Multi-Subject CST Mathematics Preparation Competency 0001- Number Systems and Quantity 1.1 Number Systems and Vector and Matrix Quantities February 2016 © NYC Teaching Fellows 2014 Agenda Introduction to Competency Content Review and Sample Problems Additional Practice / 2 Introduction to Competency 0001- NUMBER AND QUANTITY Performance Expectations The New York State Grade 7-12 Multi-Subject teacher • Demonstrates knowledge of the properties of numbers, number systems, and how number systems are extended. • Demonstrates understanding of real and complex numbers and understands the complex number system as an extension of the real number system. • Applies properties of complex numbers, works accurately with real numbers, and uses them to solve mathematical and real-world problems. • Works with vectors and matrices. • Has a deep understanding of ratios and proportional relationships, and applies connections between multiplication and division and ratios and rates. • Analyze relationships between ratios and fractions, solves problems involving ratios and rates, and demonstrates the ability to work accurately with ratios and proportional relationships. New York State Education Department / 3 Introduction to Competency 0001- Number Systems and Vector and Matrix Quantities 1.1 Number Systems and Vector and Matrix Quantities • Applies properties of signed rational numbers, ordering, and absolute value of rational numbers • Solves mathematical and real-world problems involving the four basic operations with rational numbers • Applies and extends understanding of arithmetic and the order of operations to algebraic expressions, equations, and inequalities • Uses rational approximations of irrational numbers (e.g., comparing the size of irrational numbers, locating irrational numbers on the number line) • Knows properties of repeating decimal expansions and converts between repeating decimal expansions and rational numbers • Performs operations with numbers expressed in scientific notation • Works with radicals and exponents and rewrites expressions involving radicals and rational exponents using the properties of exponents New York State Education Department / 4 Introduction to Competency 0001- Number Systems and Vector and Matrix Quantities 1.1 Number Systems and Vector and Matrix Quantities • Uses units as a way to understand problems and to guide the solution of multistep problems, and chooses and interprets units consistently in formulas • Performs arithmetic operations with complex numbers and represents complex numbers and their operations on the complex plane • Uses complex numbers to solve quadratic equations with real coefficients that have complex solutions • Represents and models with vector quantities and performs operations on vectors both algebraically and graphically • Performs basic operations on matrices and uses matrices in applications • Demonstrates knowledge of how to analyze and interpret assessment data to inform and plan instruction that engages and challenges all students to meet or exceed the NYCCLS related to number systems and vector and matrix quantities New York State Education Department / 5 Agenda Introduction to Competency Content Review and Sample Problems Additional Practice / 6 Number Systems: Applies properties of signed rational numbers, ordering, and absolute value of rational numbers Signed rational numbers • What is a rational number? o Any number that can be expressed as a ratio or two integers. Basically, a number that can be written in fractional form. o Watch - Understanding Rational Numbers • What is a signed rational number? o A positive or negative number that can be written as a fraction. Practice- Identify rational numbers New York State Education Department / 7 Number Systems: Applies properties of signed rational numbers, ordering, and absolute value of rational numbers Ordering Rational Numbers • How to order signed rational numbers o To order rational numbers, first write them in the same form and then order the numbers. For example, convert all numbers to fractions or decimals. o Watch- Ordering Rational Numbers Practice- Ordering rational numbers New York State Education Department / 8 Number Systems: Applies properties of signed rational numbers, ordering, and absolute value of rational numbers Absolute Value • What is absolute value? o Absolute value describes the distance of a number on the number line from zero without considering which direction from zero the number lies. o The absolute value of a number is never zero. o The symbol for absolute value is two straight lines surrounding the number. o Watch- Find absolute value o Read- Absolute Value Practice-Absolute value of rational numbers New York State Education Department / 9 Number Systems: Applies properties of signed rational numbers, ordering, and absolute value of rational numbers Operations with Rational Numbers • Adding integers o Rule- If signs are the same, ADD and keep the signs. If signs are different (one positive and one negative) SUBRTRACT the values and use the sign of the number with the largest absolute value o 4 +18 = 22 o -5 + -6 = -11 o Watch- Adding integers with the same signs o Watch- Adding integers with different signs o Watch- Adding fractions with different signs • Subtracting Integers o Rule- Keep the first number the same, Change subtraction sign to addition, Change the sign of the last number to it’s opposite. Then use addition rules. o OR add the additive inverse o 3 – (-9) 3 + 9 = 12 o -8 – 7 -8 + -7 = -15 o Watch- Subtracing Integers o Watch- Subtracting fractions Practice- Add and subtract integers New York State Education Department / 10 Number Systems: Applies properties of signed rational numbers, ordering, and absolute value of rational numbers Operations with Rational Numbers • Multiplying Integers o Rule- If signs are the same the product is positive. If signs are different (one positive and one negative) the product is negative o -4 x -3 = 12 o 8 x (-2) = - 16 • Dividing Integers o Rule- If signs are the same the quotient is positive. If signs are different (one positive and one negative) the quotient is negative o -8 ÷ (-2) = 4 o -15 ÷ 3 = -5 • Watch- Multiplying and dividing integers • Practice- Multiply and divide integers New York State Education Department / 11 Number Systems: Solves mathematical and real-world problems involving the four basic operations with rational numbers Applying operations with rational numbers to mathematical situations (expressions and equations) • Simplify an expression by distributing a negative -3(x + 6) Distribute the -3 to both terms in the parentheses -3(x) + -3(6) Then multiply using integer operation rules -3x - 18 o Watch- Distributive property with integers Practice- Simplify expressions involving integers • Evaluate expressions with integers Evaluate the expression -2x + 6y if x = -5 and y = -4 Substitute -5 for x and -4 for y and then solve using order of operations -2(-5) + 6(-4) Now multiply 10 – 24 Subtract using integer operations -14 o Watch- Evaluate variable expressions with integers Practice- Evaluate variable expressions involving integers New York State Education Department / 12 Number Systems: Solves mathematical and real-world problems involving the four basic operations with rational numbers Applying operations with rational numbers to real-world problems • Calculations with rational numbers are used when recording investment transactions. o Deposits are added to an account balance; money is deposited into the account. o Withdraws are subtracted from an account balance; money is taken out of the account. o Gains are added to an account balance; they are positive returns on the investment. o Losses are subtracted from an account balance; they are negative returns. • Temperature is another way to work with rational numbers o Watch -Use addition and subtraction to solve real-world problems involving decimals o Watch- Negative number word problems Practice- Integer addition and subtraction word problems New York State Education Department / 13 Number Systems: Applies and extends understanding of arithmetic and the order of operations to algebraic expressions, equations, and inequalities Order of operations • Order of operations is a rules that defines which procedures to perform first in a given mathematical expression. o Parentheses- Perform operations in the parentheses first o Exponents- Perform any work with exponents or radicals o Multiplication and Division- working from left to right, do all multiplication and division o Addition and Subtraction- - working from left to right, do all addition and subtraction -8 2 2 + 7 ( -4 + 1) Add in parentheses first -8 2 2 + 7 (-3) Exponents -8 4 + 7 (-3) Multiplication -32 + -21 Addition -53 • Watch- Order of operations with integers Practice- Order of operations with integers New York State Education Department / 14 Number Systems: Applies and extends understanding of arithmetic and the order of operations to algebraic expressions, equations, and inequalities Equations • Solving equations with rational numbers is the same as solving basic equations. o Solving an equation means getting the variable alone on one side of the equation to find it’s value. o To get the variable alone, you use inverse operations to undo what has been done to the variable. o Addition and subtraction are inverse operations. o Whatever you do to one side of the equation, you must also do the other side to maintain the equality. x + 5.7 = 2.5 Subtract 5.7 from each side x + 5.7 – 5.7 = 2.5 – 5.7