Page 1 Building Understanding of the Number System Through Hands-On Experiences Marcia Torgrude...

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Building Understanding of the Number System Through

Hands-On Experiences

Marcia TorgrudeK-12 Math [email protected]

• Develop understanding and ideas to promote deeper understanding of the number system within the Common Core

• Develop hands-on strategies to build understanding of place value.

• 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

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.

Standards of Mathematical Practice

What does it mean to “do mathematics?”

The Standards of Mathematical Practice are descriptions of the fundamental skills needed to “do” mathematics.

What does it mean to “do mathematics?”

• Standards of Practice describe what it means for students to demonstrate proficiency in mathematics. They are our new “basic skills.”

• Content Standards are the “what” of mathematics

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”

Working with Whole Numbers

• Adding• Subtracting• Multiplying• Dividing

With Base 10 Blocks


Using Modeling to Make Sense of

Mathematical Procedures

Modeling addition with Base 10 blocks

Place It!


Using Modeling to Make Sense of

Mathematical Procedures

Modeling subtraction with Base 10 blocks

ONLY BUILD

the beginning number

302 − 178

412 - 189

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

Practice Concrete Multiplication

What does multiplication look like using base ten blocks?

Let’s Try this without the blocks

143x23

Making Connections through diagram – 23 x 143

2000 800 60

300 120 9

100 + 40 + 3

20

+

3

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

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. Did we make the connections? How?

Practice Concrete Division

What does division look like using base ten blocks?

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 and the array model to help us with division?

Understanding the abstract

Do you think that using base-10 blocks helps to give meaning to the division algorithm? How?

Another Strategy for Division

Use of friendly or “benchmark” numbers

Partial quotient division:Multiplication for division – use what we know

Partial Quotient Division

Our family took a trip and my dad told me we drove a total of 2,112 miles in 6 days. How many miles per day did we travel on the average?

What friendly numbers did you use?

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

http://illuminations.nctm.org/

http://illuminations.nctm.org/ActivityDetail.aspx?ID=75













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

Working with Fractions

• Equivalence• Addition• Subtraction• Multiplication• Division

Fraction Equivalence, Adding, and Subtracting

• Using Pattern Blocks

Modeling equivalence, adding, subtracting, multiplying, and dividing

• Using Cuisenaire Rods• http://www.learner.org/vod/vod_window.html?

pid=1853

• http://www.teachersdomain.org/assets/wgbh/rttt12/rttt12_int_cuisenaire/index.html

http://www.learner.org/vod/vod_window.html?pid=1853

http://www.learner.org/vod/vod_window.html?pid=1853

http://www.teachersdomain.org/assets/wgbh/rttt12/rttt12_int_cuisenaire/index.html




Fraction Using Cuisenaire Rods

• http://www.learner.org/courses/learningmath/number/session8/part_b/modeling.html

• If we are trying to work with fourths and thirds what will our new whole need to be?

• Using Cuisenaire rods model:• 1/3 + 1/4• 1/3 - 1/4• 1/3 x 1/4• 1/3 ÷ 1/4

http://www.learner.org/courses/learningmath/number/session8/part_b/modeling.html



Why do we invert and multiply to divide?

• How does this work?• ¾ ÷ 5/6= B• How many 5/6 are in ¾?• x 5/6 = ¾• B x 5/6 = ¾• I need to multiply 5/6 by its reciprocal to

solve for B or box.• B x 5/6 x 6/5 = ¾ x 6/5• B = ¾ x 6/5

Fractions, Decimals, and Percents

• Make sense through grids

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





http://nlvm.usu.edu/en/nav/frames_asid_274_g_3_t_1.html?open=activities&from=category_g_3_t_1.html










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

Rational Numbers

• Integers– Charge Model– Linear Model

·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

·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?


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?


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?


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?


Concrete Algebra

• Explore• Build• Add• Subtract• Multiply• Divide

Connecting Number System to Algebra

Making Connections through diagram – 23 x 143

2000 800 60

300 120 9

100 + 40 + 3

20

+

3

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.ht

ml?open=instructions&from=category_g_3_t_2.html• Pan Balance - Numbers• http://illuminations.nctm.org/ActivityDetail.aspx?id=26• Pan Balance - Expressions• http://illuminations.nctm.org/ActivityDetail.aspx?ID=10





http://nlvm.usu.edu/en/nav/frames_asid_201_g_3_t_2.html?open=instructions&from=category_g_3_t_2.html




http://illuminations.nctm.org/ActivityDetail.aspx?id=26


Thank You - Exit Card

• I am reaffirmed because I already…

• The big idea I will work on is…

• I still need help with….

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