Rounding a method for estimating a number by increasing or retaining a specific place value digit according to specific rules and changing all

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1 Unit 1 This unit bundles student expectations that address whole number estimation and computational fluency and proficiency. According to the Texas Education Agency, mathematical process standards including application, a problem-solving model, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace. During this unit, students work toward becoming fluent when solving whole number addition, subtraction, and multiplication problems and proficient when solving whole number division problems. Estimating solutions to problems involving all four operations becomes embedded within one- and two-step real-world contexts to help determine the reasonableness of solutions. The number set for multiplication grows to three-digit by two-digit factors while the number set for division becomes four-digit dividends by two-digit divisors. Using the standard algorithm becomes a focus for both multiplication and division. Vocabulary Compatible numbers a method for estimating a group of numbers by slightly adjusting some or all of the numbers to allow for easy mental computation Compensation a method for adjusting an estimate to draw closer to an exact calculation Counting (natural) numbers the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3,..., n } Dividend the number that is being divided Divisor the number the dividend is being divided by Estimation reasoning to determine an approximate value Factor a number multiplied by another number to find a product Fluency efficient application of procedures with accuracy Front-end method a method for estimating a number in which the first digit of a number is retained and all remaining digits are changed to zero Product the total when two or more factors are multiplied Proficiency the mathematical ability to reason, apply, understand, compute, and engage in meaningful mathematics Quotient the size or measure of each group or the number of groups when the dividend is divided by the divisor

2 Rounding a method for estimating a number by increasing or retaining a specific place value digit according to specific rules and changing all trailing digits to zero Strip diagram a linear model used to illustrate number relationships Whole numbers the set of counting (natural) numbers and zero {0, 1, 2, 3,..., n}

3 Unit 2 This unit bundles student expectations that address prime and composite numbers, grouping symbols, simplifying expressions, and solving equations. During this unit, students are introduced to the properties of prime and composite numbers. Students are encouraged to use previous understandings of representing products using arrays and area models to help support new learning and to identify prime and composite numbers by analyzing their factors. Students examine the meaning of grouping symbols within a numeric expression and simplify numerical expressions based on socially constructed conventions. In addition, students represent and solve multi-step problems algebraically using an equation with a letter representing the unknown. All operations within this unit are limited to whole numbers. Vocabulary: Composite number a whole number with more than two factors Counting (natural) numbers the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3,..., n } Dividend the number that is being divided Divisor the number the dividend is being divided by Equation a mathematical statement composed of algebraic and/or numeric expressions set equal to each other Expression a mathematical phrase, with no equal sign or comparison symbol, that may contain a number(s), an unknown(s), and/or an operator(s) Factor a number multiplied by another number to find a product Grouping symbols symbols to show a group of terms and/or expressions within a mathematical expression Order of operations the rules of which calculations are performed first when simplifying an expression Prime number a whole number greater than 1 with exactly two factors, 1 and the number itself Product the total when two or more factors are multiplied Quotient the size or measure of each group or the number of groups when the dividend is divided by the divisor Whole numbers the set of counting (natural) numbers and zero {0, 1, 2, 3,..., n }

4 Unit 3 This unit bundles student expectations that address managing financial resources and records effectively, defining different types of taxes, and analyzing different methods of payment. During this unit, students define different types of taxes including income tax, sales tax, property tax, and payroll tax. Students take a closer look at payroll tax when they investigate the differences between gross income and net income. They also compare and contrast various methods of payment including checks, credit cards, debit cards, and electronic payments. Students develop a system for keeping financial records with regards to income and expenses and balance a simple budget. The concept of negative values is abstractly embedded within balancing budgets as students are expected to describe the actions that might be taken should expenses exceed the income. Vocabulary Available balance the amount available in an account for a person, business, or organization to spend Balance to reconcile your budget or account statement with your check register to make sure the records match and are accurate Budget a monthly or yearly spending and savings plan for an individual, family, business, or organization Check a written document telling the financial institution to pay a specific amount of money from your account to a specific person or organization Credit card a card that can be used to borrow money from financial institutions, stores, or other businesses in order to buy products and services on credit Debit card a bankcard issued by a financial institution that is electronically linked to an individual s checking account for the purpose of making banking transactions, making payments for services, and/or making purchases Electronic payment (e-payment) payments using security features on the Internet Expense payment for goods and services Financial records a formal record of the transactions made by a person, business, or other organization Gross income the total amount of personal income prior to taxes and deductions

5 Income money earned or received Income tax money paid by an employee to the federal government based on an individual's income as required by law Net income the income that remains after taxes and other deductions are taken from an individual s gross income Payroll tax money paid by an employer to the federal government based on employee income as required by law Property tax the amount of money collected from the property owner based on the value of a property for the local government as required by law Register a small table to track deposits added to account, expenses withdrawn from account, and current available balance Sales tax the amount of money collected by a store (retailer), in addition to a good or service that was purchased, for the local government as required by law Transaction a specific time or instance when money changes hands

6 Unit 4 This unit bundles student expectations that address foundational understandings of decimals as well as adding and subtracting of decimals through the thousandths place. During this unit, students are formally introduced to the thousandths place. Students build upon the idea that our base-10 place value system extends infinitely to very small values as well as very large values, and that each place-value position is one-tenth the value of the place to its left and 10 times the value of the place to its right. Students relate previous representations of decimals to the hundredths with concrete and pictorial models to develop their conceptual knowledge of decimals through the thousandths. Students are expected to use expanded notation and numerals to represent the value of a decimal through the thousandths. Students use comparison symbols to compare and order decimals to the thousandths and round decimals to the tenths or hundredths place. Students continue to estimate solutions and extend addition and subtraction with decimals to include the thousandths place. Numerical expressions are revisited as a means for students to communicate their solution process and to solve problem situations involving decimals. Vocabulary Compare numbers to consider the value of two numbers to determine which number is greater or less or if the numbers are equal in value Compatible numbers a method for estimating a group of numbers by slightly adjusting some or all of the numbers to allow for easy mental computation Compensation a method for adjusting an estimate to draw closer to an exact calculation Counting (natural) numbers the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3,..., n } Decimal number a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part

7 Digit any numeral from 0 9 Estimation reasoning to determine an approximate value Expanded form the representation of a number as a sum of place values (e.g., 985,156, as 900,000, ,000, ,000, , , , or as 900,000, ,000, ,000, , , , ) Expanded notation the representation of a number as a sum of place values where each term is shown as a digit(s) times its place value (e.g., 985,156, as 9(100,000,000) + 8(10,000,000) + 5(1,000,000) + 1(100,000) + 5(10,000) +6(1,000) + 7(100) + 8(10) (0.1) + 8(0.01) + 2 (0.001) or as 9(100,000,000) + 8(10,000,000) + 5(1,000,000) + 1(100,000) + 5(10,000) +6(1,000) + 7(100) + 8(10) ( ) + 8( ) + 2 ( )) Expression a mathematical phrase, with no equal sign or comparison symbol, that may contain a number(s), an unknown(s), and/or an operator(s) Fluency efficient application of procedures with accuracy Front-end method a method for estimating a number in which the first digit of a number is retained and all remaining digits are changed to zero Grouping symbols symbols to show a group of terms and/or expressions within a mathematical expression Numeral a symbol used to name a number Order numbers to arrange a set of numbers based on their numerical value Order of operations the rules of which calculations are performed first when simplifying an expression Place value the value of a digit as determined by its location in a number, such as ones, tens, hundreds, one thousands, ten thousands, etc. Positive rational numbers the set of numbers that can be expressed as a fraction, where a and b are counting (natural) numbers Rounding a method for estimating a number by increasing or retaining a specific place value digit according to specific rules and changing all trailing digits to zero Standard form the representation of a number using digits (e.g., 985,156, ) Trailing zeros a sequence of zeros in the decimal part of a number that follow the last non-zero digit, and whether recorded or deleted, does not change the value of the number Whole numbers the set of counting (natural) numbers and zero {0, 1, 2, 3,..., n } Word form the representation of a number using written words (e.g., 985,156, as nine hundred eighty-five million, one hundred fifty-six thousand, seven hundred eighty-nine and seven hundred eighty-two thousandths)

8 Unit 5 This unit bundles student expectations that address determining products and quotients of decimals. During this unit, students represent multiplicative structures (multiplication and division) of problem situations with products and decimals to the hundredths with the use of concrete objects, pictorial models, and area models. These models serve as a bridge between whole-number multiplication and division and should help students to assimilate new understandings involving decimal multiplication and division. There is an emphasis on representing multiplication and division before solving for products and quotients of decimals to allow for the development of the conceptual understanding before procedural understanding. Students are expected to estimate to determine products and quotients, solve for products and quotients, and simplify numerical expressions that include multiplication and division of whole numbers and decimals. The number set within this unit is limited to products and quotients to the hundredths. Factors may include decimals through the thousandths place as long as the product is only through the hundredths place. Division is limited to four-digit dividends and two-digit whole number divisors. Vocabulary Associative property of multiplication if three or more factors are multiplied, they can be grouped in any order, and the product will remain the same Commutative property of multiplication if the order of the factors are changed, the product will remain the same Compatible numbers a method for estimating a group of numbers by slightly adjusting some or all of the numbers to allow for easy mental computation Compensation a method for adjusting an estimate to draw closer to an exact calculation

9 Counting (natural) numbers the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3,..., n } Decimal number a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part Distributive property of multiplication if multiplying a number by a sum of numbers, the product will be the same as multiplying the number by each addend and then adding the products together Dividend the number that is being divided Divisor the number the dividend is being divided by Estimation reasoning to determine an approximate value Expression a mathematical phrase, with no equal sign or comparison symbol, that may contain a number(s), an unknown(s), and/or an operator(s) Factor a number multiplied by another number to find a product Front-end method a method for estimating a number in which the first digit of a number is retained and all remaining digits are changed to zero Grouping symbols symbols to show a group of terms and/or expressions within a mathematical expression Order of operations the rules of which calculations are performed first when simplifying an expression Product the total when two or more factors are multiplied Quotient the size or measure of each group or the number of groups when the dividend is divided by the divisor Rounding a method for estimating a number by increasing or retaining a specific place value digit according to specific rules and changing all trailing digits to zero Whole numbers the set of counting (natural) numbers and zero {0, 1, 2, 3,..., n }

10 Unit 6 This unit bundles student expectations that address adding and subtracting fractions with unequal denominators and applying previous understandings of adding and subtracting decimals. During this unit, students extend previous understandings of adding and subtracting whole numbers and decimals with the expectation of developing fluency in adding and subtracting positive rational numbers, including fractions. Students continue to estimate solutions to mathematical and real-world problems to determine the reasonableness of solutions. Concrete objects, pictorial models, and properties of operations are used to represent and solve problems involving adding and/or subtracting fractions with unequal denominators referring to the same whole. Previous grade level expectations regarding adding and subtracting fractions with equal denominators using models that build to the number line serve as a bridge to adding and subtracting fractions with unequal denominators. Students also continue to simplify numerical expressions that involve adding and subtracting fractions with unequal denominators. Vocabulary Associative property of addition if three or more addends are added, they can be grouped in any order, and the sum will remain the same Commutative property of addition if the order of the addends are changed, the sum will remain the same Compatible numbers a method for estimating a group of numbers by slightly adjusting some or all of the numbers to allow for easy mental computation Compensation a method for adjusting an estimate to draw closer to an exact calculation Counting (natural) numbers the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3,..., n } Decimal number a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part Denominator the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set Estimation reasoning to determine an approximate value Expression a mathematical phrase, with no equal sign or comparison symbol, that may contain a number(s), an unknown(s), and/or and operator(s)

11 Fluency efficient application of procedures with accuracy Fraction a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division. Front-end method a method for estimating a number in which the first digit of a number is retained and all remaining digits are changed to zero Grouping symbols symbols to show a group of terms and/or expressions within a mathematical expression Improper fraction a number in the form where a and b are whole numbers and a > b where b is not equal to zero Least common denominator (LCD) the least common multiple of the denominators of two or more fractions Mixed number a number that is composed of a whole number and a fraction Numerator the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered Order of operations the rules of which calculations are performed first when simplifying and expression Positive rational numbers the set of numbers that can be expressed as a fraction, where a and b are counting (natural) numbers Proper fraction a number in the form where a and b are whole numbers and a < b where b is not equal to zero Rounding a method for estimating a number by increasing or retaining a specific place value digit according to specific rules and changing all trailing digits to zero Unit fraction a fraction in the form representing the quantity formed by one part of a whole that has been partitioned into b equal parts where b is a non-zero whole number Whole numbers the set of counting (natural) numbers and zero {0, 1, 2, 3,..., n }

12 Unit 7 This unit bundles student expectations that address representing and solving multiplication and division problems involving a whole number and a fraction. During this unit, students represent a product of whole number and a fraction referring to the same whole using concrete and pictorial models, including area models and strip diagrams. Concrete and pictorial models, including area models and strip diagrams, are also used to represent the division of a whole number by a unit fraction and the division of a unit fraction by a whole number. Students continue to estimate solutions to mathematical and real-world problems to determine the reasonableness of solutions. Students also continue to simplify numerical expressions that involve all operations with whole numbers, decimals, and fractions. This unit concludes the formal study of operations with whole numbers, fractions, and decimals in Grade 5. Vocabulary Compatible numbers a method for estimating a group of numbers by slightly adjusting some or all of the numbers to allow for easy mental computation Compensation a method for adjusting an estimate to draw closer to an exact calculation Counting (natural) numbers the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3,..., n } Decimal number a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part Denominator the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set Dividend the number that is being divided Divisor the number the dividend is being divided by Estimation reasoning to determine an approximate value

13 Expression a mathematical phrase, with no equal sign or comparison symbol, that may contain a number(s), an unknown(s), and/or an operator(s) Factor a number multiplied by another number to find a product Fraction a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division. Front-end method a method for estimating a number in which the first digit of a number is retained and all remaining digits are changed to zero Grouping symbols symbols to show a group of terms and/or expressions within a mathematical expression Improper fraction a number in the form where a and b are whole numbers and a > b where b is not equal to zero Mixed number a number that is composed of a whole number and a fraction Numerator the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered Order of operations the rules of which calculations are performed first when simplifying an expression Product the total when two or more factors are multiplied Proper fraction a number in the form where a and b are whole numbers and a < b where b is not equal to zero Quotient the size or measure of each group or the number of groups when the dividend is divided by the divisor Rounding a method for estimating a number by increasing or retaining a specific place value digit according to specific rules and changing all trailing digits to zero Strip diagram a linear model used to illustrate number relationships Unit fraction a fraction in the form representing the quantity formed by one part of a whole that has been partitioned into b equal parts where b is a non-zero whole number Whole numbers the set of counting (natural) numbers and zero {0, 1, 2, 3,..., n }

14 Unit 8 This unit bundles student expectations that address perimeter, area, volume, and converting within a measurement system, customary or metric. During this unit, students are introduced to the concept of volume as a three-dimensional measure. Students are expected to understand the concept that a cube with a side length of one unit is a unit cube having one cubic unit of volume, and the volume of a three-dimensional figure as the number of unit cubes ( n cubic units) needed to fill the figure with no gaps or overlaps. Students use objects and pictorial models to develop the formulas for volume of a rectangular prism ( V = l x w x h and V = Bh ), including the special form for the volume of a cube ( V = s x s x s ). Grade 5 is the first grade level where students are formally introduced to formulas for volume as seen on the STAAR Grade 5 Mathematics Reference Materials. Students use these formulas to represent and solve problems related to perimeter and/or area and volume. Formulas are used to establish the concept that the volume of a rectangular prism with whole number side lengths is related to the number of layers times the number of unit cubes in the area of the base. Students extend previous knowledge of classifying two-dimensional figures based on the presence or absence of parallel or perpendicular lines or angles of a specified size to formally classify two-dimensional figures into a hierarchy of sets and subsets using graphic organizers. Students extend previous work with conversions to solve problems by calculating conversions within a measurement system. Measurements for side lengths of two- or three-dimensional figures and measurement conversions may include positive rational numbers within the number system and operational limitations for the grade level.

15 Vocabulary Acute an angle that measures less than 90 Angle two rays with a common endpoint (the vertex) Angle congruency marks angle marks indicating angles of the same measure Area the measurement attribute that describes the number of square units a figure or region covers Attributes of two-dimensional figures characteristics that define a geometric figure (e.g., sides, vertices, etc.) Classify applying an attribute to categorize a sorted group Congruent of equal measure, having exactly the same size and same shape Conversion a change from one measurement unit to another measurement unit without changing the amount Irregular Figure a polygon with sides and/or angles that are not all congruent Obtuse an angle that measures greater than 90 but less than 180 Perimeter a linear measurement of the distance around the outer edge of a figure Polygon a closed figure with at least 3 sides, where all sides are straight (no curves) Properties of two-dimensional figures relationship of attributes within a geometric figure (e.g., a square has 4 congruent sides and 4 right angles, etc.) and between a group of geometric figures (e.g., a square and a rectangle both have 4 sides and 4 right angles; however, a square has 4 congruent sides but a rectangle has only opposite sides congruent; etc.) Regular figure a polygon with all sides and angles congruent Right an angle (formed by perpendicular lines) that measures exactly 90 Side a line segment that forms the boundary of a two-dimensional figure Side congruency marks side marks indicating side lengths of the same measure Three-dimensional figure a figure that has measurements including length, width (depth), and height Two-dimensional figure a figure with two basic units of measure, usually length and width Vertex (vertices) in a two-dimensional figure the point (corner) where two sides of a two-dimensional figure meet Volume the measurement attribute of the amount of space occupied by matter

16 Unit 9 This unit bundles student expectations that address additive and multiplicative numerical patterns as well as graphing in the first quadrant of the coordinate plane During this unit, students are introduced to the coordinate plane and its key attributes including the axes and origin. Students analyze the process for graphing ordered pairs in the first quadrant of the coordinate plane and relate the first number in an ordered pair to indicate the movement parallel to the x -axis starting at the origin and the second number in the ordered pair to indicate the movement parallel to the y -axis starting at the origin. Although graphing is limited to the Quadrant I of the coordinate plane, ordered pairs may include any positive rational number, including fractions and decimals. Students are expected to graph ordered pairs in the first quadrant of the coordinate plane that are generated from number patterns or an input-output table. Number patterns are examined closely as students recognize the difference between additive and multiplicative numerical patterns when given in a table or graph. Students use input-output tables and graphs to generate numerical patterns when given a rule in the form y = ax (multiplicative numerical pattern) or y = a + x (additive numerical pattern). Vocabulary Additive numerical pattern a pattern that occurs when a constant non-zero value is added to an input value to determine the output value ( y = x + a ) Axes the vertical and horizontal lines that act as a reference when plotting points on a coordinate plane Coordinate plane a two-dimensional plane on which to plot points, lines, and curves Counting (natural) numbers the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3,..., n } Decimal number a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part

17 Factor a number multiplied by another number to find a product Fraction a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division. Improper fraction a number in the form where a and b are whole numbers and a > b where b is not equal to zero Input-output table a table which represents how the application of a rule on a value, input, results in a different value, output Intersecting lines lines that meet or cross at a point Mixed number a number that is composed of a whole number and a fraction Multiplicative numerical pattern a pattern that occurs when a constant non-zero value is multiplied by an input value to determine the output value ( y = ax) Origin the starting point in locating points on a coordinate plane Perpendicular lines lines that intersect at right angles to each other to form square corners Product the total when two or more factors are multiplied Proper fraction a number in the form where a and b are whole numbers and a < b where b is not equal to zero Quadrants any of the four areas created by dividing a plane with an x- axis and y- axis Whole numbers the set of counting (natural) numbers and zero {0, 1, 2, 3,..., n }

18 Unit 10 This unit bundles student expectations that address categorical and numerical data, and discrete paired data. During this unit, students represent categorical data with bar graphs and frequency tables. Numerical data, including data sets of measurements in fractions or decimals, is represented with dot plots or stem-and-leaf plots. Students are introduced to scatterplots as a means to represent discrete paired data. Students utilize all of these graphical representations to solve one- and two-step problems. Vocabulary Bar graph a graphical representation to organize data that uses solid bars that do not touch each other to show the frequency (number of times) that each category occurs Categorical data data that represents the attributes of a group of people, events, or objects Data information that is collected about people, events, or objects Discrete data data with finite and distinct values that can be counted, not inclusive of in-between values Dot plot a graphical representation to organize small sets of data that uses dots (or Xs) to show the frequency (number of times) that each number occurs Frequency table a table to organize data that lists categories and the frequency (number of times) that each category occurs Numerical data data that represents values or observations that can be measured and placed in ascending or descending order Scatterplot a graphical representation used to display the relationship between discrete data pairs Stem-and-leaf plot a graphical representation used to analyze and compare groups or clusters of numerical data by separating one place value from another place value of a data set. The larger of the two place values is called the stem and the smaller of the two place values is called the leaf.

19 Unit 11 This unit bundles student expectations that address examining problem situations involving whole numbers, fractions, and decimals, discerning the necessary operations needed to solve the problems, and verifying and/or justifying the solutions. During this unit, students revisit representing and solving problems involving addition and subtraction of fractions with unequal denominators referring to the same whole using objects and pictorial models and properties of operations. Students also solve addition and subtraction situations involving positive rational numbers. Students revisit representing and solving problems involving the multiplication of a whole number and a fraction that refers to the same whole using objects and pictorial models, including area models. Students continue multiplication as they solve for products of decimals to the hundredths, including situations involving money, using strategies based on place-value understandings, properties of operations, and the relationship of decimal multiplication to the multiplication of whole numbers. Students revisit division as they solve for quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using strategies and algorithms, including the standard algorithm. They divide whole numbers by unit fractions and unit fractions by whole numbers. Students continue to simplify numerical expressions that do not involve exponents, including up to two levels of grouping. Vocabulary Associative property of multiplication if three or more factors are multiplied, they can be grouped in any order, and the product will remain the same Commutative property of multiplication if the order of the factors are changed, the product will remain the same Counting (natural) numbers the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3,..., n }

20 Decimal number a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part Denominator the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set Distributive property of multiplication if multiplying a number by a sum of numbers, the product will be the same as multiplying the number by each addend and then adding the products together Dividend the number that is being divided Divisor the number the dividend is being divided by Expression a mathematical phrase, with no equal sign or comparison symbol, that may contain a number(s), an unknown(s), and/or an operator(s) Factor a number multiplied by another number to find a product Fluency efficient application of procedures with accuracy Fraction a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division. Grouping symbols symbols to show a group of terms and/or expressions within a mathematical expression Improper fraction a number in the form where a and b are whole numbers and a > b where b is not equal to zero Least common denominator (LCD) the least common multiple of the denominators of two or more fractions Mixed number a number that is composed of a whole number and a fraction Numerator the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered Order of operations the rules of which calculations are performed first when simplifying an expression Positive rational numbers the set of numbers that can be expressed as a, where a and b are counting (natural) numbers Product the total when two or more factors are multiplied Proper fraction a number in the form where a and b are whole numbers and a < b where b is not equal to zero Quotient the size or measure of each group or the number of groups when the dividend is divided by the divisor Strip diagram a linear model used to illustrate number relationships Whole numbers the set of counting (natural) numbers and zero {0, 1, 2, 3,..., n } Unit fraction a fraction in the form representing the quantity formed by one part of a whole that has been partitioned into b equal parts where b is a non-zero whole number

21 Unit 12 This unit bundles student expectations that address adding and subtracting of decimals through the thousandths place and determining products and quotients of decimals. During this unit, students revisit and solidify essential understandings of decimals. Students estimate to determine the reasonableness of sums, differences, products, and quotients. They solve situations involving addition and subtraction of decimals through the thousandths. Students represent multiplicative structures (multiplication and division) involving products and quotients using concrete objects, pictorial models, and area models. Factors may include decimals through the thousandths place as long as the product is only through the hundredths place. Division is limited to four-digit dividends and two-digit whole number divisors, with quotients limited to the hundredths. Simplifying numerical expressions is revisited as a means for students to communicate their solution process and to solve problem situations involving decimals. Vocabulary Associative property of multiplication if three or more factors are multiplied, they can be grouped in any order, and the product will remain the same Commutative property of multiplication if the order of the factors are changed, the product will remain the same Compatible numbers a method for estimating a group of numbers by slightly adjusting some or all of the numbers to allow for easy mental computation Compensation a method for adjusting an estimate to draw closer to an exact calculation Counting (natural) numbers the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3,..., n } Decimal number a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part

22 Distributive property of multiplication if multiplying a number by a sum of numbers, the product will be the same as multiplying the number by each addend and then adding the products together Dividend the number that is being divided Divisor the number the dividend is being divided by Estimation reasoning to determine an approximate value Expression a mathematical phrase, with no equal sign or comparison symbol, that may contain a number(s), an unknown(s), and/or an operator(s) Factor a number multiplied by another number to find a product Fluency efficient application of procedures with accuracy Front-end method a method for estimating a number in which the first digit of a number is retained and all remaining digits are changed to zero Grouping symbols symbols to show a group of terms and/or expressions within a mathematical expression Order of operations the rules of which calculations are performed first when simplifying an expression Positive rational numbers the set of numbers that can be expressed as a fraction, where a and b are counting (natural) numbers Product the total when two or more factors are multiplied Quotient the size or measure of each group or the number of groups when the dividend is divided by the divisor Rounding a method for estimating a number by increasing or retaining a specific place value digit according to specific rules and changing all trailing digits to zero Whole numbers the set of counting (natural) numbers and zero {0, 1, 2, 3,..., n }

23 Unit 13 This unit bundles student expectations that address adding and subtracting fractions with unequal denominators, solving multiplication problems involving a whole number and a fraction, and solving division problems involving a whole number and a unit fraction. During this unit, students revisit and solidify essential understandings of fractions. Students estimate to determine the reasonableness of solutions to mathematical and real-world problems involving addition, subtraction, multiplication, and division. Students represent and solve addition and subtraction of fractions with unequal denominators using concrete objects, pictorial models, and properties of operations to build to the expectation of adding and subtracting positive rational numbers fluently. Students use concrete objects and pictorial models to multiply a whole number by a fraction and divide a whole number by a unit fraction and a unit fraction by a whole number. Throughout the unit, students simplify numerical expressions that do not involve exponents. Vocabulary Associative property of addition if three or more addends are added, they can be grouped in any order, and the sum will remain the same Commutative property of addition if the order of the addends are changed, the sum will remain the same Compatible numbers a method for estimating a group of numbers by slightly adjusting some or all of the numbers to allow for easy mental computation Compensation a method for adjusting an estimate to draw closer to an exact calculation Counting (natural) numbers the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3,..., n } Denominator the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set Dividend the number that is being divided Divisor the number the dividend is being divided by Estimation reasoning to determine an approximate value

24 Expression a mathematical phrase, with no equal sign or comparison symbol, that may contain a number(s), an unknown(s), and/or an operator(s) Factor a number multiplied by another number to find a product Fluency efficient application of procedures with accuracy Fraction a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division. Front-end method a method for estimating a number in which the first digit of a number is retained and all remaining digits are changed to zero Grouping symbols symbols to show a group of terms and/or expressions within a mathematical expression Improper fraction a number in the form where a and b are whole numbers and a > b where b is not equal to zero Least common denominator (LCD) the least common multiple of the denominators of two or more fractions Mixed number a number that is composed of a whole number and a fraction Numerator the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered Order of operations the rules of which calculations are performed first when simplifying an expression Positive rational numbers the set of numbers that can be expressed as a fraction, where a and b are counting (natural) numbers Product the total when two or more factors are multiplied Proper fraction a number in the form where a and b are whole numbers and a < b where b is not equal to zero Quotient the size or measure of each group or the number of groups when the dividend is divided by the divisor Rounding a method for estimating a number by increasing or retaining a specific place value digit according to specific rules and changing all trailing digits to zero Strip diagram a linear model used to illustrate number relationships Unit fraction a fraction in the form representing the quantity formed by one part of a whole that has been partitioned into b equal parts where b is a non-zero whole number Whole numbers the set of counting (natural) numbers and zero {0, 1, 2, 3,..., n }