"Oh! This is Cross-Multiplication....!!"

I had given them enough time to struggle and figure out how to evaluate 'x' from such equations, where variable is in the denominator:

But when I found that they were unable to do this for long, I decided to intervene. In the above problem, it's easy to visualize that 2x-1 has to be 3, so that we get quotient as 3, and for this 2x should be 4 and hence x=2.

But I was aware that this game would not be always so easy. For ex., what if it is -

It's not so easy to solve this problem using the approach above (Why?)

So the objective was to accomplish a general strategy. Meanwhile, I was also aware that they are very comfortable with equations having variable in the numerator. 

So this is how it went:

How do you solve:  x/4 = 2

They immediately said, x = 8

 Okay, but how?

"We have expressed division in the form of multiplication..."

Correct. And how else would you arrive at the 2nd equation from the 1st one -- using any rule of equality?

"Yes, we multiplied both the sides by 4.."

And why do you need to do so?

"Because we want to keep the 'x' alone...and its now divided by 4.... so only when we multiply it by 4, its division effect will get nullified."

 Fine.... If you understand this form, then lets consider this now:   

(only the 1st equation of the image below was written)

"We will multiply by x on both the sides.."

Why?

"To remove the denominator from the equation..."

And they solve it further as above. (divided the equation by 3 to get the value of 1x...)

Fine... Lets consider this equation now.... 

(only the 1st equation of the image below was written)

Immediately, they said -- "Multiply by 'x+1' on both the sides...."

So tell me what should I write....   They dictated, & I simply wrote...

Now?

"Multiply  3 and x+1 "

I am waiting for you to tell me.....

" 7 = 3x + 3 "

I paused again....... they understood.....

" 4 = 3x "

Pause...

" x = 4/3 "

( I didn't push them for the reason this time...)

Fine... let's take one more case now.....

And I just kept writing what they dictated to me.....

Finally, I drew an arrow (as you see above) and asked them to notice something....

" Oh.... It's a Cross Multiplication...."

I looked at their faces, one by one, while some were either still glued to this magical appearance or wondering about its How part  :-)

So... what happened...?

"Sir, that's how its taught in the school...."

Then why weren't you using it to solve the equation...?

"We forgot...."

And what could be the reason for this?

They laughed.... "Sir, we didn't know the reason for this rule...we had not understood this..."

So now?

"Now we understood this... We won't forget..."  this came smilingly with some shades of embarrassment...  :)

(But was it 'they' who need to be embarrassed for this?)

All this was written on our floor and they began noting this down in their note-books....

Meanwhile, I made up these problems for them on the board for practice:

Do you feel I have made these 3 problems randomly or there is some thought process invested in formulating them? (to answer this, you might like to have a re-look at all of the problems discussed above)

I gave them some time to solve..... And after few minutes, this happened....

Student-1

Student-2

They were surprised in the beginning when they arrived at the equation, 34x = 0. I think this was the first time, they were solving an equation where the value of the variable turns out to be 0. If you notice, they have also verified it by substituting the value of x in the original equation.

Student-3

Yes, as you see they solved it well, even simplified it.However my objective to give them this problem was something else....I just drew the arrow again... and waited for their response...

One of them shouted -- "Sir, its getting inverted..."

Other said -- "Inverse on both the sides.."

Without any discussion at this point, I simply, gave them this problem:


 
So how would have students done this problem? And what would have been the line of discussion further?   We will see that in Part-2 of this post :-) 

Meanwhile, I would like to know  ---

a) Your views about the trajectory followed by the teacher to make the students learn to solve Linear equations of :   a / f(x) = b / g(x)    -- What would have You done if your student/ child would have expressed his/ her inability to solve such problems? 

b)  Your experiences about the so called 'Cross multiplication' method?  - and how it was arrived here...

c) Was it a right decision/ practice of me to expose them to this Inverse taking method of simplification now? Why so? 

Waiting to hear from you,

Thanks and Regards

Rupesh Gesota

www.supportmentor.weebly.com

 "I touch the future..... I teach."