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Building Understanding of the Number System Through
Hands-On Experiences
Marcia TorgrudeK-12 Math [email protected]
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• Develop understanding and ideas to promote deeper understanding of the number system.
• Develop hands-on strategies to help promote understanding of fractions.
• Use tools to help students work fluently with rational numbers.
• Experience online tools for the number system
Outcomes for Today
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I Have….. Who Has• Let’s play!• What does this have to do with
learning?• Where does it fit the common
core standards?• What about the Standards of
Mathematical Practice?• Search I Have…Who Has online.
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Standards of Mathematical Practice
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Common Core Refresher• Person facing screen – gives clues• Person facing away – guesses the word or
phrase
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Common Core
• Reasoning• Tools• Abstract• Precision• Structure
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Common Core
• Standards of Mathematical Practice
• Persevere• Questioning• Concrete• Modeling
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What does it mean to “do mathematics?”
The Practice Standards are descriptions of the fundamental skills needed to “do” mathematics.
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What does it mean to “do mathematics?”
• Practice Standards describe what it means for students to demonstrate proficiency in mathematics. They are our new “basic skills.”
• Content Standards are the “what” of mathematics
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We must get past the idea of mathematics as a collection of algorithms, steps, or procedures.
Just getting answers, although important, is not “doing mathematics.”
“Doing mathematics”
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Working with Whole Numbers
• Adding• Subtracting• Multiplying• Dividing
With Base 10 Blocks
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“Doing mathematics”
Using Modeling to Make Sense of
Mathematical Procedures
Modeling addition with Base 10 blocks
Race to 100
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“Doing mathematics”
Using Modeling to Make Sense of
Mathematical Procedures
Modeling subtraction with Base 10 blocks
302 − 178
412 - 189
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Multiplication and Division
• Identify strategies that individuals can use to solve multi-digit multiplication and division problems in sense-making ways
• Connect concepts to “standard algorithms”
• Discuss teaching strategies that enhance a child’s understanding
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Practice Concrete Multiplication
What does multiplication look like using base ten blocks?
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Let’s Try this without the blocks
143x23
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Making Connections through diagram – 23 x 143
2000 800 60
300 120 9
100 + 40 + 3
20
+
3
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Making Connections –Getting to the algorithm
300+20+6X 10+9 3000 200 60
2700 180 54 6194
326 x 19 54 180 2700 60 200 3000 6194
326 x 19 2934 326 6194
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Understanding the abstract
Do you think that using base-10 blocks helps to give meaning to the multiplication algorithm? How?
One common concern when using models is that students will not make connections between the concrete models, their representations, and the mathematical concept. Base-10 blocks as an area model emphasize the distributive property and provide a visual representation to the partial products of the multiplication algorithm.
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Practice Concrete Division
What does division look like using base ten blocks?
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American Idol is back! If they travel to 11 different cities and can only take a total of 132 people to Hollywood, how many people can be selected from each city?
How can we use the base ten block array model to help us with division?
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Understanding the abstract
Do you think that using base-10 blocks helps to give meaning to the division algorithm? How?
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Another Strategy for Division
Use of friendly or “benchmark” numbers
Partial quotient divisionMultiplication for division – use what we know
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Virtual Whole Number Tools• http://illuminations.nctm.org/
– Activities– Calculation Nation
• Tens Frame - http://illuminations.nctm.org/ActivityDetail.aspx?ID=75• Grouping and Grazing - http://
illuminations.nctm.org/ActivityDetail.aspx?ID=218• Adding with base 10 Blocks - http://
nlvm.usu.edu/en/nav/frames_asid_154_g_1_t_1.html?from=category_g_1_t_1.html
• Subtracting with Base 10 Blocks - http://nlvm.usu.edu/en/nav/frames_asid_155_g_1_t_1.html?from=category_g_1_t_1.html
• Primary Krypto - http://illuminations.nctm.org/ActivityDetail.aspx?ID=173• Product Game - http://illuminations.nctm.org/ActivityDetail.aspx?ID=29• Times Table -http://
illuminations.nctm.org/ActivityDetail.aspx?ID=155
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What were the goals of the activities?
What common core standards have we been working on?
What Standards of Mathematical Practice were present during the activities?
Small Group Discussion
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Working with Fractions
• Equivalence• Addition• Subtraction• Multiplication• Division
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Fraction Equivalence, Adding, and Subtracting
• Using Pattern Blocks• Using Cuisenaire Rods
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Fraction Multiplication
• Using Cuisenaire Rods• Using Arrays
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Fraction Division
• Using Cuisenaire Rods• Using Sets
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http://www.learner.org/courses/learningmath/number/session8/part_b/modeling.html
http://www.learner.org/courses/learningmath/number/session8/part_b/try.html
Another Strategy for Division
http://www.learner.org/courses/learningmath/number/session8/video.html
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Virtual Fractions• Equivalent Fractions -
http://illuminations.nctm.org/ActivityDetail.aspx?ID=80• Fraction Models -
http://illuminations.nctm.org/ActivityDetail.aspx?ID=11• Fraction Game - http://
illuminations.nctm.org/ActivityDetail.aspx?ID=18• Fraction Pieces -
http://nlvm.usu.edu/en/nav/frames_asid_274_g_3_t_1.html?open=activities&from=category_g_3_t_1.html
• Fraction Adding - http://nlvm.usu.edu/en/nav/frames_asid_106_g_3_t_1.html?from=category_g_3_t_1.html
• Fraction Comparing - http://nlvm.usu.edu/en/nav/frames_asid_159_g_3_t_1.html?from=category_g_3_t_1.html
• Fraction Equivalence - http://nlvm.usu.edu/en/nav/frames_asid_105_g_3_t_1.html?from=category_g_3_t_1.html
• Fraction Rectangle Multiplication - http://nlvm.usu.edu/en/nav/frames_asid_194_g_3_t_1.html?from=category_g_3_t_1.html
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What were the goals of the activities?
What common core standards have we been working on?
What Standards of Mathematical Practice were present during the activities?
Small Group Discussion
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Rational Numbers
• Integers– Charge Model– Linear Model
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·Charge Model
Use your positive/negative counters to represent the following numbers using at least the number of tiles listed.
You can challenge yourself by using more than the minimum number of tiles.
Be prepared to share and prove your solution.
Ways to build understanding of Integers
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·Linear Model
Matt earns merits and demerits at his school. One day he earned 3 merits for his math game, 2 demerits for being late to class, 1 merit for being courteous, 5 demerits for arguing with his teacher, and 2 merits for helping another student. If he began the day with 4 merits, how many did he have at the end of the day?
Ways to build understanding of Integers
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Model the following problems with your counters and sketch your work using a plus sign for positive and a negative sign for negative counters:
3 + 5 +3 + (-5) -3 + 5 -3 + (-5)
What do you notice? Make some generalizations about the rules for adding integers.
Now consider: -3 - (-5)
What generalization can you make?
Ways to build understanding of Integers
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Charge and Linear Model
Solve this problem using both methods:Heather started the month with $12. She spent $5 on a game, but realized that she forgot to pay her annual club dues so she wrote a check for $15 because her dad said he would loan her enough money to cover the check. How much does Heather have to borrow from her dad?
Ways to build understanding of Integers
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How is this different from the way students built their understanding of positive/negative integers in the past?
What common core standard have we been working on?
What Standards of Mathematical Practice were present during the activity?
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Concrete Algebra• Explore• Build• Add• Subtract• Multiply• Divide
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Connecting Number System to Algebra
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Making Connections through diagram – 23 x 143
2000 800 60
300 120 9
100 + 40 + 3
20
+
3
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Virtual Algebra
• Illuminations Algebra Tiles - http://illuminations.nctm.org/ActivityDetail.aspx?ID=216
• NLVM algebra Tiles - http://nlvm.usu.edu/en/nav/frames_asid_189_g_3_t_2.html?open=activities&from=category_g_3_t_2.html
• NLVM Scales -Positives http://nlvm.usu.edu/en/nav/frames_asid_201_g_3_t_2.html?open=instructions&from=category_g_3_t_2.html
• NLVM Scales – Negatives• http://nlvm.usu.edu/en/nav/frames_asid_201_g_3
_t_2.html?open=instructions&from=category_g_3_t_2.html
• Pan Balance - Numbers• http://illuminations.nctm.org/ActivityDetail.aspx?i
d=26• Pan Balance - Expressions• http://illuminations.nctm.org/ActivityDetail.aspx?I
D=10
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Exit Card• 3 Ideas you will use with
your learners
• 2 Aha’s you had today
• 1 Question you still have – Please provide your email for a response
