Unit-2.-PArt-1.-Teaching-Numbers-and-Number-Sense.docx

***Teaching Numbers and Number Sense*** **The big ideas or major concepts in Number Sense and Numeration are the following: counting, operational sense, quantity, relationships and representation.** ***Counting.* The following list of concepts is presented to assist teachers understand the compone...

***Teaching Numbers and Number Sense*** **The big ideas or major concepts in Number Sense and Numeration are the following: counting, operational sense, quantity, relationships and representation.** ***Counting.* The following list of concepts is presented to assist teachers understand the components embedded in the skill of counting. It is not intended to represent a lockstep continuum that students must follow faithfully.** a. **Stable order -- the idea that the counting sequence stays consistent; it is always 1, 2, 3, 4, 5, 6, 7, 8,... , not 1, 2, 3, 5, 6, 8.** b. **Order irrelevance -- the concept that counting objects can begin with any object in a set and the total will remain constant.** c. **Conservation -- the concept that the count for a particular collection of objects remains constant regardless of how far apart they are or how close they are.** d. **Abstraction -- the concept that a quantity can be represented by a variety of objects (e.g., 5 can be represented by 5 like objects, by 5 different objects, by 5 invisible things \[5 ideas\], or by 5 points on a line). Abstraction is a difficult subject to grasp, although most students quickly grasp it.** e. **One-to-one correspondence - refers to the principle that each object being counted should only receive one count. It is beneficial for children to tag each thing as they count it and to move the object out of the way as it is counted in the early stages.** f. **Cardinality - The notion that the last count of a group of objects represents the total number of objects in the group is known as cardinality. When asked how many candies are in the set that he or she has just counted, a youngster who recounts does not grasp cardinality.** g. **Movement is magnitude - The idea that as one moves up the counting sequence, the quantity increases by one (or whatever number is being counted by), and as one moves down or backwards in the sequence, the quantity decreases by one (or whatever number is being counted by) (e.g., in skip counting by tens, the amount goes up by ten each time).** h. **Unitizing -- the idea that in the base ten system, objects are grouped into tens once the count exceeds 9 (and tens of tens once the count 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 of a number once the count exceeds 99).** +-----------------------+-----------------------+-----------------------+ | **Grade 1** | **Grade 2** | **Grade 3** | +=======================+=======================+=======================+ | providing | | | | opportunities to | | | | experience counting | | | | in engaging and | | | | relevant situations | | | | in which the meaning | | | | of the numbers is | | | | emphasized and a link | | | | is established | | | | between the numbers | | | | and their visual | | | | representation as | | | | numerals. Especially | | | | important is the | | | | development of an | | | | understanding that | | | | the numeral in the | | | | decades place | | | | represents 10 or a | | | | multiple of 10 (e.g., | | | | 10, 20, 30, 40,... | | | | ). | | | +-----------------------+-----------------------+-----------------------+ | **using songs, | | | | chants, and stories | | | | that emphasize the | | | | counting sequences of | | | | 1's, 2's, 5's, and | | | | 10's, both forward | | | | and backwards and | | | | from different points | | | | within the sequence, | | | | especially beginning | | | | at tricky numbers | | | | (e.g., 29);** | | | +-----------------------+-----------------------+-----------------------+ | **providing | **providing | **providing | | opportunities to | opportunities to | opportunities to | | engage in play-based | engage in problem | engage in problem | | problem solving | solving that involves | solving in contexts | | that** | counting | that encourage | | | strategies;** | students to use | | **involves counting | | grouping as a | | strategies (e.g., | | counting strategy | | role-playing a bank; | | (e.g., grouping | | shopping for | | objects into 2's, | | groceries** | | 5's, 10's, 25's);** | | | | | | **for a birthday | | | | party);** | | | +-----------------------+-----------------------+-----------------------+ | **providing | | | | opportunities to | | | | participate in games | | | | that emphasize | | | | strategies for | | | | counting (e.g., games | | | | that involve moving | | | | counters along a line | | | | or a path and keeping | | | | track of the counts | | | | as one moves forward | | | | or backwards). These | | | | games should involve | | | | numbers in the | | | | decades whenever | | | | possible (e.g., games | | | | using two-digit | | | | numbers on a hundreds | | | | carpet);** | | | +-----------------------+-----------------------+-----------------------+ | **building counting | | | | activities into | | | | everyday events | | | | (e.g., lining up at | | | | the door; getting | | | | ready for home);** | | | +-----------------------+-----------------------+-----------------------+ | **using counters and | | | | other manipulative | | | | materials, hundreds | | | | charts or carpets, | | | | and number lines | | | | (vertical and | | | | horizontal) in | | | | meaningful ways, on | | | | many different | | | | occasions;** | | | +-----------------------+-----------------------+-----------------------+ | **continuing to build | **providing support | | | up their | to help students | | | understanding of 5 | recognize the various | | | and 10 as anchors for | counting strategies** | | | thinking about all | | | | other numbers;** | **for counting larger | | | | numbers (e.g., | | | | counting by 100's | | | | from 101, 201, 301,. | | | |.. ).** | | +-----------------------+-----------------------+-----------------------+ | **providing support | | **providing support | | to help students | | to help students | | recognize the various | | sketch a blank number | | counting | | line that will | | strategies.** | | facilitate counting | | | | to solve a problem | | | | (e.g., to solve | | | | 23+36, they count** | | | | | | | | **23, 33, 43, 53 on | | | | the number line and | | | | then add the | | | | remaining 6 from | | | | the** | | | | | | | | **36 to make 59).** | +-----------------------+-----------------------+-----------------------+ ***Operational Sense.* Addition, subtraction, multiplication, and division are among operations that students with operational sense understand. They are able to perceive the connections between these activities and use them successfully in real-life circumstances.** **Understanding the Properties of the Operations** **Teachers must recognize the properties of operations while teaching them to pupils, which they may explain with examples and which kids at this grade level intuitively comprehend. Students in these grades do not need to know the names of the properties. Rather, these are inherent qualities that youngsters employ when combining numbers.** **The properties of addition include:** ** the commutative property (e.g., 1+2=2+1)** ** the associative property \[e.g., (8+9)+2 is the same as 8+(9+2)\]** ** the identity rule (e.g., 1+0=1)** **The properties of subtraction include:** ** the identity rule (1 -- 0=1)** **The properties of multiplication include:** ** the commutative property (e.g., 2 x 3=3 x 2)** ** the associative property \[e.g., 5 x (2 x 6) is the same as (5 x 2) x 6\]** ** the identity property of whole-number multiplication (e.g., 3 x 1=3)** ** the zero property of multiplication (e.g., 2 x 0=0)** ** the distributive property \[e.g., (2+2) x 3=(2 x 3)+(2 x 3)\]** **The properties of division include:** ** the identity property (e.g., 5÷1=5)**