Proportional and Linear

Grade

Duration

45 minutes

Class size

25 students

Curriculum framework

International Baccalaureate Middle Years Programme (MYP) Mathematics
Focus: MYP Criterion A (Knowing and understanding), Criterion B (Investigating patterns), Criterion C (Communicating)

Learning Objectives

By the end of this lesson, students will be able to:

Identify and graph proportional relationships and interpret the unit rate as a slope (aligning with TEKS 8.4.B).
Distinguish between proportional and non-proportional linear situations, including direct variation.
Identify functions from ordered pairs, tables, mappings, and graphs (TEKS 8.5.G), supporting matrices, relations, and function notation (TEKS 8.5.A, 8.5.B).
Solve systems of linear equations using graphical and algebraic methods (TEKS 8.9.A).

These objectives directly align with MYP mathematics focus on both conceptual understanding and application.

Key Concepts & Related Concepts

Key concepts: Relationships, Change
Related concepts: Proportionality, Functions, Rate of Change

Prior Knowledge

Students should be familiar with:

Ratios and rates
Basic graphing skills
Simple linear equations
Function notation

Materials Needed

Whiteboard and markers
Student notebooks or digital device for blended learning notes
Printable or digital quick notes handout (fill-in-the-blank style)
Graph paper or graphing app (e.g. Desmos or similar)
Projector for visuals
Exit ticket slips or digital quiz

Lesson Sequence

1. Introduction & Engagement (5 minutes)

Begin with a quick warm-up question projected on the board:
"If 3 apples cost $6, what is the cost of 1 apple? How would you show this relationship as a graph?"
Encourage 2-3 brief student answers, emphasize the idea of unit rate and proportional relationships.
Connect to today’s objectives and relevance: "Today, we will explore how these rates connect to lines on a graph and how to decide if a relationship is proportional or not."

2. Mini-Lecture + Interactive Notes (15 minutes)

Using a blended learning approach, distribute quick, fill-in-the-blank notes for students to complete together. Display the notes on the projector, and students fill them in either on paper or digitally.

Sample Notes Outline:

Quick Notes: Proportional and Non-Proportional Linear Relationships

1. Proportional Relationship:
A relationship is proportional if the ratio between quantities is __________.
The graph of a proportional relationship is a straight line that passes through the point (__, ____).

2. Unit Rate and Slope:
The unit rate is the amount of change in the ____ variable for one unit of change in the ____ variable.
On a graph, this is represented by the ____, or steepness, of the line.

3. Identifying Functions:
A relation is a function if every input (____) has exactly one ____.
Functions can be represented using ____________, ___________, ____________, and ____________.

4. Direct Variation:
A type of proportional relationship where y = kx. Here, k is the constant of ________________.
The graph always passes through ________________.

5. Non-proportional Linear Relationship:
Linear relationships that do not pass through (0,0) are called ______________ relationships.
Their equations look like ______________, where b ≠ 0.

6. Systems of Equations:
A system is two linear equations solved ________________.
Methods include graphing, substitution, and _____________.

Answers for teacher:

constant
(0,0)
dependent, output, ordered pairs, tables, mappings, graphs
variation
direct variation, origin
non-proportional
simultaneously, elimination

Teacher notes during mini-lecture:

Use examples for each fill-in (e.g., show graphs with/without passing through origin).
Demonstrate how to find slope (unit rate) on the board with an ordered pair example.
Highlight the difference between proportional and non-proportional visually.

3. Guided Practice: Explore Functions and Graphs (15 minutes)

Distribute graph paper or use an online graphing app like Desmos (if devices available).
Give students 3 sets of data:
1. Proportional relationship (e.g., y = 2x)
2. Non-proportional linear relationship (e.g., y = 2x + 3)
3. A system of two equations (e.g., y = x + 1 and y = -x + 5)
In pairs, students:
- Plot the graphs.
- Identify if each relationship is proportional or not.
- For the system, find the intersection point (solution).
Circulate and support, asking probing questions:
- "Does the line pass through (0,0)? How do you know it’s proportional?"
- "What does the intersection point mean in the context of solving a system?"

4. Independent Reflection and Assessment (7 minutes)

Distribute an Exit Ticket (print or digital):

Exit Ticket Questions:

What is the unit rate in a proportional relationship that passes through the points (0,0) and (4,8)?
Is the equation y = 3x + 4 proportional? Why or why not?
Given the system: y = 2x + 1 and y = -x + 4, what is the solution?
How can you tell if a relation is a function?

Collect and quickly scan for understanding.

5. Wrap-up and Connection (3 minutes)

Recap key points: proportional = passes through the origin with constant ratio, functions have unique outputs, systems solved by intersection.
Preview next lesson: more practice with solving systems and deeper function analysis.
Encourage students to use the fill-in notes for review and the graphing tool to experiment at home.

Differentiation

Students needing extra support will have access to graph templates and guided questions.
Advanced students will be challenged to create their own systems and explain solutions verbally or in writing.
Visual and kinesthetic learners engage deeply through graphing and interactive notes.

Reflection for Teacher

Assess student responses on exit tickets for misconceptions on slope and identifying functions.
Adjust next lessons based on students’ ability to solve systems and interpret graphs accurately.
Consider extending blended tools with interactive quizzes for homework (e.g., Kahoot or Google Forms).

This lesson plan integrates key IB MYP approaches by emphasizing conceptual understanding, real-world application, and communication using multiple representations of functions and equations. The fill-in-the-blank notes engage students actively, supporting both face-to-face and digital learning environments.

Teaching Instructions