## Decimals and Number Lines

My third grade class is learning about decimals. Students have been identifying place value positions up the hundredths place. So far students have been successful in decomposing numbers into expanded form and using base-ten blocks to compare decimals.

Comparing decimals between the tenths, hundredths and thousandths proved challenging.  I was finding that some student were perceiving that a larger number indicates a greater value (0.1 compared to 0.09). I asked students to place decimals on an actual number line. This was where we ran into a few problems. There was a disconnect between comparing decimals with symbols and comparing them on an actual number line. Students understood how to use the greater, less than and equal sign but became confused once hundredths were introduced. After running into this issue multiple times, I was starting to find that some students could compare decimals, but didn’t understand where to place them on a number line.  Then maybe they didn’t understand the value in relation to a number line?

The next morning I ate my breakfast and paged through Teaching Student-Centered Mathematics, a book that I’ve been using this year.  This resource is a gem and I highly recommend any middle school or even upper elementary teachers to add it to their inventory. After reviewing a few a few different options I came across an activity from NCTM (page 151) that was placed in the book. I thought this might be a worthwhile activity for my third graders. The project asked students to place decimals on a number line between 0 – 1. The project also asked students to explain why they placed each number in a specific location.  I thought this might be a good way to assess whether students can translate their value of a decimal to a number line.   That morning I asked students to use dice, create a number line and explain why they picked each point on the line.

I collected the projects and had to reevaluate whether to proceed with the next lesson. Some students knocked it out of the park with some fabulous answers, while others needed some work. Regardless, it was the high-quality feedback that I was looking for and I was able to quickly address misconceptions.  That was an #eduwin situation.

Next week students will be adding and subtracting decimals.  That should be interesting as my students use the partial-sums and traditional algorithms.

## The Real Number Line

`Image by Winnond`

Approximately two weeks have passed since the new school year has started and I’m finding that the traditional number line (that many teachers have become accustomed to) needs an upgrade.  My math students are benefiting from the number line, but true understanding of numbers doesn’t come from a number line alone.  For the past seven years I’ve used a “typical” number line from -10 to 100 in my classroom.

Don’t get me wrong … the number line is helpful in teaching many number sense concepts.  In my opinion, the number line offers students a visual/spatial representation of the number system.  I  believe many numeracy concepts are built from understanding the system of numbers.  What is often missed, or not necessarily taught, while utilizing the number line are numbers that don’t fit the category of being whole.  For example, I generally don’t see pi or irrational numbers being part of a number line.

Recently I found a “Real Number Line” poster.  I was fortunate enough to find this poster and have utilized it to teach elementary students about the number system. I think it’s important to communicate that square roots, fractions, percentages, mixed numbers, etc.  should be included on a number line.

I actually created a practical follow up activity in response to this post here.

Instead of purchasing a poster, you could have the students create their own.  A few examples are found below:

4/25/12

I believe that Wolfram Alpha does an excellent job of emphasize the importance of a number line in the answer it provides.  The answer can be represented on a number line.  See the example below.

8/14/12

I’ve been reading How the Brain Learns Mathematics by David Sousa.

David emphasis the importance of the mental number line.  All humans have number sense.  For example:  studies indicate that the brain can decide that 60 is larger than 12, but it takes the brain a longer time to distinguish that 76 is less than 79.  It seems that when the digits are closer in value the response time of the human increased.  Visualizing many different forms of number lines would be beneficial and assist in developing better number sense skills at a young age.

I thought this quote was beneficial:

“The increasing compression of numbers on our mental number line makes it more difficult to distinguish the larger of a pair of numbers as their value gets greater.  As a result the speed and accuracy with which we carry out calculations decrease as the numbers get larger”

– David Sousa