Course Description In Grade 4, instructional time should focus on five critical areas: (1) attaining fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends; (2) developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; (3) analyzing and classifying geometric figures based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry; (4) solving problems involving measurement and conversion of measurements from a larger unit to a smaller unit; (5) collecting, representing, and interpreting data. Scope And Sequence Timeframe Unit Instructional Topics 4 Week(s) 8 Week(s) 7 Week(s) 4 Week(s) 7 Week(s) Course Rationale In alignment with Common Core State Standards, the Park Hill School District's Mathematics courses provide students with a solid foundation in number sense while building to the application of more demanding math concepts and procedures. The courses focus on procedural skills and conceptual understandings to ensure coherence and depth in mathematical practices and application to real world issues and challenges. Understanding numerical expressions builds the relationship between numbers. Measurement describes the attributes of objects and events. Extending understanding of base 10 notation is the basis for our number system. Fractions are different representations of numbers Representing and interpreting data helps analyze information and develops critical thinking skills Describing and analyzing objects develops a foundation for understanding our physical environment. Key Resources Pearson Envision Board Approval Date January 10, 2013 Operations and Algebraic Thinking Numbers and Operations Base 10 Numbers and Operations- Fractions Geometry Measurement and Data Unit: Operations and Algebraic Thinking 1. Represent and Solve Problems 2. Arithmetic Pattern 3. Factors and Multiples 1. Place Value Strategies to Add and Subtract 2. Place Value 3. Multiply Whole Numbers 4. Divide Whole Numbers 1. Compare and Order Fractions 2. Add and Subtract Fractions 3. Multiply Fractions 4. Compare Decimals and Fractions 1. Classify Shapes 2. Measurement of Angles 3. Lines of Symmetry 1. Customary Measurement and Conversions 2. Area and Perimeter 3. Metric Measurement and Conversions 4. Represent and Interpret Data Course Details Duration: 4 Week(s) Page 1
This unit focuses on using operations and algebraic thinking to solve problems, understanding factors and multiples and generating and analyzing patterns. The four basic arithmetic operations are interrelated, and the properties of each may be used to understand the others. Analyzing patterns increases mathematical understanding of whole numbers. Flexible methods of computation involve grouping numbers in strategic ways. The distributive property is connected to the area model and/or partial products method of multiplication. Multiplication and division are inverse operations. There are three different structures for multiplication and division problems: area/arrays, equal groups, and comparison, and the unknown quantity in multiplication and division situations is represented in three ways: unknown products, group size unknown, and number of groups unknown. Analyzing patterns increases mathematical understanding of whole numbers. Patterns are generated by following a specific rule. A whole number is a multiple of each of its factors. How do the four operations relationships help to solve problems? How can patterns and properties be used to find some multiplication facts? How are multiplication and division related? What are different models for multiplication and division? How can unknown multiplication facts be found by breaking them apart into known facts? How can unknown division facts be found by thinking about a related multiplication fact? What are efficient methods for finding products and quotients, and how can place value properties aid computation? Why do number patterns repeat? What strategies can be used to find rules for patterns and what predictions can the pattern support? How can patterns be used to describe how quantities are related? How can a relationship between two quantities be shown using a table? How do I determine the factors of a number? What is the difference between a prime and composite number? How are factors and multiples related? Topic: Represent and Solve Problems The student will interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. The student will solve word problems involving comparisons using drawings or equations. The student will solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. The student will use mathematical language to justify your reasoning Arithmetic Pattern The student will generate a number or shape pattern that follows a given rule. use mathematical language to describe the features of a number or shape pattern, including those that were not explicit in the rule itself. Factors and Multiples The student will determine whether a given whole number in the range 1-100 is prime or composite. The student will find all factor pairs for a whole number in the range 1-100. The student will recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Page 2
Unit: Numbers and Operations Base 10 This unit focuses on the understanding of place value when solving multi-digit arithmetic with all four operations. As digits progress from right to left, their individual value increases ten times. Understanding place value aids in reading, writing, rounding, and comparing multi-digit numbers. Place value is based on groups of ten and the value of a number is determined by the place of its digits. Whole numbers are read from right to left using the name of the period; commas are used to separate periods. A number can be written using its name, standard, or expanded form. Rounding numbers can be used when estimating answers to real-world problems. Duration: 8 Week(s) Place value understanding and properties of operations are necessary to solve multi-digit arithmetic. The standard algorithm for addition and subtraction relies on adding or subtracting like base-ten units. Place value understanding and properties of operations are necessary to solve multi-digit arithmetic. Understanding place value and properties of operations is necessary to perform multi-digit multiplication. There are three different structures for multiplication and division problems: area/arrays, equal groups, and comparison, and the unknown quantity in multiplication and division situations is represented in three ways: unknown products, group size unknown, and number of groups unknown. Place value understanding and properties of operations are necessary to solve multi-digit arithmetic. Understanding place value and properties of operations is necessary to perform multi-digit division. There are three different structures for multiplication and division problems: area/arrays, equal groups, and comparison, and the unknown quantity in multiplication and division situations is represented in three ways: unknown products, group size unknown, and number of groups unknown. Some division situations will produce a reminder, but the remainder should always be less than the divisor. If the remainder is greater that the divisor, that means at least one more can be given to each group or at least one more group of the given size may be created. When suing division to solve word problems, how the remainder is interpreted depends on the problem situation. How does the value of a digit change within a number? How can place value understanding help us with comparing, ordering, and rounding whole numbers? How can the value of digits be used to compare two numbers? In what ways can numbers be composed and decomposed? How are greater numbers read and written? How can my understanding of place value explain the process of addition and subtraction? How are addition and subtraction related to one another? How does understanding place value help you solve multi-digit addition and subtraction problems, and how can rounding be used to estimate answers to problems? What are standard procedures for adding and subtracting numbers? How can my understanding of place value explain the process of multiplication? How can products be found mentally? How can products be estimated? What is a standard procedure for multiplying multi-digit numbers, and how do place value properties aid computation? What real-life situations require the use of multiplication? What are different models for multiplication (arrays) and division? How are multiplication and division related? How can my understanding of place value explain the process of multiplication? How are multiplication and division related? What real-life situations require the use of division? What are different models for multiplication and division (repeated subtraction)? How are dividends, divisors, quotients, and remainders related? How can a remainder affect the answer in a division word problem? What are the different meanings of division? How can mental math and estimation be used to divide? What is the standard procedure for dividing multi-digit numbers? Topic: Place Value Strategies to Add and Subtract Description The student will fluently add and subtract multi-digit whole numbers using the standard algorithm. The student will describe and justify the processes used to add and subtract. Page 3
Place Value The student will read and write multi-digit whole numbers using base-ten numerals, number names and expanded form. The student will compare multi-digit numbers based on meanings of the digits in each place, using <,>,= symbols to record results of the comparison. The student will round whole numbers up to 1,000,000 to the nearest ten, hundred, thousand, ten thousand, hundred thousand, million using place value understanding. Multiply Whole Numbers The student will multiply two 2-digit numbers using strategies based on place value and properties of operations. The student will illustrate and use mathematical language to explain the calculations using equations, rectangular array and area models. The student will estimate and reason when multiplying whole numbers. Divide Whole Numbers The student will demonstrate how to solve whole number quotients with remainders with up to four digit dividends and one digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. The student will illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. The student will estimate and reason when dividing whole numbers. Unit: Numbers and Operations- Fractions Duration: 7 Week(s) Page 4
This unit focuses on understanding and ordering fractions and their equivalence; adding and subtracting fractions and mixed numbers; solving word problems with fractions, and understanding how fractions and decimal notation compare. Use comparing, ordering, and equivalent fractions to extend understanding of fractions. Fractions can be represented visually and in written form. Comparisons are only valid when the two fractions refer to the same whole. Fractions and mixed numbers are composed of unit fractions and can be decomposed as a sum of unit fractions. Improper fractions and mixed numbers express the same value. Using students previous knowledge of the properties of whole numbers in addition and subtraction will aid in teaching of addition and subtractions of fractions. Improper fractions and mixed numbers express the same value. Addition and subtraction of fractions involves joining and separating parts referring to the same whole. Multiplying a fraction by a whole number is a logical step after multiplication of whole numbers. Improper fractions and mixed numbers represent the same value. A product of a fractions times a whole number can be written as a multiple of a unit fraction. Decimal notation is another way to represent a fraction. Fractions with denominators of 10 can be expressed as an equivalent fraction with a denominator of 100. Fractions with denominators of 10 and 100 may be expressed when using decimal notation. When comparing two decimals to hundredths, the comparisons are only valid if they refer to the same whole. How does finding equivalent fractions help you compare? How are fractions used in problem-solving situations? How are fractions composed, decomposed, compared and represented? Why is it important to identify, label, and compare fractions as representations of equal parts of a whole or of a set? How can the same fractional amount be named in different ways using symbols? How can fractions be compared and ordered? Why does the numerator change, but the denominator stay the same when adding and subtracting fractions with like denominators? What does it mean to add and subtract fractions and mixed numbers with like denominators? What is a standard procedure for adding and subtracting mixed numbers with like denominators? How can fractions and mixed numbers be added and subtracted on a number line? How/why does the whole number become smaller when you multiply a whole number by a fraction? How can multiplying a whole number by a fraction be displayed as repeated addition (as a multiple of a unit fraction)? How can a fraction be represented by a decimal? How can visual models be used to help with understanding decimals? How can visual models be used to determine and compare equivalent fractions and decimals? How would you compare and order decimals through hundredths? How is decimal numeration related to whole number numeration? Topic: Compare and Order Fractions The student will explain and compare using mathematical language how two fractions, e.g., 2/8 and 4/16 are equivalent fractions through the use of a visual model or through multiplying by 1 whole (which can be represented whenever the numerator and denominator are the same). Show 2/8 x (2/2) = 4/16 or 2/8 x (3/3) = 6/24. The student will compare two fractions through comparing both to a benchmark fraction such as 1/2. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. The student will use models, benchmarks (0, 1/2 and 1) and equivalent forms to judge the size of fractions. Add and Subtract Fractions The student will add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. The student will decompose a fraction into a sum of fractions with the same denominator in more than one way, such as an equation or a fraction model. Example: 3/8 = 1/8 + 1/8 + 1/8 or 3/8 = 1/8 + 2/8 The student will solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. The student will understand addition and subtraction of fractions as joining and separating parts referring to the same whole. Page 5
Multiply Fractions The student will apply and extend previous understandings of multiplication to multiply a fraction by a whole number: o understand a fraction a/b as a multiple of 1/b, e.g. ¾ = ¼ +1/4 +1/4 or 3 x (1/4) = ¾ (4.NF.4a) o understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 x (2/5) as 6 x (1/5). The student will solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. Compare Decimals and Fractions The student will compare two decimals to hundredths by reasoning by their size by recording the comparisons with the symbols >, <, =, and justify conclusions using a visual model with mathematical language. The student will express a fraction with denominator 10 as an equivalent fraction with denominator 100 and use this technique to add two fractions. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. The student will use decimal notation for fractions with denominators 10 to 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. Unit: Geometry This unit focuses on drawing and identifying lines, rays, and angles, and classifying shapes by their properties, as well as measuring and identifying angles by these measurements. Lines, rays, and angles are used to identify two-dimensional figures. Two-dimensional figures are classified based on type of lines (parallel/perpendicular) and size of angles. Angles can be measured and these measurements are additive. Angles are measured in the context of a central angle of a circle. Angles are composed of smaller angles A line of symmetry for a two-dimensional figure can be found by folding the shape into two congruent parts. What are examples of two-dimensional figures in everyday life? How/why are geometric shapes constructed from different types of lines and angles? How are parallel and perpendicular lines used in classifying two-dimensional shapes? How can lines, angles, and shapes be described, analyzed, and classified? Duration: 4 Week(s) How can angles be composed or decomposed to form larger or smaller angles? How are angles measured, added, or subtracted? What are the types of angles and the relationships? How are angles applied to the context of a circle? How are protractors used to measure and aid in drawing angles and triangles? How can an addition or subtraction equation be used to solve a missing angle measure when the whole angle has been divided into two angles and only one measurement is given? Why do some shapes have more than one line of symmetry? How do the measurement of angles and sides relate to the number of lines of symmetry a shape has? Topic: Classify Shapes The student will construct and classify points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular/parallel lines in twodimensional figures Measurement of Angles Page 6
The student will classify two-dimensional figures based on the absence or presence of their parallel or perpendicular lines. The student will measure angles in whole-number degrees using a protractor and sketch angles of a specific measure, realizing that a circle contains 360 1- degree angles and that all angles are measured with reference to a circle. The student will understand the angle measure of the whole is the sum of the angle measures of the parts and solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure. The student will classify two dimensional figures based on the absence or presence of angles and their specific size. The student will recognize angles as geometric shapes that are formed where two rays share a common endpoint, and understand concepts of angle measurement. Lines of Symmetry The student will identify line-symmetric figures and draw lines of symmetry. The student will represent a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Unit: Measurement and Data This unit focuses on solving problems by converting standard units of measurement, as well as finding area and perimeter of rectangles. Measurement units can be converted within a single system of measurement. Use the four operations aids in solving word problems involving measurement. When converting measurements within one system, the size, length, mass, volume, of the object remains the same. Measurement units can be converted within a single system of measurement. Use the four operations aids in solving word problems involving measurement. When converting measurements within one system, the size, length, mass, volume, of the object remains the same. Converting from larger to smaller units of measurement in the metric system is done by multiplying by powers of ten. The area and perimeter of objects can be found in the real world and mathematical problems. Perimeter is a real life application of addition and subtraction. Area is a real life application of multiplication and division. Using addition and subtraction aids in solving word problems involving measurement. Data sets can be organized in a variety of ways, including line plots. How can you estimate, measure, and change customary units of length, volume, and mass? How can use the four operations to help solve word problems in measurement? Why does the size, length, mass, volume of an object remain the same when converted to another unit of measurement? What are the customary units for measuring length, capacity, and weight/mass, and how are they related? How can you estimate, measure, and change metric units of length, volume, and mass? How can use the four operations to help solve word problems in measurement? Why does the size, length, mass, volume of an object remain the same when converted to another unit of measurement? What are the metric units for measuring length, capacity, and weight/mass, and how are they related? Why would you need to find the area and perimeter of something? What do area and perimeter mean and how can each be found? How can the formulas for area and perimeter help you solve real-world problems? Duration: 7 Week(s) How can you use addition and subtraction to help solve world problems in measurement? How can line plots and other tools help to solve measurement problems? Topic: Customary Measurement and Conversions The student will know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb., oz.; l, ml; h, min, sec. The student will represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. Page 7
The student will express (convert) measurements in a larger unit in terms of a smaller unit within a single system of measurements. The student will construct and record measurement equivalents in a two column table. The student will use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. Topic: Duration: 0 Day(s) Area and Perimeter The student will apply the area and perimeter formulas for rectangles in real world and mathematical problems Metric Measurement and Conversions The student will know relative sizes of measurement units within one system of units including km, m, cm. The student will express (convert) measurements in a larger unit in terms of a smaller unit within a single system of measurements. The student will construct and record measurement equivalents in a two column table. The student will represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. The student will use the four operations to solve word problems involving distance, intervals of time, liquid volumes, masses of objects, including problems involving simple fractions or decimals and problems that require expressing measurements given in a larger unit in terms of a smaller unit (e.g., milliliters to liters, grams to kilograms, meters to centimeters). Represent and Interpret Data The student will make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). The student will solve problems involving addition and subtraction of fractions by using information presented in line plots. Page 8