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Math Toolkit Time!
Using Manipulatives to Help Students Develop
Common Core Math Mastery
Jacqueline Burns, Global Mathematics Consultant
11 March 2017
Bahrain
Today’s Objectives
• Increase awareness of the purpose and benefits of using math manipulatives for
students to develop conceptual understanding
• Work with tools and explore teaching and learning strategies to help students
understand numbers less than one
• Broaden awareness of and access to a variety of high quality resources to support
teaching and learning of mathematics
Session Agenda
9:00am – 9:45am Math Toolkit Time
Using Manipulatives to Help Students Develop Common Core Mastery
9:45-10:30am Alpha Math
10:30-10:45am Coffee Break
10:45-11:15am e-Alpha
11:15-12:00pm Digging Deeper: Numbers less than One
Teaching and learning experiences with quantity and numbers less than one
12:00pm – 1:00pm Lunch
1:00pm – 2:45pm Resources and Tools for 21st Century Teaching & Learning
Exemplar tasks, websites, and strategies for today's classroom
2:45pm – 3:00pm Closing and Certificates
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.
Math Facts about Bahrain
Bahrain is actually an archipelago of 33 islands.
The largest, on which many people live, is 55km long by
18km wide and areas such as Sitra and Muharraq are
actually separate islands.
Bahrain’s flag used to be the largest flag in the world,
setting a Guinness world record in 2004 at 169.5m long and
97.1m wide. The five red points signify the five pillars of
Islam. The oldest known Bahraini flags were plain red. In
1820, the island signed a treaty with the UK and a white
stripe was added to the flag to indicate the truce. In 1932,
a serrated edge was added to distinguish Bahrain’s flag
from those of its neighbours. The flag originally had 28
white points, but this was reduced to eight in 1972 and five
in 2002.
Source: http://www.timeoutbahrain.com/aroundtown/features/40006-25-amazing-bahrain-facts
More Math Connections
in Bahrain
Bahrain World Trade Centre is the first skyscraper in
the world to integrate wind turbines into its design.
Each turbine is 29m in diameter and their capacity is
675kw of wind power production. The 240m-tall tower
has won several international awards including the
LEAF (Leading European Architects’ Forum) award for
best use of technology in a large scheme.
The British School of Bahrain holds the world record
for the largest simultaneous coin toss. As part of
World Maths Day in 2010, 1,117 staff and students
took part in the toss which made it into the Guinness
Book of Records.
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
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 = ?
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 5 groups of 7 counters and
counts them all.
7, 14, 21, 28, 35 or
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.
Source: https://catalyst.uw.edu/gopost/conversation/ekazemi/117145
Children’s Strategies for Solving Basic Facts
Create a Mini-Lesson
Components for the Mini-Lesson
• at least one math manipulative
• CCSS-M (math standard)
• Standard(s) for Mathematical Practice
• Interdisciplinary connection/s
• Independent/pair/small group/whole
group?
• How will students communicate
mathematically?
Manipulatives for
Consideration
• Counters
• Connecting cubes
• Base ten counters
• Number line
• Random number generator
(number cubes)
• Spinners
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
“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
What are some of the biggest challenges
students face with fractions?
In One Minute…
• write down everything that comes to mind when you think about or see
5
8
Now, in one minute…
• write down everything that comes to mind when you think about or see
9
Compare Your Responses
• What similarities do you see?
• What differences do you see?
• Any surprises or insights?
The Progression of Fractions
NF Standards, Grades 3-5
Prior knowledge through the lens of geometry
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.
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 .
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 .
Other Helpful Considerations for Understanding
Fractions
• 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.
• 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.
Equivalent Fractions
• The meaning of fraction equivalence
• The equivalence of whole numbers and fractions
• Explaining fraction equivalence in general
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,
• ………..?
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 5/16
It is half of 1¼
……………?
Which is bigger . . .
𝟏
𝟑
𝒐𝒓
𝟏
𝟖
?
Source - https://www.youtube.com/watch?v=g0nuomCCu9A
It’s Your Turn!
Create a Lesson that supports, develops, reinforces, applies
or extends understanding of fractions
Components for the
Mini-Lesson
• at least one math manipulative
• CCSS-M (math standard)
• Standard(s) for Mathematical
Practice
• Interdisciplinary connection/s
• Independent/pair/small
group/whole group?
• How will students communicate
mathematically?
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
Interactive Resources and eResources
• http://nrich.maths.org/
• www.illustrativemathematics.org
• http://achievethecore.org/category/854/mathematics-lessons
• National Library of Virtual Manipulatives
• https://illuminations.nctm.org
• K-2 Algebra Placement Assessments
• http://www.mathplayground.com/math_manipulatives.html
• https://www.pbslearningmedia.org/asset/mgbh_int_balance/
• Plickers
• Alpha Math Grade 1, Grade 2, Grade 3, Grade 4, and Grade 5
Closing
• Processing, Next Steps, Reflections, Questions
"The essence of mathematics is not to make simple things
complicated, but to make complicated things simple."
– Stan Gudder, American mathematician
• Written reflection: Write one action you will take as a result
of today’s exploration of algebraic thinking.
Questions?
Thank you!
EM: JacquelineBurns@gmail.com
WhatsApp: +971 56 4166150
Twitter: @GlobalMath411

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Math Toolkit Time

  • 1. Math Toolkit Time! Using Manipulatives to Help Students Develop Common Core Math Mastery Jacqueline Burns, Global Mathematics Consultant 11 March 2017 Bahrain
  • 2. Today’s Objectives • Increase awareness of the purpose and benefits of using math manipulatives for students to develop conceptual understanding • Work with tools and explore teaching and learning strategies to help students understand numbers less than one • Broaden awareness of and access to a variety of high quality resources to support teaching and learning of mathematics
  • 3. Session Agenda 9:00am – 9:45am Math Toolkit Time Using Manipulatives to Help Students Develop Common Core Mastery 9:45-10:30am Alpha Math 10:30-10:45am Coffee Break 10:45-11:15am e-Alpha 11:15-12:00pm Digging Deeper: Numbers less than One Teaching and learning experiences with quantity and numbers less than one 12:00pm – 1:00pm Lunch 1:00pm – 2:45pm Resources and Tools for 21st Century Teaching & Learning Exemplar tasks, websites, and strategies for today's classroom 2:45pm – 3:00pm Closing and Certificates
  • 4. 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.
  • 5. Math Facts about Bahrain Bahrain is actually an archipelago of 33 islands. The largest, on which many people live, is 55km long by 18km wide and areas such as Sitra and Muharraq are actually separate islands. Bahrain’s flag used to be the largest flag in the world, setting a Guinness world record in 2004 at 169.5m long and 97.1m wide. The five red points signify the five pillars of Islam. The oldest known Bahraini flags were plain red. In 1820, the island signed a treaty with the UK and a white stripe was added to the flag to indicate the truce. In 1932, a serrated edge was added to distinguish Bahrain’s flag from those of its neighbours. The flag originally had 28 white points, but this was reduced to eight in 1972 and five in 2002. Source: http://www.timeoutbahrain.com/aroundtown/features/40006-25-amazing-bahrain-facts
  • 6. More Math Connections in Bahrain Bahrain World Trade Centre is the first skyscraper in the world to integrate wind turbines into its design. Each turbine is 29m in diameter and their capacity is 675kw of wind power production. The 240m-tall tower has won several international awards including the LEAF (Leading European Architects’ Forum) award for best use of technology in a large scheme. The British School of Bahrain holds the world record for the largest simultaneous coin toss. As part of World Maths Day in 2010, 1,117 staff and students took part in the toss which made it into the Guinness Book of Records.
  • 7. 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
  • 8. 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 = ?
  • 9. 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 5 groups of 7 counters and counts them all. 7, 14, 21, 28, 35 or 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. Source: https://catalyst.uw.edu/gopost/conversation/ekazemi/117145 Children’s Strategies for Solving Basic Facts
  • 10. Create a Mini-Lesson Components for the Mini-Lesson • at least one math manipulative • CCSS-M (math standard) • Standard(s) for Mathematical Practice • Interdisciplinary connection/s • Independent/pair/small group/whole group? • How will students communicate mathematically? Manipulatives for Consideration • Counters • Connecting cubes • Base ten counters • Number line • Random number generator (number cubes) • Spinners
  • 11. 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
  • 12. “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
  • 13. What are some of the biggest challenges students face with fractions?
  • 14. In One Minute… • write down everything that comes to mind when you think about or see 5 8
  • 15. Now, in one minute… • write down everything that comes to mind when you think about or see 9
  • 16. Compare Your Responses • What similarities do you see? • What differences do you see? • Any surprises or insights?
  • 17. The Progression of Fractions NF Standards, Grades 3-5
  • 18. Prior knowledge through the lens of geometry
  • 19. 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.
  • 20. 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 .
  • 21. 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 .
  • 22. Other Helpful Considerations for Understanding Fractions • 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. • 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.
  • 23. Equivalent Fractions • The meaning of fraction equivalence • The equivalence of whole numbers and fractions • Explaining fraction equivalence in general
  • 24. 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, • ………..?
  • 25. 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 5/16 It is half of 1¼ ……………?
  • 26. Which is bigger . . . 𝟏 𝟑 𝒐𝒓 𝟏 𝟖 ? Source - https://www.youtube.com/watch?v=g0nuomCCu9A
  • 27. It’s Your Turn! Create a Lesson that supports, develops, reinforces, applies or extends understanding of fractions Components for the Mini-Lesson • at least one math manipulative • CCSS-M (math standard) • Standard(s) for Mathematical Practice • Interdisciplinary connection/s • Independent/pair/small group/whole group? • How will students communicate mathematically?
  • 28. 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
  • 29. Interactive Resources and eResources • http://nrich.maths.org/ • www.illustrativemathematics.org • http://achievethecore.org/category/854/mathematics-lessons • National Library of Virtual Manipulatives • https://illuminations.nctm.org • K-2 Algebra Placement Assessments • http://www.mathplayground.com/math_manipulatives.html • https://www.pbslearningmedia.org/asset/mgbh_int_balance/ • Plickers • Alpha Math Grade 1, Grade 2, Grade 3, Grade 4, and Grade 5
  • 30. Closing • Processing, Next Steps, Reflections, Questions "The essence of mathematics is not to make simple things complicated, but to make complicated things simple." – Stan Gudder, American mathematician • Written reflection: Write one action you will take as a result of today’s exploration of algebraic thinking.
  • 32. Thank you! EM: JacquelineBurns@gmail.com WhatsApp: +971 56 4166150 Twitter: @GlobalMath411