At a Glance...

Skill: Recalling Addition Facts for Numbers Up To 20 and Related Subtraction Facts

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There are many strategies that can help students develop their sense of arithmetic operations. Not all students use the full range of strategies; some use one, for example, the doubles strategy. Students abandon strategies that they no longer need as they discover and create even more effective ones.

In the primary and junior divisions, students first use objects or their fingers to represent problems. Then, building on these experiences, they begin to use more advanced counting strategies.

Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 6^e année, p. 11-12.

Representation of Basic Number Facts

Using models to represent basic number facts can help students understand the meaning of basic operations and lessen their abstract nature. Many models can be developed using the materials below to build students' understanding of addition and subtraction:

• objects, such as tokens or tiles;
• visual material, such as illustrations;
• a five frame;
• a ten frame;
• meeting mats;
• two-coloured tiles;
• a number line;
• number grids.

Models help students to make connections and thus better understand what the symbols represent in operations. For example, students can use an addition or subtraction algorithm, depending on how they think about the problem. Students can be helped to understand the problem first with manipulatives and then with symbols such as =, + and -.

The models help students make connections between drawings or illustrations, symbols and words. Below are different models that can represent the basic 6 + 3 fact for students. Each of the four representations is appropriate. However, students will find it less necessary to use visual representations as they develop some automaticity in solving basic number facts.

Strategies for Learning Basic Number Facts

Teachers can help students develop effective strategies for identifying basic number facts by using their reasoning and encouraging them to look for patterns and relationships among numbers. The basic number fact strategies described below build on knowledge that students already have to determine unknown facts. For example, knowing what 2 + 2 is helps students figure out what 2 + 3 is (since 3 is 1 more than 2, the answer must also be 1 more), indicating an important reasoning skill. Strategies should be taught in a problem-solving context. In this regard, one should:

• select problems that lend themselves to the use of the strategies taught;
• provide opportunities for students to model the strategy under study;
• have students practice the strategy in a meaningful context.

Students need many experiences to become familiar with combining by tens and to establish relationships between the number 10 and other numbers before they can develop strategies for basic number facts. Recognizing the number 10 as an anchor and defining the relationships between this benchmark and the numbers 0-10 is a core concept for all students. A solid understanding of how numbers relate to 10 (for example, 8 is 2 less than 10 and 13 is 3 more than 10) will enable them to discover patterns and relationships between numbers when presented with strategies focused on basic number facts. Knowing the ways in which numbers are related to each other will make it easier for them to learn the strategies based on decomposing numbers.

A ten frame is an important tool that allows students to represent numbers from 0 to 10 in concretely and gives them the opportunity to relate 10 to other numbers. By filling in some of the boxes in the frame, it is possible to see the empty boxes and recognize the number represented overall. Students can use one or two ten frames and tokens to illustrate numbers from 0 to 20 (two filled ten frames). Using different coloured tokens, students can identify combining relationships and see how they relate to the number 10. For example, if there are 8 blue tokens on one ten frame and 6 red tokens on another ten frame, students can grasp that 8 blue tokens plus 2 red tokens add up to 10, plus another 4 red tokens add up to 14. This allows students to concretely see the decomposition of the number 6 into 2 + 4 and to add more easily (8 + 2 = 10 and 10 + 4 = 14).

The strategies presented below are designed to help students learn basic number facts. Students can build on their prior knowledge (combining by tens, using doubles, 10 as an anchor) when presented with these strategies and given opportunities to apply them. Activities and games provide students with interactive learning opportunities while providing a framework for practicing a particular strategy. By participating in a game, they are using strategies and building them into their repertoire.

The following strategies are not presented in any particular order. Some students may find some strategies more useful than others, or may ignore some strategies in favour of their own. Others may find it easier to memorize facts than to rely on a strategy. Whatever the case, the primary goal is to get students to fully understand addition and subtraction.

Strategies

"One-More-Than" and "Two-More-Than "

In the basic number facts \(5 + 1 = 6\), \(5 + 2 = 7\), \(7 + 1 = 8\), \(7 + 2 = 9\), one of the terms in each addition is 1 or 2. The strategy is based on the assumption that students can easily remember the number that follows and the one that comes after.

Facts with 0(\(1 + 0 = 1\), \(1 - 0 = 1)\)

In these basic number facts, 0 is one of the terms in each addition and subtraction. Students often generalize that the answer to addition is necessarily larger and the answer to subtraction is necessarily smaller. Understanding of the concept of zero neutrality in addition and subtraction can be reinforced with games involving flash cards and numbered cubes.

Doubles

In these basic number facts, both terms are the same. It is beneficial for students to recognize and learn about doubles. There are only ten number facts related to doubles. Simple memorization tricks can be used to learn several doubles:

Double of 3 : an insect has 3 legs on one side and 3 legs on the other (6)

Double of 4 : a spider has 4 legs on one side and 4 legs on the other (8)

Double of 5 : the 5 fingers of one hand and the 5 fingers of the other hand (10)

Double of 6: a box containing a dozen eggs cut in half, 6 on one side and 6 on the other (12)

Double of 7 : in a calendar, there are 7 days in a week and 14 days in two weeks

Double of 8: There are 8 crayons in one row of crayon boxes and 16 crayons in two rows.

Near-Doubles or Doubles Plus One (\(5 + 4\) can be seen as \(4 + 4\), and 1 more)

In these basic number facts, one term of the addition is 1 more than the other term. Students can learn to recognize that in these additions, the answer is the same as the double of the lesser number plus 1 (\(5 + 4 = 4 + 4 + 1\)). Students need to be proficient with doubles before they can use this strategy effectively.

"Make-Ten" Facts (8 + 4 = 8 + 2 + 2 = 12), 9 + 5 = 9 + 1 + 4 = 14)

In these basic number facts, one of the terms is 8 or 9. As appropriate, students add 1 or 2 from the other term to get 10, and add what is left from that other term. Students should be given many opportunities to use a ten frame so that combining becomes automatic.

In a number sentence like 8 + 5, for example, students can get 10 by adding 8 plus 2 (taken from 5), and then adding the remaining 3 to make 13. Once these facts are firmly established, it becomes much easier for students to use larger numbers, like 18 + 5. Knowing that 18 + 2 is 20, all they have to do is add the remaining 3 to make 23. (This property of addition is called associativity.)

Commutative Property (1 + 2 = 2 + 1)

Students who recognize the commutative property of addition can cut the amount of number facts to be learned in half. Visual representation of facts such as (3 + 2) and (2 + 3) helps students grasp this relationship.

Subtraction as Think-Addition for Facts With sums to 20

Students who are familiar with addition and understand that subtraction is the inverse operation of addition can use this knowledge to master the related subtractions (if 5 + 2 = 7, it follows that 7 - 5 = 2). For example, consider the following problem: Julian has 6 marbles in his bag. George gives him some more. Julian now has 14 marbles. How many marbles did George give to Julian? Seeing this problem, the student thinks: What number added to 6 gives 14? Thus, the addition facts that are familiar to them help them find the unknown quantity in the number sentence.

Subtraction as Adding Up

Students have more difficulty subtracting large numbers. It may be helpful for them to realize that facts like (17 - 13) can be solved by counting from 13 to 17.

"One-Less-Than" and "Two-Less-Than" Facts

These subtraction facts have one term of 1 or 2 less than the other term. Students can usually count backwards for the easiest number facts, such as any number from which 0, 1, or 2 is subtracted.

Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 6^e année, p. 19.