Variable Relationships

What is a variable?

A variable is a letter that stands for a number. In d = 60t, the letter t stands for time and d stands for distance. The letter is not a mystery. It is a label for a quantity that can change.

If the letters look strange, read them as words. d = 60t means "distance equals 60 times time."

Independent and dependent variables

Two quantities change together. One of them you choose. It is called the independent variable. The other one follows it. It is called the dependent variable, because its value depends on the first one.

Here is the example your kid will see in the module. A car drives at a steady 60 miles per hour. The rule is:

distance = speed × time, or d = 60t

Time is the independent variable. You decide how long the car drives. Distance is the dependent variable. It follows from the time.

• Drive 2 hours: d = 60 × 2 = 120 miles.
• Drive 3 hours: d = 60 × 3 = 180 miles.
• Drive 4 hours: d = 60 × 4 = 240 miles.

The names can feel backwards. Independent means you set it. Dependent means it depends on the other one.

What is this module?

The module shows one relationship three ways at once. The equation gives the rule, the table lists values, and the graph draws them. They all say the same thing.

Move the slider or drag the point along the line. All three views update together. The highlighted table row matches the marked point.

There are three situations to pick from:

• Speed: d = 60t, distance in miles over time in hours.
• Cost: c = 4q, cost in dollars for q pounds of apples.
• Wages: w = 15h, wages in dollars for h hours of work.

What the Swap axes toggle does

The toggle swaps which variable goes on the horizontal axis. Normally the independent variable is on the horizontal axis and the dependent variable is on the vertical one.

Turn Swap axes on and the dependent variable moves to the horizontal axis. The independent variable moves to the vertical one. The graph, the axis labels, the marked point, and the table all follow.

The rule itself does not change. d = 60t still describes the same relationship. Only the picture is turned around.

What they're learning

• How to tell the independent variable from the dependent one.
• How to read an equation as a rule between two quantities.
• How a table lists matching values of the two variables.
• How a graph shows the same rule as a line.
• That swapping the axes changes the view, not the relationship.

Aliases

• Independent and dependent variables
• Equations and graphs
• Function tables
• Input and output

Refresher

Two quantities change together. The one you choose is the independent variable. The one that follows is the dependent variable. An equation, a table, and a graph all show the same relationship in different ways.

Try this together

Two or three minutes at the table is enough.

1. Open Speed mode. Point at the equation d = 60t and ask which letter is the independent variable.
2. Set t = 2 and read the distance together. Then set t = 5 and ask what changed.
3. Find the table row that matches the marked point on the graph.
4. Switch to Cost mode and ask what c = 4q means in words.
5. Turn Swap axes on. Ask what moved and what stayed the same.

If they are comfortable, ask them to predict a value before moving the slider. For example, what is the cost of 6 pounds of apples?

Words you might hear

• Variable: a letter that stands for a number, like t or d.
• Independent variable: the quantity you choose or set. It goes on the horizontal axis.
• Dependent variable: the quantity that follows from the independent one. It goes on the vertical axis.
• Equation: a rule written with an equals sign, like d = 60t.
• Table: rows of matching values for the two variables.
• Graph: a picture of the rule, drawn as a line.
• Swap axes: switch which variable is on the horizontal axis. The rule stays the same.

Deeper dive

For d = 60t, time is the input. Pick t = 2 and the distance is 120. Pick t = 3 and it is 180. Read the equation as a sentence: distance equals 60 times time.

The independent variable usually goes on the horizontal axis and the dependent one on the vertical. Swapping the axes turns the picture around, but the rule d = 60t does not change, because it still describes the same relationship.