This post explores some ideas for reviewing concepts for TEKS A.9D.

graph exponential functions that model growth and decay and identify key features, including y- intercept and asymptote, in mathematical and real-world problems

### Staar Performance

On recent STAAR tests, here is how students across the state of Texas have performed.

**2023** #9 - 14% full credit, 59% partial credit, 27% no credit

**2023** #42 - 55% correct

### Active, Playful Learning

The activities shared in this post are designed to follow the six principles of __Active, Playful Learning__:

Active

Engaging

Meaningful

Social

Iterative

Joyful

These six principles, together with a clear learning goal, help students learn.

Students learn through active, engaged, meaningful, socially interactive, iterative and joyful experiences in the classroom and out. When we add a learning goal or engage in guided play we achieve Active Playful Learning.

In other words, math review doesn't have to be boring STAAR prep or mindless worksheets. Instead, students' learning is enhanced when playing with numeracy and algebraic concepts in a guided context. Who says math can't be fun? You can watch a video to learn more about Active, Playful learning __here__.

### Activities

Here's a walkthrough of all the activities on this blog post.

**Exponential Growth and Decay Scavenger Hunt**

*Learning objective: Students will find real-world examples of exponential growth and decay.*

Pair students together or put them in groups of three.

Give students a

__digital recording sheet__to document their findings.Students should search for articles or news stories that explain real-world examples of exponential growth and decay.

Simply typing in those terms into a search engine will most likely result in worksheets and practice math sites. Those should be excluded.

On the digital recording sheet, students will copy and paste the URL of the article/news story, explain the real-world context, and write out the exponential function.

*Variations*

If students struggle with finding real-world examples, provide sample topics (pictured above and on page 2 of the

__digital recording sheet__).In addition to writing out the equation, have students note the initial amount (

*a*) and the scale factor (*b*) in the equation.Have students create a table with the equation, plotting out how the function changes over various time periods (e.g., 1 year, 2 years, 3 years).

Ask students to graph the equation using a graphing calculator or Desmos.

**Interactive Exponential Graphs**

*Learning objective: Students will use an online graphing tool to manipulate the graphs of exponential functions.*

On their laptops/Chromebooks, have students open

__Desmos__.If needed, write the standard form of an exponential function on the board {f(x)=ab^x}.

Ask students to generate different exponential functions that meet various requirements. For example,

*Graph an exponential function with a y-intercept of 3*or*Graph an exponential function with a decay rate.*Either share your screen to show the correct answer or have students share their screens and have the class verify whether or not the displayed function meets the requirements.

*Variations*

As students become more comfortable manipulating the y-intercept and growth/decay rate, introduce

*q*as a constant that signifies the line of asymptote {f(x)=ab^x + q}, asking students to change its value to note the change on both the line of asymptote and the y-intercept.Create a memory match game for students to work on individually or in partners, with the matching cards being the written functions and exported images from Desmos.

Give students graphing paper and have them draw an approximation of a given function before using Desmos to check their work. If they get an element wrong, have them explain to a neighbor their mistake and how to get it correct next time.

Ask a student (or two) to take the role of the teacher, giving students clues for requirements (e.g., decay rate with a y-intercept of 3).

Instead of giving the class clues, display word problems of exponential functions and have them work with a partner to write the exponential function and graph it.

**Comparing Growth Rates**

*Learning objective: Students will compare growth rates using an online graphing tool and describe the effect of changing *b.

On their laptops/Chromebooks, have students open

__Desmos__.Have students graph multiple exponential functions with the same y-intercept (

*a*) and various growth factors (see__example__).Ask students to discuss what they notice about the steepness of the graph as the growth factor changes.

Have students write a few equations that use the same y-intercept but a different growth factor. Ask them to predict the change in the graph and then check their work using the graphing tool.

Have students write a summary statement that describes the effect of the growth rate on the graph of an exponential function.

*Variations*

As students gain understanding of the effect of various growth rates, ask them to explore what happens when the rate gets closer to 1 (e.g., What does the graph look like when the growth rate is 1.1? 1.01? 1.001? 1.0001? Hint: Students will need to zoom out on their graphs to see the effects of the latter growth rates).

Repeat the process but with decay rates, noting what happens as the rates get closer to zero.

Have students explore what happens to the graph when the growth rate stays constant but the y-intercept changes (e.g., becomes negative).

Once students are fluent in growth and decay rates, display graphs on the board and have students estimate the function (e.g., y-intercept, growth/decay rate) as a warm-up. Have students discuss in pairs their reasoning and reveal the answer at the end of class.

**Exponential Story Problems**

*Learning objective: Students will create story problems to fit the context of an exponential growth/decay scenario.*

Give students

__example real-world contexts__for exponential growth and decay.Have students generate values for the initial amount (

*a*), growth/decay factor (*b*), and the time unit (*x*).With the values and context now decided upon, have students write a story problem to fit the scenario.

The problem should give the initial amount/y-intercept (

*a*), the growth/decay factor (*b*), and the time unit (*x*). It should ask the person solving the problem to write an equation representing the function.Have students switch their problems with a partner and ask each partner to write an equation to match the new story problem.

*Variations*

Give students limitation for the values to use (e.g., make the rate a growth factor with a shallow curve, make the y-intercept greater than 10).

Have students solving their partner's problem not only write an equation but also sketch a graph to match. They should then check their graph with a graphing calculator.

Allow students to partner up and provide editing feedback for the story problems, ensuring clarity of the problem and proper conventions.

Instead of writing a matching equation from the story problem, have students give the equation and omit another component (e.g., initial amount, growth factor). Allow partners to identify the missing amount in the context from the given equation.

Take some of the best story problems and create a short extra credit assessment for the class.

**Mix and Mingle**

*Learning objective: Students will verbally describe exponential functions and their graphs.*

Have students stand and push in their chairs, ensuring clear pathways around the room.

Play music and ask them to walk around for 10 seconds or so.

When the music stops, ask students to group up with those nearby (no more than 4 in a group).

Display an exponential function on the board and have students discuss what they know about the graph.

For example, if the function is

*f(x)=45(0.3)^x*, students should be able to describe the y-intercept, direction of the slope (decreasing from left to right), it's relative shallowness, and the line of asymptote.Play the music again to regroup the students and have them discuss another exponential function.

*Variations*

After students discuss the function, display a graph and have them decide whether or not the graph matches. Ask them to provide justification.

Instead of showing a function, display a word problem that describes an exponential relationship. Have students identify the initial amount and growth/decay rate. After a consensus has been reached, display a function and ask them to confirm or deny its accuracy.

For writing practice, have students write a short statement describing why a problem situation or equation matches or doesn't match a graph or equation.

As your class improves in their ability to distinguish characteristics of exponential functions, decrease the group size of the activity (smallest group size being 2).

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