10 Plus Ways to Introduce and Practice Naming Fractions
Teaching Fractions with Cuisenaire rods begins with naming fractions. But we don’t want to just learn fraction names.
We also want to students to gain a conceptual understanding of fractions.
With the help of Cuisenaire Rods, Gattegno and fun activities, students can find naming and understanding fractions an enjoyable journey.
Reading Meaning into Fractions
The key to successful fluency in fraction depends on the ability to read meaning into fractional expressions.
What do I mean by reading meaning?
One can become a complete pro at French scrabble and not understand a lick of it. No, I am serious. See for yourself here.
We can all agree that he may have mastered the spelling of lots of French words but that doesn’t count for much in speaking and understanding French fluently.
Yet, this is a great picture of how we fail students in mathematics. Take for example the butterfly trick to remember how to multiply fractions. Sure, the student can multiply fractions like a pro, but does it mean anything to them?
Could the student draw a picture of the expression ½ x ¼? A student able to create a pictorial representation of the expression is a student able to read meaning into the expression.
This is the understanding we need to be after as educators. We need students to read meaning into expressions. So how do we get students to read meaning into fractional expressions?
Mathematical Symbols as Operators and Describers
Understanding the meaning behind mathematical symbols helps a student read meaning into an expression.
There are two ways to perceive mathematical symbols. The first is as operators. They tell specifically what to do to the numbers.
The second way to perceive mathematical symbols is as describers of a relationship. They describe the number in relation to other numbers. Think of mathematical symbols as verbs that can also become adjectives.
The following language and tasks switch between “operating” and “describing.” Each bit of language helps the student build a foundation for reading meaning into fractional expressions. Let’s look at this more closely.
Fractions as an Operator and Describer
Last post, I walked through the language and tasks to give students a thorough understanding of subtractions. This understanding is rooted in helping students perceive the relationship subtraction has with addition and division.
The language progression ended at repeated subtraction. This repeated subtraction leads to a conceptual understanding of division, but it is more than that though.
Repeated subtraction also paves the way for fractional understanding. When a student subtracts the number 2 repeatedly from 12, the student finds there is no remainder.
With Cuisenaire rods, the student perceives with their eye that 12 is made up of 6 equal parts. Each part is equal to two. Note that the parts to whole language describes the relationship of one small part to a larger part.
However, if I said, “Divide 12 by 2”, I am telling you how to operate on the number 12. But what do I mean by operate?
An operation is a specific task. If I ask for one-half of the cake, I am telling someone specifically how to cut or divide the cake. I am telling someone how to operate on the cake.
Both the operation language and describing language is important for helping the student to read meaning into fractional expressions. The tasks below switch back and forth or emphasize one over the other.
Transition Tasks that Create Fluency
The first task is a transitional task that helps the student to see the connection between operations. A transitional task is a task that manipulates the rods in the same fashion as a previous or familiar task but changes the language to emphasize a new focus.
A transitional task helps provide students with a broader view of mathematics. If the student’s eyes and hands move easily between the concepts through Cuisenaire rods, they are better able to develop the language and understanding to move between concepts.
In the previous post, several transitional tasks helped students perceive the relationship between addition and subtraction and then finally division. This following task of repeated subtraction takes it a step further by transitioning into fractional language.
The Task and Language for Repeated Subtraction
Prepare the train orange plus red (aka rorange). Ask the students to find out how many light greens fit into the train.
When the student creates a light green train the length of the rorange, ask the student how many light green parts make up the whole of the rorange train.
Repeat this task with the purple rod and ask the same follow up question. Then repeat with the dark green and red rods. (You can also use the Barnyard Fraction Play Mat Free HERE.)
As the student observes the parts to whole, it prepares them to transition into learning the names for each part. You may choose to introduce the names for each part (one-half, one-third, one-fourth and so on).
Multiplication to Fraction
Gattegno Textbook 1 (FREE here) introduces fractions as an opposite relationship to multiplication. While the previous task is great for transitioning into observing fractions, I find this task better suited for introducing fraction names.
Task the student to build solid color trains equivalent to one color rod. Ask the student to read all the trains as blue equals 3 light greens, red equals two whites, orange equals five reds, etc.
Then describe the trains in a new way. Blue is 3 times as long (big) as light green. Red is two times as long as white. Orange is five times as long as red. Allow the student opportunity to read the rest of the trains in the same fashion.
Now it is time to introduce the inverse relationship to the student. Tell the student,
“In the same way that orange is seen as five times as long as red, red is seen as smaller than orange. We describe this smallness in fractions. Red is one-fifth the length of orange. Red is one-half the length of white. Light Green is one-third the length of blue.”
This language is describing a relationship. It is not operating on a number, that is it is not telling the student what to do to the number.
A Side Note on Color Names add clarity to Fractional Language
The debate to use number names or color names is a great debate. The introduction of fractional language highlights one benefit to using only color names in the beginning.
“Two is made of two ones. One part is one-half of two. The other part is also one-half of two. Together, the two parts are two halves of two.”
Boy, that is a lot of twos and ones that I imagine a young student misses the larger picture, the fractional relationship, sorting it all out. In contrast,
“Red is made of two whites. This white is one-half of red. This other white is also one-half of red. Together, the two halves make the same length as red.”
The color names give space for the student to hear the more important language, the fractional language. It is hard for the student to distinguish the one from the one-half, but color names allow the student to distinguish with ease.
Parts to Whole to Length
You may have noticed that I switch between parts to whole language to length. Cuisenaire rods provide the context of length, but they also provide the context of parts to whole.
During a lesson, choose one over the other. Sometimes length is preferable over parts to whole language. When describing fractions as the inverse of multiplication, it seems to make more sense to use length. However, other times, it seems easy to switch between the two.
Context helps you decide. Since we want students to use the fractional language correctly in context, I believe it is important to practice using both parts to whole language and the language of lengths in these tasks.
If we talk about dividing a cake, it makes more sense to use parts to whole language. If we want to talk about the difference in length of objects, it makes sense to use length.
Remember variety in math narration is important. See this post for 5 Keys to Math Narration.
Context is Necessary for Fractions
New language takes time to learn and remember. The first hurdle with fractions is the new names: one-half, one-third, two-thirds, three-thirds, one-fourth, one-fifth, etc.
Remember how you pointed at the banana and named it for your baby. You never named it out of context. That is, you never just randomly said, “banana” and expected the child to comprehend that a banana was that yellow thing Mommy gave earlier that day.
In the same way, to learn fraction names, they need to be named in context. The following tasks use Cuisenaire rods and playful tasks to provide context for naming fractions.
White Rods in Relation to the Other Rods
This task focuses on parts to whole language.
Task the student with finding how many whites fit into the length of red, light green and purple rod. Then ask the student to tell how many white parts fit into the whole red.
Then narrate to the student the name of each white part.
White is one-half of red. Two whites is two halves of red. White is one-third of light green, two whites is two-thirds of light green, three whites is three-thirds of light green. White is one-fourth of purple, two whites is two-fourths of purple, etc.
Work through the rest of the rods on the next day after reviewing the rods of the previous day. Focus on input. That is, just immerse the student in the new language by describing the fractional relationship.
Just add this simple exercise throughout the week. Offer opportunities to repeat the language but eliminate any expectation of the student doing it alone.
(You may also use the play mat pictured from PDL’s Fraction Exploration for Cuisenaire Rods for this lesson.)
This is part of language learning. When learning another language like Spanish, a student may perceive the sweater but doesn’t remember the right word for it in Spanish. So input, input, input the language. Let the rods and the student's eyes do the rest. .
Changing the Context using the Same Fraction Language
And of course, we can’t just keep pointing to the same sweater. It is time to change it up. The next task introduces one-half by changing the context and the language.
Task the student to find all the rods that are made up of a pair of the same color rods. This exercise begins much like finding all the odd and even numbers. However, we want to segment the even rods, so the student may explore the relationship of halves.
Put aside the odd rods or the rods that cannot be made of same colored pairs. Because the student used pairs of same colored rods to find the even rods, all the rods should be available for the language part of this task.
Tell the student that you will be describing the length of the same colored rods in relationship to the one rod train beneath it.
Point and describe the pair of whites that make the length of red. “This white is half the length of red. This white is the other half of the length. “
Extend this activity by asking the student to show you another half of red and then another. Ask the student, "How many halves of red do you have?"
After the student responds with 3 or 3 halves, ask the student, "Is there another rod equal to 3 halves or red?"
Extend it again by asking, "How would you write that?"
Move to the next even rod and repeat. “This red is half the length of purple. This red is the other half.”
Then ask the student how to name the next pair of same colored rods. The rhythmic pattern of the language offers them queues that the pattern continues.
Provide these opportunities for the student to try the language out. It helps to develop student’s prediction skills in patterns of language. It isn’t a test for understanding the concept.
Building Solid Color Trains to Name Fractions
Continue the exploration of naming fractions by building solid color trains of various lengths. A solid color train is a succession of the same-colored rods like 10 white rods end to end or 4 red rods end to end.
To explore thirds, task the student to build as many 3 car (rod) solid trains as they can. Task the student to find one rod that is equal to each same colored train.
Narrate the fractional relationships. Name them as parts of a whole or as lengths. Keep the language consistent for each lesson in the beginning. If you start the lesson with language of parts to whole, continue for the whole lesson.
“Each solid color train is made of 3 parts. One light green is one of the three parts or one-third of the whole train. Two light greens are two of the three parts or two-thirds of the whole train."
Again, let the student get a feel for the rhythmic pattern of the language. Then offer the student the opportunity to narrate the rods themselves.
To explore fourths, add an additional car (rod) to each solid color train. Repeat the narration. Then offer the student an opportunity to use the language. Continue this exploration with fifths, sixths and so on.
Reduce the Burden of Naming Fractions
The student processes a lot of auditory information as they listen to you narrate these tasks. Much of the words are very unfamiliar to them. Such unfamiliarity taxes the mind. How can we help remove some of the burden?
We enjoy analogies because they help us to relate to unfamiliar ideas using what is already familiar to us. To make narrations more enjoyable, incorporate what is familiar to the student through pretend play.
I created the play mats to connect to the student’s hunger for story and play while providing familiar context to explore mathematical language.
The play mats offer choice to the student to give control and responsibility to the student over their education. It also provides real-life context for the language of math through sharing, mixing and creating.
Dividing Hamburgers to Talk Fractions
Let’s look at the play mat below (From PDL's Fraction Exploration). There are three hamburgers that need to be divided amongst these furry friends.
Introduce the task as building solid color trains. Ask the student which solid color train allows this hamburger to be divided between two foxes. What about the 3 racoons? The six hedgehogs?
After the student has worked through the task, work on naming the fractional relationships. “I see the fox has one half and the other fox has the other half. Two halves equal the whole hamburger.”
Jumping Fraction Races
Fraction races offer context to name fractions as well. In this game you are racing the rods. At the end of the game, you and the student observe and name fractions.
“I see that cow only takes two jumps to finish the race. One jump is half the race. I see the duck jumps 8 times. One jump is one-eighth of the race. “
Note that I switch between multiplicative and fractional language. We want to develop that fluid understanding that is able to move between operations and see expressions from different perspectives.
A Royal Fractional Tea Party
Sharing is a fractional concept students seem to understand at an early age. Go ahead and misappropriate those cookies. You will hear about it.
This play mat builds upon that context in fun way, a tea party with royalty. Task the student to create food with the rods that can be divided to share between to the two plates. Then have student describe the parts on each plate.
It is okay if they use the context of cakes, sandwiches, etc. "Here is half the cake for you." It isn't necessary to always use the context of the rod colors. The context of play food helps the student look for fractional relationships in the world around them.
Expand this activity by encouraging the student to use the other rods and invite more guests to the royal tea party.
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Comparing Paths thru Fractional Language
PDL’s Building Paths offer opportunity to observe and talk about fractions. In this play mat, there are three paths for the student to compare.
You can begin the discussion by talking about the difference between the paths using subtraction as the focal point. Then transition into describing the difference in fractional language. How much of the shortest path makes up the longest path?
This may require the student to overlay the current rods with larger rods the help the student count and observe the fractional relationship with ease. I suggest this activity as an extension activity for students that need a challenge.
Purchase PDL’s Building Paths for Cuisenaire rods here for several paths to explore fractions.
BINGO, Board Games and More Naming Fractions
Naming fractions takes time practice. I created this set of self-checking poke cards to offer plenty of practice naming fractions.
I love poke cards. It’s an electronic free math activity that I don’t have to supervise. I also like to print an activity once and use it over and over. The poke cards act as calling cards for BINGO and a deck of cards for a board game. Yes, 3 in 1 game fun.
I want my resources to work for you. I want you to print once and enjoy it over and over. There are 3 sets to choose from, so which set should you start with?
Gattengo saw 3 stages of sensory learning that moved a student to the abstract. Touch, Sight and Hearing. Students touch the rods in the beginning. Then they are asked to just look. Lastly, the student is asked to just listen.
Naming Fractions Game pack 1 gives students a picture of the rods to give them context for the question.
This set removes the touch sensory and forces the student to just use their sight.
It is often that manipulatives are removed too abruptly. This eases a student into the transition away from the manipulative.
The next set of Naming Fractions Game pack the student relies on just seeing the question and hearing the question. The student is forced to visualize and/or remember the rod.
The last set of Naming Fractions Game pack re-words the fraction question, but also adds more difficulty by having students find fraction of fractions.
You can purchase all 3 Game packs as a bundle here for years of fun practice naming fractions and fraction of fractions.
Progress But Never Leave Behind a Fraction
Naming fractions may get boring for the student. Yet it is necessary. It is important to change up the activities and to move between operations.
Progress through each activity but return to the previous activities again and again. The variety offers students choice and deepens their understanding.
Next post is about tasks that deepen conceptual understanding of fractions by exploring equivalency. Be sure to subscribe.
