Absolute Value Lesson 1
Use dynagraphs to visualize and represent absolute value relationships.
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).
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.]
If students struggle in making conjectures, suggest that they make a table of values while exploring the relationships.
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.
Leave your thoughts in the comments below.
Is the double number line applet linked correctly? It it showing a representation of the double number line. Should it be an interactive applet?
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.
CCSS, Content Standards to specific grade/standard
CCSS, Standards for Mathematical Practices
PtA, highlighted Effective Teaching Practice and/or Guiding Principle CCSS