Decimals and Number Lines

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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.

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

Representing Fractions with Thinking Blocks

Many classrooms in my school are in the midst of reviewing fraction concepts. Throughout the school students are finding fractional pieces, converting fractions to decimals, and identifying fractions on number lines.  For the past week students in second grade have been identifying fractional parts.  Earlier in the week students completed the page below during a math station.  Students did well on the first two pages, but struggled a bit when identifying fractions on a number line.

Representing 3/4 in many different ways
Representing 3/4 in different ways

This was a challenge for some students as many are more familiar with identifying fractions within objects (in a circle/rectangle).  Moving from identifying fraction to placing them on a number line can be a stretch.  Many students have already started to decompose numbers and have completed “fraction-of” problems.  These types of activities have helped reinforce the number line and fraction connection.  Next week students will be assessed on the fraction unit and many classrooms move into geometry concepts.  Before focusing in on geometry, I wanted to give student an opportunity to visualize fractions and use them with more complex word problems.

As I was looking for supplemental material I came across a Tweet by Paula (@plnaugle). She referenced Thinking Blocks  as a resource that she uses with an interactive whiteboard. I looked into the site and thought that it might be useful for my grades 2-3 classes since the app allows students the opportunity to solve fraction problems visually.  Specifically, I downloaded the fraction app on the school iPads.   Yesterday a second and third grade class used this app in their classroom as a guided activity.  The app was introduced to the class and I modeled the different steps involved in solving the problems.

Modeling
Modeling

The students were then asked to find a comfy place in the room and complete a minimum of three exercises.  What’s nice is that the problems are picked at random, so students aren’t on the same problem at the same time.  There’s also a feedback box that assists in guiding students towards labeling the correct parts of the fractions.

Click to enlarge
Click to enlarge

I helped the students as needed, but many were able to use the virtual manipulatives and generated feedback to stay on track.  Some students completed three problems, while some went beyond and tried out five.  After about 12 minutes the class gathered and we reflected on the perseverance that was needed and celebrated successes. This activity gave students an opportunity to make mistakes and persevere.  I’ll be keeping this app in my repertoire for the future.

Shaving Cream and Math

Image by:  Salvatore


I’m always trying to find new ways to make math interesting and relevant. Generally, the more interested the students are in the instruction, the more willing they are to apply their learning.  This past week I used one common household item to teach my elementary math class about number lines.  I’m not the only teacher who has used this strategy in the classroom, but I’ve found encouraging results by doing so, that’s why I’m sharing.  I’ve provided a few pictures for those (like me) who need a visual representation before putting a strategy into practice.

Procedure

1.)  Have all the students clear their desks.  There shouldn’t be anything on the desks, including pencils, water bottles, etc.  During this time students get a little anxious in wondering what’s going to happen next.

2.)  The teacher takes out one or two bottles of shaving cream.   I used Babaso, available at the Dollar Tree.  This works much better than some of the more expensive shaving creams.

3.)  The teacher asks the students to predict how the class will be using the shaving cream to learn about math.  You might get some interesting responses with that question.  This may also gains student interest.

4.)  Go over the ground rules.  Everyone should roll up their sleeves, don’t fling the shaving cream at anyone in the class, don’t touch the shaving cream until directed, no one gets out of their seat, etc.

5.)  Go to each desk and spray a bit of shaving cream (4-5 seconds) in the middle of each desk.

6.)  Tell the students that they will be given a few minutes to “play” with the shaving cream.  Ask the students to make different types of polygons, rays, lines, etc. with the shaving cream.

7.)  The teacher models a few number lines on the whiteboard.  Students are asked to create their own number lines.  Ask the students to create multiple number lines.  Once a student creates a number line, the teacher reviews the work (could be a great opportunity to take a picture), gives the student a bit more shaving cream and then looks for another finished project.

8.)  At the end of this project there are a lot of sticky fingers.  The teacher hands out wet wipes or wet paper towels to the students.  The students clean their own desk and hands.

9.)  Before the students leave class, or sometime in the near future, the teacher asks the students to create three additional number lines (addition, subtraction, multiplication) on paper and turn their work into the teacher.

More Examples:

Shaving Cream and Math Ideas

Greenfield Exempt Schools

Mrs. Clayton’s Class Blog – Using Shaving Cream

Disclaimer (unfortunate but necessary) : The thoughts and opinions expressed in these pages are my own, and not necessarily the opinions of my employers.

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