Chapter 7, 8, 9, 10 Math Test

Reasoning Strategies for Multiplication Basic Facts:

- Doubles (number x 2) -Fives Facts (number x 5) -Zeros and Ones (number x 0...number x 1) -Nifty Nines (number x 9) (the fingers, ages 18 and 27, count by 9s- 9 to 360) -Derived Facts: using foundational facts to find remaining facts (4x8=32...2x8+2x8...4x9-4x1)

Terminology for addition, subtraction, multiplication, division

- addend 1 + added 2 = sum -minuend-subtrahend=difference -factor1 X factor2= product -dividend / divisor = quotient

Identify 5 equivalent representations of 473

-473 ones -3 hundreds 173 ones -40 tens 73 ones -30 tens 173 ones

Examples of using foundational facts: 7x6= 42, 4x8= 32, 3x6=18, 7x7=49

-4x8=32....2x8+2x8...4x9-4x1 -7x6=42...7x5+7x1...2x6+1x6 -3x6=18...2x6+1x6...5x3+1x3 -7x7=49....7x5+7x2

What content is taught when teaching through problem solving?

-5 NCTM Content Standards: Numbers and Operations Algebra Geometry Measurement Data Analysis and Probability

Remainders are not...

"Left over"- they either are discarded to leave a smaller whole # or force the answer to next highest #

what not to do when teaching basic facts

-don't use timed tests -don't use public comparison of mastery -don't proceed through facts in order from 0 to 9. -don't work on all facts at once -dont expect automaticity too soon -don't use facts as a barrier to good mathematics -don't use fact mastery as a prereq for calculator use

Describe place value development

10 ones is 1 ten and 10 tens is 1 hundred

Maggie's chickens laid 18 eggs today. She placed the eggs into cartons. Each carton holds 6 eggs. How many cartons did Maggie use?

Measurement

Describe activities we did in class to promote place value development:

-A Daily Number routine (mentioned above) + symbol, words, place value language, and expanded form symbolic expression -Equivalent Representations: using base 10 blocks, develop equal representations of 134 and color them out on 10x10 grid. Then on recording sheet, express each representation using words. (ex: one hundred, 2 tens, and 14 ones). You should see a systematic listing. What patterns do you see? -Models for Multi-digit Numbers: (see sheet) select a row other than top row and shade it in. Construct all numbers in that row except the last number. Select one column other than the first or last and construct numbers on that column.

In the Before problem Solving the teacher should:

-Activate prior knowledge by beginning with simpler version of the task and brainstorming solutions -be sure problem is understood by being aware of possible misinterpretations/misconceptions -establish clear expectations by questioning students to determine comprehension of expectations in terms of process/product

Why should "Calendar Activities" be avoided?

-Do not emphasize foundational mathematics -based on groups of 7, so it does not support the development of mathematical relationships related to the number 10 -only engage a handful of children -does not promote problem solving/promotes repetition

Basic facts are...

-For ALL 4 operations -relationships where the two parts are less than 10

What are the features of a math problem?

-Must begin where students are -problematic aspect must be what the students are to learn -must require justification/explanation for answers and methods

Teddy bear counters

-Non proportional -The colors tell the value and each bear has a different value. Ten green bears valued at 1 will never equal one red bear valued at 10 (very difficult to grasp for young students)

Reasoning Strategies for Subtraction:

-One less than/two less than (a number -1, a number -2) -Facts with Zero (number - 0) -Doubles (8-4=4) -Near Doubles (15-7=8) -10 frame facts (10-7=3) -Think Addition- Sums Less than 10 (9-7=2) -Sums Greater than 10 (minuend greater than 10): Take from 10 and add or think addition (15-9=6)

Reasoning Strategies for Addition Basic Facts:

-One more than/two more than (a number +1, a number +2) -Facts with Zero (number + 0) -Doubles (number + itself) -Doubles Plus One/Near Doubles (number + one more than itself) -10 frame facts (add to 10) -Make a Ten (sums are between 11 and 18) -Remaining facts: Use 5 as an anchor (3+6=9 bc 3+5=8+1=9)

Base-Ten Models for Place Value

-Proportional Groupable -Proportional Pregrouped -Non proportional- cognitively demanding, teach this before money (if students struggle with money, they have not been taught place value) *proportional means the length/weight/height is consistent across. One unit is 100x smaller than a flat. When there is a multiplicative comparison, that is proportional. (Like multiplicative comparison problem)

What does it mean to teach math?

-Question students -facilitate discussion -walk around/observing what students are doing -not modeling but you can solve a simpler problem for demonstration

What is Number sense?

-Understanding of numbers and their relationships -develops gradually -results from EXPLORING numbers, VISUALIZING numbers in various contexts, and RELATING them in non-traditional algorithmic ways -knowledge of effect of operations on numbers

List the 3 stages of counting sets of objects:

-Unitary- counting by ones -name how many by counting each piece -do not think of 10 as a single unit -have to count by ones to be convinced that different sets have the same amount -Base 10-counting by tens and ones: 1,2,3,4,5 groups of 10 and 1,2,3, ones (singles) or 10, 20, 30, 40 -count a group of 10 objects as single object (unitizing) -coordiate the base ten approach with a count by ones approach to tell "how many" -Equivalent- non-standard base 10: before counting, students would trade and then count 10,20,30,40,50,51,52. -group the pieces flexibly into versions that include tens and ones but all trades have not been carried out -use these alternate groupings to relate to computation by being able to trade or regroup numbers in a variety of ways

what to do when teaching basic facts

-ask students to self monitor -focus on self-improvement -work on facts over time -involve families -make fact practice enjoyable -use technology -emphasize the importance of knowing their facts

How should elementary mathematics teachers use textbooks most effectively?

-avoid sending message to students that the textbook is the source of whether an answer is correct -as a reference to pull ideas from the margins that are not in student textbook -to retrieve appropriate numbers for grade level to be used in problems

For basic facts remediation:

-check which facts students know/don't know -diagnose strengths/weaknesses -focus on reasoning strategies

In what order should procedural knowledge and conceptual knowledge be taught? Why?

-conceptual knowledge first then procedural -By doing problem solving, conceptual knowledge develops and this will lead to them developing their own strategies (procedures). Procedural knowledge is not explicitly taught, it happens naturally. -Some procedural knowledge has to be stated, but it's after they've experienced it a lot.

How are place value concepts evident in the 1-100 chart? (look at activity)

-helps 1st graders develop base 10 understanding of adding two digit numbers with multiples of 10, noticing that jumps up or down are jumps of ten, while recognizing that jumps to the right or left are jumps of one. -There are also lots of patterns on the hundreds charts, such as: in a row, the first numbers stays the same and second number (ones digit) counts (1,2,3,) changes as you move across -you can count by tens gong down the far right-hand column -in a column, the first number (tens digit) counts or goes up by ones as you move down

Reminders for writing objectives:

-if it is bound in a classroom, it is an activity -must be specific -can't be singular -ABCD

Foundational Facts:

-number x 2 -number x 5 -number x 0 -number x 1 -number x 9 -One of the numbers has to be constant, and the other two numbers in the equation will add up to equal the number that is not constant. One of the numbers must be one of the ones above

Unifix cubes/multilink cubes

-proportional Groupable -unifix for K-1 and multilink for older kids. -unifix cubes have one face that allows to connect and multilink connects all 6 sides

Base 10 blocks

-proportional Pregrouped (cant take apart)

When teaching Basic Facts do not:

-public comparison of mastery -proceed through facts 0-9 in order at once -move to memorization too fast -use fact mastery as a barrier to good math and prereq for calculator use -lengthy timed tests

How do students best learn math?

-through problem solving -through constructivism (build new ideas from old)

How can teachers differentiate?

-use 3-phase lesson plan -teacher-directed groups to diff. Content -centers -open questions -tiered lessons and tasks -multiple entry and exit point problems -flexible grouping

activities that can help develop number sense

-using ten frames and five frames -

Explain Equivalent Representations activity

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Explain Models for Multi-Digit Numbers activity

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Explain the Flowers and Ice Cream activity

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Explain the Mickey and Friend Number Sentences activity

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Division names...

...a missing factor in terms of the known factor and the product

Addition names...

...the whole in terms of the parts

What two specific behaviors must a student be able to do in order to verbally count a set of objects accurately? Explain each behavior.

1. A child must be able to say the standard string of counting words in order. A child would be able to verbally count 1,2,3,4,5,6,7..etc in order 2. A child must be able to connect the counting sequence in a one-to-one correspondence with referents being counted in the set. A child will point to each object being counted and verbally assign it as "1," "2," or "3."

ways to count sets of objects

1. By ones- 1,2,3,4, 2. By groups and singles- 2 groups of ten and 1,2,3 singles 3. counting by tens and ones- three tens and 4 ones

Problem structures for multiplication (explain the referent of the product in terms of the referents of the factors)

1. Equal groups (table x students = students) 2. Multiplicative comparison- 2 quantities are being compared (candy x 4xcandy =candy) 3. Product of measures- the referents of factors are same, product referent is different 4. Combinations- all 3 referents are different (shirts x jeans = clothing)

Addition and Subtraction Problem Structures: (Describe referents of each quantity)

1. Joining- must have action (the person gaining must be the person asked about in question) 2. Separating- must have action (the person giving must be the person asked about in question) 3. Part-part whole- 2 types of 1 thing (monster and bang are energy drinks) no action verb 4. Compare- no action verb

Problem structures for division (explain the referent of the quotient in terms of the referents of the dividend and divisor)

1. Partition- fair sharing 2. Measurement- divisor and dividend have same measurements

The Development of Counting Skills

1. Producing the standard list of counting words in order 2. Connecting this sequence in a one-to-one manner with the items being counted 3. Assigning ONLY one item in the set a counting word.

Properties of Addition/Subtraction

1. The Commutative Property of Addison 2. The Associative Property for Addition 3. The Identity Property (Zero Property)

How to teach oral and written names for numbers

1. The Names of twenty, thirty...ninety 2. Names from twenty through ninety nine 3. The teens because they are formed backwards

Properties of Multiplication

1. The identity property 2. The distributive property 3. The associative property 4. The zero property 5. The commutative property

Why is the use of key words not a good strategy to teach children?

1. The key word strategy sends the wrong message about doing mathematics. Key words encourage children to not make sense of the problem and ignore the meaning/structure of the problem and just look for key words 2. Key words are often misleading and suggest an operation that is incorrect. (Ex: "How many in all" in multiplication problem when its really addition problem) 3. Many problems do not have key words, therefore a student who relies on key words is left with no strategy. 4. Key words do not work with multi-step problems because the key word would only assist with one step of the problem.

Stages of counting sets when developing whole-number place-value concepts:

1. Unitary (count by ones) 2. Base 10 (count by groups of tens and ones) 3. Equivalent (counting with non-standard Base 10 grouping)

All math is done in 3 languages:

1. Word problems 2. Models 3. Symbolic equations

How and why should a ten frame be used? State specific groups of Basic Facts for Addition

A ten frame should be used because it helps students learn to group numbers and count by 5s instead of counting one by one. A ten frame has 2 rows of 5, and the ten frame is always filled up starting from the left and going across the first row. When the first row is filled up, it goes to the 2nd row. How children use the ten-frame can provide the teacher with insights into children's number concept development, therefore, this activity can be used as a diagnostic overview. Ten Frame facts (such as 3+7=10) and Make a 10

The way students learn to write numbers

A. 1, 4, 7 B. 2, 3, 5 C. 6, 9, 0, 8 They start with the top numbers because they consist of straight lines, the second group of numbers because they have a straight line and curve, and the last group of numbers because they have one curved line that must attach to itself

What is a mathematical problem?

An activity/task that poses a question where students have no memorized rules/methods, nor is there a perception by students that there is a specific "correct" solution method

I bought 8 apples, 5 oranges, and 2 pears at the grocery store. My friend bought 5 apples, 2 pears, and 8 oranges. Who bought the most fruit?

Associative Property of Addition- they will learn that no matter what way they group it, they get the same thing by repeating these contextual problems 6-10 times

I'm looking at candy bars and each candy bar costs $2, and each box has 5 candy bars in it. I want to buy 3 boxes. How much will it cost? If my friend is looking at candy bars and each candy bar costs $3, and each box has 5 candy bars, how much will it cost if my friend buys 2 boxes?

Associative Property of Multiplication

Bully has 5 maroon jerseys and 4 MSU scarfs. How many outfits can he make?

Combinations

I have 2 shirts and 6 pants. How many outfits can i make?

Combinations

Tammy has 3 rows of 6 eggs. I have 6 rows of 3 eggs. Who has more eggs?

Commutative Property of Multiplication

Bully has 9 chew toys. If he has 5 more chew toys than Uga, how many chew toys does Uga have?

Compare

I have 6 cats, and my friend has 3 less cats than I do. How many cats does my friend have?

Compare

Five people went to get lunch. A burger costs $6 and a juice costs $2. If all have people ordered a burger and juice, how much money did the restuarant make for the order?

Distributive property of multiplication

Examples/non-examples of basic facts:

E: 2x3=6, 8/2=4, 15-8=7 NE: 60/12=5, 19-9=10

Peter has 3 sisters. He wants to buy 4 flowers for each sister. How many flowers does he need to buy?

Equal groups

Ryan likes to color pictures. He bought 3 boxes of crayons. Each box had 8 crayons. How many crayons did Ryan buy?

Equal groups

Facts =

Equations

The goal of place value instruction

For students to be able to create equivalent representations

I went to the pet store and saw blue fish for $3 a fish. I bought one blue fish. How much money did I spend?

Identity Property of Multiplication

Bully had 5 bones. His friend gave him 3 bones. How many bones does Bully have?

Join

Dan found 6 rocks in his yard. Then, he found 5 rocks on the street. How many rocks did Dan find?

Join

Bully is giving away 18 Pom poms to students. If he gives each student 2 Pom poms, how many students will get Pom poms?

Measurement

What does the teacher do when employing each of the three approaches to teaching Basic Facts?

Memorization- the teacher gives timed tests, and she gives repeated isolated drills. She does not dedicate time to developing strategies. She does not use story problems or contextual problems. Explicit Strategy Instruction- the teacher teaches students a strategy, such as combinations of 10, then provides students with exploration and practice of the strategy (using a ten frame to see which facts equal 10). The teacher supports student thinking instead of giving them something extra to remember. The teacher helps students see possible strategies and choose one that helps them solve the problem without counting. Guided Invention- the teacher has students select a strategy based on their knowledge of number relationships. The teacher supports student thinking by allowing each student to see the problem differently and allowing them to use number combinations and relationships to make sense of them. The teacher may explain a strategy but will carefully set up tasks where students notice number relationships.

Bully gets 3 times as many walks a week as Smokey. If Smokey gets 8 walks a week, how many walks does Bully get?

Multiplicative comparison

Jake has 6 toy cars in his toy box. His friend Sam has twice as many. How many cars does Sam have?

Multiplicative comparison

Bully has 8 beef flavored bones and 6 chicken flavored bones. How many bones does he have?

PPW

Sally buys 5 waters from the store. On Tuesday, she buys 4 powerades. How many drinks did Sally buy?

Part part whole

Tom had 3 cats. Sam had 6 cats. How many cats did Tom and Sam have?

Part part whole

A rope is 54 feet long. If 6 leases are needed, how long will each leash be, if each leash is the same length?

Partition

You have 28 bones to give fairly to 7 bulldogs. How many bones will each bulldog receive?

Partition

Developmental Phases of learning basic facts:

Phase 1: Counting Strategies Phase 2: Reasoning Strategies Phase 3: Mastery

5 NCTM Process Standards

Problem Solving Reasoning and Proof Communication Connections Representation

A frame is by 5 inches long, and it is 6 inches wide. How many square inches is the picture frame?

Product of measures

A quantity consists of two parts:

Referent (object) and number

Bully has 14 raw hide treats. He decides to give a few to Smokey. Bully now has 9 raw hide treats. How many did he give to Smokey?

Separate

Bully has a bunch of cowbells. He gave 8 to his favorite fans leaving him with 7. How many cowbells did Bully have to start with?

Separate

There were 10 fish. 5 fish swam away. How many fish are there?

Separate

How does the Number of the Day routine promote place value understanding?

The Number of the Day routine uses place value language, and it asks the questions: "One less than?" proceeded by "What place value changes? "One more than?" proceeded by '' "Ten more than?" '' "Ten less than?" ''

Subitizing

The ability to instantly "see" the number of objects in a small set without having to count them

What is mathematics?

The abstract science of number, quantity, and space (measurement and geometry)

I have 12 blue pencils and 5 purple pencils. Sam has 5 blue pencils and 12 purpose pencils. Who has more pencils?

The commutative property of addition- by presenting students with the last question, they will discover that both problems are the same answer. Repeat contextual problems in Paris 6-10 times for students to notice pattern

How should multiplication be introduced to students?

When teaching multiplication, the foundational facts should be taught first, and the foundational facts are a number x2, a number x5, a number x0, a number x1, and a numberx9. Multiplication should also be introduced by providing students with interesting contextual problems so that they can have experience with making and counting equal groups. This helps students understand that a group can be considered a single entity, even though that group contains a certain number of objects. For example, if a student were to be presented with a problem that says, "How many candy bars are there if there are 4 boxes of 6 candy bars?" students would initially solve the problem by counting out four sets of six and counting them all. The goal is for students to think of the problem as four sets of six, where six is a entity that will be counted four times, and providing students with interesting contextual problems helps them develop this understanding. 2,5,0,1,9

Last night I read 40 pages of my book before I went to sleep. Tonight, my mom told me to go to bed, and I couldn't read any pages of my book. I decided how many pages I've read of my book from both nights. How many pages did I read?

Zero (Identity property) of Addition

I went to the store. I bought 0 bags of chips for $5. How many chips did I buy?

Zero Property of Multiplication

Subtraction names...

a missing part

Explain the skills/concepts taught through the Shapes of Numbers activity

adding one to single digit numbers subtracting one from single digit numbers determining whether whole numbers less than 20 are even or odd decompose whole numbers up to 10 into pairs of addends

Cardinality

the last number in a counting sequence indicates the quantity of items in a set- meaning is attached to counting