I.               Course Title:  Algebra

II.             Unit: Graphing Lines

III.           Anticipated Grade Level:  6th or 7th grade.

IV.           Lesson Objectives:  Students will be able to graph a variety of lines given a point and a slope.  Students will also demonstrate an understanding of slopes of vertical and horizontal lines by graphing lines, with such characteristics, in class and for homework. 

V.             Prologue: My lesson is designed to use the Direct Instruction model.  My introduction grabs the students’ attention by relating the new content to prior knowledge and the real world.  The introduction also includes the use of a power point for students to visualize connections and concepts.  This will create relationships and tie information together.

I also have an extremely well organized and detailed presentation that will allow me to communicate clearly with students.  The presentation also provides a variety of examples, which allows students to stretch their comprehension and create meaningful learning rather than useless memorizations.  My questioning throughout the presentation of information allows me to monitor the students understanding throughout so I know when I need to spend more time on a specific topic, or speed the lesson up to maintain interest.  I am also able to monitor understanding by employing a thumbs up thumbs down technique that allows me to quickly scan students for understanding.

My guided practice portion of the lesson is also well constructed, with examples in ascending difficulty, while maintaining connection.  By using the lines in the guided practice along with the lines in independent practice to create a single image I have tied to parts of the lesson together to make a smooth transition as well as relate the material back to the real world.

My homework assignment is also a fair assessment of whether or not the students comprehended the material or simply memorized steps.  It is fair because the problems are within the students’ ZPD but different from the examples in class, so the students must apply their knowledge to new situations.  

 

VI.           Curricular Material:

a.     I have created an original handout homework sheet.  The homework sheet contains instructions, and ample space to do what is asked.

b.     A PowerPoint will also be paired with the introduction of the lesson to give the students a visual aid to promote learning.

c.     I also have prepared a note card for each student with a point and slope.  The students will take their card to the board and attempt the graph the line for the class and I to see.

VII.         Content:

a.     Coordinate Axis/Cartesian Plane: consists of x and y-axis which are horizontal and vertical number lines respectively.

b.     Values in the Quadrants: positive positive, negative positive, negative negative, and positive negative.

c.     Ordered Pair: a pair of numbers such that  instructing where the x coordinate is and the y coordinate is.  

d.    

e.     Rise = change in y.  Positive means up, negative means down.

f.      Run = change in x.  Positive means to the right and negative means to the left.

g.     Undefined Slope: run equals zero, zero in denominator is undefined and means we have a vertical line.

h.     Zero Slope: rise equals zero, zero in numerator equals zero, and means we have a horizontal line.

i.      Graphing Lines: starting with a point, , use the slope to create a second point from the first, connect the two points with a straight line.  (Creating a third point using the slope creates a better approximation for the line).

VIII.       Instructional Procedures:

a.     Introduction:

                                               i.     Today we are going to start with a quick review of the coordinate axis, ordered pairs and their places on the coordinate axis.  We will also discuss the concept of slope and how to apply it to graphing lines. 

1.     Graphing lines, using a point and a slope, is relevant in many real world applications.  For instance, engineers building roads, buildings, athletic fields and many other present day structures, must take into account the slope of the land they are going to build on. (Slide Two)  They need to control and use the slope of the land to make water run away from their creations, and often into water basins.  Also, creating digital images seen on websites, animated television shows, and video games are created using a coordinate axis, often a three dimensional axis, and a ton of lines and points. (Slide Three)

                                             ii.     Lesson Goals: (Slide Four) At the end of class we will be able to graph a variety of lines given a single point and a slope, including lines with undefined slopes and zero slopes. 

1.     Throughout the lesson I am going to ask you to show me your thumbs. (Slide Five)  What I mean by this, is when I ask to see your thumbs give me a thumbs up right in close to your body if you understand, and a thumbs down if you need some clarification or need to hear an explanation again.

                                            iii.     Since we are talking about graphs today I am going to hand out some graphing paper so you can follow along at your seat. 

b.     Coordinate Axis:

                                               i.     The coordinate axis, also known as the Cartesian plane, is a graph composed of the x-axis and y-axis.  Remember the x-axis is a horizontal number line and the -axis is a vertical number line. 

1.     Draw a coordinate axis on the board and ask which is the y-axis and which is the x-axis.  

a.     Label the axis.  Label your axis just like this.

2.     Conventionally we label the right and top as positive.

3.     How are we doing? Let me see your thumbs.

a.     Continue if all thumbs are up.

c.     Values in the Quadrants:

                                               i.     We label each quarter or section of the coordinate axis with a number one through four.

1.     Draw and label a coordinate axis and quadrants

                                             ii.     Now, each quadrant has specific x and y values.

1.     Quadrant One: x is positive and y is positive.

2.     Quadrant Two:  How about Quadrant two?  What do you think the values are?

3.     Quadrant Three:  And Quadrant three?

4.     Quadrant Four:  And lastly, Quadrant four?

5.     Let me see your thumbs.

a.     Continue if all thumbs are up.

d.     Ordered Pairs:

                                               i.     An ordered pair is a representation of a value on the x-axis and a value on the y-axis denoted, .

1.     Example (1,2), What’s the x value?  The y value?

a.     This means that we go to the right two and up two, because in quadrant one

2.     Example (-2, -8). What’s the x value?  The y value?

a.     This means that we go to the left

3.     Let me see your thumbs.

a.     Continue if all thumbs are up.

e.     Slope

                                               i.     Slope is defined as rise over run.  Out of convention we use the letter m to denote slope.  So we can write.

1.    

                                             ii.     Rise

1.     What is rise?

a.     It is the change in y position.

b.     Think about riding in an elevator.

                                                                                                     i.     In an elevator – going up is positive rise

1.     Write: + rise = go up

                                                                                                   ii.     Going down in negative rise.

1.     Write: - rise = go down

                                            iii.     Run

1.     What is run?

a.     It is the change in x position.

                                                                                                     i.     Going to the right is positive run.

1.     Write: + run = go right

                                                                                                   ii.     Going to the left is negative run.

1.     Write: - run = go left

                                            iv.     Let me see your thumbs.

                                             v.     Running up a hill example (Slide Six):

1.     Demonstrate rise and run with a picture of a hill.

a.     Now we are going to put the idea of rise and run together with this example. 

b.     Say we want to run up a hill. 

c.     (Slide Six) or Sonic wants to.

d.     If the top of the hill is at 15 feet and the bottom is at 0 feet what is the rise?

                                                                                                     i.     15 feet up.

e.      If Sonic has to move 20 feet horizontally to get up the hill what is the run?

                                                                                                     i.     20 feet right.

f.      Draw the coordinate axis on the hill picture.

g.     Now Sonic wants to go down the hill because he missed some coins.  So he is starting at the top of the hill at 15 feet and wants to get to 0 feet.  What is the rise now?

                                                                                                     i.     Negative 15 feet or fifteen feet down.

h.     How far did we have to move in the x direction to get back to where we started?

                                                                                                     i.     Negative 20 feet, or 20 feet.

i.      So that means the run is what?

                                                                                                     i.     20 feet left.

2.     Let me see your thumbs.

                                            vi.     Undefined Slope

1.     Now we are going to introduce a new concept, the idea of an undefined slope.

2.     Undefined, in math means that we don’t know how to compute something. 

3.     When would we not be able to compute a slope?

a.     When there is a 0 in the denominator or run is 0, we cannot compute the slope because we don’t know how to divide by zero, so we call it an undefined slope.

                                                                                                     i.     Write : 0 run = undefined

b.     As a result there is zero change in the x position.

4.     When there is no change in the x position we get a Vertical line.

a.     Think about our elevator example.  That would be an undefined slope, because in an elevator you only move up and down, not side to side.  In an elevator you would move in a vertical line.

5.     Let me see your thumbs.

                                          vii.     Zero Slope

1.     Now that we have talked about the run equaling zero, we should talk about the rise equaling zero.

a.     If the rise was zero what would our slope look like?

                                                                                                     i.     Zero in numerator.

                                                                                                   ii.     Horizontal line.

b.     We can divide zero by another number.  For example, zero divided by 8 is?

                                                                                                     i.     Zero

c.     Write: 0 run = 0 slope = horizontal line

2.     Let me see your thumbs.

f.      Graphing Lines

                                               i.     To graph lines we need to have two points.  When we connect to points we can make a line.

1.     Draw two points on the board and make a line.

                                             ii.     What can we do if we are only given one line though?

1.     Let students think about the question and accept proposed solutions.

2.     What we can do is use a starting point, or the one point we are given and then use the concept of slope to find where to next point is.

a.     Do the following example on the board If we were told the first point is (0,1) or the origin, and that the slope was m = 2/-2.  We would know that for every two spaces we go up we must also go to the left two.

                                                                                                     i.     Trace out using step method how to go up 2 and left two… Make another point.

                                                                                                   ii.     Now to make a line we simply connect the two points… Connect the points.

b.     Why do we go up and to the left instead of down and to the right?

                                                                                                     i.     Accept student solutions

                                                                                                   ii.     Because a positive numerator means a positive rise which tells us to move up.  And a negative denominator means a negative run and tells us to move left.

c.     Let me see your thumbs.

3.     Here is another example:

a.     We are given the point (0,0). 

                                                                                                     i.     Where would that be on the graph?

1.     The origin

b.     We are also given the slope .

                                                                                                     i.     What does this slope tell us to do?

1.     Go up three and right three.

c.     If we start at our original point, (0,0) and then go up three and right three we can find our next point.

                                                                                                     i.     Trace out using the step method to show how to get the next point.

                                                                                                   ii.     Draw point

                                                                                                  iii.     Connect points with line.

d.     Let me see your thumbs.

4.     Two more examples before we get you guys up here drawing lines on the board.

a.     We are given the point (-5,-3), slope is undefined.

                                                                                                     i.     What does undefined slope mean?

1.     Run is zero.

2.     No change in the x position.

                                                                                                   ii.     So like in our elevator, we go vertically up and down at our point (-5,-3).

                                                                                                  iii.     Draw a vertical line

b.     Let me see your thumbs.

c.     We are given point (5, 6), slope is zero.

                                                                                                     i.     What does zero slope mean?

1.     Rise is zero

2.     No change in y position.

                                                                                                   ii.     So if our y position stays the same we will have a horizontal line through our point     (5, 6).

1.     Draw line

d.     Let me see your thumbs.

5.     Now you guys are going to have a turn to come up on the board one at a time and graph a line given a point and a slope.  On each index card I am handing out is the information for one line.

6.     Who would like to go first?

a.     Monitor students as they attempt to graph their lines on the board. 

b.     Use scaffolding and directed questions to help a struggling student.

7.     Now that we have all of these lines on one graph I want you to look at it and see if you can find any hidden images.

a.     After taking suggestions reveal the hidden M,

b.     Remember in the beginning of class we talked about how lines could be used to make graphics, this is just one simple example.

 

IX.           Closure:

a.     The closure of the lesson will be the revealing of the hidden M on the coordinate axis containing all of the lines.  The closure is effective because it is a culmination of all the material and work of the class period as well as a relation to the real world.

b.     After the M is revealed the homework sheet will be given out.  After the students have read over the assignment they will have a chance to ask questions about the assignment. 

 

X.             Assessment and Evaluation:

a.     The students will self assess throughout the entire lesson, by constantly having to think whether or not they should have their thumbs up or thumbs down. 

                                               i.     When I ask to see their thumbs the students have to think about whether or not they understand enough to give a thumbs up or if they need to call attention to the fact that they need more instruction

                                             ii.     As a teacher this is a good, quick way to see if students are understanding and comprehending the material throughout the lesson. 

b.     I will be able to assess the students throughout my lesson by listening to and interpreting their responses to my questions.

1.     If students give appropriate and on target responses I know that the material is being comprehended.

2.     If responses are limited and off target I know more instruction is necessary. 

c.     I will also be able to assess the students individually as they come to the board and attempt to graph their lines.  This assessment will not just be a total hit or miss like the thumbs up and down assessment, but I will be able to monitor varying levels of comprehension and ability.

d.     My final form of assessment will come in terms of homework.  The homework problems given are attached.  The homework asks the students to graph lines given a point and slope similar to the ones in class but not exact replicas.  The slight differences will be important for me to be able to measure understanding, because understanding is different than being able to copy examples from class.  If the students understand they will be able to apply the concepts to new problems. 

 

 

Green = Slide Show

Blue = Action

Red = Stop for question

Black = Information

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