Rational Exponents
Learning Module
- Creator(s):
This unit is meant to be a high school mathematics unit involving rational exponents. It mostly assumes basic exponent properties have been previously covered. There is a review section (updates 4-6) that students most likely really need and offer additional challenges to the very basic problems that students are comfortable with.
This unit most likely belongs in Algebra 2, accompanying a radical unit or a unit graphing exponentials. Two other main topics would likely include: graphing square/ cube root functions and solving equations with rational exponents.
Learning Objectives:
Common Core Math Standards:
CC.8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.
CC.8.EE.2. Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
CC.8.EE.3. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.
CC.8.EE.4. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
CC.N-RN.1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
CC.N-RN.2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Mathematical Practice Standards:
CC.MP.1 Make sense of problems and persevere in solving them.
CC.MP.2 Reason abstractly and quantitatively
CC.MP.3 Construct viable arguments and critique the reasoning of others.
CC.MP.4 Model with mathematics.
CC.MP.5 Use appropriate tools strategically
CC.MP.6 Attend to precision.
CC.MP.7 Look for and make use of structure
CC.MP.8 Look for and express regularity in repeated reasoning.
Students can use the following to assess themselves on each learning target throughout the module:
This update isn't meant to take a whole class period. It might be an assignment prior to starting a unit or as an introduction. Students struggle with numeracy and recognizing different ways they might write expressions. This is one way to get students to start thinking about ways they can re-write the same value and just think about numbers differently.
Learning Objectives:I can identify and apply the properties of integer exponents to generate equivalent numerical expressions.
Numbers and values can be identified and written in numerous equivalent ways.
For example, the value 8 can be written as:
8
= 2^3
= 16/2
= Square Root (64)
= 1*8
= 2^(5-2)
= 1/4 * 32
= 4320/540
Update1: Pick a numerical value . Write it in at least 10 equivalent expressions. At least 3 of them should contain exponents or radicals. Try to be as creative and unique as possible.
Comment on at least 2 other peers' work either providing alternative different ideas or giving constructive feedback about their expressions.
Prior to getting into the thick of things, this beheads the question students always ask, "When will I need to use this?".
By having students brainstorm and look up their own ideas, there might be many more ideas than what one individual is able to come up with.
Updates 1 & 2 might go well as a single class period together, serving as an introduction to exponents.
Learning Objective: I can verbalize how we use exponents or radicals in real life.
When will I ever use this in real life?
This is a valid point - Kudos for asking the question! Now, you get to answer! When will you use exponents or radicals in real life? Or maybe not you personally, but when is it important and applicable?
Consider about the reasons, examples, and information listed here and in the two videos below. Are these valid? Do they make sense?
Update: Give one reason exponents or radicals are important. Why are we studying them? Come up with an example applicable to you or your life. “Math class requires them” doesn’t count. You may use multimedia to help you demonstrate your real-life application.
Students will be working together to apply an exponent to a combinatoric problem. It gets students to think about how quickly exponentials increase.
Students can work in groups to get the task completed. You can assign groups however you feel appropriate for your class as each class is different. I usually have pre-selected groups based on seating arrangements that students are used to working in. You can provide scaffolding to groups that need it. Some may get this right away, so have those students work on the extension question.
As a class, have groups present their group's process and solution. Calling on students randomly through a random selector keeps kids prepared because they might get called on, but reduces their frustration directed at you for calling on them. (an electronic random # generator (for instance)or using a deck of cards passed out to the groups works well!)
The learning objective here is just that students can problem solve. There is a tie to exponents but I don't like to give it away to students right away. Some groups will realize the connections right away while others might take a little longer.
Learning Objective: I can problem solve and work with my group to complete the problem.
Complete the Tic Tac Toe Task.
Group accountability: Your group needs to make a poster with at least 1 visual representation of how you came up with your answer and any other details or explanation.
Individual accountability: Each of you should be prepared to explain your group’s process and solution. You will be called on at random to represent your group.
Don't forget your group norms!
Before getting into the details of rational exponents, you need to be sure students have mastered the basic exponent properties. This is assumed to be review, however the problems become increasingly challenging (typically more challenging than what they are accustomed to). See the additional resources below in case students need additional help / explanation.
Students will work together to refresh their memories regarding exponent properties. Let students try the most basic ones first. Give each students the first full page (on student page). It is recommended to let students work ahead if groups finish ahead of others. However, do confirm that the bottom half of the first page "general rule" is correct. You can do this per group or as a whole class.
Follow up with half sheets. It is recommended to give 1 per group. This forces students to work together. As students finish and have their work checked, they can continue or you can intervene as needed.
Remind students of your group work roles or norms.
Additional Work that may be assigned as independent or extra practice:
CC Math Standards:
CC.8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.
CC.MP.1 Make sense of problems and persevere in solving them.
CC.MP.3 Construct viable arguments and critique the reasoning of others.
CC.MP.6 Attend to precision.
CC.MP.7 Look for and make use of structure
Supplemental Resources:
True/False with exponents; good to challenge common mistakes
Exponential expressions with no variables
Exponential expressions with variables
Exponential example with variables #2
Exponent properties with division (quotient property)
Learning Objectives: I can know and apply the properties of integer exponents to generate equivalent numerical expressions.
Calculate the above. Did you come up with a simple or complex answer? Why?
How did this chaotic mess of numbers become so simple? Could I have predicted that in advance?
Work with your group on these basic problems to identify standard "rules" for exponents.
Then, be prepared to work through more challenging problems. Leave your answer in its most simplified form.
Be prepared to be called on. Any individual in your group may be called on at random to explain your group's process and solution.
CC Math Standards:
CC.8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions.
CC.MP.1 Make sense of problems and persevere in solving them.
CC.MP.3 Construct viable arguments and critique the reasoning of others.
CC.MP.6 Attend to precision.
CC.MP.7 Look for and make use of structure
This lesson assumes students have already learned the basic exponent properties. The two that students tend to struggle with significantly are 0 power and negative exponent properties. This update is intended to remind students what they know about these two and why. You may use this as an intervention or require all students to participate.
It is suggested that students refresh their own memories first and try to do their own research. Some students may need particular help or intervention afterwards.
For your consideration:
Also, give a formative assessment when you and students think they are ready.
CC.8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.
Learning Objectives: I can know and apply the properties of integer exponents to generate equivalent numerical expressions.
This update serves to focus on two particular exponent properties: the 0 power property and the negative exponent properties. Make sure you are comfortable with both properties. If you already are, complete the update & comments and then see the challenge below. If you are not, you may use the below resources to help you complete the update & comments. Be sure to complete the formative assessment as provided by your teacher.
Update: Choose one of the two:
A) Explain how you know or understand that exponent to the 0 power is 1. Use visuals, use written math explanations, make a video. Use at least one source of multimedia and you may present your explanation in any way you choose.
B) Explain how you know or understand that a negative exponent is a fraction. Use visuals, use written math examples, make a video. Use at least one source of multimedia and you may present your explanation in any way you choose.
Comment on 2 peers' explanations offering additional supporting ideas, asking questions or giving constructive feedback.
Formative Assessment: Complete it and return to your teacher. Don't forget to self-assess on the learning target!
Resources:
Challenging your thinking with a base 0: What is 0 to a 0 power? What is 0 to a negative power?
This update provides an opportunity for independent learning. Students can learn from the two videos or supplemental written text. This is drawing on previous knowledge of simplifying basic radicals (assuming that has been previously covered). Some students may pick it up quicker and can help others by answering their questions.
Attached is a solution set for the update. You may need to discuss simplifying a negative radicand with either an even or odd index.
Learning Objectives:
Use the following videos to help you process and understand how to simplify radicals with higher degree indices.
Watch up until 8:51:
Update: Choose one problem from #11-32. Post your solution with written work /justification. You may use visuals like tree diagrams to show your work. If you have questions you can post those also but try to complete the problem to the best of your ability.
Comment: Comment on at least 2 others' work either asking or answering questions to assist in the learning process.
Supplemental Written explanation with a problem set & solutions:
This investigation can be done in groups or as a class and may take 2 days (3 with independent practice) with all the simplifying examples. It could be lecture, but is set up so that students can use their resources to help them figure out what rational exponents mean. There may need to be intervention for different individuals or groups.
Supplemental work for independent or extra practice:
Additional Resources :
http://patrickjmt.com/exponents-multiplying-variables-with-rational-exponents-basic-ex-1/
http://patrickjmt.com/exponents-multiplying-variables-with-rational-exponents-basic-ex-2/
Intro to: https://www.khanacademy.org/math/algebra/rational-exponents-and-radicals/introduction-to-rational-exponents-and-radicals/v/basic-fractional-exponents
Simplifying: https://www.khanacademy.org/math/algebra/rational-exponents-and-radicals/introduction-to-rational-exponents-and-radicals/v/fractional-exponents-with-numerators-other-than-1
https://www.khanacademy.org/math/algebra/rational-exponents-and-radicals/introduction-to-rational-exponents-and-radicals/v/solving-for-a-fractional-exponent
Standards: CC.N-RN.1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
CC.N-RN.2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.
What is a Rational number?
Learning Objectives:
Complete the first page in your group. Stop if you have any questions. Be sure to write your one-sentence summary.
Complete page two in your group. Be sure to include your written answers and explanations as indicated.
When your work is checked over (each individual should be prepared to explain how your group came up with their solutions) you may move onto page three simplifying expressions.
Have the students complete formative assessments when you and they are ready.
I can convert between exponential form and radical forms.
Fun Challenge: Students can try to create their own secret message through a simplifying with exponents problem.
Complete the formative assessments to show your progress on the following two learning objectives:
Be sure to give yourself an accurate self-assessment for the learning target!
A Fun Challenge: Can you make your own secret (school appropriate) message with exponents and math-speak ?
The following posts some (not all) possible solutions: http://mathforum.org/ruth/four4s.puzzle.html
You may want to provide a few different examples to get students started. Some groups or individuals may want some additional support. Others might be fine starting. For instance, you may want to show an additional example for the # 1: 4^(4-4) (note Scholar's Math does not record this well), but so students can see multiple ways of doing the same thing.
You could even start with an example that students are not asked to complete:
#47
And to scaffold you might demonstrate that 48 would be an easier number, so you can do 48 = 4*12, then just -1 to get #47.
How can you get the # 12? 4*3 for instance and three can be gained by 4-1, so you can end up with:
\(47 = 4 *(4*3) - 1\)
\(47 = 4 * (4*(4-1)) - 1\)
Now, how do you get the ones? We've already showed those ways, so just use those ways and make sure parentheses are proper.
\(47 = 4* (4*(4-(4/4))) - (4/4)\)
You could even encourage students to check their work through Wolfram Alpha, to make sure their calculations/ Order of Operations are correct!
Analyzing: All 4’s Group Work.
In your group, collectively come up with as many equivalencies for numbers using only 4’s and any operation you are familiar with (including exponents). Update: Post your group’s solutions to as many numbers as you possibly can but you should come up with at least 1 - 25. Comment on any unique or creative examples from your peers!
Example:
1 = \(4/4\)
2 = \((4+4)/4\)
3 =
4 =
5 =
** Also note they can be done in several different ways so we expect to see a variety of answers. Rewards for the most creative approach and the most amount of numbers completed.
Your last task is to make up a challenging exponential problem and demonstrate how to simplify it; it must have all exponent properties involved including at least two components be rational exponents.
You can show what you know in any number of methods including:
You need to use appropriate and accurate mathematical vocabulary.
See the attached rubric.
You will complete this project and then rehearse/ show your product to a peer. The peer will use the same rubric to "grade" you and give you constructive feedback. You will then have an opportunity to make any necessary changes or adjustments before your final product is presented.