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Building Understanding of the Number System Through Hands-On Experiences. Marcia Torgrude K-12 Math Specialist mtorgrude@tie.net. Outcomes for Today. Develop understanding and ideas to promote deeper understanding of the number system.

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slide1

Building Understanding of the Number System Through

Hands-On Experiences

Marcia Torgrude

K-12 Math Specialist

mtorgrude@tie.net

outcomes for today
Outcomes for Today
  • 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
i have who has
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.
what does it mean to do mathematics
What does it mean to “do mathematics?”

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

what does it mean to do mathematics1
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
doing mathematics
“Doing 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.”

working with whole numbers
Working with Whole Numbers
  • Adding
  • Subtracting
  • Multiplying
  • Dividing

With Base 10 Blocks

doing mathematics1
“Doing mathematics”

Using Modeling to Make Sense of

Mathematical Procedures

Modeling addition with Base 10 blocks

Place It!

doing mathematics2
“Doing mathematics”

Using Modeling to Make Sense of

Mathematical Procedures

Modeling subtraction with Base 10 blocks

302 − 178

412 - 189

multiplication and division
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
Practice Concrete Multiplication

What does multiplication look like using base ten blocks?

making connections getting to the algorithm
Making Connections –Getting to the algorithm

300+20+6

X 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
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?
practice concrete division
Practice Concrete Division

What does division look like using base ten blocks?

slide18

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?

understanding the abstract1
Understanding the abstract
  • Do you think that using base-10 blocks helps to give meaning to the division algorithm? How?
another strategy for division
Another Strategy for Division

Partial quotient division

Multiplication for division – use what we know

Use of friendly or “benchmark” numbers

virtual whole number tools
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
slide22

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
Working with Fractions
  • Equivalence
  • Addition
  • Subtraction
  • Multiplication
  • Division
modeling equivalence adding and subtracting
Modeling equivalence, adding, and subtracting
  • Using Cuisenaire Rods
  • http://www.teachersdomain.org/assets/wgbh/rttt12/rttt12_int_cuisenaire/index.html
fraction multiplication and division1
Fraction Multiplication and Division

Using Cuisenaire Rods

  • Using Cuisenaire rods model 1/3 x 1/4 to find the solution.
  • Using a similar model find 1/3 ÷ 1/4
slide29

Fraction Multiplication and Division

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

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

virtual fractions
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
slide31

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
Rational Numbers
  • Integers
    • Charge Model
    • Linear Model
slide33

Ways to build understanding of Integers

·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.

slide34

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?

slide35

Ways to build understanding of Integers

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?

slide36

Ways to build understanding of Integers

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?

slide37

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
Concrete Algebra
  • Explore
  • Build
  • Add
  • Subtract
  • Multiply
  • Divide
virtual algebra
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?id=26
  • Pan Balance - Expressions
  • http://illuminations.nctm.org/ActivityDetail.aspx?ID=10
exit card
Exit Card
  • 3 Ideas you will use with your learners
  • 2 Aha’s you had today
  • 1 Question you still have –

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