LEARNING EXPERIENCE
JUNK YARD MATH

By: Leota Crary
Madison Central School

A Third Grade Place Value Lesson

1. LEARNING CONTEXT

Standard 1: Analysis, Inquiry, and Design: Mathematical Analysis

1. Abstraction and symbolic representation are used to communicate mathematically.

• Students use basic mathematical symbols to represent situations.

Standard 3: Mathematics

1. Mathematical Reasoning: Students use mathematical reasoning to analyze mathematical situations, make conjectures, gather evidence, and construct an argument.

• Students explain to others how they went about solving a numerical problem.
• Students use a variety of strategies to understand problem situations.

2. Number and Numeration: Students use number sense and numeration to develop an understanding of the multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and the use of numbers in the development of mathematical ideas.

• Students understand the basic meaning of place value.
• Students understand that numbers are symbols used to represent large quantities or attributes of real-world objects.
• Students understand symbolic, concrete, and pictorial representations of numbers.
• Students count whole numbers.
• Students understand basic whole number relationships.
• Students use models to represent, order, and compare numbers.
• Students understand the relative magnitude of whole number…

3. Operations: Students use mathematical operations and relationships among them to understand mathematics.

• Students add and subtract whole numbers.
• Students know a variety of computational algorithms and operational methods.
• Students know single digit addition and subtraction facts.
• Students understand the inverse relationship between addition and subtraction.

Standard 2: Students will access, generate, process, and transfer information using appropriate technologies.

2.1 Information technology is used to retrieve, process, and communicate information and as a tool to enhance learning.

2.1.1 Students use a variety of equipment and software packages to enter, process, display, and communicate information in different forms.

This learning experience is a core component to the understanding of place value. It provides the students with the opportunity to manipulate quantities of numbers, to facilitate the understanding of number sense for large numbers, to facilitate the understanding of number sense for large numbers, and understand the regrouping process required for addition and subtraction.

-Student needs to perceive that sets of ten represent a single entity.

-Students will need to understand that the position of the digits in a number determines what it represents.

-Previous experiences with other bases is preferred but not required.

2. PROCEDURE

Students: The students will set up their workspace with pre-determined materials:

• Place value board (if not purchased then a sheet of paper divided into three sections labeled junk-ones, garbage cans-tens, and garbage trucks-hundreds)
• Junk box with over 100 objects
• Portion cups referred to as garbage cans
• A larger bowl or container to represent a garbage truck
• Recording sheets: strip, matrix, or addition/subtraction recording sheets (depending on lesson)

Students will record work shown on place value board onto recording sheets.

Students begin the lesson by building numbers with the junk on the place value board and recording on the place value strip. The first number recorded is 0,0,0. The students are instructed to add a piece of junk to their place value board on the junk side or ones which represents their homes or bedrooms. They record this amount onto their recording strip as 0,0,1. This represents one single item.

Additional pieces of junk are added to the place value board and are recorded until they reach 0,0,9. When the student places the tenth piece of junk onto their place value board emphasis needs to be made that on the ones side we can only have up to nine pieces. When we reach ten, we build a ten or in this case we need to pick up our junk and put it into a garbage can. The garbage can needs to be placed outside for a pick up which is the tens side of the place value board. the students record this as 0,1,0 on the recording strip.

Though out this experience students will respond to the numbers created on the place value boards. They will recognize 0,1,0 as being number ten in our number system. These type of connects are what the teacher should want the students to make. Encouragement of these comments enables other students to make the connections. Sometimes other students can be the best teachers.

Students continue to add a piece of junk to the place value board, and recording until they reach 0,9,9. At this point, when the students add one more piece of junk and build the next garbage can or ten, they then have ten tens or ten garbage cans. Since our number system doesn't allow us to have more than nine in each position, then the students need to group the garbage cans, tens, into the larger container which represents a garbage truck. they now record this number on the recording strip as 1,0,0.

At this point, the students can continue but by removing the garbage cans from the garbage truck placing them back on the blue side of the board or the ten's side. They take one garbage can and dump it back into the bedroom, the white side of the place value board, or ones and take one away. the student then records the number as 0,9,9. The students continue to remove one piece of junk and recording until they reach 0,0,0. At this point the student should look for patterns on the strip and loop the patterns with different colored crayons.

This process can continue for a few days depending on the needs of the students.

At the end of each session or few days, students should discuss and then compile a written reflection on the process used in JunkYard Math. These reflections can be recorded in their Math Journal or typed and added to their computer folders.

Students can follow the above procedure again but this time they can record on a matrix. This time they only record to 100. Then they look for and loop patterns. again, a written reflection component can be added at this point.

Additional activities before beginning addition and subtraction could include random building of numbers using dice and comparing them with a partner for greater than and less than. Students would roll two dice and arrange the dice into the largest number possible from the two digits or the smallest number possible. Then the students would build the numbers out of junk onto their place value boards.

Students can play Race to a Garbage Truck or Race to a Clean House. These games are played with a partner. Each student begins with a clean place value board. Students roll a die, preferably one labeled 4-9, and add the amount of junk to their board. The next student rolls and repeats the process on his/her place value board. The students continue rolling, building, regrouping, until one player reaches a garbage can or 100.

The Race to a Clean House game is a subtraction game where each students begins with a garbage can or 100 and rolls the die to determine the amount of junk to remove. The play continues until one student has completely cleaned the house and reaches 0.

Addition Activity:

Students will roll two dice (dice are labeled 4-9) and create the largest two digit number. Students would record that number on their addition recording sheets and build that number onto their place value board. Students then roll one die, add that number to the recording sheet and add that amount of junk to the place value boards. The process of regrouping or rebuilding should occur. Students should record on the recording sheets the regrouping process (carrying) and the sum. Students repeat this process for the number of required examples on the recording sheet.

Subtraction Activity:

Students will roll two dice and create the largest two-digit number. Students will record that number on their subtraction recording sheets and build that number onto their place value board. Beginning subtraction problems were made with students subtracting a specific number from the double-digit number rolled, i.e. 9. This is to ensure some regrouping necessary to remove that number from the ones place. Future session the students would roll one die, this number is recorded on to the subtraction-recording sheet. The students will remove that amount of junk from their place value boards regrouping as needed. Larger minuends can be created by using two dice labeled 1-6. Students need to be aware that the second number rolled needs to be smaller than the first number.

3. INSTRUCTIONAL/ENVIRONMENTAL MODIFICATIONS

This experience allows all students in the classroom to participate. It is a hands on learning experience. The pacing for some students needs to be slower i.e. the adding of a piece of junk and recording time, but sessions for those students with just adding and verbalizing the amount can be repeated as other students are recording. pairing students allows the more advanced students to tutor the others. Later, as the students become familiar with the process, they can begin to record.

The students need to be grouped with a partner to share boxes of junk if necessary. They also need to be with a partner if they play Race to a Garbage Truck or Race to a Clean House. Students can also be grouped to discuss their findings after they loop the patterns. Students can be used to check one anther's papers, searching for the correct patterns.

4. TIME REQUIRED

Planning time is preparation time for the teacher. The most time is needed assembling an assortment of junk boxes. Colored macaroni and beans can be purchased in large quantities and divided amount the boxes. Erasers can also be purchased at various stores or through catalogs. These are usually purchased by the gross thus giving the appropriate amount needed per student.

Portion cups need to collected as well as margarine containers or cool whip containers or any other type of container larger enough to be the garbage can. Students can help with this collection process through a letter sent home requesting various containers.

Place value boards can be purchased or made by the teacher or students.

Dice can be made from wooden cubes labeled with a permanent marker.

Copy the matrix, strips, and any other recording papers.

Implementation: Forty minutes of class time for a period of a week or two is needed for implementation. Length of time depends on students understanding and ability to transfer knowledge to addition and subtraction problems and understanding of the base ten system.

Assessment is ongoing. How students transfer the knowledge of regrouping with manipulates to the paper/pencil activities of addition and subtraction regrouping is the determinate for length of time for individual students. Some students require the use of manipulative for a much long time period that others. These students should be allowed to continue to use these manipulative until they reach a level of comfort without them.

5. RESOURCES

Student supplies: boxes of junk (junk boxes are small containers of assorted objects. Suggested junk box ideas include small erasers, colored macaroni, buttons, jewels, beans, centimeter cubes, pompoms, or any other small item that will fit in a small container or box plus having a quantity of over 100.), portion cups, margarine tubs, place value board (or a piece of paper divided into sections of ones, tens, and hundreds.), dice labeled 0-6 and 4-9, recording strips and matrix form. Addition and subtraction forms can be made that reflect the place value board.

Teacher supplies: a set of above materials for the overhead.

6. ASSESSMENT PLAN

Initial assessment is obtained through general observation of students as they work through the building of numbers process. Paper pencil assessment can be obtained from the recording strips and matrix and then from the addition and subtraction papers. Folders containing all of the students' work can be kept to document students' progress through the learning experience.

The Internet offers a site with flash cards for students to practice various addition and subtraction problems. Since it is on the Internet, it offers a twist to traditional flash cards or paper pencil practice.

Internet Site: www.aplusmath.com

Students can access this site at assigned computer time. An added assessment to this unit could include the student choosing one of the flashcard equation and writing a description of the process used to solve the equation through the use of junk and the place value boards. These reflective papers could be typed using Microsoft Word, creative Writer, or some other writing program on the computer. These can be posted on the bulletin board for students to review.

Various commercial companies also offer programs which use the Base Ten Blocks as manipulative on the computer providing practice with place value.

7. STUDENT WORK

As can be seen by the students' work, some students see the patterns reflected in the number system and can loop them, looking for errors in recording. Other students do not see the errors and continue to loop regardless of the patterns. Addition and subtraction work shows students the process of addition and subtraction with regards to regrouping and borrowing.

8. REFLECTION

Overall, I am pleased with the results of this learning experience. Students that were unable to understand WHY you carry or WHY you borrow could explain the process following their experience and they net with much more success on paper/pencil activities of addition and subtraction. This also lends itself to the process of multiplication and the regrouping necessary to perform division.

The use of manipulative in the classroom to teach mathematics has been on the rise for several years. "Best" classroom practice shows that students learn best and better when they understand the WHY of the process. Through the use of these manipulative and exploring the base ten system, students understand the why to regrouping and borrowing. "Oh, I don't have enough to subtract from so I need to borrow from the tens." The goal is for students to know they cannot subtract 8 from 6 and get 2.