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.
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.
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