# Discovering Absolute Value Using a Dynagraph

Lesson 1 of 3

HS Algebra

40-60 minutes

**Description**

Use dynagraphs to visualize and represent absolute value relationships.

**Materials**

### Introduce

With Dynagraphs applet, students will compare f(x)=x, f(x)=|x|, f(x)=-x among other functions. The double number line allows them to see the relationship between inputs and outputs in a visual way that is different from a graph in the coordinate plane. This will also introduce the idea of absolute value as a function for students who have only taken the absolute value of single numbers up until now.

This could be an activity where the whole class watches one double number line on a projector or where students work in pairs on personal devices.

With the function rule f(x)= ½ x hidden (Relationship 1 in the applet), start the x value at 5 and move it to the left until x = 0. Ask students if they want to see any values again and highlight any students who are using good tools like a table or diagram (SMP5). Collect predictions after some think time. Predictions might be a numeric value for f(-1), an equation, a description of the pattern or anything else students might come up with. When everyone is done sharing, move x along the negative values. Reach consensus on the function rule (emphasizing that there are many equivalent ways to write the same equation).

### Explore

Complete the Lesson 1 Exploring Functions using Dynagraphs AS.

As students work, pay attention to what is done with Relationships 3, 5, 6, and 7. For these relationships students may need help in describing what is happening. You might suggest to students to think about the relationship as two distinct (or separate) pieces. For example, Relationship 5 can be represented as y = x + 2 as long as x ≥ -2 and it can be thought of as y = -(x+2) when x < -2. [This activity connects to SMP 8.]

**Teacher Note:**

If students struggle in making conjectures, suggest that they make a table of values while exploring the relationships.

### Synthesize

Discuss the responses to the reflection questions.
If time allows, have students continue their exploration by using the "Random Problem" problem on the Dynagraph applet.

### Teacher Reflection

- How did students connect the dynagraph representation to the meaning of absolute value?
- How did the use of the dynagraph support students in making conjectures about the observed relationships?
- What modifications were needed to support students in finding and testing their conjectures?

Leave your thoughts in the comments below.