Jessica

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 * Links which align with the Common Core Standards:**

This link has station activities which align with the Common Core State Standards. It has activities for Algebra I, Geometry, and Algebra 2. You have to pay for the whole set but you can view sample pages which have some really cool activities! This is a link to a youtube video which shows examples of classrooms which have already adopted the Common Core Standards. It shows a geometry classroom and an english classroom. The video contains examples of things you can do to align your instruction with the standards. It also shows us how we can incorporate modeling into our lessons to help deepen the students' learning instead of just giving them formulas. This is a link to the National Science Digital Library which breaks up learning content according to subject and grade level. There is a specific section on the Common Core which has lesson plans which align with the standards. This link has everything about Common Core! It has common core applications for your phone, videos about the common core, and just a really good background about why we are implementing the common core and what it is all about. This link has some teaching resources for teaching trigonometry. It has applets, videos, and lesson plans which will help students understand some of the concepts in trigonometry which are sometimes hard to visualize. This link is to a specific lesson plan which has geometry students exploring rotational symmetry. The lesson plan directly aligns with the Common Core Standards and has them listed on the power point. This is an interactive lesson plan which aligns with the Common Core Standards for geometry. It has great activities for the students to do along with exploration questions that you can ask the students. > This is a podcast which talks about the implementation of the common core. This looks at the idea of using coaching as a way of helping teachers with their implementation techniques. This is a link to a place where you can discuss your questions and concerns about the common core with fellow teachers. There are links in this to questions that other teachers have already brought up which may be some questions you may be having. This is a great tool to look through to see how the common core may address some of these things. This is a link to a video which gives us a little background as to why we are going to national standards. It talks about how the common core will help students be better prepared for higher education and the work force. This is a lesson plan database in which all lesson plans published are aligned with AASL's Standards for the 21st-Century Learner and are crosswalked with the Common Core Standards. > This link provides more background about the Common Core Standards. It tells why it is important for our country, how testing will be affected, how report cards will be affected, and even covers an overlook of what other countries standards look like.
 * [|Common Core Activities for the Math class]
 * [|Common Core in the classroom video]
 * [|National Science Digital Library]
 * [|Background on the Common Core]
 * [|Common Core and Trigonometry]
 * [|Common Core Lesson- Rotational Symmetry]
 * [|Interactive Geometry Lessons]
 * [|The Common Core- Implementation Podcast]
 * [|Teachers talking with Teachers]
 * [|Why National Standards?]
 * [|Standards for the 21st-Century Learner Lesson Plan Database]
 * [|Quick Overlook of the Common Core]

This activity was used in an Honors Algebra II class and was a great way to connect mathematics and science for the students. This worksheet was made from an Mathematics Teacher article. Many students know how to multiply matrices but they do not really understand the rationale of the process. They also do not know how it relates to the real world and other contexts. This lesson corresponds with lessons in science about food webs and helps the students relate to things they are doing in other classes. The standards for this lesson can be found in the High school:Numbers & Quantity category under the Vector & Matrix Quantities tab. The following are Common Core Standards and best Mathematical Practices which align with this lesson:
 * Activities to use in the Classroom:**
 * N-VM.8. (+) Add, subtract, and multiply matrices of appropriate dimensions.
 * N-VM.9. (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
 * Model with mathematics
 * Attend to Precision
 * Make sense of problems and persevere in solving them.

Objective 2: The student will use the properties of matrix multiplication in order to multiply matrices. || Two students have a business selling handmade scarves. The scarves come in four different styles: plain, with the class year, with the school name, and with the school mascot. The costs of making each style of scarf are $10, $15, $20, and $20, respectively. The prices of each style of scarf are $15, $20, $25, and $30, respectively. For students who are having a problem understanding the concepts: I will give these students a problem where they are finding the product of two matrices. I will have the students write each matrix on a separate sheet of paper, cut or tear out each of the elements, and rearrange them in a new matrix to represent the end result of matrix mmultiplication. This method will help students be more involved in understanding how the elements in the product are obtained from the elements in the original matrices. || Answer any questions that the students may have from the previous days exit slip. Quick powerpoint to review with the students how to multiply matrices. Teacher Question: Multiplication is commutative but is matrix multiplication commutative? Students will take notes while I do a quick powerpoint review of multiplying matrices. Expected response to teacher question: Matrix multiplication is not commutative, because the dimensions of the two matrices must match up. You can multiply a 3 x 4 matrix by a 4 x 2 matrix because the inner dimensions match. If you were to change the order of those, and try to multiply the 4 x 2 matrix by a 3 x 4 matrix, you would find that it is impossible. || Explain the multiplying matrices activity by going over the first page of the handout with the class. I will then split the class up into predetermined groups. I will pair together people who seem to work well together. I will strive to pair students who are having a hard time with students who seem to be comfortable with the material as well as pairing different types of thinkers together. Teacher Questions during the activity: Closing Discussion on the Activity As you have seen, eliminating one animal or plant from an ecosystem, especially one low on the food chain has a huge impact on a biome. We can use these concepts learned today in many other areas around the world. Teacher Question: Does anyone know in which other contexts we may use matrices? Expected Student Response: In science class Expected Student Response: To study biology
 * = **Subject(s)** || Honors Algebra II, Mathematics ||
 * = **Estimated Time** || 90 minutes- February 24th, 2012 ||
 * = **Grade/Level** || Grade 10, Grade 11, Grade 12 ||
 * = **Lesson Prerequisites & Pre-assessment of Students** || Students must be able to multiply real numbers and have a beginning background in matrices. ||
 * = **Lesson Objective(s)** || Objective 1: The student will be able to multiply matrices
 * = **Summary** || * Answer any questions the students have from the previous day's exit slip.
 * Brief Power Point to review multiplying matrices
 * Multiplying Matrices Activity in groups
 * Concluding Discussion ||
 * = **Materials** || Calculators
 * Attachments:**# [|**Multiplying Matrices Activity**]
 * 1)  [|**Multiplying Matrices Answer Key**]  ||
 * = **Adaptations: Gearing Up and Gearing Down** || Challenge questions:
 * Write a 4 x 1 matrix C that gives the cost of making each style of scarf and a 4 x 1 matrix P that gives the price of teach style of scarf.
 * The sales for the first three years of the business are shown below. Write a 3 x 4 matrix S that gives the sales for the first three years
 * Year 1: 0 plain, 20 class year, 100 school name, 0 school mascot
 * Year 2: 10 plain, 100 class year, 50 school name, 30 school mascot
 * Year 3: 20 plain, 300 class year, 100 school name, 50 school mascot
 * Find SC and SP. What do these matrices represent?
 * Find SP - SC. What does this matrix represent?
 * ==BEFORE == ||
 * = **Teacher Actions** || (15-20 minutes)
 * Attachments:**# [|**Multiplying Matrices PowerPoint**]  ||
 * = **Student Actions** || Students will ask any questions they may have from the previous day's exit slip.
 * ==DURING == ||
 * = **Teacher Actions** || (70 minutes)
 * How is a "1" in a cell of the F2 matrix represented in our digraph?
 * Expected Response: This shows up as the only two-arrow route ( an arrow from mouse to rat and rat to cat)
 * Expected Response: The only time two arrows can be followed consecutively in the digraph.
 *  Notice that rows 5, 6, 7, 8, 9, 10 each contain a single 1. What does that indicate?
 * ​Expected Response: The single 1 in rows 5, 6, 7, 8, 9, 10 indicates that, according to this food web, the six types of animals listed- garter snake, praying mantis, hognose snake, rabbit, grasshopper, and spider- have only one source of food in this biome.
 * ​Column 1 contains all 0s. What does a column of all 0s indicate?
 * Expected Student response: A column of 0s indicates that the animal has no predator within this group. Here it tells us that none of the listed animals eats hawks.
 * ​Column 10 has the most 1s. What does this suggest about the food web?
 * ​Expected Student response: Of the animals listed here, spiders are food for more animals than any of the other species
 * ​What does the arrow represent between the Toad and the Hognose snake?
 * How would you represent that arrow in the matrix? ||
 * = **Student Actions** || Students will work together in their groups in order to observe the effects on an ecosystem of a wildfire that completely destroyed the grasses. They will work through the activity packet and interact with each other in small discussions on the questions asked in the packet. They will also participate in the above expected student responses. ||
 * ==AFTER == ||
 * = **Teacher Actions** || (5 minutes)

Airlines use matrices to list direct routes between cities. Teacher question: What would the squared matrices represent in this scenario? Expected Student Response: how the direct routes impact each other Expected Student Response: The paths between cities with one intermediate stop

Sports leagues also use matrices. The original matrix can provide us with the results of a competition between two teams and the squared matrix would represent how many teams that the row team defeated and that, in turn, defeated the column team.

I will then handout the worksheet for the students to do for homework. ||
 * = **Student Actions** || Students will participate in the discussion and contribute to the expected Student Responses. ||
 * = **Assessment/Rubrics** || I will collect the handout which goes along with the activity and look through the questions to see how well the students understand the concepts. They will not receive an accuracy grade for the assignment but they will receive a participation grade for the day. I will look at the number of computational errors to assess their level of understanding of multiplying matrices. I will also look at their problem solving strategies along with their analysis of the activity in order to see how they were thinking. ||

This activity was used in an AFM class as a discovery activity to get the students to understand inverse functions while making connections to previous knowledge of inverses. Many students have seen inverses in previous mathematical concepts like solving for variables and even in their everyday life. This activity has them utilizing their previous knowledge in order to discover new things about the graphs of inverses and patterns between inverses. The standards for this lesson can be found in the High school:Functions category under the building functions tab. The following are Common Core Standards and best Mathematical Practices which align with this lesson:
 * F-BF.1. Write a function that describes a relationship between two quantities.
 * F-BF.4. Find inverse functions.
 * Make sense of problems and persevere in solving them.
 * Use appropriate tools strategically

For students who finish early, I will have them show that f(x) = (3x-2)/(5x-3) is its own inverse. || Teacher Questions: I will show the first part of the video so that students get an idea of what inverses are and to make the connections to the inverses that they already know. Expected Student Responses: Teacher Questions: (5-10 min) We will go over the activity and share our answers as a group (40 min) We will go over the power point and I will ask the students some probing questions in order to get them thinking about the concepts. Teacher Questions: Expected Student Response: Teacher Questions:
 * = **Subject(s)** || AFM, Mathematics ||
 * = **Estimated Time** || 90 minutes, will be taught on March 14th, 2012 ||
 * = **Grade/Level** || Grade 10, Grade 11, Grade 12 ||
 * = **Lesson Prerequisites & Pre-assessment of Students** || Students must have an understanding of functions along with an understanding of combinations of functions. I will give exit slips to assess each of these in previous lessons. ||
 * = **Lesson Objective(s)** || * The student will be able to verify inverse functions.
 * The student will be able to find the inverse of a function.
 * The student will use the horizontal line test to determine if a function has an inverse function. ||
 * = **Summary** || * Go over previous day's exit slip
 * Inverse Discovery Activity
 * Power Point on Inverses ||
 * = **Materials** || * Calculators
 * Worksheets
 * Attachments:**# [|**2.7 Inverse Power Point**]
 * 1)  [|**Inverse Discovery Activity**]
 * Links:**# [|**Intriguing Inverses**]  ||
 * = **Adaptations: Gearing Up and Gearing Down** || I will point out to students who are having a hard time with the notation f -1 that the -1 is //not// an exponent. It does not mean 1/f it represents the inverse function of f.
 * ==BEFORE == ||
 * = **Teacher Actions** || (15 min) I will go over the exit slip from the previous day by answering any questions that the students may have. If the students do not seem to be having any questions, I will address any of the problems from the exit slip that the majority of the class seemed to have problems with.
 * Does anyone know what an inverse is?
 * Now that we know what an inverse is, What is the inverse of x2?
 * Can anyone think of any other real world examples of inverses? ||
 * = **Student Actions** || Students will ask any questions they may have about the previous day's exit slip.
 * yes or no
 * division
 * taking the square root
 * directions when driving ||
 * ==DURING == ||
 * = **Teacher Actions** || (30 min) I will introduce the inverse functions discovery assignment that the students will be spending the next 20 minutes working on. I hope that the students learn what an inverse is along with how to create algebraic inverse functions on their own. I want them to discover that inverse functions are symmetric about the y=x line by graphing the two inverse functions they created. This will be a nice introduction into the power point on inverses. I want to make sure the students have a foundation of what an inverse is before going through problems using inverses. I will walk around and ask questions as the students are working.
 * Why can we use x to represent our uknown number and the final result on the first page?
 * What do we need to know in order to graph g(x)?
 * What patterns do you notice in the outputs?
 * What will the inverse point of an ordered pair look like?
 * Does anyone know how we can find the inverse of a function algebraically? ||
 * = **Student Actions** || The students will be expected to work in partners to fill out the inverse functions worksheet. The partners will be preselected by me. I will already have the desks and groups placed where I want them to ensure that the students stay in their groups and on task.
 * You can use the same x because g(x) and f(x) are inverses and are different functions
 * We must have ordered pairs.
 * We need to plug in x-values into our g(x) equation in order to find y-values.
 * the x and g(f(x)) columns are the same.
 * the f(x) and f(g(f(x)) columns are the same.
 * g(f(x)) must be undoing what f(x) did.
 * The inverse of an ordered pair has its y and x values switched.
 * We can change the x and y and then solve for y. ||
 * ==AFTER == ||
 * = **Teacher Actions** || I will ask for any closing comments from the students and take a quick poll to see if they liked the activity.
 * Ask students to describe a situation or process and its inverse and how it relates to inverse functions. ||
 * = **Student Actions** || The students will participate in a discussion on inverses and will think about some other real world applications. ||


 * = **Assessment/Rubrics** || As we go through the power point, the students will have problems to work on individually as we progress through the slides. While they are working, I will walk around the room to make sure no one is having any problems with the material. ||