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Cutting uncut pans of brownies

 

This idea was born out of my seventh graders’ need to understand subtracting mixed numbers when regrouping was involved.

 

Description

Once our fraction unit in seventh grade gets around to adding and subtracting mixed numbers, most of the students remember the traditional algorithms and are comfortable using them.  Too comfortable.  Errors start cropping back in when subtraction of mixed numbers requires regrouping.  The procedure of borrowing from the whole number is vaguely remembered, but one of the steps will get left out or misused.  They might forget to drop the whole number to the next one smaller, they might not remember what it is they’re borrowing, they might add the borrowed quantity to the denominator instead of the numerator.  These kinds of errors tell me the students have stopped thinking about what it is they’re actually doing and are simply trying to follow a rote set of steps.  Talking about pans of brownies is an enjoyable topic, and I’ve found it to another “light bulb” experience for many of my students.

 

Needed Materials:  nothing other than a chalkboard or overhead for drawing pictures (a pan of brownies to share is optional!).

Directions: 

1.                 I ask the students to return to thinking about subtraction with pictures.  We often refer to fractions as a specific number of pieces left in a pan of brownies (the numerator) originally cut into another number of brownies (the denominator).  Whole numbers are whole, uncut, pans of brownies.

2.                 I write a problem on the board, such as , and ask a student to come to the board and represent the original quantity with pictures.  Another student comes to the board and shows how to use the pictures to represent subtraction.  The whole number part is easy—5 whole pans minus 2 whole pans leaves 3 whole pans.  The 2/5 of a pan remains as well.  Here the problem begins.  How do you take away 4 “fifths” when you have only 2 there?  You must cut one of the whole pans, of course!  The student illustrates this.  Here’s where I find I must intervene.  With pictures, the students will complete the subtraction, read the answer, and be done.  What they’re missing is how the pictorial representation relates to the traditional algorithm.

3.                 After the student illustrates cutting one of the whole pans of brownies into fifths, I interrupt him.  I tell the students I know they really don’t like doing problems by drawing pictures because it takes too much time.  Here’s the challenge.  Do you see where the graphic method relates to the analytic method?  (I introduce the importance of different representations in all my classes.  I find it provides access for students with different learning styles and helps a great deal as they progress through the high school math curriculum.)  How many new fifths did cutting the whole pan create?  How many fifths is there all together before subtracting?  What did cutting a pan of brownies do to the number of whole pans?

4.                 I rewrite the original problem on another section of the board and ask the students to compare the analytic method with what they see in the graphic illustration.  I slowly progress through the steps without talking, hesitating and looking at them at each critical step.  When finished, I ask, “Do you see it?”  If a student volunteers, I have him explain what he saw.  If not, (usually I do, but one year it took further explanation) I go back and ask questions about the critical steps.  E.g., I cross out the whole number and write the next smaller number beside it.  What did the whole number represent in the picture?  What does crossing it out mean?  What did you do in the picture method that resulted in fewer whole pans of brownies?

5.                 Essentially, what I want them to understand is that the borrowing step is decreasing the number of whole pans and adding to the number of cut pieces.  Critical to increasing the pieces is understanding that you must cut the whole pan in the same number of pieces as the partial pan was originally cut.

6.                 I usually complete the discussion with an example of mixed numbers with unlike denominators.  However, the process of getting common denominators has already been developed, and I have found few students at this point still struggling with that concept.  Once common denominators are established, the process reinforces the first example.

 

 

The mathematical goals are to think about what regrouping in subtraction really is and to understand acquiring a set of specific sized pieces when reducing a whole number.

 

 

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Activity Appraisal

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.

 

Like in “Stars and Circles”, I feel very strongly that this kind of development of fractions needs to be happening before seventh grade.  However, until the students exhibit an understanding of fractions, I will continue to use this activity at this level and higher when needed.  I believe that if students have a sound understanding of regrouping in whole number subtraction to which they can compare fractional subtraction, the process is far more easily understood.  I find, however, that students arrive in my classroom with only a rote understanding (and that sometimes shaky) of subtraction procedures.  They have no idea of “breaking apart” a larger quantity in order to get a larger number of smaller items.  I am particularly intrigued by a virtual base-ten block manipulative I read about in “What are Virtual Manipulatives?”, an article in the February 2002 Teaching Children Mathematics.  I would like to see a tool like this used in our elementary classes.  However, at this time the elementary does not have Internet access, so alternate means will need to be sought.  Our elementary school has very few manipulatives due to a prior administrative philosophy.  One of my goals for the future is to correct this situation.  It is difficult to encourage teachers to use active-learning instructional practices when there are few student-centered materials available.

 

This activity has been very successful in my math 7 classroom.  This year one of my best students—hard working, good basic skills, always wants to understand—got this incredulous look on her face when I finished the silent comparison of the analytic and graphic methods.  “Is that all there is to it?” she asked.  That simple statement spoke volumes!

 

 

_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…

 

I communicated to the students why we were doing the activity by…

…drawing attention to the difficulty they were having when subtracting mixed numbers.  “We can make this easier, I promise!”, I told them.

 

The part of the lesson that was student-centered was…

…the discussions and student participation in providing graphic representations.  Again, my role is to ask questions and give examples that prompt student thinking.  After that, I simply facilitate the discussion.  The students know I expect them to listen to each other and be prepared to discuss without my repeating everything that is said.

 

 

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