Activities to support Early Arithmetical Strategies
Where are they now?
Students:
- count-up-from and count-up-to when solving addition problems
- count-down-from and countdown-to when solving subtraction problems
- identify and recall number facts to ten
- see ten as a unit composed of ten ones and are able to use the unit to count
- know that each ten is composed of ten “ones”
- rely on the strategy of counting by ones to solve addition tasks
Where to next?
Students:
- use known facts to solve number problems
- combinations to ten | combinations to twenty | doubles | near doubles | other facts students know
- use the structure of numbers to solve number problems
- use ten as a countable unit to solve problems
Teaching point
Students may revert to counting-on-and-back as number problems become more difficult. It is important to pay close attention to students as they work mathematically, modelling how to use the efficient strate- gies they are developing as the numbers and problems increase in complexity.
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
MA1-4NA applies place value, informally, to count, order, read and represent two- and three-digit numbers
MA1-5NA uses a range of strategies and informal recording methods for addition and subtraction involving one- and two-digit numbers
How?
Teaching Point - A note about number facts
As students become proficient at counting-on-and-back, teachers need to support their development in even more efficient and sophisticated problem solving strategies. Knowing, recalling and using known facts is one way of becoming more efficient. Students need multiple and varied experiences to support them in being able to fluently use known facts. This needs to be developed through rich learning experiences rather than rote learning. As students learn to recall known facts, teachers need to provide opportunities for students to apply these facts, investigating how that helps them solve problems with greater efficiency.
3 tens in a row
Provide students with a mini whiteboard and marker each as well as a 0-9 dice. Have students draw a 3x3 grid as a game board. Students take turns to roll the dice and write the number in one of their boxes. The goal is to be able to write two numbers in each box that combine to make 10. Students continue taking turns until a student has been the first to make 3 tens in a row.
Variations
- Students complete their game board, looking for 3 doubles or 3 near-doubles in a row
- Alternatively, students use 20-sided dice, looking for friends to 20
- Students look for combinations to a different amount, for example, 50
- Students use more than 2 addends to make the target number
- Students use a 20-sided dice and subtraction to make the target number
Teaching Point
Students need strategies other than counting to solve number problems. The movement towards more efficient strategies requires fluency with the number word sequences, an understanding of numbers, flexibility of thinking and the use of known facts. Students need regular, varied and meaningful opportunities to develop these strategies along with the opportunities to share their thinking, listen to the thinking of others and explore how and when strategies work.
Number fact concentration
Provide students with a few sets of ten-frame cards. Shuffle the cards and lay them out
in a 4x4 array, face down, leaving the remaining cards in a central pile, also face down. Students take turns to turn over 2 cards, looking to match cards that demonstrate a double. If a match is found, the student collects the cards and keeps them, replenishing the 4x4 array with cards from the central pile. Play continues until all possible matches are found.
Students take their cards and record the double facts they found, scoring a point for each double. Students then work out adding the doubles and record the answer. Once their partner checks their calculations, students earn an additional point for each fact they know or worked out.
Variations
- Look for other facts such as near doubles, friends to 10, friends to 20, etc., changing the provided cards as necessary
- Look for any known fact as students turn over the cards. For example, I can match 4 + 6 as a friend of 10, 8 + 8 as a double and 2+3 as a near double
- Play as a version of snap
Doubles
Instruct the students to use two hands to demonstrate double numbers from 1 to 5. For example, “Show me double four. How many altogether?” In this example the students would raise four fingers on each hand and call out the answer.
Variation
- Instruct the students to raise their fingers for a nominated double combination and then add one more finger to find the total. Alternatively, play doubles minus one. For this activity students raise their fingers to represent a nominated double and then subtract one finger to find the total
Doubles plus one
Demonstrate the following procedure to the students.
Join two equal groups of Unifix blocks to show a double fact, such as 5 + 5. Display a number sentence to the students to describe the action of joining the two groups. Add one block to the second group of blocks. Ask the students to state the total and record the new number sentence. In the above example the new number sentence would be: 5 + 5 + 1 = 11. Separate the two groups again and remove the block just joined. Place it above the second group. Discuss the number combination now formed and its link to the previous combination of numbers, for example: 5 + 5 + 1 = 5 + 6. Explore other doubles plus one combinations.
Variations
- Demonstrate doubles minus one in the same manner, explaining that doubles plus or minus one are called near doubles
- Have students go on a “doubles and near doubles” search where they look for near doubles and near doubles that can be found in their environment. Have students photograph what they find and share back with the class
Near doubles bingo
Provide the students with a bingo board displaying a 4x4 grid. Ask the students to place the numbers 5, 7, 9, 11, 13, 15, 17 and 19 randomly into the squares of the grid. Each number will need to be written twice. Call out near doubles, for example 6 + 7, 10 + 9, in random order. The students determine the answer and place a counter onto the bingo board if they are able to match a numeral to the answer. The first player to complete a line of four counters in any direction is the winner.
Deriving facts
Provide the students with a supply of Unifix cubes. Call out an addition sum, such as 5 + 7, where the addends differ by two. Instruct the students to make the two numbers using the Unifix cubes and to record the number sentence. Ask the students to move one block from the second group (in this example, 7) and place it with the first group (the group of 5). Have the students record the two groups now. Discuss how 6 + 6 = 5 + 7.
Discuss how using known facts often means using our knowledge of numbers to find facts that are “hidden”. 5+7 was “hiding” a double fact of 6+6. Have students write some “hidden” facts and give to a partner to try to work out the number fact they were hiding.
Why?
Knowing doubles and near doubles is an effective method for solving some arithmetical tasks and building knowledge of number combinations.
Double or near double?
Provide students with a 1-10 dice, a recording sheet and a hundred chart. Students select a starting number, for example, 60, and mark it on their charts. Students take turns to the roll the dice and choose to double the number, or, work out a near double. For example, if a student rolls 8 he or she could choose to double 8 to make 16, or, decide 7 + 8 is a near double totalling 15, or, 9 + 8 is a near double that makes 17. Students tell their partner the fact they are using and the answer. If their partner agrees their thinking is correct, the students gets to move that number of spaces towards zero, from the chosen starting number. The first person to reach zero is the winner.
Variations
- Allow students to double or halve the number rolled, trying to get to a target number
- Use a 1-6 dice to support student as the learn early facts
Finding known facts
Provide students with a packet of playing cards, removing the picture cards. Students take turns to deal 4 cards to one another. Students turn over their cards and look for any known facts they see (that is, friends of 10, doubles, near doubles and friends of 20). For every known fact a student finds, they have to explain their thinking to their partner who records the information. If their partner can understand their thinking, they are given a counter. Students take turns listening and recording each others’ ideas. Play continues for several rounds, dealing out cards so each player has a total of 4 cards in each round. The student with the largest number of counters at the end is the winner. If a student cannot go, they can swap 1 or 2 cards before new cards are dealt.
Variations
- Use numeral cards or twenty frame cards
- Deal out 7 cards each
- Add an additional column and look for “hidden” facts (see Deriving facts)
Three-dice game
Prepare a set of numeral cards for the numbers three to eighteen. Lay the cards face up in a line on the desk or floor. Have the students take turns to roll three dice and add together the numbers rolled, then take a corresponding numeral card. The game continues until
all cards have been taken. If the numeral card has already been taken, the player’s turn is forfeited. Observe how each student determines the total. Encourage students to look for known facts to help them solve their problem rather than counting by ones. Students should record their thinking.
Variation
- Use a variety of dice, increasing the range of numbers and the challenge
Teaching point
Students need to have spent time investigating the commutative and associative properties of mathematics before being asked to apply their knowledge of known facts. Students need to be able to explain how the order in which you combine collections does not change the total. See activities such as Combining collections and Peggy combinations in Perceptual as well as Order of combining and Fact families in Figurative.
Applying known facts
Provide students with two different coloured counters, a game board (BLM - Applying Known Facts), a recording sheet and a 1-10 dice. Students take turns to roll the dice. The digit they role is multiplied by ten to move it into the tens place. Students then have to double the number they have, explaining their move to their partner who records their thinking. If their partner agrees with their thinking, they cover that number on the game board with a counter. Students continue play until one student has been able to get 4 in a row either horizontally, vertically or diagonally.
Variations
- Change the numbers on the game board to suit near doubles or friends to 10
- Provide students with only 4 counters each. This means that students will need to move their counters around after all 4 are placed on the board, making it more difficult to get 4 in a row
Get your balance: counting-on-and-back
Provide students with a balance scale and Unifix cubes. Have students work out, and record, a number of amounts that balance. For example:
Ask a group to make one balancing equation (such as 18 and 15 is the same as 33). ask questions such as:
- What does it mean when the balance arm is level?
- What would happen if we removed one unit from each side?
- What would happen if I removed 1 ten from the left side?
- What would happen if I added 3 tens to the right side? What might the scale look like (have the students draw and label what it might look like)?
Variations
- Have students use numeral cards to represent amounts rather than Unifix cubes
- Hide some of the Unifix cubes from view using a very light sheet of paper or cloth. Ask students to work out how many are hidden
- Provide an unbalanced scale. Ask questions such as:
- Provide an unbalanced scale. Ask what number might be on the other side, supporting students in justifying their thinking
- Why is the scale not equal?
- What do we need to do to make the scale balanced?
Race to 200
Provide students with a calculator, recording sheet (BLM - Race To 200) and a deck of playing cards, using A-9. Working with a partner, students take turns to choose to turn over 4 cards. Students use their cards as 1-digit numbers. Students examine the cards, looking for known facts they can apply. Have the students work out the total of the 4 cards in the most efficient way possible, explaining their thinking to their partner who records their strategies and checks their solution with a calculator. If correct, the student scores the corresponding number of points. The winner is the first person to reach the target of 200. Students can use the calculator to keep track of their cumulative total.
Variations
- Change the target to 100 and turn over only 3 cards
- Include the picture cards as being worth 11, 12 and 13
- Students can choose to make 2-digit numbers or a combination of 2-digit and 1-digit numbers and have a target of 500
- Start with 200 points and subtract the total. Have zero as the target number
How would you...?
Provide students with a number problem to solve, for example, 45 – 15. Ask students to think how they could solve this problem without using counting and provide think time. Model to students a range of ways you would solve the problem, using concrete materials and images to support comprehension. For example:
“I could look for the number facts that I know. I know that 5 – 5 is the same as 0. So, the first thing I would do is partition 45 into 40 and 5, and, partition 15 into 10 and 5. Now I can take 5 away from 5 which leaves me with 0. Now, I have to work out 40 – 10. I know how to do that because I can imagine the hundred chart so 10 less than 40 is 30. The answer is 30”.
“I could have thought about the problem differently and used addition to help me solve a subtraction problem. If I start at 15 and try to work out how many more I need to make 45, I would be thinking about the problem like this: 15 + ___ = 40. I need 5 more to make 20.
I know that because I know my friends to 20. Now I know I need 20 more because I know that 2 and 2 makes 4 so 2 tens and 2 tens makes 4 tens which is the same as 40.”
Teaching point
Using the split strategy is not recommended when the number you are subtracting has a ones value larger than the number you are subtracting from, e.g. 45-16.
Why?
These conversations are incredibly useful in supporting students to see there are a number of ways to solve problems and to represent them. The ability to communicate mathematically requires significant support and should be scaffolded through tools such as using of concrete materials, sentence stems, visuals and multiple opportunities to practise language development. Teachers should use these opportunities to investigate student-invented strategies and other methods, investigating the contexts in which particular strategies work and when they do not.
How does knowing...?
Write a number sentence on the board and provide a range of concrete materials that will support the question the teacher is preparing to ask. Pose questions for discussion.
For example:
Teaching points
Teachers should use these questions as a point of discussion, selecting only a few questions based upon the learning needs of students. These questions are not intended to be provided as a worksheet or an independent activity – it is the richness of conversation that these questions are hoping to draw out that makes this learning opportunity valuable. Teachers should change and adapt the questions to suit the needs of the learners.
Knowing basic number combinations that form ten allows students to use a range of strategies for addition, for example, knowing that 7+3 is the same as 8+2, and introduces the idea of compensation (one up, one down).
Why?
Students need to develop a variety of related strategies to use when solving number problems. These strategies may include applying base 5 knowledge, their knowledge of tens and ones and recall of number combinations.
Subtraction bingo
Provide each student with a blank 4x4 bingo board. Select a student to act as a caller. Students fill their bingo boards with numbers in the range of 10 - 29. Provide the caller with a 6-sided dice labelled 30, 40, 30, 40, 30, 40 and a 20- sided dice. The caller rolls the two dice and calls out the numbers, saying the largest number first. Students work out the problem and cross the corresponding number off their bingo board (if present). The caller must keep track of the number sentences and the answer. The game continues until one student covers all the numerals on his or her bingo card.
Variation
- Students play in small groups of 3 and swap roles
Five dice
Provide the students with five dice. Have the students take turns rolling the dice and finding the total of all five dice rolled. Students should be supported to first look for, and use, number facts they know rather than simply counting-on. If students roll numbers that add up to make 7, 14, 21, 28 or 35, they score an extra 3 points. The winner is the person with the largest cumulative total once time is up. Students record their totals for each roll.
Variation
- Students can choose to take their umber as a ones or a tens, e.g. 3 or 30. Students try to male their total closest to 100 in one turn
Race to the pool
Prepare an adequate supply of base boards (BLM - Race To The Pool). Organise students into pairs and provide them with two dice and two markers. Have the students place their markers at the starting position on the base board. Ask the students to take turns to roll the dice and add both numbers rolled. The student then moves the marker to the first corresponding numeral on the base board. The first player to reach the “pool” at the centre of the board wins.
Variation
- Play race from the pool
Teaching point
After setting an addition task, follow up with associated subtraction tasks. This enables students to use familiar addition facts and strategies to solve subtraction problems as well as seeing the inverse operations.
Sentence maker
Provide each student with a whiteboard, marker and a calculator. Call out a number, for example, 36. Allow two or three minutes for students to write as many number sentences that combine to make the nominated number. After the allotted time, have the students verify the additions with the use of the calculator. If needed, provide students with concrete materials in order to make various combinations.
Addition wheel
Provide students with a copy of the addition wheel (BLM - Addition Wheel). Ask the students to write a number in the centre of the wheel. Have students record number combinations within the “spokes” of the addition wheel, radiating out from the centre. Have students share their wheel with a partner. Students earn a point for every correct number sentence and a bonus point for the number combination they had but their partner did not. Play again with a different number. The person with the most points at the end is declared the winner.
Variations
- Students write subtraction sentences that equal the centre number. For example, if the number is 15 a student could write 40 - 20 - 5
- Increase or decrease the number of circles along the spokes
- Students use a mixture of operations to reach the target number
Teaching point
Students need to develop a variety of strategies to use when solving number problems. Knowing the relationship between addition and subtraction can help students solve number problems.
Balancing numbers
Construct number sentence cards where the addends are selected from numbers between 11 and 20. Prepare a chart with a balance beam drawn on it, as shown in the diagram.
Organise the students into pairs. Ask the students to take turns to place a number sentence card on each side of the balance so that the sum of each card is equal. Allow students to confirm their partner’s answer.
Variation
- Increase the number range
- Match number sentences that involve subtraction, for example, a student could match 18 - 9 with 20 - 11. Alternatively, students could match any number sentence that has the same total
- Students make their own number sentences by using numeral and operation cards
- Students can use three numbers and more than one operation, e.g. 3 + 3 + 4 with 12 - 1- 1
Trading game
Provide the students with five dice. Have the students take turns rolling the dice and finding the total of all five dice rolled. Students should be supported to first look for, and use, number facts they know rather than simply counting-on. If students roll numbers that add up to make 7, 14, 21, 28 or 35, they score an extra 3 points. The winner is the person with the largest cumulative total once time is up. Students record their totals for each roll.
Variation
- Play in reverse. So, starting from 2 hundreds, students have to work back to be the first person to have nothing left
- Use different equipment
Fact matching
Construct two sets of cards. The first set consists of the “sum cards” and should contain one card for each numeral from 0 to 18. The second set consists of the “digit cards” and should contain four cards for each numeral from 0 to 9. Also provide cards displaying subtraction and addition symbols. Shuffle the “digit cards” and deal four cards to each student, placing them face up before the student. Shuffle the “sum cards” and place them in a pile, face up.
Ask the students to use their digit cards and the operation cards to make a number fact which will equal the numeral shown on the top “sum card” displayed. For every number combination made, students score five points. After the students have made all possible combinations, ask the students to hand in their cards, shuffle all digit cards and deal four new cards to each player. Turn over a new “sum card” and repeat the process. The winner is the student with the highest score.
Variation
- Deal six digit cards to each player. The student gains five points for combining two numbers which equal the “sum card” and ten points for correctly combining three digit cards which equal the “sum card”
Why?
Students need to develop a range of strategies for solving arithmetic problems. Automatic recall of addition and subtraction facts allows students to attend to other features when solving problems and is a more efficient strategy than counting.
Ghost busters
Prepare “Ghost busters” base boards (BLM - Ghost Busters). Prepare two dice, one displaying 10, 20, 30, 40, 50, 60 and the other displaying 0, 10, 20, 30, 40, 50.
Students take turns to roll the two dice. Students then add the dice together and then place a counter onto a ghost displaying the corresponding numeral. The winner is the first to cover all the ghosts.
Variation
- Change the task to subtraction, adjusting the numbers on the dice or the game board accordingly
Teaching point
Provide many opportunities for oral counting by 10s to support students in playing these games and provide concrete material as necessary.
Why?
Developing knowledge of multiples of ten allows students to complete 2-digit addition and subtraction tasks efficiently.
Domino duel
Provide students with a packet of dominos, a hundred chart each, markers and a target number. Place dominos face down in pile. Each student takes 1 domino, choosing which way to orientate the domino based on the target number. For example, say the target number is 63. If a student takes a domino with 4 and 2, he or she can choose to make 42 or 24. Players mark their number on their hundred chart, using the chart to help them work out the difference between their number and the target number, counting up, down, on or back to solve their problem. The player who is closest to the target number in each round wins a point. Have students record their thinking, for example:
Outdoor bean bag target
Draw a large target on the ground. Write a number which is a multiple of ten on each segment of the target. Organise the students into teams and provide each team with
two bean bags. Have the students take turns to throw the bean bags onto the target. Students call out the number that the bean bags land on and then find the total. Organise a “recorder” for each team to keep a record of the team’s score.
Variations
- Start at 500, for example, and race towards being the first person to get to zero, students must throw the exact number to reach zero
- Use 3 bean bags and add or subtract 3 numbers