Struggling with Perpendicular Lines
Concepts
Background knowledge
Positive/Negative numbers
Slope
"S1: I learned that a positive slope is y=mx + b and a negative slope is y= 1/m x + b."
Addition/Subtracton
Graphing
"Location and Movement
By the end of Grade 8, students will:
– graph the image of a point, or set of points, on the Cartesian coordinate plane after applying a transformation to the original point(s) (i.e., translation; reflec- tion in the x-axis, the y-axis, or the angle bisector of the axes that passes through the first and third quadrants; rotation of 90°, 180°, or 270° about the origin);
– identify, through investigation, real-world movements that are translations, reflec- tions, and rotations." http://www.edu.gov.on.ca/eng/curriculum/elementary/math18curr.pdf
Knowledge
Prior knowledge
Facts/information
Students were asked to think about their prior knowledge about slopes, etc prior to solving the problem.
"T: First, it is important that we consider what we already know. Can we come up with some prior knowledge that we all might have regarding perpendicular lines and slopes?"
"S2: I think that two lines intersect to make two right angles. Is that right?"
planningskills (e.g.,understandingtheproblem,makingaplanforsolvingtheproblem) processingskills (e.g.,carryingoutaplan,lookingbackatthesolution)
critical/creativethinkingprocesses (e.g.,inquiry,problemsolving) http://www.edu.gov.on.ca/eng/curriculum/secondary/math910curr.pdf
Proper use of mathematical tools
"I started by putting in the known coordinates for “m” and multiply by the “x” number."
Representations
Modeling/Showing understanding
Use of mathematical models
Graph
"mathematical modelling. The process of describing a situation in a mathematical form. See also mathematical model."http://www.edu.gov.on.ca/eng/curriculum/secondary/math910curr.pdf
"coordinate plane. A plane that contains an x-axis (horizontal) and a y-axis (vertical), which are used to describe the location of a point. Also called Cartesian coordinate grid or Cartesian plane."http://www.edu.gov.on.ca/eng/curriculum/elementary/math18curr.pdf
"Figure 1. Example of positive and negative slops on a graph. Plotting the points on the x and y axis to display the slopes intersecting. January 28, 2020
Retrieved from:
(https://www.google.com/search?q=intersecting+slopes+in+math&source=lnms&tbm=isch&sa=X&ved=2ahUKEwij8OmwsKfnAhWrhOAKHTZ-BYMQ_AUoAXoECA0QAw&biw=1280&bih=698#imgrc=vF0PmoszlJStYM)"
"Figure 2. Example of graph to plot the points on the x and y axis. Retrieved January 28, 2020 from: (https://www.algebra.com/algebra/homework/Linear-equations/Linear- equations.faq.question.102253.html)"
"Figure 3. Example of positive slope on a graph plotted on the x and y axis. Retrieved January 28, 2020 from: https://www.google.com/search?q=slope+in+math&source=lnms&tbm=isch&sa=X&ved=2ahU KEwjugYLGu6znAhXHPM0KHVX6AJoQ_AUoAXoECA4QAw&biw=1280&bih=698#imgrc =IIZqz04OEHgVPM"
"Figure 4. Example of the final graph of the equation y= -1⁄3 x + 4.33."
"slope. A measure of the steepness of a line, calculated as the ratio of the rise (vertical change between two points) to the run (hori- zontal change between the same two points)."http://www.edu.gov.on.ca/eng/curriculum/secondary/math910curr.pdf
Works Cited:
References:
Amy L. Nebesniak. (2013). Effective Instruction: A Mathematics Coach's Perspective. The Mathematics Teacher, 106(5), 354-358. doi:10.5951/mathteacher.106.5.0354
https://www-jstor-org.uproxy.library.dc- uoit.ca/stable/pdf/10.5951/mathteacher.106.5.0354.pdf?ab_segments=0%252Fbasic_SYC- 4946%252Fcontrol&refreqid=excelsior%3A46983deb17387a75b4a3c0dffa1ead50
Figure 1. January 28, 2020
Figure 2. Retrieved January 28, 2020 from: (https://www.algebra.com/algebra/homework/Linear-equations/Linear- equations.faq.question.102253.html)
Figure 3. Retrieved January 28, 2020 from: https://www.google.com/search?q=slope+in+math&source=lnms&tbm=isch&sa=X&ved=2ahU KEwjugYLGu6znAhXHPM0KHVX6AJoQ_AUoAXoECA4QAw&biw=1280&bih=698#imgrc =IIZqz04OEHgVPM
Youtube Video. Retrieved January 28, 2020 from: (https://www.youtube.com/watch?v=0671cRNjeKI)
Assignment C- Perpendicular Lines - Amanda,Brooke, Jessica (10/10)
Processes
Mathematical Processes
Reasoning
Problem solving
"develop,select,apply,andcompareavarietyofproblem-solvingstrategiesastheyposeand solve problems and conduct investigations, to help deepen their mathematical under- standing;"http://www.edu.gov.on.ca/eng/curriculum/secondary/math910curr.pdf
Communicating
"inductive reasoning. The process of reach- ing a conclusion or making a generalization on the basis of specific cases or examples." http://www.edu.gov.on.ca/eng/curriculum/secondary/math910curr.pdf
Students used prior knowledge of slopes/negative and positive numbers in order to reflect on "how" they knew what next steps to take.
"Students used prior knowledge of slopes/negative and positive numbers in order to reflect on "how" they knew what next steps to take."
Connecting
"makeconnectionsamongmathematicalconceptsandprocedures,andrelatemathematical ideas to situations or phenomena drawn from other contexts (e.g., other curriculum areas, daily life, current events, art and culture, sports);" http://www.edu.gov.on.ca/eng/curriculum/secondary/math910curr.pdf
"developandapplyreasoningskills(e.g.,recognitionofrelationships,generalization through inductive reasoning, use of counter-examples) to make mathematical conjectures, assess conjectures, and justify conclusions, and plan and construct organized mathematical arguments;" http://www.edu.gov.on.ca/eng/curriculum/secondary/math910curr.pdf
Reflecting
"demonstrate that they are reflecting on and monitoring their thinking to help clarify their understanding as they complete an investigation or solve a problem (e.g., by assessing the effectiveness of strategies and processes used, by proposing alternative approaches, by judging the reasonableness of results, by verifying solutions);" http://www.edu.gov.on.ca/eng/curriculum/secondary/math910curr.pdf
Strategies/tools
"Figure 1. Example of positive and negative slops on a graph. Plotting the points on the x and y axis to display the slopes intersecting. January 28, 2020
Retrieved from:
(https://www.google.com/search?q=intersecting+slopes+in+math&source=lnms&tbm=isch&sa=X&ved=2ahUKEwij8OmwsKfnAhWrhOAKHTZ-BYMQ_AUoAXoECA0QAw&biw=1280&bih=698#imgrc=vF0PmoszlJStYM)"
Perpendicular Lines (how to): Youtube Video. Retrieved January 28, 2020 from: (https://www.youtube.com/watch?v=0671cRNjeKI)
"Figure 2. Example of graph to plot the points on the x and y axis. Retrieved January 28, 2020 from: (https://www.algebra.com/algebra/homework/Linear-equations/Linear- equations.faq.question.102253.html)"
"Figure 3. Example of positive slope on a graph plotted on the x and y axis. Retrieved January 28, 2020 from: https://www.google.com/search?q=slope+in+math&source=lnms&tbm=isch&sa=X&ved=2ahU KEwjugYLGu6znAhXHPM0KHVX6AJoQ_AUoAXoECA4QAw&biw=1280&bih=698#imgrc =IIZqz04OEHgVPM"
"Figure 4. Example of the final graph of the equation y= -1⁄3 x + 4.33."
Computational
Relationships
Connection to understanding
y+x value
Script: The students knew that we could get -1/3 for the slope
"we know that the slope will be: - 1/3"
Ordered pairs
Properties
Property knowledge
Commute
Addition/multiplication
Identity
Addition/multiplication
Associate
Addition/multiplication
Distribute
Division