STANDARD 4: Mathematics

Mathematics Areas of Focus: Grade 5

Mission: Through mathematics, students communicate, make connections, reason, and represent the world quantitatively in order to pose and solve problems.

Standard 4.1 Number and Numerical Operations

All students will develop number sense and will perform standard numerical operations and estimations on all types of numbers in a variety of ways.

Big Idea: Numeric reasoning involves fluency and facility with numbers.
4.1.5 A. Number Sense
Descriptive Statement: Number sense is an intuitive feel for numbers and a common sense approach to using them. It is a comfort with what numbers represent that comes from investigating their characteristics and using them in diverse situations. It involves an understanding of how different types of numbers, such as fractions and decimals, are related to each other, and how each can best be used to describe a particular situation. It subsumes the more traditional category of school mathematics curriculum called numeration and thus includes the important concepts of place value, number base, magnitude, and approximation and estimation.
Essential Questions	Enduring Understandings
- How do mathematical ideas interconnect and build on one another to produce a coherent whole? (4.5C1; 4.5C6) - How can we compare and contrast numbers? (4.5A4) - How can counting, measuring, or labeling help to make sense of the world around us?	- One representation may sometimes be more helpful than another; and, used together, multiple representations give a fuller understanding of a problem. - A quantity can be represented numerically in various ways. Problem solving depends upon choosing wise ways. - Numeric fluency includes both the understanding of and the ability to appropriately use numbers.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
1. Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 5 pertain to these sets of numbers as well): All fractions as part of a whole, as subset of a set, as a location on a number line, and as divisions of whole numbers; All decimals.	It is important to note that the sets of numbers specified in this CPI also apply to the other grade 5 mathematics CPIs, including for example 4.1.5A3 and 4.1.5A6 below. *Sample Short Constructed Response (SCR) Item*: Four friends have three brownies left over from a party. They would like to split them equally. How much should each of them receive? (Answer: 75% or .75 or 3/4 of a brownie)
2. Recognize the decimal nature of United States currency and compute with money.	Assessment Focus: • The emphasis in statewide assessment is on the computation. Sample Multiple Choice (MC) Item: Debbie has a $5 bill. She wants to purchase a notebook for 75¢ and a pen for 50¢. How much money will Debbie have left after purchasing the notebook and the pen? a. $1.25 b. $2.75 * c. $3.75 d. $4.25 Sample Short Constructed Response (SCR) Item: Juliette has a $5 bill. She wants to purchase a notebook for 75¢ and a pen for 50¢. How much money will Juliette have left after purchasing the notebook and the pen? (Answer: $3.75)
3. Demonstrate a sense of the relative magnitudes of numbers.	Instructional/Assessment Focus: • Refers not only to whole numbers, but also to fractions and decimals, as specified in 4.1.5A1. Sample MC Item: If these fractions were graphed on the number line, which of them would be closest to zero? a. 3/5 b. 1/4 c. 3/20 * d. 1/10
4. Use whole numbers, fractions, and decimals to represent equivalent forms of the same number	Sample MC Item: Which of the following is equivalent to 3/4? a. .25 b. 4/3 c. .85 * d. 9/12
5. Develop and apply number theory concepts in problem solving situations: Primes, factors, multiples.	Assessment Focus: • The emphasis in statewide assessment is on application. Sample MC Item: How many numbers between 20 and 50 have no remainder when divided by 6? a. 3 b. 4 * c. 5 d. 6
6. Compare and order numbers.	Instructional/Assessment Focus: • Refers not only to whole numbers, but also to fractions and decimals, as specified in 4.1.5A1. Sample SCR Item: State a number that is between 1/3 and 0.36. Acceptable answers would include various representations of Real Numbers between 1/3 and .36 (e.g., 0.34, 0.334, 0.35, 7/20, etc.) Sample Extended Constructed Response (ECR) Item: On the number line in your answer folder, plot points for the following numbers. 4/5, 0.6 • Label each point. • Name two different rational numbers that are greater than 0.6 and less than 4/5. (Write one of your numbers in fractional form and write the other number in decimal form.) • Explain how you know that each of your numbers is greater than 0.6 and less than 4/5.
4.1.5 B. Numerical Operations
Descriptive Statement: Numerical Operations are an essential part of the mathematics curriculum, especially in the elementary grades. Students must be able to select and apply various computational methods, including mental math, pencil-and-paper techniques, and the use of calculators. Students must understand how to add, subtract, multiply, and divide whole numbers, fractions, decimals, and other kinds of numbers. With the availability of calculators that perform these operations quickly and accurately, the instructional emphasis now is on understanding the meanings and uses of these operations, and on estimation and mental skills, rather than solely on the development of paper-and-pencil proficiency.
Essential Questions	Enduring Understandings
- What makes a computational strategy both effective and efficient? (4.5D1) - How do operations affect numbers? - How do mathematical representations reflect the needs of society across cultures? (An essential question with broad applicability across multiple standards) (4.5C5)	- Computational fluency includes understanding not only the meaning, but also the appropriate use of numerical operations. - The magnitude of numbers affects the outcome of operations on them. - In many cases, there are multiple algorithms for finding a mathematical solution, and those algorithms are frequently associated with different cultures.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
1. Recognize the appropriate use of each arithmetic operation in problem situations.	Instructional/Assessment Focus: • The intent is that students not only recognize the appropriate use of arithmetic operations in the work of others, but that they also be able to appropriately use those operations themselves.
2. Construct, use, and explain procedures for performing addition and subtraction with fractions and decimals with: · Pencil-and-paper · Mental math · Calculator	"Construct" here means "develop" an algorithm or process. Sample SCR Item: Paula’s tractor holds 3 liters of gasoline. Tom’s tractor holds 2.4 liters. How much more does one tractor hold than the other? (Answer: 0.6 liters) Sample ECR Item: Joe had a pizza party. He ordered 8 pizzas, each cut into 8 slices. When his friends went home, he had 1/4 of a pepperoni pizza, 5/8 of a mushroom pizza, 1/2 of a cheese pizza, and 1/8 of a veggie pizza left over. How much pizza was left over in all? • Show one way to get the answer to this problem. Explain your method. • Show another way to get the answer to this problem. Explain your method. Sample SCR Item: A fifth-grade class will perform an act for the spring talent show. Two-thirds of the class of 24 students want to perform a skit. The rest of the students in the class want to sing a song. The teacher decided that 3/4 of the students must agree on an act before the decision will be final. How many more students would have to choose a skit before 3/4 of the students agree on it? (Answer: 2 students)
3. Use an efficient and accurate pencil-and-paper procedure for division of a 3-digit number by a 2-digit number.	Sample SCR Item: A gallon contains 128 ounces. Paul wants to divide three gallons of apple cider equally among the two dozen friends at his party. How much apple cider will each friend receive? (Answer: 16 oz.)
4. Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers.	Assessment of this CPI is generally within the context of one or more of the other content CPIs.
5. Check the reasonableness of results of computations.	Instructional/Assessment Focus: Includes • Identifying unreasonable answers obtained using a calculator; • Using inverse operations to check solutions; • Reasoning (4.5D2) and communication (4.5B2); • Solving problems (4.5A2) involving this recognition; and • Application to all fractions and decimals, as specified in 4.1.5A1. Sample ECR Item:** The fifth grade at Park Middle School is taking a field trip using buses that hold 36 passengers each.. There are three classes of 25 students each, and 5 adults (teachers or parents) will accompany each of the three classes. The Principal wants to order 2 and 1/2 buses; the Superintendent wants to order 2 buses; and the fifth-grade teachers want to order 3 buses. Which suggestion is most reasonable and why? Explain your reasoning.
6. Understand and use the various relationships among operations and properties of operations.	"Use" here means "apply." The "properties of operations" referred to include those specifically listed in 4.3.2D1, 4.3.3D1, or 4.3.4D1 (commutative properties, identity elements, associative properties, and multiplication or division by zero). Assessment Focus: • The emphasis in statewide assessment is on the "use" or "apply," rather than on the "understand."
4.1.5 C. Estimation
Descriptive Statement: Estimation is a process that is used constantly by mathematically capable adults, and one that can be easily mastered by children. It involves an educated guess about a quantity or an intelligent prediction of the outcome of a computation. The growing use of calculators makes it more important than ever that students know when a computed answer is reasonable; the best way to make that determination is through the use of strong estimation skills. Equally important is an awareness of the many situations in which an approximate answer is as good as, or even preferable to, an exact one. Students can learn to make these judgments and use mathematics more powerfully as a result.
Essential Questions	Enduring Understandings
- How can we decide when to use an exact answer and when to use an estimate?	- Context is critical when using estimation.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
1. Use a variety of estimation strategies for both number and computation.	"Number" here refers to "quantities." Assessment of this CPI is generally within the context of one or more of the other content CPIs.
2. Recognize when an estimate is appropriate, and understand the usefulness of an estimate as distinct from an exact answer.	"Understand" implies "explain," consistent with 4.5B1 and 4.5B2. This is an area of focus in grade 4 and may be assessed at a higher level of understanding in grade 5.
3. Determine the reasonableness of an answer by estimating the result of operations.
4. Determine whether a given estimate is an overestimate or an underestimate.

Standard 4.2 Geometry and Measurement All students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe and analyze phenomena.
Big Idea Geometry: Spatial sense and geometric relationships are a means to solve problems and make sense of a variety of phenomena. Big Idea Measurement: Measurement is a tool to quantify a variety of phenomena.
4.2.5 A. Geometric Properties
Descriptive Statement: This includes identifying, describing and classifying standard geometric object, describing and comparing properties of geometric objects, making conjectures concerning them, and using reasoning and proof to verify or refute conjectures and theorems. Also included here are such concepts as symmetry, congruence, and similarity.
Essential Questions	Enduring Understandings
- How can spatial relationships be described by careful use of geometric language? - How do geometric relationships help in solving problems and/or make sense of phenomena?	- Geometric properties can be used to construct geometric figures. (4.5D1; 4.5D2; 4.5E3) - Geometric relationships provide a means to make sense of a variety of phenomena.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
1. Understand and apply concepts involving lines and angles. · Notation for line, ray, angle, line segment · Properties of parallel, perpendicular, and intersecting lines · Sum of the measures of the interior angles of a triangle is 180°	"Understand and apply" here means "define, recognize, and apply."
2. Identify, describe, compare, and classify polygons. · Triangles by angles and sides · Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi · Polygons by number of sides · Equilateral, equiangular, regular · All points equidistant from a given point form a circle
3. Identify similar figures.
4. Understand and apply the concepts of congruence and symmetry (line and rotational).
4.2.5 B. Transforming Shapes
Descriptive Statement: This includes identifying, describing and classifying standard geometric object, describing and comparing properties of geometric objects, making conjectures concerning them, and using reasoning and proof to verify or refute conjectures and theorems. Also included here are such concepts as symmetry, congruence, and similarity.
Essential Questions	Enduring Understandings
- What situations can be analyzed using transformations and symmetries? (4.5E1; 4.5E2; 4.5E3)	- Shape and area can be conserved during mathematical transformations..
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
1. Use a translation, a reflection, or a rotation to map one figure onto another congruent figure.
2. Recognize, identify, and describe geometric relationships and properties as they exist in nature, art, and other real-world settings.
4.2.5 C. Coordinate Geometry
Descriptive Statement: Coordinate geometry provides an important connection between geometry and algebra. It facilitates the visualization of algebraic relationships, as well as an analytical understanding of geometry.
Essential Questions	Enduring Understandings
- How can geometric/algebraic relationships best be represented and verified? (4.5C2; 4.5D2; 4.5E1; 4.5E2; 4.5F5)	- Reasoning and/or proof can be used to verify or refute conjectures or theorems in geometry (4.5D1; 4.5D3; 4.5D4; 4.5D5; 4.5F5) - Coordinate geometry can be used to represent and verify geometric/algebraic relationships.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
1. Create geometric shapes with specified properties in the first quadrant on a coordinate grid.	Sample SCR Item: Three vertices of a parallelogram are at the points (0, 0), (2, 4), and (6, 0). What are the coordinates of the fourth vertex? (Answer: (8,4) or (-4,4) or (4, -4). Although not expected to find either of the answers out of the first quadrant, a student would not be penalized for finding such a vertex.)
4.2.5 D. Units Of Measurement
Descriptive Statement: Measurement helps describe our world using numbers. An understanding of how we attach numbers to real-world phenomena, familiarity with common measurement units (e.g., inches, liters, and miles per hour), and a practical knowledge of measurement tools and techniques are critical for students' understanding of the world around them.
Essential Questions	Enduring Understandings
- How can measurements be used to solve problems? (4.5A6)	- Everyday objects have a variety of attributes, each of which can be measured in many ways. -What we measure affects how we measure it. (4.5A4; 4.5A6) - Measurements can be used to describe, compare, and make sense of phenomena.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
1. Select and use appropriate units to measure angles and area.	Sample MC Item: What units would most likely be used to measure the area of a classroom floor? a. Square inches * b. Square feet c. Cubic feet d. Cubic yards
2. Convert measurement units within a system (e.g., 3 feet = ___ inches).	Sample ECR Item: Two students measured the same book shelf. Debbie said the measurement is 3. Tim said the measurement is 36. How can both students be correct? Explain your reasoning.
3. Know approximate equivalents between the standard and metric systems (e.g., one kilometer is approximately 6/10 of a mile).	"Know approximate equivalents" means that students should be able to recognize or produce approximate equivalents. Sample ECR Item: Carol measured her height to be 1.5. How can this be possible? Explain your reasoning.
4. Use measurements and estimates to describe and compare phenomena.	This CPI will infrequently be measured independently, but will provide a context for measuring other CPIs.
4.2.5 E. Measuring Geometric Objects
Descriptive Statement: This area focuses on applying the knowledge and understandings of units of measurement in order to actually perform measurement. While students will eventually apply formulas, it is important they develop and apply strategies that derive from their understanding of the attributes. In addition to measuring objects directly, students apply indirect measurement skills, using, for example, similar triangles and trigonometry.
Essential Questions	Enduring Understandings
- How can measurements be used to solve problems? (4.5A6)	- Everyday objects have a variety of attributes, each of which can be measured in many ways. - What we measure affects how we measure it. (4.5A4; 4.5A6)** - Measurements can be used to describe, compare, and make sense of phenomena.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
1. Use a protractor to measure angles.
2. Develop and apply strategies and formulas for finding perimeter and area. · Square · Rectangle
3. Recognize that rectangles with the same perimeter do not necessarily have the same area and vice versa.	Assessment of this CPI is generally within the context of CPI 4.2.5E2. Assessment Focus: • Students are expected to solve problems (4.5A2)** involving this recognition.
4. Develop informal ways of approximating the measures of familiar objects (e.g., use a grid to approximate the area of the bottom of one’s foot).

Standard 4.3 Patterns and Algebra All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.
Big Idea Algebra provides language through which we communicate the patterns in mathematics.
4.3.5 A. Patterns
Descriptive Statement: Algebra provides the language through which we communicate the patterns in mathematics. From the earliest age, students should be encouraged to investigate the patterns that they find in numbers, shapes, and expressions, and by doing so, to make mathematical discoveries. They should have opportunities to analyze, extend, and create a variety of patterns and to use pattern-based thinking to understand and represent mathematical and other real-world phenomena.
Essential Questions	Enduring Understandings
- How can change be best represented mathematically? (4.5C1; 4.5F1; 4.5F2; 4.5F3; 4.5F4) - How can patterns, relations, and functions be used as tools to best describe and help explain real-life situations? (4.5C1)	- The symbolic language of algebra is used to communicate and generalize the patterns in mathematics. - Algebraic representation can be used to generalize patterns and relationships.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
1. Recognize, describe, extend, and create patterns involving whole numbers. · Descriptions using tables, verbal rules, simple equations, and graphs	Sample MC Item: Last year, the cafeteria at Kyle's school recycled 100 pounds of the trash that was collected. This year was the second year of recycling, and the cafeteria recycled twice as much. If the amount of trash the cafeteria recycles doubles each year, how much will be recycled in the fourth year? a. 1600 pounds * b. 800 pounds c. 600 pounds d. 400 pounds
4.3.5 B. Functions and Relationships
Descriptive Statement: The function concept is one of the most fundamental unifying ideas of modern mathematics. Student begin their study of functions in the primary grades, as they observe and study patterns. As students grow and their ability to abstract matures, students form rules, display information in a table or chart, and write equations which express the relationships they have observed. In high school, they use the more formal language of algebra to describe these relationships.
Essential Questions	Enduring Understandings
- How are patterns of change related to the behavior of functions? (4.5F1; 4.5F2; 4.5F3; 4.5F4)	- Patterns and relationships can be represented graphically, numerically, symbolically, or verbally. (4.5E1)
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
1. Describe arithmetic operations as functions, including combining operations and reversing them.
2. Graph points satisfying a function from T-charts, from verbal rules, and from simple equations
4.3.5 C. Modeling
Descriptive Statement: The function concept is one of the most fundamental unifying ideas of modern mathematics. Student begin their study of functions in the primary grades, as they observe and study patterns. As students grow and their ability to abstract matures, students form rules, display information in a table or chart, and write equations which express the relationships they have observed. In high school, they use the more formal language of algebra to describe these relationships.
Essential Questions	Enduring Understandings
- How are mathematical models used to describe physical relationships? (4.5E2) - How are physical models used to clarify mathematical relationships? (4.5E3)	- Mathematical models can be used to describe and quantify physical relationships. (4.5E2) - Physical models can be used to clarify mathematical relationships. (4.5E3)
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
1. Use number sentences to model situations. · Using variables to represent unknown quantities · Using concrete materials, tables, graphs, verbal rules, algebraic expressions/equations
2. Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events. · Changes over time · Rates of change (e.g., when is plant growing slowly/rapidly, when is temperature dropping most rapidly/slowly)	Assessment Focus: • Students are asked to draw a graphical representation of a story.
4.3.5 D. Procedures
Descriptive Statement: Techniques for manipulating algebraic expressions - procedures - remain important, especially for students who may continue their study of mathematics in a calculus program. Utilization of algebraic procedures includes understanding and applying properties of numbers and operations, using symbols and variables appropriately, working with expressions, equations, and inequalities, and solving equations and inequalities.
Essential Questions	Enduring Understandings
- What makes an algebraic algorithm both effective and efficient? (4.5D1)