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ITQ – 3 C Number & Quantity Subtizing Compose and Decompose to 5 Terminology 1 • Subitizing means to instantly recognize a quantity without counting. – For example, if I roll the pair of dice, I instantly recognize the 4 and the 2 without counting. • Cardinalities means the number of elements in a given mathematical set. • Exactly the same/not exactly the same/the same but…(ways to analyze objects to match or sort) • Match (group items that are the same or that have the same given attribute) • Sort (group objects according to a particular attribute) • • • • • • • • • Terminology 2 “How many” (with reference to the counting quantities or sets) Hidden partners (embedded numbers) Counting path (With reference to the order of count) Number story (stories with add to or take from situations) Zero (understanding the meaning of, write and recognize) Number sentence (3 + 2 + 1) 5-group Rows/columns (linear configuration types) 5 + n pattern Number path – ***** – * ***** ** ***** *** ***** **** • 1 more (e.g., 4.1 more is 5) • 1 less (e.g., 4. 1 less is 3) ***** ***** Overview Topic A • The classification activities we are doing today allow students to analyze objects and observe their world and articulate their observations. • Reasoning and dialogue begin immediately. – “These stacks are exactly the same.” – “These stacks are the same but a different size.” Topic B • Student will recognize cardinalities as yet one more lens for classification. – “I put a pencil, a book, and an eraser, 3 things, in the backpack for school; I put 5 toys in the closet to keep home.” Main Ideas Most traditional textbooks move almost immediately from skills of counting and set-to numeral matching addition. This causes many children to count their way completely through the first and second grade. Addition and subtraction facts, mental mathematics, and a general sense of numbers should be built on a set of rich relationships for small numbers. More time need to be spent on number sense and these relationships. Effective activities will help students see these relationships. Number Sense Number sense can be described as a good intuition about numbers and their relationships. It develops 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. No substitute exists for a skillful teacher and an environment that fosters curiosity and exploration at all grade levels. Hilde Howden. Arithmetic Teacher, Feb., 1989, p.11 • From the moment students enter school, they practice the counting sequence so that when counting a set of objects, their attention can be on matching one count to one object, rather than retrieving the number words. Mental Math The foundations for mental computation begin as early as preschool and kindergarten. All preservice or practicing teachers should gain a knowledge of the groundwork that is established in the pre-K–2 grade span. They need to value what goes on in the early grades and understand the foundations for the topics they should be working on in grades 3 and above. There are two ideas most clearly connected to the early number relationships: one- and two-more-than (-less-than) is related to one more (or less) ten or the adding and subtracting of tens. The ideas of part-part-whole and anchors of 5 and 10 combine as illustrated in the following sequence: Parts of 8 leads to parts of 80 (with tens) leads to parts of 80 using two 5s (as in 35 and 45) leads to any two parts of 80 and to missing parts of 80 (37 and what makes 80) leads to 80 − 37 in a mental computation. Extensions to Numbers up to 20 It is not reasonable to simply do the same type of activities for numbers between 12 and 20 that were done for numbers 10 and less. Use approaches that can either extend some of the relationships on smaller numbers to larger numbers or develop new ones (one-more-than relationships). Use connect relationships between say 6 and 7 to 16 and 17. The anchors of 5 and 10 have obvious extensions to 15 and 20 and subsequently to all numbers ending in 5 and 0. Students can learn that teens are sets of tens and some more (a special part-part-whole idea) well before they understand ideas of base-ten place value. Counting: An Important Problem-Solving Tool continued There is a paradox here: being able to recite the numbers in order does not necessarily mean the child is able to count. A child may know the counting sequence from one to ten perfectly and still be unable to use this sequence to count a group of objects. Knowing and successfully using the sequence are two separate, but sequentially related skills. A child might look you straight in the eye and rattle off the numbers from one to ten and yet look down at a group of four objects, ouch one of them, and say, “One, two,” touch another object and say, “three,” skip over a third object entirely, and move his or her finger up and down near the last object, and say, “four, five, six.” This child has not yet put the words and objects in one-to-one correspondence. She or he needs many opportunities to experience saying one number word with one motion or one object. Counting with skill and understanding is an important problem-solving tool in mathematics. Children should start these activities by counting to one number beyond the point where they begin to have difficulty. When they become confident counting to this number in the sequence, one more number should be added to the sequence until the children build gradually to ten. There is no need to rush or push children ahead quickly to ten, for this pressure produces anxiety rather than learning. A firm foundation is laid gradually in an atmosphere of confidence and support; growth takes time and the children need adults to enjoy, not rush, their growing. As children explore relationships for numbers from 10 and less to those between 11 and 20, there are instructional approaches that can either extend relationships with smaller numbers to larger numbers or develop new relationships. The one-more-than relationship can be used to help connect the relationships between, for example, 6 and 7 to 16 and 17. The anchors of 5 and 10 have obvious extensions to 15 and 20 and subsequently to all numbers ending in 5 and 0. Young children can learn that the teens are sets of ten and some more (a special part-part-whole idea) well before they understand the comparatively sophisticated ideas of base-ten place value. The double and near-double relationships are new but important ways to think about larger numbers. Add a Unit to Your Number Write a number on the board. Now suggest some units to go with it and ask the children what they can think of that fits. For example, suppose the number is 9. “What do you think of when I say 9 dollars? 9 hours? 9 cars? 9 kids? 9 meters? 9 o’clock? 9 hand spans? 9 gallons?” Spend some time in discussion of each. Let children suggest units as well. Be prepared to explore some of the ideas either immediately or as projects or tasks to share with parents or guardians at home. Beginning Number Activities CMW List of Games and Activities 1. 2. 3. 4. 5. 6. How Many Pennies Did I Hide? Empty the Bank-Level 1 Empty the Bank-Level 2 How Many More to Ten? Egghead Back and Forth How Many Pennies Did I Hide? Materials: Players: 2 Players 1 Penny Strip Rules: A Penny Strip is laid side up in front of two players. One player covers a number of pennies (with a hand or a piece of cardboard), and asks the partner “How many pennies did I hide?” The other partner has to figure out how many were covered and then checks the answer by uncovering the hidden pennies. If they guess correctly, they get a point. Partners take turns being the one who hides the pennies in the Penny Strip. Harder Variation: When players get good at this game they can figure out how many pennies are covered. They cannot see the Penny Strip. They have to do it with their eyes closed! Ask; “I have 4 pennies left. How many did I hide? or “I hid 6 pennies. How many did I have left? Counting Game: Empty the Bank: Level 1 Materials: Players: 2 to 4 Players 1 Game board One die 1 marker for each player 75 beans and Game Cards Rules: Distribute 5 beans to each player and place the rest of the beans on the Bank. Shuffle the Game Cards and place them face down on the Card slot. Players roll the dice to decide who goes first. All players start at the square. Each player rolls the dice and moves that many circles as the dice says. 2 If the Player stops on a circle with a number they get that many beans FROM the bank. If the Player stops on a circle with a square inside they GET A CARD and count how many objects has. They player keep the card. At the end of each turn the player has to count how many things they have. Decide before you start how many times you will go around or stop when the bank runs out of beans. The winner is the player with the most things (beans and object on cards) that can be counted at the end of the game Counting Game: Empty the Bank: Level 2 Materials: Players: 2 to 4 Players 1 Game board (White and Gray Squares) One die 1 marker for each player 75 beans and Game Cards Rules: Distribute 5 beans to each player and place the rest of the beans on the Bank. Shuffle the Game Cards and place them face down on the Card slot. Players roll the dice to decide who goes first. All players start at the square. Each player rolls the dice and moves that many circles as the dice says. 2 If the Player stops on a circle with a number they get that many beans FROM the bank. If the Player stops on a circle with a square inside they GET A CARD and count how many objects has. They player keep the card. 5 If the Player stops on a gray circle with a number inside they GIVE CARD that many beans TO the bank. At the end of each turn the player has to count how many things they have. Decide before you start how many times you will go around or stop when the bank runs out of beans. The winner is the player with the most things (beans and object on cards) that can be counted at the end of the game. How Many More to Ten Activity Anchors to 5 and 10 Materials: Players: 2 Players 1 2 X 5 Playing Board 10 Objects Number Cards with numbers 0-10 Rules: The Playing Board, the objects, and the number cards are placed in front of two players. One player chooses a card and puts that amount of objects on the Playing Card and says, “I put 6 objects on the Playing Board. How many more do I need to complete the Playing Board?” The other Player has to find the answer (They can count the empty squares). They get one point for a correct answer. Egg Head Materials: Players: 2 Players 1 Playing Board 2 Dice 1 Marker for each player Rules: The object of the game is to move the each marker from egg to egg by finding the sum of the dice. Put the markers on START. Roll the dice to see who begins. The greater sum starts. Roll the dice. If the sum of the two numbers rolled is in the egg, move your marker to the egg; otherwise, stay on START. Take turns rolling the dice. After each roll, move your marker to the next egg by finding the sum of the numbers rolled. You may only move your marker if the sum of numbers rolled is in the next egg. The first player to reach the last egg is the winner. Back and Fourth Materials: Players: 2 Players 1 Playing Board 1 Die 1 Marker for each player Rules: The object of the game is to move the each marker from your starting number to 9 by finding the sums and differences. Roll the die to see who begins. The higher number beings the game and chooses to be Player 1 or Player 2. Player 1 begins on 0 Player 2 starts on 18. Take turns rolling the die. You may either add or subtract the number rolled from the number you are on. For example, if you are on 12 and you roll 6, you may move your marker to 6 (12-6) or to 18 (12 + 6). On each turn you must move your marker, but you may not move it off the number line. That means that sometimes you won’t have a choice whether to add or subtract. Continue playing until a player land on 9. The first player to land on 9 is the winner.