# Counting - Cardinality unit planning framework

Young learners

### Overview

*Overview*:

This page provides a sequences of activities to facilitate counting and the conceptualization of cardinality. Activities to find the quantity and value of objects and create a procedure to count groups of objects, which will eventually lead to the development of cardinality:

Cardinality - Counting objects results in a cardinal result (numbered value). Beginning learners will repeat or emphasize the last word in a sequence to mean the total value of a set of objects. one two three four five six seven = seven (x x x x x x x)

## Mathematics content - mathematical discoveries

Includes enduring understanding, big ideas, land marks, structure, concepts, and generalizations for mathematical content (number value and operations, algebra, geometry, measurement, data analysis and probability).

Supporting ideas from unpacking the big idea (facts, concepts, generalizations) to include as concepts and outcomes when teaching counting can be selected from the following sources depending on the abilities of the learners. A general outcome like:

*Count groups of objects to twenty.*

Seems fairly simple. However, if it is unpacked to include accurate representation and cardinality, then it requires many supporting ideas like the ones identified below:

- Mathematical knowledge base for number value concepts, misconceptions, & outcomes
- Developmental ideas for pre number sense

## Processes - how mathematicians know

Includes enduring understanding, big ideas, land marks, structure, concepts, and generalizations for mathematical processes (problem solving, representation, communication, connections, reasoning and proof). Mathematical knowledge base

Use of *reasoning and proof *to verify mathematical ideas.

- Numbers can be listed in order and matched to objects to determine how many or a total value.

*Problem solving* is a way of understanding and communicating ideas.

*Steps (procedure) can be used to solve problems.*- Counting can solve the problem of how many.
- Strategies can be used to solve problems. I can count objects by touching them and saying a number. I need to keep track of where to start and stop and make sure I count all the objects.

Use of *representation* to understand mathematical ideas.

- Numbers in order
*represent*counting.

*Communication* of mathematical ideas

- Numbers communicate the mathematical idea of how many, quantity. I

*Connections* of mathematical ideas

- Numbers connect mathematics to objects in the world

## Habits of Mind - how mathematicians think better

Includes enduring understanding, big ideas, land marks, structure, concepts, and generalizations for a state of mind that encourages and assists in doing mathematics (mathematizing). Examples of attitudes, dispositions, values, ... concepts in the mathematical knowledge base.

- Persistence pays off.

## Resources and materials

Pencil, sequence of numbers, objects to count, activities, math notes, worksheets

## Pedagogical ideas

*Focus questions*

- How do we know how many?

*Exploration*

- Provide concrete objects to sort, classify, sequence, match to numbers, problems to solve, by exploring the matherials, questioning, and discussing the processes and p=opertions used.
- Learners share and demonstrate their explorations

*Invention*

- Teacher guides learners to explain their explorations, demonstrations, and actions to determine their solutions to
*justify*them with mathematical*reasoning*. - Teacher uses the learners ideas to suggest
*bridges*that match concrete examples (objects of different valued sets) with numerals (orally - saying 1, 2, ..., visually writing or showing a card with numerals 1, 2, ...) by physically putting them side by side or pointing to show one-to-one correspondence.

*Exploration, discovery, application, extension*

- Repeat activities with larger number values to gain skill in accuracy and efficiency.
- Develop strategies for working with larger numbers: tallies, groups of ten, using five as anchor, ten frames, ...

### Vocabulary

*total*- cardinality*count*- process of matching objects in a one to one coorespondence with a sequence of numbers in a number system.*number*- a unit of value in a number system*how many -*cardinality of a group of objects*reason*- explanation*solving problems*- use a heuristic that includes a strategy to communicate a desired explanation.*strategy*- a process used to solve a problem*represent, representation*- a communication that connects objects, ideas, and actions to a mathematical way of explaning the world.

### Activity sequence

While these are sequenced, to add structure, the order isn't meant to imply one process or skill needs to be mastered before moving to the next. Learners will construct structures and procedures for each of these ideas simultaneously and some interaction of differnt ideas, evne if inaccurate, is helpful in the eventual development of counting and cardinality. Since this is a framework see also specific activities, plans, and materials.

*Classification*

- Classification (same, different)

*Counting*

- Sequence of sounds, words… & Recognition (Numerals, words…
- Count orally
- Count objects
- Count objects with motion. Count and move hand down, count and show finger, (rate student moves finger helps the teacher to visualize the student's understanding)
- Count pointing to numerals or write a numeral and count
- Count pointing to or writing number words
- Make a number roll: A roll of paper with five or ten big dots evenly spaced on it, roll it up. Ask a student to unroll the number roll with you and have them count the numbers as each appears.
- Count without starting on one
- Count visible items
- Count items not visible, but with sensory input - hear objects as they drop onto something, feel objects inside a sock
- Count claps patterns. Vary the rate and pattern of clapping. (See hierarchical inclusion)
- Count items hidden from view. Show five hide three ask how many hidden. Or told three in box and two in other box, how many in both boxes.
- What is the next number? What was the number before 5…? What is the number after…?
- Count using counting-on or count-down.

*Number recognition*

- Point to a numeral and ask what the number is.
- Put these cards in order.
- Put these plastic numerals in order.
- Arrange those cards: Make decks of cards (or hands of cards) with numbers missing from the sequence (3, 4, 7, 9, 11, 13), randomize the deck, pass them out to the students and have students arrange them in order. Have students point to numbers and tell their numeral name. Have a partner point and tell.

*Subitizing*

- Use a non-counting strategy to count. Additive, (construct, combine) subtractive, (destruct, partition) compensation, using a know result, use multiples, five or ten as an anchor, commutative property, inverse property, or a, combination of these.
- Pattern recognition instead of counting: dominoes, die, ten frame, playing cards, regular plane figures, rectangles, arrays, finger patterns, FLASH dots or any of the above, have students air count, visualize without air counting…
- Roll six dice and arrange them in order without counting, how fast can you do it?
- Roll a die and air draw it plane figure 1 – dot, 2- line, 3 – triangle, 4 – square or rectangle, 5 – star (okay it's not a plane figure, but it's more fun air drawing than a pentagon), and 6 a hexagon.

- Make number roll. A roll of paper with numbers 1 – 10 (20) on it evenly spaced, roll it up. Ask a student to unroll the number roll with you and have them count the numbers as each appears.
- FLASH any of the above and have students hold up a card with the numeral that represents the number. (let a student FLASH the class while the teacher talks to individual students to see if there is a strategy they can use to better recall certain patterns and numbers (composite groups, connecting die with others or dominoes with die…)
- Sort and classify all of the above into groups with 1, 2, 3, 4, 5, …

*Beyond Ten*

- Ten strips
- FLASH ten frames with different arrangements in the frames. How many were occupied and how many were not. Fill with different colored counters (five blue on top, two red and three green on bottom) how many…?
- Number cover up: Cut a 12x18 piece of construction paper in half length wise and fold one of the pieces in half length wise. Divide it into ten equal rectangles. Inside the folded part with the crease on the top write the numbers 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 one the bottom so that top will fold down and cover the numbers. Cut slits so that a flap can be lifted and the number below can be seen. Use this number teen chart by calling out a number and have students find the number.
- Dots in a row: point to a dot in a row, give it a value (23) point to a dot three away and ask what that dot is. Repeat before after, could also do later with skip counting.

*Numbers to 100*

- Hundred squares or hundred charts Randomize the numbers and pass one to each child. Challenge the students to come one at time and place the numbers in its location on the chart.
- Hundred squares or hundred charts Number cover up with a completed 100 chart cover one number and as students what number is covered.
- Hundred squares or hundred charts puzzles complete and incomplete with one number to start.
- Hundred squares or hundred charts Arrow math problems give a starting number and then arrows as code to a secret number. (43 ٨ ٨ > > > ) find the number or (43 ٧ ٧ ٧ < < )

- Number roll: A roll of paper with numbers (1 – 100) evenly spaced on it, roll it up. Ask a student to unroll the number roll with you and have them count the numbers as each appears.
- Can Use the Hundred squares or hundred charts and the Number roll to do any of the counting, skip counting and point to the numbers. Make screens of different sizes to place to challenge students even more. Point to where the number is screened if a screened number is in the sequence.

*Skip Counting*

- Skip counting Alternating counting:
- (counting in multiples), arrange objects, oral, written, ten strips, number dots, 100 squares, other rectangles and squares, Add cover to get help students remember patterns through visualization or voice pattern or other mnemonic device.
- Most ideas from counting can also be used for skip or alternate counting forward and backwards.
- Arrange those cards: Make decks of cards for different multiples (2, 4, 6…; 5, 10, 15…; 10 , 20, 30…) Randomize a deck of multiple cards and have students arrange them in order.
- Unitary group: recognize a group of three only as a group of three. Repeated addition or repeated subtraction (skip counting…)
- Every one make bunny ear fingers, how many bunny ears are there? Make a W with your fingers, how many w's? …
- Count objects in different groups.
- Provide larger groups and ask to find five, 10, 15, 21…
- Watch me count… arrange before counting and count by touching, or not touching but pointing, not touching or pointing, arrange in a pattern and count.
- Composite group: (unitize) – recognize a group of three as a group of three, and a group of one. Six groups of three can think of the six groups and multiply 6*3 as well as 6 collections of three
- Watch me count… arrange in a pattern and count the pattern or skip count.
- Watch me count… arrange objects into two rows and count by twos.
- Counting-on use screen cover, 8, 9, 10, 11. (8+3)
- Teacher count students repeat (1…3; 4..6…) forward backward starting and stopping with different numbers, can have student repeat again in their head or out loud. All together, or out loud by self…

- Arrange manipulatives with a recognizable pattern one color and rest another pattern. Ask how many in recognizable pattern and see if they will count-on.
- Use number strips, ten frames….
- Count and turn, Counting off in a line, Counting off in chairs, Counting in the circle game, Objects in the box game, piggy bank game, Spill the beans game, Pendulum game, Jump rope count, Ball bounce count, Number lines, Counting on game put out 5 blocks cover up two, count on, Silent count to rhythm, _ _ 2 3 4 5 _ _ 2 3 4 5; Counting backwards. (Use variations of all above), Put pictures in sequence (baking cookies,...), Snap and clap, Stand up sit down count...; Double circle, walk in opposite directions, count with hand slaps;
- Put number words in order on cards together, in groups, by self on desk, one card to each child have line up according to the numbers. Have different groups do it at the same time…

*Counting Backwards*

- Counting backwards is more difficult for children. Use same ideas above. Provide enough thinking time, hints, visual prompts, what ever and patience.
- All above only alternate/ take turns… teacher one student two….
- All above only up and down….
- Count stop and have child say next number (in order: 1, 2, 3, ___ 4, 5, 6 ____; out of sequence 1, 2, 3, _____ 6, 7, 8 ______ )
- All above only do backwards.
- If I start counting at six and count three, where would I be?
- Measuring as length or counting random length units
- Mark a jar as scoops of rice are added to it.
- Number value as length or counting random length units
- One more one less
- Two more two less

*One-to-one correspondence*

- Do Dot flash or dot plates.
- Move them off the grid. Have students stand on a grid. Have them take steps from one square to the next by telling them how many squares to move relative to two points marked at two different points on the grid 3 from the x, 2 from the polka-dot. Until they have all moved off of the grid.
- Making equal shares, groups, sets… Have collections in bags, boxes, on cards, or inside loops of yarn… with the number of objects predetermined and a collection of more objects for the students to make a second collection that matches. Use similar objects and use different objects. Challenge them to try and convince the class and themselves that they have equal shares.
- Have the same type of activity as above only place two cards or … and one collection of predetermined amount of objects and have students divide it into two equal collections. Challenge them to try and convince the class and themselves that they have equal shares.
- Magic Sack: Give each student a magic sack with less than 9 or 10-20 objects depending on the student. Ask the student to remove the objects and count them. Have them put the objects in the sack and tell how many objects are in the sack. Dump them out and count them again. Repeat as desired.
- From sack to number trail: Provide students with a number trail with numbers 0- 9 or 0-20 and have students remove a set of up to 20 objects placing each object beside a number on the trail. Keeps track of the count and double-check the total as necessary. Students trade sacks and repeat the procedure. If students can skip count or count on have students line up the objects on the counting number on the number trail. (If count 12 by 5, 10, 11, 12; then line up five objects on the number trail at the numbers 5 and 10 with one at 11 and one at 12.
- Repeat the activities with a pocket chart, placing numerals as count...
- Use a 100 chart
- Use of tallies...

See more activities, plans, and materials.

### Assessment

Provide students with a problem and score their work with the Cardinality scoring guide.

Recording materials or scoring guides

- Rote counting record sheet
- Counting objects record sheet 1
- Counting objects record sheet 2
- Conservation of numbers record sheet
- Recognize patterns record sheet
- Cardinality record sheet
- Add scoring guide for problem solving
- Add scoring guide for representation
- Add scoring guide for reasoning and proof

*Need to add more scoring guides with levels for other dimensions or categories like cardinality.*

### Student work sheets

See specific activities, plans, and materials.