Activities to support Early Arithmetical Strategies
Where are they now?
Students:
- may say some number words in the correct sequence and relate some number words to the process of counting
Where to next?
Students:
- count visible items to find the total count
- say one number word for each item counted (one-to-one correspondence)
- label collections by understanding that the last word we say when counting tells us how many (or how much) we have
Outcomes
The following activities provide opportunities for students to demonstrate progress towards the following outcomes. A student:
MAe-1WM describes mathematical situations using everyday language, actions, materials and informal recordings
MAe-2WM uses objects, actions, technology and/or trial and error to explore mathematical problems
MAe-3WM uses concrete materials and/or pictorial representations to support conclusions
MAe-4NA counts to 30, and orders, reads and represents numbers in the range 0-20
How?
A note about daily counting opportunities for students at the emergent and perceptual stages
Teaching Point
Please note: These activities are almost identical in both the emergent and perceptual sections as daily, meaningful experiences with counting are incredibly important. These learning opportunities could also be adapted to support students as they progress towards more efficient strategies by adapting the number range and complexity of questions asked.
In order to retain new learning, students need multiple exposures to concepts and regular, frequent opportunities to practice their skills. As such, teachers need to provide a broad range of learning experiences that ask students to practice their skills and apply their understanding. Learning the forward and backward number word sequence and counting with understanding are imperative to becoming efficient users of mathematics. This selection of activities provides suggestions for incorporating opportunities to count each day. It is not the intention that each activity is conducted each day.
1. Daily counting: How many people are here today?
- Count how many people are away by counting down from the total number of students in the class.
- Count how many students are at school and use that information to work out how many students are away by modelling how to count down to work out the difference. For example, “We counted 16 people who are here today. We have 18 people in our class. If we count down from 18, we can work out how many people are missing.” Teachers should ask students if there is another way to work out how many people are away and model a range of ways to use counting to solve problems.
- Decide upon a category and have the students sort themselves into their two groups (e.g. boys and girls, long hair and short hair, black shoes and not black shoes, etc.) Count how many students are in each group and then work out how many people there are altogether.
2. Daily counting: Which collection has...?
Show the students two or more collections of objects (you could use photographs or objects shown on the interactive whiteboard, real collections of objects, collections students have made, etc.) and ask questions such as:
- Which collection has the fewest objects? How do you know?
- Which collection of objects has the largest amount? How do you know?
- Are there any collections of objects that have more than 3 items? How do you know?
- Are there any collections of items that have less than 5 items? How do you know?
- Are there any collections of objects that have the same amount? How do you know?
- How do you know which collection is...?
3. Daily counting: What do you think?
Pose a question to the students and graph student responses by having them move their name card into the appropriate column on a class column or picture graph blank organiser (this could be an interactive whiteboard file or drawn on a large sheet of cardboard). Discuss the data collected with students by adding the appropriate labels to the graph and asking questions such as:
- How many people chose a particular category?
- What would happen if...? (for example, 4 more people had chosen a specified category)
- How else could we have represented the information we collected?
- How can we interpret the information we collected? What can we say as a conclusion? (The teacher should model how to interpret the information presented in a display by thinking out aloud, providing examples of the sorts of language students can use when communicating mathematically)
This activity allows students to practice counting strategies for addition and subtraction as well as the forward number word sequence.
Do you like ice-cream cake more than normal cake?
4. Daily counting: Counting circles
This activity is designed to develop reasoning and communicating skills in students and requires rich classroom dialogue to build understanding.
Explain:“Let’s count around the class circle. If we count by 1s, start from 6 (for example), what number might (choose a student somewhere in the circle, for example Sara). Sara say?”
“Let’s use our skills as mathematicians to estimate and reason before we count. That means we need to use information to help us work out what number Sara might say. We can’t count just yet because we want to use our skills in reasoning as mathematicians. So let’s look at where Sara is standing. She is about half way around the circle. Since there are 20 people in our class, I think the number Sara might say will be a teen number, that is, a number between 10 and 20. I think that because we are starting at 6 and when we get to Sara it will be about half the way around the circle. I think that means it will still be less than 20. What do you think and why? Do you agree? Do you disagree? What do you estimate?”
The teacher should ask students to share their thinking with someone standing next to them and then select some students to share their ideas with the whole class. This could provide opportunities to listen to students’ understanding about the magnitude of numbers, their reasoning skills and the language they use to explain their thinking as well as clarify ideas. Estimations and reasoning could be recorded.
After the discussion, the teacher leads the students in discovering what number word Sara (for example) will say when we count by 1s starting from 6.
Discuss if the estimations made were reasonable. It is important to avoid saying that estimations were “wrong” or “right” and should instead be “rejected” or “confirmed” based on their validity.
Variations
- Have the same conversation about what number word a particular student may say when counting backwards
- The teacher records each number on an empty number line as students say the number words
Developing one-to-one
Lay out a row of double-sided counters (within an appropriate number range), all showing the same colour. Ask students to watch what is being done in order to count the number of items in the row. Model one-to-one correspondence by saying one number word as each counter is turned over until the entire row has been counted. Ask the students to recall the last number word spoken. Explain that the last number word spoken, tells us how many counters we have. Discuss with students the process undertaken to count, that is, that one number word was said for each item that was counted. Repeat a number of times with different amounts, asking students to explain the actions taken to find out how many counters are in the collection. Finally, discuss with students how each time counting takes places, the number words are said in the same order. Provide students with matching rows of counters, and together, practice by saying number words together and turning counters at the same time.
Teaching Point
Teachers should regularly model the same process with counting backwards from 10 to 0. Share with students that the same counting principles apply regardless of the things we count or the direction we are counting in.
Feather drop
Show the students a row of canisters, such as empty cups (or individual sections of an egg carton). Begin with three cups and build up to five or beyond. Tell the students you have three feathers and then place them, one at a time, into the cups, saying one number word for each item dropped into the cup. The students repeat the process of placing the feathers into the canisters.
Teaching Point
All of these learning experiences can be modified to support learning in backward counting. For feather drop, start with feathers in the canisters and say a number word for each feather that is removed from a canister.
Why?
It is important to build the process of assigning one number word to each object in both forward and backward counting.
Teaching Point
These activities are used to develop one-to-one correspondence. Initially, students could directly match items to amounts shown using dots or images to develop one-to-one matching as their understanding of numerals develops. As skills develop, introduce the sequence of number words when completing the activity.
Egg carton drop
This activity is similar to Feather drop. For this activity replace the feathers with counters and ask students to drop them into an egg carton, matching one counter to each cup. Cut the egg carton into parts, one part containing three cups, one containing four cups and one with five cups.
Collect me
Provide each student with a dot card. Have the students collect the correct number of objects from around the room to match their card. For example, a student with the dot card “three” might collect three pencils. Students could be asked to collect as many collections of different objects as possible, using zip lock bags to keep their collections in. To prove their thinking, students could check their collections before asking a partner to check again by counting the collections made.
Egg game
Provide each student with a base board (BLM - Egg, BLM - Egg Pieces) displaying an outline of an egg. Cut a second egg outline into pieces to create a jigsaw. The first student rolls a dice with a standard dot pattern and selects a piece of the “egg” jigsaw displaying a corresponding dot pattern. This piece is placed on top of the game board. Continue the game until all children have completed their egg.
I feel
This activity is designed for a pair of students. Instruct one partner to sit opposite a row of numeral or dot cards. The other student stands behind the partner and taps him or her on the shoulder a certain number of times. The student who is sitting counts the number of taps and picks up the numeral card which corresponds to the number of taps.
Mothers and babies
Duplicate and cut out cards displaying a set of bear cubs in the range one to ten (BLM - Mother Bears, BLM - Baby Bears, or alternatively use Ten Frames and Numeral Cards 1-6). Construct a second set of mother bear cards displaying numerals in the range 1-10. Have the students select a “cub card”, count the cubs and match the card to a corresponding mother bear card. Students continue until all of the cards have been matched.
Beehive
Construct base boards displaying beehives with numerals written on them (BLM - Beehive, BLM - Bees). Create a supply of cut-out bees. Have the students state the numeral written on the hive and collect the correct corresponding number of bees. They then attach the bees to the hive, using paperclips or fold-back clips. Other students in the group should count the bees to confirm that the number of bees matches the numeral on the hive.
Ten pegs
Provide each student with ten pegs and a length of cardboard displaying ten dots (BLM - Ten Strips). Have the students take turns to roll a dice and count the dots on the dice. After counting the dice pattern the student then takes a corresponding number of pegs and attaches them to the cardboard strip, matching each peg to a dot. Play continues until the students have attached pegs to all the dots on their strip of cardboard. They need to roll the exact number needed to finish. The activity could be extended by using 2 ten strips to count to 20.
Alternatively, start with ten pegs clipped on to a ten strip. Students take turn to roll a dice and remove the corresponding number of pegs from their ten strip. Play continues until the students have removed all of the pegs. They need to roll the exact number needed to finish. Students could be supported to record each turn by taking photos using a mobile device, narrating each photo which could later be shared using the interactive whiteboard.
Variations
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Change the cardboard strip to show ten teddies in a line (BLM - Ten Teddies). The student rolls the dice, counts the dots and collects the correct number of plastic teddies to place along the cardboard strip of teddies
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Provide each student with a coat hanger and ten clothes pegs. The students take turns to roll a dice displaying dot patterns and attach the corresponding number of pegs to the coat hanger. They continue until all ten pegs are attached to the hanger. The exact number needed to form ten must be rolled to finish. Alternatively, have the students start with 10 pegs on their coat hanger. Students remove the corresponding number of pegs after rolling a dice. They continue until all of their pegs have been removed. The exact number must be rolled to finish. Students should be encouraged to record their game. They could be provided a graphic organiser to record each move (see example below)
Ten frames
Provide each student with a ten frame (BLM - Ten Frames) and ten counters. Students take turns to roll a dice displaying dot patterns, count the dots and place the corresponding number of counters onto the ten frame. The exact number needed to complete the ten frame must be rolled to finish.
Why?
Instant recognition of dot patterns can lead to strong visualisation or mental images for students. This visualisation will assist them in counting and problem-solving tasks.
Teaching Point
Have students fill the ten frame horizontally across the top row first. This emphasises “fives” in the ten frame. Use the interactive whiteboard or a document camera to introduce and model ten frame activities.
Variation
- Substitute paperclips for counters. Students roll a dice and collect the corresponding number of paperclips. They then slide them onto the ten frame squares.
Ten strips
Construct ten strips (BLM - Ten Strips). Have partners take turns to select a numeral card and place a corresponding number of counters on the ten strip board
Variation
- Give students a pile of empty ten strips. The students roll a dice, collect the correct number of counters and place them on the ten strip. Attach numeral tags and/or number word tags to the end of each ten strip before rolling a dice and creating a new strip. After a given length of time, students could take a photo of the collections. Then have students order their various ten strips from the most to least, or least to most, and take a second photo. Photos can be shared using an interactive whiteboard. Alternatively, students could record their work by drawing their collections
Teddy bear race
Construct playing boards for each pair of students (BLM - Teddy Bear Race). Line up plastic teddies at the start of the playing board, so that one teddy is on each numeral. Ask the students take turns to roll a dice and move a teddy one space each time its corresponding numeral is rolled. Play continues until all teddies reach “home” on the playing board.
Cupcake papers
Present a collection of cupcake or muffin papers to the students. Write a numeral in the range one to ten on the inside of each paper. Instruct the students to place the correct number of items, such as counters, shells, beans or rocks, into each cupcake paper according to the numeral that is written on the bottom.
Alternatively, take the counters out of the containers, practising saying one number word for each item removed until 0 counters remain.
Candle holders
Place candle holders upright in six containers. Each container should hold a different number of candle holders within the range of one to six. Students take turns to roll a standard dice. After counting the dots on the dice, students count out a corresponding number of candles. Students then find a container with the same number of candle holders and place the correct number of candles in the candle holders. Students continue until candles have been placed in all of the holders.
Teaching Point
Containers could be made from strawberry punnets or plastic food containers filled with foam, plasticine or play dough to enable the candle holders to remain upright.
Why?
Students need to be able to match a number word to an object in order to count perceived items. They also need to be able to recognise and identify numerals in order to record totals.
The number train
Construct a train from Lego® blocks or cut-off milk cartons. Display a numeral on each carriage of the train. Have the students place the correct number of items, such as Lego® people, counters or blocks, into each carriage. Instruct partners to count the items in each carriage to confirm that the collection of items corresponds with the numeral.
Make a zoo
Construct clear plastic containers, such as strawberry punnets, displaying numerals in the range one to five and collections of zoo animals for each number indicated on the containers. For example, one elephant, two camels, three tigers, four zebras and five monkeys. Direct the students to sort the animals and place each collection into a plastic container, ensuring that the number of animals matches the numeral card on the container.
Alternatively, take the animals out of the containers, practising saying one number word for each item removed until 0 animals remain.
Flowers in the vase
Label small plastic orange-juice containers with numerals. Ask the students to place a corresponding number of plastic flowers into each vase. Students should count the flowers in each vase to verify the count.
Alternatively, take the flowers out of the containers, practising saying one number word for each item removed until 0 flowers remain. Straws or pencils could also be used to complete this activity.
Colourful clowns
Construct clown baseboard (BLM - Clown, or use empty Ten Frames, Ten Strips or double 5 domino card). Students roll a dice and collect a corresponding number of counters. Students use different coloured counters for each roll of the dice. Instruct the students to place the counters onto the circles. On the next roll the student repeats the process, using counters of a different colour. The process continues until all circles are covered. To finish, students must roll the exact number needed to cover all the circles. When all circles are covered, students make statements about their clown’s juggling circles. For example, “My clown has three red circles, five green circles and two yellow circles.”
Note: The BLM may need to be enlarged depending on the size of the counters
Variations
- As a variation, students could start with all of the circles covered by the same coloured counters. Have the students roll a dice and remove the corresponding number of counters. Have students record their turn by using a scaffolding organiser that has the following:
“I had ___ counters. Then I removed ___ counters. Now I have ___ counters.”
“I had ___ counters. Then I removed ___ counters. Now I have ___ counters.”
“I had ___ counters. Then I removed ___ counters. Now I have ___ counters.” - Students could also write addition number sentences to match their board e.g. 3 and 2 and 5 is 10
Popstick designs
Distribute a pile of popsticks to the students. Each student is asked to count out five popsticks and use them to make a design. Students continue making different designs with five popsticks. Have the students record and/or share their designs by using an iPad or other tablet device. Each design should be checked by using counting to ensure 5 popsticks are used. Teachers should model counting forward and counting backward to check. Teachers can use this activity to discuss how many designs were created and how different they can be from one another despite them all using the same number of popsticks.
Variation
- Use other material such as tiles, coloured paper squares, toothpicks, straws, pattern blocks or connecting cubes to make patterns
Investigating collections
Show students a collection of 5 large cubes (large dice with the faces covered could be used as well as blank pocket dice). Together, count how many large cubes there are, explicitly modelling how we say one number word for each item counted. Take a photo of the cubes in the arrangement they are in. If immediately able to, share the photo on an interactive whiteboard and write the numeral 5 next to the image). Rearrange the cubes (such as stacking them in a tower) and ask students to count how many cubes are present now. Model how we say one number word for each item we count. Take a second photo of the cubes stacked as a tower, adding to the projected screen if possible, again recording the numeral 5. Rearrange the cubes again (for example, spreading them out with a significantly larger distance between them), counting the cubes once more to work out how many cubes there are altogether. Take a third photo. Repeat a number of times, arranging the large cubes so that they are stacked in different combinations, sometimes placing them close together and sometimes placing them further apart. Examine the photos on display and explain to the students that you notice that regardless of how the cubes were arranged, the total number of cubes stayed the same. Reinforce the role of number words in telling us how many we have and that the arrangement of the objects does not matter.
Why?
Students need to know that a set number of objects has the same numerical quantity, no matter how they are arranged.
A counting investigation
Ask 6 students to come to the front of the room to make a group. Explain to all of the students that we are going to work out how many students we have in the group. Model how to count each student using one-to-one correspondence, saying one number word for each student counted. Write the corresponding numeral on the whiteboard so it can been seen by students.
Explain to the students that you are going to count the number of students in the group again, but this time, in a different order, to see what happens. Repeat the process again.
Explain to the students that you noticed something whilst you were counting and wonder if they noticed something too. Explain: “I realised when I was counting the students, I always said the number words in the same order. Did you notice that too?”
Count again to demonstrate and ask students if they noticed the same thing. Then explain:
“I also noticed something else. I have counted the number of students in the collection a few times now, and no matter what order I counted them in, the total was always the same. Did you notice that too?”
Demonstrate by counting the group again, using a different order each time. Explain:
“That makes me wonder something. I wonder if that is always true... that it does not matter what order I count them in? I think we should investigate this idea.”
Provide pairs of students with a small amount of objects to count (within an appropriate range for each pair). Ask the students to count the number of items they have and
record the last number word they say (“because this tells us how many objects are in our collection in total”) on a mini whiteboard or tablet device. Ask the students to share with the class how many items they have in their collection. Ask the students to count again, counting the items in a different order and recording the total number of objects they have. Repeat a third time.
Ask the students to discuss what they noticed with their partner and share back with the class. Elicit from students that the order we count items in does not make a difference to the total number of items we have.
Variation
- Using a tablet or video camera, have the students record each count as a video, capturing their one-to-one correspondence, order of count and number words they are using. Students should then be asked to share this with the class when they explain what they investigated
Tower battles
Using a spinner (in the range of 0-3, showing numerals and dice patterns) and connecting blocks, have students work in pairs in a race to build a tower taller than their partner’s tower. Students take turns to spin the spinner and build a secret tower. With each spin, students take corresponding number of blocks from the central pile, counting out aloud to prove to their partner that they are collecting the correct number of pieces. Students take turns until each player has had 5 turns. Students then discuss who they think might have the tallest tower, record their predictions and then reveal their towers. Students directly compare the two towers to work out whose tower is the tallest. Students can then count to see how many blocks they collected in total. Play best out of three.
What’s in the square?
On a large sheet of cardboard construct a 5 x 6 grid. Along the top row of the grid write the numerals one to five , starting from the second column. Down the first column, starting from the second row, draw a different shape in each square.
Provide the students with cut-out shapes, pattern blocks or attribute blocks, corresponding to those drawn in the first column. The students count out the correct number of shapes and place them appropriately on the grid.
Alternatively, use other classroom objects in place of the shapes such as pencils (in various colours), different shaped counters, beads, etc.
Pick up chips
Construct a deck of dot pattern cards for numbers one to six with four of each pattern. Distribute five counters to each player and place 100 counters in a central pile. Shuffle the cards and place them face down in the centre of the table. Have the students take turns to take a card from the pile and pick up a corresponding number of counters from the central pile to add to their collection. That is, a student who draws a “three card” collects three counters from the central pile. The activity continues until all cards have been drawn. The student with the most counters wins. Vary the game by adding “magic numbers”. For example, if a “magic two” is drawn, the student takes two counters from all other players. If a “crazy five” is drawn, the player puts five counters back into the central pile. A player who runs out of counters is out of the game.
Variation
- Each student starts with 31 counters each. Students take a numeral card and remove the corresponding number of counters from their collection. The player who runs out of counters first is the winner
Classify and count
Provide students with a small range of objects (within their counting range) that have at least one different attribute (such as pattern blocks, transport-shaped counters, buttons, teddies etc.) Ask the students to sort their objects into categories. Have the students order their collections from the largest to the smallest number of items.
Provide students with numeral cards and ask them to find the appropriate numeral card that matches their categorised collections. Have the students take a photo of their work and using the interactive whiteboard, ask students to share their collections, the categories they sorted their objects into and the order they placed them into. As a class, discuss the similarities and differences between the collections.