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Counting Stars & Circles
This idea was born out of my seventh graders’ need for clarification of when to use common denominators and when not to.
Description
Each year in seventh grade as we begin talking about fractions, I ask the students to tell me what they remember about computing with fractions. Invariably, the argument about when common denominators are needed will surface. I have found this guided discussion to clarify the topic for most students.
Needed Materials: nothing other than a chalkboard or overhead for drawing pictures.
Directions:
1. First I lead the students to discuss addition. I ask the students to think about their first recollections of addition and how they did it. I have never had a class that didn’t tell me they started adding by counting. I use this opportunity to reinforce the fact that addition is simply a shortcut for counting, “invented” to make the process quicker. I draw pictures on the board of stars and circles mixed around in small groups—3 circles here, 2 stars here, 4 stars there, another circle there, etc. and try to make a theatrical rendition of a kindergarten classroom. (I’m not much of an artist; that’s why I used stars and circles! Any other pictures take too long and elicit too much humor from the students!) I ask the students to add. While some will start counting all of what they see, usually at least one student will ask, “add what?” That is the lead-in to discussing common denominators.
2. Aha! Add what? You need me to tell you whether you’re adding stars or adding circles. That’s because stars and circles are different, and addition is usually counting “like” things. We see that there are 2 stars here and 4 stars there which gives us a total of 6 stars because 2 + 4 = 6. If I tried to add 3 circles to 2 stars, I would have to call my answer something else—maybe 5 shapes—because the objects weren’t alike. We have to find a “more general” word that can be used to describe both objects. (Be aware—calling it “5” shapes makes me a little nervous because I’m afraid it will reinforce the common mistake of simply adding the numerators and renaming the denominator. That hasn’t seemed to be a problem, but I am constantly vigilant for that misconception.)
3. We do a little grammar review. (I love doing that because students usually think you shouldn’t do any “English” in math class!) I write a simple sentence, such as “I see 6 stars.” on the board and ask the class to identify the sentence parts. They get around to identifying “6” as an adjective—a modifier that simply tells how many of the item of focus. Then I write some fractions on the board and ask the students to say them—and listen to the words their mouths say as they say the fraction names. When you say “two fifths” what is the modifier and what is the item of focus? Many, many times this is the moment at which I start to see light bulbs go on in students’ eyes. I can tell this may be the first time they have understood the meaning of the numerator and denominator. The numerator and denominator tell us “how many” “of what”, respectively.
4. I ask the students to draw pictorial representations of several fractions and then compare them with students sitting around them. I circulate to assess understanding at this point, then have students put several representations on the board and discuss them as a class. (I have had classes that had no idea of how to draw a picture of a fifth or a third.) Finally, I pull it all together by writing a couple fraction problems on the board and reiterating the main points of previous discussion.
What are we counting? (fifths)
Are they alike? (yes) How many of each? (2 & 1) What will the
answer tell us? (That there are 3 of
the same thing—3 fifths.
What are we counting? (fifths and fourths) Are they alike? (no) What does that tell
you? (That we’ll have to call the
answer something else, because we’re not counting the same thing. We have to find a “more general” word that
can be used to describe both fifths and fourths.) Aha! We were adding and
you’re telling me I need to find a different descriptor before counting—that
would be a common denominator. I don’t
emphasize how to find that common denominator at this point—that’s another
lesson—my focus is WHEN the common denominator is needed.
5. What about multiplication? Does it need a common denominator? This usually draws a mixed response, so I prompt them to recall their first experiences with multiplication. What did they do? These answers do not come as quickly as did the addition discussion answers, but usually someone expresses the “using groups of like things” to find a total. A total? That sounds like addition. Yes—multiplication is repeated addition, so multiplication is another shortcut developed to make using math quicker. But the critical part of the definition of multiplication was the “groups of like things”. When you’re multiplying, you already have like things. One part of the problem tells you how many groups and the other part tells you what the groups are. Since you already have “like things”, you don’t need to get a common denominator. The traditional algorithm for multiplication is usually presented as procedural in nature and doesn’t do much to reinforce this idea of “how many” “of what”. Later we use fraction “pictures” to develop the concept of multiplication and compare it to the traditional algorithm they “learned” (at least some of them remember it) to find where the two are the same.
What does the problem say? (that there are 4 groups of 2/3) What are we counting? (thirds)
How many are there? (4 groups of
2, which would be 8) Eight what? (8 thirds)
What does this problem say? (that you only have part of ½) If it’s only part of ½, what do you expect
about the answer? (that it will be
smaller than ½) If the answer is going
to be smaller than ½, is the denominator going to change? (yes)
So, are you getting common denominators? (no, just the answer is going to be something different because
you only have a portion of the original number [½]. You don’t have to change the denominators; they just change
themselves!) I’m not completely
satisfied with the “just change themselves” part, but I know we can find a
better way to describe it later on after we’ve developed what common
denominators really are. The key part
to the discussion is that multiplying fractions is finding a part of a given
quantity, as opposed to having multiple groups of the same thing. To find a part, you must split the original
quantity into parts, then you can go about collecting the required number of
those parts. Splitting the original
makes the answer smaller than what you started with.
The mathematical goals are to think about what addition is in comparison to multiplication. An understanding of the operations should lead to valid decisions about whether common denominators are needed for calculation.
Comment on the overall effectiveness of this activity. Then use the form below to reflect on the activity from the perspective of both the teacher and the student. Give the lesson points based on how well it went--each question is worth up to 4 points.
I feel very strongly that this
kind of development of fractions needs to be happening before seventh
grade. However, until my students
exhibit an understanding of fractions, I will continue to use this activity at
this level and higher when needed.
Those “light bulb” moments are what let me know I’ve done my job as a
teacher. One problem I have is a
resistance from students who memorize well.
They have learned the procedural algorithm and remember it. Often they don’t want to THINK about why it
works. I feel the “like terms” metaphor
is one of those themes that carries all the way through mathematics and is very
helpful when students reach algebra. I
have successfully compared adding algebraic terms (like 4x and 7x versus 4x and
7y) to adding fractions. 4x and 7x are
counting like quantities and so can be “pushed” together and counted as a
single unit. Since x and y are unknown,
and possibly varying, it isn’t possible to find a “common” descriptor for 4x
and 7y, so they must be left as separate quantities.
For students to whom the
traditional algorithms made no sense, I have found this activity/discussion to
be a lifesaver. I find these students
referring to pictures to help their understanding on homework and in small group
discussions. For many, the collective
feeling of relief is almost a tangible thing in the classroom. Many students have asked me why their
previous teachers didn’t explain it this way.
_4_ Was the purpose of the activity clear to me?
_4_ Was the purpose of the activity clear to my students?
_3_ Did the activity support a range of learning styles?
_4_ Were the students engaged during this activity?
_15_ Total points (out of 16)
The purpose (objective) of the activity was…
- Assess the students’ understanding of addition and
multiplication
- Provide conceptual clarification of when common denominators
are needed for calculation
I communicated to the students why we were doing the activity by…
…when the opening discussion to
“what we remember about computing with fractions” leads to the confusion over
common denominators, I tell the students we need to get the matter settled in
their minds. I tell them that their
understanding of the concept is my goal for the day.
The part of the lesson that was student-centered was…
…the discussions in small and large groups. My role is to ask questions and give examples that prompt student thinking. After that, I simply facilitate the discussion. By this time of the year, the students have learned that I expect them to listen to each other analytically.