 # Curriculum

## Middle School Math Modeling Curriculum For In-person, Digital, or Hybrid learning

Topics covered: Ratios & Proportions, Slope & Speed, and Linear Equations

### Curriculum Overview

In order to understand linear equations, students need to have an understanding of the meaning of all the parts of the equation as well as how changes to the equation affect the graph of the equation and the meaning that the graph conveys. By starting with Ratios and Proportions, students are able to differentiate between these two very important mathematical concepts and determine which they are dealing with when they get to linear equations. Thay can begin to internalize what the ratios are representing  in the real world and how changing the ratio changes the graph of the line of the graph and what it represents.

In the next section, students connect the proportional relationship of constant speed with the slope of the line. They can use the slope of a line to create the equation for the graph of their real world data and then use slopes to graph other lines and begin to understand the story of the graph even without having to collect the data themselves. At this point, students will have an understanding of “m,” the first part of the linear equation.

In Part 3, students will again use motion to give them context for the Y-intercept part of the linear equation. By knowing that the y-intercept can mean the startinging point, students can combine their understanding of the ratio as the slope of the line, model the proportional relationship on the graph and choose an appropriate starting point for data they are given, data they collect, or a graph they are provided with.

They can tell the story of what is happening in the graph by understanding the numbers in the equation and create an equation that mathematically explains the motion described by the graph.

#### Options for In-Person and Digital Learning

Throughout this curriculum, we’re offering options for conducting classroom activities or discussions both in-person and online, which we hope will be especially useful given the uncertainty of teaching with the ongoing COVID-19 pandemic. We want to provide options for each step of the activities that typically appear in this curriculum.

The steps in each lesson usually take the following form:

1. Conducting a lab
2. Creating a whiteboard in small groups
3. Commenting and discussing the class’ whiteboards
4. Having a class-wide whiteboard meeting

Here, we break down what each step looks like in-person or digitally.

#### Options for Digital Whiteboards

• Whiteboard option 1 – Jamboard
• Each group can make one slide and click through to view other slides. Students can write, draw, upload an image, change the background, etc.
• Jamboard is entirely free with no limits on functionality
• Whiteboard option 2 – https://www.whiteboard.chat/
• Slightly fancier features than Jamboard but may require a slightly more preparation. Every group can make their own whiteboard and click out to see other boards. Includes teacher tools like a grid view of everybody’s board, option to project boards to the class
• In the free version, teachers have ads displayed to them (but students do not). Free version is limited to 10 boards at a time (which should suffice for group work)
• Whiteboard option 3 – Miro’s Web White Board
• Slightly improved features from those of Jamboard but not everything is unlocked in the free version. Every group can work in the same space and then zoom in to other people’s boards when they’re ready for a gallery walk.
• Requires upgrades for certain features

#### Options for Digital Discussions

• Option 1: https://yoteachapp.com/
• The website serves as a kind of chat room that allows students to send messages, respond to specific messages, and even draw/graph/upload responses. While the interface is a little messy, it’s a nice tool for keeping track of discussions and their responses.
• You can create a google sheet to have students type their questions. Responses can be in the form of comments or indented/bulleted responses.
• Option 3: Jamboard
• Have students write their questions in one color sticky note and their answers in another color. You can organize questions in columns or clusters. Here’s a sample of a discussion board in Jamboard.

### Instructional Goals

Note: Full instructional goals are listed in lessons

1. Use ratio and rate reasoning to solve mathematical problems and problems in real-world context (e.g., by reasoning about data collected from measurements, tables of equivalent ratios, tape diagrams, double number line diagrams, or equations).
2. Solve unit rate problems including those involving unit pricing and constant speed.
3. Use variables to represent two quantities that change in relationship to one another to solve mathematical problems and problems in real-world context.

1. Recognize and represent proportional relationships between quantities. a. Decide whether two quantities are in a proportional relationship b. Identify the constant of proportionality (unit rate) c. Represent proportional relationships by equations.
2. Explain what a point (x, y) on the graph of a proportional relationship means in the situation, with special attention to the points (0, 0) and (1, r) where r is unit rate.

1. Graph proportional relationships interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.
2. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.
3. Given a description of a situation, generate a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or a graph. Track how the values of the two quantities change together. Interpret the rate of change and initial value of a linear function in terms of the situation it models, its graph, or its table of values.

### Lesson Sequence

#### Section 1.0: Perfect Purple Paint (Ratios and Proportions)

1.1 Introduction

1.2 Create a Model

1.3 Refine Your Model (Whiteboard Discussion)

Suggested Assignment: Intro to Ratios

Suggested Assignment: Help Nico with Ratios

1.4 Practice Help Giving & Review Talk Moves

1.7 Integrate Feedback on Model

Suggested Assignment: Proportional Relationships

Suggested Assignment: Solving Proportions

#### Section 2.0: Buggy Lab (Slope and Speed)

2.1 Proportion Problems Brainstorm

2.2 Accuracy with a Stopwatch

2.3 Buggy Lab: Collect Data

2.4 Buggy Lab: Create a Whiteboard

2.5 Buggy Lab: Discussion

2.6 Buggy Lab: Board Meeting

Suggested Assignment: Khan Academy Unit Rate

#### Section 3.0: Row Boats (Linear Equations)

3.1 Row Boat Lab: Collect Data and Make Whiteboards

3.2 Row Boat Lab: Discuss Your Models

3.3 Row Boat Lab: Whiteboard Meeting

3.4 Row Boat Lab: Takaways

Suggested Assignment: Help with Functions Assignment

Suggested Assignment: Slope-Intercept Form

Section 0.0 Materials Checklist

Links labeled with lesson numbers are specific to those lessons. Links without a number are used throughout or are suggested materials.

#### Physical Materials:

• Whiteboards
• Sticky Notes
• Computers (can be one per student, per group, etc. use your best judgment)
• Unifix cubes (can also be digital if you do not have a physical set)
• Tumble Buggies
• Meter Sticks
• Markers
• Erasers
• Stopwatch (students can also use their cell phone stopwatch)