Putting the Pieces Together: Literacy, Modeling, and Problem-Solving for Fraction Instruction (Grades 3 – 5) Milligan College, 2015 ITQ Grant Program Dr.
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Putting the Pieces Together: Literacy, Modeling, and Problem-Solving for Fraction Instruction
(Grades 3 – 5)Milligan College, 2015 ITQ Grant Program
Dr. Lyn HowellDr. Angela Hilton-Prillhart
Dr. Jamie Price
June 15-19, 2015
Monday, June 15 8:30 AM – 11:30 AM: Morning Session (Whole Group) 11:30 AM – 1:00 PM: Lunch (on your own) 1:00 PM – 4:00 PM: Accountable Talk/Lesson Planning
Introduction Tuesday, June 16 – Thursday, June 18
8:30 AM – 11:30 AM: Morning Sessions 8:30 AM – 10:00 AM: Math Workshop (Group A/B) 10:00 AM – 11:30 AM: Literary Workshop (Group A/B) 11:30 AM – 1:00 PM: Lunch (on your own) 1:00 PM – 4:00 PM: Lesson Planning/Afternoon math
workshop
Agenda for the Week
Friday, June 19 8:30 AM – 10:30 AM – Post test/Concept Maps 10:30 AM – 11:30 AM – Teacher Group
Presentations 11:30 AM – 1:00 PM – Lunch (on your own) 1:00 PM – 4:00 PM – Teacher Group
Presentations
Agenda for the Week
Keep students at the center.
Be present and engaged.
Monitor air time and share your voice.
Challenge with respect.
Stay solutions oriented.
Risk productive struggle.
Balance urgency and patience.
BE OPEN TO NEW IDEAS!
Workshop Expectations
1. Concrete: At this stage, students are introduced to a new concept with the aid of manipulatives/hands-on work
2. Pictures (Representational): At this stage, students are able to draw pictures to explain their reasoning used to solve a problem. Students may draw pictures to indicate what they would have done with manipulatives.
3. Symbols (Abstract): This is the most abstract stage. Students are able to use symbols (numbers, operation signs, algorithms, etc.) to solve the problem.
Stages for Learning Mathematics
Stages for Learning Mathematics
While the stages for learning should progress in order as students learn a concept, once students reach the symbol (abstract) stage, they should understand the relationship between the symbols and the previous two stages.
Adding It Up Framework(Adding It Up: Helping Children Learn
Mathematics, NRC 2001)
Five Strands of Mathematical Proficiency
Conceptual Understanding—comprehension of mathematical concepts, operations, and relations Students know more than just isolated facts and
methods Students understand why a mathematical idea is
important and the kinds of contexts in which it is useful
Students have organized their knowledge into a coherent whole
Conceptual understanding supports retention; students can reconstruct facts and methods that are forgotten when needed (p. 118)
Five Strands of Mathematical Proficiency
Procedural Fluency – knowledge of procedures, when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently Students need to be efficient and accurate in
performing basic mathematical computations Students need to be able to estimate the result
of a procedure Students use a variety of mental strategies to
solve various problems (p. 121)
Five Strands of Mathematical Proficiency
Strategic Competence – ability to formulate mathematical problems, represent them, and solve them Students must first understand the situation and
determine the key features Generate a mathematical representation of the
problem that captures the key features and ignores irrelevant ones (drawing, equation, graph, etc.)
Students come up with multiple approaches to solving the problem and choose flexibly among various approaches (reasoning, algebraic, guess and check) (p. 124)
Five Strands of Mathematical Proficiency
Adaptive Reasoning – capacity to think logically about the relationships among concepts and situations Adaptive Reasoning is the glue that holds
everything together Includes not only informal explanation and
justification of a solution, but also intuitive and inductive reasoning based on pattern, analogy, and metaphor
Ability to justify one’s work (p. 124)
Five Strands of Mathematical Proficiency
Productive Disposition – refers to the tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics If students are to develop in any of the other
strands of proficiency, then they must possess a productive disposition towards mathematics
A productive disposition develops when the other strands do and helps each of them develop (p. 131)
Five Strands of Mathematical Proficiency
Examining Student Work
Consider the student responses to the problem on the back of the Adding It Up Handout.
What mathematical strands of proficiency are shown in each student’s response?
What mathematical strands of proficiency appear to be missing?
Reflecting On Your Practice
Take a few minutes to reflect on your own practice, especially related to teaching fractions.
Within your own practice, do you feel that you help students develop all strands of mathematical proficiency related to understanding fractions?
Are there particular strands that you feel that you develop more than others? Why do you think this is so?
Are there particular strands that you feel you never really develop? Why do you think this is so?
Other comments/thoughts?
Keisha receives her paycheck for the month. She spends 1/6 of it on food. She then spends 3/10 of what remains on her mortgage payment. She spends 3/7 of what is now left for her other bills, and 5/8 of what now remains for entertainment. This leaves her with $300. What was her original monthly take-home pay?
Keisha’s Paycheck
The rectangle represents the paycheck (our whole).One of the six parts is shaded yellow to represent the amount Keisha pays for food.5 equal size pieces remain.
Cut each piece in half to get 10 white pieces (12 pieces now in the whole).
The yellow (food) section is now divided into two pieces and represents 1/6 or 2/12.
food
Rent
Food
Of the 7 pieces remaining (white pieces), Keisha spends 3/7 on other bills. Shade 3 white pieces pieces to represent other.
What fraction of the whole paycheck is represented by food? What fraction of the whole paycheck is represented by rent?
Food
Rent
Other
4 white pieces remain. But, 5/8 of the remaining is spent on entertainment. What do I do?
Rent
Food
Fun
Other
I need 8 pieces, so each piece is cut in half again.Five are shaded blue. There are 3 white pieces remaining.
How many pieces are now in the whole? What fraction of the whole paycheck is represented by food? Rent? Other? Fun?
$300
Rent
Food
Fun
Other
The 3 white pieces represent $300 left from her paycheck. How much is each white piece?
Keisha’s Paycheck and Fractions
Consider the Keisha’s Paycheck problem. What concepts related to fractions are
addressed in this single problem?
Keisha’s Paycheck and Mathematical Proficiency
Consider the various solutions presented to the Keisha’s paycheck problem.
What strands of mathematical proficiency are addressed in each solution?
Understanding Fractions
Suppose you have a fraction of the form A . B
We call the value of A the numerator of the fraction and we call the value of B the denominator of the fraction.
What do the numerator and denominator represent in a given fraction problem?
Types of Fraction ModelsTypes of Fraction ModelsModel Example Description
Area/Region
2/5 of the picture is blue
Set
2/5 of the counters are red
Length
The object is 2/5 of a unit long
1 0
Area Model vs. Set Model
What are some similarities to an area model for fractions and a set model for fractions?
What are some differences between these two models?
Which model was represented by the picture solution to the Keisha’s paycheck problem?
Fraction Hexagon Task Handout
Fractions as Parts of Sets
Lesson One from Lessons for Introducing Fractions by Marilyn Burns (Teaching Arithmetic Series)
For Tuesday, read selections from Chapter 7 of Adding it Up: Helping Children Learn Mathematics Read pp. 231 – 241 (stop at Proportional
Reasoning) Read pp. 246- 247 (section titled Beyond
Whole Numbers) Be ready to have a discussion of the reading in
your math session on Tuesday
A mathematical task is a problem or set of problems that focuses students’ attention on a particular mathematical idea and/or provides an opportunity to develop or use a particular mathematical habit of mind.
from http://commoncoretools.me
What is a math task?
A high-quality math task has the following characteristics: Aligns with relevant mathematics content
standard(s) Encourages the use of multiple representations Provides opportunities for students to develop
and demonstrate the mathematical practices Involves students in an inquiry-oriented or
exploratory approach
What are the characteristics of a high-quality math task?
Allows entry to the mathematics at a low level (all students can begin the task) but also has a high ceiling (some students can extend the activity to higher-level activities)
Connects previous knowledge to new learning Allows for multiple solution approaches and
strategies Engages students in explaining the meaning of
the result Includes a relevant and interesting contextfrom Putting Essential Understanding of Fractions
into Practice (3-5), p. 8
What are the characteristics of a high-quality math task?
Work with the members of your team to create a task-based lesson plan which satisfies the following: Addresses at least one Math Common Core State
Standard from your grade level related to fractions Addresses multiple Mathematical Practices Problem for the task must relate to a chosen piece
of literature selected by your team Problem for the task must be original work Complete the template provided Be prepared to present your task with the members
of your team on Friday morning or afternoon
Your Task-Based Lesson Plan
Requirements/Expectations
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