```PROBLEM SOLVING
Erie 1 BOCES
May 22, 2013
Facilitators: Andrea Tamarazio & Steve Graser
Pre-Assessment
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K – 5 Module 1 is currently available
6 – 12 Overview of Module 1 is currently available
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Curriculum Maps for Grades 6 – 12 are available
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Changes have been made to the K – 5 Curriculum Map
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 Module Titles

Link to K – 5 Materials
Link to 6 – 12 Materials
K – 5 Modules
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Modules released during May NTI
Kindergarten Module 1
 Grade 2 Modules 1 & 2
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Other Modules also available
Kindergarten Module 5
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Recommended Instructional Minutes
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Elementary
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60 minutes
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Fluency
Application Problems (RDW)
Concept Development (includes Problem Set)
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Math Models
Student Debrief
MS / HS
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45 minutes
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Fluency
Application Problems (RDW)
Concept Development

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Math Models
Student Debrief
Primary Resource

Districts have choices regarding the modules
 Delete

(Ignore)
If adapting or ignoring, it is highly recommended to
implement a Balanced Math Program that will
include the following:
 Daily
Fluency
 Problem Solving
 Student Reflection
Lessons within a Unit Activity


and sequence of lesson titles (6 – 10) for facilitation
of instruction on multiplication.
Title of the Unit is Multiplication
 Modify
Lessons within a Unit

Title: Units of Measure
 Lesson
1:
 Lesson 2:
 Lesson 3:
 Lesson 4:
 Lesson 5:
 Lesson 6:
 Lesson 7:
Converting Customary Units of Length
Converting Customary Units of Capacity
Converting Customary Units of Weight
Converting Metric Units of Length
Converting Metric Units of Capacity
Converting Metric Units of Weight
Problem Solving: Multi-Step Problems
What is Problem Solving?

Solving problems is not only a goal of learning
mathematics but also a major means of doing so…
Problem solving is an integral part of all
mathematics learning, and so it should not be an
isolated part of mathematics program. Problem
solving in mathematics should involve all the five
content areas described in these standards …
Good problems will integrate multiple topics and
will involve significant mathematics.
-NCTM, Principles and Standards for School Mathematics. (2000), p.52
Math SHIFTS
Mathematical Practices
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Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning
of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning
Questions….


How are you providing an opportunity for your
students to apply and explain the mathematical
practices?
How are you including writing in your math
program?
Steal Time from ELA
Steal Time from ELA
Writing for Problem Solving in Math
Class
“When students write, their papers provide a window
into their understandings, their misconceptions, and
M. Burns, “Writing in Math,” Education Leadership, 62(2)
(October 2004), p.30
Rationale
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Organize and consolidate their mathematical
thinking through communication
Communicate mathematical thinking coherently and
clearly to others
Analyze and evaluate mathematical thinking and
strategies of others
Use the language of mathematics to express
mathematical ideas precisely.
NCTM, Principles and Standards for School Mathematics. (2000)
Writing for Understanding
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Sentence 1 summarize in 15 words or less
Summarize sentence 1 and 2 in 15 word or less
Summarize sentence 1, 2 and 3 in 15 words or less
Continue process for paragraph
Problem: Writing for Understanding
Activity
As I watch the snow pile up outside, I’m dreaming of the sunny
weather in California. Planning an imaginary trip across the
country from Allentown, PA, to Los Angeles, CA, will help keep me
warm. I could use your help.
According to MapQuest, the total driving distance is 2,688.2
miles. With side trips and allowing for getting lost we will
actually cover 2,800 miles. At times we will be on highways
where we can travel from 50-70 miles per hour. At other times we
will be on city streets with a speed limit of 35 miles per hour.
Problem Continued
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To help with calculation use the data below:
Travel 35 miles per hour for five hours
Travel 50 miles per hour for fourteen hours
Travel 65 miles per hour for fourteen hours
Travel 70 miles per hour for the remaining distance
If we spend 8 hours a day driving, how many days
will the trip take? How long will we drive the last day?
Bonus
Following the MapQuest route, name the eleven states
in order from Pennsylvania to California that you
would travel through on this imaginary trip
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Does this problem promote application of the
mathematical ideas presented in the current instructional
focus or unit of study?
Does this problem match students’ current instructional
level?
Is this problem accessible to all students?
Is the problem relevant and engaging to students?
Does this problem require students to “stretch” their
mathematical reasoning abilities?
(cont)
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Does this problem involve more than one strand or
standard of mathematics?
Is there more than one way to solve the problem?
Could the problem be extended or enriched?
Do I fully understand the mathematics in this
problem, so that I can better facilitate student
understanding?
Goal for Problem Solving at the

Ultimate goal for Intermediate grade students is for
them to be able - by the end of the school year –
to solve a multistep problem and to communicate
verbally and in writing the process they used.
Scaffolds to Meet Goal
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Model the process for the whole class until familiar
Student solve the selected problems in cooperative
groups or selected teams
Partner students to solve a given problem together
We expect students to be able to complete the
entire process independently
Problem Solving Sequence

Whole class problem solving at beginning of the
year
 Gain
experience and capacity for problem solving
 Learn the writing
 Interact with peers
 Communicate mathematically

Transition to cooperative learning groups, partners,
and individual practice
Cooperative Teams

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Teacher arranges the students into small
cooperative groups
Guided by teachers, students attempt to solve a
problem
Using words, pictures, and/or numbers, the students
complete the Data Sheet
The students then complete then work onto Chart
Paper
Students share out to the rest of the class
Benefits of Cooperative Teams
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Different teams will describe their process
differently and thus add to everyone’s
understanding of the problem
Examples around the room will provide references
for the next solving activity
Math team posters are updated regularly which
keeps student interest in the problem solving process
Find all the different ways you can divide equally
144 marbles into bags.
Data Sheet:
 Show work in words, drawings, numbers …..
 Write up:
 Paragraph
1: Problem Statement
 Paragraph 2: Work Write Up
Problem Solving Task Write Up Guide
The total bill for Cassie and Brooke’s dinner was
\$18.40. They want to leave a 15% tip. How should
they determine how much tip to leave? What is the
total amount that Cassie and Brooke spent for dinner?
Problem Solving Task Write Up Guide
Directions
 Write your name on a piece of paper
 Solve the problem using words, pictures, and / or
numbers
 Number each step as you work to solve the problem
 Write a number sentence to match your problem
 Now write three paragraphs describing the problem,
steps used to solve the problem, and explaining your
 Use math vocabulary
Assessing Student’s Problem Solving
What are the……

Strengths

Challenges
of problem solving?
Assessing: Elements to Consider…

What particular elements do you want to see the
students include in their problem solving?
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Calculation Errors
Math Vocabulary
Strategies used to solve
Spelling and Conventions
Fluency of Explanations
Guidelines for Rubrics

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Performance levels – how many?
Publish and distribute to students prior to them
beginning the problem solving process
Problem Solving Rubric Activity
Problem: Mario has a wall in his room that measures
13 feet long and 8 ½ feet high and is freshly
painted. He wants to hang his favorite posters on the
wall. Each poster measures 3 feet long and 2 feet
high. What is the greatest number of posters that he
can hang on the wall so that the posters do not
overlap?
Problem Solving Rubric Activity (cont)


Create a rubric or guidelines for scoring
Include levels and criteria for each level (first draft)
Problem Solving Rubric Activity (cont)
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
4 levels
 Exemplary
 Proficient
 Progressing
 Beginning
 Self
Evaluation
 Teacher Evaluation
Let’s Create
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

Find or create a word problem that meets the
criteria discussed
Create an appropriate data sheet and write up
sheet for the problem
Create an appropriate rubric, include levels and
criteria for each level
Questions or Concerns

If you have any questions or concerns, please feel
Andrea Tamarazio
[email protected]
Steve Graser
[email protected]
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