Ratios & Proportions, Modeling, Number & Quantity

1. Ratios & Proportions, Modeling, Number & Quantity by Chris Pollard, Stephanie Myers, & Tom Morse

2. Ratios & Proportions • Students work with ratios as early as early as third grade when they first are introduced to fractions. • They first see the beginnings of proportions in fourth grade when looking at fraction equivalence. • The major focus on ratios and proportions begins in sixth grade with a 35-day module!! • Grades 6-8 is called a story of ratios, while 9-12 is called a story of functions.

3. Ratios & Proportions continued • Students begin their sixth grade year investigating the concepts of ratio and rate. They use multiple forms of ratio language and ratio notation, and formalize understanding of equivalent ratios. Students apply reasoning when solving collections of ratio problems in real world contexts using various tools (e.g., tape diagrams, double number line diagrams, tables, equations and graphs). Students bridge their understanding of ratios to the value of a ratio, and then to rate and unit rate, discovering that a percent of a quantity is a rate per 100. The 35 day module concludes with students expressing a fraction as a percent and finding a percent of a quantity in real world concepts, supporting their reasoning with familiar representations they used previously in the module.

4. Ratios & Proportions continued • In seventh grade, students embark upon a bold and adventurous 30-day module to build upon sixth grade reasoning of ratios and rates to formally define proportional relationships and the constant of proportionality.  Students explore multiple representations of proportional relationships by looking at tables, graphs, equations, and verbal descriptions.  Students extend their understanding about ratios and proportional relationships to compute unit rates for ratios and rates specified by rational numbers. The module concludes with students applying proportional reasoning to identify scale factor and create a scale drawing.

5. Examples in Ratios & Proportions • We paid $75 for 15 hamburgers. What is the rate per hamburger? • It takes 7 hours to mow 4 lawns. At that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? Notice the use of modeling & number quantity within these examples!! 

6. What did I learn/what should people know to better teach curriculum?? • The state is continuing to work on the modules and curriculum just like you should be!! • Use your own personal touch! • Modules are NOT meant to just copy and hand out as classroom materials. • Lessons in the modules may not be created with your classroom in mind.

7. Modeling • Modeling links classroom mathematics and statistics to everyday life, work, and decision-making. • Modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions. • When making mathematical models, technology is valuable for varying assumptions, exploring consequences, and comparing predictions with data. • Diagrams of various kinds, spreadsheets and other technology, and algebra are powerful tools for understanding and solving problems drawn from different types of real-world situations.

8. Modeling continued • In descriptive modeling, a model simply describes the phenomena or summarizes them in a compact form. Graphs of observations are a familiar descriptive model • Analytic modeling seeks to explain data on the basis of deeper theoretical ideas • Graphing utilities, spreadsheets, computer algebra systems, and dynamic geometry software are powerful tools that can be used to model purely mathematical phenomena (e.g., the behavior of polynomials) as well as physical phenomena.

9. Modeling Standards • Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol

10. Examples of Models • Estimating how much water and food is needed for emergency relief in a devastated city of 3 million people, and how it might be distributed. • Planning a table tennis tournament for 7 players at a club with 4 tables, where each player plays against each other player. • Designing the layout of the stalls in a school fair so as to raise as much money as possible. • Analyzing stopping distance for a car. • Modeling savings account balance, bacterial colony growth, or investment growth. • Engaging in critical path analysis, e.g., applied to turnaround of an aircraft at an airport. • Analyzing risk in situations such as extreme sports, pandemics, and terrorism. • Relating population statistics to individual predictions.