EDE 4123 Test 1
What is a traditional algorithm?
A rule or procedure for solving a problem; if standard algorithms are "taught" or discovered, base-10 models should be used to prove that the algorithm results in a correct answer
What is a basic fact?
For addition and multiplication are the number combinations where both addends or both factors are less than 10. For subtraction and division are the corresponding combinations (e.g. 15-8=7).
What is number sense?
Howden- "A good intuition about numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms."
How can number sense be developed?
It can be developed gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms
What addition strategy? sum is greater than 10 9+3=11
Make a ten/up over 10
What subtraction strategy? double + 1 9-5=4; 15-8=7
Near doubles
What multiplication strategy? 1x9; 2x9; 3x9....
Nifty nines
What addition strategy? 1 addend is always 1 (9 facts) 1+1; 1+2; 1+3
One more than
place value development (ideas of 10 of one thing is ONE of something else)
Pre Base Ten Concepts -based on a one-more-than or count-by-ones approach to quantity Basic Ideas of Place Value -equivalent representations; fewer than the maximum tens- not the base ten representation Developing Place-Value Concepts -count as ones and tens and ones and discuss why the result is the same **"ten ones make one ten" two meanings of ten, three meanings of hundred
What is groupable proportional?
When the ten can actually be made or grouped from the single pieces Ex.- when students put 10 beans in a cup, the cup of 10 beans literally is the same as the 10 single beans
What is an equivalent representation?
When you can represent a number in as many ways as possible except the base 10 representation
What multiplication strategy? teach through word problems 6 sets of nothing; no sets of 6 sizes
Zeros and ones
What is a proportional model of place value?
a model for ten is physically 10 times larger than the model for a one, and a hundred model is 10 times larger than the ten model.
What is front end estimation?
compute with first digits, then adjust 480 x 7 is 400 x 7
compare (addition and subtraction structure)
difference between two amounts - how much less - how much more
What subtraction strategy?
ten frame facts
What is computational fluency?
when students can complete computations mentally and quickly
part-part-whole (addition and subtraction structure)
(put together and take apart problems) one part is unknown, two parts are unknown, or answer is unknown
equal groups (multiplication and division structure structure)
(repeated addition problems) how many group and sets on ONE item
what NOT to do when remediation for basic facts is needed
-Drill is not an intervention because if students do not know their facts they likely lack number sense and reasoning strategies -use lengthy timed tests -use public comparisons of mastery -proceed through facts in order from 0 to 9 -move to memorization too soon
what to do when remediation for basic facts is needed
-Explicitly teach reasoning skills -Provide hope -Inventory the known and unknown facts -Build in success -Provide engaging activities
Skills necessary for accurate counting
-Producing the standard list of counting words in order -Connecting this sequence in a one-to-one correspondence with the items in the set being counted -Assigning each item counted one and only one counting word
How should a place value mat be used in teaching addition and subtraction of multi-digit numbers?
-Require an understanding of regrouping -Begin with base 10 materials on place value mats -Always have students model and record so the written procedure becomes apparent For addition, both numbers will be modeled on the place value mat, then pushed together. For subtraction, only the top number will be modeled, then the second number will be removed, leaving two parts- the "answer" and what remains of the whole.
Why should ten frames be used?
-Ten frames are used in order to help children relate a given number to other numbers, specifically 5 and 10 to support thinking about relationships with various combinations of numbers. -Helps develop computation skills
the three ways in which remainders can be handled
-The remainder is discarded, leaving a smaller whole-number answer -The remainder can "force" the answer to the next highest whole number -The answer is rounded to the nearest whole number for an approximate result
Progression of number development
-subitizing -development of counting skills -cardinality -factors of counting
What subtraction strategy? 4-1=3; 4-3=1
1 less than
What are the three developmental phases towards computational fluency?
1. Direct modeling 2. Invented strategies 3. Traditional algorithms
3 strategies for computational estimation
1. Front end 2. Rounding 3. Compatible numbers
Differences between invented strategies and traditional algorithms
1. Invented strategies are number-oriented rather than digit-oriented (245 is seen as 2 hundreds, 4 tens, 5 ones) 2. Invented strategies are left-handed rather than right-handed (computation starts with the leftmost digit (largest value), not the rightmost digit (smallest value) 3. Invented strategies are flexible rather than rigid (the strategy is based on the numbers)
4 addition and subtraction structures
1. Join 2. Separate 3. Part-part-whole 4. Compare
benefits of invented strategies
1. Students make fewer errors 2. Less re-teaching is required 3. Students develop number sense 4. Are the basis for mental computation and estimation 5. Flexible methods are often faster than the standard algorithms 6. Algorithm invention is itself a significantly important process of "doing mathematics" 7. Serve students well on standardized tests
why "key words" should NOT be taught
1. The key word strategy sends a terrible wrong message about doing mathematics. The most important approach to solving any contextual problem is to analyze it and make sense of it using ALL the words. 2. Key words can be misleading. 3. Many problems do not have key words. 4. Key words do not work with two-step problems or more advanced problems
4 multiplication and division structures
1. equal groups 2. Multiplicative comparison 3. array 4. combination
What subtraction strategy? 6-2=4; 6-4=2
2 less than
What multiplication strategy? use a clock 1-9 5x1=5; 5x4=20
5 facts
How is the remainder handled? The rope is 25 feet long. How many 7-foot jump ropes can be made?
Answer: 3 jump ropes (discarded)
How is the remainder handled? The ferry can hold 8 cars. How many trips will it have to make to carry 25 cars across the river?
Answer: 4 trips (Forced to next whole number)
How is the remainder handled? Six children are planning to share a bag of 50 pieces of bubble gum. About how many pieces will each child get?
Answer: about 8 pieces for each child (rounded, approximate result)
What is an invented strategy?
Any strategy other than the traditional algorithm that does not involve using physical materials such as base-10 blocks or drawings, but does include written recordings and some mental methods, when appropriate
What property? Adding zero to a number doesn't change its identity
The Zero/Identity Property of Addition
What is pregrouped proportional?
Commonly shown in textbooks and are often used in instructional activities Ex.- base-ten materials (they cannot be taken apart or put together; when 10 single pieces are accumulated, they must be exchanged or traded for a ten)
What multiplication strategy? using known "helping" facts to derive remaining facts 8x3=24... 8x2=16+8=24
Derived facts
What addition strategy? strategy uses images (7 facts) 2 hands= 5+5; spider= 4+4
Doubles
What multiplication strategy? rewrite addition facts (factor is always 2) 4+4=8=4x2=8
Doubles
What subtraction strategy? 6-3=3; 8-4=4
Doubles
What addition strategy? one addend is one more than the other (6 facts) 3+4=7; 2+3=5
Doubles plus 1
What addition strategy? one addend is two more than the other
Doubles plus 2
What subtraction strategy? 3-3=0; 4-0=4
Facts with zero
activities that can be used to develop number sense that were completed in class
Promoting number sense- Fill in the blanks activity. -Story about MSU celebrating 132nd anniversary -Some numbers were Ordinal (place) and others were Cardinal (set) The shapes of numbers -Given colored paper of squares 1-10 -Had to find 2 ways to group them -Represented addition sentences
What addition strategy? using 5 as an anchor
Remaining facts
What is mental math and what does it produce?
Students do not use paper and pencil for their calculations. Mental math produces exact answers.
What subtraction strategy? make a 10 facts become think addition
Sums greater than 10
What addition strategy? creates new basic facts other than ones we already know 2+8=10; 3+7=10
Ten frame facts
How should ten frames be used?
Ten frames are to be laid out in front of the students horizontally and be provided with about 10 or so counters. Students are then to construct their own understandings and relationships between numbers. saying the number of spaces on the card instead of the number of dots, say one more than the number of dots, saying the ten fact, adding the flashed card to a card they have at their desk (for enrichment) For example some students will learn to adjust numbers by adding on or taking off only what is required while others will clear the ten frame and construct a new number from a blank frame.
What property? multiplying 3 factors in any order results in the same product
The Associative Property
What property? Doesn't matter whether the first pair is added first or if you start with any other pair of addends Adding three addends in several different orders
The Associative Property of Addition
What property? 4 groups of 2 and 2 groups of 4 result in the same number of items
The Commutative Property
What property? Doesn't matter the order of the addends- stays the same answer Pair contextual problems that have the same addends in different orders
The Commutative Property of Addition
What property? a(b+c)= a times b plus a times c; add first and then multiply OR multiply each and then add
The Distributive Property
What property? multiplying by 1 does not change the identity of the number
The Identity Property
What property? 0 sets of anything is 0 OR any number of sets of size 0 is still 0
The Zero Property
What is direct modeling?
The use of base-10 models (manipulatives or drawings) along with counting to directly represent the meaning of an operation or story problem
comparison (multiplication and division structure structure)
There are TWO different sets or groups; the unknown can be the product, the group size, and the number of groups.
What addition strategy? 1 addend is always 2 (8 facts) 6+2; 5+2
Two more than
3 ways of counting sets of objects (listing)
Unitary: counting by ones Base 10: counting by groups of tens and ones Equivalent: counting with non-standard base 10 groupings
join (addition and subtraction structure)
initial amount + a change in the amount = the result
combination (multiplication and division structure structure)
involve counting the number of possible pairings that can be made between two or more sets (things or events)
What is computational estimation?
it involves some mental computation with at least two estimated quantities. It is not a guess.
separate (addition and subtraction structure)
largest amount - change amount = smaller result than largest amount
What is a nonproportional model of place value?
models where the ten is not physically 10 times larger than the one; are not used for introducing place-value concepts; used when students already have a conceptual understanding of the numeration system and need additional reinforcement, or by older students who may need to return to place-value concepts
What is compatible numbers estimation?
particularly with division, adjust numbers so a whole number answer results 413 x 24 is 400 x 25
What is rounding estimation?
round each factor and then compute 7 x 485, round 485 to 500
array (multiplication and division structure structure)
row and column structure; determines area, length x width; square units