Using Reasoning and Sense

Making to Teach Ratios

2013 Math Summit August 8, 2013 SAS Campus

Cary, NC

Dr. Vincent T. Snipes, Director Center for Mathematics, Science, and Technology

Education (CMSTE) Winston-Salem State University

snipesv@wssu.edu

Focus in High School Mathematics:

Reasoning and Sense Making (NCTM)

Co-author of the publication

This document proposes that all

high school mathematics programs

should focus on reasoning and

sense making (NCTM, 2009).

Definitions

Reasoning: The process of drawing conclusions on the basis of evidence or stated assumptions.

Sense making: Developing understanding of a situation, context, or concept by connecting it with existing knowledge.

Reasoning and sense making is an evolution of NCTM’s position that problem solving should be the emphasis of mathematics teaching and learning.

Chapter 2 -- Reasoning Habits

Reasoning and sense making should be a part of the mathematics classroom every day.

Reasoning habits: ◦ “Productive way[s] of thinking that

becomes common in the processes of mathematical inquiry and sense making” (p. 24)

◦ Should not be approached as a new list of topics to be added to the curriculum.

Reasoning Habits

Analyzing a problem, for example…

Implementing a strategy, for

example…

Seeking and using connections…

Reflecting on a solution to a problem,

for example…

Reasoning as a Part of All

Mathematical Activity Reasoning is interwoven with the Process

Standards

◦ Reasoning and Proof

◦ Problem solving

◦ Representation

◦ Connections

◦ Communication

Inherent in mathematical proficiency and

procedural fluency

Importance of mathematical modeling

Statistical reasoning as a part of mathematical

reasoning

Common Core State Standards for

Mathematics (CCSSM)

CCSSM (2010) begins with Mathematical

Practices that emphasize reasoning and

sense making

Standards for Mathematical Practice (MP):

MP.1. Make sense of problems and

persevere in solving them.

MP.2. Reason abstractly and quantitatively.

MP.3. Construct viable arguments and

critique the reasoning of others.

Tips for Implementing Reasoning and

Sense Making into the Classroom

Provide tasks that require students to figure things out for themselves.

Ask students questions that will press their thinking – for example, “Why does this work?” or “How do you know?”

Provide adequate wait time after a question for students to formulate their own reasoning.

Resist the urge to tell students how to solve a problem when they become frustrated; find other ways to support students as they think and work.

Expect students to communicate their reasoning to their classmates and the teacher, orally and in writing, through using proper mathematical vocabulary.

RATIO

A ratio is a way of comparing two quantities. It can be represented in multiple forms such as:

3 to 4 ¾ 3 : 4

Ex. There are 3 cats and 5 dogs in the pet hospital. The ratio of the numbers of cats to dogs is 3 to 5, 3/5, or 3:5.

EXAMPLE OF RATIO

A 16 oz. can of Hunt’s tomatoes costs

$0.64. A 24 oz. can of Del Monte

tomatoes costs $1.14. Which is the

better buy?

Hunt’s Del Monte

Cost $0.64 = $0.04/ oz. $1.14 = $0.0475/

oz.

Ounces 16 24

Hunt’s tomatoes are the better buy.

EQUAL RATIOS

If a, b, c, and d are integers with b

not equal to 0 and d not equal to 0,

two ratios a/b and c/d are equal if

and only if ad = bc.

EXAMPLE OF

EQUAL RATIOS When making iced tea, mix 2 cups of water with

6 scoops of iced tea mix. How many scoops are needed to mix with 10 cups of water?

2 CUPS WATER = 10 CUPS WATER

6 SCOOPS TEA S

2S = 10 X 6

2S = 60

S = 30

30 scoops of tea are needed.

PROPORTION

A proportion shows two ratios that have the same value; that is, the fractions representing the ratios are equivalent. Proportional reasoning is mathematical thinking in which students can recognize proportional versus nonproportional situations and can use multiple approaches for solving problems about proportional situations (Lanius & Williams, 2003). If the cross products are equal, then the two ratios form a proportion.

a = c

b d

ad=bc

a, b, c, and d are terms

a and d terms are extremes

b and c terms are means

Why is Proportional Reasoning

Important?

Proportional reasoning is evident in our

everyday activities. We utilize proportional

reasoning to calculate the best prices, taxes,

to work with drawings and maps, to perform

measurement conversions, to modify recipes,

and to create solutions.

PROPORTION CONT’D

Which of the following form a

proportion?

3 and 27 3 and 24

8 72 8 56

3 x 72 = 8 x 27 3 X 56 = 8 X 24

YES NO

SOLVING A PROPORTION

A mason mixes bags of cement and sand using a ratio of 2:5. Twelve bags of cement will be used. How much sand is needed?

2 = 12 5 S 2 x S = 5 x 12 2S = 60 S = 30 30 bags of sand are needed.

Activities

Miles Per Gallon

Shopping Disagreement

Proportional Reasoning Tips for

Teachers Provide students with proportional situations from a variety of

contexts that they can relate to

Help students distinguish between proportional and non-

proportional situations

Encourage discussion about predicting and comparing ratios

Help students relate proportional reasoning to what they

already know

(http://www.edu.gov.on.ca/eng/teachers/studentsuccess/Prop

ortionReason.pdf)

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