Innovative Number-Line Manipulatives Help Kids Grasp Concept of Rounding

EducationWorld is pleased to present this article contributed by Jim Franklin, an inclusion special education teacher at Elm Street Elementary School in Rome, Georgia. He designed low vision and Braille math manipulatives that are recommended by the American Printing House for the Blind. Franklin maintains the Web site www.slidearoundmath.com, where schools can purchase the manipulatives pictured and described below.

Number lines are valuable visual aids for students to understand the concept of rounding whole numbers for all students. Most students have achieved some success with the strategy of drawing number lines to master their rounding standards. However, would students have achieved success faster and retained the process of the rounding standards longer if other innovative methods had been available? Would challenging classrooms with student behavior problems improve their academic growth if they had an alternate strategy to hold students accountable for their work?

At the beginning of the 2011-2012 school year, my assistant special education director asked to observe my fourth-grade math inclusion class with a math coordinator from a local college. I welcomed the upcoming visit but wondered if any new strategies/interventions had been successfully implemented by other teachers with the concept of rounding whole numbers. I asked other math teachers in my school and searched for ideas on the Internet.

Honestly, there were not a whole lot of options. I only saw blocks, dry erase markers and boards, and number lines. Other than those options, paper and pencil were the last resort. The last thing I wanted my visitors to observe were towers being built out of blocks or off-task drawings on dry erase boards. I could not use a number line in my lesson because the longest one available only goes to 100; we were working with numbers greater than 100. Although all four options have been used for years and have had some success, I wanted math manipulatives that could make an immediate impact on educational performance and not be considered a “toy” by my students. Then I had an idea.

Incorporating movable, interchangeable slides, I created a number line system that can round whole numbers up to 10,000,000. It can round numbers to the nearest 10, 100, 1,000, 10,000, 100,000 and 1,000,000. When I began to show this concept to my colleagues, the response was overwhelmingly positive! Teachers began to ask me to help create manipulatives to address other mathematical standards. Therefore, I also developed manipulatives that involve weight, elapsed time, decimals/ money and fractions.

During this process, I consulted with math teachers and specialists, administrators, parents and students from several different schools and school systems. I also consulted with an occupational therapist, a hearing specialist and vision-impaired specialist. Of all of the stakeholders with whom I have worked throughout the initial part of the developmental process, I most value the student input. After all, they are the ones who will use these manipulatives as a vital part of their classroom instruction.

The ability to draw a number line is a struggle for some students. Students struggle to draw a number line that is complete and accurate, such as putting the appropriate numbers at the beginning and end of the number line. Other students lack the ability to put the numbers in the appropriate position on the number line.

One of the biggest obstacles for some students is the ability to draw the number line and listen to their teachers simultaneously. They become so focused on drawing the number line that they do not understand the directions and/or how to correctly round the whole numbers. If the number line is not drawn perfectly in the mind of a student, he can become very frustrated and begin to write and then erase over and over and eventually cause a disruption in the classroom. Other students with attention deficits draw pictures and shapes as well as waste valuable class time because they need to sharpen their pencils or need a new eraser.

When students use dry erase boards and markers to draw number lines, students often exhibit the same problems when they use this strategy as opposed to paper and pencil. Some students waste valuable instructional time by coloring their clothes, fingernails and other parts of their body. Because markers can cause a mess if they are not used properly, instructional time is wasted due to students needing to wash their hands; additional time is lost due to distributing erasers and dry erase boards, and replenishing lost and dried markers. Even though the strategy of drawing a number line is very common, students who do not benefit from drawing number lines are low-vision and blind students as well as those with fine motor deficits; these students need assistive technology to help master the rounding math standards.

Rounding whole numbers is an essential standard for students to master by fifth grade and is considered a valuable life skill. With the rigor of the Common Core, teachers and students must be prepared to move quickly in order to get in all of the other standards. Because the math standards build on one another, it is imperative that they learn how to round to the nearest 10 or 100. If students are unable to round to the nearest 10 or 100, they exhibit more and more difficulties as they round to the nearest 1,000, 10,000, 100,000 or 1,000,000. A deficit in rounding whole numbers also adversely impacts the understanding of more challenging math reasoning concepts. Students with poor number sense increasingly lose a concrete understanding of numbers as numeric values increase and problems become increasingly abstract.

Visual aids are necessary to aid in all phases of instruction. In most cases, classes have students that can be placed in low, medium and high groups. There are also situations when there is a very high and/or low-functioning student in a class. In a third-grade classroom, for example, the class is expected  to achieve mastery of the standard rounding whole numbers to the nearest 10,000. In this classroom, there are some students who cannot round to the nearest 10,000 because they have not mastered the standards of rounding to the nearest 100 or 1,000.

In addition to having below-grade-level and grade-level students in a class, there are gifted students who have already mastered the current standard and are at the point of needing to begin instruction on rounding to the nearest 100,000. In some instances, gifted students are required "go through the motions" because their peers are currently working on the rounding to the nearest 10,000 standard. Teachers are responsible for meeting the needs of all their students and face increasing pressure to have all students meet district, state and federal assessment standards. As a result, differentiation of instruction must be implemented so that all students are challenged and can build a strong math foundation.

Below is a sample classroom that requires the teacher to differentiate instruction by using math manipulatives to meet the needs of all her students. The teacher will have each group, while snapping her fingers for each number, state in unison the numbers on the slides in the correct order, beginning with 0, as well as the numbers under the semi-circles. This strategy will improve the students' awareness and sequencing of larger numbers and familiarity of the numbers on the number lines. The teacher will also mention the visual aids on the manipulatives, especially the important line in the middle. Students use the line to determine the answer when rounding, based on whether the number is to the left or right of it and proximity to the number in the window. Some students use the arrows to help find their answers.

Group 1

Group 1, which is the significantly below grade-level group, consists of three students that exhibit moderate-severe processing and language deficits and require manipulatives whenever possible to help achieve mastery of any given math standard. These students are far below grade level, receiving intensive interventions. Group 1 students often need to have their assignments modified, receive extra time up to one hour, and questions and answers read to them on classroom and standardized tests in a small-group setting. Because all three students have Individualized Education Plans (IEPs), the teacher must provide "Visual aids and use of manipulatives to support instruction," as stated on their IEPs' "Accommodations" pages. They are currently at the progressing level toward mastery of rounding whole numbers to the nearest 100.

Problem: Round 823 to the nearest 100

Directions for 100X Manipulative:

1. Move slide on the left to 800

2. Move slide on the right to 900

3. State the number slowly (823) and place finger on 23

4. Determine answer in three possible ways:

a. Proximity of finger placement to 800

b. The red arrow indicating 49 or less

c. Number is to the left of the red line in the middle

Note: The manipulative will illustrate that the 3 in the ones place is not important in determining the correct answer.

Group 2

Group 2, which is slightly below grade level group, consists of four students who have mastered rounding whole numbers to the nearest 1000. These students are at the emerging level of mastery of rounding to the nearest 1,000 because they lack consistency and demonstrate weak number sense of larger numbers. Group 2 is similar to Group 1 because they require extended time due to below-average process skills and need to have differentiated class assignments because they are below grade level.

Problem: Round 5,863 to the nearest 1,000

Directions for 1,000X Manipulative:

1. Move slide on the left so that 5,000 shows in the window

2. Move slide on the right so that 6,000 shows in the window

3. State entire number slowly (5,863) and place finger on 800

4. Determine answer in three possible ways:

a. Proximity of finger placement to 6,000

b. The sky blue arrow indicating 500 or more

c. Number to the right of the red line in the middle

Note: The manipulative will illustrate that the 6 in the tens place and 3 in the ones place are not important in determining the correct answer.

Group 3

Group 3, which is the at- grade level group, consists of 10 students. These students are currently progressing toward mastery of rounding whole numbers to the nearest 10,000.  Because several of these students have siblings who participate in activities after school and many of their parents work second shift, they occasionally have difficulty finding help with their homework.  As a result, they take their manipulatives home to solve their math problems and check their work.

Problem: Round 58,753 to the nearest 10,000

Directions for 10,000X Manipulative:

1. Move slide on left so that 50,000 is in the window

2. Move slide on right so  that 60,000 is in the window

3. State entire number slowly (58,753) and place finger on the 8,000

4. Determine answer in three possible ways:

a. Proximity of finger placement to 60,000

b. The gold arrow indicating 5,000 or more

c. Number is to the right of the red line in the middle

Note: The manipulative will illustrate that the 7 in the hundreds place, 5 in the tens place, and 3 in the ones place are not important in determining the correct answer.

Group 4

Group 4, which is the above grade level group, consists of 2 students. These students are usually the first to finish their assignments in math as well as in all of their other classes. As a result, one of the two students usually begins to talk to his peers and exhibit other off-task behaviors. The student's off-task behaviors often create a distraction, causing the other students to lose focus and negatively impact their academic progress. The teacher will differentiate instruction and challenge her brightest students by rounding whole numbers to the nearest 100,000.

Problem: Round 843,592 to the nearest 100,000

Directions for 100,000X Manipulative:

1. Move slide on the left so that 800,000 shows in the window

2. Move slide on right so that 900,000 shows in the window

3. State entire number slowly (843,592) and place finger on the 43,000

4. Determine answer in three possible ways:

a. Proximity of finger placement to 800,000

b. The lavender arrow indicating 49,999 or less

c. Number is to the left on the red line in the middle

Note: The manipulative will illustrate that the 3 in the thousands place, 5 in the hundreds place, 9 in the tens place, and 2 in the ones place are not important in determining the correct answer.

All students learn in their own unique way, and it is our jobs as teachers to find our students' strengths and improve their weaknesses. Some students' strengths are noticeably visual, auditory or kinesthetic. Other students might favor a combination of all three learning styles. Most classes have one or two students that require the teacher to analyze assessments, teacher and parent observations and work samples.

The number line system to 10,000,000 has consistency in a non-stigma design, such as the line in the middle, arrows and semi-circles to show "which way to go." Students are able to point with their fingers, manipulate the slides to find the beginning and end numbers on the number line, and learn by listening to their teacher and peers in a collaborative setting. Most importantly, however, when the students are given time to check their work, they can discover whether their answer is correct by using the manipulative to check their own work. Teachers can explain and teach why an answer is incorrect to a student, but self-discovery will help retain the information, determine why their answers are correct or incorrect and hold students accountable for their work.

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