Author: lonesc

Negative and Zero Exponents 8.EE.A.1

lonesc CCSS-Math Learning Progressions, HS Algebra

Image result for negative and zero exponents

Addressed CCSS for Math:

Expressions and Equations Work with radicals and integer exponents.

CCSS.MATH.CONTENT.8.EE.A.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.

Addressed Mathematical Practices:

CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.

CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.

Overview:

This learning progression takes place within a high school Algebra class of 25 students comprised of sophomores and freshman. The lesson is supplemented by material from the high school math textbook, “Algebra 2” by Holt, Rinehart, and Winston. The students have prerequisite knowledge on the basic mathematical propeties of exponents. The class itself has access to Chromebooks, a whiteboard, document camera, markers, scratch paper, and calculators.

The learning progression and lesson plan associated can be viewed at the link below:

Learning Progression

Lesson Plan

HSN.CN.A.1: Perform Arithmetic Operations with Complex Numbers

lonesc Assessment of CCSS-Math, CCSS-Math Learning Progressions, Common Core State Standards Mathematics, HS Algebra, HS Number and Quantity, Uncategorized

Image result for complex numbers

 

Addressed CCSS for Mathematics:

CCSS.MATH.CONTENT.HSN.CN.A.1
Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real.

CCSS.MATH.CONTENT.HSN.CN.A.2
Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

CCSS.MATH.CONTENT.HSN.CN.A.3
(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

Addressed Mathematical Practices:

CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.

CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.

Learning Progression Overview:

The lesson will open up with a brief recap to refresh the students on the prerequisite knowledge that is required. After this, the instructor is to present the concept of an imaginary number. The presentation follows the same format as described in the textbook, “Algebra 2”. To put imaginary numbers into context, the instructor will provide a quadratic equation, such that an imaginary number is produced, for the students to solve. Upon deriving the complex number, the instructor will introduce the properties of the imaginary number, i and complex numbers. The students will then transition into their first entry task of performing operations in complex numbers, beginning with addition and subtraction. Succeeding this, the students will learn proceed to their second entry task of multiplying complex numbers, employing their knowledge of the distributive property.  The third entry task will center on the utilization of the conjugate to simplify instances of a number being divided by a complex number. Throughout, the instructor is to prompt the class with short “checks” in the form of verbal questions to assess the progress of the class. Finally, the students will be administered an assessment to gauge their overall mastery of the material. The learning progression can viewed at the link below.

Complex Numbers Learning Progression

Calculating Cost of Wheels HSG.GMD.A.3

lonesc Assessment of CCSS-Math, HS Geometry, Uncategorized

roadroller

A construction company is designing new road rollers. The wheels of each road roller are made out of steel. The wheels are designed in a cylindrical shape in addition to having a smaller cylinder cut out of the center for the axel to fit through as shown in the picture below. The radius of the larger cylinder measures 2 ft., the radius of the cut-out cylinder measures 0.5 ft., and the length of the wheel measures 6 ft. From the diagram, this would mean R = 2 ft., r = 0.5 ft., and h= 6 ft. If the cost of steel is $2.40 per cubic foot, what is the cost of the steel to construct one wheel? Show your calculations and round your answer to two decimals.

cylinder

To view an assessment commentary and solution, follow the link below.

math-325-assessment-item-blog

Rush Hour HSF.IF.A.2

lonesc Picture Problems

Image result for Car driving on freeway

The most difficult aspect of being a teacher is finding a way to get students interested and active in the learning. One solution for this problem in regards to teaching mathematics is to present students with practical issues that they can relate to. With this in mind, we create a story and a problem involving real situations that an individual may find themselves in. For instance, students will one day find themselves in a situation similar to the picture above. By this, a problem can be posed:
Jesse is a college student who takes the freeway on his way to school. Normally, Jesse leaves at 7:00 A.M. to arrive 15 minutes before his 8 A.M. class.When he drives down the freeway to get to school, he averages a speed of 60 miles per hour. Suppose that Jesse wakes up late one day and ends up leaving for class 10 minutes later than he normally does. How fast must he drive to arrive at campus at his usual time?

For a problem like this, students will have to first decipher the details. This means that they will have to sort and organize the information to discern the useful details. Following this, students will have to set up equations using rates and distances. First, they will need to find the total distance that Jesse travels on the freeway. Using this information, the students will have to then find the speed he must travel to make up for his later departure.  In other words, we are looking at how a dependent variable (his driving speed) changes when an independent variable (his driving time) is altered.  Seeing as how a vast majority of students will be driving to school or work in the future, this particular problem is all too relative to situations that they will, or even already have found themselves in.

This example problem is aligned to the following Common Core State Standard for Math:

CCSS.MATH.CONTENT.HSF.IF.A.2
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

The Next Great Question? HSF.IF.B.4

lonesc HS Modeling, Technology Teaching Issues

From calculating the size of the sun, to discovering the force of gravity on Earth, mathematics is a curious world that allows us to explore the greatest questions ever posed by humans. Today, do we find ourselves looking to solve the next great question? Look no further.
For in this math lesson, we will utilize the power of modern technology to explore the wondrous mysteries of gravity and how its unceasing force can drain the kinetic energy of an elastic object via contact force to ultimately convert all of the kinetic energy to potential form and thereby, halting the motion of the object altogether. Or simply put, we will deal with combining modeling technology with a hands-on activity involving a basketball to model the behavior of a basketball as it bounces when dropped from a given height.

vernier-activity

For this activity, students will be separated into groups of 3 or 4. They will each be given one basketball and one Vernier software package (which includes the motion trackers and calculators) to work with. Using the graphing software, students will create a model that depicts the behavior of a basketball when dropped from a given height. Using their graphs, they will then analyze it mathematically using their knowledge of quadratic functions. This activity requires students to have demonstrated mastery of quadratic functions.

Image result for graph of bouncing a ball

Common Core State Standards:

CCSS.MATH.CONTENT.HSF.IF.B.4
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*
CCSS.MATH.CONTENT.HSF.IF.C.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*
CCSS.MATH.CONTENT.HSF.IF.C.7.A
Graph linear and quadratic functions and show intercepts, maxima, and minima.

This math activity requires physical and mental teamwork for adequate completion. Furthermore, this activity is designed to acknowledge students from all learning styles (visual, auditory, kinesthetic) in addition to creating an abundance of student discourse. The engaging nature of this activity makes it effective as it puts into physicality what students already have worked with and know about quadratic functions. To see the details of the activity, follow the link.

math-325-vernier-probe-modeling-activity-math-blog

The Population Pandemic HSF.LE.A1

lonesc HS Functions, HS Modeling, ML Functions, ML Modeling, Technology Teaching Issues

Image result for population graph

As the modern medicine and effective food methods become increasingly efficient, humans are able to populate more and more of the Earth. However, as we approach a net population of just over 7 billion and the Earth only holding a population threshold of over 12 billion, we must ask ourselves how long do we have before we overcome Earth’s population threshold?

With the help of a trusty TI-83 calculator teachers and students alike can find the answer to that question and more. Just follow the link lesson plan below to start an interactive and real-world activity that the students are sure to find relevant and engaging.

modeling-activity-lesson-plan