Georgia Performance Standards Framework
Fifth Grade Mathematics · Unit 2 · 1st Edition
Grade 5 Mathematics FrameworksUnit 2
Decimal Understanding and Operations
Unit 2
DECIMAL UNDERSTANDING AND OPERATIONS
(7 weeks)
TABLE OF CONTENTS
Overview 3
Key Standards & Related Standards 4
Enduring Understandings 6
Essential Questions 6
Concepts & Skills to Maintain 7
Selected Terms and Symbols 7
Classroom Routines 10
Strategies for Teaching and Learning 10
Evidence of Learning 11
Tasks 12
· High Roller Revisited 13
· Patterns-R-Us 21
· Base Ten Activity 26
· How Much Money? 33
· Super Slugger Award 38
· Number Puzzle 43
· What’s My Rule? 48
· Do You See an Error? 52
· Road Trip 56
Culminating Tasks
· Teacher for a Day 61
· Bargain Shopping 66
OVERVIEW
In this unit students will:
· Understand place value from hundredths to one million
· Model and explain multiplication and division of decimals
· Multiply and divide decimals
· Understand the rules for multiplication and division of decimals
· Use formulas to represent the relationship between quantities
· Use variables for unknown quantities
Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as add/subtract decimals and fractions with like denominators, whole number computation, angle measurement, length/area/weight, number sense, data usage and representations, characteristics of 2D and 3D shapes, and order of operations should be addressed on an ongoing basis.
To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks listed under “Evidence of Learning” be reviewed early in the planning process. A variety of resources should be utilized to supplement, but not completely replace, the textbook. Textbooks not only provide much needed content information, but excellent learning activities as well. The tasks in these units illustrate the types of learning activities that should be utilized from a variety of sources.
STANDARDS ADDRESSED IN THIS UNIT
Mathematical standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics.
KEY STANDARDS
M5N2. Students will further develop their understanding of decimals as part of the base-ten number system.
- Understand place value.
- Analyze the effect on the product when a number is multiplied by 10, 100, 1000, 0.1, 0.01, and .001.
- Use <, >, or = to compare decimals and justify the comparison.
M5N3. Students will further develop their understanding of the meaning of multiplication and division with decimals and use them.
- Model multiplication and division of decimals.
- Explain the process of multiplication and division, including situations in which the multiplier and divisor are both whole numbers and decimals.
- Multiply and divide with decimals including decimals less than one and greater than one.
- Understand the relationships and rules for multiplication and division of whole numbers also apply to decimals.
M5A1. Students will represent and interpret the relationships between quantities algebraically.
a. Use variables, such as n or x, for unknown quantities in algebraic expressions.
b. Investigate simple algebraic expressions by substituting numbers for the unknown.
c. Determine that a formula will be reliable regardless of the type of number (whole numbers or decimals) substituted for the variable.
RELATED STANDARDS
M5M1. Students will extend their understanding of area of geometric plane figures.
a. Estimate the area of geometric plane figures.
d. Find the areas of triangles and parallelograms using formulae.
M5D2 Students will collect, organize, and display data using the most appropriate graph.
M5P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
M5P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
c. Develop and evaluate mathematical arguments and proofs.
d. Select and use various types of reasoning and methods of proof.
M5P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
M5P4. Students will make connections among mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
M5P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
ENDURING UNDERSTANDINGS
· Students will understand that the location of a decimal determines the size of a number.
· Students will understand that the placement of the decimal is determined by multiplying or dividing a number by 10 or a multiple of 10.
· Students will understand that multiplication and division are inverse operations of each other.
· Students will understand that rules for multiplication and division of whole numbers also apply to decimals.
ESSENTIAL QUESTIONS
· How does the location of digit in the number affect the size of a number?
· Why does placement or position of a number matter?
· How is place value different from digit value?
· How can we use models to demonstrate decimal values?
· How can we use models to demonstrate multiplication and division of decimals?
· What happens when we multiply decimals by powers of 10?
· How do the rules of multiplying whole numbers relate to multiplying decimals?
· How are multiplication and division related?
· How are factors and multiples related to multiplication and division?
· What happens when we multiply a decimal by a decimal?
· What happens when we divide a decimal by a decimal?
· What are some patterns that occur when multiplying and dividing by decimals?
· How do we compare decimals?
· How do we find the highest batting average?
· How do we best represent data in a graph?
· How can we efficiently solve multiplication and division problems with decimals?
· How can we multiply and divide decimals fluently?
· What are some patterns that occur when multiplying and dividing by decimals?
· What strategies are effective for finding a missing factor or divisor?
· How can we check for errors in multiplication or division of decimals?
· What are the various uses of decimals?
· How do we solve problems with decimals?
· How are multiplication and division related?
CONCEPTS/SKILLS TO MAINTAIN
It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.
· Number sense
· Order of Operations
· Whole numbers and decimal computation
· Add and subtract decimals
· Add and subtract common fractions with like denominators
· Angle measurement
· Length, area, and weight
· Characteristics of 2-D and 3-D shapes
· Data usage and representation
SELECTED TERMS AND SYMBOLS
The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them.
The definitions below are for teacher reference only and are not to be memorized by the students. Teachers should present these concepts to students with models and real life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers.
Array: A rectangular arrangement of objects or numbers in rows and columns.
Associative Property of Multiplication: The product of a set of numbers is the same regardless of how the numbers are grouped.
Example: If (3 x 5) x 2 = 15 x 2 = 30, and 3 x (5 x 2) = 3 x 10 = 30, then (3x5) x 2 = 3 x (5 x 2).
Commutative Property of Multiplication: The product of a group of numbers is the same regardless of the order in which the numbers are arranged.
Example: If 8 x 6 = 48 and 6 x 8 = 48, then 8 x 6 = 6 x 8.
Distributive Property: A product can be found by multiplying the addends of a number separately and then adding the products.
Example: 4 x 53 = (4 x 50) + (4 x 3) = 200 + 12 = 212
Dividend: A number that is divided by another number.
Example: dividend ÷ divisor = quotient
Division: An operation in which a number is shared or grouped into equal parts.
Divisor:
(1) In a fair sharing division problem, the divisor is the number of equal groups. In a measurement (repeated subtraction) division problem, the divisor indicates the size of each group.
(2) A number by which another number is to be divided.
Example: dividend ÷ divisor = quotient
Factor:
(1) n. A number that is multiplied by another number to get a product
(2) v. To “factor" means to write the number or term as a product of its factors
Hundred Thousands: The digit that tells you how many sets of one hundred thousand are in the number.
Example: The number 432,895 has four hundred thousands.
Hundreds: The digit that tells you how many sets of one hundred are in the number. Example: The number 784 has seven hundreds.
Hundredths: The digit that tells you how many sets of hundredths there are in the number.
Example: The number 0.6495 has four hundredths.
Identity Property of Multiplication: Any number that is multiplied by 1 results in the number itself.
Example: 1 x 5 = 5 x 1 = 5
Measurement Division (or repeated subtraction): Given the total amount (dividend) and the amount in a group (divisor), determine how many groups of the same size can be created (quotient).
Millions: The digit that tells you how many sets of one million are in the number. Example: The number 3,901,245 has three millions.
Multiple: The product of a given whole number and an integer.
Multiplier: The number in a multiplication equation that represents the number of (equal-sized) groups.
Ones: The digit that tells you how many sets of ones are in the number.
Example: The number 784 has four ones.
Partial Products: The products that result when ones, tens, or hundreds within numbers are multiplied separately.
Example: When multiplying 63 x 37 = 1800 + 420 + 90 + 21 = 2,331
60 x 30 = 1800
60 x 7 = 420
30 x 3 = 90
3 x 7 = 21
The resulting partial products are 1800, 420, 90, and 21.
Partition Division (or fair-sharing): Given the total amount (dividend) and the number of equal groups (divisor), determine how many/much in each group (quotient).
Place Value: The use in number systems of the position of a digit in a number to indicate the value of the digit.
Product: A number that is the result of multiplication.
Quotient: A number that is the result of division (without remainders)
Example: dividend ÷ divisor = quotient
Remainder: The number left over when a number cannot be divided evenly.
Ten Thousands: The digit that tells you how many sets of ten thousand are in the number.
Example: The number 43,987 has four ten thousands.
Tens: The digit that tells you how many sets of ten are in the number.
Example: The number 784 has eight tens.
Tenths: The digit that tells you how many sets of tenths are in the number.
Example: The number 0.6495 has six tenths.
Thousands: The digit that tells you how many sets of thousand are in the number. Example: The number 5,321 has five thousands.
CLASSROOM ROUTINES
The importance of continuing the established classroom routines cannot be overstated. Daily routines must include such obvious activities such as taking attendance and lunch count, doing daily graphs, and daily question and calendar activities as whole group instruction. They should also include less obvious routines, such as how to select materials, how to use materials in a productive manner, how to put materials away, and how to access classroom technology such as computers and calculators. An additional routine is to allow plenty of time for children to explore new materials before attempting any directed activity with these new materials. The regular use of routines is important to the development of students’ number sense, flexibility, fluency, collaborative skills, and communication. All of which will support students’ performances on the tasks in this unit and throughout the school year.
STRATEGIES FOR TEACHING AND LEARNING
· Students should be actively engaged by developing their own understanding.
· Mathematics should be represented in as many ways as possible by using graphs, tables, pictures, symbols, and words.
· Appropriate manipulatives and technology should be used to enhance student learning.
· Students should be given opportunities to revise their work based on teacher feedback, peer feedback, and metacognition which includes self-assessment and reflection.
· Students need to write in mathematics class to explain their thinking, talk about how they perceive topics, and justify their work to others.
Math Literature Connections
Millions of Cats. (2006/1928) by Wanda Ga’g