Activities to support Early Arithmetical Strategies
Where are they now?
Students:
- visualise amounts to solve number problems where items are not visible
- add collections together where one group of items is not visible, starting back at one
- subtract items from a visible collection when the collection or the amount being removed is not visible,often starting back at one
- partition and combine numbers in a variety of ways
- do not see ten as a countable unit. The student’s focus is on the individual items that form a unit of ten
- recognise and recall some number combinations to ten
Where to next?
Students:
- count-up-to and count-up-from a given number to find the total of two groups
- count-down-to and count-down-from a given number to solve subtraction problems
- explain why counting-on from the largest number is a more efficient strategy
- see ten as a unit composed of ten ones and are able to use the unit to count
- recalls number facts to ten
Teaching Point
Monitor the strategies students use as they solve problems. If a student drops back to using fingers or counting by ones, model how to find the total using counting-on-and- back, demonstrating why this strategy is more efficient. Students may need learning opportunities in numeral identification and the number word sequences to support their fluency and understanding.
Outcomes
The following activities provide opportunities for students to demonstrate progress towards the following outcomes. A student:
MA1-1WM describes mathematical situations and methods using everyday and some mathematical language, actions, materials, diagrams and symbols
MA1-2WM uses objects, diagrams and technology to explore mathematical problems
MA1-3WM supports conclusions by explaining or demonstrating how answers were obtained
MA1-5NA uses a range of strategies and informal recording methods for addition and subtraction involving one- and two-digit numbers
How?
Rabbit ears: figurative
Instruct the students to make two fists and rest them on their heads, so that their hands are out of their direct line of sight. Ask the students to raise a given number of fingers on each hand and to add them together. Students may bring their hands down to confirm the answer.
Variations
- Students make rabbit ears of their choosing. The teacher then shows a number in the range of 11-20 using two ten frames. Students then subtract their number from the number they can see. Have students record their thinking on a mini whiteboard before sharing with a partner. For example,
- Have students share their thinking with the class and use this in an opportunity to discuss how counting-back and counting-up can be used to solve subtraction problems
- Use sentence stems to scaffold the language students need in order to communicate their thinking
Build a tower
Organise the students into pairs. Provide each student with ten Unifix blocks as well as an additional pile of blocks, such as 20, for each pair of students. Prepare “direction cards” showing either addition or subtraction tasks, for example: + 3. Have the students take turns to flip a “direction card” and follow the instruction by adding or subtracting the correct number of blocks to their tower. Students should work out their answer using mental strategies, sharing their thinking with their partner who records it. Students use concrete materials to check their thinking. The winner is the first to make a tower of twenty blocks.
Teaching Point
Having students record their thinking can reveal errors in counting that might otherwise go unnoticed. For example, this student may be counting-on from the incorrect number.
Variations
- Start with a tower of 20 blocks and the first student to reach zero
- Limit the range of direction cards from +/-0 to 3 before increasing the range of numbers to increase the challenge
Race to 100
Provide students with two 0-9 sided dice and a game board (BLM - Race to 100). Students take turns to roll the 2 dice. If a combination of ten is rolled, students race to “snap” the dice, explaining what number combination they saw and how they know it adds to ten. If their partner agrees with their thinking, they complete one ten frame, recording the two numbers they rolled and the number sentence. The first person to complete their game board is declared the winner.
Variation
- Students roll 3 dice, looking for combinations to 10 using any combination of dice. For example, if they rolled 9, 3 and 1 students could say that 9+1 makes 10 and discard the 3. Alternatively, they could roll 3, 6 and 1 and use all three dice to make ten
Why?
This activity provides an opportunity for students to develop awareness of number patterns and to develop strategies for solving problems other than counting by ones.
Subtraction teddies
Provide each student with twenty plastic teddies, a “double decker bus” baseboard (BLM - Bus) and a strip of paper. Have the students place the twenty teddies on the bus baseboard. Instruct the students to take turns to roll a dice and subtract the corresponding number of teddies from the collection of teddies on the bus. The student then records the number of remaining teddies. The activity continues until one student reaches zero. If a student is unable to go, they miss a turn.
Teaching Point
Counting-down-to
The student counts backwards from the larger number when solving problems where there is a missing addend. For example, when solving 9 - ( ) = 6, the students would count backwards from nine knowing they are counting to the number six and say “eight, seven, six.” Students typically hold up fingers as they count and recognise they have said three number words and the answer is 3.
Teaching Point
Counting-down-from
The student counts backwards from the larger number when solving subtraction problems. For example, when solving 9 - 3, the student counts backwards from nine saying “eight, seven, six...six!” In this case, the last number word said tells us how many are remaining.
Put in, take out
Prepare a set of “start with” cards displaying the numerals from eleven to twenty on coloured card, and a set of “put in” cards displaying the numerals from zero to nine on a different coloured card. Students will also require a large container and a supply of items, such as counters, and a mini whiteboard or tablet device.
Ask the first student to take a “start with” card from the pack, read the numeral and put a corresponding number of items into the container. Document this move by taking a photo or drawing a picture. The student then takes a “put in” card from the other pile and collects the corresponding number of items to add to the container. Have the student record what the total will be before they check. Once the estimation is recorded, students count on as each additional item is dropped into the container.
Students could record their counting by filming or annotating photos taken using a tablet device. They could also record their work by explaining out aloud what they are doing in their head and have their partner record it. Empty the container and start over.
Teaching point
Recording the number words spoken provides an opportunity to explain how we can use counting-on-and-back to solve number problems.
Variations
- Modify the activity to suit counting-back by having larger “start with” cards and an appropriate range of “take out” cards
- Have students choose a target they want to reach (for example, 34). Students take turns flipping over “put it” and “take out” cards, to try to get as close as possible to the target. The first person to reach the target number, or, the person closest when time is up, is declared the winner
- When appropriate, add an additional column where students record the number sentence
The beanstalk
Prepare beanstalk base board (BLM - Beanstalk) and a pack of instruction cards. The instruction cards should state the direction in which the student moves along the beanstalk, either count-on or count-back, and the number of spaces to move, for example, “count- back three spaces.”
Commence the activity by instructing each student to place a marker at position 10 on the beanstalk. In turns, students take an instruction card, follow the directions and move their marker accordingly along the beanstalk. The winner is the first person to reach the castle at the top or the bottom of the beanstalk. Have the students record their moves.
Variation
- Use the blank Beanstalk baseboard and have the students choose a different number range (for example, from 20-40, 18 - 38, etc)
Add two dice
Construct a set of numeral cards in the range of zero to eighteen. Place them face up on a table, or on the floor. Have the students take turns to roll two 0-9 number dice and find the total. Encourage the students to use counting in the most efficient way to solve their problem. After adding the two dice, instruct the student to take the numeral card corresponding to the total. The activity continues until all the cards have been taken. If a student’s sum is a number that has already been taken, his or her turn is forfeited. These numeral cards can be pegged to a number line.
Variation
-
Use a variety of dice to extend the range of numbers. For example, use a dodecahedron (12 faces) or 20-sided dice. Modify the set of numeral cards to the appropriate range of numbers
-
Provide the students with three dice and numeral cards within the corresponding range. Have the students roll the three dice and find the total. This provides opportunities for introducing strategies other than counting by ones to solve addition tasks e.g. near doubles
-
Allow the students to construct their own dice and work out the corresponding numeral cards that would need to accompany their game. A calculator may be used to confirm their calculations
-
Have the students write five numbers, within a nominated range, on a strip of paper. Students then take turns to roll two dice, and tell the group the total. As the totals are called, students cross off any corresponding numerals on their paper strip. The activity continues until one student has crossed off all five numerals on his or her paper
Why?
Students need to be encouraged to use increasingly efficient strategies to solve addition tasks. Teachers should take time to investigate with students why strategies are more effective and under which circumstances.
Subtract two dice
Construct a set of numeral cards in the range of zero to eighteen. Place them in a central pile. Provide students with a set of 0-9 answer cards, laid out in front of them and a 0-9 sided dice. Students take turns to select a card from the central pile and roll the 0-9 number dice to work out the difference between the two numbers. Encourage the students to use counting in the most efficient way to solve their problem. After working out the difference, instruct the student to take the corresponding number card. The activity continues until all the cards have been taken. If a student’s difference is a number that has already been taken, his or her turn is forfeited. Students may need a hundred chart to assist in calculations.
Variation
-
Students use an empty number line to help them work out the difference
-
Ask the students to subtract the number on the dice from the number on the numeral card. If they are unable to take a card from the central pile, they forfeit their turn. Students should record their thinking
-
Use a variety of dice to extend the range of numbers. For example, use a dodecahedron (12 faces) or 20-sided dice. Modify the set of numeral cards to the appropriate range of numbers
-
Have the students write five numbers within a nominated range, on a strip of paper. Students then take turns to roll two dice and tell the group the difference. As the totals are called, students cross off any corresponding numerals on their paper strip. The activity continues until one student has crossed off all five numerals on his or her paper
Teaching point
These activities help develop the concept that addition may be an appropriate strategy for solving a subtraction problem.
Allow students to discover that some numbers cannot be subtracted if they do not have enough counters to begin with.
Why?
Students need to be encouraged to use increasingly efficient strategies to solve subtraction tasks. They need support in learning how to countdown-to, count-down- from, count-up-to and count-up-from to solve subtraction problems.
Making 20
Give students two sets of 0-20 twenty frames. Place the cards face down on a table between pairs of students. Have the students take turns to turn over two cards and add the two cards together. If the total is “twenty”, the student keeps the two cards. If the cards do not equal “twenty” they are returned to the table.
Encourage students to count on from the larger number as it is a more efficient strategy when counting by ones. Have students record their partner’s move.
Variations
- Play as a version of concentration where students are looking for combinations that add to make 20
- Students flip two cards and work out the difference between them. If they can accurately work out the answer (their partner checks), they get to take the corresponding number of counters. Students swap turns and the person with the largest collection of counters at the end is declared the winner
- Student seach draw two cards, adding them together to find the total. If wanted, students can choose to take a third card which they have to add to their total. The student with the total closet to 20 wins the round and collects a counter. The student with the largest collection of counters at the end is the winner
- Use student recordings to discuss combinations to 20, particularly focussing on drawing out the connection to combinations to 10
Windows: figurative
Use a screen that students cannot see through. Have one student place an arrangement of counters on both sides of the screen. The student then tells their partner how many counters there are in total. The second student uses counting-on or counting-back to work out how many counters are hidden.
Model how to work out how many counters are on the other side using the information they know. For example:
I know that the total number of counters is 15. I know that I can see 11 counters. I know there are counters that I cannot see. I know I can use counting to solve number problems. I can count up from 11 to 15, keeping track of the number words I say using my fingers.
Starting from 11...12, 13, 14, 15. You have 4 counters on your side of the screen.
Have students record their thinking before checking their work by counting the screened counters E.g.
I could see ___ counters.
I knew there were ____ counters altogether.
I worked out that there were ____ counters hidden.
Variations
- Have students work together in pairs by asking one student to flips two numeral cards to form a 2-digit number
The second student makes a collection that corresponds to that number. The first student chooses where to place the screen and asks their partner to work out how many items are on both sides. The second student tells their partner who records their thinking before both students using counting to check. Students swap roles - Use two or more screens to separate the given number and explore how many ways students can separate a given amount. Discuss the various combinations they found that add to make a given total and the fact that no matter how they partitioned the amount, when they combined it back together, the answer is always the same
Posting counters
Provide the students with a container they can cover. Have students roll a 12-sided dice
and “post” the corresponding number of counters into the container and label it. Instruct the students to flip over a 2-digit number card within the range of 15-32 and work out how many more counters are needed to make the collection inside their container match the number card. Students record their thinking and then use counting to check.
Variation
- Change the range of numbers
Three-dice toss
Provide each pair of students with three 6-sided dice and a pile of 50 counters each. Have the students take turns to throw the dice and add the total, encouraging the use of efficient counting-on strategies. Students then take the corresponding number of counters from their partner’s pile. The game continues until one player had lost all of his or her counters. Have students record their work using a table and an empty number line.
Order of combining: An investigation
Gather three 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 counted first, the total number of items, the sum, was the same. We worked out this works with two collections but I just realised it works with three!”
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 commutative property of addition, the “turn-around facts” we discovered. Use this as the launch of two investigations:
-
Why does applying the commutative property help us to work as more efficiently?
-
Can I combine 3 or more groups of items in any order I like? How can I use that to be a more efficient in my counting?
-
Explain to the students that: “I think it has something to do with the number of counting words you have to say. For example, say I need to combine 5 and 8. If I start at 5, I have to say 8 number words to work out the total (6, 7, 8, 9, 10, 11, 12, 13) with the last number word telling us how many there are in total. If I start at 8, I only have to say 5 number words (9, 10, 11, 12, 13). “ I said less number words when counting-on from the larger number. I wonder if that is always true - does counting on from the bigger number mean I am more efficient at solving addition problems?” Have students investigate a range of combinations within their number range, presenting their work and conclusions to the class.
-
Explain to the students that: “I think I can because I know it works for two collections of objects. For example, say I need to combine 5 and 8 and 3. 5 and 8 makes 13 (count- on to demonstrate, using concrete materials or interactive whiteboard resources to aid comprehension). 13 and 3 more makes 16 (count on the demonstrate). Record on the board. But, what if I combined 5 and 3 first? That makes (count-on to demonstrate) 8. And then 8 more makes 16 (count-on to demonstrate). Record on the board. Both ways of counting-on came to make 16. Which was the most efficient? Is there another way
I could have combined the amounts to be even more efficient?” The teacher should repeat the same process, recording the number words spoken and steps taken. Have students investigate a range of combinations within their number range, presenting
Have students investigate a range of combinations within their number range, presenting their work and conclusions to the class. Discuss why when we apply this knowledge, it can help us be more efficient at solving number problems.
Variation
- Link to activities such as Combining collections (Perceptual), Add two dice or Three-dice toss
Investigating fact families
Ask students to tell you a number fact they know, in other words, a fact that they remember without having to work out the answer. For example, “I remember that 5 and 1 more makes 6. I remember that because when I see a full row in a ten frame and 1 more, I know that shows 6.” Elicit other examples from students, asking them to explain how they know that fact, drawing upon their visual images as much as possible. Explain to the students that someone told you a way to work more efficiently. “They told me about fact families”. For example:
Teaching Point
Students need to have spent time investigating the commutative and associative properties of mathematics before being asked to investigate fact families. See activities such as Combining collections and Peggy combinations in Perceptual as well as Order of combining in Figurative.
Have students investigate a range of fact families based on a number fact they already know. Have students record their work using an iPad or other tablet device, taking and annotating photos of their work to share with the class.
Teaching Point
It is important to support students in using what they already know to work out things that they do not. This strategy of making connections from the known to the unknown requires reasoning and problem solving skills which students will need support in developing. Connecting addition and subtraction facts can also provide an opportunity to discuss how we can use addition to solve subtraction problems.
Fishing addition
Construct six sets of four cards, with each card in any one set displaying a basic addition fact for the same number in the range one to ten. Each set of cards should have a unique sum. Shuffle the cards and deal five cards to each player. Place the remainder of the cards face down on the floor. Instruct the students to look at their dealt cards and take turns
to discard any pairs of cards that add up to the same total. After discarding any pairs of cards, the first player asks his or her partner for a card which equals a specific number. For example: “Pass me a sum of eight”. If the partner holds a card totalling the nominated number, it must be handed over. If the partner does not have the nominated card, the person who asked takes a card from the central pile. In both cases, if a pair of cards is formed, the student discards them. The winner is the player who discards all of his or her cards first.
