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?

6. A Few Words About Nothin (i.e. Zero)

In a previous post, we dabbled a bit in ratios.  However, I failed to mention at that time that the denominator, the number below the line, the second number of a ratio, shan't be a Zero. Many word problems dealing with ratios may provide a word problem having a ratio of a/b and then quickly add where b is not equal to zero.

The reason is this:  you cannot divide a number by Zero.  It's not allowed, it's against the law of mathematics, I'm sure it violates some federal law and maybe even the Constitution, and besides, it just ain't fittin.  Only terrorists divide by zero.

If you find yourself faced with a ratio or fraction type problem where the denominator is zero...STOP!   Put down your pencil.  And call the authorities immediately.

5. Equivalence Tables & Coordinate Graphs

Well, back in MY day, we converted word problems directly to equations...and we liked it.  (well, maybe we didn't like it...but we did it).  Now, the Common Core wants us to solve word problems using a variety of visual methods such as Equivalence Tables and Coordinate Graphs.

This post will explore the concepts of each of these alternative problem solving methods.  

Equivalence Tables: 

Equivalence tables are one option for solving ratio problems.  The way this works is you set up two columns (which actually represents the ratio).  In the left column, put one of the ratio numbers, and place the other number in the right column.  

For example:  Three boys can eat 2 pizzas.  There are 15 boys.  How many pizzas will be required?

Set up two columns with five or six rows.  Make the Boys column be the one on the left (although which side doesn't really matter) and the Pizza column on the right.

In the first row, place the number of boys in the left column (i.e. "3"), and the number of pizzas in the right (“2”).  This is the basic ratio 3:2.

We need to find the number of pizzas to feed 15 boys.  So what we are going to do is walk down the column adding “3" to each number in the left column, and “2" to the numbers in the right column.  The numbers we use [3 and 2] are whatever the numbers are in the very first row. 

We continue to add the number 3 to the left column and 2 to the right column.  When the number in the left column finally reaches the target (i.e. 15), stop.  The answer to the quantity of Pizzas required will be in the right column (i.e. 10).

Good thing we don't have 150 boys...we'd be here all night.

How this works (if you care to know):  In the left Boys column where we keep adding 3’s, the second row is actually two 3’s, the third row is three 3’s, the fourth row is four 3’s, and the fifth is five 3’s.  The right Pizza column: the second row is two 2’s, then three 2’s, then four 2’s, and finally five 2’s.  If we kept on going, then the next row would be six 3’s and six 2’s, hence 18 boys would require 12 pizzas. Each row represents the same, identical ratio:  3:2, 6:4, 9:6, 12:8, 15:10 ...they are all 3:2.

Woe be us if the total number of boys is not easily divisible by 3.  If the total number of boys were, say, 16, then this method breaks down.


Coordinate Graphs

Well, now that we have mastered THAT, let's move right into applying the same problem in a Coordinate Graph.  Applying the table to a Coordinate Graph is very straight forward.  Using the table  from the previous problem, assign one column as the "X" axis, and the other column of numbers as the "Y" axis.  Now draw a graph with X axis going to the right, and the Y axis going up assigning numbers on each axis.  

Now plot each pair from the table on to the graph:

First row pair 3,2 : go to 3 on the X axis and then go up 2 on the Y axis & place a point.

Second row pair 6,4:  Find 6 on the X axis and go up to 4 on the Y axis & place another point.

Repeat for all points on the table and it should like the one below.

You know you've done this correctly when all of the points line up in a perfectly straight line.  This means that each pair of numbers represent the same ratio.  If any point does not lie on that line...or the line curves for some strange reason, you best double check the table.

You can also use the graph to extrapolate for other numbers.  For example, if there were 17 boys, then you go to 17 on the X axis, and go straight up to where the line of points would intersect with 17 from the X axis.  From this point, make a straight line over to the Y axis.  Where that line intersects the Y axis is the # of Pizzas required.   

From the graph, it looks like you would need 11 1/2 pizzas.  Of course, you would order 12 whole pizzas, and take the extra 1/2 pizza home for dinner.  Then the School Principle finds out you absconded with the extra pizza for your own personal gain without reimbursing the school and you end up in the Principle's office the next day trying to explain yourself.  I don't have a graph for that...you're on your own.

What would a ratio graph look like if you made a mistake while setting up the table?  Let's have a look.



Zach set up a table of Equivalences...but he made a mistake on one of the entries.  Plot the number pairs on a graph and circle the mistake.  


As you can tell from the chart below, the fourth number on the graph is not really in line with the first three.  You can also tell by following the path along the first three points.  

 Starting at 0,0, you go over to 4 and up 3 to get to the first point...then over another 4, and up another 3 to get point #2.  Go over 4 and up 3 to get point #3.  As you go over 4 and up 3 again, you realize that the fourth point is not where it should be.  It is out of place.  Once this happens, then the other points follow along and the whole thing is just a Dogs Dinner.








If we go back to the table, we see that the fourth number in the X column is a 15 which should have been 16 (12 + 4 = 16).  


Boy, that Zach....what a card.

Here are a few problems to try.  The answers will follow on another post...some day.

Table Q1:  The ratio of boys to girls in the classroom is 3:5.  You count 18 boys.  How many girls are there?  Use an equivalence table.

Table Q2:  Plot the following table on a Graph.

3. Ratios: 4 Out of 5 Agree

I'll bet the odds are 3:1 that you are just as confused as I am when to use the term "rate" and when to use the term "ratio".  They seem similar, if not the same.  No matter...the common core will be asking both types of questions, so we must be prepared for both.

Ratios can be described in three basic ways:

"for every x of these, there are y of those", for example, for every 3 girls there are 2 boys.
...or... the ratio of girls to boys is 3:2 (this is the most common description of a ratio)
...or... the ratio of girls to boys is 3/2  (but this is also common...so who knows)

Like rates, a ratio can be described as x:y, or as y:x ...and both are valid.  In the girls-to-boys example, the ratio can be described as 3:2 (ratio of girls to boys), or as 2:3 (boys to girls). Both ways are perfectly valid depending one what you are trying to solve.


SAFETY WARNING:    Many word problems will describe a ratio one way, and then ask you to answer for the opposite.  For example,

"for every 5 boys, there are 3 girls.  What is the ratio of girls to boys."

See what they did here?  They offered a ratio with girls first and boys second....then reversed the order for the ratio answer asking for boys first and girls second to see if you are paying attention.  Although the ratio was first described as 5 boys to 3 girls, the question is asking for a girls-to-boys ratio, which is 3:5.


Ratio Tables
Tables seem to prevalent in many common core studies.  They set up a table of numbers and expect you (or the student) to interpret ratios based ion the table...or perhaps completing the table.  For example:

There are three school classrooms of 6th graders.  They each have the same exact ratio of boys to girls in their class.

What is the ratio of boys to girls in these classrooms?

How many girls are in class #3?

Now think....the problems is asking for the quantity of girls.  You can set the ratios up any way you want.  Since we are solving for girls, let's show the ratios with girls on top:

The ratio of Girls to Boys is 2:3.  Hence, the ratio of Boys-to-Girls is the opposite, or 3:2.

The ratio of girls-to-boys in classroom #3 must be the same ratio as classrooms #1 and #2, which is 2:3.  Another way to think about it is:  "? is to 12 as 2 is to 3."  The two relationships must be the same.  Set up an equation which shows that the two ratios are equal to each other:

Now solve for ?, which yields the answer 8 girls.


This seems pretty straight forward, so let's try a few ratio problems (answers to follow in the next post):

Ratio Q1:  Three boys can eat 2 pizzas.  What is the ratio of boys to pizzas?  What is the ratio of pizzas to boys?

Ratio Q2:  The ratio of pieces of sausage to pieces of pepperoni on these pizzas is  3:1.

For every ____ pieces of pepperoni, there are ____ pieces of sausage.


Ratio Q3:  The three grade 6 classrooms have the same ratio of Boys to Girls.

 What is the ratio of Boys-to-Girls?

How many Girls are in Classroom #2?

How many Boys are in Classroom #3?

4. Answers to Ratio Problems

Ratio Q1:  Three boys can eat 2 pizzas.

The ratio of boys to pizzas is 3:2

The ratio of pizzas to boys is 2:3

These seem kinda obvious, so I am not sure what more to add.  If you have any questions, post a comment and I'll think of something clever.


Ratio Q2:  The ratio of pieces of sausage to pieces of pepperoni on these pizzas is 3:1.

For every 1   piece of pepperoni there are  3   pieces of sausage.


Ratio Q3:  Classroom #1 has 15 boys to 20 girls, or 15:20.  Break this down to its simplest form:

For Classroom #2, we need to solve for girls.  So flip the ratio so that girls "?" is on top.  In this case, we also flip the 3/4 ratio to 4/3.  So:  ? is to 12 as 4 is to 3.

 Now solve for ?, which is 16 girls in classroom 2.

For boys in classroom 3, describe the ratio with the boys "?" on top, which is now 3/4. So ? is to 12 as 3 is to 4.  Solve for the ?, which is 9 boys in classroom 3.

The final matrix is:

One important check to make sure you got the matrix correct:  The ratio of Boys on top to Girls on the bottom row is 3:4 (there are more girls than boys).  Hence, the top row must always be smaller than the bottom row.  If they are not, then you forgot to flip the ratio, or something.

1. Rates Rule...Ratios Drool

Rates can come in many different forms, and can be quite confusing.  A Rate is a fraction describing how many of one thing compared to another thing, such as miles per gallon.

A "Ratio" is a rather simplified form of Rates where the fraction is rather generic, such as 3 to 1, or 3:1.  We'll be discussing ratios later on.

English phrases used to describe a rate or ratio in a word problem can be: 

"per" -  as in miles per gallon

"for every" -  as in one cup of sugar for every two cups of flour

"compared to" - as in I am only 4 feet tall compared to Billy who is 5 feet tall

"in"or "in about" - as in I can eat 4 of those in about 3 minutes

There is no magic as to how describe a rate: one thing compared to another.  Miles per gallon and gallon per miles are both valid, and can be used depending on what you are trying to solve.  

So the key is, whatever you are solving for, put THAT number on top in your rate.  For example:

Your car gets 30 miles per gallon, and you have 12 gallons in your tank.  How far (i.e. how many miles) will you be able to drive until the tank is empty?  The problem wants to solve for distance, or miles.  Therefore, describe the rate with miles on top:

However, what if the question was how many gallons would you need to drive 500 miles?  In this case, you are solving for gallons not miles…so describe the rate with gallons on top.

Cancelation of Units
An important thing to remember when setting up equations from word problems is to pay attention to the units "above the line" and "below the line", and set up the equation so that the units cancel out yielding the answer you want.  For example, in the sample problem above, we set the rate with gallons on top and miles below so when we multiply by the number of miles, the two "miles" (one "above the line", and the one "below the line" cancel out...and only gallons are left.

Let’s now try a few rate-type word problems (you'll find the answers in the next blog post below):

Rate Q1: The 90 minute math test has 25 word problems.   You know it takes you about 15 minutes to solve 3 word problems.  At that rate, how many word problems can you solve in the 90 minutes?  Are you going to get through all of the problems?  At what rate are you solving word problems?

Rate Q2:  The teacher has to grade all of these math tests.  By watching the clock, she noticed that she graded 5 word problems in about 12 minutes.   Since there are 25 word problems per test, how long will it take her to grade one test?  What is the rate of which she is grading the word problems?

If there are 30 students taking the tests, how long will it take her to grade all of the tests?  At what rate is she grading each test?

Rate Q3:  It is 1:20 pm on a Friday afternoon.  The school bell will ring exactly at 2 o’clock, so you have 40 minutes remaining until the weekend starts.  However, the teacher is planning to hand out a pop math quiz before the end of the school day.  You and your friends decide that if you ask enough questions to keep the teacher busy, maybe there won’t be enough time for the test!   Yesterday, you noticed that 3 math questions took about 8 minutes to ask and answer.  How many questions do you and your friends need to come up with to take up the 40 minutes left in the day?  What is the rate of questions to be asked?

Rates as Percentages

Percentages are simply rates with "100" as the denominator, for example 33% is equal to 33 out of 100, or 30/100.  If you have a different "total" amount, such as 500 and you want to find 33%, then you set up the rates to be equal.

In this case above, the rate of 33 to 100 is the same as ? will be to 500...both are 33%.  Hence the "?" will be 33% of the 500.  So, if you solve for "?", 

OK, got it?  Let's try a few percentage type problems.  Again, the answers are presented in the next blog post below.

Rate Q4:  The Math test has 30 word problems.  30% are hard, and 40% are medium.  How many are going to be easy?

Rate Q5:  There are 30 students in the class.  60% are boys.  How many are girls?

Rate Q6:  The girls in the class tend to get 80% of the math answers correct.  However, the boys only get 60% of the answers correct.  Using your answers from Rate Q5 above, and assuming there are 30 word problems in each test, how many total word problems will the girls get correct?  How many total word problems will the boys get correct?   Which group wins with the most total correct answers?

2. Answers to Rates Problems

Answer to Rate Q1:  

The question is asking for “how many word problems”, hence we need to show the rate with the number of word problems on top

Knowing the rate, we use it and the total time allowed in the test to determine the quantity of problems:

Answer to Rate Q2:

The rate we are looking for is "5 word problems in about 12 minutes.  This can be described as 5 word problems/12 minutes, or 12 minutes to grade 5 word problems.  The problem is asking for time - how long will it take.  So use the rate with minutes on top.

Use this rate to now solve for the time to do 25 problems.

We know the time for grading one test (which is a rate of 60 minutes/1 test)...now use that time to grade 25 tests.  In this case, I converted 60 minutes to one hour...

30 hours....Yikes!

Answer to Rate Q3

The problem is asking for the total number of questions required.  Hence, specify the rate as so many questions/minute

Knowing the time (minutes) before the end of class, solve for the quantity of questions needed to fill up the remaining time:

Answer to Rate Q4:  

There are two ways to solve this one.  The first method is brute force:  find the number of hard questions, then the number of medium questions, then subtract them both from the total (30) to find the remaining easy questions.  This method is the more traditional approach, and probably the method Common Core is looking for.

The alternative approach (which you can do nearly in your head) is to find the percentage of easy problems.  Percentages always add up to 100.  If we have 30/100 hard and 40/100 medium, then that leaves 30/100 remaining to make up the total 100/100, which are the easy ones.  Since the percentage of easy ones (30%) are the same percentage as the hard ones (30%), then the answers are the same:  9.

Answer to Rate Q5:  

Again, two ways to solve this one.  Find the quantity of boys (60% of 30), then subtract that from the total to find the remaining quantity of girls...or...find the percentage of girls (since the total percentage must add up to be 100), then calculate for the quantity of girls.

Answer to Rate Q6:  

This one is tricky.  The problem is asking for the total quantity of word problems solved correctly by the 18 boys versus the 12 girls. There are more boys than girls, so the total number of correct answers may be higher for the boys...except percentage of correct answers is lower than the girls.

So first, let's find the total quantity of problems being attempted by the girls, then figure out how many of those are correctly solved (in this case 60% for the girls).

Since there are 30 problems in the test, and 12 girls taking the test, then the total quantity of problems being attempted is 30 word problems x 12 girls = 360 girl word problems.

Now, if only 60% of them were solved correctly:  60% x 360 = 216 correct girl answers

Now the boys:  Total boy word problems = 30 word problems x 18 boys = 540

Quantity of Boys correct answers = 40% x 540 = 216 correct boy answers

It’s a Tie!!