ADAPTED FROM A WORKSHOP PREPARED FOR

THE RHODE ISLAND DEPARTMENT OF EDUCATION

BY KRISTINA SPARFVEN

The Rigors of Ratio and Proportional Reasoning in the Common Core State Standards

Agenda

Examine the progression of ratio and proportion in the middle grades.

Highlight the connection between content and relevant Standards for Mathematical Practice.

Explore instructional strategies and models that promote understanding of ratio and proportion.

Work on activities that support ratio and proportional reasoning.

Wrap-up/Questions.

How are the math practices incorporated?

Reason Abstractly and Quantitatively (MP2) Contextualize and Decontextualize

Construct Viable Arguments and Critique the Reasoning of Others (MP3) Use multiple models as solution paths, understand multiple

modelsModel with Mathematics (MP4)

Represent situations mathematicallyAttend to Precision (MP6)

Precise use of language when describing and interpretingLooking for structure (MP7)

Recognizing rates, ratios, and proportional relationships

What is a ratio?

An essential understanding for today, is to internalize the concept of ratio.

Turn and talk to your neighbor about what you think of when your hear the word ratio.

What is a ratio?

A relationship between 2 quantities. We write ratios to decontextualize a situation. (MP2)

A rate is a ratio of quantities with different units.

Ratios may be written in a variety of ways which must be interpreted accurately by a student. 3 to 2 (3 feet to 2 seconds) 3 for every 2 (3 cups of flour for every 2 eggs) 3 out of every 5 (3 cups of flour out of every 5 cups of

dry ingredients) 3:2

Tape Diagrams

Visual modelMeasuring in the same units.

This tape diagram shows the ratio of two juices in a fruit punch.

What different ratios could be expressed by the tape diagram?

Tape Diagrams

Double Number Line

Measuring with different units.

Each ratio pair is the same distance from 0 on their respective lines.

Double number lines have a variety of uses: to find unit rates other equivalent ratios missing value in a proportion including missing values in

percent problems

Using a Double Number Line to Find a Unit Rate

Unit Rate in the Real World

Unit rates are often expressed as a comparisons with no reference to the number oneo 65 miles per houro $2.49 per pound

In grade 7 students are able to convert a rational number to a decimal using long division (7.NS.2d)

When students create ratios to find a unit rate expressed as a decimal, it is important that they fully understand the comparison they are making

Using a Double Number Line for Percent Problems

Finding a Missing Value in a Proportion

What method would you use to solve for the value of x?

Reflect on how you might teach your students to solve for the value of x.

3   X___ = ___

21   35

The “Why” of Cross Multiplicationthe Cross Product Property

Ratio structure in TABLES

Additive structure.

Multiplicative structure

Grade 7 PARCC Item

Ratio structure in GRAPHS

Additive structure

Multiplicative structure

Correspondence between Tables and Graphs

Additive

Multiplicative

Ratio structure in E QUATIONS

Constant of proportionality (unit rate)y=mx

m is the unit rate (grade 6), constant of proportionality (grade 7), slope (grade 8)of a graph

An equation can be used to generate inputs and outputs in a table (beginning function work)

Session Summary

Today we examined… The nature of ratio and proportional reasoning and its

progression through the middle grades. Strategies for solving ratio, rate, and proportional

reasoning problems. Unit rate and its evolution through the grades to

constant of proportionality and slope. Ratio structure in tables, graphs, and equations. Integration of the mathematical practices with the

content of ratio and proportional reasoning.