Thinking Like a Mathematician
Constantina “Dina” Burow walks her students through logarithmic functions by breaking down a word problem from a mathematics textbook and then showing them how their previous knowledge of exponential functions helps them graph and solve logarithmic functions.
Teacher: Constantina “Dina” Burow
School: Health Sciences High and Middle College, San Diego, CA
Discipline: Mathematics (Algebra 2)
Lesson Topic: Logarithms and logarithmic functions
Lesson Month: February
Number of Students: 34
Other: Health Sciences High and Middle Colleges is a health-focused charter school.
Featured Lesson’s Student Goals:
- Content objectives – Use knowledge of exponential/graphing functions and apply them to logarithmic functions
- Literacy/language objectives – Use appropriate mathematics vocabulary while explaining one’s thinking; reading a complex mathematical text
- Engagement/interaction objectives – Work collaboratively using questioning and listening techniques while explaining the problem-solving process
Common Core State Standards for Mathematics
Attend to precision.
Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
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.
The focus of this unit was polynomial functions; it was taught at the beginning of the second semester. The lesson on logarithms and logarithmic functions occurred at the beginning of the unit.
Before the Video
Leading up to this lesson, students had graphed exponential functions and started solving exponential equations and inequalities.
During the Video
Ms. Burow began the class with a read-aloud on asteroids that explained the Palermo scale. Because she was teaching logarithms, which are abstract, Ms. Burow wanted to connect what her students were learning to something in the real world. Ms. Burow based her instruction on two questions: What do we remember from graphing f(x)=2^x? and What do we remember about inverse relationships? She showed linear and absolute transformation of graphs and went back and forth between modeling and guided practice.
For the guided practice, Ms. Burow had her students graph y = 2^x and x=2^y using t-tables to show graphs and the inverse relationship. Then, they folded the graph paper to show the y=x line, and discussed that inverse functions all reflect this line. The class looked at f(x)=log base b (x) and the points (1/b, -1), (1,0), (b, 1) for both situations of b>1 and 0<b<1. At this point, Ms. Burow went back to modeling transformations of familiar graphs so students could make connections to log graph transformation. Ms. Burow then discussed f(x)= log base b (x-h)+k and showed students what transformation happens with a log function. Next, students did an independent practice on communication, problem-solving skills, and time management. Independent practice in Ms. Burow’s class is working in at least pairs. (The only time students truly work alone is when they are not on campus or during an individual test.)
After the Video
After the lesson, Ms. Burow focused on graphing logarithmic functions.
Ms. Burow searched the NASA website for an article on asteroids and the likelihood of one striking Earth.
Students needed to have a basic understanding of exponents and be able to work together.
Ms. Burow used gradual release of responsibility, specifically the modeling component. Ms. Burow did the problem and then had her students do a problem with a partner. Modeling helped students understand what they were going to do and why they were going to do it as well as allowed them to see how Ms. Burow was thinking through the problem. It also helped Ms. Burow identify places of potential struggle.
Students worked in groups to explain their problem solving and how they thought through a specific part of the problem. Students also had a group competency, for which part of the test was taken together in groups of four.
Resources and Tools
- The Palermo Technical Impact Hazard Scale by NASA (National Aeronautics and Space Administration)
- NASA Information Sheet
- Glencoe Algebra 2 (Glencoe/McGraw-Hill) textbook
Ms. Burow walked around the room and asked questions of students to check for understanding while they worked together.
After the group competency, students did an online self-reflection about individual performance. They wrote two to three paragraphs and described two things they thought they did well, two things to work on, and a self-reflection about their group interaction. This self-reflection was worth five points on the final unit test.
On an exit slip, students wrote any questions they still had about the lesson and one thing they had learned about functions