Essential Understandings for Number and Numeration: PreK-2 (NCTM, 2010)

#1 Number is an extension of more basic ideas about relationships between quantities.

• Quantities can be compared without assigning numerical values to them.

• Physical objects are not in themselves quantities. All quantitative comparisons involve selecting particular attributes of objects or materials to

compare.

• The relation between one quantity and another quantity can be an equality or inequality relation.

• Two important properties of equality and order relations are conservation and transitivity.

• The equality relation between two quantities remains unchanged when one or both quantities are decomposed into parts and when one of the quantities is combined with another quantity to form a larger quantity.

#2 The selection of a unit makes it possible to use numbers in comparing quantities.

• Using numbers to describe relationships between or among quantities depends on identifying a unit.

• The size of a unit determines the number of times that it must be iterated to

count or measure a quantity.

• Quantities represented by numbers can be decomposed (or composed) into

part-whole relationships.

#3 Meaningful counting integrates different aspects of number and sets, such as sequence, order, one-to-one correspondence, ordinality, and

cardinality.

• The number-word sequence, combined with the order inherent in the natural

numbers, can be used as a foundation for counting.

• Counting includes one-to-one correspondence, regardless of the kind of

objects in the set and the order in which they are counted.

• Counting includes cardinality and ordinality of sets of objects.

• Counting strategies are based on order and hierarchical inclusion of numbers.

(NOTE: Number items are for workshop purposes and do not indicate priority)

More or Less Materials

• Ten Connecting Cubes per Student• Recording Chart• Spinner – More or Less

Directions

1. Put students in partners.

2. Each student takes ten cubes, connects them, and hides the rowbehind his/her back.

3. Break the cubes apart. Student brings out one hand of cubes andshows partner.

4. Spin the spinner. If the spinner says MORE, the student with moregets a point. If the spinner says LESS, the student with less gets apoint.

5. If students have the same amount both get a point.

6. Repeat steps 1-4 for 10 spins.

7. Once students the recording sheet, each partner counts her/his totalpoints.

8. Spin the spinner to find out which partner won!

© 2015 National Council of Teachers of Mathematicswww.nctm.org/profdev

NAME _____________________

More or Less Recording Sheet

Spin Partner 1 Partner 2 1

2

3

4

5

6

7

8

9

10

More or Less Additional Spinners

NAME _____________________

Hundred Chart

Numeral Comparing Game

Materials:

Deck of numeral cards 1-50 https://docs.google.com/file/d/0B-

v3m4ZgPmyCZTlhNWVmMzUtODdmMi00YTU2LWI5ZmItY2E5ODE2MW

U3ZTQz/edit?authkey=CLaqxYYG&hl=en# Number line from 1 to 100, Number line from 1 to 50, and/or Hundred

Chart Counters

Grade Level:

Kindergarten (use only numerals 1 – 25)

First and Second Grade (use all numerals 1 – 50)

Directions:

Kindergarten and First Grade Single Version:

a. Cards are shuffled and distributed between the two players face down.

b. Each player turns ONE card face up. The player who has the largest

number collects both cards IF they can prove it by finding both numbers on the hundreds board OR the number line.

c. The player with the most cards at the end wins the game.

First and Second Grade Double Version:

a. Cards are shuffled and distributed between the two players face down.

b. Each player turns TWO cards face up. The player who has the largest TOTAL number collects both cards IF he/she can prove it by finding

both totals on the hundreds board OR the number line.

c. The player with the most cards at the end wins the game.

Example from Second Grade

Player ONE - 35 and 22

Player TWO – 28 and 7

Player ONE has the larger total.

Number Line (1-50) (Numeral Comparing Game)

0 10 20 30 40 50

© 2015 National Council of Teachers of Mathematics www.nctm.org/profdev

Number Line (1-100) (Numeral Comparing Game)

0 10 20 30 40 50 60 70 80 90 100

Counting and Data

Materials:

Small index cards (3 x 5)

Poster paper for dichotomous graph

Large number line representation for numbers 0 to 100,

Crayons or drawing material.

Grade Level:

Dichotomous graph PreK – 2nd grade

Number Line representation (K to 2nd grade)

Directions:

1. Write your name on a small index card and add it to the

dichotomous graph with the question, “Have your ever taught 1st grade?”

2. Select a second index card. Draw a picture of one button you have on your clothing. Write a numeral to tell the total number of buttons you have on your clothing. Place it on the large number line in the most accurate place.

Days in School

NOTE: This activity is an alternative to calendar activities.

Materials:

Large 100s board grid (empty or complete with numbers); Poster paper (for manipulatives)

Index cards (for book representations) Manipulatives - ten frames (plastic or paper); counters; popsicle

sticks, dominoes; connecting tiles for arrays; paper grids or representations

Scissors and glue

Grade Level: PreK – 2nd grade, differentiated with a variety of

representations; PreK – K – use manipulative representations; 1st and 2nd grade - use book representations and manipulatives of your choice.

Directions:

This is an activity to be used everyday children spend in school. Beginning with the first day of school, a child or group of children

make representations for the specific day in school, e.g., on the tenth day of school, students would be making representations for ten.

These representations can be made with manipulatives and/or a more abstract model such as paper or drawings. In addition, these

representations should be displayed on a grid, in a book, or some other visual.

The representations should eventually include the following:

o Multiple ten frames o Popsicle sticks to represent tallies

o Base ten blocks o Dominoes

o Arrays shown with connecting square tile o Handprints

1. Select one day from the hundreds board and represent it with one

manipulative and one paper example.

2. Place the manipulative example on the large piece of butcher paper

and label it.

3. Place the paper example on an index card, label it, and post it on the

large 100s chart.

Connecting the representation of 100 to literacy and the word wall

Bean Out Directions for the Teacher

The Task

Either working independently or with a partner, students will match a numeral

with a number, a number with a number, a numeral with a number word, or a

numeral with the tally marks that represent this amount.

Directions

1. This early learning activity is meant to give students practice with

identifying (recognizing) numerals, numbers, number words, and tally marks

as representations of quantities. Students will be given a sheet with one of

these representations on it. Then they toss either a decahedron die or a

hexahedron die (number/numeral cube) and place a bean over the top of the

number or numeral that matches.

2. If this is an independent activity, model what students are to do by tossing

the number generator, identifying what is on the die, and placing a bean

somewhere on the game board. Students play the game until all of their

numbers or numerals are covered by beans.

3. If this is a paired activity, students can use bi-colored beans or bi-colored

discs. They take turns with the number generators and see who has the most

of their color once every numeral or number has been covered. If a student

tosses the number generator and cannot cover a place on the game board,

the student loses his or her turn and gives the number generator to his or

her partner.

4. This can also be a teacher-directed activity with a small group of students

who may need additional supervision or assistance.

Materials

• Bean Out game board (either numerals, numbers, tally marks, or number

words)

• Kidney beans (if played independently)

• Bi-colored beans or discs (if played with a partner)

• Number or numeral die (hexahedron or decahedron)

• Cup with a lid to put the die inside (so it doesn’t fall to the floor)

May be copied for classroom use. © 2007 by Honi J. Bamberger and Christine Oberdorf from Introduction to Connections:

Grades PreK–2 (Heinemann: Portsmouth, NH).

Talk About It

• Before playing the game, ask students to look at the numeral/number

generators and think of things to say about these.

• Have them do the same thing with the Bean Out game board.

• Ask students whether they see the same numerals/numbers more than one

time.

• Ask them how many total places there are on the game board.

• Ask them to talk about how they will take turns with their partner (if they

are playing this as pairs)

Write About It

• Because this game is geared more toward prekindergarten and kindergarten

students, it is unlikely that students will be writing about this activity.

• Instead of a written exercise, ask students to create a “Learning Experience

Story” to share how they liked playing the game and what they learned while

they played it.

Tiered Learning

• This activity is differentiated by virtue of the fact that the different game

boards work on different levels of understanding. If you have students who

just need to match a numeral with a numeral or a number with a number, give

them the game board that has the same things on it as the dice have on

them. If the die has “pip” arrangements, the game board should also have

“pip” arrangements. This is also true if you want students to match a numeral

with another numeral.

• At the second level of difficulty, students will match the numeral with a

number. So the die could have “pips” and the game board could have

numerals.

• At the third level of difficulty, students match numerals or “pips” with

number words.

• For a fourth level of difficulty, use a decahedron die and ask students to

match the numeral on the die with the tally mark arrangements.

© 2007 by Honi J. Bamberger and Christine Oberdorf from Introduction to Connections: Grades PreK–2 (Heinemann: Portsmouth, NH).

Bean Out

© 2007 by Honi J. Bamberger and Christine Oberdorf from Introduction to Connections: Grades PreK–2 (Heinemann: Portsmouth, NH).

Bean Out

six

three

five

one

6

two

one

four

six

Sorting Dominoes Turn the set of dominos over so you cannot see the pips. Take one domino and place it on start. Take turns picking dominos and placing them in the LESS than Start, EQUAL to Start, and MORE than Start columns.

START

LESS than Start EQUAL to Start MORE than Start

NAME _____________________

Finding Dominoes For each clue, find a domino whose total matches the clue. Draw a picture of the domino.

The total number of pips is an even number.

The total number of pips is an odd number.

The total number of pips is less than 8.

The total number of pips is more than 10.

Use three dominos to make a sum that is more than 30. ! ! !

Use three dominos to make a sum between 15 and 24

Use three dominos to make a sum that is less than 30 but more than 20.

!

Before and After

Crossing the decade, the century, or the millennium presents problems for many students. Students know the pattern of counting 1-10 but then often skip numbers.

Decide whether students need to focus on two-digit numbers,

three-digit numbers, or four digit numbers. This will determine

how many dice to use.

Materials:

2 - 4 Dice or place value dice (see above)

Recording sheet

Directions:

Roll the dice and create a number. Write the number in the

box.

Write the three numbers that come before your number on the blanks preceding the box. Write the three numbers that come after your number on the blanks following the box.

Before and After Recording Sheet

1. ______ ______ ______ ______ ______ ______

2. ______ ______ ______ ______ ______ ______

3. ______ ______ ______ ______ ______ ______

4. ______ ______ ______ ______ ______ ______

5. ______ ______ ______ ______ ______ ______

6. ______ ______ ______ ______ ______ ______

7. ______ ______ ______ ______ ______ ______

8. ______ ______ ______ ______ ______ ______

9. ______ ______ ______ ______ ______ ______

10. ______ ______ ______ ______ ______ _____

Out for the Count by Ian Thompson

Counts objects accurately

Associated

knowledge and

skills

Errors and

misconceptions

Questions to identify errors

and misconceptions

Teaching to address the errors and

misconceptions

Next steps in moving towards the

Key Objective

Count along and back

on a number track to

and from a given

position.

Count objects set out

in different

arrangements;

begin to recognize

small numbers without

counting and that the

number of objects is

not affected by their

position.

Count objects that are

out of reach.

1 YR

Can only begin counting

at one; inaccurately

counts objects when

rearranged; has no

consistent recognition of

small numbers of

objects; lacks systematic

approaches.

1 YR +/-

Ask the child to choose a starting

number.

What are the next two numbers?

What number comes after the

third number?

What comes before your starting

number?

The contents panel says there are

twelve. Let's check. Tip them out

and put them back. How many

are there now?

Can you tell me how many

sweets there are here without

counting them?

How many spots are in this

picture?

Throw a small number of objects

onto the table.

Can you count them without

touching them?

How did you do it?

How do you know you're correct?

Children walk forwards and backwards

along a large number track, along a number

line, on a snakes and ladders board,

counting aloud. Start from different

positions; use digit cards or dice to select

start and to move one forwards, two

backwards, etc.

Using sets of mixed objects to count and

rearrange, ask children to estimate and

check after each rearrangement.

Put small numbers of objects in familiar and

unfamiliar patterns and compare with known

patterns such as spots on dice, displays on

wall, etc.

Count sounds such as drum beats or coins

dropping into a money box; provide

counters or pencils to record marks as they

count; count counters or their marks on a

sheet; compare ways to systematically

count particular arrangements, for example

window panes, squares on a grid, chairs

around a table.

Cover up selected numbers for children to

identify as they count; move forward and

backwards from different starting points;

count aloud and silently to determine

position after a move.

Arrange a known number of objects into two

or more groups to establish that the total

remains the same; count in twos; count

objects arranged in pairs; use recognisable

patterns of small numbers, such as 3 and 4,

to introduce counting in threes and fours.

Estimate the number of objects that can be

counted reliably; check by counting, first

touching objects then without touching.

http://www.teachfind.com/national-strategies/errors-and-misconceptions-reception-addition-and-subtraction

Finding one more or one less than a given number

Associated

knowledge and

skills

Errors and

misconceptions

Questions to identify errors

and misconceptions

Teaching to address the errors and

misconceptions

Next steps in moving towards the

Key Objective

Find one more and one

less than a given

number.

2 YR

Misunderstands

meaning of ‘one more’

and ‘one less’; does not

consistently identify the

number before or after a

given number.

2 YR +/-

Here are four counters. How

many will you have if I give you

one more?

There are six spots showing on

my dice. Imagine there is one less

spot. How many spots would

there be?

What is one more than seven? …

one less than seven?

The child counts aloud from a given starting

number, stopping at particular numbers.

What number comes next? What number

comes before?

Relate counting to a set of objects, add one

more and ask children to identify how many,

similarly one less; arrange objects alongside

a number track and keep taking one away

to link one less to the number before and

similarly one more to the next number.

What is one less than six? …one more than

four?

Ask me another pair of these questions with

the same answer.

Count every other number; identify odd and

even numbers; count in twos; add and take

away two objects and find two more and two

less than a given number; build on pattern

recognition to introduce three more, three

less, etc.