MATH EXAM 2 Verified Questions and Accurate Answers (100 out of 100 Correct).

MATH EXAM 2 Verified Questions and Accurate Answers (100 out of 100 Correct). Counting tells you... ...how many things are in a set. Numbers relate through... ...comparisons of quantities. Number concepts are intimately tied to... ...operations with numbers based on situations in the world around us. Number sense means that... ... you can think about different-sized quantities and use numbers and relationships in multiple ways to estimate and solve problems. Benefits of High-quality Learning Activities (6) 1. Enhance children's natural interest in mathematics 2. Build on children's experience and knowledge. 3. Base mathematics curriculum and teaching practices on solid mathematical knowledge and child development. 4. Use formal and informal experiences to strengthen children's problem-solving and reasoning processes. 5. Provide opportunities for children to explain their thinking as they interact with mathematical ideas. 6. Assess children's mathematical knowledge, skills, and strategies through observation and other informal practices. Three Foundational Areas in Mathematics Content for Early Learners 1. number core 2. relations core 3. operations core Four Categories of Early Number Relationships 1-10 1. Spacial Relationships 2. Anchors to 5 and 10 3. One More/Two More/ One Less/ Two less 4. Part-Part-Whole -can be used for + or - Subitize ex. the ability to 'see' a small amount of objects and know how many there are without counting. Dot plates (use as flashcards) Early & Verbal Counting - rote memorization, list of counting words -connect the sequence in a one-to one correspondence with objects Meaning Attached to Counting - the meaning attached to counting is the key conceptual idea on which all other number concepts are developed -each number is one more than the previous number Cardinality Principle -cardinality of a set is a measure of the "number of elements of the set" -the last number counted is the total Change Problem (+ &-) An action occurs -Result unknown -Change unknown - Start unknown Part-Part-Whole Two groups combined -Whole Unknown -One Part Unknown -Both Parts Unknown * can be solved with addition or subrtaciton Compare Problem How many more? (+) / How many fewer? (-) -Difference Unknown -Larger Quantity Unknown -Smaller Quantity Unknown Sequence of Indicators to Watch For -repeats the patter w/o counting -recount -one-to one correspondence -reach the correct answer after counting -repeat the pattern w/ counting Numeral Writing and Recognition Similar to teaching the alphabet -clay -trace in shaving cream -interactive whiteboard -calculator keypad Relations Core: -more than, less than, and equal to -closer to ___ or ___? -estimation - making comparisons -identifying which is more Number Relationships 10-20 -ex. Three Level Progression set of 10 plays a major role -ten and some more: a ten and some ones 1. Child understands ten as ten ones and does not see ten as a unit 2. Child understands ten as a unit but relies on materials to complete tasks involving ten 3. A facile concept of ten- solves tasks without using materials Estimation and Measurement -use non-confirmative "units" of measurement (feet etc...) -measure it -estimate how many it will take - skip count to find the estimated length Data Collections and Analysis -encourage children to generate ideas for what data to gather (visual) -use the graphed data to compare number relationships rather than only referencing more and less Addition and subtraction are... connected. Multiplication involves... counting groups of equal size and determining how many in all. Multiplication and division are... related. Models can be used to... solve contextual problems for all operations and to figure out what operation is involved in a problem regardless of the size of the numbers. Contextual Problems (Word Problems) -a primary teaching tool to help children construct rich understanding of the operations -contexts activate problem-solving strategies -solving problems with words, pictures and numbers -explaining what they did and why it makes sense with the context Additive Problem Structures (6) 1. Join problems (addition problem) -change being "added to" the initial 2. Separate problems -change is being removed from initial 3. Part-part-whole problems -either missing the whole or one of the parts must be found 4. Compare problems -there are three ways to present compare problems, corresponding to which quantity is unknown (smaller, larger, or difference). 5. Contextual problems 6. Model-Based problems (*separate card) Teaching Addition and Subtraction Through Contextual Problems - lessons built on context or stories that are connected to children's lives - derived from recent experiences in the classroom; field trip, children's literature, art, science discussion Model-Based Problem (6) (addition and subtraction) 1. Action situations - join and separate 2. No-action situations -part-part-whole (larger numbers in grades 3-5) 3. Bar diagrams -strip or tape generate "meaning-making space" precursor to use of number lines 4. Number lines -shift from counting number of objects in a collection to length units 5. think-addition vs. take away -how much is missing to make this whole? 6. Comparison situations -two distinct sets and the difference between them Think-addition a method of using addition facts to find solutions for subtraction problems Properties of Addition and Subtraction (2) 1. Commutative Property -order of addends does not change the answer -essential for problem solving -mastery of basic facts -mental mathematics 2. Associative Property -when adding three or more numbers, it does not matter which numbers are added first -mental mathematics Operation sense Children's Understanding of 10 (3) 1. An initial concept of ten - ten ones 2. An intermediate concept of ten -ten as a unit, relies on manipulatives 3.A facile concept of ten -w/o materials, mentally and think about two-digit numbers as groups of tens and ones Multiplication and Division Structures (4) 1. Equal-group problems -one number or factor counts how many sets, groups, or parts of equal size -the other factor tells the size of each set, group, or part -the third number is the whole or product -when number size is known the problem in multiplication -when either group size or number of groups is unknown the problem is division 2. Comparison Problems -two different sets and groups -the comparison is based on one group being a particular multiple of the other -three possibilities for the unknown (the product, the group size, the number of groups) -similar to part-part-whole 3. Area and Array Problems -area is the product of measure -product is a different type of unit from the two factors -array is an equal group situation w/ number of rows and an equal number in each column 4. Model-Based Problems -varied models help to focus on the meaning of the operation and the associated symbolism Teaching Multiplication and Division -use interesting contextual problems instead of more sterile story problems -focus on sense making and student thinking, solving a few problems using tools such as physical materials, drawings, and equations -Introducing symbolism as a way to record children's thinking -Remainders, and what to do w/ them, is central to teaching division (discard or round to the next nearest whole number, should not be thought of as left overs) Models for Properties of Multiplication and Division (2) 1. Commutative property -the array provides a clear picture that the two represent equivalent products (3x6=6x3) 2. Associative property -this property allows that when you multiply three numbers in an expression you can multiply either the first pair of numbers or the last pair and the product remains the same 3. Zero and Identity -contextual examples help children use reason with 0 and 1 4. Distributive- factors can be split (DECOMPOSED***) on an array to show how to partition factors. compose build quantities - both addends unknown missing-part activity -part unknown, sum known Analyzing context problems (3 steps) 1. Focus on the problem and the meaning of the answer instead of on numbers. the numbers are not important in thinking about the structure of the problem. 2. With a focus on the structure of the problem, identify the numbers that are important and unimportant. 3. The thinking leads to a rough estimate of the answer and the unit of the answer. Cautions about Key Word Strategy (3) 1. Sends a wrong message about doing mathematics -children ignore the meaning and structure of the problem -mathematics is about reasoning and sense making 2. Key words are often misleading -many times the key word or phrase suggests an operation that is incorrect 3. Key words do not work with two-step problems -Using this approach with simpler problems sets students up for failure on more complex problems TRY IDENTIFYING WHAT STRUCTURE IS REPRESENTED IN EACH PROBLEM. SLIDE 6 of Chpt. 9 Semantic Equation the equation for the start problem is unknown Computational Equation the unknown is isolated on one side of the equation 3 phrases in developing fluency with basic facts. Phase 1. Counting Strategies - using object counting or verbal counting to determine the answer Phase 2. Reasoning strategies -using known information to logically determine an unknown combination Phase 3. Mastery -producing answers efficiently Number relationships provide... the foundation for strategies that help students remember basic facts. When students are not fluent with the basic facts... they often need to drop back to earlier phases; more drill is not the answer. Approaches to Teaching basic facts: (3) 1. Memorizing facts: -4 operations total over 300 facts- strong evidence that is does not work- ineffective, inappropriate applications, and inflexibility 2. Explicit strategy instruction: -support student thinking by giving them a choice of strategies to use in recall of facts -think about list of strategies 3. Guided invention: -students select a strategy based on their knowledge of number relationships - based on knowledge of number relationships, create new ways to solve the problem Teaching Basic Facts Effectively -Story problems provide a context that helps students understand the situation and apply flexible strategies for doing the computation -Example of a problem that students could use the making 10 strategy. -Explicit teaching of reasoning strategies is for students to make use of known facts and relationships to derive unknown facts Basic Facts The addends must be less than 10. The sum can be greater than 10. The factors must be less than 10. (Division and subtraction can be greater than 10) Examples: 13-8= ? 8+5=13 15÷5=? 5x3=? Separating TO Separating FROM - result unknown -change unknown Why not to use timed tests: (3) 1. Do not assess the four elements of fluency (flexibility, accuracy, efficiency and appropriately solve problems.) 2. Negatively affect students' number sense and recall of facts. 3. Take up time that could be used in more meaningful learning experiences. Effective strategies for Assessing Basic Facts 1. Appropriate strategy selection 2. Flexibility 3. Efficiency 4. Accuracy Master edition BEFORE... subtraction "friendly numbers" OR "nice numbers" VS. Benchmark numbers ends in fives or tens end in ten or multiples of five (money related numbers) Sets of tens can... ...be perceived as single entities. The positions of digits in numbers determine... ...what they represent and which size group they count. There are patterns to the way that... ...numbers are formed. the groupings of ones, tens, and hundreds can be... ...taken apart in different equivalent ways. "Really big" numbers are best understood in terms of... ...familiar real-world referents. Pre-Place Value Concepts (3) 1. one-to-one value 2. count by ones 3. then comes grouping 3 Levels of Understanding Place Value, Grouping stage 1. Unitary -Count by ones -Tell how many 2. Base Ten -Count by groups of tens and ones -Count 10 as a single item -Base ten approach when counting ones to tell how many 3. Equivalent -Non-standard base ten -Trade and regroup numbers in a variety of ways Unitize- to form into a single unit by combining parts into a whole. Integrated Place Value Model - Groupings of ten matched with numerals with manipulatives -Numerals put in place value -Written in standard form Models for Place Value -Groupable -Pregrouped -counters, cubes, sticks -base ten blocks, strips and squares, ten-frame Oral and Written Names for Numbers -Place value mats or ten frames to organize materials eg. 7 tens and 8 ones = 78 Five Levels of Place Value Understanding 1. Single Numeral -The student writes 36 but views it as a single numeral. The individual digits 3 and 6 have no meaning by themselves. 2. Position Names -The student correctly identifies the tens and ones positions but still makes no connections between the individual digits and the blocks. 3. Face Value -The students matches 6 blocks with the 6 and 3 blocks with the 3. 4. Transition to Place Value -The 6 is matched with 6 blocks and the 3 with the remaining 30 blocks but not as 3 groups of 10 5. Full Understanding -The 3 is correlated with 3 groups of ten blocks and the 6 with single blocks Patterns and Relationships with Multidigit numbers -The hundreds chart helps with the development of place value -Recognizing place-value related patterns -Relationships with landmark numbers -Number relationships for addition and subtraction Numbers Beyond 1000 -Multiplicative structure of numbers: ten in any position makes a single thing (group) in the next position to the left -Conceptualizing large numbers -Reading numbers in triples and then naming the unit (don't use the word and) Flexible methods of addition and subtraction computation involve... ... taking apart and combining numbers in a wide variety of ways. Inverted strategies are flexible methods of computation require a deep understanding of the... ...operations (commutative and associative properties). The standard algorithms are elegant strategies for computing that are based on... ...performing the operation on one place value at a time with transitions to an adjacent position. Multidigit numbers can be... ...built up or taken apart in a variety of ways to make the numbers easiest to work with. Computational estimations involve using easier-to-handle part of numbers or substituting difficult-to-hard numbers with... ...close compatible number so that the resulting computations can be done easily. Three types of Computational Strategies 1. Direct Modelling -Students who consistently count by ones need support to develop base-ten grouping concepts. -The goal is to move them from direct modeling to invented strategies derived from number sense and the properties of operations. 2. Student-Invented Strategies -Any strategy other than the standard algorithm or that do not involve the use of physical materials or counting by ones. -1st and 2nd grade -Strategies based on place value, properties of operations, and/or relationships between addition and subtraction -Develop, discuss an use efficient, accurate, and generalizable methods to compute sums and differences of whole numbers in base-ten notation. 3. Standard Algorithms Contrasts of Invented Strategies with Standard Algorithms (3) Invented strategies are- 1. number oriented rather than digit oriented 2. left-handed rather than right-handed 3. range of flexible options rather than "one right way" Benefits of Invented Strategies (7) 1. Students make fewer errors 2. Less reteaching is required 3. Invented strategies are the basis for mental computation and estimation 4. Students develop number sense 5. Flexible methods are often faster than the traditional algorithm 6. Algorithm invention is itself a significantly important process of "doing mathematics" 7. Serves students well on standardized tests Standard Algorithms Must Be Understood -If you use it, you must understand why it works and be able to explain it -Standard algorithm (once understood) are one more strategy for the students' toolbox of methods -Latin and European countries' "add tens to both" method -"equal equation" Three Common Types of Invented Strategies 1. Split Strategy 2. Jump Strategy 3. Shortcut strategy (compensation) Standard Algorithms for Addition -Begin with models only -Provide place-value mats and base-ten materials -Develop a written record Standard Algorithm for Subtraction -Begin with models only -Anticipate difficulties with zeros -Develop the written record Computational Estimation Estimation is to be able to flexibly and quickly produce an approximate result that will work for the situation and be reasonable. -Not to be taught as a guess -approximately how much, how many times Three Types of Estimation 1. Measurement-determine and approximate measure 2. Quantity- approximating the number of items in a collection 3. Computational- determining a number that is an approximation of a computation Computation Estimation Strategies 1. Front-End Methods- Front (leftmost) numbers is used and computation is then done as if there were zeros in the other positions 2. Rounding Methods- Numbers can be rounded to the same place-value for addition 3. Compatible Numbers- using friendly number to help round Making 10 Strategy Students use their known facts that equal 10 and then add the rest of the number into 10