# Bits and Pieces IC E by Le907F8Z

VIEWS: 15 PAGES: 10

• pg 1
```									                                            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
 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),
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
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
(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

```
To top