This document outlines an agenda for a workshop on using manipulatives to help students develop mastery of Common Core math standards. The agenda includes an introduction, discussing the importance of manipulatives, examining PISA test results, exploring manipulatives for operations like addition and multiplication, creating mini-lessons, discussing fraction sense, and having a question and answer period. Participants will explore different manipulatives and create mini-lessons incorporating them. The document provides examples of how students can use manipulatives to model solutions for basic math facts and strategies for fractions. Guidelines are given for implementing the concrete-representational-abstract approach to teaching math concepts.

1. Math Toolkit Time!
Using Manipulatives to Help Students Develop
Common Core Math Mastery
Jacqueline Burns, Global Mathematics Consultant

2. Session Agenda
• Introduction & Norms
• Importance of Manipulatives
• PISA
• Whole number computation and math tools (+, −, ×, ÷)
• Mini-lesson creation
• Fraction sense
• Math tool exploration
• Mini-lesson creation
• Debrief
• Q&A

3. Our Norms
• Be present both physically and
mentally.
• Listen to, and make room for, the
ideas of others.
• Share your knowledge and
wisdom.
• Table side topics.
• Exercise mobile phone etiquette.

4. Why use manipulatives?
Concrete.
The “doing” stage using
concrete objects to model
problems
Representational.
The “seeing” stage using
representations of the objects
to model problems
Abstract.
The “symbolic” stage using
abstract symbols to model
problems

5. What PISA15 Says About Math
Globally, and in Jordan
From http://www.keepeek.com/Digital-Asset-Management/oecd/education/pisa-2015-results-volume-i_9789264266490-en#page178
Mathematics Performance among PISA 2015 participants,
at national and subnational levels
Country Mean performance score
Singapore 564
Jordan 380
Highest mean performance score – 564
Lowest mean performance score -328

6. What PISA15 Says About Math
Globally, and in Jordan
From http://www.keepeek.com/Digital-Asset-Management/oecd/education/pisa-2015-results-volume-i_9789264266490-en#page181
Mean
Performance
2012
Mean
Performance
2015
386 380

7. Solving Basic Facts: Partner Work
The Student’s Hat
Work on the problems, using
manipulatives to concretely
model the solution.
The Teacher’s Hat
Consider student abilities:
Direct modeling,
counting, derived
facts, recall
PROBLEMS
5 + 7 = ?
12 – 5 = ?
4 + ? = 11
5 x 7 = ?
56 ÷ 8 = ?

8. PROBLEM DIRECT MODELING COUNTING DERIVED FACTS RECALL
5 + 7 = ?
Join Result
Unknown
Makes a set of 5 counters and a set
of 7 counters. Pushes the two sets
together and counts all the
counters.
Counts “5 [pause], 6, 7, 8, 9, 10, 11,
12,” extending a finger with each
count. “The answer is 12” [The
counting sequence may also begin
with the larger number]
“Take 1 from the 7 and give
it to the 5. That makes 6 + 6,
and that’s 12.”
5 plus 7 is 12.
12 – 5 = ?
Separate Result
Unknown
Makes a set of 12 counters and
removes 5 of them. Then counts
the remaining counters.
Counts back “12, 11, 10, 9, 8
[pause], 7. It’s 7.” Uses fingers to
keep track of the numbers of steps
in the counting sequence.
“12 take away 2 is 10, and
take away 3 more is 7.”
12 take away 5 is 7.
4 + ? = 11
Join Change
Unknown
Makes a set of 4 counters. Makes
a second set of counters, counting
“5, 6, 7, 8, 9, 10, 11,” until there is
a total of 11 counters. Counts the
7 counters in the second set.
Counts “4 [pause], 5, 6, 7, 8, 9, 10,
11,” extending a finger with each
count. Counts the 7 extended
fingers. “It’s 7.”
“4 + 6 is 10 and 1 more is 11.
So it’s 7.”
4 and 7 make 11.
5 x 7 = ?
Makes 7 groups of 5 counters and
counts them all.
5, 10, 15, 20, 25, 30, 35 5 times 5 is 25 and 10 more
is 35.
5 times 7 is 35.
56 ÷ 8 = ?
Counts out 56 counters. Pulls out
groups of 8 until 7 groups are
made.
8, 16, 24, 32, 40, 48, 56 8 times 8 is 64. 8 less is 56.
So that’s 7.
8 x 7 is 56.
Children’s Strategies for Solving Basic Facts

9. With, or Without Manipulatives?…
That is the Question

13. Create a Mini-Lesson
Create a mini-lesson that involves the use of at least
one math manipulative
• CCSS-M
• Standard(s) for Mathematical Practice
• Interdisciplinary connection/s
• Independent/pair/small group/whole group?
• How will students communicate mathematically?
• Identify the manipulative/s and other tools.
Manipulatives
for
Consideration
• Counters
• Connecting cubes
• Base ten counters
• Number line
• Random number
generator
(number cubes)
• Spinners

14. 1. Make sense of problems and
persevere in solving them.
2. Reason abstractly and
quantitatively.
3. Construct viable arguments and
critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express regularity in
repeated reasoning.
Learning through the
Standards for Mathematical Practice

15. Create a Mini-Lesson
Create a mini-lesson that involves the use of at least
one math manipulative
• CCSS-M
• Standard(s) for Mathematical Practice
• Interdisciplinary connection/s
• Independent/pair/small group/whole group?
• How will students communicate mathematically?
• Identify the manipulative/s and other tools.
Manipulatives
for
Consideration
• Counters
• Connecting cubes
• Base ten counters
• Number line
• Random number
generator
(number cubes)
• Spinners

16. In exactly one minute write down all the numbers
you can think of…….
Fraction Challenge
between
0 and 1

17. “The difficulty with fractions (including decimals and
percents) is pervasive and is a major obstacle to
further progress in mathematics. . .”
—Report of the National Math Panel, March 2008
Fraction Sense

18. What are some of the biggest challenges
students face with fractions?

19. In One Minute…
• write down everything that comes to mind when you think about or see
7
8

20. Now, in one minute…
• write down everything that comes to mind when you think about or see
9

21. Compare Your Responses
• What similarities do you see?
• What differences do you see?
• Any surprises or insights?

22. The Progression of Fractions
NF Standards, Grades 3-5

23. Prior knowledge through the lens of
geometry

24. Develop understanding of fractions
• 3.NF1 Understand a fraction 1/b as a quantity formed by 1 part
when a whole is portioned into b equal parts: understand a
fraction a/b as the quantity formed by a parts of size 1/b.

25. TASK:
Locate 1 on the number line. Label the point.
Be exact as possible.
While it is not necessary to name all of the intervals on the number line, many students
may do so .

26. TASK:
Locate 1 on the number line. Label the point.
Be exact as possible.
While it is not necessary to name all of the intervals on the number line,
many students may do so .

27. Other Helpful Considerations for Understanding
Fractions
• Where do we see and use fractions
in our daily lives?
• Support Strategy: The use of
pictures provides students with
realia for understanding
fractions
• Wholes vs. Holes
• Use visual images that support
all learners for understanding
this concept.
• Fraction Basic Vocabulary (denominator)
• 2 is pronounced “half”
• 3 is pronounced “third”
• 4 is pronounced “fourth” (or “quarter”)
• 5 is pronounced “fifth”
• 6 is pronounced “sixth”
• 7 is pronounced “seventh”
• 8 is pronounced “eighth”
• 9 is pronounced “ninth”
• 10 is pronounced “tenth,” and so on.

28. Equivalent Fractions
• The meaning of fraction equivalence
• The equivalence of whole numbers and fractions
• Explaining fraction equivalence in general

29. Adjectives vs. Nouns
(Adapted from Kathy Richardson, NCTM 2008)
• Young children initially consider numbers as adjectives or descriptors
• 9 bears
• 6 cookies
• 20 students
• Eventually, they come to understand numbers as nouns or concepts
• 9 is half of 18,
• It is 1 less than 10,
• It is 4.5 doubled,
• It is 3 squared,
• It is the square root of 81,
• ………..?

30. Adjectives vs. nouns (continued)
• Students need opportunities to transition from considering fractions as adjectives
1/2 of a pizza
3/4 of an hour
2/3 of a cup
• to considering them as nouns
5/8 is…
a little more than 1/2, but less than 1
It is 3/8 less than 1
It is equivalent to 10/16
It is twice or double 5/16
It is half of 1¼
……………?

31. Which is bigger . . .
𝟏
𝟑
𝒐𝒓
𝟏
𝟖
?
Source - https://www.youtube.com/watch?v=g0nuomCCu9A

32. It’s Your Turn!
Create a FRACTIONS Lesson that supports, develops,
reinforces, applies or extends understanding of fractions
• CCSS-M
• Standard(s) for
Mathematical Practice
• Interdisciplinary
connection/s
• Independent/pair/small
group/whole group?
• How will students
communicate
mathematically?
• Identify the
manipulative/s and
other tools.
Math Problem Solving Strategies
Act It Out Give an Opinion
Choose an Operation Make a Graph
Choose a Strategy Make a List
Draw a Diagram Make a Model
Draw a Picture Make a Table
Draw Conclusion Solve a HOD Problem
Find a Pattern Solve a Simple Problem
Find/Create/Make a Clue Write a Math Story

33. Guidelines for Implementing the CRA Approach
• Choose the math topic to be taught. Plan what is to be taught ahead of time. Sequence the
lessons so they start basic and gradually introduce new topics.
• Review abstract steps to solve the problem. Ask, what is the desired math outcome of the
group of lessons? Determine the procedural goal of the combination of math skills. List out
the steps or procedures. Adjust the steps to eliminate notation or calculation tricks. Change or
modify steps to create the most logical and sequential set of procedures.
• Match the abstract steps with an appropriate concrete manipulative. Initial understanding of
content will be based on interactions with concrete objects, so be careful which ones you
choose. The conceptual effectiveness of the manipulative object should be noted in
accordance to the math skill being taught. Avoid concrete objects that only cover a few skills.
• Arrange concrete and representational lessons. Practice concrete manipulations. The same
questions that you encounter you can be certain your students will as well. Practice how to
mark pictorial representations that appear similar to concrete manipulations. Make certain
that your language throughout instruction matches the language required for the desired
outcome.

34. Guidelines, continued
• Teach each concrete, representational, and abstract lesson to student mastery
(accuracy without hesitation). Model and guide students in their use of
manipulative objects and pictorial representations. Teach students step by step
gradually introducing mathematical vocabulary. Allow students to name or invent
their stepwise procedures within instruction. Move from concrete to
representational to abstract learning levels only after students show accuracy
without hesitations in manipulations or drawings.
• Assess each level of learning according to stepwise procedures. Help students
generalize learning through word problems and problem solving events.
Incorporate word problems throughout a lesson to help show social relevance as to
why a math skill is important to learn.
Source - http://nycdoeit.airws.org/pdf/Concret-Representational-Abstraction%20Approach%20for%20Expressions.pdf

35. What is fraction sense?
“Fraction sense implies a deep and flexible understanding of fractions that is not
dependent on any one context or type of problem. Fraction sense is tied to
common sense: Students with fraction sense can reason about fractions and don’t
apply rules and procedures blindly; nor do they give nonsensical answers to
problems involving fractions.”
Julie McNamara

36. Promoting Student Learning and Motivation
Characteristics associated with effective mathematics instruction:
• Students are actively engaged in doing mathematics.
• Students are solving challenging problems.
• Interdisciplinary connections and examples are used to teach mathematics.
• Students are sharing their mathematical ideas while working in pairs and groups.
• Students are provided with a variety of opportunities to communicate mathematically.
• Students are using manipulatives and other tools.
Note: Adapted from “What Does Good Math Instruction Look Like?” by Nancy Protheroe, September/October 2007, Principal. Retrieved from
https://www.naesp.org/resources/2/Principal/2007/S-Op51.pdf

37. An Effective Classroom for Mathematics
In effective classrooms, teachers:
• demonstrate acceptance of students’ divergent ideas.
• influence learning by posing challenging and interesting questions.
• project a positive attitude about mathematics and about students’ ability to
“do” mathematics.
Note: Adapted from “What Does Good Math Instruction Look Like?” by Nancy Protheroe, September/October 2007, Principal. Retrieved from
https://www.naesp.org/resources/2/Principal/2007/S-Op51.pdf

38. The Critical Language of Learning
*Taken from Teaching the Critical VOCABULARY of the Common Core: 55 Words That Make or Break Student Understanding
by Marilee Sprenger (2013)
· Kindergarten: compare, contrast, describe, distinguish, identify, retell
· 1st: demonstrate, determine, draw, explain, locate, suggest, support
· 2nd: comprehend, develop
· 3rd: organize, refer
· 4th: infer, integrate, interpret, paraphrase, summarize
· 5th: analyze
· 6th: articulate, cite, delineate, evaluate, trace

39. 1. Make sense of problems and
persevere in solving them.
2. Reason abstractly and
quantitatively.
3. Construct viable arguments and
critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express regularity in
repeated reasoning.
Learning through the
Standards for Mathematical Practice

40. Need Interactive and eResources?
• http://www.debbiewaggoner.com/math.html
• http://http://www.readtennessee.org/
• http://www.corestandards.org
• https://ccgpsmathematicsk-5.wikispaces.com/
• http://www.k-5mathteachingresources.com/
• http://www.insidemathematics.org

41. Questions?

42. Thank you!
EM: JacquelineBurns@gmail.com
WhatsApp: +971 56 4166150
Twitter: @GlobalMath411