Comparing Two Linear Relationships: Teacher Notes
Overview
In this activity students explore the conditions under which one option is better than another option.
Important Mathematical Ideas
· The point of intersection between two linear relations is where two relations have the same value for the dependent and independent variables.
· A graphical representation is useful to compare two options.
· Two options represented graphically as lines have the same values for the dependent and independent variables where the two lines cross.
Prior Knowledge
· Graphing relationships.
· Creating a story from information conveyed by a graph.
· Creating equations to represent situations.
· Solving equations.
· Direct and partial variations. (Unit 5 Activity 4).
Common Misconceptions
· Interchanging dependent and independent variables when creating representations; numerical, graphical and algebraic.
Curriculum Notes
· Interchanging dependent and independent variables when creating representations; numerical, graphical and algebraic.
Information to Support/ Enhance/ Extend Learning
· Students are asked to keep a journal for each unit in the course. It should contain notes of important mathematical ideas with examples and new vocabulary.
· ePortfolio may be used for these journal entries.
· Students can make individual choices whether this is a paper or digital personal resource.
· Consider a variety of formats as alternatives to journal entries (e.g., student note, pair/share, group discussion, exit card, poster, electronic posting).
· Develop a Word Wall and continue it throughout the unit as new vocabulary and terms arise that require clarification (e.g., point of intersection, direct variation, partial variation).
· Students may benefit from sketching graphs by hand.
Minds On
Task 1: Investigation 1 - Cat and Mouse Gizmo
· Students will:
o adjust graphs to investigate the effects on speed and distance
o connect real-world meaning to the steepness of a line, initial value and the intersection of lines
o predict whether lines intersect based on speed and distance
o check solutions with feedback provided
· To use the ministry-licensed Gizmos, teachers will need to set up a Class Code to create an account and give students a password.
· A Coach and Be Coached strategy can be used for this investigation.
Task 2: Investigation 2 – Distance-Time Graphs
· Students will:
o adjust the graph to investigate the effect on the speed and the distance
o use details of graphs to describe speeds and distances of the runners
o connect real-world meaning to the steepness of a line and initial value
o interpret the intersection of lines from the given context
o share their answers in the Discussion
· Solutions available in the Unit.
· A Coach and Be Coached strategy can be used for this investigation.
Discussion Prompt and Notes
Share your answers to theDistance-Time Graphs (Two Runners)activity.
· Common Errors:
o omitting the total distance ran and the time it took
o omitting units: distance (m), time (s)
o not understanding that horizontal lines on the graph mean the runner has stopped moving
o incorrect or not including direction (e.g., the runner is heading for the starting point)
o incorrect or not including time between data point
o incorrect or not including distance between data points
o not recognizing different starting positions
o graph does not match the scenario given
o incorrect interpretation of distance or time
Action
Task 3: Erica and Alex Travel to School
· Students interpret the graph to answer contextual questions.
· A Think/Pair/Share strategy can be used for this activity.
Discussion Prompt and Notes
Use the graph to describe in words each of their journeys to school. You may wish to include:
o Who was walking faster and how you know?
o What happened at the point where the lines crossed?
o Did they arrive at school at the same time? How do you know?
o Who lives closer to school? How do you know?
Solutions should include:
o Erica was walking faster because her line was steeper.
o At the point where the lines crossed they both had travelled for the same amount of time (18 minutes) and were the same distance from Erica's house (7.5 km).
o Erica arrived at the school before Alex. It took her between 28 and 29 minutes to get to school. Her distance-time graph reached 12 km first. It took Alex 36 min to get to school.
Common Errors:
o not recognizing different starting points
o not realizing that the point of intersection means that the students will meet each other
o not recognizing that a steeper rate of change means the speed is greater
Task 4: Check Your Understanding
Students will:
o answer the questions about Erica and Alex's trip to school
o check their answers with the sample ones provided
Task 5: Cell Phone Problem
Students will:
o create algebraic models for two cell phone plans
o watch the Desmos Cell Phone Problem video
o use Desmos to make graphs for the cell phone problem
o Demos Graphing Calculator is a free online graphing tool used in this course to graph linear relations in the first quadrant
o use their graphs to make recommendations about which plan to choose
o compare their solution with the one provided
The teacher:
o may lead a demonstration of Desmos on an overhead or an Interactive White Board followed by further independent student use or students working in pairs
Task 6: The Landscaping Problem
· A Think/Pair/Share or a Coach and Be Coached strategy can be used for this activity.
· Students will:
o watch the Landscape Architecture video and record mathematical terms and concepts in their journal
o create algebraic and graphical models for two landscape companies
o interpret the point of intersection in the graphical model
o make recommendations about which landscape company to choose
o check their answers with provided solutions
Journal Prompt and Sample Response
What mathematical terms and concepts did you hear him use?
He talks about designing the landscape as solving a problem. They do a site analysis which involves measuring and calculating areas. The design process involves deciding on the shape and size of the different areas of the yard.
Task 7: Check Your Understanding
Students will:
o answer eight questions about comparing two linear relationships
o can check their answers with the sample ones provided
Consolidation
Task 8: Are You Ready to Explain My Point?
· OERB Resource ID: ELO1241270
· Students will:
o practice interpreting graphs and choosing the best option
o repeat activity if they do not achieve at least 11/15
Task 9: Assignment 1 – Splash World
· Posted with unit.
· See sample solution in the Teacher Notes posted on the vLE.
· Consider co-developing an Anchor Chart that describes and provides an example for identifying the best choice given two scenarios.
· Students may use Mindomo to develop a Mind Map or a web to summarize the process for identifying the best choice given two scenarios.
Task 10: Student Reflection
· Students are asked to reflect on their understanding of this topic.
· These reflections can be used as assessment for learning to help determine next steps for individual students.
Grade 9 Applied Blended Learning: Unit 6 Activity 6 Page 4 of 4