Instructional Ideas or Activities for Number Sense & Place value

Classification

Before anything is counted an act of classification must take place. A person must group or classify the objects in the set to be counted. Cookies on a plate, fingers on a hand, toes in a shoe, toys on the floor, and so on. See classification concepts & misconceptions.

Counting

Counting begins with the memorization of numerals in their counting order. However, concepts of sequence, one more, one less, more than, less than, one-to-one-correspondence, cardinality, and conservation of number are developed slowly as children think about what numbers are and their relationships. See prenumber sense, counting, and number sense.

  1. Echo count, listen and repeat number names in order.
  2. Sequence of sounds, words… &
  3. Recognition activities of numerals, words …
  4. Count orally, from memory, by ones forward and backward
  5. Count stop and have child say next number: 1, 2, 3, ___ (4, 5, 6 ) ___; starting with numbers not one: 2, 3, ___6, 7, 8 ___
  6. Count visible objects
  7. Count objects with motion.
  8. Count and move hand up or down,
  9. Count and show number value on fingers, (rate student moves finger helps the teacher to visualize the student’s understanding)
  10. Count pointing to numerals or write a numeral and count
  11. Count pointing or writing number words
  12. 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.
  13. Count without starting on one
  14. Count items not visible, but with sensory input hear (claps, beat of drum ... ) , feel (items in a sock ... )
  15. Count clap patterns. Vary the rate and pattern of clapping. (See hierarchical inclusion)
  16. 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. Or have count items, place them in a container with a lid and ask to count how many are in the container (object permanence, conservation of number).
  17. What is the next number? What comes next? 1, 2, 3, _.
  18. What was the number before 5…?
  19. What is the number after…?
  20. Count and turn, Counting off in a line, Counting off in chairs, Counting in the circle game,
  21. Pendulum count
  22. Jump rope count
  23. Ball bounce count
  24. Polka-dot numbers in a line count
  25. Silent count to rhythm, _ _ 2 3 4 5 _ _ 2 3 4 5;
  26. Snap and clap
  27. Stand up sit down count...
  28. Double circle, walk in opposite directions, count with hand slaps
  29. Count using counting-on or count-down. Place five objects on a tray or table, ask the student to count them, when the student counts three objects, stop, cover the three objects with a hand, ask how many are under the hand, then ask to continue and count the rest (two).

Counting Backward

Counting backwards is more difficult for children. Use same ideas above. Provide enough thinking time, hints, visual prompts, whatever, and patience.

Ideas from above only alternate/ take turns… teacher one student two….

All above going 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.

(Use variations of all above)

Numeral and number recognition

  1. Point to a numeral and ask what the number is.
  2. Give students a randomized set of cards, have them turn a card over and say read the numeral.
  3. Have students point to numbers and tell their numeral name. Have a partner point and tell.
  4. Dice bingo (junior version) role a die and match the die to the numeral on a 2x3 grid (1, 2, 3, 4, 5, 6).
  5. Match dot cards with numeral cards, word cards…
  6. Match dot plates with numeral, ordinal, & number cards for students.

Number sequence

  1. Put these cards in order. Give students a set of cards. Numerals, words, combination and ask them to put them in order. Source of cards for students.
  2. Put plastic numerals in order.
  3. Deal a deck of number cards, one card to each child and have them line up according to the numbers. Have different groups do it at the same time…
  4. Not playing with a full deck: Form decks of cards (or hands of cards) with numbers missing from a sequence (3, 4, 7, 9, 11, 13), randomize the deck, give a deck to a student or pair and have students arrange them in order.
  5. Roll six dice and arrange them in order without counting, how fast can you do it?
  6. Roll a die and air draw it
  7. Show a 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.
  8. 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.
  9. Screened number roll. Show students a number roll and as it roles hold a screen to block a few numbers as they roll past.
  10. 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…)
  11. Sort and classify all of the above into groups with 1, 2, 3, 4, 5, …
  12. 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…?
  13. Number cover up: Cut a 12x18 piece of construction paper in half length wise (6 x 18) 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 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 for beginning students and 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 for more experienced students. Then numbers will be on the bottom so that top will fold down and cover them. Cut slits so a flap can be lifted and the number below can be seen. Use this number ten or number teen chart by calling out a number and have students find the number.
  14. Number cover ups can be made for larger numbers. Make covered number lines for higher numbers ( 41, 42, 43, 44, 45, … 50) and do as above.
  15. Dots with numerals on the wall. Have students close their eyes, say a number that is on a dot, ask students to point to where they think it is, have them open their eyes and see what numbered dot they pointed at.
  16. Dots in a row. Put a number of dots in a row without numbers on them. Say a number and have students point to a dot in a row that could represent that number.
  17. More dots in a row. Put a number of dots in a row without numbers on them. Point to a dot and assign it a value (8), ask students to point to a dot two away. Ask what that dot might be. Repeat before after, could also do with skip counting.
  18. Dots on a number roll. Students can count each number or skip count as the numbers roll past. Can point to the dots or not point. Can screen some of the dots as they roll past. Could challenge students even more by asking them to wait and point to a specific dot when its number rolls to the center. Could challenge more by adding a screen.

Zero

  1. Use bags of objects to sequence numbers starting with an empty bag for zero. Ask students to put the number of objects in the bag that is written on the bag. Ask how many objects are in the zero labeled bag.
  2. Later combine objects in one bag to zero objects in another bag. Suggest several problems with zero as an addend and include both kinds of problems 3 + 0 = 3 and 0 + 3 = 3. Have others make and share problems.
  3. Have students snap Unifix cubes together to make towers, trains, … for numbers (1-10). Call out a number and have them hold the tower or what ever up. Ask students to hold up zero.
  4. Ask how many elephants are in the room… zebras…

Subitizing

Subitizing is being able to identify the value of a group of objects by looking at it without actually counting the objects. See development & additional information and resources.

  1. Pattern recognition instead of counting display patterns from the following and instantly try to recognize a value: dominoes, die, ten frame, playing cards, regular plane figures, rectangles, arrays, finger patterns,
  2. FLASH dot plates or electronic dots or any of the above, have students air count, visualize without air counting …

Skip Counting or alternate counting

  1. Skip counting or counting in multiples. Arranged objects (eyes, legs, dog legs ... ) ten strips, number dots, 100 squares, shapes: triangles, rectangles and squares (count sides or angles). Count orally and written. Add a cover to 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.
  2. 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.
  3. Composite group: (unitize and composite) – recognize a group of three as a group of three and a group of one. Six groups of three can be thought of and worked with as six groups and multiply 6*3 as well as 6 collections of three Watch me count… arrange in a pattern (objects into six rows with three in each row) and count by ones and skip count by threes. Unitary group, is when a group of three is recognized or worked with as one (unitary) group (three). Repeated addition or repeated subtraction (skip counting…) can be thought of in two ways: single groups of objects and groups of groups. This becomes important when students get to the operations (addition, subtraction, multiplication & division).
  4. Every one make bunny ear fingers, how many bunny ears are there?
  5. Make a W with your fingers, how many w’s? …
  6. Count objects in different groups in different multiples.
  7. Counting-on with a screen cover, 8, 9, 10, 11. (8+3) Teacher count students repeat (1…3; 4..6…) forward and 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…
  8. 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.
  9. Use number strips, ten frames….
  10. 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.

Counting-on

  1. Count using counting-on or count-down. Place five objects on a tray or table, ask the student to count them, when the student counts three objects, stop, cover the three objects with a hand, ask how many are under the hand, then ask to continue and count the rest (two).
  2. Counting on game put out 5 blocks cover up two, count on, Silent count to rhythm, _ _ 2 3 4 5 _ _ 2 3 4 5;
  3. 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…
  4. Arrange manipulatives with a recognizable pattern (one color) and additional similar manipulatives in a random pattern (a different color). Ask how many in the recognizable pattern and see if they will count-on.

One-to-one-correspondence & matching

Matching, would seem to be a natural procedure for students to determine if groups are equal. However, it is rarely used.

One-to-one-correspondence
Connect each counting numeral with a one-to-one-correspondence to a cardinality.

  1. Use a flip ring with cards numbered 0, 1, 2, through the number you want to use in the game. Have students start at zero. Then when the bell rings have them flip a card and place a cube. As students flip and add ask them to point to the cubes represented by the current number.
  2. Giant number line: Have students play the preceding game. As they go through the chart have them record all numbers on a class grid and place the number of Unifix cubes beside it. Look for patterns (inclusion).
  3. Repeat the above and put the numbers on a matrix (4 - 4X4, 5 - 5X5 ...).
  4. Matrix patterns: Have students arrange chairs in a grid. Have them put students on the matrix in a pattern. How do they compare to the numbered squares?
  5. I’ll say a number you FLASH a pattern (1x1; 1x2; 1x3; 2x2; ... )
  6. Move on 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. For example move 3 from the x, 2 from the polka-dot. Background for hundred chart and arrow math.

Cover - up one

Materials: Large foam die, path with squares or other shapes or grid of shapes, and collection of objects to use as markers to place on the dots of the die.

Procedure

  1. Roll the large foam die.
  2. Match one of the objects to each dot on die by placing an object on each dot of the die.
  3. When all the dots are covered transfer each object one at a time to the squares on the path or grid.
  4. Repeat the steps until the path or grid is completely covered.

Cover - up two

Materials: Die, staircase of squares from one to ten (55 squares in all), 55 multilink cubes, that will fit on the squares in the staircase.

Procedure

  1. Roll die
  2. Select the number of cubes that matches the number on the die.
  3. Place the cubes on the staircase.
  4. Continue until the staircase is filled.
  5. Students could use large foam die for one-to-one-correspondence like in Cover-up 1. Students that can count can place the squares and tell how many more are needed for each stair (hierarchical inclusion and addition).

Snacks for all?

Materials: snacks and two colors of objects, one to represent students and another to represent snacks (multilink cubes green and yellow, or cutouts of children and milk cartons…).

Procedure:

  1. Ask the students if there are enough snacks (milk boxes, juice containers, cookies, crackers…) for every person in the class.
  2. Have them match yellow cubes to students and green cubes to snacks.
  3. Then ask them how they can tell.
  4. Could line up the blocks or snap them together and measure to see if there was enough.
  5. Same activity can be used to help students construct an understanding of odd and even.

Cardinality with one-to-one-correspondence

  1. Use number strips, ten frames….
  2. Randomly place students and have them find how many were placed by: Count and turn, Counting off in a line, Counting off in chairs, Counting in the circle game,
  3. Count objects. Objects in the box game, piggy bank game, spill the beans game, Pendulum game, Jump rope count, Ball bounce count,
  4. Number in a polka-dot line ,
  5. Counting backwards. (Use variations of all above),
  6. Put pictures in sequence (baking cookies,...), Snap and clap, Stand up sit down count...;
  7. 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…
  8. Have students snap Unifix cubes together to make towers, trains … for numbers (1-5) or (1-10). Call out a number and have them hold the tower or whatever up.
  9. Make number books. Records of toothpicks, tiles, pattern blocks
  10. Concentration: Take some boxes of the same size and put pairs of different amounts of objects under all the boxes. Have the students turn a pair of boxes over, if the pairs match they can remove the boxes and the objects. If the pairs do not match they cover the objects with the boxes and it is the next person's turn.
  11. Object cost grid. Make a grid with numerals from 1 - 50. Challenge students to locate an objects that cost the amount as each numeral on the grid. Pictures of the object or the object itself can be added to the grid.
  12. Bean toss and record results in Bean book and on a class chart.
  13. Estimate and count: Bring in bags of objects. Beans, M&M's, peanuts, safety pins, Q - tips, sugar cubes, crackers, cookies, macaroni ... Students estimate, out number on slip of paper, count items by putting them into cubs of ten.
  14. Count jars of objects. Use the same kinds of objects above, place them in jars, have the students guess and check. Switch jars, objects, and repeat.

Conservation of number

The activities below are often assigned as counting activities. However, young students will not conserve number as they learn to count. To be able to conserve number a host of concepts related to prenumber sense and counting must be conceptualized before number sense can include conservation of number and then other conservation skills.

  1. Arrange [(1-5), (1-10), or (0-20)] objects (toothpicks,pattern blocks, tiles, jewels, wooden blocks, beans, junk box objects, unifix cubes, pattern blocks, ... ) in different patterns to find patterns with equivalent values (cardinality) and record quantities by gluing squares, toothpicks, or other objects into a learning log.

Greater Than and Less Than

Young students are easily confused with the < and > symbols. This does not mean they have not conceptualized a concept of more, greater, less, and equal. It is helpful to have students invent a rule and nemonic. Then, display it so they can refer to it from time to time. Example, nemonic for symbol, the symbol is tricky and can be used for both, but it is greedy and tries to eat the larger number. Therefore, when it is like 10 > 2 it is read as greater than. When it is like 2 < 10, then it is read as less than.

  1. Make a more or less book. More pencils less windows.
  2. Comparing length of names on graph paper (first, last).
  3. Record height, compare each child, use cord arrange in height
  4. Compare mass with teeter totter (one to another)
  5. Compare handfuls. How little is a handful?
  6. Play tic-tac-toe and keep score with Unifix cubes
  7. Play squares and keep score with Unifix cubes
  8. Compare one jar’s capacity to another jar and pair as greater and/ or less..
  9. Put objects on a balance and mark pairs as greater and/ or less
  10. Order by volume: Put progressive amounts of rice in a collection of jars and mark each jar with a marker.

Ordering sets

Have the students grab a handful of objects (selected so the numbers would range 15-30) and put them on a tray so that they could convince another person how many there are without counting them by ones. Have all students in the class bring their tray to the front and set it on tables or the floor. Without talking each student will place their tray so that the trays are in order according to the number of objects on each. If a tray has the same number of objects as another, then the tray will be placed perpendicular to the others. If students question the placement of a tray have them convince each other how they know where it belongs. Students should settle disputes without counting or simply referring to numbers.

Activity two

Have students grab another handful of objects and place them on trays. This time place the students in pairs and have them decide which tray has more or less. Have them come to the front of the class and share their trays and tell how they know which has more or less (again without counting or just by referring to numbers). Repeat until all students have convinced each other which of the pairs is more and which is less or equal. Then have students place number and word cards to write a mathematics sentence to tell what they showed. (24 is greater than 22) or (Twenty-four is greater than Twenty-two). Students draw pictures and write a sentence of their inequalities into their journal.

Activity three

Have students repeat activity one and have them put word cards (less than, equal to, and greater than) between the trays. Either during or after completing the activity the students will notice that only the cards less than and maybe equal to were used (if the trays were placed in sequential order from small to large). Challenge them to tell you what could be done so that greater than cards could be used. Rearrange or re-grab and repeat. Then challenge students to figure how to put the trays so both greater than and less than cards could be used. Do so.

Activity four

Repeat activity one-three with the use of symbols <>= (could have students invent eat larger rule).

Activity five

Have students grab a handful and make a right or wrong equality on the trays, share with the rest of the students, and let the other students tell how they know if it is right or wrong.

Activity six

Give students number cards and ask them to stand in line in order (ascending, descending) use numbers 0-110, could use random numbers or multiples of 5, 10, or others.

Activity seven

Worksheets with equality problems

Hierarchical inclusion - counting - on and skip counting

  1. Clap patterns clap a 1-2 pattern and have students count (1, 2,3) also for same 1-2 clap pattern count (1, 1,2). Or reverse it. Clap 1-2 pattern and students count (1, 1,2) and ask them how claps first (1), how many second (2) and how many all together (3) Other patterns 2-1, 2-2, 3-1, 1-3, 2-3, 3-3, 3-2.
  2. Have the student count a specific number of beans into your hand. Hide some in the other hand, show the student how many are left and ask how many am I hiding?
  3. The hand game: Take a known amount of beans and shake them in two hands, then open one hand and ask how many beans in that hand, then ask how many beans in the second hand, and slowly open it.
  4. Peek through the wall game: Put a specific number of beans in a line. Then take a piece of cardboard cut like a picture frame and insert it somewhere in the line and say how many are on either side of the frame. 3 and 2.
  5. Lift the bowl game: Put a specific number of beans under a bowl. Take some out and put them on top. Ask or say how many are on top. Then ask how many beans are under the bowl and slowly lift the bowl. Also Cave or cover up game.
  6. Put a row of objects on a card. Say the total of objects on the card and take away a number. The teacher then covers that number with another card and asks, How many are left?
  7. Two handed FINGER FLASH two fingers on left hand and three on right…
  8. Two colored dots (3 red and 2 black)
  9. Put (1, 2, or 3) counters or blocks on a table for students to see, cover them with a card, put another group of (1, 2, or 3) counters or blocks in a line next to the previous and cover them. Ask how many? Repeat adding more … Start over and use different amounts… Can vary by covering some and leaving some uncovered.
  10. FLASH ask how many dots or other… they saw. Can anyone describe it another way? Another way?

Card game More (war)

Materials: Deck of cards with the suit symbol beside the number blacked in so that only the numeral remains and the face cards removed, ten bags with matching numbers 1 - 10 on the outside.

Procedure

  1. Deal all 40 cards to two students face down one stack for each player.
  2. Have each turn over a card.
  3. Both cards go into the bag with number of the largest card. (six beats four, so goes into the six bag).
  4. Continue until all cards have been turned over.
  5. Then explore the bag to see what is in each.
  6. Could also play by turning two cards and adding them.

Cave or cover up game

Materials: Blocks that equal the largest number to Cover-up

Procedure:

  1. Set the number of blocks that you want the student to count up to in a row on a table.
  2. Ask them how many blocks there are (five blocks).
  3. Cover up an amount of blocks with your hand (three) or butter dish (see two).
  4. Ask how many are under your hand or dish.
  5. Then lift your hand or the cave and repeat the combination (3 + 2).
  6. Repeat with the same amount of total objects, but different amounts in and out of the hand.
  7. The teacher can vary the game by saying how many to hide.
  8. Must be able to count on and have hierarchical inclusion.

Role to empty the bowl

Put a number of objects in a bowl that you want students to count back from.
Have them roll a die or dice, count or subitize the number of dots, and remove that many objects from the bowl. Continue to roll till all objects are out of the bowl.
Record number of rolls.
Record in pairs and save so the class can discuss results. ...

How many ornaments fit in a box?

What kinds of boxes and different assortments could be made?
Cubes, packages, and boxes.

Going beyond ten and moving toward operations

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.

  1. Ten and more or teen numbers worksheet and Fold book: Ten and more .PDF
  2. Blank hundred square or empty hundred pocket chart. Randomize a set of the numbers and pass one to each child. Challenge students to come one at time and place their number where they think it belongs on the chart.
  3. Number cover up with a completed 100 chart cover one number and ask students what number is covered.
  4. 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.
  5. Hundred squares or hundred charts puzzles complete and incomplete with one number to start.
  6. Hundred squares or hundred charts
  7. Flash Arrays and have students tell the number of squares in the array. If students rely on counting squares in arrays, screen part of the array, but be sure to show at least two sides. Have students tell how many squares. Later show top and side separately.
  8. 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 < < )

 

Place Value

 

 

Dr. Robert Sweetland's notes
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