Algebra S2 – Unit 2 Week # (link to class resources) Exponential Functions

Multiplication & Division Properties of Exponents Nebraska Standards for Mathematics: SS12.1.1b; 12.1.3b; 12.1.3c; 12.3.3b; DK1-3 Common Core Standards for Mathematics: A.SSE.2 Objective: Students will understand how to simplify exponential expressions.

Students will know to add the exponents when multiplying like bases. … subtract the exponents when dividing like bases

Students will be able to use the multiplication and division properties of exponents to simplify an expression. Overview Students will use desmos.com (or any graphing calculator) to explore the multiplication and division properties of

exponents, specifically 𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛 𝑎𝑛𝑑 𝑎𝑚

𝑎𝑛 = 𝑎𝑚−𝑛. The exploration begins with making observations of odd

power and even power functions as well as with a review of recognizing when two graphs are the same. Students will then graph given expressions where like bases are multiplied or divided. Based on the shape of the graph, students can determine if the expression is even or odd. By graphing the originally provided even/odd expressions at the beginning, students will determine the simplest form of each expression by looking for overlap in the graphs. By looking for a pattern in the exponents, students will be able to generalize the multiplication and division properties. Prior Knowledge

Definition of Exponents e.g. 𝑎3 = 𝑎 ∙ 𝑎 ∙ 𝑎

Identity Property of Exponents 𝑎1 = 𝑎

Zero Exponent Property 𝑎0 = 1

Negative Exponent Property 𝑎−𝑛 =1

𝑎𝑛 and 1

𝑎−𝑛 = 𝑎𝑛

Working Knowledge of Desmos. Particularly, how to enter functions and zoom. Being able to turn functions on/off may be helpful, but not required.

Be able to graphically recognize two functions are the same when they completely overlap one another, giving the appearance of only one line on the graph.

Assessment

Monitor and check that students are filling in the properties correctly. o Engage with students during work/explore time to explain how they came to their conclusion(s).

Practice problems on bottom of student handout. o Check while circulating, looking for incorrect answers to find and flush out misconceptions

Did they change their bases? Did they combine their exponents properly? If necessary, direct them back to the exploration and learn how they completed the activity.

Class discussion at conclusion of period, or following day.

Quiz over exponent properties (about 4 days after) Materials

Student handout for each student (attached)

Access to desmos.com for each student, or pairs of students if necessary. Desmos is available online, desmos.com, and as a free mobile app for Android and Apple devices.

Algebra S2 – Unit 2 Week # (link to class resources) Exponential Functions Lesson Outline Getting Started

1. Teacher and students should go to desmos.com or open the mobile app. 2. Set window -5 to 5 for x and y values.

To do this, click the settings wrench and type in the ranges for each axis.

If using a mobile device, students may want to use -2 to 2 range. 3. Graph the functions listed with problem #1 on the handout. 4. Have students write their observations about how the graphs are similar. (all graphs make a “U” shape; even power) 5. Repeat with functions listed for problem #2. (all graphs are a sideways “S” shape; odd power) 6. Have students share their observations. These may be the similar observations or how the even/odd functions

are different. Make certain students recognize the even/odd grouping of the functions. 7. Give students a minute to answer problem #3.

Call on a student to share their answer. (you only see one graph)

Call on another student to explain why. (graphs overlap completely/in all spots) 8. Explain that students will be using these facts from #1-3 to explore and generalize 2 properties of exponents.

Explore Activity

9. Teacher and students should enter #4 into desmos.

It would be a good idea to have students always start fresh, clearing out previous functions, or turning off functions by clicking the colored dot next to each function.

10. Prompt students:

Is this going to be an even or odd power? (odd)

How do you know? (the shape is like a “S”)

11. Have students graph the odd power functions provided at the beginning until they find the one that matches (overlap). They should write the power in the ( ).

12. Direct students to continue this process for each of the remaining expressions. Have students work individually. 13. As students complete each property, they should look for a pattern so that they can fill in the properties.

Students may check their answers with a neighbor after they’ve completed each property. 14. Monitor and circulate during work time, engaging with students concerning their generalizations.

Ask students to explain their process.

Ask how are they filling in their properties? o Is there a pattern in these problems? (pointing and referring to the activity problems they’ve

completed) 15. As the class finishes their exploration, they may move to the practice problems.

For students who are working on the practice problems and successfully completed the activity… o How does this relate to the properties we learned about [yesterday]? o What would this problem look like expanded out? Simplified? Does it match? o Why do these properties ‘work’? (algebraic proof, expanding expressions and simplifying)

Summary

16. When the class has completed #1-13, have them check their answers (teacher discretion at this process) 17. Conclude with a discussion of the properties.

How did they determine the properties?

Why does this work? (expand an expression out as an algebraic proof)

Does it matter which order we add/subtract the exponents? Why? 18. Depending on the class, students may complete the practice problems before the end of the period or take

home to finish. Independent Practice

Practice problems on handout and/or

Additional practice worksheet or book problems to teacher discretion.

Algebra S2 – Unit 2 Week # (link to class resources) Exponential Functions

Activity – Properties of Exponents Name: 2 Standard: SS12.1.1b; 12.1.3b; 12.1.3c; 12.3.3b; DK1-3 Date: Period:

Enter the following functions in desmos and answer the questions.

𝑦 = 𝑥2 1. How are all these graphs similar? Consider shape,

direction, size, function (equation), etc.

𝑦 = 𝑥4

𝑦 = 𝑥6

𝑦 = 𝑥3 2. How are all these graphs similar? Consider shape,

direction, size, function (equation), etc.

𝑦 = 𝑥5

𝑦 = 𝑥7

3. How can you tell graphically, if two functions are the same?

Enter the expression into desmos and determine the simplified form. Each answer will be one of the functions listed above. A function may be used more than once.

4. 𝑥2𝑥3 = 𝑥( )

MULTIPLICATION PROPERTY

When multiplying expressions with the same base,

____________ the exponents of each base.

𝑎𝑚𝑎𝑛 = 𝑎( )

5. 𝑥3𝑥3 = 𝑥( )

6. 𝑥 ∙ 𝑥4 = 𝑥( )

7. 𝑥 ∙ 𝑥3 = _________

8. 𝑥2𝑥4 = _________

Algebra S2 – Unit 2 Week # (link to class resources) Exponential Functions

9. 𝑥5

𝑥3= 𝑥( ) DIVISION PROPERTY

When dividing expressions with the same base,

_____________________ the exponents of each base.

𝑎𝑚

𝑎𝑛= 𝑎( )

10. 𝑥6

𝑥3= 𝑥( )

11. 𝑥4

𝑥2= 𝑥( )

12. 𝑥7

𝑥2= _________

13. 𝑥4

𝑥= _________

PRACTICE Use the properties of exponents to simplify each expression without using a graphing calculator.

1. 𝑑5𝑑7 2. 𝑥2𝑥3 3. 𝑦4𝑦7

4. 𝑘7

𝑘4 5.

𝑐6

𝑐4 6.

𝑥8

𝑥3

7. 𝑦0 8. 𝑚5𝑚3

9. 𝑥8𝑥−3

10. 𝑤−3 11. 𝑟4

𝑟9 12.

𝑑12

𝑑9