Easy Ways to Solve Difficult Math Problems

In mathematics, word problems have been known to pose challenges for elementary school students, middle school students and even some high school students. In addition, a vast majority of students also have difficulties with solving problems with fractions. If we mix a word problem with a problem with fractions, then we end up getting an even tougher problem to solve. How can we expect those students who have not yet mastered language to make meaning of word problems? Let's dive right into a math word problem which will illustrate this. 
Problem: Tashira has a piece of lace material that is 3/5 yard long. She used 2/3 of the material to make a quilt. How much did she use to make the quilt?
When a student reads this problem one of the questions she/he may ask is, "Where do I start?" The student may have difficulty with translating the word problem into its mathematical representation.

The next difficulty is that if the student decides to use the traditional method of solving this, he/she may need to know how to correctly multiply fractions. This is inherently confusing for some elementary school students, so I have a way to avoid this confusion.

For those of you who have issues with multiplying fractions or translating the "of" to its correct math operation: If I were to tell you that you can solve this problem without even doing any multiplication, then you would be really excited. Yes, you can! Let me show you how to go about solving this with absolutely no multiplication. This solution depends on the power of drawing a picture. I will use a number line student invented strategy. The trick is to represent our distances with a drawing and our answer will be easily visible from the picture.
Student Invented Number Line Solution:
The material is 3/5 yard long. On the number line this is represented by the top distance:
0    1     2     3     4    5

She uses 2/3 of this 3/5 yard long material. On the number line this is represented by the top-most distance:
0    1     2     3     4    5
The top-most distance which is 2/5 is indeed 2/3 of the 3/5 distance. Viola! She used 2/5 yard to make the quilt! We have our answer without doing any multiplication. You may argue that this is essentially division, but the beauty of this strategy is that we did not have to do any explicit division. Who would have known that you can solve a fraction problem in such an easy and stress-free way? I can only predict where your mind is headed. It's about time that some of us who love doodling during class can also thrive in math! This method is good for spatial learners and visual learners. I find that many students with ADHD and ADD love this strategy.
Besides being easy to comprehend for students with disabilities, this strategy works like a charm for ESOL/ESL students or students who have not mastered the english language enough to interprete the meaning of certain words in a word problem. Learning by immersion in concert with these student invented strategies work miracles.
The trick is that you have to work in context. First you draw your 3/5 distance. Now we start from 0 and we go 2 units up out of the total of 3 units and this is our 2/3rds of the 3/5ths distance. We now look at this distance in context to our whole drawing and this is a distance of 2/5 yard. The magic here is in drawing a picture.
Now let's look at the traditional way of solving this problem. We will notice that the traditional strategy is error prone if the student is not good in multiplying fractions. Also the student needs to remember to use multiplication instead of division for the correct meaning of "Used 2/3 OF the material."
Traditional Solution:
With a word problem we always need to get the mathematical meaning.
"Used 2/3 OF the material" means we multiply 2/3 to the number we started with in order to get what she used.
She uses 2/3 of the material and the material is 3/5 yard long. The "of" tells us we need to multiply the two numbers.
Now 2/3 * 3/5 yard
The 3s cancel out and we get 2/5.
So 2/3 * 3/5 =  2/5 yard
She used 2/5 yard to make the quilt.
Feel free to share any comments or even other student invented strategies that would work in this context.


This method for multiplying fractions is brilliant, Charles! Thanks so much for sharing!
I am an OVER 60 yr old educator (language arts/esl/ specifically OG trained for one to one tutoring... BUT who STILL hides under my desk if asked ANY math questions beyond a second grade calculation! YET! you took the BITE out of this! I will read all your blog material and hope for more EYE-opening, reason, logic, drawing based info! Also, I shall refer math based inquiries to you!
Liz, I appreciate the kind comments. Kudos to you for your efforts on homeschooling. One thing I've noticed is that home-schooling is just as challenging to teach since you still have to master your student's learning styles and enforce those routines for best performance. What is your experience?
Diana, thanks for the kudos. I try to find out-of-the-box and easier ways to teach notoriously difficult subjects such as math, science, and even computer programming. I see you are doing great things tutoring language arts. Do you think my approach would also work for students with dyslexia.


Charles C.

Effective English, math, coding Tutor who taught thousands of students

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