This post explores some ideas for reviewing concepts for TEKS A.11B.
simplify numeric and algebraic expressions using the laws of exponents, including integral and rational exponents
Staar Performance
On recent STAAR tests, here is how students across the state of Texas have performed.
2023 #10 - 38% correct
2023 #35 - 4% full credit, 38% partial credit, 58% no credit
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.
What if?
Learning objective: Students will verbally simplify an algebraic expression to a partner using the laws of exponents.
Have students count off A - B - A - B in a large circle. Have the As take a step in and turn around, forming an inner circle. Each student should have a partner. If an odd number, the teacher can partner up with a student.
Show a basic expression on the interactive whiteboard, such as x^(3/7).
Ask the class a series of questions about what would happen if you manipulated the expression. Students talk it out with their partner. Some examples are: What if I multiplied it by x^(4/7)? What if I divided it by x^(4/7)? What if I raised it to the (14/3) power? What if I squared it?
After each question, debrief as a class and discuss the correct answer and the related laws of exponents.
Before asking the next What if?, have either the inside or outside circle rotate a certain number of places to the left or the right.
Repeat as needed.
Variations
In addition to verbally sharing the manipulation (e.g., What if you squared it?), also write the new expression on the interactive whiteboard (e.g., (x^(3/7)^2).
Allow students paper or personal whiteboards as needed.
In addition to sharing the correct answer, have students name the laws of exponents needed to simplify the expression.
Have an anchor chart with all the laws of exponents available for student reference.
As students progress, use expressions with different bases (e.g., (x^6)(y^3).
Simplify Around the World
Learning objective: Students will simplify expressions while explicitly naming the laws of exponents.
Pair students up and display an algebraic expression that needs to be simplified (see examples from released STAAR).
Working in three columns, have each partner list out the step, the relevant law(s) of exponents, and the partially simplified expression (per row).
Work the problem out for the class, step by step, having students check their work against yours.
Each person earns 3 points for having the correct simplified expression and 1 point for each step correctly labeled (action and law).
If time allows, pair students up with another partner to simplify another expression. If not, do so another day but students carry their points with them from partner to partner. Check to see who has the most points by the end of a week or month.
Be flexible in awarding points, as most steps can be done in a different order.
Variations
Instead of solving the problem for a class, have a pair of students come to document camera and share their work. They earn 2 extra points for sharing if one or more of their steps are incorrect and 4 extra points for sharing if every step is correct.
Have students in groups of three (rather than two) for additional support.
For students that need it, provide a scaffold that provides the actions and laws needed for each step.
Use this as a daily routine to encourage timeliness to class. Allow students the first three minutes of class to simplify the daily expression. Students who are more than three minutes late cannot participate for that day.
Allow students to create and submit their own expressions for the class. To do so, they earn the maximum number of points possible for that problem but must include a completed example with each step listed out.
Rolling Exponents
Learning objective: Students will generate and simplify expressions using the laws of exponents.
Pair students together and distribute 2 dice to each pair. Ideally they are blank, erasable dice (pictured below) with the operations written on them. If those are not available, standard dice will work as long as you display a key.
Display a simple base and exponent on the board (e.g., x^4).
Students roll both dice. The first die will dictate which operation to use and the second die describes the new exponent (same base).
Students work together to simplify the new expression.
Variations
Add a timer for each expression, giving only 45 seconds to simplify it.
Allow students to decide on the initial expression (x^4) and then roll the dice for the class to solve.
Add an extra die for negative exponents. If the die lands on an odd number, the new expression will have a negative exponent.
Let students roll die 1 and die 2 twice in order to create a larger expression. Be sure to clarify any use of parentheses as that will change the simplified expression.
Add more dice to represent additional bases [(x^4)(y^2)]. Be sure to clarify any use of parentheses as that will change the simplified expression.
Work backwards. Give students a starting expression (x^4) and a simplified expression (e.g., x^64) and ask them to figure out what was rolled (i.e., operation and exponent) were used to get from the starting expression to the simplified expression. For example, to get from x^4 to x^64, the teacher could have rolled a 5 for the first die (multiply the two bases and square the product) and a 4 for the second die (x^4).
Spot the Mistake
Learning objective: Students will analyze worked examples of simplifying expressions, identify the error, and simplify the expression correctly.
Share an example of a simplified expression with a mistake with students. Here are some examples from Simplify Around the World with mistakes highlighted.
Students work in partners or small groups to analyze the problem, step-by-step, to spot the mistake.
Students are looking for a mistake in the arithmetic, the wrong law of exponent being applied, or both.
When a group spots the mistake, have them come to the front of the class to explain. Give every other group 30 - 45 more seconds to try and spot the mistake before the first group explains.
After hearing the explanation, give groups a minute or two to simplify the expression correctly. Have a group volunteer to come up to the front and explain how to simplify the expression correctly.
Variations
Instead of putting a time limit on it, share the example with the mistake at the beginning of class. Allow students to work on it, alone or with a partner, throughout class. Go over it during the last three minutes before the bell rings.
Every once in a while, share a worked example with no mistakes.
Have students create their own examples, complete with highlights, and submit them for use in the future.
For a slightly scaffolded version, create examples where every law is listed correctly (middle column) and the only mistakes are in the application of the law.
DIY Simplification
Learning objective: Students will create and simplify expressions using the laws of exponents.
Have students work in pairs or groups of four.
Each partner/pair will create an expression for the other pair/partner to solve.
Include limitations on the expression to focus the creation. Some examples are: You must include the product law; You may only use 2 or 3 bases; You must NOT use a fraction; You must use a fractional exponent.
Before sharing their work, each partner/pair must solve their expression themselves.
If students struggle to simplify the expression, the partner/pair that created it can show them their worked example.
If any questions arise about the correct simplified version, have students use a radical equation calculator such as Symbolab.com.
Variations
As students succeed with this activity, give multiple limitations for them to work with.
Write 6 limitations on the board and number them 1 - 6. Give each group a die to roll before making their expressions. Whatever they roll will be the limitation.
Instead of randomly grouping students, break the class into those that struggle with this concept and those that excel at it. You can give each group different limitations to best support their learning.
Turn this into a whole class review activity. After each group has created, exchanged, and simplified their expressions, save them for future use. For a bell ringer, display one expression and have the class simplify it individually or in pairs.
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