2010 Mathematics Institute Patterns, Functions, and Algebra Grades 3-5

Fall 2010

2010 Mathematics Institute2010 Mathematics InstitutePatterns, Functions, and Patterns, Functions, and

AlgebraAlgebraGrades 3-5Grades 3-5

2010 Mathematics Institute2010 Mathematics InstitutePatterns, Functions, and Patterns, Functions, and

AlgebraAlgebraGrades 3-5Grades 3-5

Fall 2010

AgendaAgendaAgendaAgenda

• IntroductionsIntroductions

• Big TopicsBig Topics

• PatternsPatterns

• PropertiesProperties

• LunchLunch

• Equalities and InequalitiesEqualities and Inequalities

• IntroductionsIntroductions

• Big TopicsBig Topics

• PatternsPatterns

• PropertiesProperties

• LunchLunch

• Equalities and InequalitiesEqualities and Inequalities

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Big TopicsBig TopicsBig TopicsBig Topics

I.I. PatternsPatterns• Using Number LinesUsing Number Lines

II.II. PropertiesProperties• Building VocabularyBuilding Vocabulary

III.III. Equations and InequalitiesEquations and Inequalities• Keeping it BalancedKeeping it Balanced

I.I. PatternsPatterns• Using Number LinesUsing Number Lines

II.II. PropertiesProperties• Building VocabularyBuilding Vocabulary

III.III. Equations and InequalitiesEquations and Inequalities• Keeping it BalancedKeeping it Balanced

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Unpacking Unpacking PatternsPatterns

Unpacking Unpacking PatternsPatterns

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Multiplication shown on a Number Line

2009 SOL 3.6

Multiplication shown on a Number Line

2009 SOL 3.6

1 2 4 5 7 8 10 11 1314 16 170

Write the multiplication number sentence that matches the hops “Factor Frog” made.

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Multiplication on a Number Line 2009 SOL 3.6

Multiplication on a Number Line 2009 SOL 3.6

http://illuminations.nctm.org/LessonDetail.aspx?ID=L316

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Least Common Multiple2009 SOL 4.5a

Least Common Multiple2009 SOL 4.5a

1 2 4 5 7 8 10 11 13 14 16 170 3 9 15

LCM

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Primes and Composites 2009 SOL 5.3

Primes and Composites 2009 SOL 5.3

Welcome to the Welcome to the Bubble Gum Bubble Gum

FactoryFactory

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Primes and Composites 2009 SOL 5.3

Primes and Composites 2009 SOL 5.3

Bubble Gum Factory Investigation

At the Bubble Gum Factory, lengths of gum are stretched to larger lengths by putting them through stretching

machines.

There are 99 stretching machines, numbered 2 through 100.

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Primes and Composites 2009 SOL 5.3

Primes and Composites 2009 SOL 5.3

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Primes and Composites 2009 SOL 5.3

Primes and Composites 2009 SOL 5.3

Machine 3 triples the length and so forth.

=

So, machine 23, for example, will stretch apiece of gum to 23 times its original length.

= Well…you get the point.

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Primes and Composites 2009 SOL 5.3

Primes and Composites 2009 SOL 5.3

Now It Is Your Job!Now It Is Your Job!

An order has just come in for a piece of An order has just come in for a piece of bubble gum 24 inches in length. bubble gum 24 inches in length.

The factory has pieces of gum that are only The factory has pieces of gum that are only 1 inch in length, and machine number 24 is 1 inch in length, and machine number 24 is broken. broken.

*Is there any way to create a piece of bubble *Is there any way to create a piece of bubble gum 24 inches in length by using other gum 24 inches in length by using other machines? machines?

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Primes and Composites 2009 SOL 5.3

Primes and Composites 2009 SOL 5.3

Figure out whichmachines are

actually necessary.

Do we need all of them?

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Primes and Composites 2009 SOL 5.3

Primes and Composites 2009 SOL 5.3

We know that 2 is a necessary machine, but every even number has 2 as a factor…

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Primes and Composites2009 SOL 5.3

Primes and Composites2009 SOL 5.3

We also know that 3 is a necessary machine, but every third number has 3 as a factor.

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Primes and Composites2009 SOL 5.3

Primes and Composites2009 SOL 5.3

…and we also know that 5 is a necessary machine, but every fifth number has 5 as a factor.

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Primes and Composites2009 SOL 5.3

Primes and Composites2009 SOL 5.3

We know that 7 is a necessary machine and every seventh number has 7 as a factor.

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Primes and Composites2009 SOL 5.3

Primes and Composites2009 SOL 5.3

After exploring divisibility rules for 2, 3, 5, 7, 11, and 17, the prime numbers under 100 are revealed.

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Think/Pair/ShareThink/Pair/Share

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Properties VocabularyProperties VocabularyProperties VocabularyProperties Vocabulary

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Properties VocabularyProperties VocabularyProperties VocabularyProperties Vocabulary

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Now, they also have to name it

-Students have always had to understand the property

Addition to the Standard:

Commutative Property2009 SOL 3.20

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Commutative Property2009 SOL 3.20

Commutative Property2009 SOL 3.20

If students know:

4 + 5

Then they know :5 + 4

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Commutative Property2009 SOL 3.20

Commutative Property2009 SOL 3.20

“It is not intuitively obvious that 3 x 8 is the same as 8 x 3 or that, in general, the order of the numbers makes no difference (the commutative or order property). A picture of 3 sets of 8 objects cannot immediately be seen as 8 piles of 3 objects. Eight hops of 3 land at 24, but it is not clear that 3 hops of 8 will land at the same point.

The array, by contrast, is quite powerful in illustratingthe order property. Students should draw or build arraysand use them to demonstrate why each array represents two different multiplications with the same product.”

Van de Walle (2001)

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Commutative Property2009 SOL 3.20

Commutative Property2009 SOL 3.20

- Given experiences with arrays, If students know

3 x 7

Then they can see that it is equal to -

7 x 3

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Commutative Property2009 SOL 3.20

Commutative Property2009 SOL 3.20

6 x 2 = 2 x 6

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AutomaticityAutomaticity

If I asked you to multiply 56 x 36 using mental math, would you be able to do that with automaticity?

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Associative Property2009 SOL 4.16b

Associative Property2009 SOL 4.16b

Given a problem -

(41 + 25) + 75

How can you make it an easier problem?

41 + (25 + 75)

- Looking for friendly numbers

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Associative Property2009 SOL 4.16b

Associative Property2009 SOL 4.16b

Solving a volume problem -

5

227

(27 x 5) x 2

Becomes - 27 x (5 x 2)

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Distributive Property2009 SOL 5.19

Distributive Property2009 SOL 5.19

Partial Products

3 x 24 = 3 x 20 + 3 x 4

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http://www.glencoe.com/sites/common_assets/mathematics/ebook_assets/vmf/VMF-Interface.html

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Distributive Property2009 SOL 5.19

Distributive Property2009 SOL 5.19

Slice It

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Think/Pair/ShareThink/Pair/ShareThink/Pair/ShareThink/Pair/Share

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Equations and InequalitiesEquations and InequalitiesEquations and InequalitiesEquations and Inequalities

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What does the equal sign mean?

What does the equal sign mean?

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Equalities2009 SOL 4.16a Equalities

2009 SOL 4.16a

http://illuminations.nctm.org/LessonDetail.aspx?ID=L18335

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Equalities2009 SOL 3.20 Equalities2009 SOL 3.20

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Inequalities2009 SOL 3.20 Inequalities2009 SOL 3.20

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Equalities2009 SOL 3.20 Equalities2009 SOL 3.20

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

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Equalities2009 SOL 4.16aEqualities

2009 SOL 4.16a

8 = 1 + 7

3 + 5 = 5 + 3

9 = 9

2 + 3 = 2 x 3True or False?

7 x 4 = 4 + 4 + 4 + 4

What will the students say?

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Equalities2009 SOL 4.16a

Equalities2009 SOL 4.16a

True or False?

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Examples/Non-Examples

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Modeling One-step Linear Equations2009 SOL 5.18c

Modeling One-step Linear Equations2009 SOL 5.18c

Using your cups and candy corn, construct a

model for

J = 6

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Modeling One-step Linear Equations2009 SOL 5.18c

Modeling One-step Linear Equations2009 SOL 5.18c

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Modeling One-step Linear Equations2009 SOL 5.18c

Modeling One-step Linear Equations2009 SOL 5.18c

Using your cups and candy corn, construct a

model for

J + 4 = 7

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Modeling One-step Linear Equations2009 SOL 5.18c

Modeling One-step Linear Equations2009 SOL 5.18c

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Modeling One-step Linear Equations2009 SOL 5.18c

Modeling One-step Linear Equations2009 SOL 5.18c

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Modeling One-step Linear Equations2009 SOL 5.18c

Modeling One-step Linear Equations2009 SOL 5.18c

B + 2 = 9

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Modeling One-step Linear Equations2009 SOL 5.18c

Modeling One-step Linear Equations2009 SOL 5.18c

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Modeling One-step Linear Equations2009 SOL 5.18c

Modeling One-step Linear Equations2009 SOL 5.18c

B + 4 = 11

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Modeling One-step Linear Equations2009 SOL 5.18c

Modeling One-step Linear Equations2009 SOL 5.18c

http://illuminations.nctm.org/ActivityDetail.aspx?id=33

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Modeling One-step Linear Equations2009 SOL 5.18c

Modeling One-step Linear Equations2009 SOL 5.18c

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http://illuminations.nctm.org/ActivityDetail.aspx?id=10

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Think/Pair/ShareThink/Pair/ShareThink/Pair/ShareThink/Pair/Share

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Questions?

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