MathNotations

Posted by Dave Marain at 6:58 AM 0 comments

Labels: fractions, mixed numerals

What middle schooler doesn't get that warm and fuzzy feeling when we tell them we're going to play with fractions! Here's a small investigation to capture that mood of euphoria...

RULES OF THE GAME:

• No Calculators - No decimals!
  
• All fractions must be expressed in lowest terms
  

Posted by Dave Marain at 10:38 AM 0 comments

Labels: fractions, middle school

Now that our family health crisis has abated (my daughter is doing well), I guess it's time to jump back in with both feet. A math program leader in a district with which I am consulting, asked for my opinion on an important issue of curriculum and instruction.

How much time should middle school teachers spend on the traditional vertical algorithm for adding and subtracting mixed numerals vs. converting to improper fractions immediately?

I assumed that both methods are still commonly taught with about equal time given to each, but I wasn't all that sure about how that was across the country. This is where I need the help of my informed readers.

First, my thoughts. From a practical perspective of those who utilize fractions in their occupation, I would guess that the mixed numeral form is most commonly employed. Whether it's the carpenter taking measurements to see how many board feet of wood must be ordered (or for precise measurement to the nearest sixteenth of an inch) or someone following a recipe in the kitchen, I can't imagine that converting to improper fractions would be their first choice. On the other hand when I personally need to add fractions in a math problem, I usually use improper. I took an informal survey of one of the groups I'm working with and the majority stated they were taught both methods and some preferred working with mixed numerals and others said it's more complicated that way.

Are the number of steps roughly the same?

Mixed Numerals Algorithm for Subtraction:
(1) Convert the proper fractions to common denominator form.
(2) If needed, regroup, i.e., "borrow" 1 from the whole number part of the larger mixed number, convert the 1 into common denominator form and combine this with the other fraction (of course students are shown short-cuts for this which they blithely and mechanically follow without much thought).
(3) Subtract the whole numbers and the proper fractions.
(4) If the resulting fraction is improper convert it and add the whole number part to the previous result.

Improper Fraction Algorithm:
(1) Convert each mixed numeral to an improper fraction by the traditional algorithm (again blithely and mechanically without much thought).
(2) Determine a common denominator (or the lcd) and convert each fraction.
(3) Subtract the fractions.
(4) Convert the answer to mixed numeral form by the traditional division algorithm.

Now I may have combined steps or there are oversights but essentially they appear to be roughly the same number of steps. However, the difficulty or complexity level of the steps
may not be equivalent.

I also feel that the mixed numeral form requires somewhat more conceptual understanding even if the child does it routinely. It may also prepare the youngster for working with algebraic expressions like A + B/C, but that's debatable. Further there seems to me to be a strong connection between the Mixed Numerals Algorithm and adding and subtracting denominate numbers. For example:
Subtract
15 hr 37 min
9 hr 46 min
I doubt that we would encourage students to convert both to minutes first, subtract, then convert back to hrs and min. I could be wrong there!

I feel there are arguments on both sides here. My instinct is that both need to be taught but it's not clear to me how much time should be spent on each method. Certainly some youngsters could handle both with facility while some would struggle mightily with at least one of these methods.

Further, I suspect there are some youngsters who convert mixed numerals to improper fractions procedurally without full conceptual understanding that a mixed numeral is an addition problem!

Your experiences and thoughts...

Posted by Dave Marain at 4:44 PM 13 comments

Labels: fractions, improper fractions, middle school, mixed numerals, pedagogy

Totally Clueless sent me an email reminding me of some famous 'fractured fraction' examples like the one in the title. Can you think of a couple of other two-digit examples of the same type that 'reduce' this way? Note that 10/30 = 1/3 doesn't qualify (the digits have to 'cancel' diagonally!).

Here is TC's version:
Note that the product 16 x 4 can be obtained by deleting the '1' and the 'x'!

READER/STUDENT CHALLENGE

(a) Find the other two instances of this 'weird' multiplication. The two factors have to be of the same type as in the example, i.e., a 2-digit number by a 1-digit number and the tens' digit of the 2-digit number must be 1.

(b) Most would find the other instances by guess-test. Here's a more significant challenge. Verify algebraically that there are exactly three such solutions.

(c) Is this problem equivalent to the 'easy way' to reduce fractions mentioned in the title of this post? Why or why not?

Posted by Dave Marain at 3:35 PM 9 comments

Labels: advanced algebra, fractions, number tricks

From various Google searches reaching this site I often discover some wonderful problems. These often inspire me to extend the question further. Part (a) below was from the Google search. I modified it slightly and then developed it. Thank you to whomever typed in the original search!

Note: For all of these questions, p/q represents an "improper fraction" in which p>q>0 and p,q are positive integers.

(a) How many values of p/q are there in which the sum of the numerator and denominator is 29 and 2 1/3> p/q> 2 1/4.
Notes: The numerical quantities in the inequality are mixed numerals.
How might one approach this without algebra?

(b) Show there are exactly 16 values of p/q such that p+q = 500 and 5> p/q> 4.
Is algebra the preferred method here?

(c) Show there are exactly 33 values of p/q such that p+q = 1000 and 5> p/q> 4.

(d) For Advanced Algebra students: Derive a formula for the number of values of p/q such that p+q = N and B> p/q> A.
N is a positive integer greater than 2 and A,B are positive reals with B>A>2 . Express your formula in terms of A, B and N. The greatest integer function may be needed.

Posted by Dave Marain at 1:45 PM 1 comments

Labels: algebra, fractions, math challenge, middle school

Update as of 6-2-07: All solutions to the problem below are now posted in the Comments section. Also, read Eric Jablow's astute comments and his challenge to students!

The following problem is well-known, however it is an excellent exercise for middle schoolers and as a challenge for older students as well.
Target Audience: Grades: 6-12
Prerequisite skills: Basic understanding of fractions and simple operations
Develops: Logical thinking, Systematic Counting/Listing, Fraction Concepts, Structure of Mathematical Proof
Recommended Classroom Organization for this activity: Students working in groups up to 4.
Online Resource: Here's one of the best sites on Egyptian Fractions I have found on the web. The problem is discussed but not solved!

Introduction for student:
A unit fraction is defined as 1/n, where n is a positive integer greater than 1.
The number 1 can be written as a sum of 3 unit fractions in 3 ways:
1 = 1/2 + 1/3 + 1/6
1 = 1/2 + 1/4 + 1/4
1 = 1/3 + 1/3 + 1/3
No other ways (other than rearrangement) can be found using the following reasoning:
The largest of the 3 fractions could not be less than 1/3. Why?
If 1/3 is the largest fraction, then the larger of the remaining fractions could not be less than 1/3. Why?

Here's your challenge:

There are 14 ways to write 1 as a sum of four unit fractions of the form:
1 = 1/a + 1/b + 1/c + 1/d, where a ≤ b ≤ c ≤ d.
Make a list of all of these ways. Have fun! Uh, no calculators please!

Posted by Dave Marain at 6:10 AM 5 comments

Labels: egyptian fractions, fractions, investigations, middle school, proof, unit fractions

[Update: I'm adding an additional problem at the end. This question seems to be of interest to some since I've seen it in a Google search for awhile now. Answers and solutions to some of the problems will shortly appear in the comments.]


The issue for me as a math educator has always been:
How do we enable children to think conceptually?

Here are some standardized types of questions each of which can be solved by a variety of methods.

In each of the problems below, there is a conceptual approach that requires skill, knowledge and some insight. We all know as educators we can do something about the first two (provided students are given enough practice and they do it!), but how do we develop insight? I can only tell you how my insight improves: When I tackle harder problems or those requiring me to 'think differently'. I am sure there are those out there who are able to invent these methods on their own, but, as for me, I have to work at it and think about it!

The first 3 questions involve ratios. Since we know how the current generation feels about fractions, these may cause students to feel some frustration!

See if you can find an 'insightful' or conceptual approach. Also, ask yourself how most middle or secondary students would approach these:

1. Background Terminology: For those of the younger generation who may not have heard of the term proper fraction, it means a ratio of positive integers in which the denominator is larger than the numerator. Thus, 4/3 and 3/3 are improper; 2/3 and 1/3 are proper. Also, the phrase 'in lowest terms' means that the greatest common factor of the numerator and denominator is 1. (But everyone knew that, of course!)

If 10/n, 14/n and 15/n are proper fractions in lowest terms, what is the least possible positive integer value of n which is not prime?

2. For how many integer values of n is 11/n between 1/9 and 1/10?
Note: Is this an algebra problem? A guess-test 'plug-in' problem? A calculator problem? Or just a fraction 'exercise'?

3. Fahrenheit temperature is related to Celsius by the equation: F = (9/5)C + 32.
An increase of 36 degree F. is equivalent to an increase of how many degrees Celsius?
Note: Many students struggle with an approach here. Some try it algebraically, most plug in some initial temperature, virtually none I have observed think conceptually about the meaning of ratios.

4. A cube 6 inches on each edge is sliced 'horizontally' to form 2 congruent rectangular solids. If these 2 solids are joined to form a rectangular solid which is not a cube, the surface area of this resulting solid is how many more square inches than the surface area of the original cube?
Note: Some youngsters simply 'see' this with little or no calculation! I guess you could say, they really 'think outside the box!' (sorry 'bout that...)

5. What is the smallest positive integer having exactly seven factors?

Posted by Dave Marain at 10:56 PM 11 comments

Labels: factors, fractions, ratios, SAT-type problems, slope, spatial sense