Absolute Value Lesson 1

• # 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?

## Related Material

Mathematics Teacher, Mark W. Ellis and Janet L. Bryson

Mathematics Teaching in the Middle School, Michael Todd Edwards and Jennifer Nickell

Mathematics Teaching in the Middle School, Margaret Kidd

## Other Lessons in This Activity

Comparing the graphs of functions and related absolute value functions using reflection.
Solving absolute value equations graphically and algebraically.

• Is the double number line applet linked correctly? It it showing a representation of the double number line. Should it be an interactive applet?

• ## Ratings

•  Average 4.5 out of 5
• ### Essential Question(s)

• Given a set of inputs and outputs, how can we write a rule describing the relationship for the function?  How is the range affected by an absolute value function?
Teacher note: The lesson uses exploration to discover the absolute value function.  Share the essential question at the end of the lesson.

### Standards

CCSS, Content Standards to specific grade/standard

• HSF.IF.A.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
• HSF.BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

CCSS, Standards for Mathematical Practices

• SMP 5 Use appropriate tools strategically.
• SMP 8 Look for and express regularity in repeated reasoning.

PtA, highlighted Effective Teaching Practice and/or Guiding Principle CCSS

• Use and connect mathematical representations.
• Implement tasks that promote reasoning and problem solving.
• Support productive struggle in learning mathematics.