Activities to support Early arithmetical strategies
Where are they now?
Students:
- count objects they can see, hear or feel but not those in concealed collections
- count perceived items but are not able to visualise the set when the concrete material is removed
- count with understanding
Where to next?
Students:
- count items without relying on visual or concretere presentations apply the cardinal principle when counting
- understand the order we count objects in a collection in is irrelevant
- add visible collections together
- subtract items from a visible collection demonstrate the difference between two groups of objects by using the language of comparison
- partition and combine numbers in a variety of ways to support their knowledge of numbers
- understand “is the same as” and “is equal to”
- begin to recognise and recall number combinations to ten
Teaching Point
As a preliminary activity concentrate on numbers one to five. Many students will already have a strong sense of the number, however, they may not have a true understanding of the number five. Investing time exploring part-part-whole partitioning and pattern structures up to 5 will support students in developing strong mental images and understanding
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
MAe-5NA combines, separates and compares collections of objects, describes using everyday language, and records using informal methods
HOW?
A note about daily counting opportunities for students at the emergent and perceptual stages
Teaching Point
Perceptual Stage
Please note: these activities are almost identical in both the emergent and perceptual sections as daily meaningful experiences with counting 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 word sequences 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 work out how many people are away (such as counting up, counting back, etc.)
- 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 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 whiteboards, 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 collection of objects that have 3 or more items? How do you know?
- Are there any collection of objects that have less than 5 objects? How do you know?
- Are there any collection of objects that have the same amount? How do you know?
- How do you know which collection is...?
- What would a collection of zero look like?
- Do any of the collections have the same amount? How can you prove it?
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.
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: “Lets count around the class circle. If we count by 1s, starting from 6 (for example), what num- ber 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 exam- ple) 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
- Discuss counting in composite numbers, such as counting by 2s. In this case, discuss how when we count by 2s, we say every second number word. Ask the students to discuss which of the numbers recorded on the number line should be said if we counted by 2s around the circle and whether a particular student (such as Sara for example) would still say their number word. Have the students share their thinking. Circle the numbers they suggest. Before starting the count again, explain to students that if we are counting by 2s, the 2 things we are count- ing should be grouped together so they should link arms with a partner standing beside them and instead of both people saying a number word, only the second person should say their number word. This could be modified to show counting by 2s from an odd number as well as counting by 2s staring with an even number
Ten Frame Patterns
Model the use of ten frames on an interactive whiteboard. Explain how we fill a ten frame (from left to right, top row and then bottom row) and can use them to investigate numbers and counting. Explain why a ten frame is called a ten frame (“because when it is full, we know that is shows 10”). Show a few different amounts using the ten frames and discuss
and record the number combinations that students see using mathematical language. Ask students questions such as; “How many dots do you see? How do you know? How many more to make ten?” Provide students with their own ten frames and a set of ten counters (all the same colour). Practise filling the ten frame, discussing observations that can be made (i.e. “When the top row is full, there are always 5 or more counters in the ten frame”, etc.)
Variations
- Ask the students to close their eyes and imagine a number pattern in their mind. Then have them make the pattern with counters on individual ten frames
- Ask the students to close their eyes and imagine a number pattern on a ten frame. Have them describe the pattern to a partner who has to make it on their individual ten frame which the first student is unable to see. Have the students check the ten frame pattern
- Using an interactive whiteboard, arrange a nominated number of counters into different patterns on the ten frame. Discuss which number patterns were the easiest to identify and why they think that
- Complete subtraction tasks using the ten frames. For example, arrange nine counters on the ten frame. Ask students to determine how many counters would need to be taken away to leave six. Demonstrate the strategies of counting down to and counting down from a given number
- Use two ten frames to work with numbers to twenty
- Using an interactive whiteboard, display a pattern for a number in the range of one to ten on a ten frame. Have students close their eyes, remove or add one counter to the ten frame. Show the ten frame again and ask the students to state how many counters there are now and how many were removed
Teaching point
Note: By modelling this learning activity first, the teacher is able to demonstrate the language students could use to describe the patterns created in their ten frames. Recording the appropriate mathematical languages that it is visible for students will support them in using mathematical language.
Why?
Ten frames provide students with a visual structure for a number. This encourages students to establish and work with visual images of numbers. Ten frames emphasise doubles and five as part of a number. For example, using a ten frame, 9 can be seen as 5+4, one less than double 5 or one more than double 4.
Making ten
Make 10, using the same coloured side of the counters and discuss that there are (for example), 10 red counters and no yellow counters. Reinforce through discussion how we know that when we see a full ten frame, the total number of counters is always ten. Investigate a few times in order to “prove” this fact. Turn over one counter and discuss how many counters there are in total, how many red and how many yellow. Continue this a few times, discussing and recording the total number of counters, the number of red counters and the number of yellow counters. It is important to model how we fill the ten frame (left to right, top row and then bottom row) and how we can see things in the ten frame such as when a number is bigger or smaller than 5, when it is odd, when it is even, etc.
Provide individual ten frames for students and allow them to make their own combinations to ten. Have the students discuss in pairs the patterns they have made on their ten frame - how many counters in total, how many red and how many yellow. Students could record their combinations by copying their patterns onto a paper ten frame.
Ask students to share back the various combinations they made, recording their thinking on the whiteboard. Teachers should use this information to have a discussion about the pairs of numbers that combined together to make ten (such as 10 and 0, 8 and 2, 6 and 4, etc.) and the pairs of numbers that do not combine together to make ten (such as 4 and 1, 2 and 6, etc.)
Teaching point
This could form the beginning of a conversation about “friends of ten” and how we know that when we see these pairs of numbers (and only these pairs - the “friends of ten”), they always combine to make ten.
Extend this activity to focus on other “friends to ten” e.g. the combinations for the number 9 Remove one counter from the ten frame and repeat the above activity to make the combinations for 9. Repeat again for friends of 6, 7 or 8
Peggy combinations
Provide pairs of students with a particular numeral card and piles of pegs, a mini whiteboard and marker. Ask students to place the corresponding number of pegs on the numeral card, counting to check they have the correct amount. Challenge students to place their pegs in as many different ways as possible to show two different amounts that could be combined together to make the total. Demonstrate how to record their combinations. Share various ways of partitioning the given number and discuss any patterns that may emerge.
Ask students how many ways they can partition the given number if they could partition the total into two or more collections of pegs.
Why?
Developing a strong understanding of part-part-whole partitioning (separating and combining) can support students in developing flexibility of thinking in mathematics.
Domino patterns
Display a domino on the interactive whiteboard. Discuss the two sets of dots that can
be seen on either side of the bar and the total number of dots on the domino. E.g. 2 and 1. Label the two collections and the total. Pose the question: “I wonder if there are other dominos that have 3 dots in total? That means, that they have a sum of 3 dots.” Give students a range of dominos (or domino pattern cards) and ask them to find dominos that have 3 dots in total. As each one is found, discuss the combination and record it on the whiteboard. Discuss the different combinations of amounts that could be combined to make a total of 3, introducing (or reinforcing) the concept of “is the same as”.
Have students work in pairs to find the dominos that make a particular total, recording the various combinations they find to make that sum. If you have access to iPads (or other tablet devices), students could take a photo of the dominos they found that add to a given total and narrate their explanations. These digital artefacts could be shared on the interactive whiteboard and discussed as a class.
Variations
- Show a domino pattern to the students. Cover half of the dots and ask the students to find a dot card to match the hidden dots
- Tell the students to look carefully at both sides of a domino as you show it to them. Cover half of the dots. Ask the students to work out the total number of dots, briefly flashing the covered section if needed. The students could write their response on a mini whiteboard before sharing with the class
- Tell the students to look carefully at both sides of a domino as you show it to them. Cover half of the dots. Tell the students the total number of dots on the domino and ask them to work out the missing number of dots. The students could write their response on a mini whiteboard before sharing with the class
Why?
Frequent practice with combining and separating collections, describing and visualising number structures, can benefit students in developing deep conceptual understanding. This assists the understanding of number relations and the movement towards more efficient problem solving strategies.
Apple turnovers
Construct four sets of cards with dot patterns (BLM - Apple Dot Patterns) for the numerals 1 to 6 (or use dominos or playing cards within an appropriate range).
Give fifty counters to each pair of students. Shuffle the cards and place them face down on the table. Have both players take a card from the pile of cards. The players use counting to determine the difference between the two numbers on the apple cards. The player with the larger number takes the difference in counters from his or her partner’s pile of counters. Continue until all of the cards have been turned over. The player with the most counters is the winner.
Variations
- Students use numeral cards rather than dot cards to encourage them to use their own mental images of an amount in order to calculate the difference
- The first student flips over 1 numeral card and 1 dot card. The second player must add the numbers together. He or she records the total. Students work together to check. The second student then takes that number of counters from his or her partner’s pile. Continue until one player has no counters left
Diffy towers
Organise the students into pairs and provide each pair with a dice and a supply of connecting blocks. Have the first student roll a dice, take a corresponding number of blocks from a central pile and build them into a tower. The second student then rolls the dice and repeats the process. Students then compare their two towers to see who has the most blocks and determine the difference between the two towers. The player with the larger number of blocks keeps the difference and all other blocks are returned to the central pile. The activity continues until one student accumulates a total of ten blocks.
Variation
- Use a range of dice to increase the level of challenge to suit the learning needs of students
Teaching point
The word and concept of “difference” is often difficult for students to understand, particularly its connection to subtraction. Many students only see it as meaning “different” e.g. “Your tower is tall and mine is short, or, yours has blue blocks and mine has red”. Students are not seeing the comparative aspect of difference, “How much taller is your tower than your partner’s?” Teachers need to discuss the language and the associated concept of difference regularly when teaching subtraction.
Combining collections: An investigation
Gather two different collections of items and show them to the students. Explain to students that you noticed something when you were counting how many items you had altogether – “That it didn’t matter which collection you counted first, the total number of items, the sum, was the same”.
Demonstrate by using counting to combine the collections, starting from a different group each time and labelling the total amount when combined. Explain to the students that this reminds you of the way we count items – “That when we count to find out how many we have, it doesn’t matter what order we count objects in as the last number word we say is the same”. (Teachers could also link this to Peggy combinations, for example, if students have experienced this learning activity).
Explain to the students that: “When I realised it didn’t matter what order I combined my two collections in, it made me wonder if that is always true. If I am combining or adding amounts together, does it matter what order I do my calculation? I think we should investigate!”
Record the questions and steps you might take to work out a solution. Demonstrate how students could record their thinking and provide students with a range of addition scenarios to investigate, working towards finding a conclusion to the investigatory question. For example:
Have students investigate a range of combinations within their number range, presenting their work and conclusions to the class. Share with students that this is called the commutative property of mathematics and now that we know that, we can apply it to help us be more efficient at solving number problems.
Variations
- Ask students to investigate what happens if we are combining more than 2 amounts together, for example, what if we are working out 7 + 2 + 5? Or 3 + 8 + 4?
- Increase the challenge by increasing the number range
- Link to activities such as Peggy combinations, Making ten, Domino patterns as well as concepts such as “friends to ten”, acknowledging what we sometimes call “turn around facts”
- Investigate whether the same property of mathematics works when subtracting
- Use a picture book with a mathematical focus as a stimulus into wondering about the commutative property and drawing students into investigating whether “it always works” when combining amounts and when removing or subtracting items