Elementary Mathematics Pacing Guide Grade 4 FCPS Program of Studies (POS): The state of Virginia provides schools with a standards based curriculum to support student learning. These Essential Standards, Benchmarks, and Indicators (SBIs) are measurable learning goals that are embedded in Fairfax County Public School’s Program of Studies. FCPS has ensured that the content in the FCPS Mathematics Program of Studies is organized within and across grade levels to support students’ mathematical understanding by providing Expected SBIs which provide a context, a bridge or an enhancement to the Essential SBI’s. Extended SBIs were also created to be used to differentiate instruction for advanced learners. Elementary Mathematics Instructional Sequence (EMIS): The PreK‐12 Mathematics Office designed an Elementary Mathematics Instructional Sequence (EMIS) that is aligned with the new standards. The new instructional sequence provides teachers with the tools needed to offer students the opportunity to access an advanced mathematics curriculum as they develop the cognitive skills to be successful. Advanced mathematics extensions will be available for students as early as kindergarten and will build a strong foundation in mathematics. Tools, such as the Mathematics POS Inventory, the Mathematics Behaviors Rubric, and performance tasks, will be available in eCART to assist with the differentiation process. Assessments, such as unit assessments and standards assessments will be available in Horizon for teachers to use as formative assessments throughout the year. An eLEARNit course is available to provide more information about the Elementary Mathematics Instructional Sequence. FCPS Pacing Guides: The State of Virginia recognizes five student goals to support the mathematics content. They include: becoming mathematical problem solvers, communicating mathematically, reasoning mathematically, making mathematical connections, and using mathematical representations to model and interpret practical situations. These five goals support students’ learning of the mathematics content and appear in conjunction with the SBI’s in FCPS mathematics pacing guide documents. The FCPS Mathematics office used research‐based resources and the support from focus groups of FCPS teachers to create instructional units within the mathematics pacing guides. There are nine weeks in each quarter ‐ eight weeks are designated for instruction, leaving one week (five days) for teachers to integrate assessment throughout the quarter. Six weeks of instruction are included in quarter four due to SOL testing. There are two or three units per quarter and each unit has a suggested sequence and time table. In general, Number and Number Sense, Computation, and Algebra concepts are taught during the first half of the school year. These concepts are embedded, reinforced, and built upon in the second half of the school year when students are working on Measurement, Geometry, and Probability and Statistics concepts. Advanced Mathematics Pacing Guides: There is no separate pacing guide for Advanced Mathematics. The pacing guide teachers will use to teach Advanced Mathematics is the grade level pacing guide. Elementary Mathematics Pacing Guide Grade 4 This pacing guide contains four tools for teacher use: Unit Overview Chart: This chart provides a year‐long look at the grade‐level units to be taught in each quarter. Correlated Standards Chart: This chart shows the correlation of the standards taught within a given unit. Grade level standards are listed as well as correlated above and below grade level standards. Correlated Indicators Chart: This chart provides a more in depth look at the correlations within a given unit. All indicators (essential, expected, and extended) are listed for a given unit. The above grade level indicators are included. This chart should be used to provide differentiation during mathematics instruction. POS Framework Document: This document provides teachers with information to understand the standard. Teacher notes and vocabulary related to each standard are listed. Pacing Guide Features: Essential indicators: Indicators from the VDOE Standards of Learning document. Expected indicators: Indicators added by the FCPS Program of Studies to provide a context, a bridge, or an enhancement to the Essential SBIs. Extended Indicators: Indicators added by the FCPS Program of Studies generally used to differentiate instruction for advanced learners (Honors). Directly Correlated Standards: Standards that build on above and below grade level standards. Indirectly Correlated Standards: Standards that do not directly connect to or build on above and below grade level standards. Indirectly correlated standards are indicated on the pacing guide with the following symbol, . Resources: There are two types of resource materials which can be used for instructional use. They include: 1. eCART Resources ‐ These are public domain or FCPS‐produced resources that can be found in eCART. These resources include lessons and other teacher tools to support EMIS. Search by indicator to locate eCART resources. 2. Print Resources ‐ These are resources that have been provided to schools to support instruction. These resources cannot be included in eCART due to copyright laws. A chart listing alignments between print resources and grade‐level standards may be found in eCART. Alignments between textbook and grade‐level standards are included in this chart. Complete a grade level search in eCART to locate the print resource alignment. Table of Contents Grade 4 Grade 4 Unit Overview……………………………………………………………………………………………………………………………………………………………………………………………1 Quarter 1 Units Unit 1: Working with Patterns………………………………………………………………………………………………………………………………………………………………………………..3 Unit 2: Place Value………………………………………………………………………………………………………………………………………………………………………………………………….5 Unit 3: Working with Rational Numbers………………………………………………………………………………………………………………………………………………………………….7 Quarter 2 Units Unit 4: Computing with Whole Numbers………………………………………………………………………………………………………………………………………………………………11 Unit 5: Computing with Rational Numbers……………………………………………………………………………………………………………………………………………………………14 Quarter 3 Units Unit 6: Working with Measurement……………………………………………………………………………………………………………………………………………………………………..17 Unit 7: Elapsed Time……………………………………………………………………………………………………………………………………………………………………………………………..21 Unit 8: Working with Geometry…………………………………………………………………………………………………………………………………………………………………………….23 Quarter 4 Units Unit 9: Graphing……………………………………………………………………………………………………………………………………………………………………………………………………26 Unit 10: Working with Probability…………………………………………………………………………………………………………………………………………………………………………29 Mathematics POS Framework Grade 4………………………………………………………………………………………………………………………………………………………………….31 Mathematics POS Framework Grade 5………………………………………………………………………………………………………………………………………………………………...58 *Use the bookmark button to the left of your PDF document to navigate to different parts of the electronic version of the pacing guide. Quarter 1
Unit Name
Suggested Time Frame
Unit 1
Working with Patterns
1.5 weeks
Unit 2
Place Value 1.5 week
Unit 3
Working with Rational Numbers
5 weeks
4 weeks
Quarter 2
Unit 4
Computing with Whole Numbers
Grade 4 Unit Overview
Standard
MTH.G4.15 Recognize, create and extend numerical and geometric patterns
MTH.G5.17 Describe the relationship found in a number pattern and express the relationship
MTH.G4.1 Identify, compare, and round whole numbers expressed through millions MTH.G4.2 Compare and order fractions, represent equivalent forms
MTH.G4.3 Read, write, represent, identify, round, and compare decimals
MTH.G5.3 Identify and describe prime and composite numbers/ even and odd numbers
MTH.G5.1 Round decimals to the nearest whole number, tenth, or hundredth
MTH.G5.2 Recognize, name, compare, and order fractions in decimal form and vice versa
Assessment: eCART Window 1
MTH.G4.4 Estimate and find sums, differences, MTH.G5.4 Create and solve problems involving operations with whole numbers
products, and quotients of whole numbers
MTH.G5.7 Evaluate whole number expressions, MTH.G4.16 Demonstrate the meaning of equality and investigate associative property for using order of operations addition and multiplication
MTH.G5.18 Investigate, describe, and model the concept of variable; model one‐step linear equations; and create problems/write open sentence to represent a relationship MTH.G5.19 Investigate and recognize the distributive property of multiplication over addition
MTH.G4.5 Solve problems involving addition and subtraction of fractions and decimals and determine common multiples and factors
Unit 5
Computing with Rational Numbers
Correlated Standard
4 weeks
MTH.G5.5 Find sum, difference, product, and quotient of decimals through thousandths and create and solve problems involving decimals
MTH.G5.6 Solve single‐step and multistep practical problems involving addition and subtraction with fractions and mixed numbers Assessment: MOY (optional)
1
Unit Name
Quarter 3
Unit 6
Working with Measurement
Suggested Time Frame
3.5 weeks
Unit 7
Elapsed Time
1 weeks Unit 8
Working with Geometry
3.5 weeks
Grade 4 Unit Overview
Standard
MTH.G4.6 Estimate and measure weight/mass MTH.G5.8 Differentiate among and find and identify equivalent measurements between perimeter, area, and volume; identify units within U.S. Customary and metric systems equivalent measurements within the metric MTH.G4.7.Estimate and measure length and system; estimate, measure, and choose an identify equivalent measurements between appropriate unit of measure to solve problems
units within U.S. Customary and metric systems MTH.G5.9 Identify and describe the diameter, MTH.G4.8 Estimate and measure liquid volume radius, chord, and circumference of a circle and identify equivalent measurements between units within U.S. Customary and metric systems
MTH.G4.9 Determine elapsed time in hours and MTH.G5.10 Determine an amount of elapsed minutes within a 12‐hour period
time in hours and minutes within a 24‐hour period
MTH.G4.10 Identify and describe points, lines, MTH.G5.11 Measure right, acute, obtuse, and line segments, rays, angles, and intersections. straight angles
MTH.G4.11 Investigate congruence and MTH.G5.12 Classify angles and triangles recognize images after transformations
MTH.G5.13 Develop definitions, combine, and MTH G4 12 D fi
MTH.G4.12 Define polygon and identify l
d id if
subdivide plane figures (square, rectangle, bdi id l
fi
(
l
polygons with 10 or fewer sides
triangle, parallelogram, rhombus, and trapezoid)
Use an extended indicator at the current grade level to deepen student(s) mathematical knowledge or spend additional time teaching concepts already reviewed
Quarter 4
Assessment: eCART Window 2
MTH.G4.14 Collect, organize, display, and interpret data from a variety of graphs
Unit 9
Graphing
3 weeks
Unit 10
Working with Probability
3 weeks
Correlated Standard
MTH.G5.15 Given a problem situation, collect, organize, and interpret data in a variety of forms, using stem‐and‐leaf plots and line graphs MTH.G5.16 Describe and find the mean, median, mode, and range of a set of data
MTH.G4.13 Predict the likelihood of an outcome 5.14 Make Predictions and determine the and represent probability as a number between probability of an outcome by constructing a 0 and 1, inclusive
sample space
Assessment: Grade level SOL
2
Mathematics Grade 4 Unit 1: Working with Patterns Suggested Time Frame: 1.5 weeks Correlated Standards Grade Level Below Standards Current Grade Level Standards MTH.G3.19‐The student will MTH.G4.15‐The student will recognize and describe a variety of patterns formed recognize, create, and extend numerical and using numbers, tables, and pictures, and extend the geometric patterns.
patterns, using the same or different forms.
Indirect Correlation 2011 Grade Level Above Standards MTH.G5.17‐The student will describe the relationship found in a number pattern and express the relationship.
3 Mathematics Grade 4 Unit 1: Working with Patterns Suggested Time Frame: 1.5 weeks Correlated Indicators Grade 4 Indicators Essential MTH.G4.15.a.1 Describe geometric and numerical patterns, using tables, symbols, or words. MTH.G4.15.a.2 Create geometric and numerical patterns, using concrete materials, number lines, tables, and words. MTH.G4.15.a.3 Extend geometric and numerical patterns, using concrete materials, number lines, tables, and words. Expected Extended MTH.G4.15.a.4 Find function rules using a combination of two of the four operations (e.g., times 2, add 1) given input and output data (e.g., input 5, output 11; input 7, output 15). Grade 5 Indicators Essential MTH.G5.17.a.1 Describe numerical and geometric patterns formed by using concrete materials and calculators. MTH.G5.17.a.2 Describe the relationship found in patterns, using words, tables, and symbols to express the relationship. Expected Extended 2011 4 Mathematics Grade 4 Unit 2: Place Value Suggested Time Frame: 1.5 weeks Grade Level Below Standards MTH.G3.1‐The student will a) read and write six‐digit numerals and identify the place value and value of each digit; b) round whole numbers, 9,999 or less, to the nearest ten, hundred, and thousand; and c) compare two whole numbers between 0 and 9,999, using symbols (>, <, or =) and words (greater than, less than, or equal to). Correlated Standards Current Grade Level Standards MTH.G4.1‐The student will a) identify orally and in writing the place value for each digit in a whole number expressed through millions; b) compare two whole numbers expressed through millions, using symbols (>, <, or =); and c) round whole numbers expressed through millions to the nearest thousand, ten thousand, and hundred thousand.
Grade Level Above Standards MTH.G5.3‐The student will a) identify and describe the characteristics of prime and composite numbers; and b) identify and describe the characteristics of even and odd numbers. Indirect Correlation 2011 5 Mathematics Grade 4 Unit 2: Place Value Suggested Time Frame: 1.5 weeks Correlated Indicators Grade 4 Indicators Essential MTH.G4.1.a.1 Identify and communicate, both orally and in written form, the place value for each digit in whole numbers expressed through the one millions place. MTH.G4.1.a.2 Read whole numbers through the one millions place that are presented in standard format, and select the matching number in written format. MTH.G4.1.a.3 Write whole numbers through the one millions place in standard format when the numbers are presented orally or in written format. MTH.G4.1.b.1 Identify and use the symbols for greater than, less than, and equal to. MTH.G4.1.b.2 Compare two whole numbers expressed through the one millions, using symbols >, <, or =. MTH.G4.1.c.1 Round whole numbers expressed through the one millions place to the nearest thousand, ten thousand, and hundred‐thousand place. Expected MTH.G4.1.a.4 Represent whole numbers on a number line. Extended Grade 5 Indicators Essential MTH.G5.3.a.1 Identify prime numbers less than or equal to 100. MTH.G5.3.a.2 Identify composite numbers less than or equal to 100. MTH.G5.3.a.3 Explain orally and in writing why a number is prime or composite. MTH.G5.3.b.1 Identify which numbers are even or odd. MTH.G5.3.b.2 Explain and demonstrate with manipulatives, pictorial representations, oral language, or written language why a number is even or odd.
Expected Extended MTH.G5.3.a.4 Represent numbers as the products of their prime factors (prime factorization). 2011 6 Mathematics Grade 4 Unit 3: Working with Rational Numbers Suggested Time Frame: 5 weeks Grade Level Below Standards MTH.G3.3‐The student will a) name and write fractions (including mixed numbers) represented by a model; b) model fractions (including mixed numbers) and write the fractions' names; and c) compare fractions having like and unlike denominators, using words and symbols (>, <, or =). No correlated standard. Correlated Standards Current Grade Level Standards MTH.G4.2‐The student will a) compare and order fractions and mixed numbers; b) represent equivalent fractions; and c) identify the division statement that represents a fraction.
MTH.G4.3‐The student will a) read, write, represent, and identify decimals expressed through thousandths; b) round to the nearest whole number, tenth, and hundredth; c) compare and order decimals; and d) given a model, write the decimal and fraction equivalents. Grade Level Above Standards MTH.G5.2‐The student will a) recognize and name fractions in their equivalent decimal form and vice versa; and b) compare and order fractions and decimals in a given set from least to greatest and greatest to least. MTH.G5.1‐The student, given a decimal through thousandths, will round to the nearest whole number, tenth, or hundredth. MTH.G5.2‐The student will a) recognize and name fractions in their equivalent decimal form and vice versa; and b) compare and order fractions and decimals in a given set from least to greatest and greatest to least. Indirect Correlation 2011 7 Mathematics Grade 4 Unit 3: Working with Rational Numbers Suggested Time Frame: 5 weeks Correlated Indicators Grade 4 Indicators Grade 5 Indicators Essential MTH.G4.2.a.1 Compare and order fractions having denominators of 12 or less, using manipulative models and drawings, such as region/area models. MTH.G4.2.a.2 Compare and order fractions with like denominators by comparing number of parts (numerators) (e.g., < ). MTH.G4.2.a.3 Compare and order fractions with like numerators and unlike denominators by comparing the size of the parts (e.g., < ). MTH.G4.2.a.4Compare and order fractions having unlike denominators of 12 or less by comparing the fractions to benchmarks (e.g., 0, , or 1) to determine their relationships to the benchmarks or by finding a common denominator. MTH.G4.2.a.5 Compare and order mixed numbers having denominators of 12 or less. MTH.G4.2.a.6 Use the symbols >, <, and = to compare the numerical value of fractions and mixed numbers having denominators of 12 or less. MTH.G4.2.b.1 Represent equivalent fractions through twelfths, using region/area models, set models, and measurement models. MTH.G4.2.c.1 Identify the division statement that represents a fraction (e.g., means the same as 3 divided by 5). MTH.G4.3.a.1 Investigate the ten‐to‐one place‐value relationship for decimals through thousandths, using Base‐10 manipulatives (e.g., place‐value mats/charts, decimal squares, Base‐10 blocks, money). MTH.G4.3.a.2 Represent and identify decimals expressed through thousandths, using Base‐10 manipulatives, pictorial representations, and numerical symbols (e.g., relate the appropriate drawing to 0.05). MTH.G4.3.a.3 Identify and communicate, both orally and in written form, the Essential MTH.G5.2.a.1 Represent fractions (halves, fourths, fifths, eighths, tenths, and twelfths) in their equivalent decimal form. MTH.G5.2.a.2 Represent decimals in their equivalent fraction form (halves, fourths, fifths, eighths, tenths, and twelfths). MTH.G5.2.a.3 Recognize and name equivalent relationships between decimals and fractions with denominators up to 12. MTH.G.5.2.b.1 Compare and order from least to greatest and greatest to least a given set of no more than five numbers written as decimals, fractions, and mixed numbers with denominators of 12 or less. MTH.G5.1.a.1 Round decimal numbers to the nearest whole number, tenth, or hundredth. Expected MTH.G5.2.a.4 Represent fractions (thirds and sixths) in their equivalent decimal form. MTH.G5.2.a.5 Represent decimals in their equivalent fraction form (thirds and sixths). MTH.G5.2.a.6 Distinguish between terminating and repeating decimals and write repeating decimals using the appropriate notation.
2011 8 position and value of a decimal through thousandths. For example, in 0.385, the 8 is in the hundredths place and has a value of 0.08.
MTH.G4.3.a.4 Read and write decimals expressed through thousandths, using Base‐10 manipulatives, drawings, and numerical symbols. MTH.G4.3.b.1 Round decimals to the nearest whole number, tenth, and hundredth. MTH.G4.3.c.1 Compare the value of two decimals, using the symbols <, >, =. MTH.G4.3.c.2 Order a set of decimals from least to greatest or greatest to least. MTH.G4.3.d.1 Represent fractions for halves, fourths, fifths, and tenths as decimals through hundredths, using concrete objects (e.g., demonstrate the relationship between the fraction and its decimal equivalent 0.25). MTH.G4.3.d.2 Relate fractions to decimals, using concrete objects (e.g., 10‐by‐
10 grids, meter sticks, number lines, decimal squares, decimal circles, money [coins]). MTH.G4.3.d.3 Write the decimal and fraction equivalent for a given model (e.g., = 0.25 or 0.25 = ). Expected MTH.G4.3.a.5 Represent decimal numbers on a number line. Extended MTH.G4.2.a.7 Use partitioning to determine "how much more" when comparing fractions. MTH.G4.2.b.2 Generate equivalent expressions for the same quantity [e.g., 1 case of cola = 24 cans = 2 (12‐packs) = 4 (6‐packs) = 3 (8‐packs)]. MTH.G4.3.a.6 Use and represent numbers through millions in various contexts, including estimation of relative sizes of amounts or distances (e.g., 310,989,822 is the population of the United States of America). 2011 Extended MTH.G5.2.a.7 Solve problems involving fractions by reasoning up and down (e.g., 8 ladybugs = of the ladybugs on the tree. How many lady bugs are on the tree? Think: 8 ladybugs = , so 2 ladybugs = , so 10 ladybugs = or 1). MTH.G5.2.a.8 Solve comparison and missing value problems by reasoning up and down (e.g., If it takes four people three days to wash windows at the office building, how long would it take eight people to do the job?). 9 2011 10 Mathematics Grade 4 Unit 4: Computing with Whole Numbers Suggested Time Frame: 4 weeks Grade Level Below Standards MTH.G3.2‐The student will recognize and use the inverse relationships between addition/subtraction and multiplication/division to complete basic fact sentences. The student will use these relationships to solve problems. MTH.G3.4‐The student will estimate solutions to and solve single‐step and multistep problems involving the sum or difference of two whole numbers, each 9,999 or less, with or without regrouping. MTH.G3.5‐The student will recall multiplication facts through the twelves table, and the corresponding division facts. MTH.G3.6‐The student will represent multiplication and division, using area, set, and number line models, and create and solve problems that involve multiplication of two whole numbers, one factor 99 or less and the second factor 5 or less.
MTH.G3.20‐The student will a) investigate the identity and the commutative properties for addition and multiplication; b) identify examples of the identity and commutative properties for addition and multiplication; and c) write number sentences to represent equivalent and nonequivalent mathematical relationships. Correlated Standards Current Grade Level Standards MTH.G4.4‐The student will a) estimate sums, differences, products, and quotients of whole numbers; b) add, subtract, and multiply whole numbers; c) divide whole numbers, finding quotients with and without remainders; and d) solve single‐step and multistep addition, subtraction, and multiplication problems with whole numbers.
MTH.G4.16‐The student will a) recognize and demonstrate the meaning of equality in an equation; and b) investigate and describe the associative property for addition and multiplication. Grade Level Above Standards MTH.G5.4‐The student will create and solve single‐step and multistep practical problems involving addition, subtraction, multiplication, and division with and without remainders of whole numbers. MTH.G5.7‐The student will evaluate whole number numerical expressions, using the order of operations limited to parentheses, addition, subtraction, multiplication, and division. MTH.G5.18‐The student will a) investigate and describe the concept of variable; b) write an open sentence to represent a given mathematical relationship, using a variable; c) model one‐step linear equations in one variable, using addition and subtraction; and d) create a problem situation based on a given open sentence, using a single variable. MTH.G5.19‐The student will investigate and recognize the distributive property of multiplication over addition. Indirect Correlation 2011 11 Mathematics Grade 4 Unit 4: Computing with Whole Numbers Suggested Time Frame: 4 weeks Correlated Indicators Grade 4 Indicators Grade 5 Indicators Essential MTH.G4.4.a.1 Estimate whole number sums, differences, products, and quotients. MTH.G4.4.a.2 Refine estimates by adjusting the final amount, using terms such as closer to, between, and a little more than. MTH.G4.4.b.1 Determine the sum or difference of two whole numbers, each 999,999 or less, in vertical and horizontal form with or without regrouping, using paper and pencil, and using a calculator. MTH.G4.4.b.2 Estimate and find the product of two whole numbers when one factor has two digits or fewer and the other factor has three digits or fewer, using paper and pencil and calculators. MTH.G4.4.c.1 Estimate and find the quotient of two whole numbers, given a one‐digit divisor and a two‐ or three‐digit dividend. MTH.G4.4.d.1 Solve single‐step and multistep addition, subtraction, and multiplication problems using whole number operations. MTH.G4.4.d.2 Verify the reasonableness of sums, differences, products, and quotients of whole numbers using estimation. MTH.G4.16.a.1 Recognize and demonstrate that the equal sign (=) relates equivalent quantities in an equation. MTH.G4.16.a.2 Write an equation to represent equivalent mathematical relationships (e.g., 4 × 3 = 2 × 6). MTH.G4.16.a.3 Recognize and demonstrate appropriate use of the equal sign in an equation. MTH.G4.16.b.1 Investigate and describe the associative property for addition as (6 + 2) + 3 = 6 + (2 + 3). Essential MTH.G5.4.a.1 Select appropriate methods and tools from among paper and pencil, estimation, mental computation, and calculators according to the context and nature of the computation in order to compute with whole numbers. MTH.G5.4.a.2 Estimate the sum, difference, product, and quotient of whole‐
number computations. MTH.G5.4.a.3 Solve single‐step and multistep problems involving addition, subtraction, multiplication, and division with and without remainders of whole numbers, using paper and pencil, mental computation, and calculators in which ‐ sums, differences, and products will not exceed five digits; ‐ multipliers will not exceed two digits; ‐ divisors will not exceed two digits; or ‐ dividends will not exceed four digits. MTH.G5.4.a.4 Use two or more operational steps to solve a multistep problem. Operations can be the same or different. MTH.G5.4.a.5 Create single‐step and multistep problems involving the operations of addition, subtraction, multiplication, and division with and without remainders of whole numbers, using practical situations. MTH.G5.7.a.1 Simplify expressions by using the order of operations in a demonstrated step‐by‐step approach. MTH.G5.7.a.2 Find the value of numerical expressions, using the order of operations. MTH.G5.7.a.3 Given an expression involving more than one operation, describe which operation is completed first, which is second, etc. MTH.G5.18.a.1 Describe the concept of a variable (presented as boxes, letters, or other symbols) as a representation of an unknown quantity. 2011 12 MTH.G4.16.b.2 Investigate and describe the associative property for multiplication as (3 x 2) x 4 = 3 x (2 x 4). Expected MTH.G4.4.c.2 Interpret the remainder in problem‐solving settings. MTH.G4.4.d.3 Solve single‐step and multistep division problems using whole number operations. Extended MTH.G4.4.d.4 Evaluate the traditional multiplication and division algorithms and invented algorithms used by peers for efficiency and accuracy. MTH.G5.18.b.1 Write an open sentence with addition, subtraction, multiplication, or division, using a variable to represent a missing number. MTH.G5.18.c.1 Model one‐step linear equations using a variety of concrete materials such as colored chips on an equation mat or weights on a balance scale. MTH.G5.18.d.1 Create and write a word problem to match a given open sentence with a single variable and one operation. MTH.G5.19.a.1 Investigate and recognize the distributive property of whole numbers, limited to multiplication over addition using diagrams and manipulatives. MTH.G5.19.a.2 Investigate and recognize an equation that represents the distributive property, when given several whole number equations, limited to multiplication over addition. Expected Extended MTH.G5.4.a.6 Evaluate alternative multiplication and division algorithms used in other countries for efficiency and accuracy. MTH.G5.4.a.7 Evaluate alternative multiplication and division algorithms used by peers for efficiency and accuracy. MTH.G5.4.a.8 Investigate, develop, and use rules for divisibility. 2011 13 Mathematics Grade 4 Unit 5: Computing with Rational Numbers Suggested Time Frame: 4 weeks Grade Level Below Standards MTH.G3.7‐The student will add and subtract proper fractions having like denominators of 12 or less. *This standard correlates to MTH.G4.5.a and MTH.G4.5.b. There is no correlated standard for MTH.G4.5.c and MTH.G4.5.d. Correlated Standards Current Grade Level Standards MTH.G4.5‐The student will a) determine common multiples and factors, including least common multiple and greatest common factor; b) add and subtract fractions having like and unlike denominators that are limited to 2, 3, 4, 5, 6, 8, 10, and 12, and simplify the resulting fractions, using common multiples and factors; c) add and subtract with decimals; and d) solve single‐step and multistep practical problems involving addition and subtraction with fractions and with decimals.
Grade Level Above Standards MTH.G5.5‐The student will a) find the sum, difference, product, and quotient of two numbers expressed as decimals through thousandths (divisors with only one nonzero digit); and b) create and solve single‐step and multistep practical problems involving decimals. MTH.G5.6‐The student will solve single‐step and multistep practical problems involving addition and subtraction with fractions and mixed numbers and express answers in simplest form. Indirect Correlation 2011 14 Mathematics Grade 4 Unit 5: Computing with Rational Numbers Suggested Time Frame: 4 weeks Correlated Indicators Grade 4 Indicators Grade 5 Indicators Essential MTH.G4.5.a.1 Find common multiples and common factors of numbers. MTH.G4.5.a.2 Determine the least common multiple and greatest common factor of numbers. MTH.G4.5.b.1 Use least common multiple and/or greatest common factor to find a common denominator for fractions. MTH.G4.5.b.2 Add and subtract with fractions having like denominators whose denominators are limited to 2, 3, 4, 5, 6, 8, 10, and 12, and simplify the resulting fraction using common multiples and factors. MTH.G4.5.b.3 Add and subtract with fractions having unlike denominators whose denominators are limited to 2, 3, 4, 5, 6, 8, 10, and 12, and simplify the resulting fraction using common multiples and factors.
MTH.G4.5.b.4 Solve problems that involve adding and subtracting with fractions having like and unlike denominators whose denominators are limited to 2, 3, 4, 5, 6, 8, 10, and 12, and simplify the resulting fraction using common multiples and factors. MTH.G4.5.c.1 Add and subtract with decimals through thousandths, using concrete materials, pictorial representations, and paper and pencil. MTH.G4.5.d.1 Solve single‐step and multistep problems that involve adding and subtracting with fractions and decimals through thousandths. Expected MTH.G4.5.b.5 Estimate the sum and difference of fractions with like and unlike denominators which are limited to 2, 3, 4, 5, 6, 8, 10, and 12. MTH.G4.5.c.2 Estimate the sum and difference of decimals to thousandths by using models, pictorial representations, and paper and pencil. Essential MTH.G5.5.a.1 Determine an appropriate method of calculation to find the sum, difference, product, and quotient of two numbers expressed as decimals through thousandths, selecting from among paper and pencil, estimation, mental computation, and calculators. MTH.G5.5.a.2 Estimate to find the number that is closest to the sum, difference, and product of two numbers expressed as decimals through thousandths. MTH.G5.5.a.3 Find the sum, difference, and product of two numbers expressed as decimals through thousandths, using paper and pencil, estimation, mental computation, and calculators. MTH.G5.5.a.4 Determine the quotient, given a dividend expressed as a decimal through thousandths and a single‐digit divisor. For example, 5.4 divided by 2 and 2.4 divided by 5. MTH.G5.5.a.5 Use estimation to check the reasonableness of a sum, difference, product, and quotient. MTH.G5.5.b.1 Create and solve single‐step and multistep problems. A multistep problem needs to incorporate two or more operational steps (operations can be the same or different). MTH.G5.6.a.1 Solve single‐step and multistep practical problems involving addition and subtraction with fractions having like and unlike denominators. Denominators in the problems should be limited to 12 or less (e.g., ) and answers should be expressed in simplest form. MTH.G5.6.a.2 Solve single‐step and multistep practical problems involving addition and subtraction with mixed numbers having like and unlike denominators, with and without regrouping. Denominators in the problems should be limited to 12 or less, and answers should be expressed in simplest form.
2011 15 Extended MTH.G4.5.b.6 Estimate, add, and subtract mixed numbers with like and unlike denominators limited to 2, 3, 4, 5, 6, 8, 10, 12. MTH.G4.5.d.2 Use given information to create story problems with fractions and with decimals. Explain solution strategies. 2011 MTH.G5.6.a.3 Use estimation to check the reasonableness of a sum or difference. Expected Extended MTH.G5.6.a.4 Use given information to create story problems involving fractions and mixed numbers. Explain solution strategies. 16 Mathematics Grade 4 Unit 6: Working with Measurement Suggested Time Frame: 3.5 weeks Grade Level Below Standards MTH.G3.9‐The student will estimate and use U.S. Customary and metric units to measure a) length to the nearest inch, inch, foot, yard, centimeter, and meter; b) liquid volume in cups, pints, quarts, gallons, and liters; c) weight/mass in ounces, pounds, grams, and kilograms; and d) area and perimeter. Correlated Standards Current Grade Level Standards MTH.G4.6‐The student will a) estimate and measure weight/mass and describe the results in U.S. Customary and metric units as appropriate; and b) identify equivalent measurements between units within the U.S. Customary system (ounces, pounds, and tons) and between units within the metric system (grams and kilograms). MTH.G4.7‐The student will a) estimate and measure length, and describe the result in both metric and U.S. Customary units; and b) identify equivalent measurements between units within the U.S. Customary system (inches and feet; feet and yards; inches and yards; yards and miles) and between units within the metric system (millimeters and centimeters; centimeters and meters; and millimeters and meters). MTH.G4.8‐The student will a) estimate and measure liquid volume and describe the results in U.S. Customary units; and b) identify equivalent measurements between units within the U.S. Customary system (cups, pints, quarts, and gallons). Grade Level Above Standards MTH.G5.8‐The student will a) Find perimeter, area, and volume in standard units of measure; b) Differentiate among perimeter, area, and volume and identify whether the application of the concept of perimeter, area, or volume is appropriate for a given situation; c) identify equivalent measurements within the metric system; d) estimate and then measure to solve problems, using U.S. Customary and metric units; and e) choose an appropriate unit of measure for a given situation involving measurement using U.S. Customary and metric units. *MTH.G5.8.a and MTH.G5.8.b indirectly correlate to MTH.G4.7 MTH.G5.9‐Identify and describe the diameter, radius, chord, and circumference of a circle. Indirect Correlation 2011 17 Mathematics Grade 4 Unit 6: Working with Measurement Suggested Time Frame: 3.5 weeks Correlated Indicators Grade 4 Indicators Grade 5 Indicators Essential MTH.G4.6.a.1 Determine an appropriate unit of measure (e.g., ounce, pound, ton, gram, kilogram) to use when measuring everyday objects in both metric and U.S. Customary units. MTH.G4.6.a.2 Estimate and measure objects in both metric and U.S. Customary units (e.g., ounce, pound, ton, gram, or kilogram) to the nearest appropriate measure, using a variety of measuring instruments. MTH.G4.6.a.3 Record the mass of an object including the appropriate unit of measure (e.g., 24 grams). MTH.G4.6.b.1 Identify equivalent measurements between units within the U.S. Customary system (ounces, pounds, and tons). MTH.G4.6.b.2 Identify equivalent measurements between units within the metric system (grams and kilograms). MTH.G4.7.a.1 Determine an appropriate unit of measure (e.g., inch, foot, yard, mile, millimeter, centimeter, and meter) to use when measuring everyday objects in both metric and U.S. Customary units. MTH.G4.7.a.2 Estimate the length of everyday objects (e.g., books, windows, tables) in both metric and U.S. Customary units of measure. MTH.G4.7.a.3 Measure the lengths of objects in both metric and U.S. Essential MTH.G5.8.a.1 Determine the perimeter of a polygon, with or without diagrams, when – the lengths of all sides of a polygon that is not a rectangle or a square are given; – the length and width of a rectangle are given; or – the length of a side of a square is given. MTH.G5.8.a.2 Estimate and determine the perimeter of a polygon, and area of a square, rectangle, and right triangle following the parameters listed above, using only whole number measurements given in metric or U.S. Customary units, and record the solution with the appropriate unit of measure (e.g., 24 square inches). MTH.G5.8.a.3 Estimate and determine the area of a square, with or without diagrams, when the length of a side is given. MTH.G5.8.a.4 Estimate and determine the area of a rectangle, with or without diagrams, when the length and width are given. MTH.G5.8.a.5 Estimate and determine the area of a right triangle, with or without diagrams, when the base and the height are given. MTH.G5.8.a.6 Develop a procedure for finding volume using manipulatives (e.g., cubes). MTH.G5.8.b.1 Differentiate among the concepts of area, perimeter, and Customary units, measuring to the nearest inch , , , foot, yard, mile, volume. millimeter, centimeter, or meter, and record the length including the appropriate unit of measure (e.g., 24 inches). MTH.G5.8.b.2 Describe practical situations where area, perimeter, and volume are appropriate measures to use, and justify their choices orally or in writing. MTH.G4.7.a.4 Compare estimates of the length of objects with the actual measurement of the length of objects. MTH.G5.8.b.3 Identify whether the application of the concept of perimeter, area, or volume is appropriate for a given situation. MTH.G4.7.b.1 Identify equivalent measurements between units within the U.S. Customary system (inches and feet; feet and yards; inches and yards; yards MTH.G5.8.c.1 Identify equivalent measurements within the metric system for and miles) and between units within the metric system (millimeters and 2011 18 centimeters; centimeters and meters; and millimeters and meters). MTH.G4.8.a.1 Determine an appropriate unit of measure (cups, pints, quarts, gallons) to use when measuring liquid volume in U.S. Customary units. MTH.G4.8.a.2 Estimate the liquid volume of containers in U.S. Customary units of measure to the nearest cup, pint, quart, and gallon. MTH.G4.8.a.3 Measure the liquid volume of everyday objects in U.S. Customary units, including cups, pints, quarts, and gallons, and record the volume including the appropriate unit of measure (e.g., 24 gallons). MTH.G4.8.b.1 Identify equivalent measures of volume between units within the U.S. Customary system (cups, pints, quarts, and gallons). Expected 2011 length: millimeters, centimeters, meters, and kilometers. MTH.G5.8.c.2 Identify equivalent measurements within the metric system for mass: grams and kilograms. MTH.G5.8.c.3 Identify equivalent measurements within the metric system for liquid volume: milliliters, and liters. MTH.G5.8.d.1 Solve problems involving measurement by selecting an appropriate measuring device and a U.S. Customary or metric unit of measure for length: part of an inch , , , inches, feet, yards, millimeters, centimeters, meters, and kilometers. MTH.G5.8.d.2 Solve problems involving measurement by selecting an appropriate measuring device and a U.S. Customary or metric unit of measure for the following: – weight: ounces, pounds, and tons; – mass: grams and kilograms. MTH.G5.8.d.3 Solve problems involving measurement by selecting an appropriate measuring device and a U.S. Customary or metric unit of measure for liquid volume: cups, pints, quarts, gallons, milliliters, and liters. MTH.G5.8.d.4 Solve problems involving measurement by selecting an appropriate measuring device and a U.S. Customary or metric unit of measure for area: square units. MTH.G5.8.d.5 Solve problems involving measurement by selecting an appropriate measuring device and a U.S. Customary or metric unit of measure for temperature: Celsius and Fahrenheit units. MTH.G5.8.d.6 Estimate the conversion of Celsius and Fahrenheit units relative to familiar situations: ‐ Water freezes at 0°C and 32°F. ‐ Water boils at 100°C and 212°F. ‐ Normal body temperature is about 37°C and 98.6°F. MTH.G5.8.e.1 Choose an appropriate unit of measure for a given situation involving measurement using U.S. Customary and metric units. MTH.G5.9.a.1 Identify and describe the diameter, radius, chord, and 19 Extended MTH.G4.8.a.3 Explore volume using nonstandard units. circumference of a circle. MTH.G5.9.a.2 Describe the relationship between – diameter and radius; – diameter and chord; – radius and circumference; and – diameter and circumference. Expected MTH.G5.8.a.7 Develop formulas for perimeter and area. Extended 2011 20 Mathematics Grade 4 Unit 7: Elapsed Time Suggested Time Frame: 1 week Grade Level Below Standards MTH.G3.11‐The student will a) tell time to the nearest minute, using analog and digital clocks; and b) determine elapsed time in one‐hour increments over a 12‐hour period.
Indirect Correlation 2011 Correlated Standards Current Grade Level Standards MTH.G4.9‐The student will determine elapsed time in hours and minutes within a 12‐hour period.
Grade Level Above Standards MTH.G5.10‐The student will determine an amount of elapsed time in hours and minutes within a 24‐hour period. 21 Mathematics Grade 4 Unit 7: Elapsed Time Suggested Time Frame: 1 week Correlated Indicators Grade 4 Indicators Essential MTH.G4.9.a.1 Determine the elapsed time in hours and minutes within a 12‐
hour period (times can cross between a.m. and p.m.) MTH.G4.9.a.2 Solve practical problems in relation to time that has elapsed. Expected Extended 2011 Grade 5 Indicators Essential MTH.5.10.a.1 Determine elapsed time in hours and minutes within a 24‐hour period. Expected Extended 22 Mathematics Grade 4 Unit 8: Working with Geometry Suggested Time Frame: 3.5 weeks Grade Level Below Standards MTH.G3.15‐The student will identify and draw representations of points, line segments, rays, angles, and lines.
MTH.G3.16‐The student will identify and describe congruent and noncongruent plane figures. MTH.G3.14‐The student will identify, describe, compare, and contrast characteristics of plane and solid geometric figures (circle, square, rectangle, triangle, cube, rectangular prism, square pyramid, sphere, cone, and cylinder) by identifying relevant characteristics, including the number of angles, vertices, and edges, and the number and shape of faces, using concrete models. Correlated Standards Current Grade Level Standards MTH.G4.10‐The student will a) identify and describe representations of points, lines, line segments, rays, and angles, including endpoints and vertices; and b) identify representations of lines that illustrate intersection, parallelism, and perpendicularity MTH.G4.11‐The student will a) investigate congruence of plane figures after geometric transformations, such as reflection, translation, and rotation, using mirrors, paper folding, and tracing; and b) recognize the images of figures resulting from geometric transformations, such as translation, reflection, and rotation. MTH.G4.12‐The student will a) define polygon; and b) identify polygons with 10 or fewer sides. Grade Level Above Standards MTH.G5.11‐The student will measure right, acute, obtuse, and straight angles. There is not a correlated standard. Use an extended indicator at the current grade level to deepen student(s) mathematical knowledge, or spend additional time reviewing concepts already taught.
MTH.G5.12‐The student will classify a) angles as right, acute, obtuse, or straight; and b) triangles as right, acute, obtuse, equilateral, scalene, or isosceles MTH. G5.13‐The student, using plane figures (square, rectangle, triangle, parallelogram, rhombus, and trapezoid), will a) develop definitions of these plane figures; and b) investigate and describe the results of combining and subdividing plane figures. Indirect Correlation 2011 23 Mathematics Grade 4 Unit 8: Working with Geometry Suggested Time Frame: 3.5 weeks Correlated Indicators Grade 4 Indicators Grade 5 Indicators Essential MTH.G4.10.a.1 Identify and describe representations of points, lines, line segments, rays, and angles, including endpoints and vertices. MTH.G4.10.b.1 Understand that lines in a plane can intersect or are parallel. Perpendicularity is a special case of intersection. MTH.G4.10.b.2 Identify practical situations that illustrate parallel, intersecting, and perpendicular lines. MTH.G4.11.a.1 Recognize the congruence of plane figures resulting from geometric transformations such as translation, reflection, and rotation, using mirrors, paper folding and tracing. MTH.G4.11.b.1 Recognize the images of figures resulting from geometric transformations such as translation, reflection, or rotation. MTH.G4.12.a.1 Define and identify properties of polygons with 10 or fewer sides. MTH.G4.12.b.1 Identify polygons by name with 10 or fewer sides in multiple orientations (rotations, reflections, and translations of the polygons). Expected MTH.G4.12.a.2 Identify a figure as open or closed and identify plane figures as polygons or nonpolygons. Essential MTH.G5.11.a.1 Identify the appropriate tools (e.g., protractor and straightedge or angle ruler as well as available software) used to measure and draw angles and triangles. MTH.G5.11.a.2 Measure right, acute, straight, and obtuse angles, using appropriate tools, and identify their measures in degrees. MTH.G5.11.a.3 Recognize angle measure as additive. When an angle is decomposed into non‐overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. MTH.G5.11.a.4 Solve addition and subtraction problems to find unknown angle measures on a diagram in practical and mathematical problems, (e.g., by using an equation with a symbol for the unknown angle measure). MTH.G5.12.a.1 Classify angles as right, acute, obtuse, or straight. MTH.G5.12.b.1 Classify triangles as right, acute, and obtuse. MTH.G5.12.b.2 Classify triangles as equilateral, scalene, or isosceles. MTH.G5.13.a.1 Develop definitions for squares, rectangles, triangles, parallelograms, rhombi, and trapezoids. MTH.G5.13.b.1 Investigate and describe the results of combining and subdividing plane figures. Expected Extended MTH.G4.10.a.2 Compare and contrast points, line segments, rays, and angles including endpoints and vertices. Extended MTH.G4.10.a.3 Sort angles and describe sorting rule. MTH.G4.11.b.2 Build tessellations with congruent shapes using transformations. 2011 24 2011 25 Mathematics Grade 4 Unit 9: Graphing Suggested Time Frame: 3 weeks Grade Level Below Standards MTH.G3.17‐The student will a) collect and organize data, using observations, measurements, surveys, or experiments; b) construct a line plot, a picture graph, or a bar graph to represent the data; and c) read and interpret the data represented in line plots, bar graphs, and picture graphs and write a sentence analyzing the data.
Correlated Standards Current Grade Level Standards MTH.G4.14‐The student will collect, organize, display, and interpret data from a variety of graphs.
Grade Level Above Standards MTH.G5.15‐The student, given a problem situation, will collect, organize, and interpret data in a variety of forms, using stem‐and‐leaf plots and line graphs. MTH.G5.16‐The student will a) describe mean, median, and mode as measures of center; b) describe mean as fair share; c) find the mean, median, mode, and range of a set of data; and d) describe the range of a set of data as a measure of variation.
Indirect Correlation 2011 26 Mathematics Grade 4 Unit 9: Graphing Suggested Time Frame: 3 weeks Correlated Indicators Grade 4 Indicators Grade 5 Indicators Essential MTH.G4.14.a.1 Collect data, using, for example, observations, measurement, surveys, scientific experiments, polls, or questionnaires. MTH.G4.14.a.2 Organize data into a chart or table. MTH.G4.14.a.3 Construct and display data in bar graphs, labeling one axis with equal whole number increments of 1 or more (numerical data) (e.g., 2, 5, 10, or 100) and the other axis with categories related to the title of the graph (categorical data) (e.g., swimming, fishing, boating, and water skiing as the categories of “Favorite Summer Sports”). MTH.G4.14.a.4 Construct and display data in line graphs, labeling the vertical axis with equal whole number increments of 1 or more and the horizontal axis with continuous data commonly related to time (e.g., hours, days, months, years, and age). Line graphs will have no more than 10 identified points along a continuum for continuous data. For example, growth charts showing age versus height place age on the horizontal axis (e.g., 1 month, 2 months, 3 months, and 4 months). MTH.G4.14.a.5 Title or identify the title in a given graph and label or identify the axes. MTH.G4.14.a.6 Interpret data from simple line and bar graphs by describing the characteristics of the data and the data as a whole (e.g., the category with the greatest/least, categories with the same number of responses, similarities and differences, the total number). Data points will be limited to 30 and categories to 8. MTH.G4.14.a.7 Interpret the data to answer the question posed, and compare the answer to the prediction (e.g., “The summer sport preferred by most is swimming, which is what I predicted before collecting the data.”). MTH.G4.14.a.8 Write at least one sentence to describe the analysis and interpretation of the data, identifying parts of the data that have special characteristics, including categories with the greatest, the least, or the same. Essential MTH.G5.15 a.1 Formulate the question that will guide the data collection. MTH.G5.15.a.2 Collect data, using observations (e.g., weather), measurement (e.g., shoe sizes), surveys (e.g., hours watching television), or experiments (e.g., plant growth). MTH.G5.15.a.3 Organize the data into a chart, table, stem‐and‐leaf plots, and line graphs. MTH.G5.15.a.4 Display data in line graphs and stem‐and‐leaf plots. MTH.G5.15.a.5 Construct line graphs, labeling the vertical axis with equal whole‐number, decimal, or fractional increments and the horizontal axis with continuous data commonly related to time (e.g., hours, days, months, years, and age). Line graphs will have no more than six identified points along a continuum for continuous data (e.g., the decades: 1950s, 1960s, 1970s, 1980s, 1990s, and 2000s). MTH.G5.15.a.6 Construct a stem‐and‐leaf plot to organize and display data, where the stem is listed in ascending order and the leaves are in ascending order, with or without commas between leaves. MTH.G5.15.a.7 Title the given graph or identify the title. MTH.G5.15.a.8 Interpret the data in a variety of forms (e.g., orally or in written form). MTH.G5.16.a.1 Describe the mean of a group of numbers representing data from a given context as a measure of center. MTH.G5.16.a.2 Describe the median of a group of numbers representing data from a given context as a measure of center. MTH.G5.16.a.3 Describe the mode of a group of numbers representing data from a given context as a measure of center. 2011 27 MTH.G5.16.a.4 Describe the impact on measures of center when a single value of a data set is added, removed, or changed. MTH.G5.16.b.1 Describe mean as fair share. MTH.G5.16.c.1 Find the mean of a group of numbers. MTH.G5.16.c.2 Find the median of a group of numbers. MTH.G5.16.c.3 Find the mode of a group of numbers. MTH.G5.16.c.4 Find the range of a group of numbers. MTH.G5.16.d.1 Describe the range of a group of numbers representing data from a given context as a measure of variation. Expected Extended Extended MTH.G4.14.a.10 Describe the difference between representing categorical data MTH.G5.15.a.9 Create a possible set of data points from a given mean, median, and representing numerical data. mode, and/or range. MTH.G4.14.a.11 Collect, display, and analyze data to answer a student‐
MTH.G5.15.a.10 Construct and analyze double stem‐and‐leaf plots and generated question. multiple line graphs. MTH.G4.14.a.12 Construct and analyze double bar graphs. Expected MTH.G4.14.a.9 Collect, organize, display, and interpret data on line plots with no more than 30 data points. 2011 28 Mathematics Grade 4 Unit 10: Working with Probability Suggested Time Frame: 3 weeks Grade Level Below Standards MTH.G3.18‐The student will investigate and describe the concept of probability as chance and list possible results of a given situation. Correlated Standards Current Grade Level Standards MTH.G4.13‐The student will a) predict the likelihood of an outcome of a simple event; and b) represent probability as a number between 0 and 1, inclusive.
Grade Level Above Standards MTH.G5.14‐The student will make predictions and determine the probability of an outcome by constructing a sample space. Indirect Correlation 2011 29 Mathematics Grade 4 Unit 10: Working with Probability Suggested Time Frame: 3 weeks Correlated Indicators Grade 4 Indicators Essential MTH.G4.13.a.1 Model and determine all possible outcomes of a given simple event where there are no more than 24 possible outcomes, using a variety of manipulatives, such as coins, number cubes, and spinners. MTH.G4.13.b.1 Write the probability of a given simple event as a fraction, where the total number of possible outcomes is 24 or fewer. MTH.G4.13.b.2 Identify the likelihood of an event occurring and relate it to its fractional representation (e.g., impossible/0; equally likely/ ; certain/1). MTH.G4.13.b.3 Determine the outcome of an event that is least likely to occur (less than half) or most likely to occur (greater than half) when the number of possible outcomes is 24 or less. MTH.G4.13.b.4 Represent probability as a point between 0 and 1, inclusively, on a number line.
Expected Extended MTH.G4.13.b.5 Compare theoretical probability to experimental probability of simple events. Grade 5 Indicators Essential MTH.G5.14.a.1 Construct a sample space, using a list or chart to represent all possible outcomes of a single event. MTH.G5.14.a.2 Construct a sample space, using a tree diagram to identify all possible outcomes of a single event. MTH.G5.14.a.3 Predict and determine the probability of an outcome by constructing a sample space. The sample space will have a total of 24 or less possible outcomes. Expected Extended 2011 30 MTH.G4.1‐The student will a) identify orally and in writing the place value for each digit in a whole number expressed through millions; b) compare two whole numbers expressed through millions, using symbols (>, <, or =); and c) round whole numbers expressed through millions to the nearest thousand, ten thousand, and hundred thousand. ESSENTIAL KNOWLEDGE AND SKILLS UNDERSTANDING THE STANDARD (Teacher Notes) The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G4.1.a.1 Identify and communicate, both orally and in written form, the place value for each digit in whole numbers expressed through the one millions place. MTH.G4.1.a.2 Read whole numbers through the one millions place that are presented in standard format, and select the matching number in written format. MTH.G4.1.a.3 Write whole numbers through the one millions place in standard format when the numbers are presented orally or in written format. MTH.G4.1.b.1 Identify and use the symbols for greater than, less than, and equal to. MTH.G4.1.b.2 Compare two whole numbers expressed through the one millions, using symbols >, <, or =. MTH.G4.1.c.1 Round whole numbers expressed through the one millions place to the nearest thousand, ten thousand, and hundred‐thousand place. Expected MTH.G4.1.a.4 Represent whole numbers on a number line. Extended State Notes: • The structure of the Base‐10 number system is based upon a simple pattern of tens, in which the value of each place is ten times the value of the place to its right. • Place value refers to the value of each digit and depends upon the position of the digit in the number. For example, in the number 7,864,352, the eight is in the hundred thousands place, and the value of the 8 is eight hundred thousand or 800,000. • Whole numbers may be written in a variety of formats: –Standard: 1,234,567 –Written: one million, two hundred thirty‐four thousand, five hundred sixty‐seven –Expanded: (1 x 1,000,000) + (2 x 100,000) + (3 x 10,000) + (4 x 1,000) + (5 x 100) + (6 x 10) + (7 x 1) • Numbers are arranged into groups of three places called periods (ones, thousands, millions, …). Places within the periods repeat (hundreds, tens, ones). Commas are used to separate the periods. Knowing the place value and period of a number helps students find values of digits in any number as well as read and write numbers. • Reading and writing large numbers should be meaningful for students. Experiences can be provided that relate practical situations (e.g., numbers found in the students’ environment including population, number of school lunches sold statewide in a day, etc.). Concrete materials such as Base‐10 blocks and bundles of sticks may be used to represent whole numbers through thousands. Larger numbers may be represented by digit cards and place value charts. • Mathematical symbols (>, <) used to compare two unequal numbers are called inequality symbols. • A procedure for comparing two numbers by examining place value may include the following: –Compare the digits in the numbers to determine which number is greater (or which is less). –Use a number line to identify the appropriate placement of the numbers based on the place value of the digits. –Use the appropriate symbol > or < or words greater than or less than to compare the numbers in the order in which they are presented. –If both numbers have the same value, use the symbol = or words equal to. • A strategy for rounding numbers to the nearest thousand, ten thousand, and hundred 2011 31 thousand is as follows: –Use a number line to determine the rounded number (e.g., when rounding 4,367,925 to the nearest thousand, identify the ‘thousands’ the number would fall between on the number line, then determine the thousand that the number is closest to): –Look one place to the right of the digit to which you wish to round. –If the digit is less than 5, leave the digit in the rounding place as it is, and change the digits to the right of the rounding place to zero. –If the digit is 5 or greater, add 1 to the digit in the rounding place and change the digits to the right of the rounding place to zero. Essential Understanding: All students should • Understand the relationships in the place‐value system in which the value of each place is ten times the value of the place to its right. • Use the patterns in the place‐value system to read and write numbers. • Understand that reading place value correctly is essential when comparing numbers. • Understand that rounding gives a close number to use when exact numbers are not needed for the situation at hand. • Develop strategies for rounding. Vocabulary: whole number 2011 32 MTH.G4.2‐The student will a) compare and order fractions and mixed numbers; b) represent equivalent fractions; and c) identify the division statement that represents a fraction. ESSENTIAL KNOWLEDGE AND SKILLS UNDERSTANDING THE STANDARD (Teacher Notes) The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G4.2.a.1 Compare and order fractions having denominators of 12 or less, using manipulative models and drawings, such as region/area models. MTH.G4.2.a.2 Compare and order fractions with like denominators by comparing number of parts (numerators) (e.g., < ). MTH.G4.2.a.3 Compare and order fractions with like numerators and unlike denominators by comparing the size of the parts (e.g., < ). MTH.G4.2.a.4 Compare and order fractions having unlike denominators of 12 or less by comparing the fractions to benchmarks (e.g., 0, , or 1) to determine their relationships to the benchmarks or by finding a common denominator. MTH.G4.2.a.5 Compare and order mixed numbers having denominators of 12 or less. MTH.G4.2.a.6 Use the symbols >, <, and = to compare the numerical value of fractions and mixed numbers having denominators of 12 or less. MTH.G4.2.b.1 Represent equivalent fractions through State Notes: • A fraction is a way of representing part of a whole (as in a region/area model or a measurement model) or part of a group (as in a set model). A fraction is used to name a part of one thing or a part of a collection of things. • In the area/region and length/measurement fraction models, the parts must be equal. In the set model, the elements of the set do not have to be equal (i.e., "What fraction of the class is wearing the color red?"). • The denominator tells how many equal parts are in the whole or set. The numerator tells how many of those parts are being counted or described. • When fractions have the same denominator, they are said to have "common denominators" or "like denominators." Comparing fractions with like denominators involves comparing only the numerators. • Strategies for comparing fractions having unlike denominators may include –comparing fractions to familiar benchmarks (e.g., 0, , 1); –finding equivalent fractions, using manipulative models such as fraction strips, number lines, fraction circles, rods, pattern blocks, cubes, Base‐10 blocks, tangrams, graph paper, or a multiplication chart and patterns; and –finding a common denominator by finding the least common multiple (LCM) of both denominators and then rewriting each fraction as an equivalent fraction, using the LCM as the denominator. • A variety of fraction models should be used to expand students' understanding of fractions and mixed numbers: –Region/area models: a surface or area is subdivided into smaller equal parts, and each part is compared with the whole (e.g., fraction circles, pattern blocks, geoboards, grid paper, color tiles). –Set models: the whole is understood to be a set of objects, and subsets of the whole make up fractional parts (e.g., counters, chips). –Measurement models: similar to area models but lengths instead of areas are compared (e.g., fraction strips, rods, cubes, number lines, rulers). • A mixed number has two parts: a whole number and a fraction. • Equivalent fractions name the same amount. Students should use a variety of models to 2011 33 twelfths, using region/area models, set models, and measurement models. MTH.G4.2.c.1 Identify the division statement that represents a fraction (e.g., means the same as 3 divided by 5). Expected Extended MTH.G4.2.a.7 Use partitioning to determine "how much more" when comparing fractions. MTH.G4.2.b.2 Generate equivalent expressions for the same quantity [e.g., 1 case of cola = 24 cans = 2 (12‐packs) = 4 (6‐
packs) = 3 (8‐packs)]. identify different names for equivalent fractions. • Students should focus on finding equivalent fractions of familiar fractions such as halves, thirds, fourths, sixths, eighths, tenths, and twelfths. • Decimals and fractions represent the same relationships; however, they are presented in two different formats. The decimal 0.25 is written as . When presented with the fraction , the division expression representing a fraction is written as 3 divided by 5. FCPS Notes: • It is important for children to think flexibly about quantities. • Looking at one quantity as a variety of different sized pieces (chunks) is referred to as unitizing. For example, a case of soda could be 24 individual cans, two 12‐packs, or four 6‐
packs. "In mathematics, the ability to conceive of a commodity in terms of more than one size chunk frequently adds convenience, simplicity, speed, and sophistication to one's mathematical reasoning" (Lamon, 2005, p. 80). • Students can use notation that tells how they are "chunking" the quantity. Here is a convenient notation: # of chunks (size of chunks). For example, let's think about the case of soda. 24 cans = 2 (12‐packs) = 4 (6‐packs). Let's think about 24 eggs. 24 eggs = 2 (dozen) = 4 (6‐
packs) = 1 (18‐packs) = 12 (pairs). Let's think about 32 crayons. 32 crayons = (64‐box) = 2 (16‐
boxes) = 16 (pairs) = 64 ( crayons). Notice that, in all three examples, the quantity did not change, just the way we thought about it changed. • Partitioning is the act of dividing up a whole into smaller parts of equal size. • Partitioning activities build on students' knowledge of fair sharing. • Using partitioning to determine "how much more" helps to develop the idea of using common denominators to compare fractions. • When using partitioning to determine "how much more" it is important that the fractions being compared are from the same size unit (whole). • To compare to , draw a measurement model of each fraction, using the same size whole. • Essential Understanding: All students should Then, partition each of the fractions further until they both have the same size parts. 2011 34 Now you can see that is larger than . Reference: Lamon, S. J. (2005). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (2nd ed.). Mahwah, NJ: Lawrence Erlbaum Associates, Inc. Essential Understanding: All students should • Develop an understanding of fractions as parts of unit wholes, as parts of a collection, and as locations on a number line. • Understand that a mixed number is a fraction that has two parts: a whole number and a proper fraction. The mixed number is the sum of these two parts. • Use models, benchmarks, and equivalent forms to judge the size of fractions. • Recognize that a whole divided into nine equal parts has smaller parts than if the whole had been divided into five equal parts. • Recognize and generate equivalent forms of commonly used fractions and decimals. • Understand the division statement that represents a fraction. ). • Understand that the more parts the whole is divided into, the smaller the parts (e.g. ,
Vocabulary: equivalent 2011 35 MTH.G4.3‐The student will a) read, write, represent, and identify decimals expressed through thousandths; b) round to the nearest whole number, tenth, and hundredth; c) compare and order decimals; and d) given a model, write the decimal and fraction equivalents. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G4.3.a.1 Investigate the ten‐to‐one place‐value relationship for decimals through thousandths, using Base‐10 manipulatives (e.g., place‐value mats/charts, decimal squares, Base‐10 blocks, money). MTH.G4.3.a.2 Represent and identify decimals expressed through thousandths, using Base‐10 manipulatives, pictorial representations, and numerical symbols (e.g., relate the appropriate drawing to 0.05). MTH.G4.3.a.3 Identify and communicate, both orally and in written form, the position and value of a decimal through thousandths. For example, in 0.385, the 8 is in the hundredths place and has a value of 0.08. MTH.G4.3.a.4 Read and write decimals expressed through thousandths, using Base‐10 manipulatives, drawings, and numerical symbols. MTH.G4.3.b.1 Round decimals to the nearest whole number, tenth, and hundredth. MTH.G4.3.c.1 Compare the value of two decimals, using the symbols <, >, =. MTH.G4.3.c.2 Order a set of decimals from least to greatest or 2011 UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • The structure of the Base‐10 number system is based upon a simple pattern of tens, where each place is ten times the value of the place to its right. This is known as a ten‐to‐one place value relationship. • Understanding the system of tens means that ten tenths represents one whole, ten hundredths represents one tenth, ten thousandths represents one hundredth. • A decimal point separates the whole number places from the places that are less than one. Place values extend infinitely in two directions from a decimal point. A number containing a decimal point is called a decimal number or simply a decimal. • To read decimals, –read the whole number to the left of the decimal point, if there is one; –read the decimal point as “and”; –read the digits to the right of the decimal point just as you would read a whole number; and –say the name of the place value of the digit in the smallest place. • Any decimal less than 1 will include a leading zero (e.g., 0.125). • Decimals may be written in a variety of forms: –Standard: 26.537 –Written: twenty‐six and five hundred thirty‐seven thousandths –Expanded: (2 x 10) + (6 x 1) + (5 x 0.1) + (3 x 0.01) + (7 x 0.001). • Decimals and fractions represent the same relationships; however, they are presented in two different formats. The decimal 0.25 is written as . • Decimal numbers are another way of writing fractions. When presented with the fraction , the division expression representing a fraction is written as 3 divided by 5. The Base‐10 models concretely relate fractions to decimals (e.g., 10‐by‐10 grids, meter sticks, number lines, decimal squares, money). • The procedure for rounding decimal numbers is similar to the procedure for rounding whole numbers. • A strategy for rounding decimal numbers to the nearest tenth and hundredth is as follows: –Look one place to the right of the digit you want to round to. 36 greatest to least. MTH.G4.3.d.1 Represent fractions for halves, fourths, fifths, and tenths as decimals through hundredths, using concrete objects (e.g., demonstrate the relationship between the fraction and its decimal equivalent 0.25). MTH.G4.3.d.2 Relate fractions to decimals, using concrete objects (e.g., 10‐by‐10 grids, meter sticks, number lines, decimal squares, decimal circles, money [coins]). MTH.G4.3.d.3 Write the decimal and fraction equivalent for a given model (e.g., = 0.25 or 0.25 = ). Expected MTH.G4.3.a.5 Represent decimal numbers on a number line. Extended MTH.G4.3.a.6 Use and represent numbers through millions in various contexts, including estimation of relative sizes of amounts or distances (e.g., 310,989,822 is the population of the United States of America). 2011 –If the digit is 5 or greater, add 1 to the digit in the rounding place, and drop the digits to the right of the rounding place. –If the digit is less than 5, leave the digit in the rounding place as it is, and drop the digits to the right of the rounding place. • Different strategies for rounding decimals include: –Use a number line to locate a decimal between two numbers. For example, 18.83 is closer to 18.8 than to 18.9. –Compare the digits in the numbers to determine which number is greater (or which is less). –Compare the value of decimals, using the symbols >, <, = (e.g., 0.83 > 0.8 or 0.19 < 0.2). –Order the value of decimals, from least to greatest and greatest to least (e.g., 0.83, 0.821, 0.8). • Decimal numbers are another way of writing fractions (halves, fourths, fifths, and tenths). The Base‐10 models concretely relate fractions to decimals (e.g., 10‐by‐10 grids, meter sticks, number lines, decimal squares, decimal circles, money). • Provide a fraction model (halves, fourths, fifths, and tenths) and ask students for its decimal equivalent. • Provide a decimal model and ask students for its fraction equivalent (halves, fourths, fifths, and tenths). Essential Understanding: All students should • Understand the place value structure of decimals and use this structure to read, write, and compare decimals. • Understand that decimal numbers can be rounded to an estimate when exact numbers are not needed for the situation at hand. • Understand that decimals are rounded in a way that is similar to the way whole numbers are rounded. • Understand that decimals and fractions represent the same relationship; however, they are presented in two different formats. • Understand that models are used to show decimal and fraction equivalents. Vocabulary: decimal number 37 MTH.G4.4‐The student will a) estimate sums, differences, products, and quotients of whole numbers; b) add, subtract, and multiply whole numbers; c) divide whole numbers, finding quotients with and without remainders; and d) solve single‐step and multistep addition, subtraction, and multiplication problems with whole numbers. ESSENTIAL KNOWLEDGE AND SKILLS UNDERSTANDING THE STANDARD (Teacher Notes) The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G4.4.a.1 Estimate whole number sums, differences, products, and quotients. MTH.G4.4.a.2 Refine estimates by adjusting the final amount, using terms such as closer to, between, and a little more than. MTH.G4.4.b.1 Determine the sum or difference of two whole numbers, each 999,999 or less, in vertical and horizontal form with or without regrouping, using paper and pencil, and using a calculator. MTH.G4.4.b.2 Estimate and find the product of two whole numbers when one factor has two digits or fewer and the other factor has three digits or fewer, using paper and pencil and calculators. MTH.G4.4.c.1 Estimate and find the quotient of two whole numbers, given a one‐digit divisor and a two‐ or three‐digit dividend. MTH.G4.4.d.1 Solve single‐step and multistep addition, subtraction, and multiplication problems using whole number operations. MTH.G4.4.d.2 Verify the reasonableness of sums, differences, products, and quotients of whole numbers using estimation. State Notes: • A sum is the result of adding two or more numbers. • A difference is the amount that remains after one quantity is subtracted from another. • An estimate is a number close to an exact solution. An estimate tells about how much or about how many. • Different strategies for estimating include using compatible numbers to estimate sums and differences and using front‐end estimation for sums and differences. ‐Compatible numbers are numbers that are easy to work with mentally. Number pairs that are easy to add or subtract are compatible. When estimating a sum, replace actual numbers with compatible numbers (e.g., 52 + 74 can be estimated by using the compatible numbers 50 + 75). When estimating a difference, use numbers that are close to the original numbers. Tens and hundreds are easy to subtract (e.g., 83 ‐ 38 is close to 80 ‐ 40). ‐The front‐end strategy for estimating is computing with the front digits. Front‐end estimation for addition can be used even when the addends have a different number of digits. The procedure requires the addition of the values of the digits in the greatest of the smallest number. For example: 2367 2300 243 200 + 1186 +1100 3600 • Front‐end or leading‐digit estimation always gives a sum less than the actual sum; however, the estimate can be adjusted or refined so that it is closer to the actual sum. • Addition is the combining of quantities; it uses the following terms: addend 45,623 addend + 37,846 sum 83,469 • Subtraction is the inverse of addition; it yields the difference between two numbers and uses the following terms: minuend 45,698 subtrahend – 32,741 difference 12,957 2011 38 Expected MTH.G4.4.c.2 Interpret the remainder in problem‐solving settings. MTH.G4.4.d.3 Solve single‐step and multistep division problems using whole number operations. Extended MTH.G4.4.d.4 Evaluate the traditional multiplication and division algorithms and invented algorithms used by peers for efficiency and accuracy. 2011 • Before adding or subtracting with paper and pencil, addition and subtraction problems in horizontal form should be rewritten in vertical form by lining up the places vertically. • Using Base‐10 materials to model and stimulate discussion about a variety of problem situations helps students understand regrouping and enables them to move from the concrete to the pictorial, to the abstract. Regrouping is used in addition and subtraction algorithms. In addition, when the sum in a place is 10 or more, it is used to regroup the sums so that there is only one digit in each place. In subtraction, when the number (minuend) in a place is not enough from which to subtract, regrouping is required. • A certain amount of practice is necessary to develop fluency with computational strategies for multidigit numbers; however, the practice must be meaningful, motivating, and systematic if students are to develop fluency in computation, whether mentally, with manipulative materials, or with paper and pencil. • Calculators are an appropriate tool for computing sums and differences of large numbers, particularly when mastery of the algorithm has been demonstrated. • The terms associated with multiplication are factor 376 factor x 23 product 8,648 • One model of multiplication is repeated addition. • Another model of multiplication is the "Partial Product" model. 24 x 3 12 Multiply the ones: 3 x 4 = 12 + 60 Multiply the tens: 3 x 20 = 60 72 • Another model of multiplication is the "Area Model" (which also represents partial products) and should be modeled first with Base‐10 blocks. (e.g., 23 x 68) • Students should continue to develop fluency with single‐digit multiplication facts and their related division facts. • Calculators should be used to solve problems that require tedious calculations. • Estimation should be used to check the reasonableness of the product. Examples of estimation strategies include the following: ‐The front‐end method: multiply the front digits and then complete the product by recording the number of zeros found in the factors. It is important to develop understanding of this process before using the step‐by‐step procedure. 523 500 x 31 x 30 15,000 ‐This is 3 x 5 = 15 with 3 zeros. 39 ‐Compatible numbers: replace factors with compatible numbers, and then multiply. Opportunities for students to discover patterns with 10 and powers of 10 should be provided. 64 64 x 11 x 10 x 11
• Division is the operation of making equal groups or equal shares. When the original amount and the number of shares are known, divide to find the size of each share. When the original amount and the size of each share are known, divide to find the number of shares. Both situations may be modeled with Base‐10 manipulatives. • Multiplication and division are inverse operations. • Terms used in division are dividend, divisor, and quotient. dividend ÷ divisor = quotient quotient
divisor dividend
• Opportunities to invent division algorithms help students make sense of the algorithm. Teachers should teach division by various methods such as repeated multiplication and subtraction (partial quotients) before teaching the traditional long division algorithm. Essential Understanding: All students should • Develop and use strategies to estimate whole number sums and differences and to judge the reasonableness of such results. • Understand that addition and subtraction are inverse operations. • Understand that division is the operation of making equal groups or equal shares. When the original amount and the number of shares are known, divide to find the size of each share. When the original amount and the size of each share are known, divide to find the number of shares. • Understand that multiplication and division are inverse operations. • Understand various representations of division and the terms used in division are dividend, divisor, and quotient. dividend ÷ divisor = quotient quotient
divisor dividend
nd • Understand how to solve single‐step and multistep problems using whole number operations. Vocabulary: a little more than, between, closer to, dividend, divisor, quotient, remainder 2011 40 MTH.G4.5‐The student will a) determine common multiples and factors, including least common multiple and greatest common factor; b) add and subtract fractions having like and unlike denominators that are limited to 2, 3, 4, 5, 6, 8, 10, and 12, and simplify the resulting fractions, using common multiples and factors; c) add and subtract with decimals; and d) solve single‐step and multistep practical problems involving addition and subtraction with fractions and with decimals. ESSENTIAL KNOWLEDGE AND SKILLS UNDERSTANDING THE STANDARD (Teacher Notes) The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G4.5.a.1 Find common multiples and common factors of numbers. MTH.G4.5.a.2 Determine the least common multiple and greatest common factor of numbers. MTH.G4.5.b.1 Use least common multiple and/or greatest common factor to find a common denominator for fractions. MTH.G4.5.b.2 Add and subtract with fractions having like denominators whose denominators are limited to 2, 3, 4, 5, 6, 8, 10, and 12, and simplify the resulting fraction using common multiples and factors. MTH.G4.5.b.3 Add and subtract with fractions having unlike denominators whose denominators are limited to 2, 3, 4, 5, 6, 8, 10, and 12, and simplify the resulting fraction using common multiples and factors. MTH.G4.5.b.4 Solve problems that involve adding and subtracting with fractions having like and unlike denominators whose denominators are limited to 2, 3, 4, 5, 6, 8, 10, and 12, and simplify the resulting fraction using common multiples and factors. State Notes: • A factor of a number is an integer that divides evenly into that number with a remainder of zero. • A factor of a number is a divisor of the number. • A multiple of a number is the product of the number and any natural number. • A common factor of two or more numbers is a divisor that all of the numbers share. • The least common multiple of two or more numbers is the smallest common multiple of the given numbers. • The greatest common factor of two or more numbers is the largest of the common factors that all of the numbers share. • Students should investigate addition and subtraction with fractions, using a variety of models (e.g., fraction circles, fraction strips, rulers, linking cubes, pattern blocks). • When adding or subtracting with fractions having like denominators, add or subtract the numerators and use the same denominator. Write the answer in simplest form using common multiples and factors. • When adding or subtracting with fractions having unlike denominators, rewrite them as fractions with a common denominator. The least common multiple (LCM) of the unlike denominators is a common denominator (LCD). Write the answer in simplest form using common multiples and factors. • Addition and subtraction of decimals may be explored, using a variety of models (e.g., 10‐by‐
10 grids, number lines, money). • For decimal computation, the same ideas developed for whole number computation may be used, and these ideas may be applied to decimals, giving careful attention to the placement of the decimal point in the solution. Lining up tenths to tenths, hundredths to hundredths, etc. helps to establish the correct placement of the decimal. • Fractions may be related to decimals by using models (e.g., 10‐by‐10 grids, decimal squares, money). Essential Understanding: 2011 41 MTH.G4.5.c.1 Add and subtract with decimals through thousandths, using concrete materials, pictorial representations, and paper and pencil. MTH.G4.5.d.1 Solve single‐step and multistep problems that involve adding and subtracting with fractions and decimals through thousandths. Expected MTH.G4.5.b.5 Estimate the sum and difference of fractions with like and unlike denominators which are limited to 2, 3, 4, 5, 6, 8, 10, and 12. Extended MTH.G4.5.b.6 Estimate, add, subtract mixed numbers with like and unlike denominators limited to 2, 3, 4, 5, 6, 8, 10, and 12. MTH.G4.5.c.2 Estimate the sum and difference of decimals to thousandths by using models, pictorial representations, and paper and pencil. MTH.G4.5.d.2 Use given information to create story problems with fractions and with decimals. Explain solution strategies. 2011 All students should • Understand and use common multiples and common factors for simplifying fractions. • Develop and use strategies to estimate addition and subtraction involving fractions and decimals. • Use visual models to add and subtract with fractions and decimals. Vocabulary: factor, greatest common factor, least common multiple, multiple, simplify 42 MTH.G4.6‐The student will a) estimate and measure weight/mass and describe the results in U.S. Customary and metric units as appropriate; and b) identify equivalent measurements between units within the U.S. Customary system (ounces, pounds, and tons) and between units within the metric system (grams and kilograms). ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G4.6.a.1 Determine an appropriate unit of measure (e.g., ounce, pound, ton, gram, kilogram) to use when measuring everyday objects in both metric and U.S. Customary units. MTH.G4.6.a.2 Estimate and measure objects in both metric and U.S. Customary units (e.g., ounce, pound, ton, gram, or kilogram) to the nearest appropriate measure, using a variety of measuring instruments. MTH.G4.6.a.3 Record the mass of an object including the appropriate unit of measure (e.g., 24 grams). MTH.G4.6.b.1 Identify equivalent measurements between units within the U.S. Customary system (ounces, pounds, and tons). MTH.G4.6.b.2 Identify equivalent measurements between units within the metric system (grams and kilograms). Expected Extended 2011 UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • Weight and mass are different. Mass is the amount of matter in an object. Weight is determined by the pull of gravity on the mass of an object. The mass of an object remains the same regardless of its location. The weight of an object changes depending on the gravitational pull at its location. In everyday life, most people are actually interested in determining an object’s mass, although they use the term weight (e.g., “How much does it weigh?” versus “What is its mass?”). • Balances are appropriate measuring devices to measure weight in U.S. Customary units (ounces, pounds) and mass in metric units (grams, kilograms). • Practical experience measuring the mass of familiar objects helps to establish benchmarks and facilitates the student’s ability to estimate weight/mass. • Students should estimate the mass/weight of everyday objects (e.g., foods, pencils, book bags, shoes), using appropriate metric or U.S. Customary units. Essential Understanding: All students should • Use benchmarks to estimate and measure weight/mass. • Identify equivalent measures between units within the U.S. Customary and between units within the metric measurements. Vocabulary: ton 43 MTH.G4.7‐The student will a) estimate and measure length, and describe the result in both metric and U.S. Customary units; and b) identify equivalent measurements between units within the U.S. Customary system (inches and feet; feet and yards; inches and yards; yards and miles) and between units within the metric system (millimeters and centimeters; centimeters and meters; and millimeters and meters). ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G4.7.a.1 Determine an appropriate unit of measure (e.g., inch, foot, yard, mile, millimeter, centimeter, and meter) to use when measuring everyday objects in both metric and U.S. Customary units. MTH.G4.7.a.2 Estimate the length of everyday objects (e.g., books, windows, tables) in both metric and U.S. Customary units of measure. MTH.G4.7.a.3 Measure the lengths of objects in both metric and U.S. Customary units, measuring to the nearest inch , , , foot, yard, mile, millimeter, centimeter, or meter, and record the length including the appropriate unit of measure (e.g., 24 inches). MTH.G4.7.a.4 Compare estimates of the length of objects with the actual measurement of the length of objects. MTH.G4.7.b.1 Identify equivalent measurements between units within the U.S. Customary system (inches and feet; feet and yards; inches and yards; yards and miles) and between units within the metric system (millimeters and centimeters; centimeters and meters; and millimeters and meters). UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • Length is the distance along a line or figure from one point to another. • U.S. Customary units for measurement of length include inches, feet, yards, and miles. Appropriate measuring devices include rulers, yardsticks, and tape measures. Metric units for measurement of length include millimeters, centimeters, meters, and kilometers. Appropriate measuring devices include centimeter ruler, meter stick, and tape measure. • Practical experience measuring the length of familiar objects helps to establish benchmarks and facilitates the student's ability to estimate length. • Students should estimate the length of everyday objects (e.g., books, windows, tables) in both metric and U.S. Customary units of measure. • When measuring with U.S. Customary units, students should be able to measure to the nearest part of an inch ( , , , inch, foot, or yard. Essential Understanding: All students should • Use benchmarks to estimate and measure length. • Understand the relationship between U.S. Customary units and the relationship between metric units. Vocabulary: mile, millimeter Expected Extended 2011 44 MTH.G4.8‐The student will a) estimate and measure liquid volume and describe the results in U.S. Customary units; and b) identify equivalent measurements between units within the U.S. Customary system (cups, pints, quarts, and gallons). ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G4.8.a.1 Determine an appropriate unit of measure (cups, pints, quarts, gallons) to use when measuring liquid volume in U.S. Customary units. MTH.G4.8.a.2 Estimate the liquid volume of containers in U.S. Customary units of measure to the nearest cup, pint, quart, and gallon. MTH.G4.8.a.3 Measure the liquid volume of everyday objects in U.S. Customary units, including cups, pints, quarts, and gallons, and record the volume including the appropriate unit of measure (e.g., 24 gallons). MTH.G4.8.b.1 Identify equivalent measures of volume between units within the U.S. Customary system (cups, pints, quarts, and gallons). Expected Extended MTH.G4.8.a.4 Explore volume using nonstandard units. 2011 UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • U.S. Customary units for measurement of liquid volume include cups, pints, quarts, and gallons. • The measurement of the object must include the unit of measure along with the number of iterations. • Students should measure the liquid volume of everyday objects in U.S. Customary units, including cups, pints, quarts, gallons, and record the volume including the appropriate unit of measure (e.g., 24 gallons). • Practical experience measuring liquid volume of familiar objects helps to establish benchmarks and facilitates the student’s ability to estimate liquid volume. • Students should estimate the liquid volume of containers in U.S. Customary units to the nearest cup, pint, quart, and gallon. Essential Understanding: All students should • Use benchmarks to estimate and measure volume. • Identify equivalent measurements between units within the U.S. Customary system. Vocabulary: 45 MTH.G4.9‐The student will determine elapsed time in hours and minutes within a 12‐hour period. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G4.9.a.1 Determine the elapsed time in hours and minutes within a 12‐hour period (times can cross between a.m. and p.m.) MTH.G4.9.a.2 Solve practical problems in relation to time that has elapsed. Expected Extended 2011 UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • Elapsed time is the amount of time that has passed between two given times. • Elapsed time should be modeled and demonstrated using analog clocks and timelines. • Elapsed time can be found by counting on from the beginning time to the finishing time. –Count the number of whole hours between the beginning time and the finishing time. –Count the remaining minutes. –Add the hours and minutes. For example, to find the elapsed time between 10:15 a.m. and 1:25 p.m., count 10 minutes; and then, add 3 hours to 10 minutes to find the total elapsed time of 3 hours and 10 minutes. Essential Understanding: All students should • Understanding the “counting on” strategy for determining elapsed time in hour and minute increments over a 12‐hour period from a.m. to a.m. or p.m. to p.m. Vocabulary: 46 MTH.G4.10‐The student will a) identify and describe representations of points, lines, line segments, rays, and angles, including endpoints and vertices; and b) identify representations of lines that illustrate intersection, parallelism, and perpendicularity. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G4.10.a.1 Identify and describe representations of points, lines, line segments, rays, and angles, including endpoints and vertices. MTH.G4.10.b.1 Understand that lines in a plane can intersect or are parallel. Perpendicularity is a special case of intersection. MTH.G4.10.b.2 Identify practical situations that illustrate parallel, intersecting, and perpendicular lines. Expected Extended MTH.G4.10.a.2 Compare and contrast points, line segments, rays, and angles including endpoints and vertices. MTH.G4.10.a.3 Sort angles and describe sorting rule 2011 UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • A point is a location in space. It has no length, width, or height. A point is usually named with a capital letter. • A line is a collection of points going on and on infinitely in both directions. It has no endpoints. When a line is drawn, at least two points on it can be marked and given capital letter names. Arrows must be drawn to show that the line goes on in both directions infinitely (e.g., , read as "the line AB"). • A line segment is part of a line. It has two endpoints and includes all the points between those endpoints. To name a line segment, name the endpoints (e.g., , read as "the line segment AB"). • A ray is part of a line. It has one endpoint and continues infinitely in one direction. To name a ray, say the name of its endpoint first and then say the name of one other point on the ray (e.g., , read as "the ray AB"). • Two rays that have the same endpoint form an angle. This endpoint is called the vertex. Angles are found wherever lines and line segments intersect. An angle can be named in three different ways by using ‐three letters to name, in this order, a point on one ray, the vertex, and a point on the other ray; ‐one letter at the vertex; or ‐a number written inside the rays of the angle. • Intersecting lines have one point in common. • Perpendicular lines are special intersecting lines that form right angles where they intersect. • Parallel lines are lines that lie in the same plane and do not intersect. Parallel lines are always the same distance apart and do not share any points. • Students should explore intersection, parallelism, and perpendicularity in both two and three dimensions. For example, students should analyze the relationships between the edges of a cube. Which edges are parallel? Which are perpendicular? What plane contains the upper left edge and the lower right edge of the cube? Students can visualize this by using the classroom itself to notice the lines formed by the intersection of the ceiling and walls, of the floor and wall, and of two walls. FCPS Notes: 47 • Students should work with a variety of angles and should decide how to sort the set of angles. • Students can describe their sorting methods orally or using models or pictures. Essential Understanding: All students should • Understand that points, lines, line segments, rays, and angles, including endpoints and vertices are fundamental components of noncircular geometric figures. • Understand that the shortest distance between two points on a flat surface is a line segment. • Understand that lines in a plane either intersect or are parallel. Perpendicularity is a special case of intersection. • Identify practical situations that illustrate parallel, intersecting, and perpendicular lines. Vocabulary: end point, intersecting, parallel, perpendicular 2011 48 MTH.G4.11‐ The student will a) investigate congruence of plane figures after geometric transformations, such as reflection, translation, and rotation, using mirrors, paper folding, and tracing; and b) recognize the images of figures resulting from geometric transformations, such as translation, reflection, and rotation. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G4.11.a.1 Recognize the congruence of plane figures resulting from geometric transformations such as translation, reflection, and rotation, using mirrors, paper folding and tracing. MTH.G4.11.b.1 Recognize the images of figures resulting from geometric transformations such as translation, reflection, or rotation. Expected Extended MTH.G4.11.b.2 Build tessellations with congruent shapes using transformations. 2011 UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • The van Hiele theory of geometric understanding describes how students learn geometry and provides a framework for structuring student experiences that should lead to conceptual growth and understanding. • ‐Level 0: Pre‐recognition. Geometric figures are not recognized. For example, students cannot differentiate between three‐sided and four‐sided polygons. • ‐Level 1: Visualization. Geometric figures are recognized as entities, without any awareness of parts of figures or relationships between components of a figure. Students should recognize and name figures and distinguish a given figure from others that look somewhat the same. (This is the expected level of student performance during grades K and 1.) • ‐Level 2: Analysis. Properties are perceived but are isolated and unrelated. Students should recognize and name properties of geometric figures. (Students are expected to transition to this level during grades 2 and 3.) • ‐Level 3: Abstraction. Definitions are meaningful, with relationships being perceived between properties and between figures. Logical implications and class inclusion are understood, but the role and significance of deduction is not understood. (Students should transition to this level during grades 5 and 6 and fully attain it before taking algebra.) • Congruent figures are figures having exactly the same size and shape. Opportunities for exploring figures that are congruent and/or noncongruent can best be accomplished by using physical models. • A translation is a transformation in which an image is formed by moving every point on a figure the same distance in the same direction. • A reflection is a transformation in which a figure is flipped over a line called the line of reflection. All corresponding points in the image and preimage are equidistant from the line of reflection. • A rotation is a transformation in which an image is formed by turning its preimage about a point. • The resulting figure of a translation, reflection, or rotation is congruent to the original figure. Essential Understanding: All students should 49 • Understand the meaning of the term congruent. • Understand how to identify congruent figures. • Understand that the orientation of figures does not affect congruency or noncongruency. Vocabulary: reflection, rotation, transformation, translation 2011 50 MTH.G4.12‐The student will a) define polygon; and b) identify polygons with 10 or fewer sides. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G4.12.a.1 Define and identify properties of polygons with 10 or fewer sides. MTH.G4.12.b.1 Identify polygons by name with 10 or fewer sides in multiple orientations (rotations, reflections, and translations of the polygons). Expected MTH.G4.12.a.2 Identify a figure as open or closed and identify plane figures as polygons or nonpolygons. Extended 2011 UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • A polygon is a closed plane geometric figure composed of at least three line segments that do not cross. None of the sides are curved. • A triangle is a polygon with three angles and three sides. • A quadrilateral is a polygon with four sides. • A rectangle is a quadrilateral with four right angles. • A square is a rectangle with four sides of equal length. • A trapezoid is a quadrilateral with exactly one pair of parallel sides. • A parallelogram is a quadrilateral with both pairs of opposite sides parallel. • A rhombus is a quadrilateral with 4 congruent sides. • A pentagon is a 5‐sided polygon. • A hexagon is a 6‐sided polygon. • A heptagon is a 7‐sided polygon. • An octagon is an 8‐sided polygon. • A nonagon is a 9‐sided polygon. • A decagon is a 10‐sided polygon. Essential Understanding: All students should • Identify polygons with 10 or fewer sides in everyday situations. • Identify polygons with 10 or fewer sides in multiple orientations (rotations, reflections, and translations of the polygons). Vocabulary: heptagon, hexagon, nonagon, octagon, parallelogram, pentagon, quadrilateral, rhombus, trapezoid 51 MTH.G4.13‐The student will a) predict the likelihood of an outcome of a simple event; and b) represent probability as a number between 0 and 1, inclusive. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G4.13.a.1 Model and determine all possible outcomes of a given simple event where there are no more than 24 possible outcomes, using a variety of manipulatives, such as coins, number cubes, and spinners. MTH.G4.13.b.1 Write the probability of a given simple event as a fraction, where the total number of possible outcomes is 24 or fewer. MTH.G4.13.b.2 Identify the likelihood of an event occurring and relate it to its fractional representation (e.g., impossible/0; equally likely/ ; certain/1). MTH.G4.13.b.3 Determine the outcome of an event that is least likely to occur (less than half) or most likely to occur (greater than half) when the number of possible outcomes is 24 or less. MTH.G4.13.b.4 Represent probability as a point between 0 and 1, inclusively, on a number line. Expected Extended MTH.G4.13.b.5 Compare theoretical probability to experimental probability of simple events. 2011 UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • A spirit of investigation and experimentation should permeate probability instruction, where students are actively engaged in explorations and have opportunities to use manipulatives. • Probability is the chance of an event occurring. • The probability of an event occurring is the ratio of desired outcomes to the total number of possible outcomes. If all the outcomes of an event are equally likely to occur, the probability of the event = . • The probability of an event occurring is represented by a ratio between 0 and 1. An event is "impossible" if it has a probability of 0 (e.g., the probability that the month of April will have 31 days). An event is "certain" if it has a probability of 1 (e.g., the probability that the sun will rise tomorrow morning). • When a probability experiment has very few trials, the results can be misleading. The more times an experiment is done, the closer the experimental probability comes to the theoretical probability (e.g., a coin lands heads up half of the time). • Conduct experiments to determine the probability of an event occurring for a given number of trials (no more than 25 trials), using manipulatives (e.g., the number of times "heads" occurs when flipping a coin 10 times; the chance that when the names of 12 classmates are put in a shoebox, a name that begins with D will be drawn). • Students should have opportunities to describe in informal terms (i.e., impossible, unlikely, as likely as unlikely, equally likely, likely, and certain) the degree of likelihood of an event occurring. • Activities should include practical examples. • For an event such as flipping a coin, the equally likely things that can happen are called outcomes. For example, there are two equally likely outcomes when flipping a coin: the coin can land heads up, or the coin can land tails up. • For another event such as spinning a spinner that is one‐third red and two‐thirds blue, the two outcomes, red and blue, are not equally likely. This is an unfair spinner (since it is not divided equally), therefore, the outcomes are not equally likely. Essential Understanding: All students should • Understand and apply basic concepts of probability. 52 • Describe events as likely or unlikely and discuss the degree of likelihood, using the terms certain, likely, equally likely, unlikely, and impossible. • Predict the likelihood of an outcome of a simple event and test the prediction. • Understand that the measure of the probability of an event can be represented by a number between 0 and 1, inclusive. Vocabulary: 2011 53 MTH.G4.14‐The student will collect, organize, display, and interpret data from a variety of graphs. ESSENTIAL KNOWLEDGE AND SKILLS UNDERSTANDING THE STANDARD (Teacher Notes) The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G4.14.a.1 Collect data, using, for example, observations, measurement, surveys, scientific experiments, polls, or questionnaires. MTH.G4.14.a.2 Organize data into a chart or table. MTH.G4.14.a.3 Construct and display data in bar graphs, labeling one axis with equal whole number increments of 1 or more (numerical data) (e.g., 2, 5, 10, or 100) and the other axis with categories related to the title of the graph (categorical data) (e.g., swimming, fishing, boating, and water skiing as the categories of “Favorite Summer Sports”). MTH.G4.14.a.4 Construct and display data in line graphs, labeling the vertical axis with equal whole number increments of 1 or more and the horizontal axis with continuous data commonly related to time (e.g., hours, days, months, years, and age). Line graphs will have no more than 10 identified points along a continuum for continuous data. For example, growth charts showing age versus height place age on the horizontal axis (e.g., 1 month, 2 months, 3 months, and 4 months). MTH.G4.14.a.5 Title or identify the title in a given graph and label or identify the axes. MTH.G4.14.a.6 Interpret data from simple line and bar graphs by describing the characteristics of the data and the data as a whole (e.g., the category with the greatest/least, categories with the same number of responses, similarities and differences, the total number). Data points will be limited to State Notes: • Data analysis helps describe data, recognize patterns or trends, and make predictions. • Investigations involving practical data should occur frequently, and data can be collected through brief class surveys or through more extended projects taking many ways. • Students formulate questions, predict answers to questions under investigation, collect and represent initial data, and consider whether the data answer the questions. • Line graphs are used to show how two continuous variables are related. Line graphs may be used to show how one variable changes over time. If this one variable is not continuous, then a broken line is used. By looking at a line graph, it can be determined whether the variable is increasing, decreasing, or staying the same over time. –The values along the horizontal axis represent continuous data on a given variable, usually some measure of time (e.g., time in years, months, or days). The data presented on a line graph is referred to as “continuous data,” as it represents data collected over a continuous period of time. –The values along the vertical axis are the scale and represent the frequency with which those values occur in the data set. The values should represent equal increments of multiples of whole numbers, fractions, or decimals, depending upon the data being collected. The scale should extend one increment above the greatest recorded piece of data. –Each axis should be labeled, and the graph should be given a title. –A line graph tells whether something has increased, decreased, or stayed the same with the passage of time. Statements representing an analysis and interpretation of the characteristics of the data in the graph should be included (e.g., trends of increase and/or decrease, and least and greatest). • Bar graphs should be used to compare counts of different categories (categorical data). Using grid paper ensures more accurate graphs. –A bar graph uses parallel, horizontal or vertical bars to represent counts for several categories. One bar is used for each category, with the length of the bar representing the count for that category. –There is space before, between, and after the bars. –The axis that displays the scale representing the count for the categories should extend one increment above the greatest recorded piece of data. • Fourth‐grade students should collect data that are recorded in increments of whole numbers, usually multiples of 1, 2, 5, 10, or 100. –Each axis should be labeled, and the graph should be given a title. –Statements representing an analysis and interpretation of the characteristics of the data in 2011 54 30 and categories to 8. MTH.G4.14.a.7 Interpret the data to answer the question posed, and compare the answer to the prediction (e.g., “The summer sport preferred by most is swimming, which is what I predicted before collecting the data.”). MTH.G4.14.a.8 Write at least one sentence to describe the analysis and interpretation of the data, identifying parts of the data that have special characteristics, including categories with the greatest, the least, or the same. Expected MTH.G4.14.a.9 Collect, organize, display, and interpret data on line plots with no more than 30 data points. Extended MTH.G4.14.a.10 Describe the difference between representing categorical data and representing numerical data. MTH.G4.14.a.11 Collect, display, and analyze data to answer a student‐generated question. MTH.G4.14.a.12 Construct and analyze double bar graphs. the graph (e.g., similarities and differences, least and greatest, the categories, and total number of responses) should be written. FCPS Notes: • A line plot shows the frequency of data on a number line. Line plots are used to show the spread of the data and quickly identify the range, mode, and any outliers. • When data are displayed in an organized manner, the results of the investigations can be described and the posed question answered. • Recognition of appropriate and inappropriate statements begins at this level with graph interpretations. Essential Understanding: All students should • Understand the difference between representing categorical data and representing numerical data. • Understand that line graphs show change over time (numerical data). • Understand that bar graphs should be used to compare counts of different categories (categorical data). • Understand how data displayed in bar and line graphs can be interpreted so that informed decisions can be made. • Understand that the title and labels of the graph provide the foundation for interpreting the data. Vocabulary: horizontal axis, vertical axis 2011 55 MTH.G4.15‐The student will recognize, create, and extend numerical and geometric patterns. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G4.15.a.1 Describe geometric and numerical patterns, using tables, symbols, or words. MTH.G4.15.a.2 Create geometric and numerical patterns, using concrete materials, number lines, tables, and words. MTH.G4.15.a.3 Extend geometric and numerical patterns, using concrete materials, number lines, tables, and words. Expected Extended MTH.G4.15.a.4 Find function rules using a combination of two of the four operations (e.g., times 2, add 1) given input and output data (e.g., input 5, output 11; input 7, output 15). 2011 UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • Most patterning activities should involve some form of concrete materials to make up a pattern. – Students will identify and extend a wide variety of patterns, including rhythmic, geometric, graphic, numerical, and algebraic. The patterns will include both growing and repeating patterns. • Reproduction of a given pattern in a different representation, using symbols and objects, lays the foundation for writing the relationship symbolically or algebraically. • Tables of values should be analyzed for a pattern to determine the next value. Essential Understanding: All students should • Understand that patterns and functions can be represented in many ways and described using words, tables, graphs, and symbols. Vocabulary: 56 Mathematics POS Framework Grade 4 MTH.G4.16‐The student will a) recognize and demonstrate the meaning of equality in an equation; and b) investigate and describe the associative property for addition and multiplication. ESSENTIAL KNOWLEDGE AND SKILLS UNDERSTANDING THE STANDARD (Teacher Notes) The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G4.16.a.1 Recognize and demonstrate that the equal sign (=) relates equivalent quantities in an equation. MTH.G4.16.a.2 Write an equation to represent equivalent mathematical relationships (e.g., 4 × 3 = 2 × 6). MTH.G4.16.a.3 Recognize and demonstrate appropriate use of the equal sign in an equation. MTH.G4.16.b.1 Investigate and describe the associative property for addition as (6 + 2) + 3 = 6 + (2 + 3). MTH.G4.16.b.2 Investigate and describe the associative property for multiplication as (3 x 2) x 4 = 3 x (2 x 4). Expected Extended State Notes: • Investigating arithmetic operations with whole numbers helps students learn about several different properties of arithmetic relationships. These relationships remain true regardless of the numbers. • The commutative property for addition states that changing the order of the addends does not affect the sum (e.g., 4 + 3 = 3 + 4). Similarly, the commutative property for multiplication states that changing the order of the factors does not affect the product (e.g., 2 x 3 = 3 x 2). • The associative property for addition states that the sum stays the same when the grouping of addends is changed [e.g., 15 + (35 + 16) = (15 + 35)+ 16]. The associative property for multiplication states that the product stays the same when the grouping of factors is changed [e.g., 6 x (3 x 5) = (6 x 3) x 5]. Essential Understanding: All students should • Understand that mathematical relationships can be expressed using equations. • Understand that quantities on both sides of an equation must be equal. • Understand that the associative property for addition means you can change the groupings of three or more addends without changing the sum. • Understand that the associative property for multiplication means you can change the groupings of three or more factors without changing the product. Vocabulary: associative property 2011 57 MTH.G5.1‐The student, given a decimal through thousandths, will round to the nearest whole number, tenth, or hundredth. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G5.1.a.1 Round decimal numbers to the nearest whole number, tenth, or hundredth. Expected Extended 2011 UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • The structure of the Base‐10 number system is based upon a simple pattern of tens in which each place is ten times the value of the place to its right. This is known as a ten‐to‐one place value relationship. • A decimal point separates the whole number places from the places less than one. Place values extend infinitely in two directions from a decimal point. A number containing a decimal point is called a decimal number or simply a decimal. • To read decimals, –read the whole number to the left of the decimal point, if there is one; –read the decimal point as "and"; –read the digits to the right of the decimal point just as you would read a whole number; and –say the name of the place value of the digit in the smallest place. • Decimals may be written in a variety of forms: – Standard: 23.456 – Written: Twenty‐three and four hundred fifty‐six thousandths – Expanded: (2 x 10) + (3 x 1) + (4 x 0.1) + (5 x 0.01) + (6 x 0.001) • To help students identify the ten‐to‐one place value relationship for decimals through thousandths, use Base‐10 manipulatives, such as place value mats/charts, decimal squares, Base‐10 blocks, and money. • Decimals can be rounded to the nearest whole number, tenth or hundredth in situations when exact numbers are not needed. • Strategies for rounding decimal numbers to the nearest whole number, tenth and hundredth are as follows: –Look one place to the right of the digit to which you wish to round. –If the digit is less than 5, leave the digit in the rounding place as it is, and change the digits to the right of the rounding place to zero. –If the digit is 5 or greater, add 1 to the digit in the rounding place and change the digits to the right of the rounding place to zero. • Create a number line that shows the decimal that is to be rounded. • The position of the decimal will help children conceptualize the number's placement relative for rounding. An example is to round 5.747 to the nearest hundredth: 58 Essential Understanding: All students should • Understand that decimals are rounded in a way that is similar to the way whole numbers are rounded. • Understand that decimal numbers can be rounded to estimate when exact numbers are not needed for the situation at hand. Vocabulary: 2011 59 MTH.G5.2‐The student will a) recognize and name fractions in their equivalent decimal form and vice versa; and b) compare and order fractions and decimals in a given set from least to greatest and greatest to least. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G5.2.a.1 Represent fractions (halves, fourths, fifths, eighths, tenths, and twelfths) in their equivalent decimal form. MTH.G5.2.a.2 Represent decimals in their equivalent fraction form (halves, fourths, fifths, eighths, tenths, and twelfths). MTH.G5.2.a.3 Recognize and name equivalent relationships between decimals and fractions with denominators up to 12. MTH.G5.2.b.1 Compare and order from least to greatest and greatest to least a given set of no more than five numbers written as decimals, fractions, and mixed numbers with denominators of 12 or less. Expected MTH.G5.2.a.4 Represent fractions (thirds and sixths) in their equivalent decimal form. MTH.G5.2.a.5 Represent decimals in their equivalent fraction form (thirds and sixths). MTH.G5.2.a.6 Distinguish between terminating and repeating decimals and write repeating decimals using the appropriate notation. Extended MTH.G5.2.a.7 Solve problems involving fractions by reasoning 2011 UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • Students should recognize, name, and focus on finding equivalent decimals of familiar fractions such as halves, fourths, fifths, eighths, and tenths. • Students should be able to determine equivalent relationships between decimals and fractions with denominators up to 12. • Students should have experience with fractions such as , whose decimal representation is a terminating decimal (e. g., = 0.125) and with fractions such as , whose decimal representation does not end but continues to repeat (e. g., = 0.222…). The repeating decimal can be written with ellipses (three dots) as in 0.222… or denoted with a bar above the digits that repeat as in 0. 2. • To help students compare the value of two decimals through thousandths, use manipulatives, such as place value mats/charts, 10‐by‐10 grids, decimal squares, Base‐10 blocks, meter sticks, number lines, and money. • A procedure for comparing two decimals by examining may include the following: –Line up the decimal numbers at their decimal points. –Beginning at the left, find the first place value where the digits are different. –Compare the digits in this place value to determine which number is greater (or which is less). – Use the appropriate symbol > or < or the words greater than or less than to compare the numbers in the order in which they are presented. – If both numbers are the same, use the symbol = or words equal to. Two numbers can be compared by examining place value and/or using a number line. • Decimals and fractions represent the same relationships; however, they are presented in two different formats. Decimal numbers are another way of writing fractions. Base‐10 models (e.g., 10‐by‐10 grids, meter sticks, number lines, decimal squares, money) concretely relate fractions to decimals and vice versa. FCPS Notes: • It is important for children to explore units of different types when learning about fractions (Lamon, 2005). If region models, such as pizzas, are all they ever use, they will try to use pizzas even when other models are easier to work with. For example, cutting a rectangular cake into 60 thirds is much easier than cutting a circular pizza into thirds. Another reason for children to work with units of different types is because different units are psychologically different. How many lady bugs are on the tree? Think: 8 ladybugs = , so Dividing a set of hard candies into equal groups seems very different from cutting a cake into 2 ladybugs = , so 10 ladybugs = or 1. equal size pieces. • A unit can be a single thing, such as a pizza or a sandwich. It can also be a collection of things that are unitized, or mentally "chunked" together. MTH.G5.2.a.8 Solve comparison and missing value problems • Vary fraction problems by involving units of the following types (Barnett‐Clark, Fisher, Marks, by reasoning up and down (e.g., If it takes four people three days to wash windows at the office building, how long would it & Ross, 2010; Lamon, 2005): – one continuous item, such as one pie; take eight people to do the job?). – more than one continuous item, such as three pizzas or three yellow hexagons; – one or more continuous objects that are perforated or prepartitioned, such as a candy bar; – discrete objects (separate things, distinct parts), such as a group of hard candies or counters; – discrete objects that typically come arranged in a special way, such as a six‐pack of soda or eggs in a carton; and – composite units (i.e., units consisting of single packages that have multiple objects inside) such as a package of cupcakes or a pack of gum. • Fraction problems should clearly define the unit or provide enough information for students to determine the unit. • Reasoning up and down (Lamon, 2005) can be used to solve fraction problems. Reasoning up and down involves reasoning down from a fraction to a unit fraction, then up from the unit fraction to the unit, than back down to another fraction. For example, if 8 lady bugs equals of the lady bugs on a tree, how many ladybugs are on the tree? Reason down to the unit fraction: If 8 = , then 2 = . Then reason up to the unit. If 2 = , then 10 = or 1. Another example, if 15 up and down (e.g., 8 ladybugs = of the ladybugs on the tree. sticks of gum is of the package, how many sticks of gum would be in 1 packages? Reason down to the unit fraction. If 15 = , then 3 = . Then reason up to the unit. If 3 = , then 18 = or 1. Then reason to the target fraction. If 18 = 1, then 27 =1 . • There are two types of proportional reasoning problems: comparison problems, and missing value problems (Lamon, 2005). In comparison problems, four quantities forming two ratios are given. The task is to find out whether the two ratios are equivalent. For example, the Jones family went to the Multiplex Theater and all 6 of them got in for $10. The Miller family went to the Fairfax Theater and all 4 of them got in for $7. Which theatre has better prices? In missing value problems, three quantities are given and the fourth quantity is missing. For example, if it takes 4 people 3 days to wash the windows at an office building, how long will it take 8 people to do the job? Students can solve these problems by reasoning up and down. Students can solve these types of problems by reasoning up and down before they are formally introduced to ratios. References 2011 61 Barnett‐Clark, C., Fisher, W., Marks, R., & Ross, S. (2010). Developing essential understanding of rational numbers for teaching mathematics in grades 3‐5. Reston, VA: NCTM. Lamon, S. J. (2005). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (2nd ed.). Mahwah, NJ: Lawrence Erlbaum Associates, Inc. Essential Understanding: All students should • Understand the relationship between fractions and their decimal form and vice versa. • Understand that fractions and decimals can be compared and ordered from least to greatest and greatest to least. Vocabulary: 2011 62 MTH.G5.3‐The student will a) identify and describe the characteristics of prime and composite numbers; and b) identify and describe the characteristics of even and odd numbers. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G5.3.a.1. Identify prime numbers less than or equal to 100. MTH.G5.3.a.2 Identify composite numbers less than or equal to 100. MTH.G5.3.a.3 Explain orally and in writing why a number is prime or composite. MTH.G5.3.b.1 Identify which numbers are even or odd. MTH.G5.3.b.2 Explain and demonstrate with manipulatives, pictorial representations, oral language, or written language why a number is even or odd. Expected Extended MTH.G5.3.a.4 Represent numbers as the products of their prime factors (prime factorization). 2011 UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • A prime number is a natural number that has exactly two different factors, one and the number itself. • A composite number is a natural number that has more than two different factors. • The number 1 is neither prime nor composite because it has only one factor, itself. • The prime factorization of a number is a representation of the number as the product of its prime factors. For example, the prime factorization of 18 is 2 x 3 x 3. • Prime factorization concepts can be developed by using factor trees. • Prime or composite numbers can be represented by rectangular models or rectangular arrays on grid paper. A prime number can be represented by only one rectangular array (e.g., 7 can be represented by a 7 x 1 and a 1 x 7). A composite number can always be represented by more than two rectangular arrays (e.g., 9 can be represented by a 9 x 1, a 1 x 9, or a 3 x 3). • Divisibility rules are useful tools in identifying prime and composite numbers. • Students should use manipulatives (e.g., Base‐10 blocks, cubes, tiles, hundreds board, etc.) to explore and categorize numbers into groups of odd or even. • Students should use rules to categorize numbers into groups of odd or even. Rules can include: –An odd number does not have 2 as a factor or is not divisible by 2. –The sum of two even numbers is even. –The sum of two odd numbers is even. –The sum of an even and an odd is odd. –Even numbers have an even number or zero in the ones place. –Odd numbers have an odd number in the ones place. –An even number has 2 as a factor or is divisible by 2. Essential Understanding: All students should • Understand and use the unique characteristics of certain sets of numbers, including prime, composite, even, and odd numbers. Vocabulary: composite, prime 63 MTH.G5.4‐The student will create and solve single‐step and multistep practical problems involving addition, subtraction, multiplication, and division with and without remainders of whole numbers. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G5.4.a.1 Select appropriate methods and tools from among paper and pencil, estimation, mental computation, and calculators according to the context and nature of the computation in order to compute with whole numbers. MTH.G5.4.a.2 Estimate the sum, difference, product, and quotient of whole‐number computations. MTH.G5.4.a.3 Solve single‐step and multistep problems involving addition, subtraction, multiplication, and division with and without remainders of whole numbers, using paper and pencil, mental computation, and calculators in which ‐sums, differences, and products will not exceed five digits; ‐multipliers will not exceed two digits; ‐divisors will not exceed two digits; or ‐dividends will not exceed four digits. MTH.G5.4.a.4 Use two or more operational steps to solve a multistep problem. Operations can be the same or different. MTH.G5.4.a.5 Create single‐step and multistep problems involving the operations of addition, subtraction, multiplication, and division with and without remainders of whole numbers, using practical situations. Expected Extended MTH.G5.4.a.6 Evaluate alternative multiplication and division 2011 UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • An example of an approach to solving problems is Polya's four‐step plan: –Understand: Retell the problem; read it twice; take notes; study the charts or diagrams; look up words and symbols that are new. –Plan: Decide what operation(s) to use and what sequence of steps to use to solve the problem. –Solve: Follow the plan and work accurately. If the first attempt doesn't work, try another plan. –Look back: Does the answer make sense? • Estimation gives a rough idea of an amount. Strategies such as front‐end, rounding, and mental computation may be used to estimate addition, subtraction, multiplication, and division of whole numbers. • Examples of problems to be solved by using estimation strategies are encountered in shopping for groceries, buying school supplies, budgeting allowance, and sharing the cost of a pizza or the prize money from a contest. • Estimation can be used to check the reasonableness of the results. Essential Understanding: All students should • Understand the meaning of mathematical operations and how these operations relate to one another when creating and solving single‐step and multistep word problems. Vocabulary: 64 algorithms used in other countries for efficiency and accuracy MTH.G5.4.a.7 Evaluate alternative multiplication and division algorithms used by peers for efficiency and accuracy. MTH.G5.4.a.8 Investigate, develop, and use rules for divisibility. 2011 65 MTH.G5.5‐The student will a) find the sum, difference, product, and quotient of two numbers expressed as decimals through thousandths (divisors with only one nonzero digit); and b) create and solve single‐step and multistep practical problems involving decimals. ESSENTIAL KNOWLEDGE AND SKILLS UNDERSTANDING THE STANDARD (Teacher Notes) The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G5.5.a.1 Determine an appropriate method of calculation to find the sum, difference, product, and quotient of two numbers expressed as decimals through thousandths, selecting from among paper and pencil, estimation, mental computation, and calculators. MTH.G5.5.a.2 Estimate to find the number that is closest to the sum, difference, and product of two numbers expressed as decimals through thousandths. MTH.G5.5.a.3 Find the sum, difference, and product of two numbers expressed as decimals through thousandths, using paper and pencil, estimation, mental computation, and calculators. MTH.G5.5.a.4 Determine the quotient, given a dividend expressed as a decimal through thousandths and a single‐digit divisor. For example, 5.4 divided by 2 and 2.4 divided by 5. MTH.G5.5.a.5 Use estimation to check the reasonableness of a sum, difference, product, and quotient. MTH.G5.5.b.1 Create and solve single‐step and multistep problems. A multistep problem needs to incorporate two or more operational steps (operations can be the same or different). State Notes: • Addition and subtraction of decimals may be investigated using a variety of models (e.g., 10‐
by‐10 grids, number lines, money). • Decimal computation uses similar procedures as those developed for whole number computation and applies them to decimal place values, giving careful attention to the placement of the decimal point in the solution. • Multiplication of decimals follows the same procedure as multiplication of whole numbers. The only difference is that a decimal point must be correctly placed in the product giving careful attention to the placement of the decimal point in the solution. • The product of decimals is dependent upon the two factors being multiplied. • In cases where an exact product is not required, the product of decimals can be estimated using strategies for multiplying whole numbers, such as front‐end and compatible numbers, or rounding. In each case, the student needs to determine where to place the decimal point to ensure that the product is reasonable. • Division is the operation of making equal groups or shares. When the original amount and the number of shares are known, divide to find the size of each share. When the original amount and the size of each share are known, divide to find the number of shares. Both situations may be modeled with Base‐10 manipulatives. • The fair‐share concept of decimal division can be modeled, using manipulatives (e.g., Base‐10 blocks). • Division with decimals is performed the same way as division of whole numbers. The only difference is the placement of the decimal point in the quotient. • The quotient can be estimated, given a dividend expressed as a decimal through thousandths (and no adding of zeros to the dividend during the division process) and a single‐digit divisor. • Estimation can be used to check the reasonableness of a quotient. • Division is the inverse of multiplication; therefore, multiplication and division are inverse operations. • Terms used in division are dividend, divisor, and quotient. dividend divisor = quotient quotient
divisor
dividend
nd
• There are a variety of algorithms for division such as repeated multiplication and subtraction. Experience with these algorithms may enhance understanding of the traditional long division 2011 66 algorithm. • A multistep problem needs to incorporate no more than two operational steps (operations can be the same or different). Essential Understanding: All students should: • Use similar procedures as those developed for whole number computation and apply them to decimal place values, giving careful attention to the placement of the decimal point in the solution. • Select appropriate methods and tools from among paper and pencil, estimation, mental computation, and calculators according to the context and nature of the computation in order to compute with decimal numbers. • Understand the various meanings of division and its effect on whole numbers. • Understand various representations of division, i.e., dividend ÷ divisor = quotient, quotient
divisor dividend
Expected Extended = quotient Vocabulary: 2011 67 MTHG5.6‐The student will solve single‐step and multistep practical problems involving addition and subtraction with fractions and mixed numbers and express answers in simplest form. UNDERSTANDING THE STANDARD (Teacher Notes) ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G5.6.a.1 Solve single‐step and multistep practical problems involving addition and subtraction with fractions having like and unlike denominators. Denominators in the problems should be limited to 12 or less (e.g., ⅕ +¼) and answers should be expressed in simplest form. MTH.G5.6.a.2 Solve single‐step and multistep practical problems involving addition and subtraction with mixed numbers having like and unlike denominators, with and without regrouping. Denominators in the problems should be limited to 12 or less, and answers should be expressed in simplest form. MTH.G5.6.a.3 Use estimation to check the reasonableness of a sum or difference. Expected Extended MTH.G5.6.a.4 Use given information to create story problems involving fractions and mixed numbers. Explain solution strategies. State Notes: • A fraction can be expressed in simplest form (simplest equivalent fraction) by dividing the numerator and denominator by their greatest common factor. • When the numerator and denominator have no common factors other than 1, then the fraction is in simplest form. • Fractions having like denominators mean the same as fractions having common denominators. • Equivalent fractions name the same amount. To find equivalent fractions, multiply or divide the numerator and denominator by the same nonzero number. • Addition and subtraction with fractions and mixed numbers can be modeled using a variety of concrete materials and pictorial representations as well as paper and pencil. • To add, subtract, and compare fractions and mixed numbers, it often helps to find the least common denominator. The least common denominator (LCD) of two or more fractions is the least common multiple (LCM) of the denominators. • To add or subtract with fractions having the same or like denominators, add or subtract the numerators and write in simplest form. • To add or subtract with fractions that do not have the same denominator, first find equivalent fractions with the least common denominator. Then add or subtract and write the answer in simplest form. • A mixed number has two parts: a whole number and a fraction. The value of a mixed number is the sum of its two parts. • To add or subtract with mixed numbers, students may use a number line, draw a picture, rewrite fractions with like denominators, or rewrite mixed numbers as fractions. Essential Understandings: All students should • Develop and use strategies to estimate and compute addition and subtraction of fractions. • Understand the concept of least common multiple and least common denominator as they are important when adding and subtracting fractions. • Understand that a fraction is in simplest form when its numerator and denominator have no common factors other than 1. The numerator can be greater than the denominator. Vocabulary: simplest form 2011 68 MTHG5.7‐The student will evaluate whole number numerical expressions, using the order of operations limited to parentheses, addition, subtraction, multiplication, and division. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G5.7.a.1 Simplify expressions by using the order of operations in a demonstrated step‐by‐step approach. MTH.G5.7.a.2 Find the value of numerical expressions, using the order of operations. MTH.G5.7.a.3 Given an expression involving more than one operation, describe which operation is completed first, which is second, etc. Expected Extended 2011 UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • An expression, like a phrase, has no equal sign. • Expressions are simplified by using the order of operations. • The order of operations defines the computation order to follow in simplifying an expression. • The order of operations is as follows: –First, complete all operations within grouping symbols. If there are grouping symbols within other grouping symbols, do the innermost operation first. –Second, evaluate all exponential expressions. –Third, multiply and/or divide in order from left to right. –Fourth, add and/or subtract in order from left to right. Essential Understanding: All students should • Understand that the order of operations describes the order to use to simplify expressions containing more than one operation. Vocabulary: numerical expression, order of operations, parentheses 69 MTH.G5.8‐The student will a) find perimeter, area, and volume in standard units of measure; b) differentiate among perimeter, area, and volume and identify whether the application of the concept of perimeter, area, or volume is appropriate for a given situation; c) identify equivalent measurements within the metric system; d) estimate and then measure to solve problems, using U.S. Customary and metric units; and e) choose an appropriate unit of measure for a given situation involving measurement using U.S. Customary and metric units. UNDERSTANDING THE STANDARD (Teacher Notes) ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G5.8.a.1 Determine the perimeter of a polygon, with or without diagrams, when –the lengths of all sides of a polygon that is not a rectangle or a square are given; –the length and width of a rectangle are given; or –the length of a side of a square is given. MTH.G5.8.a.2 Estimate and determine the perimeter of a polygon, and area of a square, rectangle, and right triangle following the parameters listed above, using only whole number measurements given in metric or U.S. Customary units, and record the solution with the appropriate unit of measure (e.g., 24 square inches). MTH.G5.8.a.3 Estimate and determine the area of a square, with or without diagrams, when the length of a side is given. MTH.G5.8.a.4 Estimate and determine the area of a rectangle, with or without diagrams, when the length and width are given. MTH.G5.8.a.5 Estimate and determine the area of a right triangle, with or without diagrams, when the base and the height are given. 2011 State Notes: • Perimeter is the distance around an object. It is a measure of length. Area is the number of square units needed to cover a surface. Volume is a measure of capacity and is measured in cubic units. • To find the perimeter of any polygon, add the lengths of the sides. • Students should label the perimeter, area, and volume with the appropriate unit of linear, square, or cubic measure. • Area is the number of square units needed to cover a surface or figure. • Students should investigate, using manipulatives, to discover the formulas for the area of a square, rectangle, and right triangle; and volume of a rectangular solid. –Area of a rectangle = Length x Width –Area of a square = Side x Side –Area of a right triangle = Base x Height –Volume of a rectangular solid = Length x Width x Height • Length is the distance along a line or figure from one point to another. • U.S. Customary units for measurement of length include inches, feet, yards, and miles. Appropriate measuring devices include rulers, yardsticks, and tape measures. Metric units for measurement of length include millimeters, centimeters, meters, and kilometers. Appropriate measuring devices include centimeter ruler, meter stick, and tape measure. • When measuring with U.S. Customary units, students should be able to measure to the nearest part of an inch ,
, , foot, or yard. • Weight and mass are different. Mass is the amount of matter in an object. Weight is determined by the pull of gravity on the mass of an object. The mass of an object remains the same regardless of its location. The weight that an object changes is dependent on the gravitational pull at its location. In everyday life, most people are actually interested in determining an object's mass, although they use the term weight (e.g., "How much does it weigh?" versus "What is its mass?"). • Appropriate measuring devices to measure mass in U.S. Customary units (ounces, pounds) 70 MTH.G5.8.a.6 Develop a procedure for finding volume using manipulatives (e.g., cubes). MTH.G5.8.b.1. Differentiate among the concepts of area, perimeter, and volume. MTH.G5.8.b.2 Describe practical situations where area, perimeter, and volume are appropriate measures to use, and justify their choices orally or in writing. MTH.G5.8.b.3 Identify whether the application of the concept of perimeter, area, or volume is appropriate for a given situation. MTH.G5.8.c.1 Identify equivalent measurements within the metric system for length: millimeters, centimeters, meters, and kilometers. MTH.G5.8.c.2 Identify equivalent measurements within the metric system for mass: grams and kilograms. MTH.G5.8.c.3 Identify equivalent measurements within the metric system for liquid volume: milliliters, and liters. MTH.G5.8.d.1 Solve problems involving measurement by selecting an appropriate measuring device and a U.S. Customary or metric unit of measure for length: part of an and metric units (grams, kilograms) are balances. • U.S. Customary units to measure liquid volume (capacity) include cups, pints, quarts, and gallons. Metric units to measure liquid volume (capacity) include milliliters and liters. • Temperature is measured using a thermometer. The U.S. Customary unit of measure is degrees Fahrenheit; the metric unit of measure is degrees Celsius. • Practical experience measuring familiar objects helps students establish benchmarks and facilitates students' ability to use the units of measure to make estimates. Essential Understanding: All students should • Understand the concepts of perimeter, area, and volume. • Understand and use appropriate units of measure for perimeter, area, and volume. • Understand the difference between using perimeter, area, and volume in a given situation. • Understand how to select a measuring device and unit of measure to solve problems involving measurement. Vocabulary: inch , , , inches, feet, yards, millimeters, centimeters, meters, and kilometers. MTH.G5.8.d.2 Solve problems involving measurement by selecting an appropriate measuring device and a U.S. Customary or metric unit of measure for the following: –weight: ounces, pounds, and tons; –mass: grams and kilograms. MTH.G5.8.d.3 Solve problems involving measurement by 2011 71 selecting an appropriate measuring device and a U.S. Customary or metric unit of measure for liquid volume: cups, pints, quarts, gallons, milliliters, and liters. MTH.G5.8.d.4 Solve problems involving measurement by selecting an appropriate measuring device and a U.S. Customary or metric unit of measure for area: square units. MTH.G5.8.d.5 Solve problems involving measurement by selecting an appropriate measuring device and a U.S. Customary or metric unit of measure for temperature: Celsius and Fahrenheit units. MTH.G5.8.d.6 Estimate the conversion of Celsius and Fahrenheit units relative to familiar situations: ‐Water freezes at 0°C and 32°F. ‐Water boils at 100°C and 212°F. ‐Normal body temperature is about 37°C and 98.6°F. MTH.G5.8.e.1 Choose an appropriate unit of measure for a given situation involving measurement using U.S. Customary and metric units. Expected MTH.G5.8.a.7 Develop formulas for perimeter and area. Extended 2011 72 MTH.G5.9‐The student will identify and describe the diameter, radius, chord, and circumference of a circle. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G5.9.a.1 Identify and describe the diameter, radius, chord, and circumference of a circle. MTH.G5.9.a.2 Describe the relationship between –diameter and radius; –diameter and chord; –radius and circumference; and –diameter and circumference. Expected Extended 2011 UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • A circle is a set of points on a flat surface (plane) with every point equidistant from a given point called the center. • A chord is a line segment connecting any two points on a circle. Students will benefit from understanding that a chord goes from one side of the circle to the other, but does not need to pass through the center. • A diameter is a chord that goes through the center of a circle. The diameter is two times the radius. A radius is a segment from the center of a circle to any point on the circle. Two radii end‐to‐end form a diameter of a circle. • Circumference is the distance around or perimeter of a circle. The circumference is about 3 times larger than the diameter of a circle. Essential Understanding: All students should • Understand that a chord is a line segment that extends between any two unique points of a circle. • Understand that a diameter is also a special chord that goes through the center of a circle. • Understand the relationship between the measures of diameter and radius and the relationship between the measures of radius and circumference. • Understand that a radius is a line segment that extends between the center and the circumference of the circle. • Understand that the circumference is the distance around the circle. Perimeter is the measure of the circumference. Vocabulary: chord, circumference, diameter, radius 73 MTH.G5.10‐The student will determine an amount of elapsed time in hours and minutes within a 24‐hour period. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G5.10.a.1 Determine elapsed time in hours and minutes within a 24‐hour period. Expected Extended 2011 UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • Elapsed time is the amount of time that has passed between two given times. • Elapsed time can be found by counting on from the beginning time to the finishing time. –Count the number of whole hours between the beginning time and the finishing time. –Count the remaining minutes. –Add the hours and minutes. For example, to find the elapsed time between 10:15 a.m. and 1:25 p.m., count on as follows: from 10:15 a.m. to 1:15 p.m., count 3 hours; from 1:15 p.m. to 1:25 p.m., count 10 minutes; and then add 3 hours to 10 minutes to find the total elapsed time of 3 hours and 10 minutes. Essential Understanding: All students should • Understand that elapsed time can be found by counting on from the beginning time to the finishing time. Vocabulary: 74 MTH.G5.11‐The student will measure right, acute, obtuse, and straight angles. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G5.11.a.1 Identify the appropriate tools (e.g., protractor and straightedge or angle ruler as well as available software) used to measure and draw angles and triangles. MTH.G5.11.a.2 Measure right, acute, straight, and obtuse angles, using appropriate tools, and identify their measures in degrees. MTH.G5.11.a.3 Recognize angle measure as additive. When an angle is decomposed into non‐overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. MTH.G5.11.a.4 Solve addition and subtraction problems to find unknown angle measures on a diagram in practical and mathematical problems, (e.g., by using an equation with a symbol for the unknown angle measure). Expected Extended 2011 UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • Angles are measured in degrees. There are up to 360 degrees in an angle. A degree is of a complete rotation of a full circle. There are 360 degrees in a circle. • To measure the number of degrees in an angle, use a protractor or an angle ruler. • A right angle measures exactly 90°. • An acute angle measures less than 90°. • An obtuse angle measures greater than 90° but less than 180°. • A straight angle measures exactly 180°. • Before measuring an angle, students should first compare it to a right angle to determine whether the measure of the angle is less than or greater than 90°. • Students should understand how to work with a protractor or angle ruler as well as available computer software to measure and draw angles and triangles. Essential Understanding: All students should • Understand how to measure acute, right, obtuse, and straight angles. Vocabulary: acute angle, angle ruler, obtuse angle, protractor, right angle, straight angle 75 MTH.G5.12‐The student will classify a) angles as right, acute, obtuse, or straight; and b) triangles as right, acute, obtuse, equilateral, scalene, or isosceles. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G5.12.a.1 Classify angles as right, acute, obtuse, or straight. MTH.G5.12.b.1 Classify triangles as right, acute, and obtuse. MTH.G5.12.b.2 Classify triangles as equilateral, scalene, or isosceles. Expected Extended 2011 UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • A right angle measures exactly 90°. • An acute angle measures greater than 0° but less than 90°. • An obtuse angle measures greater than 90° but less than 180°. • A straight angle forms an angle that measures exactly 180°. • A right triangle has one right angle. • An obtuse triangle has one obtuse angle. • An acute triangle has three acute angles (or no angle measuring 90° or greater). • A scalene triangle has no congruent sides. • An isosceles triangle has two congruent sides. • To facilitate the exploration of relationships, ask students whether a right triangle can have an obtuse angle. Why or why not? Can an obtuse triangle have more than one obtuse angle? Why or why not? What type of angles are the two angles other than the right angle in a right triangle? What type of angles are the two angles other than the obtuse angle in an obtuse triangle? Essential Understanding: All students should • Understand that angles can be classified as right, acute, obtuse, or straight according to their measures. • Understand that a triangle can be classified as either right, acute, or obtuse according to the measure of its largest angle. • Understand that a triangle can be classified as equilateral, scalene, or isosceles according to the number of sides with equal length. Vocabulary: acute triangle, equilateral triangle, isosceles triangle, obtuse triangle, right triangle, scalene triangle 76 MTH.G5.13‐The student, using plane figures (square, rectangle, triangle, parallelogram, rhombus, and trapezoid), will a) develop definitions of these plane figures; and b) investigate and describe the results of combining and subdividing plane figures. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G5.13.a.1 Develop definitions for squares, rectangles, triangles, parallelograms, rhombi, and trapezoids. MTH.G5.13.b.1 Investigate and describe the results of combining and subdividing plane figures. Expected Extended 2011 UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • A triangle is a polygon with three sides. Triangles may be classified according to the measure of their angles, i.e., right, acute, or obtuse. Triangles may also be classified according to the measure of their sides, i.e., scalene (no sides congruent), isosceles (at least two sides congruent) and equilateral (all sides congruent). • A quadrilateral is a polygon with four sides. • A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. Properties of a parallelogram include the following: –A diagonal (a segment that connects two vertices of a polygon but is not a side) divides the parallelogram into two congruent triangles. –The opposite sides of a parallelogram are congruent. –The opposite angles of a parallelogram are congruent. –The diagonals of a parallelogram bisect each other. To bisect means to cut a geometric figure into two congruent halves. A bisector is a line segment, line, or plane that divides a geometric figure into two congruent halves. A sample of a bisected parallelogram is below. • A rectangle is a parallelogram with four right angles. Since a rectangle is a parallelogram, a rectangle has the same properties as those of a parallelogram. • A square is a rectangle with four congruent sides. Since a square is a rectangle, a square has all the properties of a rectangle and of a parallelogram. • A rhombus is a parallelogram with four congruent sides. Opposite angles of a rhombus are congruent. Since a rhombus is a parallelogram, the rhombus has all the properties of a parallelogram. • A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases, and the nonparallel sides are called legs. If the legs have the same length, then the trapezoid is an isosceles trapezoid. • Two or more figures can be combined to form a new figure. Students should be able to identify the figures that have been combined. • The region of a polygon may be subdivided into two or more regions that represent figures. Students should understand how to divide the region of a polygon into familiar figures. 77 Essential Understanding: All students should • Understand that simple plane figures can be combined to make more complicated figures and that complicated figures can be subdivided into simple plane figures. Vocabulary: 2011 78 MTH.G5.14‐The student will make predictions and determine the probability of an outcome by constructing a sample space. UNDERSTANDING THE STANDARD (Teacher Notes) ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G5.14.a.1 Construct a sample space, using a list or chart to represent all possible outcomes of a single event. MTH.G5.14.a.2 Construct a sample space, using a tree diagram to identify all possible outcomes of a single event. MTH.G5.14.a.3 Predict and determine the probability of an outcome by constructing a sample space. The sample space will have a total of 24 or less possible outcomes. Expected Extended 2011 State Notes: • Probability is the chance of an event occurring. • The probability of an event occurring is the ratio of desired outcomes to the total number of possible outcomes. If all the outcomes of an event are equally likely to occur, the probability of . the event = • The probability of an event occurring is represented by a ratio between 0 and 1. An event is "impossible" if it has a probability of 0 (e.g., the probability that the month of April will have 31 days). An event is "certain" if it has a probability of 1 (e.g., the probability that the sun will rise tomorrow morning). • When a probability experiment has very few trials, the results can be misleading. The more times an experiment is done, the closer the experimental probability comes to the theoretical probability (e.g., a coin lands heads up half of the time). • Students should have opportunities to describe in informal terms (i.e., impossible, unlikely, as likely as unlikely, as likely as, equally likely, likely, and certain) the degree of likelihood of an event occurring. Activities should include practical examples. • For any event such as flipping a coin, the equally likely things that can happen are called outcomes. For example, there are two equally likely outcomes when flipping a coin: the coin can land heads up, or the coin can land tails up. • A sample space represents all possible outcomes of an experiment. The sample space may be organized in a list, chart, or tree diagram. • Tree diagrams show all possible outcomes in a sample space. The Fundamental Counting Principle describes how to find the number of outcomes when there are multiple choices. For example, how many different outfit combinations can you make from 2 shirts (red and blue) and 3 pants (black, white, khaki)? The sample space displayed in a tree diagram would show that there are 2 x 3 = 6 (Fundamental Counting Principle) outfit combinations: red‐black; red‐
white; red‐khaki; blue‐black; blue‐white; blue‐khaki. • A spinner with eight equal‐sized sections is equally likely to land on any one of the sections, three of which are red, three green, and two yellow. Have students write a problem statement involving probability, such as, "What is the probability that the spinner will land on green?" Essential Understanding: All students should • Understand that the basic concepts of probability can be applied to make predictions of 79 outcomes of simple experiments. • Understand that a sample space represents all possible outcomes of an experiment. Vocabulary: outcome, sample space 2011 80 MTH.G5.15‐The student, given a problem situation, will collect, organize, and interpret data in a variety of forms, using stem‐and‐leaf plots and line graphs. ESSENTIAL KNOWLEDGE AND SKILLS UNDERSTANDING THE STANDARD (Teacher Notes) The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G5.15.a.1 Formulate the question that will guide the data collection. MTH.G5.15.a.2 Collect data, using observations (e.g., weather), measurement (e.g., shoe sizes), surveys (e.g., hours watching television), or experiments (e.g., plant growth). MTH.G5.15.a.3 Organize the data into a chart, table, stem‐
and‐leaf plots, and line graphs. MTH.G5.15.a.4 Display data in line graphs and stem‐and‐leaf plots. MTH.G5.15.a.5 Construct line graphs, labeling the vertical axis with equal whole‐number, decimal, or fractional increments and the horizontal axis with continuous data commonly related to time (e.g., hours, days, months, years, and age). Line graphs will have no more than six identified points along a continuum for continuous data (e.g., the decades: 1950s, 1960s, 1970s, 1980s, 1990s, and 2000s). MTH.G5.15.a.6 Construct a stem‐and‐leaf plot to organize and display data, where the stem is listed in ascending order and the leaves are in ascending order, with or without commas between leaves. MTH.G5.15.a.7 Title the given graph or identify the title. MTH.G5.15.a.8 Interpret the data in a variety of forms (e.g., State Notes: • The emphasis in all work with statistics should be on the analysis and the communication of the analysis, rather than on a single correct answer. Data analysis should include opportunities to describe the data, recognize patterns or trends, and make predictions. • Statistical investigations should be active, with students formulating questions about something in their environment and finding quantitative ways to answer the questions. • Investigations can be brief class surveys or more extended projects taking many days. • Through experiences displaying data in a variety of graphical representations, students learn to select an appropriate representation. • Line graphs are used to show how two continuous variables are related. Line graphs may be used to show how one variable changes over time. If one variable is not continuous, then a broken line is used. By looking at a line graph, it can be determined whether the variable is increasing, decreasing, or staying the same over time. –The values along the horizontal axis represent continuous data on a given variable, usually some measure of time (e.g., time in years, months, or days). The data presented on a line graph is referred to as “continuous data” because it represents data collected over a continuous period of time. –The values along the vertical axis are the scale and represent the frequency with which those values occur in the data set. The values should represent equal increments of multiples of whole numbers, fractions, or decimals depending upon the data being collected. The scale should extend one increment above the greatest recorded piece of data. –Each axis should be labeled and the graph should have a title. –A line graph tells whether something has increased, decreased, or stayed the same with the passage of time. Statements representing an analysis and interpretation of the characteristics of the data in the graph should be included (e.g., trends of increase and/or decrease, and least and greatest). A broken line is used if the data collected is not continuous data (such as test scores); a solid line is used if the data is continuous (such as height of a plant). • Stem‐and‐leaf plots allow the exact values of data to be listed in a meaningful array. Data covering a range of 25 numbers are best displayed in a stem‐and‐leaf plot and are utilized to organize numerical data from least to greatest, using the digits of the greatest to group data. –The data is organized from least to greatest. –Each value should be separated into a stem and a leaf [e.g., two‐digit numbers are separated into stems (tens) and leaves (ones)]. –The stems are listed vertically from least to greatest with a line to their right. The leaves are 2011 81 orally or in written form). Expected Extended MTH.G5.15.a.9 Create a possible set of data points from a given mean, median, mode, and/or range. MTH.G5.15.a.10 Construct and analyze double stem‐and‐leaf plots and multiple line graphs. 2011 listed horizontally, also from least to greatest, and can be separated by spaces or commas. Every value is recorded regardless of the number of repeats. –A key is often included to explain how to read the plot. Essential Understanding: All students should • Understand how to interpret collected and organized data. • Understand that stem‐and‐leaf plots list data in a meaningful array. It helps in finding median, modes, minimum and maximum values, and ranges. • Understand that line graphs show changes over time. Vocabulary: 82 MTH.G5.16‐The student will a) describe mean, median, and mode as measures of center; b) describe mean as fair share; c) find the mean, median, mode, and range of a set of data; and d) describe the range of a set of data as a measure of variation. ESSENTIAL KNOWLEDGE AND SKILLS UNDERSTANDING THE STANDARD (Teacher Notes) The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G5.16.a.1 Describe the mean of a group of numbers representing data from a given context as a measure of center.
MTH.G5.16.a.2 Describe the median of a group of numbers representing data from a given context as a measure of center.
MTH.G5.16.a.3 Describe the mode of a group of numbers representing data from a given context as a measure of center.
MTH.G5.16.a.4 Describe the impact on measures of center when a single value of a data set is added, removed, or changed. MTH.G5.16.b.1 Describe mean as fair share. MTH.G5.16.c.1 Find the mean of a group of numbers. MTH.G5.16.c.2 Find the median of a group of numbers. MTH.G5.16.c.3 Find the mode of a group of numbers. MTH.G5.16.c.4 Find the range of a group of numbers. MTH.G5.16.d.1 Describe the range of a group of numbers representing data from a given context as a measure of State Notes: • Statistics is the science of conducting studies to collect, organize, summarize, analyze, and draw conclusions from data. • A measure of center is a value at the center or middle of a data set. Mean, median, and mode are measures of center. • The mean, median, and mode are three of the various ways that data can be analyzed. • Mean represents a fair share concept of the data. Dividing the data constitutes a fair share. This is done by equally dividing the data points. This should be demonstrated visually and with manipulatives. The arithmetic way is to add all of the data points then divide by the number of data points to determine the average or mean. • The median is the piece of data that lies in the middle of the set of data arranged in order. • The mode is the piece of data that occurs most frequently in the data set. There may be one, more than one, or no mode in a data set. Students should order the data from least to greatest so they can better find the mode. • The range is the spread of a set of data. The range of a set of data is the difference between the greatest and least values in the data set. It is determined by subtracting the least number in the data set from the greatest number in the data set. An example is ordering test scores from least to greatest: 73, 77, 84, 87, 89, 91, 94. The greatest score in the data set is 94 and the least score is 73, so the least score is subtracted from the greatest score or 94 ‐ 73 = 21. The range of these test scores is 21. • Students need to learn more than how to identify the mean, median, mode, and range of a set of data. They need to build an understanding of what the number tells them about the data, and they need to see those values in the context of other characteristics of the data in order to best describe the results. Essential Understanding: All students should • Understand how to determine the mean, median, mode, and range of a set of data. • Understand that the mean is the numerical average of a data set; the median is the number in the middle of a set of data; the mode is the piece of data that occurs most often; and the 2011 83 variation. Expected Extended 2011 range is the spread of a set of data. Vocabulary: mean, median, mode, range 84 MTH.G5.17‐The student will describe the relationship found in a number pattern and express the relationship. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G5.17.a.1.Describe numerical and geometric patterns formed by using concrete materials and calculators. MTH.G5.17.a.2 Describe the relationship found in patterns, using words, tables, and symbols to express the relationship. Expected Extended UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • There are an infinite number of patterns. • The simplest types of patterns are repeating patterns. In such patterns, students need to identify the basic unit of the pattern and repeat it. • Growing patterns are more difficult for students to understand than repeating patterns because not only must they determine what comes next, they must also begin the process of generalization. Students need experiences with growing patterns. • Sample numerical patterns are 6, 9, 12, 15, 18, ...; 5, 7, 9, 11, 13, ...; 1, 2, 4, 7, 11, 16, ...; 2, 4, 8, 16, 32, ...; 32, 30, 28, 26, 24…; and 1, 5, 25, 125, 625,... • An expression, like a phrase, has no equal sign. • When the pattern data are expressed in a T‐table, an expression can represent that data. An example is: X Y 6 9 7 10 11 14 15 18 This example defines the relationship as x + 3. • Expressions are simplified by using the order of operations. • A verbal quantitative expression involving one operation can be represented by a variable expression that describes what is going on. Numbers are used when they are known; variables are used when the numbers are unknown. For example, "a full box of cookies and four extra" can be represented by b + 4; "three full boxes of cookies" by 3b; "a full box of cookies shared among four" by . • A mathematical expression contains a variable or a combination of variables, numbers, and/or operation symbols and represents a mathematical relationship. An expression cannot be solved. 2011 85 • Essential Understanding: All students should • Understand that patterns and functions can be represented in many ways and described using words, tables, and symbols. • Understand the structure of a pattern and how it grows or changes using concrete materials and calculators. • Understand that mathematical relationships exist in patterns. • Understand that an expression uses symbols to define a relationship and shows how each number in the list, after the first number, is related to the preceding number. • Understand that expressions can be numerical or variable or a combination of numbers and variables. Vocabulary: 2011 86 MTH.G5.18‐The student will a) investigate and describe the concept of variable; b) write an open sentence to represent a given mathematical relationship, using a variable; c) model one‐step linear equations in one variable, using addition and subtraction; and d) create a problem situation based on a given open sentence, using a single variable. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G5.18.a.1 Describe the concept of a variable (presented as boxes, letters, or other symbols) as a representation of an unknown quantity. MTH.G5.18.b.1 Write an open sentence with addition, subtraction, multiplication, or division, using a variable to represent a missing number. MTH.G5.18.c.1 Model one‐step linear equations using a variety of concrete materials such as colored chips on an equation mat or weights on a balance scale. MTH.G5.18.d.1 Create and write a word problem to match a given open sentence with a single variable and one operation. Expected Extended 2011 UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • A variable is a symbol that can stand for an unknown number or object. • A variable expression is like a phrase: as a phrase does not have a verb, so an expression does not have an equals sign (=). • A verbal expression involving one operation can be represented by a variable expression that describes what is going on. Numbers are used when they are known; variables are used when the numbers are unknown. For example, "a full box of cookies and four extra" can be represented by b +4; "three full boxes of cookies" by 3b; "a full box of cookies shared among four" by . • An open sentence contains a variable and an equals sign (=). For example, the sentence, "A full box of cookies and four extra equal 24 cookies." can be written as b + 4 = 24, where b stands for the number of cookies in one full box. "Three full boxes of cookies equal 60 cookies." can be written as 3b = 60. • Another example of an open sentence is b + 3 = 23 and represents the answer to the word problem, "How many cookies are in a box if the box plus three more equals 23 cookies, where b stands for the number of cookies in the box? • At this level, discuss how the symbol used to represent multiplication can often be confused with the variable . Students can minimize this confusion by using parentheses [e.g. 4(x) = 20, or 4x = 20 ] or a small dot raised off the line to represent multiplication 4 ·
20 . • By using story problems and numerical sentences, students begin to explore forming equations and representing quantities using variables. • An open sentence containing a variable is neither true nor false until the variable is replaced with a number. Essential Understanding: All students should • Understand that a variable is a symbol that can stand for an unknown number or object. • Understand that a variable expression is a variable or combination of variables, numbers, and 87 symbols that represents a mathematical relationship. • Understand that verbal expressions can be translated to variable expressions. • Understand that an open sentence has a variable and an equal sign (=). • Understand that problem situations can be expressed as open sentences. Vocabulary: open sentence, variable 2011 88 Mathematics POS Framework Grade 5 MTH.G5.19‐The student will investigate and recognize the distributive property of multiplication over addition. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Essential MTH.G5.19.a.1 Investigate and recognize the distributive property of whole numbers, limited to multiplication over addition using diagrams and manipulatives. MTH.G5.19.a.2 Investigate and recognize an equation that represents the distributive property, when given several whole number equations, limited to multiplication over addition. Expected Extended 2011 UNDERSTANDING THE STANDARD (Teacher Notes) State Notes: • The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products [e.g.,3(4 + 5) = 3 x 4 + 3 x 5, 5 x (3 + 7) = (5 x 3) + (5 x 7); or (2 x 3) + (2 x 5) = 2 x (3 + 5)]. • The distributive property can be used to simplify expressions [e.g., 9 x 23 = 9(20 + 3) =180 + 27 = 207; or 5 x 19 = 5(10 + 9) = 50 + 45 = 95)]. Essential Understanding: All students should • Understand that the distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products. Vocabulary: distributive property 89