ITQ C 3 Number & Quantity Subtizing Compose and Decompose
to 5
Slide 2
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)
Slide 3
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)
Slide 4
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.
Slide 5
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.
Slide 6
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
Slide 7
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.
Slide 8
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-K2 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.
Slide 9
Slide 10
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.
Slide 11
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.
Slide 12
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.
Slide 13
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 oclock? 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.
Slide 14
Beginning Number Activities CMW
Slide 15
List of Games and Activities 1.How Many Pennies Did I Hide?
2.Empty the Bank-Level 1 3.Empty the Bank-Level 2 4.How Many More
to Ten? 5.Egghead 6.Back and Forth
Slide 16
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?
Slide 17
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. 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 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.
Slide 18
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. 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. 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.
Slide 19
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.
Slide 20
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.
Slide 21
Slide 22
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 wont 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.