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Math Toolkit Time!
Using Manipulatives to Help Students Develop Common Core Math Mastery
Jacqueline Burns, Global Mathematics Consultant11 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 Agenda9:00am – 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 OneTeaching 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 & LearningExemplar 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 BahrainBahrain is actually an archipelago of 33 islands.The largest, on which many people live, is 55km long by18km wide and areas such as Sitra and Muharraq areactually 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 and97.1m wide. The five red points signify the five pillars ofIslam. The oldest known Bahraini flags were plain red. In1820, the island signed a treaty with the UK and a whitestripe was added to the flag to indicate the truce. In 1932,a serrated edge was added to distinguish Bahrain’s flagfrom those of its neighbours. The flag originally had 28white points, but this was reduced to eight in 1972 and fivein 2002.
Source: http://www.timeoutbahrain.com/aroundtown/features/40006-25-amazing-bahrain-facts
More Math Connections in BahrainBahrain World Trade Centre is the first skyscraper inthe world to integrate wind turbines into its design.Each turbine is 29m in diameter and their capacity is675kw of wind power production. The 240m-tall towerhas won several international awards including theLEAF (Leading European Architects’ Forum) award forbest use of technology in a large scheme.
The British School of Bahrain holds the world recordfor the largest simultaneous coin toss. As part ofWorld Maths Day in 2010, 1,117 staff and studentstook part in the toss which made it into the GuinnessBook 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 + ? = 11Join 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-LessonComponents 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 tofurther 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 FractionsNF 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 adjectives1/2 of a pizza3/4 of an hour2/3 of a cup
• to considering them as nouns5/8 is…
a little more than 1/2, but less than 1It is 3/8 less than 1It is equivalent to 10/16It is twice 5/16It is half of 1¼ ……………?
Which is bigger . . . 𝟏
𝟑𝒐𝒓
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?
