Traffic Jam
Task
You are stuck in a big traffic jam on the freeway and you are wondering how long it will take to get to the next exit, which is 1 \frac12 miles away. You are timing your progress and find that you can travel \frac23 of a mile in one hour. If you continue to make progress at this rate, how long will it be until you reach the exit? Solve the problem with a diagram and explain your answer.
IM Commentary
It is much easier to visualize division of fraction problems with contexts where the quantities involved are continuous. It makes sense to talk about a fraction of an hour. The context suggests a linear diagram, so this is a good opportunity for students to draw a number line or a double number line to solve the problem. Linker cubes are also an appropriate tool to solve this problem. The linker cube solution suggests an algorithm for dividing fractions using a common denominator. The context of this problem would also work in the case where the dividend is smaller than the divisor, e.g. \frac14 \div \frac23.
Solutions
Solution: Number Line
Using a double number line where one line is measured in miles and the other one is measure in hours we get the following diagram.
![Sol_1_85467de2c0283f11dcdd76b43b264f42](http://s3.amazonaws.com/illustrativemathematics/images/000/000/236/max/Sol_1_85467de2c0283f11dcdd76b43b264f42.jpg?1327542750)
In order to measure both \frac12 miles and \frac13 miles, we divide the 1 mile into \frac16 mile pieces. This way we can find 1 \frac12 miles and \frac23 miles. Driving two \frac23 mile stretches takes two hours. That leaves \frac16 mile, which will take \frac14 hour to drive. Therefore, it takes 2 \frac14 hours to drive 1 \frac12 miles.
Since we are asking "How many \frac23 are there in 1\frac12?" this is a "How many groups?" division problem: 1\frac12\div\frac23 = ?
Solution: Linker Cubes
Using linker cubes we need at least 6 linker cubes to represent 1 mile in order to divide the mile into thirds and halves at the same time. (Note: Any multiple of 6 would also work.) So 1 \frac12 miles is represented by 9 linker cubes and \frac23 of a mile is represented by 4 linker cubes. Now the question becomes: How many times do the 4 linker cubes (\frac23 mile) fit into the 9 linker cubes (1 \frac12 miles)? The answer is 1+1+\frac14 = 2 \frac14. (see photo below)
Since we are asking "How many \frac23 are there in 1\frac12?" this is a "How many groups?" division problem: 1\frac12\div\frac23 = ?
Note: This problem could lead into a discovery of a “common denominator” procedure for dividing fractions: Find a common denominator of both fractions, then just divide the numerators: 1 \frac12 \div \frac23 = \frac96 \div \frac46 = 9 (\frac16) \div 4 (\frac16) = 9 \div 4 = \frac94 = 2 \frac14.
![Sol_2_34a36e1250e11f78df0f649f407c3bdc](http://s3.amazonaws.com/illustrativemathematics/images/000/000/237/max/Sol_2_34a36e1250e11f78df0f649f407c3bdc.jpg?1327542764)
Solution: Number line solution
![Number_line_solution_53d55f47c13b95634afea5a6c6fa3960](http://s3.amazonaws.com/illustrativemathematics/images/000/000/538/max/number_line_solution_53d55f47c13b95634afea5a6c6fa3960.jpg?1332084762)
Since 1\frac12=\frac96 and it takes an hour to travel \frac23 = \frac46 miles, we can look at the number lines above and see that it will take 2\frac14 hours to travel the distance to the exit.
Since we are asking "How many \frac23 are there in 1\frac12?" this is a "How many groups?" division problem: 1\frac12\div\frac23 = ?
Traffic Jam
You are stuck in a big traffic jam on the freeway and you are wondering how long it will take to get to the next exit, which is 1 \frac12 miles away. You are timing your progress and find that you can travel \frac23 of a mile in one hour. If you continue to make progress at this rate, how long will it be until you reach the exit? Solve the problem with a diagram and explain your answer.