5th Grade Mathematics
| 5th Grade Mathematics - Investigations |
|Unit 4: Decimals |
|Teacher Resource Guide |
|2012-2013 |
In Grade 5, instructional time should focus on four critical areas:
1. Developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions (limited to unit fractions divided by whole numbers and whole numbers divided by unit fractions);
Students apply their understanding of fractions and fraction models (set model, area model, linear model) to represent addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators (¼ + 2/8 = ¼ + ¼). They develop fluency in calculating sums and differences of fractions, and make reasonable estimates of them. Students also use the meaning of fractions and the relationship between multiplication and division to explain why the procedures for multiplying and dividing fractions make sense.
2. Extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations;
Students develop understanding of why division procedures work based on place value and properties of operations. They are fluent with multi-digit addition, subtraction, multiplication, and division. They apply their understandings of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. Students use the relationship between decimals and fractions, and the relationship between decimals and whole numbers (i.e., a decimal multiplied by anpower of 10 is a whole number) to understand and explain why procedures for multiplying and dividing decimals make sense. They compute products and quotients of decimals to hundredths efficiently and accurately.
3. Developing understanding of volume.
Students understand that volume is an attribute of three-dimensional space and can be measured by finding the total number of same-size units of volume required to fill the space without gaps or overlaps. They understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume. They decompose three-dimensional shapes and find volumes of right rectangular prisms by viewing them as decomposed into layers of arrays of cubes.
5th Grade Mathematics 2012-2013
| |Unit |Time Frame |Testing Window |
|TRIMESTER|1: Multi-Digit Multiplication and Division |7 weeks |8/22 – 10/12 |October 12 |
|1 | | | | |
| |2: Measurement/Geometry |4 weeks |10/15 – 11/9 |November 9 |
|TRIMESTER|3: Addition and Subtraction of Fractions |8 weeks |11/12 – 1/18 |January 18 |
|2 | | | | |
| |4: Decimals |8 weeks |1/22 – 3/14 |March 14 |
|TRIMESTER| | | | |
|3 | | | | |
| |5: Multiplication and Division of Fractions |9 weeks |3/25 – 5/30 |May 30 |
|Big Ideas |Essential Questions |
|Place value helps us understand the size of a number. |Why is place value important? |
|With whole numbers and decimals, addition and subtraction is based on like position values. |How is adding and subtracting decimals like adding and subtracting whole numbers? |
|With whole numbers and decimals, multiplication and division will produce the same digits. Estimation |How is multiplying and dividing decimals like multiplying and dividing whole numbers? |
|helps determine where to place the decimal in the answer. | |
|Identifier |Standards |Mathematical Practices |
|PRIO|5.NBT.3 |Read, write, and compare decimals to thousandths. |1) Make sense of problems and persevere in solving them. |
|RITY| |Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g.,| |
| | |347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 ×(1/10) + 9 × (1/100) + 2 × (1/1000). |2) Reason abstractly and quantitatively. |
| | |Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < | |
| | |symbols to record the results of comparisons. |3) Construct viable arguments and critique the reasoning of others. |
| | | | |
| | | |4) Model with mathematics. |
| | |Convert among different-sized standard measurement units within a given measurement system (e.g., | |
| |5.MD.1 |convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems. |5) Use appropriate tools strategically. |
| | | | |
| | |Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it |6) Attend to precision. |
| | |represents in the place to its right and 1/10 of what it represents in the place to its left. | |
| |5.NBT.1 | |7) Look for and make use of structure. |
| | | | |
| | | |8) Look for and express regularity in |
| | | |repeated reasoning. |
| |5.NBT.4 |Use place value understanding to round decimals to any place. | |
| |5.NBT.7 |Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and | |
| | |strategies based on place value, properties of operations, and/or the relationship between addition | |
| | |and subtraction; relate the strategy to a written method and explain the reasoning used. | |
| | | | |
| | |Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and | |
| |5.NBT.2 |explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a | |
| | |power of 10. Use whole-number exponents to denote powers of 10. | |
|Identifier |Standards |Bloom’s |Skills |Concepts |
|PRIO|5.NBT.3 |Read, write, and compare decimals to thousandths. |Understand (2) |Read, write and compare|decimals |
|RITY| |Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g.,| |(decimals to |tenths |
| | |347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 ×(1/10) + 9 × (1/100) + 2 × (1/1000). | |thousandths) |hundredths |
| | |Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < | | |thousandths |
| | |symbols to record the results of comparisons. |Understand (2) |Compare (two decimals |expanded form |
| | | | |to thousandths using |greater than > |
| | | | |symbols (< , > , = ) |less than < |
| | |Convert among different-sized standard measurement units within a given measurement system (e.g., | | |equal = |
| |5.MD.1 |convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems. | | | |
| | | | | | |
| | |Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it | | | |
| | |represents in the place to its right and 1/10 of what it represents in the place to its left. | | | |
| |5.NBT.1 | | | | |
| |5.NBT.4 |Use place value understanding to round decimals to any place. |Understand (2) |Round (decimals) | |
| |5.NBT.7 |Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and |Apply (3) |Add, subtract, multiply| |
| | |strategies based on place value, properties of operations, and/or the relationship between addition | |or divide (decimals to | |
| | |and subtraction; relate the strategy to a written method and explain the reasoning used. | |hundredths) | |
| | | | | | |
| | |Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and | | | |
| |5.NBT.2 |explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a | | | |
| | |power of 10. Use whole-number exponents to denote powers of 10. | | | |
|Instructional Strategies for ALL Students |
|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. (Adapted from Van de Walle, 2006) |
| |
|In this first year of transition to the new Iowa Core, activities to support conceptual understanding of decimal numeration from the 4th grade curriculum will be included in 5th grade.|
|The activities are designed to help students develop a deeper understanding of decimals which will benefit them in their work with decimal computation. |
| |
|Activities that utilize these instructional strategies to support students’ conceptual understanding for decimals are listed on p. 6-7 of this guide. The activities are also provided |
|on the Wiki. |
| |
|Decimal computation – Traditionally decimal computation has been dominated by the following rules: line up the decimal points (addition and subtraction), count the decimal points |
|(multiplication), and shift the decimal point in the divisor and dividend so that the divisor is a whole number (division). These rules have often proved confusing for students and |
|many times teachers find that reteaching is necessary each year. In order for a rule to make sense, conceptual understanding has to be developed first. Contrary to traditional |
|instruction, students should become adept at estimating decimal computations well before they learn to compute with pencil and paper. For many decimal computations, rough estimates can|
|be made easily by rounding the numbers to nice whole numbers or simple base-ten fractions. When the focus is estimation, students’ thinking turns to the meaning of the numbers and the |
|operations and not on counting decimal places. Many times students who are taught to focus only on the rules do not even consider the actual values of the number much less estimate. |
|Therefore, a good place to begin decimal computation is with estimation. Not only is it a highly practical skill, but it also helps students look at answers in ballpark terms and make |
|sense of the computation. A good time to begin decimal computation is as soon as a conceptual background in decimal numeration has been developed. (Adapted from Van de Walle, 2006) |
| |
|Activities that emphasize estimation as a beginning instructional strategy for decimal computation are listed on p. 6-7 of this guide. The activities are also provided on the Wiki. |
| |
| |
| |
|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. |
|Concepts taught through Meaningful Distributed Practice during Unit 4: |
| |
|Skill |
|Standard |
|Resource |
| |
|Convert among different-sized standard measurement units |
| |
| |
|( |
|5.MD.1 |
| |
| |
|Place Value |
|5.NBT.1 |
| |
| |
|Multiplication of multiples of powers of 10 |
|5.NBT.2 |
| |
| |
|Multi-digit multiplication |
|5.NBT.5 |
| |
| |
|Whole-number division |
|5.NBT.6 |
| |
| |
|Addition and Subtraction of Fractions (word problems) |
|5.NF.2 |
| |
| |
|Other skills students need to develop based on teacher observations and formative assessments. |
| |
|Investigations Resources for Unit 4 – 5th grade Decimals |
|Instructional Plan |Resource |Standards |
| | |Addressed |
|Unit 6: Decimals on Grids and Number Lines |Investigations |5.NBT.1 (imbedded) |
|Investigation 1 | |5.NBT.3a |
|Do 1.5A before 1.5 (ICCSS, CC92-CC96, C64-C66) |Investigations and the Common|5.NBT.3b |
|Investigation 2 |Core State Standards (ICCSS) |5.NBT.4 |
|Do 2.5A before 2.5 (ICCSS, CC97-102, C67-C69) | |5.NBT.7 |
|Investigation 3A (ICCSS) | |5.NBT.2 |
|3A.1: CC103-CC108, C70, C71 | |5.MD.1 |
|3A.2: CC109-CC114, C72-C74 | | |
|3A.3: CC115-CC118, C75-C78 | | |
|3A.4: CC119-CC123, C79-C81 | | |
|3A.5: CC124-CC129, C82-C84, C98 | | |
|3A.6: CC130-CC135, C85-C86, C99 | | |
|3A.7: CC136-CC141, C87-C90, C100 | | |
|3A.8: CC142-CC146, C91-C93 | | |
|3A.9: CC147-CC151, C94-C97, C101, C102 | | |
|Supplemental Lessons – Use as needed |
|Lessons |Teacher Directions |Standards |
| | |Addressed |
|Base-Ten Fractions |Complete activities multiple times and continue them throughout the unit as needed. |5.NBT.3 |
|Wiki: Base-Ten Fraction Models | | |
|Wiki: Multiple Names & Formats | | |
|Extending the Place Value System |Complete activities multiple times and continue throughout the unit. This is important, so |5.NBT.1 |
| |students will need to continue practicing throughout the unit. | |
|Wiki: A Two-way Relationship | | |
| | | |
|Making the Fraction-Decimal Connection |Complete activities multiple times and continue them throughout the unit. The activities can be |5.NBT.3 |
| |slightly modified and used as a center activity. | |
|Wiki: Base-Ten Fractions to Decimals | | |
|Wiki: Calculator Decimal Counting | | |
|Developing Decimal Number Sense |Complete activities multiple times and continue them throughout the unit. The activities can be |5.NBT.3, 5.NBT.4 |
| |slightly modified and used as a center activity. | |
|Wiki: Friendly Fractions to Decimals | | |
|Wiki: Estimate Then Verify | | |
|Wiki: Decimals on a Friendly Fraction Line | | |
|Approximations with a Nice Fractions |Complete activities multiple times and continue them throughout the unit. The activities can be |5.NBT.4 |
| |slightly modified and used as a center activity. | |
|Wiki: Close to a Friendly Fraction | | |
|Wiki: Best Match | | |
|Ordering Decimal Numbers |Complete activities multiple times and continue them throughout the unit. The activities can be |5.NBT.3 |
| |slightly modified and used as a center activity. | |
|Wiki: Line ‘Em Up | | |
|Wiki: Close “Nice Numbers” | | |
|Wiki: Zoom In Number Line | | |
|Rounding |“Rounding Decimals” on the Wiki describes the rationale for teaching rounding through using |5.NBT.4 |
| |“nice numbers” instead of the standard algorithm. The Iowa Core states (5.NBT.4) “Use place | |
| |value understanding to round decimals to any place.” Therefore some instruction on rounding | |
| |decimals to a given place may be necessary. In Lesson 15 students practice rounding to a given | |
| |place using a number line. Rounding should be taught as mini-lessons then continued throughout | |
| |the unit as this is a priority standard. | |
|Wiki: Rounding Decimals | | |
|Wiki: Rounding Decimals Number Line | | |
|Supplemental Lessons – Use as needed |
|Lessons |Teacher Directions |Standards |
| | |Addressed |
|Computation with Decimals - Addition and Subtraction |Read the Instructional Strategies for more information about Teaching computation with decimals |5.NBT.7 |
|Wiki: The Role of Estimation | | |
|Wiki: Addition and Subtraction | | |
|Wiki: Exact Sums and Differences | | |
|Computation with decimals - Multiplication |Once the students have learned how to add and subtract with decimals, continue to present word |5.NBT.7 |
| |problems that require these operations as you move forward to multiplication. | |
|Wiki: Multiplication (with decimals) | | |
|Wiki: Where Does the Decimal Go? (Multiplication) | | |
|Computation with decimals- Division |Once the students have learned how to add, subtract and multiply with decimals, continue to |5.NBT.7 |
| |present word problems that require these operations as you move forward to division. | |
|Wiki: Division (with decimals) | | |
|Wiki: Where Does the Decimal Go? (Division) | | |
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