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 NimTM, 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 30 27

26 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 30 27

26 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:

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:

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:

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

 

 

 

 

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.