Comparing Fractions - The Fourth Grade Discovery

I have been collaborating with teachers as we plan to make learning accessible to all.  When it comes to fractions, this is a real challenge.  I've really been grappling with why fractions are so challenging for our students and what we can do to support this learning.

The other day a teacher I was working with asked her class, "Are fractions numbers?"  To our complete surprise about a third of the students did not think they were numbers, but that there were numbers in fractions.  So with that misconception in mind, our fraction lessons have centered on concrete visual models and discovery.

In the world of nine and ten year olds, fraction exposure is so limited.  Yes, they hear about a 1/2 price sale or 3/4 of a mile, but their world thus far has been filled with whole numbers.  Their fraction understanding is in its infantile stages.

Hence - concrete models.  The students' tools include fraction bars, compliments of Math Coach's Corner and fraction number lines.

Our plan was to start by introducing the comparison of fractions with the same denominator.  We thought this would be a quick review.  Not!  The students actively used their tools to determine the size of 2/6 as compared to 5/6.  Yes, some quickly grasped the concept, but the majority of the children were relying heavily on their tools.

So we decided to take the approach of coming up with a theory and then testing this theory out.  The students noticed that the denominators were the same.  Then after several trials, they were ready to write a rule:

Next we posed comparisons with the same numerator:  3/4 compared to 3/10.  Again we took the same approach.  We asked the students what they noticed.  They developed a theory, tested it out several times using their tools, and eventually created a rule:

Yes, this took some time, but time so well spent.  The children really persevered and made the discoveries on their own.  As they realized their theories were actually strategies for comparing fractions, an air of confidence filled the room.

At the climax of this two day lesson, a student came up to the teacher and said that she use to not like math, but now she likes it.  Priceless!!!

I am learning so much from the kids - they need visual models, time to explore, discuss, and discover, and lots of repeated practice.  What do you do to foster fraction fundamentals?  Please share your ideas.  We all learn from each other.

Bedtime Math - There's an app for that!

We've all heard countless research on the importance of parents reading to their children.  The question is what about math?  NPR just published a captivating article on just that:  Where the Wild Fractions are: The Power of a Bedtime (Math) Story.

A University of Chicago study suggests that parents doing math with their kids can help to strengthen their kids' math achievement, even if it's as little as once or twice a week.  The article goes on to highlight  the Bedtime Math app, as the place to go for parents to do some math with their children.

So I checked it out and here's the run down:

1.  It's a FREE app!

2.  The problems are organized into levels:

• "Wee Ones" for preschool kids
• "Little Kids" for Kindergarten through Grade 2
• "Big Kids" for Grades 2 and onward (in my opinion it's mainly Grades 3 -5)
• "Sky's the Limit" for really big kids and grown-ups

3.  Each math problem starts with a little story to set the stage, many with illustrations.

 4.  There are three buttons to choose at the bottom of the screen:  Wee Ones, Little Kids, or Big Ones.  All the problems connect with the story scenario and the questions build off each other.  This is perfect for the parent that is sitting with children of different ages - the kindergartner gets to work on the Wee One questions and the 2nd graders tackles the Little Kids questions, all enjoying math together.

Wee Ones

Little Kids

Big Kids

5.  The answers are given - just click on the star!
6.  Sometimes, when the answer is given animations or a bonus question appears.
7.  On the average, there are three new math problems posted per week.  Plus, parents can opt to receive an email for new daily Bedtime Math Problems (just click on the envelop on the home page).
8.  By clicking on the "sorting" link, problems can be sorted by date or by title.
9.  If you click on the star listed before the title of the math problem, it goes to your favorites.  This makes it easy to go mark the problems you want to revisit another time.
10.  Lastly, you can sort the problems by categories.  Animals, nature, sports are just a few.

Of course, the best part of this "bedtime" math routine is sparking a discussion about the math. Asking an open ended question strengthens the child's reasoning skills.  For example:  What the problem asking?  How did you decide to solve the problem?  Does your answer make sense?  What are some other strategies you might try?

Bottom line - It's so easy to use, it's an incredibly engaging way to do some math with your kiddos, and it's free!  So, parents, add some paper and a pencil with this app to your bedtime routine and watch your children's math muscles grow.

By the way, if apps are not your thing, there is a series of three Bedtime Math books available.  So start a new routine and do the math!

The Size of the Whole Matters

How to deepen students' fraction sense has been a real challenge for me.  Our fourth graders just started a new fraction unit.  Yesterday they focused on brushing out the cob webs by making some simple visual representations of a given fraction.

When students were posed with this question:

                 Kyle read 2/3 of his independent reading book.  Justin also read 2/3 of his

                 independent reading book.  Kyle said he read more pages than Justin did.  

                 Justin said they both read 2/3 of their books so they read the same amount.  

                 Who is correct?  

95% of the students said Justin was right.  The one student that questioned that said she didn't think he was right, but also added, "I really don't know."  

So the teacher and I put our heads together.  We used a LearnZillion Lesson as a spring board for our lesson.  

As we thought about posing this inquiry scenario, our goal was to make this learning accessible to all. It's one thing to see this visual illustration, but we thought we could really reach students by having them simulate the planning of these two gardens.  Manipulates were the answer!

As the students used the hexagons and triangles, the partnerships instantly embarked in engaging conversations about the value of the triangle in each garden.  

 

We, the teachers, really gave the students control of their learning and boy, did they run with it.  Themes included:

The students even went on to share that Meg's garden would need to have 2/12 planted to equal 1/6 of Sam's garden - and we have not even introduced equivalent fractions yet.  

Of course, the true test is when students can explain their reasoning independently.  

So as I reflect on this lesson I know that making these fraction lessons as hands-on as possible will make the learning as accessible as possible for all learners.

Stay tuned as we continue our journey to deepen fraction understanding!