CCSS.Math.Content.HSN.Q. Work it harder, Make it better, Do it faster, Makes us stronger.

Alignment to Content Standards

CCSS.MATH.CONTENT.HSN.Q.A.1
Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

CCSS.MATH.CONTENT.HSN.Q.A.2
Define appropriate quantities for the purpose of descriptive modeling.

CCSS.MATH.CONTENT.HSN.Q.A.3
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

daft punk

Tasks

Both of the guys from Daft Punk are supposed to go onstage in 20 minutes but first they need to clean their helmets. It would take one of them 50 minutes to clean both helmets by himself and it would take the other 35 minutes to clean them both by himself. Using your knowledge of algebra, ratios, and unit analysis, determine how long it will take them to clean the helmets if they work together. Will their concert be able to start on time?

See the following file for the assessment task, commentary, and solution for this problem:

IM Writing Task Template-1 (1)

N.RN – Rational Exponents

By Lisa Flores

N-RN Rational Exponents

Alignment: HSN-RN.A.

1. Use the Product of a Power Property to simplify these expressions

(you may further simplify using the Negative Exponents Property):

a. (32a) · (ab3c)

b. (2d5g4) · (3gr) · (dr3)

c. (4ad) · (b-2c) · (2a-4b6)

2. Use the Power of a Product Property to simplify these expressions

(you may further simplify using the Negative Exponents Property):

a. (a2b3d)5

b. (3f4g5k)2 · (m0)4

c. (3r2t-2w)2 · (2-1s3t5)-3

3. Use the Product of a Power Property and the Power of a Product Property to simplify these expressions with rational exponents:

(you may further simplify using the Negative Exponents Property):

a. (a1/2bc3) · (ab1/3c)

b. (2d3fg)1/2 · (3e2f2g4)

c. (4-1rs5)1/5 · (5s3t-2/3)1/3

Commentary

This task combines skills in using the Product of a Power Property, Power of a Product Property, and Negative Exponents Property separately, and then together, in a gradually challenging manner. Each skill is to be used separately first so that the exponent property to be used is straightforward. Then, the expressions gradually become more challenging in order to prepare for the more-complicated expressions in section 3. Each of the expressions will help to develops skills in simplifying the next, more challenging expression. Ultimately, with practice in using each of the other three properties of exponents in the task, the students will be prepared to reach the math standard HSN-RN.A. Extend the properties of exponents to rational exponents.

In section 3, each expression has rational exponents. The three exponent properties must be used in order to simplify the expressions which may be left as a negative exponent (like 4-1/5) or as a reciprocal of an exponent. The expressions in section 3 require the use of all properties while working with rational exponents which aligns with the content standard HSN-RN.A. Extend the properties of exponents to rational exponents.

The tasks that help to reach the standard HSN-RN.A. “Extend the properties of exponents to rational exponents” support student understanding of separate exponent properties. With practice in using all of the properties of exponents, students will be prepared to proceed on to applying the properties in mathematical problems, like solving problems with functions that have multiple bases and exponents, for example.

Solution

1. Problem: Use the Product of a Power Property to simplify these expressions

(you may further simplify using the Negative Exponents Property):

a. (32a) · (ab3c)

b. (2d5g4) · (3gr) · (dr3)

c. (4ad) · (b-2c) · (2a-4b6)

Answer:

a. (32a)(ab3c) = 9a2b3c

The exponents are added when the bases are the same

b. (2d5g4)(3g2)(dr3) = 6d6g6r3

Although bases 2 and 3 are not the same base, they are multiplied because they have the   same power

c. (4ad)(b-2c)(2a-4)(b6) = 8a-3b4cd or

2. Problem: Use the Power of a Product Property to simplify these expressions

(you may further simplify using the Negative Exponents Property):

a. (a2b3d)5

b. (3f4g5k)2 · (m0)4

c. (3r2t-2w)2 · (2-1s3t5)-3

Answer:

a. a10b15d5

The power outside the quantity is distributed to each base by multiplying the power of each base by the power of the quantity

b. 9f8g10k2m0 or 9f8g10k2

Any base to the power of 0 is equal to 1

c. (32r4t-4w2)(23s-9t-15) = 9 · 8r4s-9t-19w2  or

The answer may be left as a negative exponent or as a reciprocal of the exponent and its base, coefficients may be multiplied out or left as a product (9 times 8 or 72)

3. Problem: Use the Product of a Power Property and the Power of a Product Property to simplify these expressions with rational exponents:

(you may further simplify using the Negative Exponents Property):

a. (a1/2bc3) · (ab1/3c)

b. (2d3fg)1/2 · (3e2f2g4)

c. (4-1rs5)1/5 · (5s3t-2/3)1/3

Answer:

a. a3/2b4/3cor

Exponents are added using the Product of a Power Property

A whole number exponent and rational number exponent in the form of a fraction are added and left as a fraction or in radical form

b. (21/2d3/2f1/2g1/2)(3e2f2g4) = 3 · 21/2d3/2e2f5/2g9/2  or 3

Since bases 2, d, f, and g have their powers with the same denominator, they may be under the same radical

c. (4-1/5r1/5s1)(51/3s1t-2/9) = 4-1/551/3r1/5s2 t-2/9   or      or

Answers may be left with exponents in the form of a fraction with negative exponents, as exponents in the form of a fraction with reciprocals of bases and their powers, or as exponents in the form of a radical with reciprocals.

Each answer given above is simplified further in each version.

 

Here is the worksheet with just the problems (title and CCSS-math alignment included): Assessment Item-just problems

 

N.RN-Using the iPad for Peer Review with Rational Exponents

Attached is an article explaining how the MyScript Calculator ipad app can be used to check open-ended student responses. Student can show mathematical understanding by writing a math expression or equation of their choosing with their finger and see if the app interprets their notation and solution as intended. Teachers can use this app as a teaching tool by requiring students to suggest a math problem with a worked solution that they feel demonstrates their mastery of a learning target and have the app check their work. The MyScript Calculator can be used to check both the correctness of their notation and solution.

The teacher can use this app for whole class instruction can use this resource by putting the iPad under a document camera or by using screen share software. Teachers can also using this resource for small groups or individual students by giving an iPad to each group or individual. Internet access is not needed because the MyScript Calculator app does not use the internet when operating.

Article and Worksheet Rational Exponents with MyScript Calculator App

Rational Exponents Worksheet

Technological advances for the class by going online

The use of technology has become a growing requirement in mathematics curriculum. As technology became more used in people’s lives it was decided that technology should see more use in schools. One very useful technology that can be adapted and integrated into any mathematics class is Edmodo. Edmodo is a social media style program where teachers can set up online classes where students can join. In this online class students can post questions and receive answers to those questions by the teacher and fellow students alike. This allows students to receive more immediate feedback to questions while they are at home or away from school, instead of having to wait till the following day. Students are able to reflect on what they know by helping their fellow students, and by what they don’t know from the questions they can ask. The teacher can also have a more close interaction with each individual student who has questions, while also allowing the teacher to help expand the students learning.

Edmodo can be accessed at www.edmodo.com where the teacher can sign up for free. The teacher can create a group that their students can join by using the group code. By being connected like this it can be especially useful for CCSS N-RN.3 allowing students to explain to each other and to the teacher why the sums or products of rational or irrational numbers are rational or irrational. This technology allows the students communication to be organized and recorded so that students can review what they talked about and can verify their understanding.

Using MyScript Calculator ipad App to teach CCSS Rational Exponents

I had a lot of fun playing with the free ipad app MyScript Calculator. This app allows the user to write a mathematical statement or equation with their finger on the tablet; the app then translates the handwriting into math typescript; and finally the app evaluates the math statement. A math statement will be evaluated by producing an equation, with an equal sign and correct value to make the new equation correct. If you want to create an equation with a placeholder variable, create an equation with an empty parentheses for the variable. In both cases the users’ input is in black and the ipad response is in gray.

I have an example of how a teacher could use this app in the class to enhance teaching of the CCSS MATH. To teach the following CCSS Math:
CCSS.Math.Content.HSN-RN.A.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 5 to the 1/3 power to be the cube root of 5 because we want (5 to the 1/3 power) to the power of 3 = 5, so (5^1/3)^3 must equal 5.
The following math activity illustrates how technology can be used to provide feedback to students in an open-ended modeling activity. To meet this standard students must be able to explain how properties of integer exponents can be applied to make sense of rational exponents. Ideally students should be able to express math statement using correct math syntax and explain the evaluation of the math statement using properties of exponents.

Math Activity:
Each student would be given a worksheet that would be used to write down three examples that use at least one rational exponent to model the following integer exponent properties (a to power of m) to the power of n , a power of m multipled by a to the power of n, and a to the power of m divided by a to the the power of n. Student will work in pairs with an ipad to write and explain 2 original rational exponent examples for each property. The teacher would model the process by saying, “to model (am)n an exponent of an exponent, chose (43)1/3 that this would equal 4 because the exponents are multiplied resulting in 41, this also shows that the inverse operation of 4 to the 3rd power is the cube root of 4. Then students have your partner check your work and their understanding of your example by writing it with your finger on the ipad using MyScript Calculator. power outside a power If there is a discrepancy work it out, if you cannot raise your hand I will come to help. You are not allowed to try out your example on the ipad before you write it down, we learn from our what doesn’t work as much as what works.”
Reason for using the ipad:
I know the students will want to play with the ipads so I will let them each try writing math statement for 5 minutes before I give them the activity. For this activity the ipad acts as peer assessment for the both the person suggesting the model example and the partner writing the example in math syntax.

Number and Quantity Domain

Teaching Standards Washington State – Secondary Mathematics

Preparation for teaching the Common Core State Standards – Mathematics 5-12

Number and Quantity Domain

Overview

Mathematics teacher candidates must be able to use and describe multiple number systems and operations.  They must be able to use the number system to model real world situations.  They must be able to describe the problem solving process and justify their solution.  They will develop this understanding through successful completion of Linear Algebra, Abstract Algebra, Discrete Mathematics, and the Calculus series (Lab-based Sciences – Chemistry, Physics).

Numbers & Number Systems

Use, explain, and operate on integers, rational numbers, real numbers, and complex numbers.  Understand operations include addition, subtraction, multiplication, and division and that the communicative, distributive, and associative properties are consistent with their previous meanings.  Use, understand, and explain properties of exponents, including rational exponents and represent in radical form.  Understand the workings of matrix, vector, and complex number algebra.  Use technology, including calculators, spreadsheets, and computer algebra programs to manipulate these number systems.

 Quantities

Make sense of real world problems by reasoning quantitatively.  Use and explain measurements, unit conversions, and label solution with correct units.  Justify the problem solving process and solution in the context of the problem.

 Learning Targets (Indicators)

Teaching candidates will be able to:

  • Use, explain, and operate on integers, rational numbers, real numbers, and complex numbers
  • Model operations including addition, subtraction, multiplication, and division
  • Demonstrate connections between their previous knowledge of the communicative, distributive, and associative properties and how they are consistent in more complex number systems
  • Apply and explain properties of exponents, including rational exponents and represent in radical form
  • Use matrices, vectors, and complex number algebras to solve problems
  • Use and explain quantitative reasoning to solve problems including units of measure
  • Use structures from many branches of mathematics to represent and manipulate number systems
  • Use technology to explore and represent number systems

 

-Katelyn, Jenn, Casey