**BUTLER R-5 SCHOOL
DISTRICT**

**MATHEMATICS
CURRICULUM**

**MATH**

**LEVEL 4**

**Approved by the Board of Education, September 2004**

** **

**DESCRIPTION:**

The goal of fourth grade mathematics is to
further develop skills in computation, problem solving, critical thinking, and
basic math knowledge. Methods used to focus on these skills include practice on
traditional computational skills, manipulatives,
calculators, computers, modeling skills, and creating situations that require
students to use problem solving and critical thinking.

**RATIONALE:**

Mathematic skills are needed to function in
the real world. Children must learn to collect, understand, interpret, apply,
and evaluate information in order to be a good problem solver. The fourth grade
math curriculum reinforces the development of mathematical concepts. The
curriculum enhances problem solving skills and critical thinking skills through
real-life situations.

**Butler R-V School District**

**Curriculum**

**Fourth Grade**

**I. Problem Solving**

**Content Overview**

*In order to make sound decisions, one must be able to
recognize and pose problems as well as develop skills for investigating
possible solutions. The central focus of
the mathematical curriculum should be problem solving. The mathematics curriculum must provide
students the opportunity to solve relevant problems which require students to
work cooperatively, to use technology, and to experience the power and
usefulness of mathematics.*

* A given problem can be solved using
a variety of strategies. Many problems
can be derived from everyday situations.
Problem-solving is an activity which can be approached individually or
as a group. The process used to solve a
problem, not just the solution, is an important part of mathematics. A solution should make sense when restated
with the original problem. Having a
command of various problem-solving strategies helps develop confidence in the
use of mathematics.*

**All fourth grade
students should know**

1. A variety of problem-solving
strategies (e.g., make a list, draw a picture, look
for a pattern, act out the problem).

2. Computational strategies with whole numbers
(addition, subtraction, multiplication, and division)

3. When to use concrete objects, calculators,
computers, charts, graphs, etc., to organize and solve problems.

4. That mathematical problem-solving strategies can
apply to all disciplines and real-world problems.

**All fourth grade
students should be able to**

** **A. Work
individually and with others to use problem-solving approaches to investigate
and understand mathematical content (NCTM Standard 1; MO 1.6, 3.5, 3.6, 4.6)

B. Use problem-solving strategies to construct
meaning from mathematical tasks (NCTM Standard 1; MO 1.6, 3.7)

C. Recognize and define theoretical and actual
problems encountered in everyday life, mathematical situations, and the various
disciplines (NCTM Standard 1; MO 3.1, 3.4)

D. Develop and apply strategies to predict, prevent,
and solve a wide variety of problems (NCTM Standard 1; MO 3.2, 3.3)

E. Verify, interpret, and evaluate whether or not a
solution addresses the original problem (NCTM Standard 1; MO 2.2, 3.6, 3.7,
3.8)

F. Select and apply appropriate mathematical tools
and technology to solve problems (NCTM Standard 1; MO 2.7)

**CORE
COMPETENCY:**

G Apply
problem-solving strategies

**KEY
SKILLS:**

** G1 Restate, illustrate,
or dramatize the meaning of a problem.**

** ****G2 Create and solve word problems.**

** G3 Search for and evaluate alternative
methods of solving a given **

**CONCEPT
ANALYSIS:**

Problem solving should be the
central focus of the mathematics curriculum at all levels. Problem solving is
not only a goal of any mathematics curriculum, it is
also the process that should be used to develop the concepts and skills
necessary to mathematical literacy.

Problem situations should arise
naturally from the activities and interests of the students. A *problem*
is defined as any situation in which the solution is not readily known. If the
"problem" fits the same algorithm as all the others on a given page,
it may provide good computation practice, but it does not meet the definition
of a problem.

While many skills are valuable in
solving problems, these skills do not need to be mastered prior to attempting
problems; rather, they should be practiced through problem solving. Solving
problems, especially with the help of others through small-group activities,
helps students develop confidence in their mathematical power. It also replaces
the drudgery of mathematics with fun and excitement.

**INSTRUCTION:**

** ****G1** Help the class
understand a given word problem through the use of putting the problem in own
words, drawing pictures, or acting the problem out.

**G2** Have students bring in
problems from their personal experiences. Permit students to use whatever tools
(manipulatives, books, calculators, etc.) are
necessary for them to solve the problem.

**G3** Discuss the following
strategies:

a) guess
and test. b) draw a picture or diagram. c) make an organized list. d) make a
table or graph. e) work backwards. f) look for a pattern. g) use objects.
h) act out the problem.

**ASSESSMENT
STRATEGIES:**

Constantly monitor students' skills
and attitudes toward solving problems. Keep a variety of problems before
students at all times-some with relatively quick solutions and others requiring
longer thought processes.

Encourage students to work at
problems. Reward perseverance. Real problems are seldom ever solved in three
minutes.

Use games such as Nim^{TM},
dominoes, checkers, backgammon, etc., to build thinking strategies and logical
processes. Discuss "winning" strategies with the class.

**G1** Put the students in
groups. Each group will restate, illustrate, or dramatize the meaning of a
given problem.

**G2** Each group will create a
word problem, present it to the class, and help the class solve the problem.

**G3** Give the students a
problem than can be solved using more than one strategy. The students will work
in groups and discover the various strategies.

**RESOURCES:**

**G2** Manipulatives;
calculators.

** ****G3
**Graph
paper; markers; ruler.

**CORE
COMPETENCY:**

H Solve
problems in consumer situations

**KEY
SKILLS: *State
Tested**

** *H2 Solve problems involving money
management.**

**CONCEPT
ANALYSIS:**

Problem solving should be the
central focus of the mathematics curriculum. It is a primary goal of all
mathematics instruction as well as an integral part of all mathematical
activity. Problem solving is not a distinct activity but a process through
which all concepts and skills can be learned.

A major goal of problem-solving
instruction is to enable children to develop and apply strategies to solve
problems. Strategies include using manipulative materials, trial and error,
making an organized list or table, drawing a diagram, looking for a pattern,
and acting out a problem. Recognizing and applying a
computational algorithm should not be the only way to arrive at a
"correct" answer and should not be valued over other more creative
means. Students should be exposed to problems that do not lend themselves
to an algorithmic solution, as well as to problems that provide computational
practice within context.

**INSTRUCTION:**

Review objective H-1. Present money
management problems modeling the strategy used to solve each problem. Put
students in groups. Give each group a problem to solve. Each group will
demonstrate their solution for the class.

**ASSESSMENT
STRATEGIES:**

Give each student a money management
problem. The student will solve on paper.

Test - File H-2.

**TEST
CONTENT SPECIFICATION:**

__Component
Skills:__

A. Solve money problems which may be
solved with computational algorithms.

B. Solve money problems which may
require nonstandard strategies.

__Specification:__

Students are given a word problem
involving money. Addition problems should have sums less than $1,000 and fewer
than four addends. Subtraction problems should involve minuends and subtrahends
of less than $1,000. Divisors and multipliers will be one digit. Problems
involving dollars and cents should be written as $xx.xx, and problems
involving cents should be written using the cents sign. The two forms may be
mixed within a problem. Students may solve the problem by drawing, counting, or
other methods.

__Sample
Item:__

1. Terry has $1.10 in coins. What is
the greatest number of coins he can have if he has at least two different kinds
of coins?

A. 2

*B. 106

C. 109

D. 110

**RESOURCES:**

Real or play money; File - H-2.

**CORE
COMPETENCY:**

H Solve
problems in consumer situations

**KEY
SKILLS: *State
Tested**

** *H4 Solve standard multistep problems.**

**CONCEPT
ANALYSIS:**

Problem solving should be the
central focus of the mathematics curriculum. It is a primary goal of all
mathematics instruction as well as an integral part of all mathematical
activity. Problem solving is not a distinct activity but a process through
which all concepts and skills can be learned.

A major goal of problem-solving
instruction is to enable children to develop and apply strategies to solve
problems. Strategies include using manipulative materials, trial and error,
making an organized list or table, drawing a diagram, looking for a pattern,
and acting out a problem. Recognizing and applying a
computational algorithm should not be the only way to arrive at a
"correct" answer and should not be valued over other more creative
means.

**INSTRUCTION:**

** **Review objective G-1.
Model solving multistep problems. Working
in groups, the students practice solving multistep problems.

**SUGGESTED
ASSESSMENT:**

** **Test
- File H-4.

**TEST
CONTENT SPECIFICATION:**

__Component
Skills:__

A. Solve multistep word problems
using only one operation.

B. Solve multistep word problems
requiring more than one operation.

__Specification:__

Students are given a multistep word problem
which may be solved with addition, subtraction, or multiplication, or some
combination of those operations. Some steps may require only mental
computation. Numbers are restricted to no more than four digits; regrouping may
be required. Addition problems should have no more than three addends. Factors
in multiplication problems should be single digit numbers. Students may solve
the problem with drawing, counting, or other methods.

__Sample
Item:__

1. Jamie bought a pen for $1.29 and
a note pad for $1.79. She gave the clerk $5. How much change should she
receive?

*A. $1.92

B. $1.98

C. $2.92

D. $3.01

**RESOURCES:**

** **File H-4.

**CORE
COMPETENCY:**

H Solve
problems in consumer situations

**KEY
SKILLS: *State
Tested**

** *H5 Solve nonstandard problems.**

**CONCEPT
ANALYSIS:**

Problem solving should be the
central focus of the mathematics curriculum. It is a primary goal of all
mathematics instruction as well as an integral part of all mathematical
activity. Problem solving is not a distinct activity but a process through
which all concepts and skills can be learned.

A major goal of problem-solving
instruction is to enable children to develop and apply strategies to solve
problems. Strategies include using manipulative materials, trial and error,
making an organized list or table, drawing a diagram, looking for a pattern,
and acting out a problem. Students should practice solving problems that do not
lend themselves to an algorithmic solution.

**INSTRUCTION:**

Help students solve nonstandard
problems by drawing pictures, counting, reasoning, and other methods. A good
source for problems is __Problem-Solving Experiences__ by Addison Wesley.

**SUGGESTED
ASSESSMENT:**

Test - File H-5.

**TEST
CONTENT SPECIFICATION:**

__Component
Skills:__

Same as Key Skill.

__Specification:__

Students are given a word problem
that does not require a computational solution. Situations should be readily
understandable by grade 4 students. They may solve the problem by drawing,
counting, reasoning, or other methods.

__Sample
Item:__

1. If eggs were sold in cartons of
10 rather than 12, how many more cartons would be needed to pack 60 eggs?

*A. 1

B. 2

C. 3

D. 12

**RESOURCES:**

File - H-5; __Problem-Solving
Experiences__ by Addison Wesley.

**Sample Learning Activities
for Problem Solving**

** **Given objects and/or pictures that have a variety of attributes such as
shapes, colors, and sizes, devise a rule for sorting and sort the objects
and/or pictures using that rule. Write a paragraph to explain the reasoning used
to select a particular rule.

Given a menu from a fast food
restaurant, list five ways that a friend and you could eat for $5.00. Compute the cost of each of the five ways.

Devise a strategy for determining the possible
rectangles that have a distance around (perimeter) of 20 centimeters. Find the length and width for each of the
rectangles. Organize and explain your
results.

If the teacher is holding six coins worth 42 cents,
what coins is he/she holding? Explain
your reasoning.

boxes of the same height are
stacked on top of each other in a storeroom.
A first set of boxes is eight inches high, a second set of boxes is 12
inches high, and a third set of boxes is 16 inches high. design a strategy to
determine the height when the tops of all the boxes are even. The height of the storeroom is ten feet. Is there more than one time when the tops of
the boxes would be the same height? Justify
your answer.

Given an advertisement from a local toy store, write
a story problem that could be solved using both addition and subtraction. Solve the problem.

**II. Communication**

**Content
Overview**

*Communication is a major focus for mathematical
activities. Through reading, writing,
listening, viewing, and speaking, students come to understand mathematics.
Mathematical communication includes mental, verbal, written, concrete,
pictorial, graphic, and algebraic representations. The effective use of communication is
necessary for success in the classroom and workplace. "The mathematics curriculum should
include the continued development of language and symbolism to communicate
mathematical ideas." (NCTM, 1989)*

* Mathematical ideas and concepts may be communicated
verbally and non-verbally. Communication
helps relate the vocabulary of mathematics to the symbolism. Ideas and information may be expressed
through various forms, including pictures, graphs, mental representation, manipulatives, and speech.
Exploration, investigation, description, and explanation of the
occurrences within an activity promote communication. Probing questions open windows of opportunity
to engage in problem-solving strategies.*

**All fourth grade
students should know**

1. That the language of
mathematics may be used in reading, writing, listening, and speaking.

2. That mathematical ideas may
be represented by visual models.

3. That mathematical symbols
represent real-world situations.

4. That information may be
organized in a variety of ways.

**All fourth grade
students should be able to**

** **A. Relate
physical materials, pictures, and diagrams to mathematical ideas (NCTM Standard
2; MO 2.1)

B. Organize information into useful forms, such as
verbal, symbolic, or graphic (NCTM Standard 2;MO 1.8)

C. Apply information-processing skills to reflect on
and clarify thinking about mathematical ideas and situations (NCTM Standard 2;
MO 2.2)

D. Communicate the relationship between everyday
language, mathematical language and symbols (NCTM Standard 2; MO 2.3)

E. Demonstrate the ability to select and apply
appropriate strategies such as representing, discussing, reading, writing,
listening, and using technology in mathematics as a vital element of learning
and using mathematics (NCTM Standard 2; MO 2.2)

**CORE
COMPETENCY:**

A Demonstrate
an understanding of numbers

**KEY
SKILLS:**

** A5 Represent and describe mathematical
relationships.**

**CONCEPT
ANALYSIS:**

Mathematical relationships are those
numerical patterns that consistently hold true. Children often discover these
patterns, and teachers should capitalize on these discoveries. Representing
these discoveries with numbers and symbols gives learners a sense of the power
of mathematics and of their own mathematical ability. Use of properties
introduced earlier should be continued and extended. The need for and the use
of the parenthesis should be stressed at this level. The distributive property
and its role in tying the operations of addition and multiplication together
should be carefully developed. Its application in mental computation should be
demonstrated and practiced, for example, 99 x 16 = 16 x (100 - 1) = (16 x 100)
- (16 X 1). Learning the properties for the sake of being able to name them is
non-productive; learning the relationship in order to use it to simplify work
or to generalize a procedure shows students the power of mathematics.

**ASSESSMENT
STRATEGIES:**

Give students multiplication
problems and ask them to explain how to do the problem mentally or in the
"easiest" way. Be flexible in scoring by allowing for alternative
strategies.

The following is an example:
"Explain the easiest way to multiply 12 times 98 doing as much of the work
in your head as possible and give the product." Students might say 12 x
100 - 12 x 2 = 1176, or 10 x 98 + 2 x 98 = 1176.

Ask children to discuss/write about
how addition, subtraction, multiplication, and division are different.
Encourage them to give examples of properties that work one way with one
operation and a different way with another.

**CORE
COMPETENCY:**

F Use
statistical techniques and interpret statistical information

**KEY
SKILLS:**

** F1 Collect, organize,
and classify data.**

**CONCEPT
ANALYSIS:**

Collecting, organizing, and
classifying data are increasingly important skills in a society based on
technology and communication. These processes are particularly appropriate for
students because they offer opportunities for inquiry, and they can be used to
solve interesting problems and to represent significant applications of
mathematics to practical questions.

Students need to recognize that many
kinds of data come in many forms, and that collecting, organizing, classifying,
and interpreting data can be done in many ways. Along with traditional picture
graphs, bar graphs, pie charts, and line graphs, students can use a variety of
plots, such as stem-and-leaf plots and box-and-whisker plots. Statistics is
more than reading and interpreting graphs. It is describing and interpreting
the world around us with numbers; it is a tool for solving problems.

In the fourth grade, students should
be able to use scales representing units up to and including 10. Computer
graphics programs make a wide variety of explorations accessible to students. A
class or group project conducted over a period of time enables the students to
make predictions and to modify them as more data are collected. Teachers should
monitor from a distance and guide with thoughtful questions.

**INSTRUCTION:**

** **Model the following
strategies:

a) sorting
information. b) relating information. c) combining information. d) ordering
information. e) logically linking ideas together.

Make a graph from data.

**ASSESSMENT
STRATEGIES:**

Give students graphs from newspapers
or magazines. Ask them to formulate questions that can be answered by
interpreting the data on the graph.

Give students graphs from newspapers
or magazines. Ask them to consider alternative ways to display the data and to
discuss the merits of each alternative.

Ask students to submit questions of
interest to them that require data to answer. Discuss methods of collecting the
data needed, do the collection, and organize the results.

Given data, the student will
organize it and present it in a table or graph form.

Test - File F-1.

**RESOURCES:**

Graph paper; File
F-1.

**CORE
COMPETENCY:**

F Use
statistical techniques and interpret statistical information

**KEY
SKILLS: *State
Tested**

** *F2 Construct, read, and interpret
displays of data.**

**CONCEPT
ANALYSIS:**

Collecting, organizing, describing,
displaying, and interpreting data, as well as making decisions and predictions
on the basis of that information, are becoming increasingly important skills in
a society based on technology and communication. These processes are
particularly appropriate for young children because they can be used often to
solve problems that are inherently interesting, that represent significant
applications of mathematics to practical situations, and that offer rich
opportunities for mathematical inquiry.

Collecting and graphing data from
the children themselves is the way to begin the study of statistics. Graphing
the kinds of pets owned by students, lengths of little fingers, or favorite
television characters provides data meaningful to students. This data can be
used to experiment with different types of displays and to answer a variety of
questions.

**INSTRUCTION:**

** **Review strategies
taught in objective F-1.

Present graphs and tables. Help the
students read and interpret the information.

**ASSESSMENT
STRATEGIES:**

** **Given data, the
student will organize it and present the data in a table or graph.

Given a table or graph, the student
will interpret orally the data.

Test - File F-2.

**TEST
CONTENT SPECIFICATION:**

__Component
Skills:__

A. Select bar graphs accurately
representing given data.

B. Select pictographs accurately
representing given data.

C. Construct displays of data.

__Specification:__

Students are given data in an
organized or tabular form. They are asked to identify the bar graph or
pictograph which accurately represents that information. Examples of
inaccuracies include, but are not limited to, misrepresented data and unequal
increments.

**RESOURCES:**

Graphs; tables;
charts; time lines; File - F-2.

**Sample Learning
Activities for Communication**

Use manipulatives (e.g.,
color tiles, Unifix cubes) to demonstrate the meaning
of addition, subtraction, multiplication, and division.

Write, tell, or illustrate a story using mathematical language to
describe real-world situations.

Investigate the number of rectangles that can be made
using a set number of color tiles.
Communicate your results.

Design a display for time, temperature, or rainfall
amounts.

**III. Reasoning**

**Content
Overview**

*Reasoning is a process basic to all disciplines. Reasoning allows students to understand that mathematics
makes sense and is a way of thinking.
Mathematics is more than a body of facts. Mathematical reasoning cannot develop in
isolation. Students "need to know
that being able to explain and justify their thinking is important and that how
a problem is solved is as important as its answer." (NCTM, 1989)*

* Students should "justify their solutions,
thinking processes, and conjectures in a variety of ways. Manipulatives and
other physical models help (students) relate processes to their conceptual underpinnings
and give them concrete objects to talk about in explaining and justifying their
thinking." (NCTM, 1989) As the depth and complexity of content increases,
logical reasoning (deductive and inductive)becomes
more important.*

* Reasoning in mathematics involves informal
conjectures, validations, explanations and justifications of one's thinking
processes. Simple logic, a vital part of
reasoning, is an important aspect of problem solving, demonstrating that mathematics
makes sense.*

**All fourth grade students
should know**

1. That objects/numbers may be used in more than one
way to determine or construct relationships between and among them.

2. That results must be verified.

3. That data may be organized in a variety of forms
for looking for patterns.

4. Geometric and number properties.

**All fourth grade
students should be able to**

** **A. Draw
logical conclusions about mathematics (NCTM Standard 3; MO 3.5)

B. Use models, known facts, properties, and
relationships to explain their thinking (NCTM Standard 3, MO 4.1)

C. Justify answers and solution process in an
organized and convincing way (NCTM Standard 3; MO 1.8, 3.4, 3.7, 4.1)

D. Use patterns and relationships to analyze
mathematical situations (NCTM Standard 3; MO 1.6)

**CORE
COMPETENCY:**

B Apply
the basic operations in computational situations

**KEY
SKILLS:**

** B2 Informally use the
commutative and associative properties of addition and multiplication, the zero property of addition, and
the multiplication properties of zero and one.**

**INSTRUCTION:**

Discuss and demonstrate examples of
addition and multiplication properties.

1). Associative or grouping property
(3 + 2) + 4 = 9; (6 x 4) x 2 = 48; 6 x (4 x 2) = 48;

3
+ (2 + 4) = 9

2). Commutative or order property 8
x 3 = 24; 3 x 8 = 24; 7 + 5 = 12; 5 + i7 = 12

3). Identity property or zero
property 5 x 1 = 5; 3 + 0 = 3; 4 x 0 = 0

**CORE
COMPETENCY:**

C Estimate results and judge
reasonableness of solutions.

**KEY
SKILLS:**

** C1 Use estimation
strategies and mental computation to produce reasonable estimates.**

**CONCEPT
ANALYSIS:**

Children begin school as good
problem solvers and with estimation skills that have evolved from their
experiences. Formal schooling must recognize this learning and build on it. At
this grade level, students should be taught specific strategies to aid them in
computational estimation. Flexible and front-end rounding, compatible numbers,
special numbers, and clustering are examples of frequently used strategies.
Lessons devoted to estimation are necessary and should be continuously integrated
into all computation and problem-solving lessons.

Mental computation produces an exact
answer, as opposed to an estimate. Most estimation involves some mental
computation. Practice with procedures efficient for mental computation helps
students become good estimators.

**INSTRUCTION:**

Discuss and show examples of
rounding and front-digit estimation with adjustment step to produce reasonable
estimates.

Ex: 27 à 30 27

__26 ____à____ 30__ __26__

60 40 (front digits)

Since 7 + 6 is about 10, __adjust__
to 40 + 10 = 50. Actual 27 + 26 = 53.

Have students use counters on a
number line labeled 0-100, by 10's. Place counter on number line to show
indicated number, then move it to the closest 10 to illustrate rounding.

Stress that 5
always rounds up to the next higher value.

**ASSESSMENT
STRATEGIES:**

Present problems on overhead
transparencies to control the time. Ask students to respond after a
three-second exposure. Stage the problems to encourage the various strategies,
but allow student decision and justification of the "best" method.

Constantly encourage students to
estimate solutions before working a problem and to consider the reasonableness
of the result. Informally assess the skills demonstrated as they work through
problems by noting areas of strength and areas needing more development.
Consider skills in justifying their response as well as use of specific
strategies.

Do "minds only" drills to
assess mental computation skills. Give or display the problem, allow think
time, then signal pencil time and allow only a short time to write down the
answer before calling for "minds only" again.

Have students use counters on a
number line, labeled 0-100, by 10's to indicate rounded answer when given
numbers.

**RESOURCES:**

Number line;
counters.

**CORE
COMPETENCY:**

C Estimate results and judge
reasonableness of solutions.

**KEY
SKILLS:**

** C2 Use estimation
strategies and mental computation to determine if an answer is reasonable.**

**CONCEPT
ANALYSIS:**

Estimation skills must become an integral
part of every mathematics program. Research about students' thinking on
estimation has provided new insight into the learning and teaching of
computational estimation.

Mental computation skills are
necessary to be a good estimator. The fact that calculators and computers are
supplanting the need for a high degree of skill in pencil/paper computation
increases the need for good estimation skills.

**INSTRUCTION:**

List column of two
digit numbers on board. Model rounding, front-digit
with adjustment estimation, and mental computation of numbers for reasonable
answers. Use calculators to check if these answers are reasonable.

Assign a numerical value to each
letter of the alphabet (A-1, B-2) and have each child add up the value of their
first name, using each method.

**ASSESSMENT
STRATEGIES:**

Constantly encourage students to
consider and to justify the reasonableness of their answers, orally and in
writing. Informally assess the skills demonstrated as they work through
problems by noting areas of strength and areas needing more development.

Give students problems complete with
the answers and ask them to evaluate the reasonableness (rather than accuracy)
of the answer. Time some of these tests to encourage rapid response. In another
test, allow more time but ask students to justify their answers or to describe
the process by which they arrived at their conclusions. Be flexible in scoring
by allowing for a good justification of a response that might not be
"reasonable" in the strictest sense.

After a problem-solving assignment,
have students exchange papers and judge the answers of their peers for
reasonableness. Require justification, and score the results. This would be
especially effective for open-ended problems-problems with potentially
different correct answers.

Column addition
worksheets using estimation and mental computation to be verified with use of
calculators.

**RESOURCES:**

Calculators;
alphabet/numerical chart.

**Sample Learning
Activities for Reasoning**

Given the following statements,
find the "secret number":

(a) The number is odd.

(b) The number is less than 40 and greater than 30.

(c) the number is not 33.

(d) the sum of the digits is
between 5 and 10.

Prepare a set of three clues (similar to the ones presented
in activity #1) that will describe a "secret number".

What could be the next number in the sequence,
"2, 4, ..."?
Explain your answer. Can you
justify any other numbers? Explain all
possibilities you find.

Determine how much food a given pet would eat in a
year and estimate the total cost.
Explain how you reached your answer.

Play "Guess My Rule" with a set of
attribute blocks or links, where one person thinks of an attribute to label a
string loop. Determine the chosen
attribute by placing the blocks in or out of the circle.

Play card games such as "War" or
"Double War". Tell how you
decided who won the game.

Play strategy games such as "Tic-Tac-Toe," NIM games, checkers, chess, and other games
requiring the use of or development of winning strategies. Explain the winning strategies for the game.

**IV. Connections**

**Content
Overview**

*The need for all students to incorporate prior
knowledge with real-world situations and integrate concepts across the
curricular areas is vital. Furthermore,
focusing on the relationships between mathematical ideas and modeling, students
are able to realize the powerful role mathematics plays in other
disciplines. Interaction within and
among mathematics and other disciplines makes the study of mathematics relevant.*

* Mathematical connections allow construction of
bridges between the concrete and the abstract which link conceptual and
procedural knowledge. These bridges
invite students to informally explore, conjecture, and develop mathematical
generalizations.*

* The ability to make connections enables students to
view problem situations from multiple perspectives. Insightful connections make it possible to
solve problems in creative and simply elegant ways. Connections allow students to view
mathematics as dynamic and evolutionary in response to the needs of a
technological world. Connections are a
powerful tool for problem solving and enable students to develop a deeper
appreciation of the consistency and beauty of mathematics.*

* Mathematical and real-world situations can be
modeled. A problem solved yesterday
could have a relationship to a problem being solved today. Mathematics is used in all subject areas.*

**All fourth grade
students should know**

1. That problems may be looked
at in more than one way.

2. That mathematics is used in
other subject areas.

3. That mathematics is used in
the real world.

**All fourth grade
students should be able to**

** **A. Link
concepts to student-generated procedures (NCTM Standard 4; MO 1.6, 1.10, 2.2)

B. Relate various representations of concepts or
procedures to one another using a variety of methods, forms, and technologies
(NCTM Standard 4, MO 1.6, 2.7)

C. Recognize relationships among different topics in
mathematics (NCTM Standard 4, MO 1.6, 1.10)

D. Use mathematics in other curriculum areas and in
daily living (NCTM Standard 4; MO 1.10, 4.7)

**CORE
COMPETENCY:**

A Demonstrate
an understanding of numbers

**KEY
SKILLS: *State
Tested**

** *A6 Graph points representing fractions on a
number line with denominators of
2, 3, 4, 5, 8, or 10.**

**CONCEPT
ANALYSIS:**

All work with fractions in the early
grades should involve fractions that occur frequently in everyday life. Initial
instruction needs to emphasize oral language and to demonstrate the mathematical
relationship represented by one-half, one-fourth, etc. As students progress in
their mathematical thinking, emphasis should be placed on representing and
comparing relative sizes.

Graphing
fractions in a variety of formats and modes helps students establish the
concepts of size, increments, and relationships. Fraction strips, commercial or
cut from paper, help them make the transition from modeling to graphing on a
number line. The concept of a unit and its subdivision into equal parts is
fundamental to understanding fractions, whether the quantity to be divided is a
rectangular candy bar or a handful of jelly beans. Instruction and practice
should include improper fractions and fractions not in lowest terms, although
these will not be tested at the state level in grade 4.

Students need opportunities to apply
these concepts to represent and solve problems involving measurement, geometry,
and statistics. For example, they might be asked: "Where would you be if
you were one-third of the distance between Houston and St. Louis?"
"If two-thirds of the red balls is the same
amount as three-fifths of the blue balls, are there more red balls or blue
balls?"

**INSTRUCTION:**

** **Explain that
fractions can be graphed on a number line. Students are given a number line with
an indicated point and are asked to select the fraction represented by the
point or to select the point representing a given fraction. Mixed numbers may
be used, but not improper fractions. Fractions will be in lowest terms.

**TEST
CONTENT SPECIFICATION:**

__Component
Skills:__

A. Select the fraction represented
by a given point on a number line.

B. Select the point on a number line
representing a given fraction.

__Specification:__

Students are given a number line
with an indicated point and are asked to select the fraction represented by the
point or to select the point representing a given fraction. Fractions may have
denominators of 2, 3, 4, 5, 8, or 10. Mixed numbers may be used, but not
improper fractions. Fractions will be in lowest terms. Answers equivalent to
the correct answer will not be used as distractors
(2/4 will not be a distractor if 1/2 is the correct
answer).

**CORE
COMPETENCY:**

C Estimate results and judge
reasonableness of solutions.

**KEY
SKILLS:**

** C4 Use a calendar for making plans and
organizing activities.**

**INSTRUCTION:**

** **Discuss a twelve
month calendar, number of days in each month, and number of days in a week.

Give each child a calendar for a
month and have the child fill in activities, assignments etc., for the month.

Make a school year plan of
activities for the present grade including tests, field trips, parties,
birthdays, holidays, etc.

**ASSESSMENT
STRATEGIES:**

** **The student can list
in order the twelve months from memory.

Given a calendar, the student will
answer questions involving date related situations.

**RESOURCES:**

** **Lunch
menu; classroom calendar (list events, birthdays, etc.); day planner (list
assignments due).

** **

**CORE
COMPETENCY:**

C Estimate results and judge
reasonableness of solutions.

**KEY
SKILLS:**

** C5 Use the understanding of the clock to
solve time related activities.**

**INSTRUCTION:**

** **Make clocks with
index cards and use construction paper for hands. Use brads to fasten the
hands. Each child can demonstrate a specific time. Make a time schedule for the
day. Discuss how long until lunch, time for special area, time for recess, etc.

**ASSESSMENT
STRATEGIES:**

** **Use clocks made in
class. Each child will demonstrate the appropriate time that is asked by the
teacher.

**RESOURCES:**

** **Classroom clock;
clocks made by students.

**CORE
COMPETENCY:**

D Apply
the concept of measurement to the physical world

**KEY
SKILLS:**

** D1 Choose an appropriate unit of measure (metric
or English) to measure length, area, capacity, weight, or mass.**

**CONCEPT
ANALYSIS:**

Lynn Steen in *On The Shoulders Of Giants* says, "Learning how to
measure is the beginning of numeracy." Measurement shows students the
usefulness of mathematics in everyday life. It can help them develop many
mathematical concepts and skills such as fractions and decimals. By
establishing an understanding of the purpose of measurement and the various
attributes often requiring measuring, students will have a firm foundation that
will enable them to use any measurement system. The process of measuring is
identical for any attribute within any system: choose a unit with the same
attribute as that which is being measured, compare that unit to the object, and
report the number of units. The number of units can be determined by counting,
by using an instrument, or by using a formula.

Students must realize that an
appropriate unit depends upon the size of the object and/or the precision
desired. They need to develop a "feel" for the relative sizes of the
commonly used metric and English units.

**INSTRUCTION:**

** **Provide various
objects. Ask students to decide how that object would be measured. Then
demonstrate how to measure that object.

Students measure objects themselves,
determining which tool to use.

**ASSESSMENT
STRATEGIES:**

Ask students to construct/select the
type of unit/tool needed to measure a particular attribute.

Incorporate this objective into a
bigger problem-solving picture. Selecting the appropriate unit should not be an
end in itself; rather, it should be only one step in the process of using a
measurement to solve a problem.

**RESOURCES:**

Balance scale, spring scale, liquid
measurement tools, yard stick, meter stick, graduated cylinders, measuring
cups.

**CORE
COMPETENCY:**

H Solve
problems in consumer situations

**KEY
SKILLS:**

** H1 Determine the value of a set of coins
and bills in amounts up to $10.**

**CONCEPT
ANALYSIS:**

Since our society "runs"
on money, it is important for every citizen to use money effectively. This
skill is necessary for one to buy and/or sell and to make intelligent decisions
about those transactions.

It is easy to assume that since
everyone handles money and buys objects, everyone understands the value of
coins, can count change, and can use money wisely. This, unfortunately, is not
so. Care must be taken to develop these skills in students to ensure capable
functioning today and in the future.

Actual coins and bills should be
used. Textbook activities and worksheets are never adequate to develop the
understanding necessary to handle situations involving money confidently.

Our
monetary system also provides rich opportunities for problem solving and mental
arithmetic. Students can determine the number of coins needed to pay for an
item; they can use their knowledge of multiples to determine the value of a
number of nickels or dimes; or they can find coins whose value is more or less
than a given value.

**INSTRUCTION:**

Using play __or__ real money discuss the value of coins and bills. Illustrate and model
counting change.

**ASSESSMENT
STRATEGIES:**

Prepare collections of coins and
bills for each student or small group. Ask each student to determine the value
of the money in her collection and then find another student with the same
value and verify.

Given a certain set of coins, ask
the students in groups to determine which amounts cannot be made with the given
coins.

Each student will identify the value
of the coins and bills in amounts up to $10 using real or play money.

**RESOURCES:**

Play money; real money.

**Sample Learning
Activities for Connections**

** **Using only
a pair of scissors, cut a piece of paper shaped like a
square into eight equal pieces. Explain
your process. Use another square to cut
into twelve equal pieces. Can it be
done? why or why
not? Would it be possible to cut a
square into 15 equal pieces? Explain
your answer.

Given a floor plan of your school, find the shortest
walking path from your classroom to the cafeteria. then find the
shortest path to the cafeteria that goes by a rest room.

Your school's parent group has donated funds to
purchase new playground equipment.
Design a model of the equipment and the playground.

Create a timeline for the history of the school.

Collect or record data gathered in other disciplines.

Explore numeric and geometric patterns.

After attending a career/hobby day, discuss how
mathematics is used in the careers and hobbies you investigated.

Identify real-world situations in which mathematics
is used.

**V. Number Sense**

**Content
Overview**

*Through number sense it is possible to gain a solid
understanding of mathematics in the real world.
The ability to understand numbers and how they interrelate empowers
people to feel comfortable with the use of numbers. Utilizing number sense enables students to
put the world in perspective and realize mathematics is more than a
paper-and-pencil activity. Understanding
numbers allows students to select the type of answer needed and a method
necessary to arrive at a solution.*

* Numbers should make sense and be used in multiple
ways. Numbers can quantify, identify,
locate, denote a specific object in a collection, or be used to name or to
measure. Numbers have relative
magnitude. Physical models can be used
to help make sense of number operations.*

**All fourth grade
students should know**

1. Counting and grouping strategies.

2. Mental computation and estimation strategies.

3. Place value.

4. Basic computation facts (addition, subtraction,
multiplication, and division) with whole number.

5. Addition and subtraction of fractions with like
denominators.

6. U.S. customary and metric units of measure.

7. The appropriate use of calculators.

**All fourth grade
students should be able to**

** **A. Model, explore, develop and explain number operations for whole
numbers (NCTM Standard 7; MO 1.6, 2.1, 3.3)

B. Use technology to explore numbers (NCTM Standard
6; MO 1.4, 1.6, 2.7)

C. Use physical models and real-world experiences to
construct number meanings (NCTM Standard 5; MO 1.10, 2.3, 4.1)

D. Demonstrate an understanding of our numeration
system by relating counting, grouping, and place value concepts (NCTM Standard
6; MO 1.6, 3.6, 4.1)

E. Utilize number sense to develop number meanings
and explore number relationships (NCTM Standard 6; MO 1.6, 3.3)

F. Use a variety of mental computation and estimation
strategies to solve specific problems (NCTM Standard 5; MO 1.10, 3.3, 4.1)

G. Demonstrate an understanding of the attributes of
length, capacity, weight, area, volume, time, temperature, and angle (NCTM
Standard 5; MO 1.6, 4.1)

H. Make and use standard and nonstandard measurements
in problems and everyday situations (NCTM Standard 5; MO 3.2, 3.3)

I. Explore the concepts of fractions, mixed numbers,
and decimals and be able to apply them to problem situations (NCTM standard 12;
MO 1.6, 3.2, 3.3, 4.1)

**CORE
COMPETENCY:**

A Demonstrate
an understanding of numbers

**KEY
SKILLS:**

** A1 Read and write whole numbers through 7
digits.**

** A1a.
Compare numbers through five digits
using less than, greater than, and equal to.**

**INSTRUCTION:**

**A1** Write whole numbers on
board, explaining how to divide a number with commas into periods. Read whole
numbers together as a group.

Using place value charts, students
place a given whole number on chart and read the number.

** ****A1a** Basic introduction:
Use base ten blocks or other manipulatives to compare
2 (1-2 digit) numbers. Introduce <, >, = signs and how to use them.

Compare 2 written numbers placed on
a place value chart by comparing digits in each place. (2,421 - 2,241)

Write comparisons using <, >,
= signs. (2,421 > 2,241)

**ASSESSMENT
STRATEGIES:**

Have individual students read large
numbers aloud. Have students write numbers given verbally. Students place a
given number on place value chart.

**RESOURCES:**

Place value charts (student made).** **

** **

**CORE
COMPETENCY:**

A Demonstrate
an understanding of numbers

**KEY
SKILLS:**

** A2 Extend understanding
of whole numbers to fractions and decimals.**

** A2a. Read and write common fractions.**

**CONCEPT
ANALYSIS:**

Any quantity may be expressed**
**in a
variety of forms. The evolution, use, and representation of number and number
relationships is historically and practically
significant. Counting numbers were invented first, later fractions, then zero to meet people's needs as civilization progressed.
Decimals are a relatively new invention. As students are introduced to
fractions and decimals they should be encouraged to consider the need for such
numbers both currently and historically.

The ability to
generate, use, and appreciate multiple representations for the same quantity is
a critical part of developing mathematical literacy. Sometimes 1/2 is
mentally efficient, but 0.5 works better in certain situations. Students should
encounter various models for numbers, including number lines, area models,
graphs, calculators, and computers.

**INSTRUCTION:**

Use fraction strips to discover
common fractions.

Write common fractions discovered
using fraction strips.

Read the fractions together.

Discuss and demonstrate how to find
common fractions of a given fraction.

**ASSESSMENT
STRATEGIES:**

Ask students to represent given
fractions in various ways, such as diagrams, divisions of a set of objects, or
paper folding.

Create games using equivalent
numbers. A "concentration" game can be made in which students match 2 with 4/2, or 1/2 with 0.5. A
"dominoes" game can be created with cards or wooden blocks such that
1/2 would be matched with 3/6 or 4/8. Teachers should monitor play and determine the level
of understanding and proficiency displayed by each student.

Ask students to write number stories
featuring selected decimals or fractions.

Ask students to illustrate number
stories written by themselves or fellow classmates.

Student will read a given fraction.

Given a number of fractions, student
will group the common fractions.

Given a fraction, student will write
1-3 common fractions.

**RESOURCES:**

Fraction strips.

**CORE
COMPETENCY:**

A Demonstrate
an understanding of numbers

**KEY
SKILLS: *State
Tested**

** *A3 Identify place value of each digit in
whole numbers with no more than seven digits.**

**CONCEPT
ANALYSIS:**

Understanding**
**place
value is a critical step in the development of students' comprehension of
number concepts and is necessary for developing good estimation skills. Care
should be taken to emphasize the value and not just the name. A student who
understands place value knows not only that the numeral "52" can be
used to represent "how many" for a collection of fifty-two objects, but
also that the digit on the right represents two of them, and the digit on the
left represents fifty of them (five sets of ten). Extension to larger numbers
should not be left to abstraction. Continued grouping activities, use of
base-ten blocks, and other manipulatives help
students see patterns as the numbers extend to thousands, millions, and beyond.
Calculators can be used to explore place value in both small and large numbers.
Rote activities, such as writing a number in extended form, do little to develop
or verify place value understanding; instead, concentration should be focused
on physical representations and discussion.

**INSTRUCTION:**

Use a place value chart to: a)
Identify the digit in the ones', tens', hundredths', thousands', or ten
thousands' place. (May also use to identify digits up to the
millions' place).

b)
Identify the place value of a given digit by name through ten thousands'
(millions' place).

c)
Identify the value of a specified digit(s) through the ten thousands' place
(millions' place).

Explain the importance of the digit
zero.

**ASSESSMENT
STRATEGIES:**

Test 2 (see file A-3)

Chapter 1 test (see file A-3)

** **

**TEST
CONTENT SPECIFICATION:**

__Component
Skills:__** **

** **A. Identify the digit
in the ones', tens', hundreds', thousands', or ten thousands' place.

B. Identify the place value of a
given digit by name through ten thousands.

C. Identify the value of a specified
digit(s) through the ten thousands' place.

__Specification:__

Students are given a number with no
more than five digits. They are asked to identify the digit in a given place,
the place value of a given digit by name, or the value of a particular digit or
digits in the number.

__Sample
Item:__

1. What is the value of the 2 in
1,234?

A. 2

B. 20

*C. 200

D. 2000

**RESOURCES:**

Student made place value charts for
individual use.

Calculator Connection Card (see file
A-3).

Calculator games (see file A-3).

Base Ten blocks.

Dice games (making largest and
smallest numbers).

**CORE
COMPETENCY:**

A Demonstrate
an understanding of numbers

**KEY
SKILLS: *State
Tested**

** *A6 Graph points representing fractions on a
number line with denominators of 2, 3, 4, 5, 8, or 10.**

**CONCEPT
ANALYSIS:**

All work with fractions in the early
grades should involve fractions that occur frequently in everyday life. Initial
instruction needs to emphasize oral language and to demonstrate the
mathematical relationship represented by one-half, one-fourth, etc. As students
progress in their mathematical thinking, emphasis should be placed on representing
and comparing relative sizes. Graphing fractions in a variety of formats and
modes helps students establish the concepts of size, increments, and
relationships. Fraction strips, commercial or cut from paper, help them make
the transition from modeling to graphing on a number line. The concept of a
unit and its subdivision into equal parts is fundamental to understanding
fractions, whether the quantity to be divided is a rectangular candy bar or a
handful of jelly beans. Instruction and practice should include improper
fractions and fractions not in lowest terms, although these will not be tested
at the state level in grade 4.

Students
need opportunities to apply these concepts to represent and solve problems
involving measurement, geometry, and statistics. For example, they might be
asked: "Where would you be if you were one-third of the distance between
Houston and St. Louis?" "If two-thirds of the red balls is the same amount as three-fifths of the blue balls, are
there more red balls or blue balls?"

**INSTRUCTION:**

** **Explain that
fractions can be graphed on a number line. Students are given a number line
with an indicated point and are asked to select the fraction represented by the
point or to select the point representing a given fraction. Mixed numbers may
be used, but not improper fractions. Fractions will be in lowest terms.

**TEST
CONTENT SPECIFICATION:**

__Component
Skills:__

A. Select the fraction represented
by a given point on a number line.

B. Select the point on a number line
representing a given fraction.

__Specification:__

Students are given a number line
with an indicated point and are asked to select the fraction represented by the
point or to select the point representing a given fraction. Fractions may have
denominators of 2, 3, 4, 5, 8, or 10. Mixed numbers may be used, but not
improper fractions. Fractions will be in lowest terms. Answers equivalent to
the correct answer will not be used as distractors
(2/4 will not be a distractor if 1/2 is the correct
answer).

**CORE
COMPETENCY:**

A Demonstrate
an understanding of numbers

**KEY
SKILLS:**

A7 Identify fractional parts of a whole and
of a set.

**INSTRUCTION:**

** **Model
whole and fractional parts using various shapes with shaded parts representing
denominators of 2, 3, 4, 5, 8, 10. Model sets divided into parts.

**ASSESSMENT
STRATEGIES:**

** **Students draw models
of a given fraction and of a given set. Given fractional models, students can
write the corresponding fraction.

**CORE
COMPETENCY:**

B Apply the basic operations in
computational situations

**KEY
SKILLS:**

** B1 Recall
multiplication and division facts.**

**CONCEPT
ANALYSIS:**

Computation is an enabling skill,
not an end in itself. Although computation is important in mathematics, our
technological age requires a different approach to computation. Today, almost
all computations requiring an exact answer are done by calculators and
computers. This produces situations in which pencil-and-paper computation decreases
in importance, while the abilities of mental computation and estimation to
judge the reasonableness of answers become paramount. Therefore, in an age of
technology, knowledge of basic facts is more important than ever.

Helping children develop thinking
strategies for learning basic facts enables them to understand relationships
and to reason mathematically. The initial use of physical materials gives
children visual and kinesthetic experiences to help them attach meaning to the
operations. Teachers should develop creative ways to provide practice in
problem-solving situations. Students should reduce the time spent with
flashcards or other types of pure drill that tend to sacrifice conceptual
development and problem solving in an attempt to master isolated basic facts.
Operation sense and an understanding of the need for immediate recall will make
learning much easier.

**ASSESSMENT
STRATEGIES:**

Give short drills asking students to
record and keep track of the number correct and the number answered. Challenge
them to improve their own personal best. Alternate speed with
accuracy as the main criteria.

Give students a writing exercise
such as "Suppose you forgot what 7 times 9 is.
Explain how you might figure out the answer."

Ask students to select their three
hardest facts and explain ways to figure out and then to remember the answer.

Give the answers and ask students to
supply as many different problems as possible. For example, given the number
24, students could say 3 x 8, 12 x 2, 4 x 6, or 48/2.

**CORE
COMPETENCY:**

B Apply
the basic operations in computational situations

**KEY
SKILLS:**

** B3 Select and use the
appropriate operation for computing +, -, and x.**

** B3a Select and use the
appropriate method for computing from among mental math, paper and pencil,
calculator, and estimation.**

**INSTRUCTION:**

**B3** Discuss symbols,
terminology and checking methods of each operation.

1.) Addition: +, addends, sum, total
(vertical placement, up to 4 addends, addends of 1-3 digits)

2.) Subtraction: -, minuend,
subtrahend, difference (numbers up to 4 digits)

3.) Multiplication: x, multiplicand,
multiplier, product (3 digits by 1 digit)

Model examples of
each operation.

Do selected activity sheets in file
B-4 and B-5.

** **

** B3a** Discuss and
demonstrate strategies of computing using:

1.) Mental Math: use doubles, plus
one, minus one; combine addends out of order to make sums of 10 or doubles; add
same number to both subtraction terms to reach next ten; pencil/paper or
calculator might not be available

2.) Estimation: rounding;
front-digit method with adjustment

3.) Calculator: + rapid, especially
for larger number calculations; - chance of entering wrong number or operation

4.) Pencil/paper: can show work

Allow students to work in pairs,
generating problems to use each method. Be sure to check estimated answer to
mental math, calculator and pencil/paper answers for "closeness" of answers.

**ASSESSMENT
STRATEGIES:**

B3 Computation
worksheets. Assessment sheets in files B-4 and B-5.

B3a Assessments in file C-3.

**RESOURCES:**

B3 File B-4; File
B-5.

B3a Calculators;
student problems; File C-3.

**CORE
COMPETENCY:**

B Apply
the basic operations in computational situations

**KEY
SKILLS: *State
Tested**

***B4 Compute answers requiring use of addition
and subtraction through four digits.**

**CONCEPT
ANALYSIS:**

Understanding the fundamental
operations of addition and subtraction is central to knowing mathematics. An
essential component of understanding an operation is recognizing conditions in
real-world situations in which the operation is useful. This component should
be developed from the beginning while basic facts are being learned.

Students need extensive and
continual experience with problem situations and language prior to and
throughout explicit instruction and symbolic work with the operations. Time
devoted to conceptual development provides meaning and context to subsequent
work with computational skills. Estimation strategies and mental computation
should be taught and practiced in every lesson.

**INSTRUCTION:**

Demonstrate addition and subtraction
problems with up to 4 digits, requiring regrouping. Addition problems may have
up to 4 addends. Each addend may have up to 3 digits.

Use correct terminology: addends,
sum, total

minuend, subtrahend, difference

Show checking
procedures for both operations.

**ASSESSMENT
STRATEGIES:**

** **Computation tests in
file B-4.

**TEST
CONTENT SPECIFICATION:**

__Component
Skills:__

A. Add.

B. Subtract.

__Specifications:__

Students are given vertically
written addition and subtraction problems which require a decision on
regrouping. Addition problems may have up to four addends. Addends may vary
from one to three digits within a problem. Subtraction problems may contain
numbers with up to four digits.

__Sample
Item:__

1. Subtract.

1137

__-482__

*A. 655

B. 755

C. 759

D. 1619

**RESOURCES:**

File B-4.

**CORE
COMPETENCY:**

B Apply
the basic operations in computational situations

**KEY
SKILLS:**** **** *State
Tested**

***B5 Compute answers requiring use of
multiplication of whole numbers. (limited to three
digits by one digit)**

**CONCEPT
ANALYSIS:**

Understanding the fundamental
operation of multiplication is central to knowing mathematics. An essential
component of understanding an operation is recognizing conditions in real-world
situations in which the operation is useful. This component should be developed
from the beginning to help students see a reason for learning basic facts.

Students need extensive informal
experience with problem situations and language prior to explicit instruction
and symbolic work with the operations. Time devoted to conceptual development
provides meaning and context to subsequent work with computational skills.
Understanding place value is critical to efficient use of the algorithm for
multiplying with multiple-digit factors. Informal work with operations and
their properties helps children to invent techniques for solving exercises
(e.g., 28 x 15 = ___). More formal work might have students demonstrating with
base-ten blocks the solution for 28 x 15. Estimation strategies and mental
computation should be taught and practiced continuously.

**INSTRUCTION:**

Discuss examples of multiplication
property:

1.) Commutative, or order property.
2 x 4 = 8 or 4 x 2 = 8

2.) Associative, or grouping
property. (3 x 2) x 4 or 3 x (2 x 4)

6 x 4 = 24 or 3 x 8 = 24

3.) Identity property. Property of 1. 6 x 1 = 6 Zero property.
6 x 0 = 0

Show examples of multiplication
problems using up to a 3 digit multiplicand by 1 digit multiplier, using
regrouping.

Use correct terminology:
multiplicand (number of items in a group)

multiplier (number of group)

product (answer)

Stress keeping
columns straight.

**ASSESSMENT
STRATEGIES:**

** **Assessment
test in file B-5.

**TEST
CONTENT SPECIFICATION:**

__Component
Skills:__

Same as Key Skill

__Specification:__

Students are given a problem
consisting of a three-digit number multiplied by a one-digit number.

Problems will be written vertically
and may require regrouping.

__Sample
Item:__

1. Multiply.

138

__x____
8__

A. 804

B. 1004

*C. 1104

D. 1106

**RESOURCES:**

File B-5.

** **

**CORE
COMPETENCY:**

B Apply
the basic operations in computational situations

**KEY
SKILLS:**

** B6 Recall from memory multiplication and
division facts through 9 x 9.**

**INSTRUCTION:**

Give each student a copy of the 100
multiplication facts to memorize.

Review order property of
multiplication to show that this reduces the number of facts to be memorized to
55.

Let students use flash cards to
practice in pairs.

Illustrate fact families: 6 x 2 = 12 12 ÷ 6 = 2

2 x 6 = 12 12 ÷ 2 = 6

**ASSESSMENT
STRATEGIES:**

Assessment tests in file B-5.

**RESOURCES:**

Flash cards;
multiplication fact sheet; File B-5.

** **

**CORE
COMPETENCY:**

B Apply
the basic operations in computational situations

**KEY
SKILLS:**

** B8 Compute answers
requiring the use of addition and subtraction of decimals through the
hundredths place.**

**INSTRUCTION:**

Provide examples of addition and
subtraction problems using decimals through the hundredths place. Compare to
adding and subtracting money. Decimal point is to be read as "and."
Show that zeros may be annexed where necessary for computation. Stress that place value columns and decimal points must be kept
straight vertically.

Have students use graph paper to
mark off hundred square sections to color squares to match decimal problem.

Students may use beans on graph
paper to show decimal minuend and remove beans to correspond to subtrahend,
proving the decimal difference.

Use play money to demonstrate value
of decimal numbers and problems of tenths and hundredths.

**ASSESSMENT
STRATEGIES:**

Display answers to decimal addition
and subtraction problems by placing beans on graph paper.

**RESOURCES:**

Graph paper; beans; play money.

**CORE
COMPETENCY:**

B Apply
the basic operations in computational situations

**KEY
SKILLS:**

** B9 Develop the understanding of a division
problem.**

**INSTRUCTION:**

Illustrate that division is a short
way of subtracting. 15 ÷ 5 = 3 (15 - 5 = 10 - 5 = 5 - 5 = 0)

Use proper symbol and terminology: ÷ means divided by;
dividend - number being divided up; divisor - number in each group that the
dividend will become part of; quotient - number of groups; remainder - part of
a group left over.

Have students use counters to
illustrate simple division problems.

Stress that the quotient is not the answer, rather that the quotient indicates the number of
sets the size of the divisor, contained in the dividend.

**ASSESSMENT
STRATEGIES:**

Have students use counters to show
simple division solutions.

**RESOURCES:**

Counters.

**CORE
COMPETENCY:**

C Estimate results and judge
reasonableness of solutions.

**KEY
SKILLS:**

** C1 Use estimation
strategies and mental computation to produce reasonable estimates.**

**CONCEPT
ANALYSIS:**

Children begin school as good
problem solvers and with estimation skills that have evolved from their
experiences. Formal schooling must recognize this learning and build on it. At
this grade level, students should be taught specific strategies to aid them in
computational estimation. Flexible and front-end rounding, compatible numbers,
special numbers, and clustering are examples of frequently used strategies.
Lessons devoted to estimation are necessary and should be continuously
integrated into all computation and problem-solving lessons.

Mental computation produces an exact
answer, as opposed to an estimate. Most estimation involves some mental
computation. Practice with procedures efficient for mental computation helps
students become good estimators.

**INSTRUCTION:**

Discuss and show examples of
rounding and front-digit estimation with adjustment step to produce reasonable
estimates.

Ex: 27 à 30 27

__26 ____à____ 30__ __26__

60 40 (front digits)

Since 7 + 6 is about 10, __adjust__
to 40 + 10 = 50. Actual 27 + 26 = 53.

Have students use counters on a
number line labeled 0-100, by 10's. Place counter on number line to show
indicated number, then move it to the closest 10 to illustrate rounding.

Stress that 5
always rounds up to the next higher value.

**ASSESSMENT
STRATEGIES:**

Present problems on overhead
transparencies to control the time. Ask students to respond after a
three-second exposure. Stage the problems to encourage the various strategies,
but allow student decision and justification of the "best" method.

Constantly encourage students to
estimate solutions before working a problem and to consider the reasonableness
of the result. Informally assess the skills demonstrated as they work through
problems by noting areas of strength and areas needing more development.
Consider skills in justifying their response as well as use of specific
strategies.

Do "minds only" drills to
assess mental computation skills. Give or display the problem, allow think
time, then signal pencil time and allow only a short time to write down the
answer before calling for "minds only" again.

Have students use counters on a
number line, labeled 0-100, by 10's to indicate rounded answer when given
numbers.

**RESOURCES:**

Number line;
counters.

** **

**CORE
COMPETENCY:**

C Estimate results and judge
reasonableness of solutions.

**KEY
SKILLS:**

** C2 Use estimation
strategies and mental computation to determine if an answer is reasonable.**

**CONCEPT
ANALYSIS:**

Estimation skills must become an integral
part of every mathematics program. Research about students' thinking on
estimation has provided new insight into the learning and teaching of
computational estimation.

Mental computation skills are
necessary to be a good estimator. The fact that calculators and computers are
supplanting the need for a high degree of skill in pencil/paper computation
increases the need for good estimation skills.

**INSTRUCTION:**

List column of two
digit numbers on board. Model rounding, front-digit
with adjustment estimation, and mental computation of numbers for reasonable
answers. Use calculators to check if these answers are reasonable.

Assign a numerical value to each
letter of the alphabet (A-1, B-2) and have each child add up the value of their
first name, using each method.

**ASSESSMENT
STRATEGIES:**

Constantly encourage students to
consider and to justify the reasonableness of their answers, orally and in
writing. Informally assess the skills demonstrated as they work through
problems by noting areas of strength and areas needing more development.

Give students problems complete with
the answers and ask them to evaluate the reasonableness (rather than accuracy)
of the answer. Time some of these tests to encourage rapid response. In another
test, allow more time but ask students to justify their answers or to describe
the process by which they arrived at their conclusions. Be flexible in scoring
by allowing for a good justification of a response that might not be
"reasonable" in the strictest sense.

After a problem-solving assignment,
have students exchange papers and judge the answers of their peers for
reasonableness. Require justification, and score the results. This would be
especially effective for open-ended problems-problems with potentially
different correct answers.

Column addition
worksheets using estimation and mental computation to be verified with use of
calculators.

**RESOURCES:**

Calculators;
alphabet/numerical chart.

**CORE
COMPETENCY:**

C Estimate results and judge
reasonableness of solutions.

**KEY
SKILLS: *State
Tested**

*C3 Recognize a situation in which an
estimate is appropriate.

**CONCEPT
ANALYSIS:**

"Estimation interacts with
number sense and spatial sense to help children develop insights into concepts
and procedures, flexibility in working with numbers and measurements, and an
awareness of reasonable results" (from the NCTM *Standards*).
Instruction should build on the estimation skills children have as they enter
school. Mathematical knowledge should build as a logical process, not a system
of rote procedures. Estimation strategies and practice can help students see
beyond the procedures to the connections and the inherent usefulness of the
properties and relationships of numbers.

The constant stress on producing
"the right answer" detracts from the enjoyment of mathematics and
does not align with the ways mathematics is actually used in most situations.
Classroom instruction on estimation helps children develop an estimation
mind-set which will be a valuable tool in making decisions in their daily
lives.

**INSTRUCTION:**

Discuss situations and reasons to
estimate answers, for example, simple math problems, number of students in a
class, apples in a bushel, names in a phone book, etc. Evaluate which
situations would be possible to really prove by easily counting and which would
be impractical to try to count exactly but where a close guess or estimate
would be appropriate.

Pair students to
make lists of situations. Let students take turns reading a situation to
the class and let class provide reasons whether or not there should be an
estimate or an actual count.

**ASSESSMENT
STRATEGIES:**

Assessments in
file C-3.

**TEST
CONTENT SPECIFICATION:**

__Component
Skills:__

A. Recognize a situation in which an
estimate is appropriate.

B. Recognize a situation in which an
exact answer is appropriate.

__Specification:__

Students are given four situations
readily understandable by grade 4 students involving mathematics and are asked
to identify the situation in which an exact answer is most appropriate or the
situation in which an estimate is most appropriate.

__Sample
Item:__

1. For which number is an estimate
most appropriate?

A. score of
a basketball game

B. amount of medicine to take

*C. distance between two cities

D. number of inches in a foot

**RESOURCES:**

File C-3; student lists.

**CORE
COMPETENCY:**

D Apply
the concept of measurement to the physical world

**KEY
SKILLS: *State
Tested**

** *D2 Estimate measurements including area and
perimeter of regular and irregular regions.**

**CONCEPT
ANALYSIS:**

More attention is being given to
estimation in current mathematics instruction, especially with regard to
problems involving computation. Techniques of measurement estimation are
equally important. Students need to develop the ability to "see" or
mentally compute an estimate and the ability to decide whether refinement is
needed and in what direction. Measurement is usually taught by having students
perform the mechanics of measuring, but often the mechanics are divorced from
the practice of identifying a referent and estimating, and from using the
results in any meaningful way. Teaching measurement in a meaningful context
permits and encourages life-long skills in estimation.

Students should practice estimating
the measurement of many different attributes, including length, mass, capacity,
and area. Teachers should encourage students to use such strategies as
establishing a personal referent (the size of a garage door or the length of a
bedroom wall) or subdividing an object into smaller parts (chunking) when
estimating the area or perimeter of regions.

**INSTRUCTION:**

Use string to measure the distance
around a box. This demonstrates that perimeter is the distance around
something. Perimeter equals the sum of all the sides.

Have students walk around a shape in
the room. Then measure each side of the shape to discover the perimeter.

Cover a piece of construction paper
with square units to demonstrate area. Count the square units. Continue this
activity using other shapes. Explain area equals length times
width. The answer is in square units.

**ASSESSMENT
STRATEGIES:**

Assessment test: See File D-2.

**TEST
CONTENT SPECIFICATION:**

__Component
Skills:__

A. Estimate linear measure,
including perimeter.

B. Estimate area.

__Specification:__

Students are given a diagram showing
a length and a referent or nonstandard unit. They are asked to estimate the
number of units in the length or the perimeter; they are also asked to estimate
the area of the figure. Regions or illustrations of real-life situations
(fields, lawns, etc.) may be shown superimposed on grids with whole and partial
units indicated. Counting and putting parts together may be used to determine
an estimate.

**RESOURCES:**

ITV: __It Figures__ "Finding
Area by Covering."

ITV: __Math Works__
"Area."

File: D-2 Estimate measurements
including area and perimeter of regular and irregular regions.

**Sample Learning
Activities for Number Sense**

** **Using manipulatives (e.g., color tiles, Unifix
cubes), show the relationships among addition, subtraction, multiplication, and
division.

Use manipulatives to
demonstrate an understanding of place value by modeling operation on numbers.

Use manipulatives
or models to demonstrate fractional parts and the fractional notation to
represent each fraction (e.g., 1/2, 1/3, 2.4). Show fractional parts that are
equivalent. Explain how you know they
are equivalent.

Skip count using a calculator. Record the data. Examine the units place for patterns. Describe the patterns.

Using a frisbee
or paper plate, estimate how far it can be thrown. Throw and give a better estimate. Now measure the actual
distance. repeat
the experiment several times and average the data.

Estimate the number of raisins in a 1/2 oz. box. Record estimates. Open the box and examine the raisins in
view. (Do not empty the box.) Revise estimates if necessary. Count raisins. (A third estimate can be made when raisins
are dumped from the box.)

**VI. Geometric and Spatial Sense**

**Content
Overview**

*Geometry is the study of visual patterns. Geometry helps to represent and describe the
world in which we live. Spatial sense is
needed to interpret, understand, and appreciate our geometric world. Real-world investigations, experiments, and
explorations provide the basis for more formal explorations as students develop
the ability to recognize and apply geometric concepts as a means to solve
problems. Through the study of geometric
and spatial sense, students look at the world around them in a more meaningful
way. Students discover relationships and
develop spatial sense by constructing, drawing, measuring, visualizing,
comparing, transforming, and classifying geometric figures.*

* Geometric knowledge, relationships, and insights are
useful in everyday situations and connect other mathematical topics, thereby
providing links with other subject areas.
'geometric terms and relationships helps
communicate the connections between mathematics and the natural world.*

**All fourth grade students
should know**

1. Standard and nonstandard
units of measure.

2. Descriptions of two- and
three-dimensional.

3. That geometric shapes are
found in the real world.

4. The process of measurement.

**All fourth grade
students should be able to**

** **A. Describe,
model, draw, and classify shapes (NCTM Standard 9; MO 1.4, 1.6, 2.1)

B. Investigate and predict the results of combining,
subdividing, and changing shapes (NCTM Standard 9; MO 1.1, 1.6, 3.1)

C. Visualize, draw, and compare shapes (NCTM Standard
9; MO 1.8, 2.1, 3.2, 3.3)

D. Connect geometric ideas to number and measurement
ideas (NCTM Standard 9; MO 1.6, 3.5, 4.1)

E. Explore geometry in their world (NCTM** **Standard
9; MO 1.10, 2.4)

F. Investigate concepts of lines, angles, similarity,
congruence, and symmetry (NCTM Standard 9; MO 1.6, 2.5)

G. Investigate length, capacity, weight, mass, area,
volume, time, and temperature (NCTM Standard 10; MO 1.6, 2.5)

H. Use standard and nonstandard units of measure
(NCTM Standard 10; MO 1.10)

**CORE
COMPETENCY:**

C Estimate results and judge
reasonableness of solutions.

**KEY
SKILLS:**

** C4 Use a calendar for making plans and
organizing activities.**

**INSTRUCTION:**

** **Discuss a twelve
month calendar, number of days in each month, and number of days in a week.

Give each child a calendar for a
month and have the child fill in activities, assignments etc., for the month.

Make a school year plan of
activities for the present grade including tests, field trips, parties,
birthdays, holidays, etc.

**ASSESSMENT
STRATEGIES:**

** **The student can list
in order the twelve months from memory.

Given a calendar, the student will
answer questions involving date related situations.

**RESOURCES:**

** **Lunch
menu; classroom calendar (list events, birthdays, etc.); day planner (list
assignments due).

** **

**CORE
COMPETENCY:**

C Estimate results and judge
reasonableness of solutions.

**KEY
SKILLS:**

** C5 Use the understanding of the clock to
solve time related activities.**

**INSTRUCTION:**

** **Make clocks with
index cards and use construction paper for hands. Use brads to fasten the
hands. Each child can demonstrate a specific time. Make a time schedule for the
day. Discuss how long until lunch, time for special area, time for recess, etc.

**ASSESSMENT
STRATEGIES:**

** **Use clocks made in
class. Each child will demonstrate the appropriate time that is asked by the
teacher.

**RESOURCES:**

** **Classroom clock;
clocks made by students.

**CORE
COMPETENCY:**

D Apply
the concept of measurement to the physical world

**KEY
SKILLS:**

**CONCEPT
ANALYSIS:**

Lynn Steen in *On The Shoulders Of Giants* says, "Learning how to
measure is the beginning of numeracy." Measurement shows students the
usefulness of mathematics in everyday life. It can help them develop many
mathematical concepts and skills such as fractions and decimals. By
establishing an understanding of the purpose of measurement and the various
attributes often requiring measuring, students will have a firm foundation that
will enable them to use any measurement system. The process of measuring is
identical for any attribute within any system: choose a unit with the same
attribute as that which is being measured, compare that unit to the object, and
report the number of units. The number of units can be determined by counting,
by using an instrument, or by using a formula.

Students must realize that an
appropriate unit depends upon the size of the object and/or the precision
desired. They need to develop a "feel" for the relative sizes of the
commonly used metric and English units.

**INSTRUCTION:**

** **Provide various
objects. Ask students to decide how that object would be measured. Then
demonstrate how to measure that object.

Students measure objects themselves,
determining which tool to use.

**ASSESSMENT
STRATEGIES:**

Ask students to construct/select the
type of unit/tool needed to measure a particular attribute.

Incorporate this objective into a
bigger problem-solving picture. Selecting the appropriate unit should not be an
end in itself; rather, it should be only one step in the process of using a
measurement to solve a problem.

**RESOURCES:**

Balance scale, spring scale, liquid
measurement tools, yard stick, meter stick, graduated cylinders, measuring
cups.

**CORE
COMPETENCY:**

D Apply
the concept of measurement to the physical world

**KEY
SKILLS:**** **** *State
Tested**

** *D2 Estimate measurements including area and
perimeter of regular and irregular regions.**

**CONCEPT
ANALYSIS:**

More attention is being given to
estimation in current mathematics instruction, especially with regard to
problems involving computation. Techniques of measurement estimation are
equally important. Students need to develop the ability to "see" or
mentally compute an estimate and the ability to decide whether refinement is
needed and in what direction. Measurement is usually taught by having students
perform the mechanics of measuring, but often the mechanics are divorced from
the practice of identifying a referent and estimating, and from using the results
in any meaningful way. Teaching measurement in a meaningful context permits and
encourages life-long skills in estimation.

Students should practice estimating
the measurement of many different attributes, including length, mass, capacity,
and area. Teachers should encourage students to use such strategies as
establishing a personal referent (the size of a garage door or the length of a
bedroom wall) or subdividing an object into smaller parts (chunking) when
estimating the area or perimeter of regions.

**INSTRUCTION:**

Use string to measure the distance
around a box. This demonstrates that perimeter is the distance around
something. Perimeter equals the sum of all the sides.

Have students walk around a shape in
the room. Then measure each side of the shape to discover the perimeter.

Cover a piece of construction paper
with square units to demonstrate area. Count the square units. Continue this
activity using other shapes. Explain area equals length times
width. The answer is in square units.

**ASSESSMENT
STRATEGIES:**

Assessment test: See File D-2.

**TEST
CONTENT SPECIFICATION:**

__Component
Skills:__

A. Estimate linear measure,
including perimeter.

B. Estimate area.

__Specification:__

Students are given a diagram showing
a length and a referent or nonstandard unit. They are asked to estimate the
number of units in the length or the perimeter; they are also asked to estimate
the area of the figure. Regions or illustrations of real-life situations
(fields, lawns, etc.) may be shown superimposed on grids with whole and partial
units indicated. Counting and putting parts together may be used to determine
an estimate.

**RESOURCES:**

ITV: __It Figures__ "Finding
Area by Covering."

ITV: __Math Works__
"Area."

File: D-2 Estimate measurements
including area and perimeter of regular and irregular regions.

**CORE
COMPETENCY:**

D Apply
the concept of measurement to the physical world

**KEY
SKILLS: *State
Tested**

***D5 Measure lengths to the nearest half inch
and to the nearest centimeter**.

**CONCEPT
ANALYSIS:**

The process of measuring unifies
many mathematical skills and concepts, especially communication. The purpose of
a standard system of measurement is to communicate. Evidence indicates that
using instruments or formulas is superficial if either is thrust upon young
students too quickly. Young children can develop an understanding of a
measurable attribute by comparing objects based on that attribute and later by
comparing the objects to a standard unit.

Nonstandard units should be used until
students have the time and experience to understand the need for standard
units. Students may compare lengths of two pieces of paper by lining them up
side-by-side, or they might compare the areas by placing one on top of the
other. Such exploration provides the context for developing appropriate
vocabulary and leads to comparing an object to a standard unit.

**INSTRUCTION:**

** **Measure various
objects in the room to the nearest half inch and to the nearest centimeter. For
example: desk, length of child's foot, side of a book, height of a child.

Activities in File
D-5.

**ASSESSMENT
STRATEGIES:**

** **Assessment
test in File D-5.

**TEST
CONTENT SPECIFICATION:**

__Component
Skills:__

A. Measure lengths to the nearest
half inch.

B. Measure lengths to the nearest
centimeter.

__Specification:__

Students are given a picture of an
item with a ruler placed along side and are asked to
identify the correct measurement of its length to the nearest half inch or
centimeter. They are also given a measurement and asked which picture shows an
item of equal length. Rulers are zero-based with the item placed at the zero
mark.

**RESOURCES:**

Rulers, both
centimeter and inch.

ITV: __It Figures__
"Deciding How Close To Measure."

File: D-5 Measure lengths.

** **

**CORE
COMPETENCY:**

D Apply
the concept of measurement to the physical world

**KEY
SKILLS:**

** D7 Weigh objects to the nearest ounce and
determine mass to the nearest gram.**

**CONCEPT
ANALYSIS:**

Various types of scales and balances
should be available to students as they investigate weight and mass and solve
problems using these measurements. Sharing cookies equally by weight instead of
number makes an interesting problem for students to solve. Students need to
develop a feel for the relative sizes of commonly used standard and metric
units. Students gain understanding of the relationships among units through
hands-on measuring, but formal conversions between systems should be delayed.

**INSTRUCTION:**

Children will practice weighing
objects on a scale to determine the weight to the nearest ounce.

Children will practice finding the
mass of an object to the nearest gram using a balance.

**ASSESSMENT
STRATEGIES:**

Give each group of three to five
students a package of cookies to be shared equally by weight. (Homemade cookies
of inconsistent sizes or a mixture of different kinds of store-bought cookies
makes the problem more interesting.)

Give students different sizes and
types of containers and an amount of sand or water (or similar material) and ask
them to fill each container with enough material so that each weighs the same.

Children will demonstrate the
ability to weigh an object and read the weight on the scale to the nearest
ounce.

Students will demonstrate the
ability to determine the mass of an object to the nearest gram.

**RESOURCES:**

Spring scale; balance scale.

**CORE
COMPETENCY:**

E Recognize
geometric relationships

**KEY
SKILLS:**

** E1 Explore the properties of lines.**

**CONCEPT
ANALYSIS:**

A study of the properties of lines
might include straightness, parallelism, perpendicularity, intersection,
orientation (e.g., vertical or horizontal). Developing knowledge of these
properties allows children to describe the relationship of lines in
mathematically precise terms. The emphasis should be on exploring, e.g., sets
of parallel lines that form a rectangle or sets of perpendicular lines that
form the angle of a square.

Practical applications, such as the
need for parallel and perpendicular lines in construction or the interruption
of such relationships if a foundation shifts, make interesting topics for
students to investigate.

**INSTRUCTION:**

** **Demonstrate
perpendicular and parallel lines on the board. Relate perpendicular lines to
intersecting lines and to right angles.

Pupils state examples of
perpendicular and parallel lines in the classroom and the world.

**ASSESSMENT
STRATEGIES:**

Ask students to sketch a diagram of
the playground using specific colors for different relationships among lines.
For example, they might trace parallel lines with red and perpendicular lines
with green.

Ask students to write about the
results of a law prohibiting perpendicular lines.

Ask students to devise
"new" ways to cut cakes of various kinds (e.g., sheet cakes or layer
cakes).

Students can demonstrate their
understanding of perpendicular and parallel lines by making examples of each
using yarn and construction paper. Glue the yarn to show both types of lines.

**RESOURCES:**

Rulers; yarn;
construction paper.

**CORE
COMPETENCY:**

E Recognize
geometric relationships

**KEY
SKILLS: *State
Tested**

** *E2 Explore properties of angles.**

**CONCEPT
ANALYSIS:**

Geometry gives students a different
view of mathematics. As they explore patterns and relationships with models,
blocks, geoboards, graphs, and dot paper, they learn
about the properties of shapes and increase their awareness of spatial
concepts. Spatial sense is an intuitive feel for one's surroundings and the
objects in them. The vocabulary of geometry helps us precisely communicate the
location, properties, and relationships within our environment.

**INSTRUCTION:**

** **Teacher demonstrates
examples of the acute, obtuse, and right angles. Given examples, have the
pupils identify the angles.

The students can make angles using
two rays made from construction paper and a brad.

**ASSESSMENT
STRATEGIES:**

** **Assessment
Test E-2.

Student demonstrates a right angle,
an acute angle, and an obtuse angle using two rays joined at the point by a
brad.

**TEST
CONTENT SPECIFICATION:**

__Component
Skills:__

A. Identify right angles by
inspection.

B. Determine, by inspection, whether
the measure of a given angle is less than or greater than the measure of a
right angle.

__Specification:__

Students are given diagrams of
angles of various sizes and orientations. They are asked to select the angle
which appears to be a right angle, the angle whose measure appears to be less
than 90 degrees, or the angle whose measure appears to be more than 90 degrees.
The angles may appear in singular diagrams or within a polygon or other figure.

**RESOURCES:**

File E-2; angles made by students;
ITV: __Solve It__ "Geometry and Angles."

** **

**CORE
COMPETENCY:**

E Recognize
geometric relationships

**KEY
SKILLS:**

** E3 Investigate the
concepts of similarity, congruence, and symmetry.**

**CONCEPT
ANALYSIS:**

Similarity and congruence can be
introduced through comparing scale models or photographs of various sizes.
Measuring and comparing the sides and angles of similar polygons helps students
develop and understand the mathematical concept of similar figures. Congruence
can be explored as similarity with a ratio of 1.

Symmetry provides rich opportunities
for students to see geometry in the worlds of art, nature, and construction.
Butterflies, faces, flowers, arrangements of windows, reflections in water, and
some pottery designs involve symmetry. Turning symmetry is illustrated by
bicycle gears. Pattern symmetry can be observed in the multiplication table, in
numbers arrayed in charts, and in Pascal's triangle. Lynn Steen says in *On
the Shoulders of Giants,* "Learning to recognize symmetry trains the
mathematical eye."

**INSTRUCTION:**

** **Introduce each term
defining it and giving examples.

In pairs, have students draw
polygons (6 sides or less) on two sections of graph paper. The graph paper
should be folded in four sections. Partners exchange papers. On the other two
sections, students draw the same shape. One shape is to be congruent and the
other one should be similar. Students can cut the shapes out to determine
congruence. Then, students can fold shapes to test for lines of symmetry.

**ASSESSMENT
STRATEGIES:**

From a photograph, create a model of
a structure by establishing a scale and measurement.

Use paper-folding activities to identify
symmetry. Ask students to create their own designs involving a given number of
lines of symmetry.

Use mirrors to investigate symmetry.
Encourage students to discuss and write about their observations.

Challenge
students to create symmetrical and asymmetrical designs for pottery, placemats,
or other items of interest.

Pupils can draw shapes that are
congruent and shapes that are similar. They can demonstrate lines of symmetry
on the shapes.

Students name objects in the room
that are congruent or similar. They point out lines of symmetry.

**RESOURCES:**

Shapes; rulers;
graph paper.

**Sample Learning
Activities for Geometric and Spatial Sense**

** **A dog is
tied to a five-meter rope at the middle of the ten-meter-long side of a
garage. Make a sketch of the outer path
on which the dog can walk. Move the tie
post to the corner of the garage. compare and contrast the outer path of the two situations.

Given a model, use interlocking
cubes to build a three-dimensional replica of the model.

Build a shape (two-dimensional) or structure
(three-dimensional) from a description provided by another student. One student builds a shape or structure with
a set of materials which cannot be seen by the other student. Another student with an identical set of
materials tries to build the shape or structure with only verbal directions.

Copy a partner's design onto another geoboard or dot paper.

Have students make a shape (perhaps something that
can fly) on their geoboards. Have students describe how some shapes are alike
and how they are different. Students can
then sort and classify the shapes.

**VII. Data Analysis, Probability, and Statistics**

**Content
Overview**

*Collecting, organizing, and interpreting data allow
students to make informed decisions based on the information. The study of probability enables students to
understand how predictions are made once data has been analyzed. Instructional strategies in data
analysis/probability and statistics provide opportunities for students to make
connections to other academic areas and out-of-school activities. A working knowledge in data
analysis/probability and statistics enables students to function as informed
consumers and citizens to make decisions of economic impact. The volatile nature of real-world data may be
anticipated to understand and describe the relationship between predicted and
actual results. Data collection,
organization, and analysis may be employed across the curriculum to solve
problems, make reasoned decisions, communicate ideas, and make connections.*

* Data analysis/probability and statistics is used to
make predictions and solve problems. It
involves the collection or generation of data, making visual representations
(charts, graphs, and tables), and interpreting displays of data. Data analysis involves describing and
interpreting the world with numbers.*

**All fourth grade
students should know**

1. Strategies to collect data.

2. Strategies to organize data.

3. Different displays of data.

4. The appropriate display of data.

5. The appropriate use of technology.

**All fourth grade
students should be able to**

** **A. Collect,
organize, and describe data through the use of
technologies and other resources (NCTM Standard 11; MO 1.1, 1.3, 1.4, 1.8)

B. Construct, read, and interpret
displays of data through verbal, nonverbal, symbolic, and graphic forms (NCTM
Standard 11; MO 1.5, 3.3, 3.6, 4.1)

C. Solve problems which require collecting and
analyzing data (NCTM Standard 11; MO 2.3, 3.2, 3.3, 4.3)

D. Explore concepts of chance (NCTM Standard 11; MO
1.6, 1.7, 4.3, 4.7)

**CORE
COMPETENCY:**

F Use
statistical techniques and interpret statistical information

**KEY
SKILLS:**

** F1 Collect, organize,
and classify data.**

**CONCEPT
ANALYSIS:**

Collecting, organizing, and
classifying data are increasingly important skills in a society based on
technology and communication. These processes are particularly appropriate for
students because they offer opportunities for inquiry, and they can be used to
solve interesting problems and to represent significant applications of
mathematics to practical questions.

Students
need to recognize that many kinds of data come in many forms, and that
collecting, organizing, classifying, and interpreting data can be done in many
ways. Along with traditional picture graphs, bar graphs, pie charts, and line
graphs, students can use a variety of plots, such as stem-and-leaf plots and
box-and-whisker plots. Statistics is more than reading and interpreting graphs.
It is describing and interpreting the world around us with numbers; it is a
tool for solving problems.

In the fourth grade, students should
be able to use scales representing units up to and including 10. Computer
graphics programs make a wide variety of explorations accessible to students. A
class or group project conducted over a period of time enables the students to
make predictions and to modify them as more data are collected. Teachers should
monitor from a distance and guide with thoughtful questions.

**INSTRUCTION:**

** **Model the following
strategies:

a) sorting
information. b) relating information. c) combining information. d) ordering
information. e) logically linking ideas together.

Make a graph from data.

**ASSESSMENT
STRATEGIES:**

Give students graphs from newspapers
or magazines. Ask them to formulate questions that can be answered by
interpreting the data on the graph.

Give students graphs from newspapers
or magazines. Ask them to consider alternative ways to display the data and to
discuss the merits of each alternative.

Ask students to submit questions of
interest to them that require data to answer. Discuss methods of collecting the
data needed, do the collection, and organize the results.

Given data, the student will
organize it and present it in a table or graph form.

Test - File F-1.

**RESOURCES:**

Graph paper; File
F-1.

**CORE
COMPETENCY:**

F Use
statistical techniques and interpret statistical information

**KEY
SKILLS: *State
Tested**

** *F2 Construct, read, and interpret
displays of data.**

**CONCEPT
ANALYSIS:**

Collecting, organizing, describing,
displaying, and interpreting data, as well as making decisions and predictions
on the basis of that information, are becoming increasingly important skills in
a society based on technology and communication. These processes are
particularly appropriate for young children because they can be used often to
solve problems that are inherently interesting, that represent significant
applications of mathematics to practical situations, and that offer rich
opportunities for mathematical inquiry.

Collecting and graphing data from
the children themselves is the way to begin the study of statistics. Graphing
the kinds of pets owned by students, lengths of little fingers, or favorite
television characters provides data meaningful to students. This data can be
used to experiment with different types of displays and to answer a variety of
questions.

**INSTRUCTION:**

** **Review strategies
taught in objective F-1.

Present graphs and tables. Help the
students read and interpret the information.

**ASSESSMENT
STRATEGIES:**

** **Given data, the
student will organize it and present the data in a table or graph.

Given a table or graph, the student
will interpret orally the data.

Test - File F-2.

**TEST
CONTENT SPECIFICATION:**

__Component
Skills:__

A. Select bar graphs accurately
representing given data.

B. Select pictographs accurately
representing given data.

C. Construct displays of data.

__Specification:__

Students are given data in an
organized or tabular form. They are asked to identify the bar graph or
pictograph which accurately represents that information. Examples of
inaccuracies include, but are not limited to, misrepresented data and unequal
increments.

**RESOURCES:**

Graphs; tables;
charts; time lines; File - F-2.

**CORE
COMPETENCY:**

F Use
statistical techniques and interpret statistical information

**KEY
SKILLS:**

** F3 Formulate and solve problems that
involve collecting and analyzing data.**

**CONCEPT
ANALYSIS:**

Students should be taught to look at
data the way a good statistician does. Statisticians first try to determine
whether the data are reliable by asking whether they were collected in a
reasonable manner. Whether any values are missing, whether values are in error,
and whether they are the right data for the question. The next step is to
display the data in appropriate plots. The statistician is likely to use more
than one plot to investigate the data. Finally, the statistician examines the
plots and tries to make some sense of the data.

Data is collected to answer
questions and to solve problems. Real-life examples abound. Through answering
questions and solving problems generated by the students, the full power of
mathematics is communicated.

**INSTRUCTION:**

Class Project: Poll the members of
the class to determine favorite places to eat. Graph the data. Analyze the
results.

**ASSESSMENT
STRATEGIES:**

Present the class with graphs from
newspapers or magazines on topics of interest to this age group. Ask them to
consider the information presented in the graph from a local perspective. Will
the people in this community/school respond in the same way? Ask them to design
a method for determining the answer. When possible, allow students to carry out
the process. Ask students to evaluate their design as well as answer the
question.

Take an issue of local concern, such
as the need for a new swimming pool or a change in school starting time. Ask
students to determine how the community/school feels about the issue. Arrange
for students to report their results to the appropriate authorities.

In groups, students choose a topic
to collect data on. The groups do their research and present the results in
graph or table form. Each group must explain their results.

**RESOURCES:**

Graph paper; items
for projects.

**Sample Learning
Activities for Data Analysis, Probability, and Statistics**

** **Collect
data on a given topic and classify the data based on similarities.

Use collected data to construct a variety of graphs
or charts.. discuss which
representation best displays the information.

Make a prediction on an upcoming event based on data
collected and analyzed from a past series of events.

Given a spinner that is divided into four equal
sections with two blue sections, one red section, and one green section, how
many times would you expect to get green if you spin 20 times? Explain your answer. Now spin 20 times and record your
results. Compare the results to your
prediction.

Make a spinner with four equal parts. Spin the spinner 20 times and record the
number of times each area of the spinner "comes up." Make spinners with non-congruent areas and
record the number of times each area is spun.

Make a graph of the different ways children get to
school (bus, car, walk).

Collect data from a lunch graph to complete lunch
information for the school cafeteria.

Collect and interpret information from an opinion
graph (favorite color, cartoon, movie, place to eat, etc.).

**VIII. Patterns and Relationships**

**Content
Overview**

*Recognizing patterns and relationships has been
instrumental in the development of mathematics and the study of numbers. The world around us exhibits a multitude of
patterns. Exploring patterns and
relationships allows young children to develop and understand how mathematics
applies to their environment. Organizing
and classifying leads to opportunities to discover patterns and
relationships. Representing and
analyzing patterns and relationships allows for the development of
understanding the connections between and among algebra, geometry,
trigonometry, functions, and change relationships.*

* Increased use of technology is critical to the
development and understanding of patterns and relationships. Technology reduces time-consuming tasks (such
as graphing) and allows time to study relationships and draw conclusions from
information gathered.*

* Creation of patterns through the use of physical
materials and pictorial displays may help in the recognition of a particular
relationship. Observing varied representations
of the same pattern helps in the identification of the pattern. Patterns may be represented through concrete
materials, tables, charts, graphs, or sets of numbers. Predictions can be made from observing the
patterns and relationships that result.
Patterns can be used to show relationships among various mathematical
topics. Recognizing the relationship of
addition to multiplication is an example illustrating how patterns and
relationships enhance the students learning in mathematics.*

* *

**All fourth grade
students should know**

1. That mathematical ideas may
be represented by visual models.

2. That mathematical symbols can
be used to represent real-world situations.

3. That patterns and
relationships can be represented in a variety of ways.

4. That information can be
organized to look for a pattern or relationship.

5. That patterns can be
geometric and/or numeric.

**All fourth grade
students should be able to**

** **A. Create,
recognize, describe, and extend a wide variety of patterns (NCTM Standard 13; MO
1.6, 1.8, 2.1, 3.3)

B. Represent and describe mathematical relationships
(NCTM Standard 13; MO 1.6, 1.8, 2.2, 3.3)

C. Investigate the use of variables and open
sentences in expressing relationships (NCTM Standard 13; MO 1.6, 1.8, 3.3)

**CORE
COMPETENCY:**

A Demonstrate
an understanding of numbers

**KEY
SKILLS:**

** A4 Identify and
continue patterns created with objects, pictures, and numerals.**

**CONCEPT
ANALYSIS:**

Patterns are everywhere. Students
who are encouraged to look for patterns and to express them mathematically
begin to understand how mathematics applies to their world. Looking for
patterns in numbers, geometry, and measurement helps students understand
connections among mathematical topics.

Physical
materials and displays should be used to help students recognize and create
patterns. Identifying similar patterns within different contexts helps students
to begin to look for regularity in their world. Creating their own patterns
verifies their understanding and helps give them a sense of their own
mathematical power. Operation sense in multiplication and division developed by
this level provides new avenues for creating number patterns. Remember, several
patterns may exist within one display. What a particular student sees may be
different from, but just as correct as the intended pattern.

**INSTRUCTION:**

Discuss patterns that surround us.
Use pattern strips to identify patterns and discuss the continuation of the
pattern. Use geometric shapes, pictures, objects, and numerals to create
patterns.

**ASSESSMENT
STRATEGIES:**

Using unifix
cubes, pattern blocks, or similar objects (e.g., buttons or rocks), create a
pattern consisting of at least three repetitions. Ask students to add the next
element(s) and to discuss/write why they chose this element(s).

Create or ask students to create
number patterns. Encourage the use of multiple-digit numbers. Ask students to
add the next element(s) and to explain why they chose those numbers.

Assign a project of finding patterns
within the community, nature, or the school building. Students might cut and
paste pictures in a notebook, draw diagrams of their findings, or write
narratives describing their patterns.

Ask students to create their own
wrapping design or textile design using a repeating pattern.

Integrate assessment of patterns
with science lessons concerning aspects of nature.

** **

**CORE
COMPETENCY:**

A Demonstrate
an understanding of numbers

**KEY
SKILLS:**

** A5 Represent and describe mathematical
relationships.**

**CONCEPT
ANALYSIS:**

Mathematical relationships are those
numerical patterns that consistently hold true. Children often discover these
patterns, and teachers should capitalize on these discoveries. Representing
these discoveries with numbers and symbols gives learners a sense of the power
of mathematics and of their own mathematical ability. Use of properties
introduced earlier should be continued and extended. The need for and the use
of the parenthesis should be stressed at this level. The distributive property
and its role in tying the operations of addition and multiplication together
should be carefully developed. Its application in mental computation should be
demonstrated and practiced, for example, 99 x 16 = 16 x (100 - 1) = (16 x 100)
- (16 X 1). Learning the properties for the sake of being able to name them is
non-productive; learning the relationship in order to use it to simplify work
or to generalize a procedure shows students the power of mathematics.

**ASSESSMENT
STRATEGIES:**

Give students multiplication
problems and ask them to explain how to do the problem mentally or in the
"easiest" way. Be flexible in scoring by allowing for alternative
strategies.

The following is an example:
"Explain the easiest way to multiply 12 times 98 doing as much of the work
in your head as possible and give the product." Students might say 12 x
100 - 12 x 2 = 1176, or 10 x 98 + 2 x 98 = 1176.

Ask children to discuss/write about
how addition, subtraction, multiplication, and division are different.
Encourage them to give examples of properties that work one way with one
operation and a different way with another.

**CORE
COMPETENCY:**

B Apply
the basic operations in computational situations

**KEY
SKILLS: *State
Tested**

** *B7 Use open
sentences to express mathematical relationships.**

**CONCEPT
ANALYSIS:**

Exploring open sentences should be
viewed as a natural extension of work with operations. The use of manipulatives can help students transform everyday problem
situations into number sentences. Students should have an understanding of the
equal sign, the operations, and the properties of numbers (associative,
commutative, distributive, inverse, and identity). Instruction should include
the use of parentheses and more than one operation within a sentence.

Physical materials and pictorial
displays should be used to help students recognize and create patterns and
relationships. Observing varied representations of the same pattern helps
students identify its properties. The use of letters and other symbols in
generalizing descriptions of these properties prepares students to use
variables in the future. This experience builds readiness for a generalized
view of mathematics and the later study of algebra.

**INSTRUCTION:**

** **Provide examples of
open sentences involving addition, subtraction, multiplication, and division of
whole numbers.

Do activity sheets in File B-7.

Students make up their own open
sentences and trade with a partner for solutions.

**ASSESSMENT
STRATEGIES:**

** **Assessment tests,
File B-7.

**TEST
CONTENT SPECIFICATION:**

__Component
Skills:__

A. Identify the open sentence which
represents a computational situation involving addition and/or subtraction.

B. Identify the computational
situation which represents an open sentence involving addition and/or
subtraction.

C.
Identify the open sentence which represents a computational situation involving
multiplication and/or division.

D. Identify the computational
situation which represents an open sentence involving multiplication and/or
division.

__Specification:__

Students are asked to identify the
open sentence which represents a given computational situation or the
computational situation which is represented by a given open sentence.

__Sample
Item:__

1. Mike was asked to count the total
number of students in his class. He gave a balloon to each student and then
counted 14 red, 7 blue, and 4 pink balloons. Which expression illustrates the
number of students in the class?

A. 4 + 14 + 7 + 4 = q

B. 14 + (7 x 4) = q

C. 4 x (14 + 7 + 4) = q

*D. 14 + 7 + 4 = q

**RESOURCES:**

File B-7.

**Sample Learning Activities for Patterns and
Relationships**

Given a picture of a design in a quilt, describe all
the patterns you see. Explain what could
be added to the design to generate yet another pattern.

Use square tiles to make the first four rectangles in
the design below. Count the number of
squares in each rectangle and the number of units around each rectangle. Write a rule you might use to find the number
of squares and the number of units around for the fifth rectangle.

a. q qq qqq qqqq

b. q qq qqq qqqq

qq qqq qqqq

qqq qqqq

Given
a hundreds chart with a few numbers darkened on the first two rows, complete a
pattern over the entire chart. Describe
or classify the numbers that have been darkened over the entire chart. Extension: Given a blank number hundreds
chart and counting disks, make an original pattern. Change the pattern by moving a designated
number of counting disks. Describe both
patterns and their differences.

Using the constant function on a calculator,
construct an input/output table of numbers.
Describe the relationships. Graph
the results.

Have the students repeat a rhythm pattern begun by
the teacher (e.g., clap-clap-clap-stamp-stamp-stamp-clap-clap-clap or
snap-clap-snap-clap-snap).

Have the students initiate or extend geometric
patterns or codes 9e.g., circle-square-circle-square
or AB-AAB-AAAB).

**IX. Mathematical Systems and Number Theory**

**Content
Overview**

*Mathematical systems and number theory offer many rich
opportunities for explorations and generalizations. Instruction should facilitate student
development and understanding of the underlying structure of arithmetic through
informal explorations to emphasize the reasons why various kinds of numbers
occur, commonalities of various arithmetic processes, and relationships between
and within the set of real numbers. Without an understanding of mathematical
systems and number theory, mathematics is a mysterious collection of
facts. With such an understanding,
mathematics can be seen as a beautiful, cohesive whole.*

* Mathematics is a coherent body of knowledge, not a
mere collection of isolated facts and rules.
Numbers can be expressed in multiple forms and there
is a relationship between and among the operations of addition, subtraction,
multiplication, and division.*

* *

**All fourth grade
students should know**

1. That the basic operations (addition, subtraction,
multiplication, and division) are related to each other.

2. The concepts of factors and multiples in relation
to multiplication and division.

**All fourth grade
students should be able to**

** **A. Develop
the need for whole numbers, integers, and rational numbers (e.g., fractions and
decimals) by looking for patterns and relationships to solve problems (NCTM
Standard 6; MO 1.6, 3.2, 3.3)

B. Develop and use number operations and order
relations for decimals (money) (NCTM Standard 6; MO 1.6, 3.2, 3.3)

C.
Develop an understanding of how basic arithmetic operations are related to one
another (NCTM Standard 6; MO 1.6)

D. Develop and use number theory concepts (e.g.,
factors, and multiples) in problem solving (NCTM Standard 6; MO 1.6, 3.5)

**CORE
COMPETENCY:**

A Demonstrate
an understanding of numbers

**KEY
SKILLS:**

** A5 Represent and describe mathematical
relationships.**

**CONCEPT
ANALYSIS:**

Mathematical relationships are those
numerical patterns that consistently hold true. Children often discover these
patterns, and teachers should capitalize on these discoveries. Representing
these discoveries with numbers and symbols gives learners a sense of the power
of mathematics and of their own mathematical ability. Use of properties
introduced earlier should be continued and extended. The need for and the use
of the parenthesis should be stressed at this level. The distributive property
and its role in tying the operations of addition and multiplication together
should be carefully developed. Its application in mental computation should be
demonstrated and practiced, for example, 99 x 16 = 16 x (100 - 1) = (16 x 100)
- (16 X 1). Learning the properties for the sake of being able to name them is non-productive;
learning the relationship in order to use it to simplify work or to generalize
a procedure shows students the power of mathematics.

**ASSESSMENT
STRATEGIES:**

Give students multiplication
problems and ask them to explain how to do the problem mentally or in the
"easiest" way. Be flexible in scoring by allowing for alternative
strategies.

The following is an example:
"Explain the easiest way to multiply 12 times 98 doing as much of the work
in your head as possible and give the product." Students might say 12 x
100 - 12 x 2 = 1176, or 10 x 98 + 2 x 98 = 1176.

Ask children to discuss/write about
how addition, subtraction, multiplication, and division are different.
Encourage them to give examples of properties that work one way with one
operation and a different way with another.

**CORE
COMPETENCY:**

B Apply
the basic operations in computational situations

**KEY
SKILLS:**

**INSTRUCTION:**

Discuss and demonstrate examples of
addition and multiplication properties.

1). Associative or grouping property
(3 + 2) + 4 = 9; (6 x 4) x 2 = 48; 6 x (4 x 2) = 48;

3
+ (2 + 4) = 9

2). Commutative or order property 8
x 3 = 24; 3 x 8 = 24; 7 + 5 = 12; 5 + i7 = 12

3). Identity property or zero
property 5 x 1 = 5; 3 + 0 = 3; 4 x 0 = 0

**Sample Learning
Activities for Mathematical Systems and Number Theory**

Use a set of manipulatives to
explain the relationship between and among the operations of addition,
subtraction, multiplication, and division.

Given a set of student-generated fraction or decimal
cards, place the cards in order. Explain
the process used.

Use manipulatives (e.g.,
color tiles, Unifix cubes) to represent all the
possible rectangles for the numbers 1-25.
Explore and discuss the relationships of numbers that have multiple
representations, a single representation, and can be represented by a square.

Given a hundreds chart or 0-99 chart, search for
patterns like doubles, odd numbers, even numbers, numbers with a seven in them,
numbers whose digits add to ten, etc.

Given a temperature of twenty degrees at 8:00 p.m.,
develop a model to represent the temperature at 7:00 a.m. if the temperature
drops an average of four degrees per hour.

**X. Discrete Mathematics**

**Content
Overview**

*Many applications tied to the principles in business,
to computer science, and to other real-world problems involve an area of
mathematics called "discrete mathematics." The word "discrete" is defined as
"separate or distinct."
Discrete mathematics is the study of points, ideas, and objects that are
separate from each other or distinct.
John Dossey, in his article DISCRETE
MATHEMATICS: THE MATH OF OUR TIME (NCTM Yearbook, 1991) states that
"discrete mathematics allows students to explore unique problem situations
that are not directly approachable through writing an equation or applying a
common formula." Modeling or other
forms of representation are often required to help students visualize the
situation as well as other areas of mathematics. Discrete mathematics builds upon and extends
the mathematics in the first nine strands of this document. Algorithmic thinking, graph theory,
probability and counting techniques, mathematics of social decision-making,
matrices, and recursion are all included in this strand. Discrete mathematics promotes the making of
mathematical connections, provides a setting for problem solving involving
real-world applications, takes advantage of a technological setting, and
provides the opportunities for critical thinking and mathematical reasoning.
(Kenny, NCTM Yearbook, 1991) Through discrete mathematics, students are
involved in a variety of experiences that build on mathematics taught in the
earlier strands in this document.
Connections to real-life problem solving allow students to look at
possible career choices to take them beyond the year 2000.*

* Objects may be counted or they may be compared to
determine if they are longer or shorter, larger or smaller, or identical or
different. Numbers may be placed in
sequential order and simple patterns recognized. Visual objects and print materials illustrate
both two- and three-dimensional figures.*

**All fourth grade students
should know**

1. Numbers in sequence to count
objects.

2. Definition of
"more" and "fewer."

3. Definition of
"same" and "different."

4. Definition of
"shortest" and "longest."

**All fourth grade
students should be able to**

** **A.
Determine what should be counted in a set of objects, and actually count the
objects (NCTM Standard 6; MO 1.8)

B. Predict whether the objects contain more or fewer
of one subset than the other (NCTM Standard 6; MO 2.2)

C. Illustrate or explain how the subset of objects
are the same or different (NCTM Standard 3; MO 1.8)

D. Identify and discuss overlapping subsets of
objects (Venn diagrams) (NCTM Standard 3; MO 2.2)

E. Create algorithms based on constructing meaning from
explorations (NCTM Standards 7 and 8; MO 1.6, 3.4, 3.6)

F. Determine a path through a maze, whether a street
network could be traveled going over each street one time, and the shortest
distance traveling on a network of roads or streets (NCTM Standard 9; MO 2.2,
3.3, 3.4)

G. Apply the concept of "fair division" to
real-world situations (NCTM Standard 1; MO 2.2, 3.2, 3.3, 3.4, 3.7)

**CORE
COMPETENCY:**

A Demonstrate
an understanding of numbers

**KEY
SKILLS:**

** A8 Read and write decimals through the
hundredths.**

**INSTRUCTION:**

** **Use decimal squares
to acquaint students with the part-to-whole relationship, terminology, and
notation. Using decimal squares students model a decimal and then write
corresponding decimal. Practice reading decimals aloud as a group.

**ASSESSMENT
STRATEGIES:**

** **Students write
decimals to correspond with decimal squares.

**RESOURCES:**

** **Decimal
Squares (Scott Resources).

Student-made place value charts.

**CORE
COMPETENCY:**

A Demonstrate
an understanding of numbers

**KEY
SKILLS:**

** A9 Compare decimals
through hundredths.**

**INSTRUCTION:**

** **Use decimal squares
to compare decimals.

**ASSESSMENT
STRATEGIES:**

** **Given 2 decimal
squares, student will write an equality or inequality statement.

**RESOURCES:**

** **Decimal
Squares (Scott Resources).

Student made place value charts.

**Sample Learning
Activities for Discrete Mathematics**

** **Use
concrete materials (manipulatives) and/or diagrams to
illustrate a variety of problem solving situations with emphasis on thinking
and common sense. Sequence then count a
set of objects. Group the objects to
count by 2's, 3's, 5's, and 10's. How
does this relate to the concept of multiplication? How many different ways can three counters be arranged in a row?
How do you know you have found all the ways? Explain or draw a diagram.

Investigate algorithms
in the study of mathematics to help organize and structure thinking. An algorithm is a sequence of instructions
that, if followed for an operation, will always lead to a defined result. Use concrete materials to invent algorithms
to solve addition, subtraction, multiplication, and division problems. Or sequence a given set of pictures. OR Illustrate a sequence of instructions on
how to get to their house from the school, or the order in which they do things
when they are getting ready for school in the mornings.

Examine a city map and identify two routes to get
from the elementary school to the high school.
Determine the shorter route, and then talk about or write about why one
route is better than the other.

Given a collection of objects, sort the materials
into two, three, or four sets (these sets could be placed within string
loops). record
the attributes for each set on a sheet of paper. Have another group of students observe the
sets and try to identify the attributes.

**SUGGESTED PERFORMANCE
ACTIVITIES**

__SHOPPING
TRIP__

OBJECTIVE:

MATERIALS:
Paper, pencil, price guides.

INSTRUCTIONS:
Each child has $50 to spend. Using the price guides, choose 7 items you would
like to purchase. Be careful not to spend more than $50. List the items in order
starting with the item most wanted to the item least wanted. Include the
purchase price. Write down why you want each item and what you plan to do with
it. Using your 7 items create a multi-step story problem and then solve the
problem.

RUBRIC:

4.
List of 7 items; list totals no more than $50; reasons and plans stated for
purchasing items; story problem is multi-step, problem is clear, the question
is well stated, and the problem is answered correctly; prices for items are
correct.

3.
List of 7 items; list totals no more than $50; reasons and plans stated for
purchasing items; story problem is clear, the question is well stated, and an
attempted answer; prices for items are correct.

2.
List of 7 items; list totals no more than $50; reason for purchasing is stated;
story problem written with an answer; prices given for each item.

1.
A list; prices on the list; story problem written; list has a total; favorite
item listed first.

__TRAVELING
FUN__

OBJECTIVE:

MATERIALS:
United States map, rulers, scale of miles, paper,
pencil.

INSTRUCTIONS
(Teacher): Review map skills. Review rounding numbers to
nearest hundred. Introduce $10 per 100 miles scale to be used in
figuring transportation cost. Review letter writing skills.

INSTRUCTIONS
(Student): Select a place you would like to visit within the United States.
Using a scale of miles, determine the distance from Butler. Then,
round that distance to the nearest one hundred. At a rate of $10 per 100
miles, determine the cost of transportation. Write a letter to the place you
have selected requesting visitation information.

RUBRIC:

4.
Calculate distance; round miles to nearest one hundred; calculate
transportation cost; write a letter; letter has no spelling mistakes and has
complete sentences.

3.
Calculate distance; round miles to nearest one hundred; attempt to calculate
transportation cost; write a letter; letter has complete sentences.

2.
Attempt to calculate distance; round miles to nearest one hundred; attempt to
calculate transportation cost; write a letter; letter is readable.

1.
Select a destination; attempt to calculate distance; attempt to round miles to
nearest one hundred; attempt to calculate transportation cost; attempt to write
a letter.