The different representations of proportional relationships - contexts, tables, equations, and graphs - all provide unique ways to understand and solve real-world problems, but what makes these tools even more powerful is the connections between them. Understanding how these representations relate to each other can provide a deeper deeper understanding of the problem and can often make finding solutions easier.

An image shows growth of a tree after 2 years. On year 2, the tree measure 3 feet and on year 4, the tree measures 6 feet.

\,\\\,Consider a tree that is planted from an acorn. It grows 3 feet taller every 2 years, and it starts at 0 feet tall at year 0 (when it is first planted).

Let x represent the number of years since planting and y represent the height of the tree in feet.

We can find the rate of change or growth by using the slope formula.

The change in y is the 3\text{ ft} increase. The change in x is the 2 years. So for the slope, we have:

m = \dfrac{\text{change in } y}{\text{change in } x} = \dfrac{3}{2}

x	0	2	4	6	8
y	0	3	6	9	12

\,\\\,We can create a table of values. Every 2 years, we will add 3 feet to the height. Start at the origin, (0,\,0).

\text{year}

\text{height}

\,\\\,We can create a graph from the table of values or using the slope triangle approach.

Since we know the slope is \dfrac{3}{2}, we can also create an equation to represent the situation:

y = \dfrac{3}{2} x

Each of these four representations describe our situation, and we can see the rate of change in each one is \dfrac{3}{2}.

Each of the representations has strengths and limitations.

For example, a table of values gives a snapshot of the tree's height at certain moments in time, but creating a large table that includes every time we're interested in cpuld take a very long time. It is also difficult to see overall trends like how quickly the height of the tree is changing.

The context helps make connections with the real-world application, but it can be difficult to make predictions for the pattern.

Graphs are easy to extend beyond the data we have and the slope of the graph shows us visually how quickly the height of the tree is changing.

Finally, equations help us predict future values but make it difficult to visualize what is happening.

It is important to be able to navigate between representations so we can create the best one to help us solve a given problem.

Examples

Example 1

The table shows how the number of hours a group of volunteers spend on a tree planting project is related to the number of trees they plant.

Hours	1	2	3	4	5
Trees	20	40	60	80	100

Write an equation that represents this relationship.

Worked Solution

Create a strategy

We can find the slope by finding the change in the y-values (number of trees) in the table and dividing that by the change in the x-values (hours). Then write the equation in the form y=mx, where m is the slope.

Apply the idea

The change between each coordinate pair is 1 hour and 20 trees. We can use the formula to calculate slope:

m = \dfrac{\text{change in } y}{\text{change in } x} = \dfrac{20}{1} = 20

This means that the group plants 20 trees per hour. The equation that represents this relationship is y=20x, where y is the number of trees and x is the number of hours.

Reflect and check

Equations are helpful if we want to calculate very large values, like how many trees they can plant in 120 hours. Creating tables or graphs that include very large values can take a lot of time.

Represent this relationship on a graph.

Worked Solution

Create a strategy

Since we already have a table, we will graph each as a coordinate pair where the x-value is the number of hours and the y-value is the number of trees.

Apply the idea

From the table, we get the points: (1,\,20),\,(2,\,40),\,(3,\,60),\,(4,\,80),\, and (5,\,100).

\text{trees}

100

\text{hours}

Reflect and check

Being able to create graphs is useful because we can easily see values that fall in between the ones that were in the table. For example, we can see that volunteers can plant 30 trees in 1.5 hours.

At this rate, how long will it take them to plant 175 trees?

Worked Solution

Create a strategy

The most efficient representation for answering this question is the equation because we can simply substitute 175 for y and solve for x to find the number of hours.

Apply the idea

In part (a), we wrote the equation y=20x, where y is the number of trees and x is the number of hours.

\displaystyle 175	\displaystyle =	\displaystyle 20 \cdot x	Substitute y= 175
\displaystyle 8.75	\displaystyle =	\displaystyle x	Divide both sides by 20

It would take them approximately 8.75 hours to plant 175 trees.

Example 2

It costs \$9 a month for a music streaming service.

Create a table, graph, and equation that represent this situation.

Worked Solution

Create a strategy

We can create our representations in any order. Let x be the number of months subscribed to the service and let y be the total cost.

Apply the idea

First, we can create a table. Let x be the the number of months and y be the cost of the streaming service. We can start with x=0 and y=0 because 0 months of the service costs \$ 0. For every additional month, the total cost increases by \$9.

x	0	1	2	3	4	5	6
y	0	9	18	27	36	45	54

Since the cost increase by \$9 every month, the slope is 9. We can use this to create and equation:

y = 9x

Finally, we can use the slope triangle approach or the table to create a graph:

Idea summary

Understanding the connections between tables, equations, and graphs is important when working with proportional relationships.

Each of these representations offers unique insights.

Graphs can be useful for looking at data between values in a table or visualizing the overall change in a situation.
Equations are helpful to predict very large values.
Tables are great for seeing a snapshot of the situation at a specific point.

Outcomes

7.PFA.1

The student will investigate and analyze proportional relationships between two quantities using verbal descriptions, tables, equations in y = mx form, and graphs, including problems in context.

7.PFA.1a

Determine the slope, m, as the rate of change in a proportional relationship between two quantities given a table of values, graph, or contextual situation and write an equation in the form y = mx to represent the direct variation relationship. Slope may include positive or negative values (slope will be limited to positive values in a contextual situation).

7.PFA.1e

Make connections between and among representations of a proportional relationship between two quantities using problems in context, tables, equations, and graphs. Slope may include positive or negative values (slope will be limited to positive values in a contextual situation).

2.07 Compare representations of proportional relationships

Ideas

Compare representations of proportional relationships

Examples

Example 1

Example 2

Outcomes

7.PFA.1

7.PFA.1a

7.PFA.1e

What is Mathspace

About Mathspace