Tag Archives: fractions

Multiple Methods of Explanation

Tonight a parent of one of my 6th graders e-mailed me about his math homework.  I love it when parents contact me to discuss their student’s progress.  Parental involvement has such a powerful effect on student success.  In the process of crafting a response, I was forced to really grapple with what the student’s misunderstanding was.  I think the result is interesting:

Alex* did both worksheets today, however his sense on how you’ve taught him to convert whole numbers into fractions appears to be different than what I know (keeping whole number as numerator and making the denominator 1). I think he must have been confusing whole number conversions with something else you were teaching but he was obstinate. Thus I think he had trouble with the generic rectangle calculations and I could not help him and we lost patience with each other.

Perhaps you can revisit whole number conversions with him as he would not believe me. I’ll have him do the three problems once he gets the concept correctly — or that you confirm that in this case I am smarter than a 6th grader….and if I am not, I am so sorry. It is probably better I write for a living.

*Not his real name

And then here is my response:

Don’t worry, you’re certainly smarter than a 6th grader.  Certainly more wise, too, because 6th graders have a problem not realizing what they don’t know.

I haven’t taught students that they can write any whole number as a fraction over 1 (e.g., 4 = 4/1).  As you realized, they sometimes don’t “get” why it works and end up getting more confused.  Currently we’ve just been reasoning through multiplicatively via repeated addition: for example, if you had 4 times 2/3, that just means 2/3 + 2/3 + 2/3 + 2/3.  To go further, drawing a picture of 2/3 four times and then counting the pieces to recognize you’d have 8/3.  A lot of students are still struggling with the idea, and want to say the answer is 8/12 (because they multiply top and bottom by 4).  But then you can present them with the question: Aren’t 2/3 and 8/12 equivalent fractions?  This creates the dissonance for them to see that the answer should be more than what they started with, not the same.  Put another way, multiplying by 4 changes the number of pieces you have (the numerator), but not the size of the pieces (the denominator).

He’s okay right where he is–a lot of students are still there, since we only just learned how to multiply fractions.  We’ll be doing more work multiplying fractions by both fractions and whole numbers via the mixed number generic rectangle problems.  I think with concepts like this, it’s important to foster both a continual emphasis on why the answer makes sense in addition to repeated drill, which gives students several times to make mistakes / correct them and notice the patterns themselves.

If he wants, we could do some problems together at lunch.  I often have a few kids up for extra practice / more individual instruction.

What do you think the student’s struggle was with?  What questions would you have asked him to help him resolve his misunderstanding?

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Using Pre-Assessments in a Unit on Percent

My first year of teaching I was both blessed and cursed by a lack of curriculum.  The previous 8th grade math teacher had left a collection of NY state test questions from the past several years.  But even as an unsophisticated first year teacher, I realized this was a far cry from a well-planned curriculum.  And so I spent about hours and hours every unit scouring the internet for different ways to teach a concept, interesting problems, and standards-driven activities.  This constant quest for good resources has dramatically accelerated my growth as a teacher and is a continual supply of professional development.Now that I have surveyed the landscape, I feel ready to give back to the community.  My initial contribution is a unit on percents that I planned while student teaching in an 8th grade classroom.  There are several features I’d like to highlight that I think were valuable.  In this post I will discuss the use of a pre-assessment.

This was the first time I gave a pre-assessment.  In the past I didn’t gave much thought to prerequisite skills and understanding.  By giving a pre-assessment, I forced myself to consider what students needed to know to learn the new material.  I also sent students the message that I was serious about helping them be successful, and not just blindly following some curriculum.

Following the pre-assessment I collected student work and used the document camera at the front of the room to briefly flip through student responses.  It’s important that this process is anonymous, as the point is to display the range of ideas and not single out students for their mistakes.  The teacher may also want to slip in his or her own work to make a particular point (whether it’s a common misunderstanding, a correct solution, or a controversial answer).  For example, this student’s solution prompted an interesting discussion.

After giving a pre-assessment it’s important to put the data to use.  I found that students had a lot of trouble multiplying decimals and converting between fractions, decimals, and percents.  So I planned a station activity the next day where students could practice two skills.  Each station was led by a teacher or pair of students who had demonstrated mastery of that particular skill.  In this particular instance I allowed students to decide which skill to practice, but you could also assign groups.

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