math quiz 3

Modeling Word Problems and CGI

1. Model each problem using manipulatives (counters, tiles, etc). 2. Draw a representation of the problem using either a set model or measurement model (relates to context). 3. If possible, write at least 2 different number sentences showing a sum or difference. 4. Classify each problem according to the Cognitively Guided Instruction (CGI) chart

Change 3424 (5) to a base ten representatio

480

Join (action) Result Unknown

5 kids were playing soccer. 7 more joined them. How many kids are playing soccer now?

Change 213 (5) to a base ten

58+(10)

Compare (no action) Quantity Unknown

Arthur ate 6 ice cream cones. D.W. ate 2 more ice cream cones than Arthur. How many ice cream cones did D.W. eat?

Separate (action) Result Unknown

Arthur had 12 books. He gave 6 to D.W. How many books does Arthur have now?

Separate (action) Change Unknown

Arthur had 14 coins. He spent some coins on ice cream. When he got home, he had 7 coins. How many coins did Arthur spend on ice cream?

Separate (action) Start Unknown

Arthur had some books in his room. Then, he gave D.W. 9 books. Now he has 9 books. How many books did Arthur have to start with?

Compare (no action) Different Unknown

Arthur has 16 books. D.W. has 7 books. How many fewer books does D.W. have than Arthur?

Compare (no action)

Arthur has 5 books about dogs. He has 4 fewer dog books than his teacher. How many books about dogs does Arthur's teacher have?

Test for Divisibility: 11

By 11: The difference of the sums of the digits in even and odd positions is divisible by 11. (1st odd position is the ones place.) Example: Is 42,315,690 divisible by 11? Odd positions: 0 + 6 + 1 + 2 = 9 Even positions: 4 + 3 + 5 + 9 = 21 Difference: 21-9 = 12; Not divisible by 11; therefore, 42,315,690 is not divisible by 11.

Test for Divisibility: 2, 5, and 10

By 2: Last (or units) digit is 0, 2, 4, 6, or 8. By 5: Last digit is 0 or 5. By 10: Last digit is 0.

Multiplication Activities

Circle and Stars and Cube Arrays

Five Strands of Mathematics Proficiency

Conceptual Understanding Procedural Fluency Strategic Competence Adaptive Reasoning Productive Disposition

Join (action) Change Unknown

D.W. had 4 books. Then, Mom bought her some more. Now D.W. has 8 books. How many books did Mom buy D.W.?

Join (action) Start Unknown

D.W. had some coins in her bank. Then, she found three coins on the sidewalk. Now she has 6 coins. How many coins did D.W. have to start with?

Part-Part-Whole (no action) Part Unknown

D.W. has 15 books. 7 of her books are chapter books. The rest are picture books. How many are picture books?

Cognitively Guided Instruction (CGI)

Join (action) Separate (action) Compare (no action) Part-Part-Whole (no action)

Test for Divisibility 8

Last three digits form a number divisible by 8

Test for Divisibility 4

Last two digits form a number divisible by 4

Part-Part-Whole (no action) Whole Unknown

On a walk, D.W. read 8 words. Arthur also read 8 words. How many words did they read together?

Types of Division

Partitive and Measurement

Representations for Multiplication

Set Model Measurement Model Array Model Multiplication Tree Mode Cartesian Product Model

Test for Divisibility 3

Sum of the digits is divisible by 3.

Test for Divisibility 9

Sum of the digits is divisible by 9

Strategic Competence

The ability to formulate, to represent, and to solve mathematical problems. ex: The answer is 5. What is the question?

Adaptive Reasoning

The capacity for logical thought, reflection, explanation, and justification. ex: You can represent a number between 100 and 1000 with just a few base ten blocks. What could the number be? How do you know? Activity: "Show me, convince me"

Conceptual Understanding

The comprehension of mathematical concepts, operations, and relations ex: Use 9 base-ten blocks to represent a number. What numbers can you represent?

Measurement

The divisor is the number in each group. How many groups? (size of group is known) ex: Morgan is making candles. She has 32 ounces of melted wax and each candle required 5 ounces of wax, how many candles can she make?

Partitive

The divisor is the number of groups. How many in each group? (number of groups is known) ex: Laura has 32 markers. If she only has 5 marker cases how many markers can she fit in each case?

Productive Disposition

The habitual inclination to see mathematics as sensible, useful, worthwhile, coupled with a belief in diligence and one's own efficacy. ex: You can model a number with 58 base ten blocks. What other numbers of base ten blocks could you use to model that number?

greatest common factor, GCF

The largest factor that two or more numbers have in common.

Procedural Fluency

The skill in carrying out procedures flexibly, accurately, efficiently, and appropriately ex: You subtract two numbers that are close together. What might your answer be?

least common multiple, , LCM

The smallest multiple (other than zero) that two or more numbers have in common.

Polya's Four Phases of Problem

Understand the Problem Devise a Plan Carry Out the Plan Look Back

Why Students Need to Create Their Own Representations

Vehicle for students to process their thoughts Way to organize information Avenue for communicating about mathematics.

Abundant number

any number for which the sum of its factors (other than the number itself) is greater than itself (1+2+3+4+6 = 16; 16 > 12)

Perfect number

find all the factors that divide a number evenly, except the number itself, and find the sum of the factors, which equals the number. (1+2+3 = 6)

Array Model

is formed by arranging a set of objects into rows and columns

Deficient number:

the number's factors added together are less than the number (1+2+7 = 10; 10 < 14)

Cartesian Product Model

○ Timmy has 3 tops (blue, red and green) and 2 shorts (jean and khaki) that can be mixed and matched. How many outfits does he have? § Set of tops: □ T = {b, r, g) § Set of shorts: □ S = {j, k} § Set of outfits: T X S = {(b,j), (b,k), (r,j), (r,k), (g,j), (g,k)}