Students have been learning a lot about trigonometry. They started by looking at right triangles and reviewing what they already knew about the Pythagorean Theorem. Next, students learned about the six trigonometric ratios in relation to right triangles, and used this understanding to find missing sides in right triangles when given the measures of one angle and one side. After students developed their understanding of trigonometry within right triangles, we started to look at how to apply their knowledge to angles larger than 90 degrees. To do this, students looked at angles in the coordinate plane and used reference angles to determine their trigonometric functions. Students then learned how to write angle measures using radians instead of degrees, and worked on converting between the two units. Finally, students looked at inverse trigonometric functions and used what they had learned about the trigonometric functions of certain angles to answer questions involving their inverses. Students also applied this understanding of inverse trigonometric functions to right triangles and are able to determine the angles in a right triangle given the measures of two of the sides.
Algebra 2 students are starting to learn about conic sections. Conic sections are the figures created when a double-napped cone is cut by a plane. To understand how cutting a cone differently creates different figures, students cut cones and dipped them into paint. They then stamped the cones onto paper to see what shape was created in each situation. Students are learning to identify parabolas, hyperbolas, circles, and ellipses by what the figures look like and by how a cone is cut in order to create them. Next, they will begin to look at the graphs of conic sections and also use information from their equations to explain what their graphs will look like.
Students started this unit by learning about logarithms. They learned how to rewrite logarithms using exponents, as well as how to evaluate logarithms. Next, students looked at what happened when they added and subtracted logarithms. They looked at patterns at what happened in these problems and were able to establish logarithmic properties. Using their understanding of the relationship between exponents and logarithms, as well as logarithmic properties, students then worked on solving exponential and logarithmic equations. Finally, students are using what they have learned about exponential and logarithmic equations, as well as their understanding of solving linear equations, to write the equation of an exponential or logarithmic function that goes through two given points.
Students have learning a lot about functions. In this unit, students saw how they can perform the four basic operations on functions. Students then looked at the resulting function and were able to determine what the domain of that new function was by looking at the domains of the original functions. After performing the four basic operations on functions, students learned that there is an additional operation that can be performed on functions, called the composition. Student learned how to evaluate the composition of two functions for a certain value of x. Next, students started to learn about inverse functions. They started by graphing a linear function and graphing its inverse by interchanging the x and y in the original function. Looking at this graph, we folded the paper so that the two lines overlapped. Students looked at the crease they had folded, and saw that it was the line y = x. Finally, we discussed how to verify that two functions are inverses by finding the compositions of the two functions. Students graphed inverse functions and saw that they are reflected over the line y = x.
Students have been learning about polynomial functions. They first learned how to evaluate a function for a given value of x, both with direct and synthetic substitution. Using their understanding of synthetic substitution, students were able to divide polynomials using synthetic division. They later used this knowledge to determine if a given value was a zero of a function. Using everything they learned about polynomial functions, students were able to determine how many zeros a function would have, what the possible rational zeros were, and how many positive and negative real zeros it would have. Finally, students combined this knowledge with their use of synthetic division and the quadratic formula to find all the zeros of a function.
Students have been learning how to solve quadratic equations. Using what they knew about factoring, students were able to use the zero product property to solve quadratic equations that are factorable. Students were also able to graph the quadratic functions and discuss how the resulting parabola changes with different changes to the equation. Next, students were presented with quadratic equations that were not able to be factored. They learned how to use the method of completing the square, first using algebra tiles to model it, in order to solve these equations. Using this method, we were able to derive the quadratic formula. Students enjoyed watching videos of parodies of songs in order to remember the quadratic formula. Once students worked on using the quadratic formula to solve quadratic equations, they were ready to look at what happens when taking the square root of a negative number. This began their work with imaginary numbers. Students are now able to solve equations that result in complex numbers as answers, and can perform operations on complex numbers.
Students have been spending the last couple of weeks working with polynomials. We started by adding and subtracting polynomials, and have moved on to multiplying and factoring. Students first learned how to factor polynomials using greatest common factors and difference of two squares. They then used what they knew about multiplying binomials to determine how to factor trinomials. Once students were able to factor polynomials, they were able to solve polynomial equations in factored form by using the zero product property.
Here's a game that students have played to practice factoring polynomials: Students rolled 2 dice to determine which polynomial they would factor. If they factored correctly, they would put their colored piece in the square. The object was to get 4 squares in a row. The year has begun and we're off to a great start! We started the year by discussing rules, expectations, and the Memorable Mathematicians project (take a look at the page on the website for more information). We then started off our curriculum talking about exponents and square roots. Students went over what they learned last year about powers, including multiplying and dividing powers, evaluating the power of a power, and evaluating the power of a product. We also discussed what negative exponents are, and practiced using them. Students then discussed the connection between square roots and exponents. They took a look at what happens when you perform operations on square roots, and used what they noticed about multiplication to simplify square roots. Next week, students will be applying this knowledge of square roots when they work with the Pythagorean theorem, after it's introduced with a hands on activity.
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