MTH1W
Diagnostic Thinking Tasks
Linear and Non-linear Relations
Teacher Notes
FACILITATING THE DIAGNOSTIC:
Slides with a grey background should not be shown to students.
The purpose of this diagnostic is to learn more about what students recall and what their comfort level is with various concepts. This is NOT a test.
It is HIGHLY recommended that this tool be used in the following ways:
Paper Approach: Provide each student with a paper copy of the selected questions by printing and photocopying the selected slides.
Digital Approach: Provide students with a copy of the slides (only the problem slides) via Google Classroom or your VLE. Have students show their thinking digitally or by inserting pictures onto the slides. If necessary, describe how to colour tools to digitally shade cells in tables or insert shapes on slides.
For either approach, circulate / give students the explicit option of verbally explaining/showing their thinking to you instead of recording it on paper. Avoid confusion based on running the technology. Sometimes paper / writing by hand on a whiteboard can give you more information than trying to figure out how to do something with the technology. Make notes about observations to use for future record-keeping.
ASSESSING THE DIAGNOSTIC:
Observe where students have difficulty and make notes on the MTH1W Developmental Continuum (Linear and Non-linear Relations tab) for each student. Use the look fors to help guide your observations. These notes will help you identify what you can reteach to close gaps in skills and concepts, differentiate instruction as needed, and provide intervention.
FOLLOW-UP STRATEGIES AND RESOURCES
Suggestions for how to differentiate instruction, close gaps and skills and provide intervention are provided here.
Here are figures 1, 2 and 3 of a visual pattern.
How many smiley faces would be in figure 13?
Use a tool or tools that makes sense to you.
NAME:
Figure # | Number of Smiley Faces |
0 | |
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Each hexagon represents a table.
Each black circle represents a person.
NAME:
What is the INITIAL VALUE (CONSTANT) of the pattern? How many people will there be if there are 0 tables?
What is the CONSTANT RATE (MULTIPLIER) of the pattern? Show your thinking clearly.
Describe how the pattern is growing.
Write an algebraic expression (pattern rule) to find the number of people that can sit at ANY number of tables.
Here are two patterns represented by their tables of values.
They each match one of the lines on the graph.
Which table of values matches line A?
Explain how you know.
NAME:
A
B
PATTERN 1 | |
n | T |
1 | 7 |
2 | 9 |
3 | 11 |
PATTERN 2 | |
n | T |
1 | 3 |
2 | 1 |
3 | -1 |
The graph shows information about Hayden’s bank account balance.
What information does the black circle (⚫) on the graph tell you about the situation?
Describe how Hayden’s bank account balance is changing. Be specific.
How many weeks will it take Hayden to save $200?
NAME:
Here are two pattern rules. They each match one of the lines on the graph.
T = -2n + 10 T = -1.5n + 10
Klassen says that Line A matches the rule
T = -2n + 10.
Funk says that Line B matches the rule
T = -2n + 10.
Who is correct? Explain why.
NAME:
A
B
| Understanding Rate of Change and Initial Value (C3.2) | Comparing Linear and Non-Linear Relationships (C3.1, C4.1) | ||||||||||
| Determine the constant rate (multiplier) of a linear growing or shrinking pattern. | Explain the graphical significance of the constant rate (multiplier) of a linear growing or shrinking pattern. | Determine the rate of change (calculate slope) of a line by understanding that it is the ratio between rise and run (vertical change/horizontal change) | Determine the rate of change (calculate slope) of a line, given two points | Determine the initial value (constant) of a linear growing or shrinking pattern | Explain the graphical significance of the initial value (constant) of a linear growing or shrinking pattern | compare linear growing and shrinking patterns on the basis of their constant rates and initial values | Identify and compare linear and nonlinear relations by their graphs. | Determine whether a table of values represents a linear or nonlinear relation (include ordered, unordered and skipping tables). | Identify linear and nonlinear relations by identifying the degree of their equation. | Compare and make general predictions based on graphs of linear and nonlinear relations. | Compare and describe rates of change of linear and nonlinear relations and make connections to their graphs. |
Slide | 4 | 6 (describe how balance is changing) 7 (if steepness idea is mentioned) | | | 4, 6 | 6 (describe meaning of black circle) 7 (if initial value being same is mentioned) | 5, 7 | | | | | |
Look Fors (Part 1)
Prior to Grade 8 Expectation / Grade 8 Expectation / MTH1W Expectation
| Representing Linear Relations (C3.2, C4.4) | ||||||||||||
| translate (represent) linear growing and shrinking patterns, that involve decimal numbers, using tables of values and graphs | translate (represent) linear growing and shrinking patterns, that involve rational numbers, using tables of values and graphs | Translate (represent) a growing and shrinking pattern from a table of values to a graph. | Represent linear relations using concrete materials, demonstrating an understanding of rate of change and initial value. | Represent linear relations using tables of values demonstrating an understanding of rate of change and initial value. | Represent linear relations using graphs demonstrating an understanding of rate of change and initial value. | Make connections between representations of a linear relation. | translate (represent) a linear growing pattern, that involves decimals, to an algebraic expression or equation | translate (represent) a linear growing pattern, that involves rational numbers, to an algebraic expression or equation | Represent linear relations using equations, demonstrating an understanding of rate of change and initial value. | Determine the equation of a line given its table of values by determining the slope and finding the y-intercept in a variety of ways (extending the table, plotting points, etc.) | Determine an equation of a line, given the graph by finding the slope and identifying the y-intercept | Given one representation of a linear relation, create another to solve problems. |
Slide | 4 (no decimals) | 4 (no rationals) | 5 | | | | | 4 (no decimals) | 4 (no rationals) | | | | |
Look Fors (Part 2)
Prior to Grade 8 Expectation / Grade 8 Expectation / MTH1W Expectation
| Using Representations of Linear Relations to Solve Problems (C3.3, C4.4) | ||||
| use the algebraic expression of the pattern rule to make and justify predictions of a linear growing pattern that involves decimals | use the algebraic expression of the pattern rule to solve for unknown values in linear growing and shrinking patterns that involve rational numbers | Make connections between rates of change/slope and initial values/y-intercepts in a context and use the equation of a line to solve problems. | Identify and interpret the meaning of the point of intersection on a graph of two linear relations, given a context. | Compare two linear relations (y = ax + b) algebraically (e.g., by comparing steepness, y-intercepts) to interpret the meaning of the point of intersection given a context |
Slide | 6 (if algebraic expression is used) | 6 (if algebraic expression is used) | | | |
Look Fors (Part 3)
Prior to Grade 8 Expectation / Grade 8 Expectation / MTH1W Expectation
Special Note for Slide 6:
Make note of if students take the value for 5 weeks ($100) and double the length of time to 10 weeks.
This misconception will be valuable to address with respect to students’ understanding of a constant (initial value) later.
FOLLOW-UP TEACHING STRATEGIES
Instruction that follows the diagnostic task should be responsive to observations made about students’ current understanding on the continuum. The following are possible teaching strategies you might use to help students further their understanding.
Key to understanding linear relations is the ability to see and make connections between representations. Pose problems that allow for / encourage students to use representations that make sense to them, and build on that understanding to move towards more “efficient” or “sophisticated” strategies, never invalidating their chosen approaches but instead reinforcing connections that can be made between approaches.
Small Group Instruction - work with small groups of 2, 4, 6 students (so they can be working in pairs) at a time to move them forward along the continuum (the rest of the class can be working on purposeful practice or tasks associated with the theme/project in progress) Anchor Charts - co-construct anchor charts as a class once understanding about certain concepts and connections between representations has been achieved; refer to the anchor charts regularly |
MATERIALS / MANIPULATIVES
Students who struggle with linear relations will benefit greatly from using a range of representations to explore the concepts of rate of change and initial value. Beginning with concrete (e.g., snap cube or pattern block) representations as well as visual patterns can help develop this deep understanding.
Here are some materials / manipulatives that might be beneficial in supporting this development:
Snap Cubes Centicubes Visual Patterns (this site includes examples of linear and nonlinear patterns) Desmos Graphing Calculator (encourage students to use this regularly as a tool to check their thinking and see relations visually represented) Desmos Activities (here is a collection about Linear Functions) |
ADDITIONAL RESOURCES
These resources provide some sample problems and planning tools that can be used with students to develop their understanding.
Students will benefit from purposeful activities as well as opportunities to discuss and compare their thinking. Practices that encourage students to think - rather than mimic - should be emphasized when using these or any resources.
From Patterns to Algebra (each Math Dept should have a copy of this resource) WRDSB Rich Task Database (search for *WRDSB Lesson Plan along with the relevant MTH1W overall expectation) |