| Monday | Tuesday | Wednesday | Thursday | Friday | |
|---|---|---|---|---|---|
| Objectives / Standards |
Understand how exponents and logarithms behave graphically, and suggest algebraic ways in which exponents and logarithms can be manipulated. |
The goal is for the students to build a foundation on learning and understanding what the exponent laws are and why they are true. | The goal is for the students to build a foundation on learning and understanding what the logarithmic lidentities are and why they are true. | The goal is for students to begin grasping how to solve exponential equations using logarithms. | The goal is the reverse of that of Thursday's lesson: this time students will work to use exponents to solve logarithmic equations. |
| Do Now | Ensure students bring or have access to calculators. | Briefly review the definition of an exponent, as it is the foundation to start introducing exponent laws. | Briefly review the definition of a logarithm and how they behave graphically. Review the notation for the common log (base 10) and the natural log (base e). In particular, remark that logarithms and exponents are inverse operations. | Show, in equation form, that the logarithm of an exponent of some function is the original function if the exponent and logarithm have the same base. | Show, in equation form, that the exponent of a logarithm is the original function if the logarithm and exponent have the same base. |
| Introduction of New Material | Show graphs of various logarithmic and exponential functions | Go through the chain of exponent laws, using more basic laws to explain the more complicated ones. | Starting with the definition of the logarithm, explain what each logarithmic identity means. Use exponent laws as needed. Emphasize that these explanations show how exponents and logarithms are related. | No new material is needed. This is essentially an application of the exponent laws combined with a basic understanding of logarithms. | Continue to emphasize that logarithms and exponents are inverse operations. |
| Guided Practice | Demonstrate graphs of certain logarithmic and exponential functions so that students may be able to notice patterns. Consider using the guide provided. here. | Break students into teams so that there are three or four students on each team. Introduce a number of expressions involving exponents one at a time that need to be fully simplified. The first team to correctly and completely simplify the expression earns a point. Do about ten rounds of this, depending on time constraints. | There are a wide range of problems available for practice, including those posted on this website and problems that are likely available in your textbook. |
Providing applications in guided examples will allow the students to start understanding how to build exponential equations and why they are useful. http://www.opencurriculum.org/9345/populations-with-exponential-equations/ |
As is true for Thursday's lesson, applications are an important part of ensuring students' understanding the material. |
| Independent Practice | Divide class into groups of roughly 3 students. Give a set of exponential functions for the students to graph simultaneously, and do the same for a set of logarithmic functions. Ask students to write about the patterns they notice. Consider using the guide provided here. Furthermore, be sure to encourage discussion between the students within their groups--this is a collaborative effort, after all. | The activity suggested under Guided Practice should be sufficient. If not, consider using the link below, which is also a possible homework assignment. | Walk through a handful of examples with your students and then give them a chance to work through a few more on their own or in small groups. Be sure to go over these problems as a class afterward though. | Problem-solving is key here. Ensure that students have ample time to solve problems. Word problems can also be covered at the teacher's discretion. | This worksheet provides an array of problems that can be used for class work and/or homework. http://www.opencurriculum.org/9515/worksheet-solving-logarithmic-equations/. Note that the last exercise is very challenging. |
| Exit ticket | Students submit their small group work for class participation grades. Ensure that each student fills out their own paper. | Each group should hand in their scratchwork from today's activity and the homework from the previous day, if any. | Choose an array of problems from your textbook--this will be a good starting point. Make sure that some word problems are included. Any leftover problems can be used as homework. | ||
| Homework |
Have students complete the independent practice options if they did not already do so in class. Perhaps suggest other exercises in your textbook for additional practice in case the students feel they need it. Have students preview a section in their textbook related to the Laws of Exponents. |
Select problems from your textbook or the handouts provided for your students to complete. Have students preview a section in their textbook related to Logarithm Identities. |
Various handouts and textbook problems are available, including on this website. One interesting application is the Rule of 72--see here: http://www.primerica.com/public/rule-of-72.html. Have students try to prove that the Rule of 72 works for different time periods. For instance, let them find the amount of time it takes for a sum of money to double with compound interest at a rate of 8%. | ||
| Materials |
Suggested Guided Practice Options: http://www.opencurriculum.org/8504/graphing-exponential-functions-teacher-guide/ http://www.opencurriculum.org/8509/graphing-logarithmic-functions-teacher-guide/ Suggested Independent Practice Options: http://www.opencurriculum.org/8506/graphing-exponential-functions-independent-practice/ http://www.opencurriculum.org/resources/new-document/?collection=graphing-exponential-and-logarithmic-functions |
This article gives a viable progression of an order on how to introduce the most important exponent laws: http://www.opencurriculum.org/5509/exponent-laws/ Here is a set of possible homework problems on Laws of Exponents: http://www.opencurriculum.org/8513/exponent-laws-practice-problems/
|
One possible source of examples or reading assigned for this class: http://www.opencurriculum.org/5511/logarithmic-identities/ Possible Problem Set: http://www.opencurriculum.org/9341/using-logarithm-identities-practice-problems/ Problem Set Answer Key: http://www.opencurriculum.org/9340/using-logarithm-identities-practice-problems-answer-key/ |
Many examples for in-class and homework use are available at these two links. The second of these two articles also features some examples involving both exponents and logarithms. http://www.opencurriculum.org/5510/solving-exponential-equations/ http://www.opencurriculum.org/5513/solving-logarithmic-equations/ http://www.opencurriculum.org/resources/new-document/?collection=solving-equations-with-exponents-and-logarithms |
This may be the point where you wish to announce a test or quiz. Regardless, this is where high school algebra exponents and logarithms units generally end. Assign a variety of problems covering all five subtopics in this unit. |
| Modifications / Accomodations | If more time is available, feel free to throw in more examples that further hint at fundamental exponent laws. |
The exponent laws article contains more examples than would be necessary/practical to go over in one class period. Pick and choose which ones you think are the most helpful and worth your limited classroom time. As incentive for the game, consider awarding the winning team a few extra credit points. |
Review particular exponent laws with your class as needed if they are struggling to remember them. | Ensure that assigned problems make use of different logarithmic identities to help students ingrain them into their minds if they are not already. Perhaps have the next class period be a review day where students can ask questions about general concepts or problems they are struggling with. |