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Modelling Quadratic Equations



Grade 10 Academic Math MPM2D  Quadratic Relations – Modelling Quadratic Equations

Terence Attema, Joshua MacInnis, Ryan Laureault, Ezra Mulu

This is a lesson that is meant as a fun review activity for the end of a unit of study on quadratic relations. Students will throw or kick a football, and determine the equation of the path of the football.

quadratic model  – PowerPoint

ExploringQuadraticsUsingAFootball – Pedagogical tool: Football Activity

QUADRATIC graphicorganizer – Handout for the end of class

QUADRATICS – KNOWLEDGE RATINGS CHART

 

Place an X in the column that best describes what you know about each definition

DEFINITION I know this well enough to define it and apply it I have some idea what this is, but I need more practice I don’t understand this much at all
Domain
Range
y-intercept
x-intercept
roots/zeroes
parabola
vertex
axis of symmetry
expanding (i.e. FOIL)
factoring
quadratic equations
standard form
vertex form
factored form
completing the square
quadratic formula
transformations of functions
compression/expansion
phase shift
reflection

 

LESSON PLAN:

 

NAME OF COURSE: Grade 10 Academic Math MPM2D
UNIT NAME: Quadratic Relations

Teacher Candidates: Terence Attema, Joshua MacInnis, Ryan Laureault, Ezra Mulu

Overall Expectations: – determine the basic properties of quadratic relations- solve problems involving quadratic relations.
Specific Expectations:    – identify the key features of a graph of a parabola (i.e., the equation of the axis of symmetry, the coordinates of the vertex, the y-intercept, the zeros, and the maximum or minimum value), and use the appropriate terminology to describe them
-collect data that can be represented as a quadratic relation, from experiments using appropriate equipment and technology
– solve quadratic equations that have real roots, using a variety of methods (i.e., factoring, using the quadratic formula, graphing)
– solve problems arising from a realistic situation represented by a graph or an equation of a quadratic relation, with and without the use of technology
INTRO ACTIVITY We will hand out and explain the knowledge ratings chart to the students. The students will have several minutes to fill out the sheet. For each definition, pick a student who placed an ‘x’ in the first box for that topic, and ask them to define in their own words the definition.  Close attention will be made in correcting any misunderstandings that a student responds with. This activity should take no longer than 15 minutes, since it will be a review of everything learned in the unit so far.
LESSON -We will begin with a PowerPoint presentation that reviews some of the key concepts of the unit. A real life example of throwing a football will be described in the presentation. We will model the path of the football with a quadratic equation. We will demonstrate how to determine the value for ‘a’ when you know the coordinate of the maximum height along with the coordinates of the initial release of the football. (15 minutes)
– Next, we split the class into groups. We will take the class outside to the football field and have one person from each group kick 3 field goals or throw 3 footballs.  This will be videotaped using a stable base. Using a wall with a known height is another method that will work. Ideally an iphone 6 would be used as it has a slow motion feature built in.  They will need to keep track of where the ball is initially kicked or thrown from and where it lands.  They will then measure the distance between the two points. (20 minutes)
-Taking the students back to class, we will upload some of the videos to a smart board with a grid. On the smart board you can physically measure the distance between the starting point of the kick and the ending point of the kick with a yardstick.  Using the measured distance and the actual distance you can create a scale.  Using that scale, you can take the measurements of the tallest point of the kick to figure out the coordinate of the maximum of the parabola.  Utilizing the starting point, end point and maximum point you can solve the equation of the kick/throw.   Each group would then have to map out and calculate the parabola of each of their kicks/throws.  (20 minutes)
CLOSING We will provide the students with a graphic organizer that connects all of the concepts from the knowledge ratings chart into a handy chart. (5 minutes)
MATERIALS: Knowledge Ratings handout,  Graphic Organizer relating the definitions, footballs, something to videotape with, smartboard, yardstick
Assessment Tools:
The knowledge ratings is a diagnostic test, students will have an opportunity to self evaluate to gain a better understanding of what they still need to study for the unit test. The calculation of the parabola for each group will be formally assessed

PRAXIS:

 

This lesson will be most appropriate near the end of a unit on quadratics in the Grade 10 Academic mathematics class or a grade 11 college level course. We will start this lesson with a knowledge ratings chart. This will allow the students to self evaluate on what they need to study more on for the upcoming test. This will also allow the teacher to activate their prior knowledge, and determine what concepts they still need help with. The next part of the lesson will be a short PowerPoint presentation on the new lesson on modelling quadratic equations. We will be using the examples of throwing a football or kicking a football. This will help with the visual learners of the class, as we will show graphs, to show the students what the situation looks like. After this, we will determine the equation of the football path. Following this presentation we will begin with the pedagogical tool, which is an activity in which the class will go outside to the football field and be divided into groups of 3-4 students. Each student will be designated with different tasks. One student will kick or throw the football. Another student can be the holder (if kicking), another student will videotape, another student will measure the distance that the ball travels. This activity will serve to reinforce the theoretical concepts that are studied in this unit, by applying them to an authentic activity. If this is done well, kinesthetic learners who may have been intimidated by the in-class instruction might be able to understand important concepts that they may not have been able to otherwise grasp. We will also make effective use of technology, by uploading the video to a smart board with a grid. This will allow the students to scale the diagram, so that they can find the maximum height of the ball. From here, they can find the equation of the path, by using a method that was explained in the PowerPoint presentation. The use of this authentic problem-solving activity will help solidify the concepts learned in class. At the conclusion of the class, another graphic organizer will be given to the students. This graphic organizer describes how many of the definitions in the knowledge ratings chart relate to each other. This can be a helpful aid to use when studying for the test, and it is something that the visual learners will appreciate.