Subject |
Mathematics |

Level: |
Year 3 |

Topic |
Graphs of Exponential Functions |

Macro-concept |
Model |

Enduring Understanding |
Models communicate mathematical patterns in quantitative relationships. |

Essential Question |
Can all patterns in quantitative relationships be represented? |

Curricular or Pedagogical Focus / Lens |
Differentiation |

A class of 28 students has completed their study on rules of indices before introducing them to the graphs of exponential functions. Based on the students’ performance in the previous formative assessments and assignments, part of the lesson preparation involves splitting the class into 2 groups according to their perceived readiness in manipulating algebraic expressions by using rules of indices.

After establishing the basic knowledge of exponential graphs, the process of differentiated instruction is demonstrated in two ways: student choice and grouping by readiness.

__Establishing basic knowledge__

The key understanding for the lesson is: *Families of exponential graphs have a characteristic shape and are described by a unique function expression*. Using a series of simple-to-complex questions, the class is facilitated in activating their prior knowledge as they define a unique function expression, *y* = *a*e* ^{x}*, for the exponential function under study. They are reminded to pay particular attention to the variables x and y, the parameter a, and the domain and range of the function.

- In what ways do you
*x*^{2}and 2are different?^{x} - In general, how is the function rule of the exponential function different from that of the polynomial function?
- Are the rules of indices applicable only to exponential functions? Why or why not?

__Student Choice__

Working in pairs, students explore the graphical behaviour of the exponential function by selecting a suitable numerical (inductive) approach or analytical (deductive) approach. Students not only check the precision of their graphs using an online graphing tool, they also summarize their exploration using the acronym STAIRS (Symmetry, Turning point, Asymptote, Intercepts, Range, Shape) to systematically describe the characteristics of exponential functions. The following questions are used to guide their discussion:

- By considering the algebraic solution to
*y*=*a*= 0, explain why the graph does not intersect the^{x}*x*-axis. - Do the graphs of exponential functions have a horizontal asymptote?
- In the case when a > 1, describe the behavior of the values of
*y*as*x*approaches ¥ and -¥. Explain whether this behavior is generalizable to different values of*a*for 0 <*a*< 1? - From the graphs, identify the (three) key features of the exponential functions. How different are they compared to those when a > 1?

To summarize the exploratory activity, the class participates to add on to the description on the graphs’ commonalities in the (*decreasing*) shape, *y*-intercept, empty region below the *x*-axis and horizontal asymptote. The relationship between the exponential graphs, as * reflection of each other* about the y-axis is highlighted for the cases when a > 1 and 0 < a < 1.

__Grouping by readiness__

Using the * example in the worksheet, the group of students who have demonstrated higher proficiency in applying rules of indices are invited to independently attempt the more challenging question. The other group is attended to by using more teacher-directed approach on the less complex problem. The following questions are used for guiding the students in completing the task

- Are the graphs of {
*y*= 8and y = 2^{x}^{3x}} and {*y*= 1/8and y = 2^{x}^{-3x}} the same or similar? Justify your answer. - To what extent are the rules of indices important in helping us identify the characteristics of their graphical representation?

It was observed that students who had difficulty responding to the more challenging question turned to listen to the explanation for the less challenging problem before attempting the assigned question. On the other hand, those who have completed the less challenging problem were encouraged to proceed to the more challenging question before it was time for the class to consolidate the key learning points.

__Conclusion__

The lesson concluded with students’ reflection on the essential questions: What could the variables x and y represent in real-life? To what extent is the exponential function a realistic model in this real life situation?

** *Note:** Examples that are marked with a (+) indicate opportunity for teacher to suitably apply differentiation strategies, or for students to be challenged by choice.

**Posted by****：**** Mrs Chew Meek Lin**

Lead Teacher

Centre for Pedagogical Research and Learning (RGS PeRL)

Raffles Girls’ School