Bits and Pieces IC E by Le907F8Z


									                                            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
 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
 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number
 Convert fractions to decimals and percents and use these representations in estimations, computations, and
 Interpret and use ratios in different contexts.
 Compute and perform multiplication and division of fractions and decimals and apply these procedures to
    solving problems.

                                                 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
 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.

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

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.

                     Essential                                                 Nonessential
                      decimal                                              base ten number system
                   denominator                                                   benchmark
                 equivalent fraction                                             unit fraction

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:

       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)

      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
1.3                   Fraction strips from Problem 1.2                           8 ½” fraction strips for the overhead
1.4                                                                              8 ½” fraction strips for the overhead
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

                                    Investigation 2: Comparing Fractions
                                               Student Pages p. 19-30
                                         Teaching the Investigation 30a-30K

      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)
2.3                   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

       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

                                  Investigation 3: Cooking with Fractions
                                            Student Pages (p. 31-38)
                                      Teaching the Investigation (p. 38a-38g)

      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
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

       Problem 3.2: Baking Brownies p. 32-33
       Challenges students to adjust a recipe to make enough brownies to serve a give number of

       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)

      Problem                           For Students                                     For the teacher
All                  Calculators, grid paper (provided as a blackline           Transparencies 4.1 to 4.4(optional)
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.

       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)

      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.

       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)

      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
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

       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

       Applications, Connections, Extensions (ACE) pg. 77-82

       Mathematical Reflections p. 83


To top