Date post: | 22-Dec-2015 |
Category: |
Documents |
Upload: | juniper-mitchell |
View: | 216 times |
Download: | 2 times |
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.