This is the infamous paint problem, and I remember Jo Boaler talking about it in her summer math course she offered a few years back. My purpose wasn't for them to figure it out with their own method, or for them to sue the work formula. One person paints a house in 3 hours, then other one in 5 hours. How long would it take for them to paint it together?
So first we defined the variable x. What are we trying to find out? The time it takes to paint the house together. I asked them how much of the house would the first person paint in one hour? (1/3) So, that's 1/3x. The second person would paint 1/5 of the house in 1 hour, so 1/5x. Then I asked how much of the house they'd want to paint together? The whole thing, so those added together should equal 1. Then I asked them to solve it (1/3x+1/5x=1) I also asked what are you dividing by when you multiply by 1/3? (3) So this equation can also look like x/3 + x/5 = 1.
Some students realized the LCD of 3 and 5 was 15, so they converted both fractions to fifteenths. Some didn't realize they could then add the terms together. Others realized you could, and did. Then they were stuck at 8/15x=1. Some realized you could multiply both sides by 15 to get you 8x=15. Some put the 1 over 1 to make it look like a proportion and then to use cross products. Few students realized they could divide by 8/15 on both sides and multiply both sides by 15/8.
A few students in my first 2 classes remembered a method called Fraction Busters to eliminate the fractions in the equation that they were introduced to in 7th grade. Many had forgotten this method. I highlighted this method by using the Google Drive app on my iPhone to take a picture to my Google Drive and view it with the whole class and have that student explain their method.
Then students practiced on some of the problems in the section of CC3 section 5.1.2:
This student did not eliminate the fractions.
This student saw it as a proportion to solve for x at the end.
This student remembered the fraction busters method without me telling them.
In this practice problem, students realized all the terms had decimals. So, all terms on both sides were multiplied by 10 to make them whole numbers. The Estimation 180 we did at the beginning was really helpful because they estimated the capacity of a soda can. Most students estimated 8 or 12 ounces. The answer was 12.5, so the percent error ratio was 0.5/12.5. I asked students how to write the fraction without decimals. Some students said multiply by 10 to get 5/125. Others multiplied by 2 to make it 1/25. My colleague expected students to multiply by 8 to make a denominator of 100 but I didn't expect any of my students to do that and none did.
Here it wasn't completely necessary, but here all terms could be divided by 10 to make it simpler.
Here was a 2 step equation that they figured out.
We had a great conversation in all classes about remembering to multiply all terms by 5. Some students forgot to multiply the 1 by 5 on the left side which was a great discussion.