Unit Plan

Unit Plan Breakdown

Topic of the unit

Linear systems can be as simple as solving for one unknown, or variable given other relevant pieces of information to being as complicated as solving for thousands of variables using matrices and technology such as computers. This topic is an introduction to higher levels of mathematics, mainly higher levels of linear algebra. Linear algebra is useful in many areas of life, from a business perspective it is useful when trying to maximize available resources and minimize your cost, i.e. maximizing your profit, it is also especially useful in terms of the Internet, in particular the usefulness and relevance (order) in which a search engine, i.e. Google, which uses “google matrices”, returns search items. To be able to understand the mathematics behind these concepts you need to first understand how to solve and manipulate systems of linear equations and inequalities with one, two or three variables in a number of different ways.

There are three main methods looked at in this unit that students will be expected to learn: solving graphically, solving by substitution and solving by elimination. These three methods are useful to know how to use when doing almost any kind of algebra.

Breakdown of the Big Topic

This topic will be broken down into two major subtopics: linear equalities (equations) and linear inequalities, each subtopic will be broken down further into smaller topics that can be handled in a single class. The linear equalities subtopic will be looked at first, as the notion of equality is more common and easier to understand for novice students. It will be used to build off of for the second half of the unit, linear inequalities. The first subtopic is broken neatly into three main concepts that are introduced and looked at individually and then combined. There are three methods of solving linear equalities that will be looked at: solving by graphing, solving by substitution, and solving by elimination. These three methods will all be used later to solve systems of linear equations in three variables. Once these techniques are known and understood they will be used to look at linear inequalities in a very similar manner.

Pedagogy of the Unit

The lessons I give will be used mainly to guide the students through examples and questions that they will be encouraged to work out individually, in groups or as a class before I present the solution. The goal of doing this is to help students focus and engage more fully in the lesson and to give them more responsibility over their own learning.  To keep students engaged I will often teach from different areas in the room and will ask students to occasionally show some of their own solutions to the class, in particular when doing homework review at the beginning of the class. I will also use various activities and projects to supplement the textbook that I will be using and to give ample opportunity for various learning styles.

Assessment and Evaluation

Students will be assessed and evaluated mainly through a number of short quizzes and one larger unit test. Students will be assigned homework questions to do following each lesson. These questions will not be handed in for marks, but will be looked at and used as a gauge of student understanding and learning and for the degree of difficulty for the quizzes and test. Students will be given the opportunity to increase their scores on quizzes by doing corrections only after they have shown that they have done the majority of the assigned homework. Student learning will also be assessed through class discussion, one on one questioning and discussion and individual and group work. The pace at which materials are covered will be somewhat determined through the observations made during the above assessment tools. Lastly, there will be a project to go along with this unit that will allow students to work either individually or in groups of up to four. The project is designed to allow for student creativity and initiative.

Unit Plan:

Math 11

Solving Systems of Linear Equations and Inequalities

Textbook being used: Mathpower 11, Western Edition

Lesson no. & topic	Learning and teaching objectives	Connections made	Activities of students and teacher	Materials needed	Assessment and evaluation
1.  §1.1 Solving Systems of Linear Equations Graphically	L.O. To be able to solve linear eqn’s graphically, and to identify the number of solutions: 1, 0, or infinitely many. T.O. To get the students to work in small groups and explore systems of linear equations	Graphing linear equations of the form y = mx + b	Etch-a-Sketch activity; students work in small groups looking at how the behaviour of the slope of a line changes by turning knobs in different directions – help understanding of negative slope.	10-15 Etch-a-Sketch	Look at how the students are working together, ask questions to see if they are on task and to find what they have discovered about systems of linear equations.
2. §1.2 Problem Solving Using a Diagram	L.O. To be able to understand a word problem involving linear eqn’s and to put it into a mathematical expression and to use a diagram to help solve the question. T.O. To show a variety of problem solving approaches.	Setting up mathematical expressions given a diagram/word problem.	Individual/group work on given problems at desks and on the boards around the room; sharing solutions to the class. Board work, examples of different situations.	Word problems for students to work on and examples.	Get students to give explanations of answers to given problems and get students to work in groups solving problems together on the boards.
3. §1.3 Solving Systems of Linear Equations by Substitution	L.O. To be able to solve a system of linear eqn’s by the method of substitution. T.O. To show a variety of problem solving approaches.	Often we face some sort of system of linear eqn’s where we have some given information and two pieces of missing information, so we must first solve for on unknown in terms of the other then we can solve for the second.	Review questions from homework at the beginning of class. Teach the method of substitution using examples.	Examples of substitution.	Observe student work and allow time for questioning to see how much of the lesson students are understanding and use this as a pacing guide
4. §1.5 Solving Systems of Linear Equations by Elimination	L.O. To be able to solve a system of linear eqn’s by the method of elimination. T.O. To show the usefulness of the elimination method, its dis/advantages over substitution.	Connections to higher levels of mathematics, matrices, solving much larger systems with technology, internet, page rank, search engines.	Board work, examples of solving by elimination, time for students to practice a few problems on their own and to ask questions. 20 min. quiz at the end of class.	Quiz for §1.1 - §1.3	Short 20 min. quiz on §1.1 - §1.3
5. §1.6 Solving Systems of Linear Equations in Three Variables	L.O. To be able to solve a system of linear eqn’s in three variables using the methods learned previously. T.O. To engage the students by giving them problems to work out on their own or in small groups.	Uses knowledge from previous sections to solve in three variables and sheds further light on how larger systems could be solved, same as above.	Hand back quizzes. Review of previous sections in the form of a game/ competition. Give an example of solving a system of equations with three variables using each of the three previous techniques. Give more examples and questions for students to work on (Worksheet).	Cue cards with simple to harder questions, with answers, for the game (a speed mental math type game involving systems of linear equations). Worksheet	Use the review game to assess how much the students have retained so far and as a warm up/bridge for the lesson.
6. §2.1 Reviewing Linear Inequalities in One Variable	L.O. To understand the difference between inequalities and equalities T.O. To clearly explain what the different regions of the xy-plane represent when graphing inequalities	Uses prior knowledge of what an inequality is, upper and lower bounds.	Review questions from homework; get students to do these on the board. Lesson on linear inequalities.		Observe student work and allow time for questioning to see how much of the lesson students are understanding and use this as a guide for the pace of the lesson.
7. §2.3 Graphing Linear Inequalities in Two Variables	L.O. To be able to graph a system of linear inequalities by hand and using a graphing calculator and understand its meaning. T.O. To work on presentation skills, i.e. tone/pitch/expression of voice, clarity of notes/explanations etc.	Maximum and minimum type situations, upper and lower bounded regions.	Review questions from homework; get students to do these on the board. Quiz. Work on problems as a whole class.	Quiz for §1.5 - §2.1	Short 20 min. quiz on §1.5 - §2.1
8.  Project Research Class	L.O. To find useful connections to the topic of this unit T.O. To give in class time for students to work on their projects and ask questions about it.	The project is designed to help students make connections with this topic outside of the classroom in areas of personal interest.	Class period to be used for working on unit project in library.	Library resources, i.e. books, magazines, computers with internet access, etc.	Are the connections being made valid or useful connections?
9. §2.5 Solving Systems of Linear Inequalities	L.O. To be able to graph a system of linear inequalities by hand and using a graphing calculator in two or three variables. T.O. To present the lesson in an engaging manner	Can be related to optimization problems, i.e. optimize resources, cost etc. due to given restrictions. Has same uses as equalities but gives more freedom of parameters.	Hand back quizzes and review it and any questions from the homework. Quick review of last days lesson and give more examples of graphing linear inequalities and give a worksheet for students to work on.	Graphing calculator Overhead projector Worksheet.	Observe student work and allow time for questioning to see how much of the lesson students are understanding and use this as a pacing guide.
10. §2.6 Problem Solving Looking for a Pattern	L.O. To be able to recognize patterns to help solve linear systems T.O. To promote group problem solving techniques and to give more attention to each individual group and to lead the class discussion from different areas of the classroom.	Repeating patterns, generalizations of patterns observed, inductive and deductive thinking.	Students work in groups of 4 to solve problems using patterns and linear systems, both questions from the textbook and from a worksheet. On going informal class discussion while students run into problems and have questions.  Remind students to study for unit test.	Textbooks and worksheet	Watch how the students are working together while moving about the room, ask questions to see if they are on task and to find what they are learning and what they are struggling most with.

Lesson Plans:

Lesson 1:

§1.1 Solving

Systems of Linear Equations Graphically

	What?	How Long?	Materials
Bridge	Review/remind students about graphing equations of the form y=mx+b by allowing them to work in groups of 2 – 3 by using an Etch – a – Sketch and by giving specific instructions for them to try out	10 – 15 min. (includes time for students to play around individually on etch – a – sketches)	10 – 15 Etch – a – Sketch
Learning Objectives	To be able to solve linear eqn’s graphically, and to identify the number of solutions: 1, 0, or infinitely many.
Teaching Objectives	To get the students to work in small groups and explore systems of linear equations
Pre – Test	Walk around the classroom to look at student work and ask questions to see what they remember about graphing equations of the form y=mx+b or plotting data points on a grid
Participation	Students will work in groups and follow instructions on using an Etch – a – Sketch to review linear graphs. Work out a number of different examples on the board to illustrate how to solve a system of linear equations graphically given various situations. Time given to work on homework	10 – 15 min. 20 – 25 min. 20 min.	Textbook Homework question sheet Graphing calculator
Post – Test/Summary	What did we learn today? How can we interpret, or use, a graph in order to find a solution to a system of linear equations? How many solutions are possible?	5 min.

Lesson 5:

§1.6 Solving

Systems of Linear Equations in Three Variables

	What?	How Long?	Materials
Bridge/Pre – Test	Play a game to review concepts learned so far and to get students to start thinking of math. Use it also to introduce the next part of the topic.	15 min.	Cue cards with a range of questions, with answers, for the game (a speed mental math type game involving systems of linear equations).
Learning Objectives	To be able to solve a system of linear eqn’s in three variables using the methods learned previously. To engage the students by giving them problems to work out on their own or in small groups.
Teaching Objectives	To engage the students by giving them problems to work out on their own or in small groups.
Participation	Review game where students will “buzz” in when the question is finished being read and when they think that they have the correct answer. One example of each method learned so far in three variables, allowing time for questions in between. Students work on worksheet in groups	15 min. Small prizes for correct answers during the game, i.e. small candies. 20 min.  20 – 25 min.	Examples using the three methods of graphing, substitution and elimination to solve systems with three variables Worksheet on Linear equations in three variables.
Post – Test/Summary	If students enjoyed the game at the beginning of the class, play it again, but ask harder questions from today’s lesson as well as a few from previous lessons to keep students on their toes. Otherwise give a brief summary of important concepts they should know and remind them to continue doing their homework and that there is a quiz coming up.	10 min.	More questions from today’s lesson for the game.

Lesson 7:

§2.3 Graphing Linear Inequalities in Two Variables

	What?	How Long?	Materials
Bridge/Pre – Test	Review homework from last day by asking a few students to come to the board and work out 2 – 3 problem questions with the help of the rest of the class as well as the teacher	10 min.
Learning Objectives	To be able to graph a system of linear inequalities in two variables both by hand and using a graphing calculator and to understand what each region represents.
Teaching Objectives	To work on presentation skills, i.e. tone/pitch/expression of voice, clarity of notes/explanations, etc.
Participation	Students work on the board solving homework problems with the help of the class. Short quiz on previous three lessons (§1.5 – 2.1). Students ask and answer questions while teacher gives today’s lesson on graphing inequalities with two variables. Time given at the end of class for students to work on projects, homework, or corrections from homework.	10 min. 15 – 20 min. 20 – 25 min. 15 min.	Textbook Quiz on §1.5 – 2.1 Graphing calculators Overhead projector Examples of linear inequalities in two variables
Post – Test/Summary	Get students to stop what they are working on, ask what questions there are from the new materials they have just been working on, do a quick review of these trouble areas and go through one last clarifying problem on the board	5 – 10 min.

Project Proposal:

Description:

This project will require students to find at least two sets of comparable data to analyze.  Students will graph both sets of data and look at the trends in their data sets to come to conclusions about the topic they've picked.

What to hand in:

Each project will include:

• 2 (or more depending on what the students have chosen to compare) tables containing the data collected.
• One scatter plot with the points for all data sets.
• Graph of the best-fit lines for each data set on the same graph as the data points.
• Answers to conclusion questions.

Grading criteria:

Tables

• Tables are included and clearly labeled 

Graphs

• Include a reasonable scale
• Titles for x and y axis included
• Data points graphed correctly
• Each set of data is distinguishable, clearly labeled
• Every drawn line is labeled with its equation
• Best fit line graphed correctly
• Graphs are neat, and there is a clear use of a ruler (if by hand)

                           

    Questions

• Answers are written in complete full sentences
• Answers reflect a valid interpretation of the data
  
  

Conclusion questions:

1. Is the general trend of the data the same for both of your data sets? List 2-3 factors you feel contribute to the similarity or difference.
2. Does it seem like a linear equation is a good fit to your data?  Would a parabola work better?  What about some other type of smooth curve?
3. Using the best-fit line equations, predict the Y value for the next point after your data ends (Next year, for example).  Are there factors that aren't included in your data that might affect the future points? Explain.
4. Use graphing, substitution or elimination (show your work) to determine the exact intersection point between your lines.  Does the point occur on your graph?  What conclusions can you draw about the relationship between your two lines and their intersection point? 

Ghostly Presence of Famous Mathematicians

Mohammed Ibn-Musa al-Khwarizmi was born around 780 and died around 850. Not much is known about his life. Yet, during the period of Islamic Dynasty in which Al-Khwarizmi lived, an academy was established in Baghdad that is called the House of Wisdom. In that academy, many scholars preserved most of the Greek mathematics and science that eventually led to the stimulation and reinforcement of learning in Europe.

Alkhwarizmi was one of these scholars. He was known as the Father of Algebra and as the mathematician who brought the concept of zero to the Western world

He founded the concept of the algorithm in mathematics and the word "algorithm" is an English translation of his name. He also made major contributions to the fields of algebra, trigonometry, astronomy, geography and cartography.

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He developed algebra in linear and quadratic equations.  His most famous - Hisab al-jabr w'al-muqabala, which means "Science of the completion and the balancing”, is from which we got the name for algebra itself.  The Al-jabr wa'l-muqabala began with a discussion of solving equations of first and second degree

 The process takes place by using the two operations of al-jabr and al-muqabala.
Here "al-jabr" means "completion" and is the process of removing negative terms from an equation. For example, using one of al-Khwarizmi's own examples,
"al-jabr" transforms x² = 40 x - 4 x² into 5 x² = 40 x.
The term "al-muqabala" means "balancing" and is the process of reducing positive terms of the same power when they occur on both sides of an equation.
For example, "al-muqabala" reduces 50 + 3 x + x² = 29 + 10 x to 21 + x² = 7 x

As we see here, the term algebra is only one half of what this process should be called. It was supposed to be called algebra and almuqabala.

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Another intervention was introducing the Hindu-Arabic numerals ( 1, 2, 3, 4, 5, 6, 7, 8, 9) to medieval Europe. The Hindu-Arabic numerals and the place value of numbers were introduced in 500s AD.  After around 20 years, Alkhwarizmi, wrote about it in his book but also included an explanation of the use of zero in the same meaning we use these days and which was still confusing to people at that point of history.

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He also used a geometric proof to solve the equation x² + 10 x + 39= 0 by completing the square. He begins with a square of side x, which therefore represents an area x² (Fig 1). To the square he  adds 10x and this is done by adding to the four sides of the square four rectangles each of width 10/4 and length x (Fig 2). Fig 2 has area x² + 10 x but from the above equation that is equal to 39. To complete the outside big square he adds four little squares of area ⁵/₂ × ⁵/₂ = ²⁵/₄ each. As a result, the outside big square in Fig 3 has an area of 4 × ²⁵/₄ + 39 = 25 + 39 = 64. Which means that the side of the outside big square is equal to 8. But the same side is of length ⁵/₂ + x + ⁵/₂ so x + 5 = 8, and hence x = 3.

2) Strengths:

-       This project is well designed for those students who enjoy the imaginative process of artistic creation. (S)

-       It is also helpful for the students to get a historical perspective of mathematics. (S)

-       It gives the students the opportunity to learn who were the mathematicians that invented certain topics, hence giving them a human connection to topics learned in mathematics. (T, S)

-       The end result of the figure is much better than a poster of some research topic. (T, S)

Weakness:

-       The project may take too long, and it may cost the students some money, if they

are not able to use the materials provided for them in their art classes. (S)

-       The end result of building a figure may be nice, but it is not so interactive. (T, S)

-       The students would have to make the figure and in addition have to do extra research to find out more about the life of the mathematician. It may be too much work. (S)

-       With a life-size model, it would be hard to transport the figure and the class space may be limited. (S)

-       The project may not involve a lot of mathematical content for the student to learn. (T, S)

-       This also may be hard to mark from the teachers view. (T)

T – Teacher’s perspective

S – Student’s perspective

3) After looking at the strengths and weaknesses of the given project we have decided to modify it from building a model of a famous mathematician to conducting an interview with a famous mathematician.

Students would work in pairs to research a famous mathematician of their choice. They would research and then present the information found to the rest of the class in a short 5-10 min interview. While conducting the interview students would pose as an interviewer and the mathematician in question. The student posing as the mathematician would be strongly encouraged to dress up as the mathematician.

As well as the interview students would be expected to make a poster with a brief summary/description of the interview/mathematician researched, which would be posted in the classroom. The poster should contain a graphic of the famous mathematician. Both the poster and the interview should answer basic questions such as what era was the mathematician alive and what were some of their most famous/well known contributions to the world of mathematics?

This project can be done at any high school grade level, however the higher the grade level the more in depth and detail the interview/poster would need to be. The purpose of this project is to acknowledge the mathematicians that have formed the mathematics that we use today and to add a human touch to the math classroom by adding some decorations to the walls. Students will be marked on the information contained in both their interview and on their poster, as well as their creativity and their presentation (a rubric would need to be developed that would be appropriate for each grade level that this project would done at).

The Project’s Marking Rubric

Students work	Mark distribution	Marking
Interview	Organization, presentation and participation of both students	5 Marks
Poster	Layout and attractiveness	5 Marks
Creativity	Bringing new and exceptional  ideas	5 marks
Informative content	For both the poster and the interview. Information about the mathematician life (5 marks) Information about the mathematician work (5 marks)	10 marks
Final score		Out of 20 Marks

By Feda, Niyaz, and Mathew

 

Reflection on Creativity, flexibility, adaptivity, and strategy use in mathematics by Christoph Selter

I found this article to be simply reiterating and reinforcing the idea of relational versus instrumental learning and teaching. The discussion about creative, flexible and adaptive problem solving, rather than routine problem solving and using problem solving techniques that are suggested, i.e. "use the compensation approach to solve the following..." extends what we have previously discussed about relational and instrumental understanding. By telling students what technique to use to solve a particular problem is limiting to their mathematical creativity on one hand, but on the other it can force them to think in a different way than they may have otherwise and get them to see another way of doing a problem. I don't, however, see this as teaching flexibility, creativity and adaptivity. I see this as being helpful for some students, in particular those who struggle with recognizing what method to use, in which case I think that it is clear that these students do not have a solid conceptual understanding of the materials yet and thus need more help. I feel that if a student can recognize a method, or be able to modify a method, that will help them get to a correct solution then they should be encouraged, and given the freedom, to use it. This is, as I believe, the most important thing.

Something else that I found to be interesting that was mentioned in this article, although not very surprising, was that students with good prior knowledge had a harder time and were less successful in being adaptive than students that had less prior knowledge. This begs the question then as to how important previous knowledge really is when teaching for good relational and conceptual understanding? A method of teaching that I found myself adopting during my short practicum was an exploratory teaching/learning method where students are strongly encouraged to attempt questions with hardly any formal instruction but rather a little bit of guidance towards the direction in which they should be going.

Problems with Problems

Math Power 11 page 512, question 37

A satellite is in an orbit 1000 km above the surface of the Earth. A receiving dish is located so that the directions from the satellite to the dish and from the satellite to the centre of the Earth make an angle of 27º. If a signal from the satellite travels at 3 x 10^8 m/s, how long does it take to reach the dish, to the nearest thousandth of a second? Assume that the radius of the Earth is 6370 km.

In analyzing the above question it seems to be a reasonably good question. It has good and memorable imagery of a satellite orbiting the Earth, along with a nice diagram, not shown here. This imagery is not necessarily the most practical, but it does the job and is not too distracting for the most part, as the question is not about the satellite orbiting. That being said the wording of the problem is not as clear or concise as it could be. In the question we are given all the information required to solve the problem but it is given awkwardly. If it were not for the diagram it may become somewhat unclear as to what the set up should look like. The question gives the angle made by the dish, the satellite and the centre of the Earth in a confusing way, instead it could just read something like, "When the satellite makes an angle of 27º with the receiving dish and the centre of the Earth it sends a signal to the dish." This would be a better way to give the necessary information and the reader wouldn't become confused about "the directions" and also, on a more literal note, the question states that the satellite is orbiting, implying that it is moving and thus the angle would be changing with time, some students could be very distracted by this inaccuracy.

Two-Column Problem Solving

Short Practicum Stories

In the past two weeks I had the chance to observe, learn from, and get to know two teachers, the teachers that I will be working with later this school year. In this short time that we have just spent together I can see that we are going to become good friends and that I will learn a lot from each of them. Something that they both mentioned to me at separate times is that they are "in the business of edutainment", and that this is something that I should seriously think about.

As someone that is new to teaching I am quite concerned about being correct in the content that I am teaching and to be clear. As a result of this I find myself being rather monotonous from time to time as am giving a lesson, so to hear my sponsor teachers tell me that they are in the business of "edutainment" is probably something that is greatly needed. I consider myself to be a funny person and to have a really good sense of humour, but for some reason or another that part of me has a hard time finding its way out when I am giving a lesson. For this reason, among many others, I am greatly looking forward to working with and observing and learning from these teachers more and more.

I can already see that I have learned how to be more animated in the class from my sponsor teachers. For example this past Friday I came to school dressed up in a costume for Halloween, this already made it easier to act the part. As I became more and more comfortable with the students the more I was able to get into character, which by the way was Chef Gordon Ramsay, and the students really responded well to it, I was able to get more answers to questions and more questions themselves from the students and therefore was able to give a much nicer lesson where students were more actively involved in their own learning. Perhaps, Chef Ramsay will have to make a few more appearances throughout the school year.