Unit 5: Fraction/Decimal Connection February 21-April 13 (7 weeks)
4th Grade MathematicsUnit 6: Decimal Fractions
Teacher Resource Guide /
2012-2013 /
In Grade 4, instructional time should focus on three critical areas:
- Developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends;
Students apply their understanding of models for multiplication (equal-sized groups, arrays, area models) and models for division, place value, and the distributive property as they discuss and use efficient methods to estimate and compute products and quotients. They develop fluency with efficient procedures for multiplying whole numbers, understand and explain why the procedures work based on place value and properties, and use them to solve problems. Students interpret remainders based upon the context of the problem.
- Developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers;
Students recognize that two different fractions can be equal (12/4 = 6/2), and they develop methods for generating and recognizing equivalent fractions. Students extend understandings about how fractions are built from unit fractions (3/4 = ¼ + ¼ + ¼), and use the meaning of multiplication to multiply a fraction by a whole number.
- Understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry.
Students describe, analyze, compare, and classify two-dimensional shapes. Through building, drawing, and analyzing two-dimensional shapes, students understand properties of two-dimensional objects and solve problems involving symmetry.
4th Grade2012-2013Page 1
Unit 6: Decimal Fractions April 22 – May 29 (6 weeks)
4th Grade Mathematics 2012-2013
Unit / Time Frame / Test ByTRIMESTER 1 / 1: Multiplication and Division Concepts / 6 weeks / 8/27 – 10/5 / October 5
2: Multi-Digit Multiplication / 6 weeks / 10/8 – 11/16 / November 16
TRIMESTER 2 / 3: Measurement/Geometry / 4 weeks / 11/19 – 12/21 / December 21
4: Multi-Digit Division / 7 weeks / 1/2- 2/22 / February 22
TRIMESTER 3 / 5: Fractions / 7 weeks / 2/25-4/19 / April 19
6: Decimal Fractions / 6 weeks / 4/22 – 5/29 / May 29
Math Wiki:
Big Ideas / Essential QuestionsEvery fraction has an infinite number of equivalent fractions because a fraction can always be split into smaller divisions. / How many equivalent fractions can be made from a given fraction?
Equivalent fractions cover the same area but with different sized parts. / What are equivalent fractions?
Benchmarks, common numerators, and common denominators, are helpful strategies in comparing fractions. / What are strategies that can be used to compare fractions?
Benchmarks and place value are helpful strategies in comparing decimals. / What are strategies that can be used to compare decimals?
Identifier / Standards / Mathematical Practices
STANDARDS / 4.NF.5 / Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.[1] For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. / 1) Make sense of problems and persevere in solving them.
2) Reason abstractly and quantitatively.
3) Construct viable arguments and critique the reasoning of others.
4) Model with mathematics.
5) Use appropriate tools strategically.
6) Attend to precision.
7) Look for and make use of structure.
8) Look for and express regularity in repeated reasoning.
4.NF.6 /
- Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.
4.NF.7 /
- Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.
4.MD.2 /
- 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.
Identifier / Standards / Bloom’s / Skills / Concepts
STANDARDS / 4.NF.5 / Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.[2] For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. / Apply (3) / Fraction equivalence (10 and 100)
Add fractions / equivalent (denominators 10 and 100)
Add fractions
4.NF.6 /
- Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.
fractions with denominators of 10 or 100
4.NF.7 /
- Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.
Remember (1) / Compare (two decimals to hundredths)
Record (comparisons w/ symbols) / decimals
tenths
hundredths
greater than >
less than <
equal =
4.MD.2 /
- 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.
intervals of time
liquid volumes
mass
money
fractions
decimals
measurement
Instructional Strategies for ALL Students
Comparing Fractions –Students have a tremendously strong mind-set about numbers that causes them difficulties with the relative size of fractions. In their experience, larger numbers mean “more.” The tendency is to transfer this whole-number concept to fractions: Seven is more than four, so sevenths must be bigger than fourths. The inverse relationship between number of parts and size of parts cannot be told but must be a creation of each student’s own thought process. Telling students that “larger bottom numbers mean smaller fractions” is not only inappropriate but also dangerous to students’ conceptual understanding. Instead students must have many opportunities to make sense of the size of fractions in comparative situations through comparison tasks over time. Many times a standard algorithm is taught for comparing fractions. The usual approach is to find a common denominator. This rule can be effective in getting correct answers but requires no thought about the size of the fractions. If students are taught the common denominator rule before they have had the opportunity to think about the relative size of various fractions, there is little chance that they will develop any familiarity with or number sense about fraction sense. Number sense about fractions is critical to students’ conceptual understanding of operations with fractions which begin in 4th grade and continue through the middle grades.For these reasons, the recommended strategies for instruction in comparing fractions are benchmark fractions, focus on numerator or denominator, and common denominators. Activities to support students’ development of conceptual understanding for comparing fractions are listed on p. 6-7 of this guide. The activities are also provided on the Wiki. (Adapted from Van de Walle, 2006.)
Equivalent fractions –The general approach to helping students create an understanding of equivalent fractions is to have them use models to find different names for a fraction. This is typically followed with instruction of an algorithm. However, models alone often do not help students develop the conceptual understanding of equivalent fractions and algorithms are often taught too soon. Rather, it is recommended that multiple tasks are given to students with a focus on finding multiple ways to create fraction equivalencies and reason about the patterns. When students understand that fractions can have more than one name, they can begin to figure out a method for finding equivalent names. Therefore, the recommended strategies for instruction in equivalent fractions are to provide multiple tasks for students to develop conceptual understanding of equivalence followed by instruction of the algorithm. Activities to support students’ conceptual understanding for equivalent fractions are listed on p. 6-7 of this guide. The activities are also provided on the Wiki.
Decimals –Typically instruction in fractions and decimals are separate. While this seems logical, linking the ideas of fractions to decimals can be very helpful for students’
conceptual understanding of both. This unit’s focus begins with fractions and finishes with decimals. The primary purpose of the decimal instruction is to help students see decimals as another way to write fractions. We want students to realize that both fractions and decimals represent the same concepts.
The primary instructional strategies are:
- to use familiar fraction concepts and models to explore decimals (tenths and hundredths)
- to examine how the base-ten system can be extended to include numbers less than 1 as well as large numbers
- To use models to make meaningful translations between fractions and decimals.
Routines/Meaningful Distributed Practice
Distributed Practice that is Meaningful and Purposeful
Practice is essential to learn mathematics. However, to be effective in improving student achievement, practice must be meaningful, purposeful, and distributed.
- Meaningful: Builds on and extends understanding
- Purposeful: Links to curriculum goals and targets an identified need based on multiple data sources
- Distributed: Consists of short periods of systematic practice distributed over a long period of time
Routines are an excellent way to achieve the mandate of Meaningful Distributed Practice outlined in the Iowa Core Curriculum.. The skills presented during routines do not necessarily reinforce the lesson concept for that day. Routines may be used to address a need for small increments of exposure to a skill or review of skills already taught. Routine activities may be repeated several days in a row, allowing for a build-up of conceptual understanding, or can be visited and re-visited over a period of time. Routines can be inserted as the schedule allows; in short intervals throughout the day or as a lesson opener or closer. Selection of the routine should be made based on informal teacher observation and formative assessments.
Skill / Standard / ResourceA digit in one place represents ten times what it represents in the place to its right. / 4.NBT.1
Draw points, lines, line segments, rays, angles, parallel and perpendicular lines
(May be taught in whole group lesson and reinforced during MDP time) / 4.G.1
Solve word problems involving distance, intervals of time, liquid volumes, masses of objects and money / 4.MD.2
Whole number multiplication / 4.NBT.5
Whole number division / 4.NBT.6
Other skills students need to develop based on teacher observations and formative assessments.
Lesson Progression
These lessons are intended to be taught in the following order.
Lesson / Time Frame / Teacher Directions / Teacher’s Edition
Pages / Standards
Addressed
DECIMALS
Decimal Models / 10-15 days
Wiki: Decimals on grids / This activity is taught as a lesson and may need to be explored for several days. / 4.NF.6
Wiki: Base-ten fractions to decimals / This activity will be repeated over the course of a few days. / 4.NF.6
Wiki: Fill Two / These activities will be taught as lessons then may be used as centers/stations. Students need lots of experiences with these activities. / 4.NF.7
Wiki: Fill Four / 4.NF.7
Wiki: Smaller to Larger / 4.NF.7
Wiki: Decimals in Between / 4.NF.7
Wiki: Capture Decimals / 4.NF.7
Wiki: Calculator Decimal Counting / Foundation for 4.NF.7
Wiki: Center Activities from 2010-2011 / Depends on the activity
Unit 11, Lesson 2, Activity 1 / 1050 / 4.NF.6
Unit 11, Lesson 1, Activities 1-2 / 1036 / 4.NF.6
Wiki: Friendly Fraction to Decimals / After teaching these activities, they may be modified to use as centers / 4.NF.6
Wiki: Decimals on a Friendly Fraction Line / 4.NF.6
Comparing & Ordering / 5-10 days
Unit 11, Lesson 3, Activity 1 / 1058 / 4.NF.7
Wiki: Line ‘Em Up / 4.NF.7
Unit 11, Lesson 4, Activity 1 / Start on page 1072- Write Numbers in Decimal Form / 1072 / 4.NF.7
Unit 11, Lesson 6, Activity 1 / 1088 / 4.NF.2
Unit 11, Lesson 7, Activities 1-2 / Do not do fraction problems with 16ths as the denominator / 1094 / 4.NF.2
Math Expressions Activities
(use as centers, re-teaching/extension support, etc.)
Activity / Standards
Activity Cards 9-3 Intervention, On-Level / 4.NF.3b
Activity Card 9-10 Intervention / 4.NF.2
Activity Cards 9-11 Intervention, On-Level / 4.NF.1
Activity Cards 11-1 Intervention, On-Level / 4.NF.6
Activity Cards 11-2 Intervention, On-Level / 4.NF.6
Activity Cards 11-3 Intervention, On-Level / 4.NF.7
Activity Cards 11-4 Intervention, On-Level / Decimal # sense-no std
Activity Cards 11-5 Intervention, On-Level, Challenge / 4.NF.7
Activity Cards 11-11 Intervention, On-Level / 4.MD.2
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