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Instructional Unit Plan School: Sunrise Elementary Teacher(s): C. E. S. Grade Level: 6th Content Area(s): Bits and Pieces I Implementation Dates: Unit Focus: Bits and Pieces I, asks students to make sense of fractions, decimals, and percents in different contexts. In this unit, students will meet several interpretations and models of fractions. Students interpret fractions as parts of a whole, fractions as measures or quantities, fractions as indicated division, fractions as decimals, and fractions as percents. New Mexico Content Standards, Benchmarks and Performances Addressed: Strand 1: Number And Operations Standard: Students will understand numerical concepts and mathematical operations. 5-8 Benchmark 1: Understand numbers, ways of representing numbers, relationships among numbers, and number systems. Compare and order rational numbers. Use equivalent representations for rational numbers (e.g., integers, decimals, fractions, percents, ratios, numbers with whole-number exponents). Use appropriate representations of positive rational numbers in the context of real-life applications. Identify greatest common factor and least common multiples for a set of whole numbers. Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers. 5-8 Benchmark 2: Understands the meaning of operations and how they relate to one another. Calculate multiplication and division problems using contextual situations. Factor a whole number into a product of its primes. Demonstrate the relationship and equivalency among ratios and percents. Use proportions to solve problems. Explain and perform: o Whole number division and express remainders as decimals or appropriately in the context of the problem o Addition, subtraction, multiplication, and division with decimals o Addition and subtraction with integers o Addition, subtraction, and multiplication with fractions and mixed numerals Determine the least common multiple and the greatest common divisor of whole numbers and use them to solve problems with fractions. 5-8 Benchmark 3: Compute fluently and make reasonable estimates. Estimate quantities involving rational numbers using various estimations. Use estimates to check reasonableness of results and make predictions in situations involving rational numbers. Determine if a problem situation calls for an exact or approximate answer and perform the appropriate computation. Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line. Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. Interpret and use ratios in different contexts. Compute and perform multiplication and division of fractions and decimals and apply these procedures to solving problems. 1 Strand 2: Algebra Standard: Students will understand algebraic concepts and applications. 5-8 Benchmark 1: Understand patterns, relations, and functions Solve problems involving proportional relationships. Explain and use symbols to represent unknown quantities and variable relationships. Explain and use the relationships among ratios, proportions, and percents. Make generalizations based on observed patterns and relationships. 5-8 Benchmark 2: Represent and analyze mathematical situations and structures using algebraic symbols. Solve problems involving proportional relationships. 5-8 Benchmark 3: Use mathematical models to represent and understand quantitative relationships Develop and use mathematical models to represent and justify mathematical relationships found in a variety of situations. Create, explain, and use mathematical models such as: o Venn diagrams to show the relationships between the characteristics of two or more sets o Equations and inequalities to model numerical relationships o Three-dimensional geometric models o Graphs, tables, and charts to interpret and analyze data 5-8 Benchmark 4: Analyze changes in various contexts. Solve problems that involve change using proportional relationships. Use ratios to predict changes in proportional situations. Use tables and symbols to represent and describe proportional and other relationships involving conversions, sequences, and perimeter. Strand 5: Data Analysis And Probability Standard: Students will understand how to formulate questions, analyze data, and determine probabilities. 5-8 Benchmark 1: Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. Use statistical representations to analyze data. Draw and compare different graphical representations of the same data. Sketch circle graphs to display data. Solve problems by collecting, organizing, displaying and interpreting data. 2 Paraphrase knowledge and skills required by the Standards: Investigation 1: Fund-Raising Fractions Students explore three components of understanding fractions: the visual model (fraction strips), word names for fractions, and symbols for fractions. The part-whole interpretation of fractions is developed. Students make fraction strips to study the progress toward a fund-raising goal. The aim is to focus on the meaning of such phrases as, “two thirds of the goal has been reached.” Investigation 2: Comparing Fractions The most important concept in understanding and using rational numbers is equivalence of fractions. This concept underlies operations with fractions, changing representations of fractions, and reasoning proportionally. The context of comparing fraction strips is used to motivate an investigation of equivalence and the creation of a number line that contains all of the information of the individual fraction strips. The idea of using benchmarks to estimate the size of fractions and to make comparisons is introduced. Investigation 3: Cooking with Fractions The context of cooking-parts of cups or other measures often called for in recipes, and the need to make multiples of a recipe, sets the stage for introducing students to different kinds of area models for fractions. The square and the rectangle are particularly useful areas because they are easy to subdivide and to shade. The circle is explored because of its use in data analysis and probability. Investigation 4: From Fractions to Decimals Students are introduced to decimal representations of fractions and explore the place-value interpretation of decimals. They investigate a 100-square grid and explore how it could continue to be subdivided to show 1000 parts or 10,000 parts. This process of subdividing and naming the new parts is very important mathematically; the underpinnings of the infinite process are met in this problem. The process will continue to help students understand equivalence of fraction and equivalence of decimals as well as too see the connections between fractions and decimals. Investigation 5: Moving Between Fractions and Decimals This investigation proposes a situation in which fractions with denominators larger than students’ fraction strips show must be compared. Students find decimal estimates for fractions using the visual model. They are asked to consider whether fractions or decimals are easier to compare. Sharing is used as a context to motivate the division interpretation of fractions, leading to a strategy for changing a fraction into a decimal. Calculators are used to do the computation, providing additional evidence that the division interpretation as a way to find decimal equivalents make sense. Investigation 6: Out of One Hundred By this time, students should feel comfortable with the meaning of fractions and decimals and be able to move back and forth between the two. Percents are now introduced as another form of representation. A database of information about cats is used as a context for understanding percent. Students are engaged in activities requiring them to move among fractions, decimals, and percents. Vocabulary Essential Nonessential decimal base ten number system denominator benchmark equivalent fraction unit fraction fraction numerator percent 3 Description of the Assessments that will be used to provide evidence of student learning that targets the standards: Description of the rubrics or criteria that will be used to assess student performance: 4 Learning Activities – Description of the learning activities that will develop the knowledge and skills required by the performance standard: Investigation 1: Fund-Raising Fractions Student Pages (p. 5-18) Teaching the Investigation (p. 18a-18k) Materials Problem For Students For the teacher All Calculators Transparencies 1.1 to 1.5 (optional) 1.2 8 ½” strips of paper (9 per student) 8 ½” fraction strips for the overhead projector 1.3 Fraction strips from Problem 1.2 8 ½” fraction strips for the overhead projector 1.4 8 ½” fraction strips for the overhead projector 1.5 Labsheet 1.5 (1 per student) 8 ½” fraction strips for the overhead projector, transparent centimeter ruler (optional; copy Labsheet 1.5 onto blank transparency film) Problem 1.1: Reporting Our Progress p. 5 Students write short reports describing the progress the sixth graders at Thurgood Marshall School have made toward their fund-raising goal. This activity lets you quickly assess your students’ understanding of fractions as parts of wholes. Problem 1.2: Using Fraction Strips p. 6-7 Students are challenged to make fraction strips by folding paper and then to use these strips to investigate the progress of the sixth-grade fund-raiser at various stages. Problem 1.3: Comparing Classes p. 8-9 Students explore comparing fractions with different wholes. Problem 1.4: Exceeding the Goal p. 10-11 Involves a fund-raiser in which the amount of money raised surpassed the goal; students must describe situations involving fractions greater than 1. Problem 1.5: Using Symbolic Form p. 12-13 Begins to develop the number-line model of fractions. Students label parts of their fraction strips and begin to think about the meaning of the symbolic representation of fractions. Applications, Connections, Extensions (ACE) pg. 14-17 Mathematical Reflections p. 18 5 Investigation 2: Comparing Fractions Student Pages p. 19-30 Teaching the Investigation 30a-30K Materials Problem For Students For the teacher All Calculators Transparencies 2.1 to 2.5 (optional), fraction strips for the overhead projector (optional; copy Labsheet 1.5 onto blank transparency film) 2.2 2.3 Labeled fractions strips from Labsheet 1.5 2.4 Index cards (optional) 2.5 Labeled fraction strips from Labsheet 1.5 A large number line to display in the classroom (see Problem 2.5) Problem 2.1: Comparing Notes p. 19 Students investigate competing claims of three teachers about their fund-raising progress. Two of the teachers’ claims turn out to be the same, raising the issue of equivalent fractions. Problem 2.2: Finding Equivalent Fractions p. 20-21 Students are asked to find other names for 2/3 and ¾ by comparing fraction strips. They use the patterns they discover to find equivalent fractions for 1/8, 2/5, and 5/6. Problem 2.3: Making a Number Line p. 22 Students transfer the fractions from all of their fractions strips onto a single number line. This helps them make sense of the number line and the numbers-fractions-used to label the points between whole numbers. Problem 2.4: Comparing Fractions to Benchmarks p. 23 Students use the benchmark values of 0, ½, and 1 to estimate the size of fractions and to compare fractions. Problem 2.5: Fractions Greater Than One p. 24-45 Students consider fractions greater than 1 on the number line. As students label points between 1 and 2, they should begin to think about the notation of density: between andy two fractions there is another fraction. Applications, Connections, Extensions (ACE) pg. 26-29 Mathematical Reflections p. 30 6 Investigation 3: Cooking with Fractions Student Pages (p. 31-38) Teaching the Investigation (p. 38a-38g) Materials Problem For Students For the teacher All Calculators Transparencies 3.1 to 3.2B (optional) 3.1 Labsheet 3.1 (1 per student) Transparencies of Labsheet 3.1 (optional) 3.2 Rulers or other straightedges Problem 3.1: Area Models for Fractions p. 31 Students explore the possible ways to cut a square pan of brownies into 15 equal-size large brownies. Problem 3.2: Baking Brownies p. 32-33 Challenges students to adjust a recipe to make enough brownies to serve a give number of students. Applications, Connections, Extensions (ACE): p. 34-37 Mathematical Reflection p. 38 Investigation 4: From Fractions to Decimals Student Pages (p. 39-52) Teaching the Investigation (p. 52a-52k) Materials Problem For Students For the teacher All Calculators, grid paper (provided as a blackline Transparencies 4.1 to 4.4(optional) master) 4.1 Labsheet 4.1 (1 per student), colored cubes or tiles(optional; 100 per group), Transparency 4.2D and transparency markers (optional; for sharing answers with class) 4.2 Labsheet 4.2 (1 per student) 4.3 Distinguishing Digit cards. (Provided as blackline master. Copy the cards, cut them out, and put them in envelopes marked with the puzzle number.) ACE Labsheet 4.ACE (1 per student) Problem 4.1: Designing a Garden p. 39-40 Students plan a 100 square-meter garden plot, arranging it to accommodate specified vegetables. In doing so, they explore representing fractional parts of a whole. 7 Problem 4.2: Making Smaller Parts p. 41-42 Students are encouraged to visualize what happens as a tenths grid is partitioned into increasingly smaller subdivisions, resulting in first a hundredths grid, then a thousandths grid, and finally a ten-thousandths grid. This promotes a sense of pattern as students think about what would be the next decimal place and how this decimal place can be represented graphically. Problem 4.3: Using Decimal Benchmarks p. 43-44 Benchmarks are revisited to relate fractions and decimals. Problem 4.4:Playing Distinguishing Digits p. 45 Students solve puzzles that help further their understanding of place value. The puzzles also provide opportunities for students to reason about digits using clues that connect to their work in Prime Time. Applications, Connections, Extensions (ACE) pg. 46-51 Mathematical Reflections p. 52 Investigation 5: Moving Between Fractions and Decimals Student Pages (p. 53-66) Teaching the Investigation (p. 66a-66K) Materials Problem For Students For the teacher All Calculators Transparencies 5.1A to 5.3(optional) 5.1 Labsheet 5.1 (1 per student), colored tiles or Transparency of Labsheet 4.2D other manipulatives (optional) 5.2 Labsheet 5.2(1 per student), straightedges, chart Transparency of Labsheet 4.2D paper, or a transparency of Labsheets 5.2 and transparency markers (optional; for recording answers to share with the class) 5.3 Labsheet 5.2(1 per student), straightedges, chart Transparency of Labsheet 5.2 paper, or a transparency of Labsheet 5.2 and (optional) transparency markers (optional; for recording answers to share with the class) Problem 5.1: Choosing the Best p. 53 Students make comparisons among three quantities that can be represented with fractions. Problem 5.2: Writing Fractions as Decimals p. 54-56 Student use fractions strips, including a hundredths strip, to estimate fraction and decimal equivalents. The goal is to help students focus on fractions and decimals as quantities that can be represented in more than one form. 8 Problem 5.3: Moving From Fractions to Decimals p. 57 This problem helps students understand why a fraction can be interpreted as an implied division and to use implied division to change fractions to decimal representations. Applications, Connections, Extensions (ACE) pg. 58-65 Mathematical Reflections p. 66 Investigation 6: Out of One Hundred Student Pages (p. 67-83) Teaching the Investigation (p. 83a-84) Materials Problem For Students All Calculators Transparencies 6.1 to 6.4B (optional) 6.1 Labsheet 6.1 6.2 Hundredths (from Labsheet 5.2), fraction strips Transparency of newspaper advertisements (optional), hundredths grids (from Labsheet 5.2) (optional) 6.3 Labsheet 6.3, hundredths strips (from Labsheet 5.2) 6.4 Hundredths grids (from Labsheet 5.1) ACE Labsheet 6.ACE Problem 6.1: It’s Raining Cats p. 68-72 Students use a database of information about cats to describe the portion of cats who possess some value of an attribute (for example, blue for eye color) as a fraction, a decimal, and a percent. Problem 6.2: Dealing with Discounts p. 73-74 Students consider different ways to express discounts. The goal is to highlight the informal language in daily use and connect it to different representations of quantities. Problem 6.3: Changing Forms p. 75 Students move among different forms of representation-sometimes starting with fractions, sometimes decimals, sometimes percents. Problem 6.4: It’s Raining Cats and Dogs p. 76 Students consider what it means to talk about a percent of a data set involving more than 100 items. Applications, Connections, Extensions (ACE) pg. 77-82 Mathematical Reflections p. 83 9 10