge of place value

1.2b Applies understanding of place value and properties of operations to round, add, subtract, multiply, and divide multidigit numbers

1.2c Analyzes decimal notation and compares decimals, decimal fractions, and fractions

1.2d Justifies computational algorithms

1.2e Analyzes and performs operations with decimals

## Practice Problems

The problem below allows you to practice solving a problem involving operations on fractions.

 Show/hide comprehension question...

This graphic above provides a mnemonic device (memory trick) for remembering the order of operations. Parentheses come first, followed by exponents, then multiplication, division, addition, and finally subtraction The memory trick is "Please Excuse My Dear Aunt Sally" with the first letters of those words matching the first letter of each operation.

Picture taken from: http://www.wikihow.com/Evaluate-an-Expression-Using-PEMDAS

# Performance Indicators

1.2h Applies properties of signed rational numbers, ordering, and the absolute value of rational numbers

1.2i Applies and extends understanding of operations with fractions to add, subtract, multiply, and divide rational numbers

1.2j Solves mathematical and real-world problems involving the four basic operations with rational numbers, including the use of the distributive law to justify properties of rational numbers

## Practice Problems

Here are two examples of multi-step fraction problems. The solutions and explanations are on the right. Try them on your own first!

Problems taken from: https://www.illustrativemathematics.org/illustrations/882

# Performance Indicators

Section 2.1: Ratios and Proportional Relationships

2.1a Solves unit rate problems, including those involving unit pricing; constant speed; and ratios of lengths, areas, and other quantities measured in like or unlike units.

2.1b Interprets percents of a quantity as a rate per 100 and solves mathematical and real world problems involving percents

2.1c Identifies the constant of proportionality in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships

## Practice Problems

This is a Khan Academy video [taken from khanacademy.org] that teaches how to construct proportions to solve application problems:

www.galleryhip.com

The box below shows a unit rate problem. Click on the slideshow link for the solution and an explanation.

 Question #2 [taken from www.illustrativemathematics.org based on the Grade 6 CCSS]     Click on Slideshow Activity for solution. There is no sound in the slideshow.

Problem from https://www.illustrativemathematics.org/illustrations/711

# Performance Indicators

2.1d Represents proportional relationships by equations

2.1e Explains and analyzes the relationships between graphs of proportional relationships in terms of the situation represented by the relationship

2.1f Uses proportional relationships to solve multistep ratio and percent problems

2.1g Analyzes the connections between proportional relationships, lines, and linear equations

## Practice Problems

These performance indicators are very similar to the previous page. The answers are on the right side of the page. Do your best to answer the questions without looking at the answers!!

 Question 1 Answer to Question 1 Question 2 Answer to Question 2 Option B is correct. The student calculates that the family travels 100 miles during this 3-hour period. Their speed is 100 miles/3 hours =33 1/3 miles per hour. Question 3 Answer to Question 3

Question 1 was taken from https://www.illustrativemathematics.org/illustrations/121

Question 2 was taken from released New York State test questions found at http://www.edinformatics.com/testing/new_york_state/grade-7-math.pdf

Question 3 was taken from http://www.nystce.nesinc.com/STUDYGUIDE/NY_SG_SRI_221_subtest2.htm

2.1h Uses similar triangles to explain why the slope is the same between any two distinct points on a nonvertical line in the coordinate plane and graphs and analyzes linear equations

Also, don't forget to laugh a little and take some breaks!

Image: http://funnychoise.com/math-funny-questions-while-see-the-first-reaction-of-equation/

# Performance Indicators

Section 2.2: Rational and Real Number Systems

2.2a Applies knowledge of numbers that are not rational and finds rational approximations of irrational numbers

2.2b Applies properties of repeating decimal expansions and converts between repeating decimal expansions and rational numbers

2.2c Analyzes and applies properties of integer exponents and extends them to rational exponents

2.2d Analyzes how the definition and meaning of rational exponents allows for extending the properties of integer exponents

2.2e Rewrites expressions involving radicals and rational exponents using the properties of exponents

2.2f Uses square roots and cube roots to represent solutions to problems and equations

2.2g Performs operations with numbers expressed in scientific notation

2.2h Uses properties of rational and irrational numbers

2.2i Uses units as a way to understand problems and to guide the solution of multistep problems and chooses and interprets unit consistently in formulas

## Vocabulary

Some vocabulary to know:

 Term Definition Examples Rational Numbers A rational number is a number that can be written as a ratio, in other words as a fraction, in which both the numerator and denominator are whole numbers   Every whole number is a rational number because any whole number can be written as a fraction. For example, 4 can be written as 4/1, 65 can be written as 65/1, and 3,876 can be written as 3,876/1 -The number 8 is a rational number because it can be written as the fraction 8/1 -3/4 is a rational number because it can be written as a fraction -Even a big, clunky fraction like 7,564/89,229 is rational, simply because it can be written as a fraction. Irrational Numbers All numbers that are not rational are considered irrational. An irrational number can be written as a decimal, but not as a fraction. An irrational number has endless non-repeating digits to the right of the decimal point. π=3.141592... √2=1.414213... Integer Exponents Any exponent raised to the power of an integer. Integer Exponent Rules 8² 74³ Rational Exponents A number raised to a fraction power. b^(m/n) = nth root of b^m 4^(1/2)=2nd root of 4^1=square root of 4=2   8^(2/3)=3rd root of 8^2=cube root of 64=4

## Practice Problems

Show/hide comprehension question...

Scroll over the image for the answers to the matching.

For the answers to the cube root questions roll your cursor over this sentence.

Sources for Table:

http://www.factmonster.com/ipka/A0876704.html

Source for questions: http://www.mathworksheetsland.com/8/3intexp/matching.pdf

# Performance Indicators

Section 3.1: Algebra

3.1a Understands the vocabulary of mathematical expression (terms, factors, coefficients) and interprets their structure

3.1b Writes expressions in equivalent forms to solve problems (factor quadratic expressions, complete the square, properties of exponents)

3.1c Performs arithmetic on polynomials and understands the relationship between zeros and factors of polynomials

3.1d Creates equations and inequalities in one variable and uses them to solve mathematical and real-world problems, including equations that arise from linear, quadratic, and simple rational and exponential functions

3.1e Creates equations in two or more variables to represent relationships between quantities and analyzes graphs of equations on coordinate axes

3.1f Uses systems of equations or inequalities to represent situations, including constraints

3.1g Analyzes solving equations as a process of reasoning, explains the reasoning, solves equations and inequalities in one variable, and solves systems of equations in two variables

## Vocabulary

Some vocabulary to know:

 Variables: A symbol for a number we don't know yet. It is usually a letter like x or y.   Example: In x + 2 = 6, x is the variable Terms: Like Terms: terms whose variables are the same. Examples: 7x, x, and -2x are all like terms because the variables are all "x" (1/3)xy², -2xy², and 6xy² are like terms because all variables are xy²   Unlike Terms: terms that are not like. Examples: -3xy, -3y, and 12y² are unlike terms because xy, y, and y² are different Coefficients: A number used to multiply a variable Example: 6z means "6" times "z", and "z" is a variable, so "6" is a coefficient   Sometimes a letter stands in for the number. Example: In ax² + bx + c, "x" is a variable, and "a" and "b" are coefficients

## Practice Problems

Systems problem:

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For more help and practice with algebra, you can work through the Patterns, Functions, and Algebra materials in Annenberg Learner here. Sessions 1-5 are fundemental underpinnings of algebra. Session 6 is good for more practice solving equations. Session 9 homework is good for practice with manipulating variables.

Term Definitions modified from: http://www.mathsisfun.com/algebra/like-terms.html

# Performance Indicators

3.1h Applies the concept of a function, identifies the range and domain of a function, and uses function notation appropriately

3.1i Interprets functions that arise in applications in terms of the context and analyzes key features of functions (intercepts, max, min, zeros, asymptotes)

3.1j Analyzes functions (linear, quadratic, square root, piecewise, polynomial, exponential, logarithmic) using different representations such as graphs, verbal, equivalent algebraic forms, and numeric tables

3.1k Constructs and compares linear, quadratic, and exponential models and distinguishes between those situations that can be modeled with linear functions and those that can be modeled with exponential functions

## Vocabulary

 Range: The set of all output values of a function. Example: For the function f(x) =x² where x={1,2,3...} then the range will be {1,4,9,...} Domain: All the values that go into a function (input). For a lesson on how to find domain and range of a function click here. Intercepts: The point at which a curve intersects an axis. x-intercept: the point where the curve crosses the x-axis y-intercept: the point where the curve crosses the y-axis Zeros: A value of "x" which makes a function f(x) equal 0. Max: The point where the height of the function at "a" is greater than the height anywhere else in that interval.   Min: The point where the height of the function at "a" is less than the height anywhere else in that interval. Asymptotes: A straight line on a graph that represents a limit for a given function. You can imagine a curve that comes closer and closer to a line without actually crossing it. Quadratic Function: A function of the form ax²+bx+c, has a "U-shaped" graph called a parabola. Example: Linear Function: A function that can be represented as a straight line of the form f(x)=mx+b Example: Piecewise Function: Functions that behave differently based on the input (x) value. Example: Polynomial Function: An expression that can have constants, variables, and exponents but no division by a variable, only positive exponents, a non-infinite number of terms. Example: Exponential Function: A function of the form y= Example: Logarithmic Function: A function of the form Example:

## Practice Problems

Here is a video about how to find the domain and range of a function:

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For more help and practice with graphs you can watch a video showing teachers learning about interpreting graphs here, and more practice problems here.

# Performance Indicators

Section 3.2: Measurement and Geometry

3.2a Analyzes attributes of shapes, including symmetry and properties of their lines and angles

3.2b Solves problems involving measurement and conversions of measurement units

3.2c Solves mathematical and real-world problems involving angle measure, perimeter, area, surface area, and volume

3.2d Solves problems involving congruence and analyzes congruence in terms of a sequence of transformations

3.2g Solves problems involving similarity and analyzes similarity in terms of scale factors and similarity transformations

## Practice Problems

The two shapes below are said to be congruent. This means they are the same shape and size.

Each box in Fig. 2 below contains congruent shapes.

The two shapes below are said to be similar. This means they are the same shape but can be different sizes.

Each box in Fig. 4 contains similar shapes.

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# Performance Indicators

3.2e Graphs points and shapes on the coordinate plane to solve mathematical and real world problems

3.2f Apples the Pythagorean theorem to solve a variety of problems, including distance problems in the coordinate plane

## Vocabulary

Some words to know:

 Coordinate Plane: the plane created by the horizontal x-axis and the vertical y-axis. Pythagorean Theorem:

## Practice Problems

The pictures above show an empty coordinate plane and a triangle showing the Pythagorean Theorem, . Click here for a link on how to solve a distance problem using the Pythagorean Theorem (Click on "student" box in upper right corner).

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Images from: http://www.mathatube.com/glo-y-axis.html

# Performance Indicators

Section 3.3: Data, Statistics, and Probability

3.3a Represents, analyzes, and solves problems with data presented in various forms (line plots, bar graphs, picture graphs)

3.3b Demonstrates knowledge of statistical variability and measures and summarizes and describes data distributions

3.3c Demonstrates knowledge of the use of random sampling to draw inferences about a population

## Practice Problems

Here is a question to try.

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# Performance Indicators

Section 4.1: Instruction in Number and Operations and Algebraic Thinking

This portion of the study guide will cover Instruction. Each performance indicator addresses math curriculum from grades K-8.

The corresponding Common Core State Standards are listed above the performance indicators.

4.1a Applies strategies for teaching properties of whole numbers, counting, methods for composing and decomposing numbers, and multiple ways of representing numbers

4.1b Demonstrates knowledge of strategies for teaching place value concepts

4.1c Demonstrates knowledge of strategies that build understanding of the equal sign and the meaning of equations

4.1d Applies strategies for developing students' fluency with number operations

4.1e Applies strategies for teaching operations and the relationship between operations

4.1f Applies methods for teaching how to represent and solve one and two step problems involving addition, subtraction, multiplication, and division

4.1g Applies methods for teaching how to round, add, subtract, multiply, and divide multidigit numbers

4.1h Applies strategies for teaching and justifying computational algorithms

4.1i Applies methods for extending students' understanding of numbers to the system of rational numbers, including concepts associated with ordering, absolute value, and negative numbers

4.1j Applies strategies for extending students' understanding of arithmetic and the order of operations to algebraic expressions

4.1k Applies strategies for teaching the meaning of equations and inequalities and how to solve them

4.1l Applies strategies for teaching how to use variables to represent and analyze relationships between dependent and independent variables

## Practice Problems

Here are some problems that help you apply methods for teaching these concepts.

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For a more in-depth study of children's thinking about math, Session 10 in the Annenberg series is exceptional. You can watch video and practice identifying children's thinking here.

Problem 1 adapted from: Hill, H., Ball, D., & Schilling, S. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers' topic-specific knowledge of students. Journal for Research in Mathematics Education. 39(4) 372-400.

Problem 2 adapted from the 2011 Praxis Conference at https://www.ets.org/s/educator_licensure/ckt_handout.pdf

# Performance Indicators

Section 4.2: Instruction in Fractions and Ratios and Proportional Relationships

4.2a Applies methods for teaching how to develop understanding of fractions as numbers

4.2b Applies strategies for extending understanding of fraction equivalence and ordering

4.2c Demonstrates knowledge of strategies for teaching how to build fractions from unit fractions by applying and extending understanding of operations of whole numbers

4.2e Demonstrates knowledge of strategies for teaching the use of equivalent fractions as a strategy to add and subtract fractions

4.2d Demonstrates knowledge of strategies for teaching decimal notation for fractions and for performing operations with decimal

4.2f Applies strategies for teaching concepts of rate, ratio, unit rates, ratio language, and ratio relationships and for teaching connections between multiplication, division, ratio, rates, and fractions

4.2g Analyzes strategies for teaching the use of ratio and rate reasoning to solve real-world and mathematical problems

4.2h Demonstrates knowledge of strategies for teaching how to use ratio reasoning to convert mathematical units

4.2i Applies techniques for teaching unit rate problems, including those involving unit pricing and constant speed, and for teaching ratios of lengths, areas, and other quantities measured in like or unlike units

## Practice Problems

Here are some problems that help you apply methods for teaching these concepts.

 Show/hide comprehension question... Show/hide comprehension question... Show/hide comprehension question...

Questions adapted from the 2011 Praxis Exam at https://www.ets.org/s/educator_licensure/ckt_handout.pdf

Questions adapted from Mathematical Knowledge for Teaching (MKT) Measures at http://sitemaker.umich.edu/lmt/files/LMT_sample_items.pdf

# Performance Indicators

Section 4.3: Instruction in Measurement and Data

4.3a Applies strategies for teaching how to describe and compare measurable attributes

4.3b Applies strategies for teaching how to classify and count objects in categories

4.3c Demonstrates knowledge of strategies for teaching how to measure indirectly by iterating length units and how to measure and estimate lengths in standard units

4.3e Applies strategies for teaching how to compare, create, and compose shapes and how to analyze attribute of shapes, including symmetry and properties of their lines and angles

4.3g Applies strategies for teaching how to classify objects and generate and represent measurement data and angles

4.3f Selects strategies for teaching how to tell and write time and work with money

4.3d Applies strategies for relating addition and subtraction to length and for relating multiplication and division to area

4.3h Applies strategies for teaching concepts of perimeter, area, and volume and their relationships to number operations=

4.3i Applies strategies for teaching how to generate and represent measurement data and solve problems with data (line plots, bar graphs, picture graphs)

4.3j Applies strategies for developing understanding of statistical concepts (statistical variability, data collection, measures of center, shapes of data distributions)

## Practice Problems

Here are some problems that help you apply methods for teaching these concepts.

Show/hide comprehension question...

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# Performance Indicators

Part 5: Analysis, Synthesis, and Application

5a Analyzes and interprets samples of a student's work and other assessment data to monitor student progress and determine a students' strengths and areas of need in mathematics

5b Demonstrates knowledge of the content by identifying and analyzing any errors or misconceptions in work samples

5c Describes appropriate and effective content-specific instructional strategies, activities, or interventions to address a student's identified needs

5d Demonstrates the ability to generate real-world scenarios that illustrate specific mathematical concepts

5e Demonstrates the ability to justify the effectiveness of selected instruction strategies, activities, or interventions for prompting a student's mathematical understanding

The indicators related to Part 5 apply to the extended response (essay) question on the Mathematics part of the Multi-Subject Content Specialty Test. You will be analyzing the mathematical reasoning of a particular student, describing the strengths and weaknesses of that reasoning and designing an instructional intervention to build on the strengths and improve the reasoning of this student. When you take the CST, you might want to do this question first, while you are fresh, so that you write enough to meet the expectations of the task.

For this question, you will receive background information about a specific class and the Common Core Learning Standard that they are working with. You will receive a description of the class activity. Then you will see a transcript of an interview with the child. Make sure that you:

• identify a significant mathematical strength related to the given standard that is demonstrated by the student, citing specific evidence from the exhibits to support your assessment;
• identify a significant area of need related to the given standard that is demonstrated by the student, citing specific evidence from the exhibits to support your assessment; and
• describe an instructional intervention that builds on the student's strengths and that would help the student improve in the identified area of need.
• Include a strategy for helping the student build a viable argument related to the given standard.

You might want to plan on four paragraphs, one for each bullet point above with at least two specific pieces of evidence in each paragraph for the first two bullets and very specific interventions and strategies for the last two bullets.

Here are examples of some interventions you could use. Make sure you explicitly state how your intervention builds on the student's strengths while helping the student improve in the area of need.

• manipulatives such as fraction strips, base ten blocks, pattern blocks, fraction circles, unifix cubes, counters
• visual and graphic representations of problems such as number lines

When you are talking about helping the student build a viable argument, you may wish to discuss:

• student think alouds
• cooperative learning
• use of math journals
• teacher questioning

You will be seeking a score of 4. Responses scored as a 4 are described in the test preparation materials as responses that:

• thoroughly address all parts of the assignment.
• demonstrate the relevant knowledge and skills with thorough accuracy and effectiveness.
• are well supported by relevant examples and details and thoroughly demonstrate sound reasoning.

Here is a link to the sample Constructed Response Item that the New York State Education Department had released.

Here are links to more resources!

Here are the resources from the New York State Education Department

Here is a link to the New York State Education Department document that outlines the Multi-Subject CST Framework for Grades 1-6.

Here is a link to the New York State Education Department sample test items. Some of these appeared in the module you just completed.

For \$29.95, you can purchase a sample test from NYSED and have it scored. Here is the link for that.

Your practice test can be submitted for scoring only once, but it will be available for your review until the end of your subscription period (30 days). After you submit your practice test for scoring, you will receive immediate results showing your objective- or competency-level performance on multiple-choice questions.

Here are some more problems to try and more content-learning sources:

For a practice page on multiplying and dividing fractions cick here.

Below is a book you may wish to purchase:

Van de Walle, J. (2013). Elementary and Middle School Mathematics: Teaching Developmentally. New York, NY: Pearson.

Your students will thank you for all of the time you have spent polishing up your mathematics teaching and learning skills. Good luck!