7. Seeing Double (Lines)

Another graphical method for deciphering and solving word problems is the use of Double Number Lines.  This technique uses the same approach which we used earlier with Equivalence Tables...only Double Number Lines go horizontally, and Tables go vertically.  (It may be helpful to review the Post on Equivalence Tables if you have not already done so).

If you remember the example problem from the post on Tables:  Three boys can eat two pizzas.  How many pizzas will be required to feed 15 boys?  Let us apply Double Number Lines to this problem.


Draw two horizontal parallel lines, and mark one "Pizzas" and the other "Boys".  Since the problem is asking to solve for Pizzas, begin with the Boys line and starting from zero, place marks on the line in increments of 3 (since we know that it will take 3 boys for every 2 pizzas) until you get to 15 Boys.

Now, on the Pizza line, place increments of 2 along the same marks.  Since we know that 3 Boys eat 2 Pizzas, put a 2 under the 3, and then add increments of 2 to each subsequent mark.

When you get to 15 Boys, we'll have the answer of 10 Pizzas.

As you see, the approach is nearly identical to how we set up Equivalence Tables.

Unit Rates
A side benefit of Double Number Lines is that we can use them to graphically determine Unit Rates.  We didn't discuss Unit Rates in earlier Posts...so here we go:

How many miles per gallon does your car get?  Don't answer, or stop and go out to check your car's specifications...I really don't care what your mpg is.  The point is that mpg is a unit rate.  It defines how many miles you can drive in one gallon (e.g. miles per gallon).  Miles per hour is another Unit Rate...it defines how many miles you travel in 1 hour.  Saying you are driving 80 miles in an hour and a half is not very helpful.

There will be word problems in Common Core asking to determine a unit rate.

Going back to our Boys/Pizza problem, we can use the Double Number Lines to determine the unit rate of Boys/Pizza.  Since we want to know how many Boys per one Pizza, find the number 1 on the Pizza line & label it.  Since we started with 2 Pizzas on the line, 1 Pizza is halfway between the 2 and Zero.

Now, make a mark on the Boys line directly above the 1 Pizza mark and label it.  Since we know that the 1 Pizza mark was halfway between the 2 mark and zero, the new Boys mark must also be halfway between the 3 mark and zero.  Hence, the number of Boys per Pizza is half of 3 which is 1 1/2 Boys per Pizza (or 1.5 Boys/pizza).

We could have found the opposite - the number of Pizzas/Boy - but the process is a little more complicated.  Because we need to find the quantity of Pizzas per one boy, find a mark for 1 boy on the Boys line, which is 1/3 distance between 0 and 3.

The corresponding Pizza mark would then be 1/3 the distance between 0 and 2...which is, um, let's see, hmm...carry the one....2/3 Pizzas.  Hence, the unit rate is now 2/3 pizzas/boy (or 0.667 Pizzas/Boy).

Now a couple of test questions:

Line Q1: There are 20 girls in the class.  The ratio of girls to boys is 4:3.  Use Double Number Lines to determine the number of boys in the class.

Line Q2:   For every handful of popcorn I grab to eat, I drop 2 kernels to the floor.  Every handful of popcorn has eight kernels in it. The bowl of popcorn gave me 9 handfuls.  How many kernels landed on the floor?