In this section, you will investigate different ways to represent an additive relationship.

For example, consider the mass of a beaker containing different amounts of water.

Use the interactive below to determine the mass of the beaker and water as more water is poured into the beaker. Record the combined mass in a table like the one that follows. Copy the following table into a word processing or spreadsheet app or program, and use the program or app to complete the table.

Use the results from the interactive to complete the table you created.

Interactive popup. Assistance may be required.

Check Your Answer

Volume of Water Added (milliliters)	Mass of Water Added (grams)	Total Mass (grams)
0 mL	0 g	90 g
25 mL	25 g	115 g
50 mL	50 g	140 g
75 mL	75 g	165 g
100 mL	100 g	190 g
125 mL	125 g	215 g
150 mL	150 g	240 g

Use your completed table and the interactive to answer the questions that follow.

• How does the total mass of the combined beaker and water compare to the mass of the water added?

  Interactive popup. Assistance may be required.

  Check Your Answer

  The total mass is always 90 grams more than the mass of the water.

• What equation could you write that relates w, the mass of the water, to t, the total mass of the water and beaker?

  Interactive popup. Assistance may be required.

  Check Your Answer

  t = w + 90

Use the interactive below to plot the points from the table of values that you just generated. Drag each point from the bank to its correct place on the coordinate grid below.

Use your completed graph to answer the questions that follow.

• If you were to connect the points with a line, what would be the equation of the line?

  Interactive popup. Assistance may be required.

  Check Your Answer

  y = x + 90

• How does this equation compare to the equation from the table of values that you generated before?

  Interactive popup. Assistance may be required.

  Check Your Answer

  This equation, y = x + 90, is the same as t = w + 90. In the graph, x represents the same independent variable, the mass of the water, as the variable w represents in the table of values. In the graph, y represents the same dependent variable, the total mass of the beaker and water, as the variable t represents in the table of values.

So far, you have used verbal descriptions, tables, graphs, and equations to represent an additive relationship. Now, take a closer look at how the relationships are related.

Click on each representation to see how the independent variable, dependent variable, and constant addend appear in that representation.

Pause and Reflect

How does the constant addend, a, appear in each representation of an additive relationship? Compare these representations.

Interactive popup. Assistance may be required.

Check Your Answer

In a verbal description, the constant addend is given as a number that is added to the independent variable each time.

In a table, the constant addend is the number added to the value of the independent variable in each row of the table.

In a graph, the constant addend is the starting point, or y-intercept, of the line.

In an equation, the constant addend is the number that is added to the independent variable.

In each representation, the value of the dependent variable is a unit greater than the independent variable.

Practice

1. A box that is used for chocolate candies weighs 6.5 ounces. Generate a table of values, a graph, and an equation that shows the relationship between c, the weight in ounces of chocolate candies being packaged, and t, the total weight in ounces of the package.

   Interactive popup. Assistance may be required.

   Need a hint?

   What value is being added to the weight in ounces of the chocolate candies being packaged?
   Interactive popup. Assistance may be required.

   Check Your Answer

   
2. The table below shows the number of electoral votes and the number of U.S. Representatives that certain states have after the 2010 census.
   

   Electoral Votes and Representatives of U.S. States
   State	California	Texas	New York	Florida	Illinois
   Number of U.S. Representatives	53	32	29	25	19
   Number of Electoral Votes	55	34	31	27	21

   
   

   Write an equation that relates r, the number of U.S. Representatives, to e, the number of electoral votes for each state.

   Interactive popup. Assistance may be required.

   Need a hint?

   What number is being added to the number of U.S. Representatives to generate the number of electoral votes?
   Interactive popup. Assistance may be required.

   Check Your Answer

   e = r + 2
3. Jacob had already collected 5 toys for a toy drive. The graph below represents the relationship between the number of additional toys that were collected and the total number of toys that Jacob had collected.
   

   Write an equation that relates x, the number of additional toys that Jacob collected, to y, the total number of toys collected in the toy drive.

   Interactive popup. Assistance may be required.

   Need a hint?

   What number is being added to the number of toys that Jacob collects in order to calculate the total number of toys collected in the toy drive?
   Interactive popup. Assistance may be required.

   Check Your Answer

   y = x + 5