Math Final Quizlet
Addition and Subtraction Conceptions
change: add to (join) or Take away (take from, leave) Part-part-whole (put together/take apart) Compare
#1 comparing, looking for variable (the difference)
#2 take away, look for total
Both look for difference, in a feature we're looking for something general enough to categorize based off of, comparing two different variables
#3 add to solve, person, just apples #4 subtract to solve (but can count to add 2 on) table, color of apples (specifies the kind of unit
#5 more than- comparing two different variables looking for specific unit (whole is students vs specified unit is boys and girls gives actual number addition
#6 finding initial unit, type out number more difficult didn't have starting number but knew 3 more hopped over subtraction
Common methods
- "make a ten" method is a general method and applies to different numbers, widely taught and used in east asia - "using doubles" method is not a general method, but widely used in the US but not in East Asia
What is the progression of a student's understanding of 10 and beyond?
- An initial concept of ten-ten as ten ones, not as a unit. Count by ones - An intermediate concept of ten, knows ten as a unit of ten ones. uses materials or representations to complete tasks involving tens - A facile concept of ten, can work with tens and ones without using materials or representations. Think of 2 digit numbers as groups of tens and ones
What categorize? There is only a finite number of categories that addition/subtraction problems consist of.
- Georgie had 5 cookies this morning, then her mom baked 5 more cookies. How many cookies does Georgie have now? - Georgie has 5 peanut butter cookies and 5 chocolate chip cookies. How many cookies does Georgie have total? - Different: time passing, problems Can be unclear so make sure to avoid that - similar 5+5 = 10 both talk about same thing: cookies, same person
misconception or misunderstanding
- believing that numbers with the same digits are of the same value - confusion regarding the value of each place
Strategies for addition
- one more than and two more than (counting on, number naming) - doubles (skip counting) - near-doubles - Using 5 as an anchor - Using length measurement - Combinations of 10 - Making 10 - Knowing and understanding place value
It is important to know arithmetic procedures, it is even more important to know and understand related rule/property
..
proving 1 dollar = 1 cent
1 dollar = 100 cents = 10 cents x 10 cents = 1 dime x 1 dime = .1 dollar x .1 dollar = .01 dollar = 1 cent
To use the "make a ten" method, students must
1) know how to find all of the partners of a given number as the first step 2) know the partners to ten for the numbers 9,8,7,6, and 5 as the second step 3) know the total 10 +n composed to be written as 1 n or know that 1 n decomposes to be 10+n
For two digit numbers beyond- teen (-ty number and some ones)
1. using base-ten blocks (some tens -ty numbers- and ones) 2. compare -ty numbers ('language names' vs 'structural names/meaning') 3. Discuss general two-digit numbers (a -ty number and some ones)
Count backwards
A child at this level can count backward by removing objects one by one or just verbally as in a countdown
Producer
A student at this level can count out objects to a certain number. If asked to give you five blocks, they can show you that amount
Representing a multi=digit number as the sum of different digits with their determined by their positions in the number (unti: one)
Additive aspect of place value
Categorizing
Without Categories more difficult than with categories
Corresponder
a child at this level can make a one-to-one correspondence with numbers and objects, stating, one number per object. If asked "how many?" at the end of the count, they may have to recount to answer
Comparison
Children need to make comparisons or choices between two or more sets or values - number comparison starts in 1st grade
counter and producer
a child who combines the two previous levels can count out objects, tell how many are in a group, remember which objects are counted and which are not, and respond to random arrangements. They begin to separate tens, and ones like 23 is 20 and 3 more
Place value is the basic principle by which we represent numbers
a digit's value is governed by the position it inhabits in a number, is rather sophisticated and was developed in India and the Arabic empire
Be careful about "unit" consistency in the selection and use of a specific feature, when counting and making some operations (eg. adding, taking away)
For example, how many eggs in the following picture?
multiplicative aspect of place value
In 28, the 2 represents 2 sets of 10 and its 2x10 and has a value of 20. the 8 represents 8 1s and 8x1
At the same time, we will talk about problem solving in number and operations. Important content topics include: Whole number, addition and subtraction, multiplication and division
In order to help students learn math and develop problem solving ability, we need to: Have an in-depth understanding of mathematics and problem structures- know and understand students' thinking, misconceptions, problem solving behavior, and their development trajectories- Develop a general understanding of possible pedagogical approaches that can facilitate students' math learning
Common error 2
a student has learned to borrow or rename, in subtraction. but the student may do so, no matter whether he/she needs to or not
What does the associative property of addition state?
a+(b+c)=(a+b)+c
What does the identity (zero) property state?
a+0=a ; 0+a=a
Relative number knowledge (2nd grade +)
Number- after : the ability to enter the sequence at any point and specify the next number instead of always counting from one, the number after is ' one more than' the number before
For each collection of three groups of addition problems below, choose one strategy from the following list that may be useful in solving each collection of three problems
One more than and two more than, doubles combinations of 10 making 10 using 5 as an anchor, near-double
Stable order: the counting sequence stays consistent
One-to-one correspondence: each object being counted must be given one count and only one count
Group recognition (subitizing: a quick recognition of a quantity without counting)
Perceptual subitizing- subitizing aids in counting on and learning combinations of numbers conceptual subitizing (advanced, happen later)
Problems are those that require you to think and have no immediate solution available, focusing on ideas and structure.
Problem solving and posing are core activities in teaching and learning mathematics
Mary and John were sitting on the grass. Some more students joined them. now there are five students. How many students joined Mary and John?
St: 2+?=5 Co:5-2=? (change add to change unknown)
Sarah had five apples. Then she ate two apples. How many apples does Sarah have now?
St: 5-2=? Co: 5-2=? Change taken from result unknown
Mary has 9 chocolate bars, and 6 bars are milk chocolate and the rest are almond chocolate. How many almond chocolate bars does Mary have?
St: 6+?=9 Co: 9-6=? Part-part-whole_addend unknown
Mary has 9 dollars, and John has 6 dollars. how many fewer dollars does John have than mary?
St: 9-6=? Co:9-6=? Compare fewer difference unknown
There were some students in the playground. Bob and Mike joined them. Now there are six students. How many students were there at the beginning?
St: ?+2=6 Co:6-2 Change add to start unknown
Mary has 6 more dollars than John. Mary has 9 dollars. How many dollars does John have?
St: ?+6=9 Co: 9-6=? Compare more smaller unknown
what does the commutative property of addition state?
a+b=b+a
What does it mean to add or subtract
adding to /joining, take away/leaving, or can be called "change"
Unitizing
The idea that , in the base ten system, objects are grouped into ten system, objects are grouped into tens once the count exceeds 9 (and into tens of tens when it exceeds 99) and that this grouping of objects is indicated by a 1 in the tens place of a number once the count exceeds 9 (and by a 1 in the hundreds place once the count exceeds 99) (See also 'relationships' and representation)
movement is magnitude
The idea that, as one moves up the counting sequence, the quantity increases by 1, and as one moves down or backwards in the sequence, the quantity decreases by 1 (or by whatever number is being counted by) (eg, in skip counting by 10's, the amount goes up to 10 each time)
Counter from any number
This child can count up starting from numbers others than one. they are also able to immediately state the number before and after a given number
counter
This student can accurately count objects in an organized display (in a line, for example) and can answer "how many?" accurately by giving the last number counted (this is called cardinality) they may be able to write the matching numeral and may be able to say the number by counting up from 1
The fundamental counting principle
To determine the number of different outcomes possible in some complex process : 1) analytically break down the process into separate stages or decisions 2) count the number of options that are available at each stage or decision 3) multiply together all of the numbers from step 2 above (for example, two stages: when there are A ways to do one thing, and B ways to do another, then there are AxB ways, of doing both)
equivalent representations are those to represent the same number in different ways
all of these ways have the same value, 28, but look quite different
movement is magnitude
as you move up the counting sequence, the quantity increases by one/x and as you move down or backwards, the quantity decreases by one/x
anchors or benchmarks of 5s and 10s
building estimation skills
Subitizing
When you look at an amount of objects sometimes you are able to just "see" how many are there
benchmark numbers: 5, 10
Why are the benchmark numbers important? Benchmark numbers are predefined numbers that assist in quantity estimations. Benchmark numbers tend to be multiples of 5 or 10 How does a given number relate to 10? The base- 10 number system is the most often used in math today. Base 10 means that the number system has actual numbers for one to nine, which start over at 10 by adding a one to one-th column of 10
Possible activity in classrooms
compare objects without measuring lengths, etc. to determine which one is longer, etc. which is longer this table or that desk who is taller, sam or maria
Common strategies
concrete materials, pictorial model, number sentence
misconception or misunderstanding
confusion about zero as a placeholder in a multidigit number - lack of clear understanding of the relationships between different units
Typical steps for students in pattern activities
copy a pattern, extend a pattern, create a pattern
rational counting
count numbers to represent an object or quantities. if a student is pointing to an object and coutnign at the same time to represent how many objects there are, it is rational counting
Typical strategies used for counting
counting on, skip counting, counting back/down
Cardinal
counting telling how many there are in a set
Mathematics for 6 years of life
enhance children's natural interest in mathematics, informal and formal experiences to strengthen children's problem-solving and reasoning processes, provide opportunities for children to explain their thinking, build on children's experience and knowledge, assess children's mathematical knowledge skills and strategies
Number naming skills (introducing tens)
for teen numbers (a ten and some ones) 1) Using base-ten blocks 2) compare teen numbers (language names vs structural names/meaning 3) discuss and name one more and one less (re: a specific teen number)
There are cows and chickens in a farm yard. there are 30 heads and 70 feet. How many cows are there? How many chickens?
head: Cows+Chicken=30 Feet: 4 cows + 2 chickens=70 Cows = 5, Chicken= 25
skip counter
here the child can skip-count with understanding by a group of a given number- tens, fives, twos, etc
Precounter
here the child has no verbal counting ability. a young child looking at three balls will answer ball when asked how many. the child does not associate a number word with a quantity
Wheat on a chessboard, 1 grain on 1st cell, 2 grains on 2nd cell, 4 grains on 3rd cell, ...double the number of grain each time for the rest of the cells. = 18,446,744,073,709,551,615 grains of wheat, weighing about 1,199,000,000,000 metric tons. This is about 1,645 times the global production of wheat in 2014. Approximately 7000 grains of wheat in one pound. There are about 25 grains of wheat in a gram. 1+2+4+8+16...(2^64-1)
how to solve 2^64-1
Unitizing
in our basten system objects are grouped into tens when counting exceeds 9 and that this is indicated by a 1 in the tens place of a number
rote counting
just ramble off numbers in order, it is the simplest number concept that children develop, and it merely consists of counting numbers sequentially, but not associating a number to a given object
1 disk = 1 move 2 disks = 3 moves 3 disks = 7 moves 4 disks = 15 moves
look at online game
Engaging ways
make numerals with clay, trace in shaving cream, write them on the interactive whiteboard, calculator keypad
What techniquest help students learn numeral names?
matching sets of objects with numerals, making sets of objects given the numeral, writing or identifying the numeral for a set of objects
typical pre-number experiences of kindergarten/first grade students
matching, sorting (rearranging objects based on an attribute) , classifying(systematically grouping of items by kind), patterns, seriation/differentiation (arranging objects in order by size), group recognition (subitizing), comparison
We can also use problem scenarios (word problems or story problems) to contextualize the meaning of addition or subtraction in a problem context
this can integrate problem solving into instruction in a real-world- like fashion and may lessen the fear of (out of context word problems
Kids must first idnetigy features to create categories if not given as teachers need to know how word problems are related
ned to identify the features explain why you identify those, and use them to classify word problems, want to make our lives easier by classifying word problems
How many pre-number experiences help kids develop a better understanding of different counting principles
no idea
the value of a digit in a number is determined by its place/position in the number
number naming
The importance of understanding the part-part-whole relationship
number relationship, teens as sets of ten and some more
Nominal
number used as a name, eg. player #8
Mental comparison
once children recognize that counting can be used to compare collections and have number-after knowledge, they can efficiently and mentally determine th larger of two adjacent or close numbers
Number relationships
one more and two more, one and two less building early addition skills
Addition or subtraction situations ar enot always joining or taking away actions, can also include
part-part-whole (decomposition, composition) comparison (taller, longer, etc.)
How many moves will it take to transfer 1,2,3,4 disks from the left post to the right post?
please remember there are four situations you need to talk about.
ordinal
position as first, second, etc
Numeral writing and recognition
reading and writing numerals is similar to teaching them to read and write letters of the alphabet, and something more
reciting
saying/recalling numbers from memory in chronological order, basically repeating a seemingly memorized sentence
Principles of counting
stable order, one-to-one correspondence, cardinality, abstraction, order irrelevance, conservation, movement is magnitude, unitizing
Common error 1
taking each position as a separate subtraction problem, and always subtracting a smaller digit from the larger one
conservation
the count for a set group of objects tays in the same no matter wheteher they are spread out or close together
order irrelevance
the counting of objects can begin with any object in a set and the total will stay the same
Abstraction
the idea that a quantity can be represented by different things (ex. 5 can be represented by 5 like objects, by 5 different objects, by 5 invisible things, 5 ideas or 5 points on a line) Abstraction is a complex concept but one that most students come to understand quite easily. Some students however, struggle with such complexity, and teachers may need to provide additional support to help them grasp the concept.
one-to-one correspondence
the idea that each object being counted must be given one count and only one count. In the early stages, it is useful for students to tag each item as they count it and to move the item out of the way as it is counted.
Conservation
the idea that the count for a set group of objects stays the same no matter whether the objects are spread out or are close together
Order irrelevance
the idea that the counting of objects can begin with any object a set and the total will still be the same
Stable order
the idea that the counting sequence stays consistent ; it is always 1,2,3,4,5,6,7,.... not 1,3,4,2,5,6,7
Cardinality
the idea that the last count of a group of objects represents thte total number of objects in the group. A child who recounts when asked how many candies are in the set that he or she has just counted does not understand cardinality
Cardinality
the last count of a group of object represents how many are in the group
Important methods
the more advanced "make a ten" method The process is facilitated in Chinese But it would need a bit more in English 14 (ten and four) yes it is a fourteen
Abstraction
the quantitiy of five large things is the same count as a quantity of five small things
Reciter
this child verbally counts using number words, but not always in the right order. sometimes they are more numbers that they have objects to count, skip objects, or repeat the same number
counting
understands that each item in the set is counted once, one-to-one correspondence with objects, and that the last number stated is the amount for the entire set (a quantity)
modeling addition and subtraction in early counting activities
we can use different concrete materials (eg counters, cubes, blocks, or straws) to model addition or subtraction computations in terms of different conceptions
Comparison without counting units
which is taller, shorter, larger, longer, etc
