At a Glance...

B2.4 Represent and solve problems involving the addition and subtraction of whole numbers that add up to no more than 10 000 and of decimal tenths, using appropriate tools and strategies, including algorithms.

Skill: Representing and Solving Addition and Subtraction Problems Using Strategies Including Algorithms

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The learning of mathematical operations takes place gradually. The starting point should be the exploration of operations in problem-solving situations. Students learn to associate situations with particular operations, which allows them to begin to make sense of the operations. In addition, students need to use strategies based on their understanding of the context, the problem, and the operations. Students become aware that there are many ways to solve a problem and even many ways to carry out the same operation. Subsequently, students are asked to solve a variety of problems in order to progress to using effective strategies.

Contrary to the traditional process where students mainly learn to apply the usual algorithms, the learning of operations should focus more on understanding operations, exploring mental calculation and using various strategies to carry out operations.

To develop effective strategies in students, it's important to offer them a variety of problem types so that they can grasp the multiple meanings of operations. A well-chosen problem and the application of a well-thought-out strategy are more beneficial than a series of exercises completed mechanically. It's important to allow sufficient time for students to understand and consolidate their strategies.

Written Problems : Addition and Subtraction

In addition and subtraction, quantities are joined, separated, combined or compared. In order for students to understand the connections between quantities in each of these cases, it is important that they be presented with a variety of problem types.

Addition and subtraction are only operations that occur in problems. Therefore, it is important to avoid referring to them as "subtraction problems" or "addition problems" because it is the understanding of the situation, as well as the understanding of the operations, that leads to the choice of a problem-solving strategy, in this case the choice of addition or subtraction. Thus, students must analyze the problem, choose a strategy and apply it, just as adults do. In this context, the teacher's role is to assist students in their analysis and understanding of operations.

It is important to note that the problems presented below appear similar because of their context. However, for students, each situation represents a particular problem. It is through mastery of these different types of problems that students gain mastery of addition and subtraction.

Joining Problem

The part-whole model can be useful for representing known and unknown values in joining problems. The set model is useful for representing the addition of a quantity.

• Change situation : Result unknown - Jamil has a bag of 600 candies. He buys 500 more pieces of candy. How much candy does Jamil have now?

• Change situation : Starting point unknown - Jamil has many candies. He buys 500 more. He now has 1,100. How many pieces of candy did Jamil have at the beginning?

• Change situation : Change unknown - Jamil has a bag of 600 candies. He buys many more. He now has 1,100 pieces. How much candy did Jamil buy?

Separating Problems

The part-whole model can be useful for representing known and unknown values in separating problems. The set model is useful for representing the removal of a quantity.

• Separating: Result unknown. Nadia has $1500. She gives $500 to her brother. How many dollars does she have left now?

• Separating : Change unknown - Nadia has $1500. She gives it to her brother. She now has $1000 left. How much money did Nadia give to her brother?

• Separating : Starting point unknown - Nadia had a certain amount of money. She gave $500 to her brother. She now has $1000 left. How much money did Nadia have at the beginning?

Combining Problems

The part-whole model can be useful for representing known and unknown parts or the known and unknown whole in combining problems.

• Combining : Part of the whole unknown - The class has 800 colored pencils, and 300 of those pencils are red. The remaining crayons are blue. How many blue crayons does the class have?

• Combining : Whole unknown - The class has a lot of colored pencils. There are 300 red crayons and 500 blue crayons. How many crayons does the class have?

Comparing Problems

The linear model can be useful for representing the difference between two numbers in comparing problems. In this example, we use the Cuisenaire rods and the double number line.

• Comparing: Difference unknown. Judith has $600 and Jeanne has $300. How many more dollars does Judith have than Jeanne? OR Judith has $600 and Jane has $300. How much less does Jeanne have than Judith?

I know that the dark green rod represents 6 (hundreds), so I place it at the top of the number line from 0. I know that the lime green rod represents 3 (hundreds), so I place it below the number line from 0. I compare the two rods and see that the lime green rod is 3 (hundreds) less than the dark green rod. I find the difference or gap between the two quantities. There is a difference of $300. Judith has $300 more than Jeanne or Jeanne has $300 less than Judith.

• Comparing: Larger amount unknown - Judith has $300 more than Jeanne. Jeanne has $300. How many dollars does Judith have? OR Jeanne has $300 less than Judith. Jeanne has $300. How many dollars does Judith have?

I know that the lime green rod represents 3 (hundreds), so I place it at the top of the number line from 0. I take another lime green rod and place it under the number line from 0 and add another lime green rod since Judith has $300 more than Jeanne. I replace the two lime green rods with the dark green rod that represents 6 (hundreds). Judith has $600.

• Comparing: Smaller amount unknown - Judith has $600 and Jeanne has $300 less than Judith. How many dollars does Jeanne have? or Jeanne has $300 less than Judith. Judith has $600. How many dollars does Jeanne have?

I know that the dark green rod represents 6 (hundreds), so I place it at the top of the number line from 0. On the number line, I count backwards three jumps to 300 to represent that Jeanne has $300 less than Judith. I take a lime green rod, which represents 3 (hundreds), and place it under the number line from 0. Jeanne has $300.

Source: Guide d'enseignement efficace des mathématiques de la maternelle à la 6^e année, p. 7-10.

Change situations are perceived by students as active situations, easier to model and "see" as a quantity changes. Combining problems, however, assume a static situation, as no action or change occurs, making them more abstract and difficult to understand. Comparing problems, on the other hand, deal with the relationship between two quantities by opposing them: thus, there is no action, but a comparison of one quantity to another.

Since students are regularly exposed to problems where the result is sought, they find it easier to solve them. However, they have more difficulty solving problems where the starting point is unknown or the change is unknown. These problems help develop a more solid understanding of addition and subtraction and the connections between these operations. For example, in joining problems where the strarting point is unknown, students can more easily see the benefits of addition (for example, \(?\; + \;12\; = \;37\)) that keeps the order of action in the problem. This allows them to use a strategy (for example, counting up or counting down) to determine the starting point quantity. These students demonstrate their understanding of the problem and their ability to use a strategy to solve it. However, they do not demonstrate an understanding of the meaning of difference (and subtraction). If they had used subtraction, that is, \(37\; - \;12\; = \;?\), they would have demonstrated a broader understanding of the relationship between quantities in relation to this operation. But when students are learning, there is no need to impose a strategy on them.

Requiring students to subtract will not help those who do not see the relevance of this strategy. However, if they are regularly exposed to a variety of problems and participate in the math exchanges that follow, they will be able to see the connections between various strategies and learn a variety of strategies. They then become more successful.

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 81-83.

Computational Strategies

Developing computational strategies offers several advantages over traditional instruction in standard algorithms, starting with the pride and confidence they provide. Students who use student-generated algorithms make fewer mistakes because they understand what is being done. In addition, students improve their knowledge and understanding of the base-ten number system, on which most computational strategies are based.

Furthermore, Van de Walle and Lovin (2006, p. 40) point out that research shows that students who have been able to develop personal computational strategies perform as well as or better than others on standardized tests.

There are size disparities between student-generated algorithms and standard algorithms. Student-generated algorithms are usually oriented to the direction of the units, according to their position (for example, in the addition 323 + 20, I add 2 tens to 323, which is 343) whereas usual algorithms tend to use the units without regard to their position (for example, in the addition \(323; + \;20\), we do : \(3\; + \;0\) is 3; \(2\; + \;2\) is 4…).

Standard algorithms usually start on the right, whereas in student-generated algorithms, students often start on the left, allowing them to maintain a sense of the size of the quantities involved. Since a student-generated algorithm is a product of each student's imagination and understanding, it remains very flexible so that it can be used in a variety of situations.

In the classroom, it is suggested that several algorithms for a single operation be examined. It is essential that students understand the reasoning behind the actions in these algorithms. Over time, this allows them to choose an effective strategy depending on the context. Teachers with culturally diverse students in their classrooms can invite them to discuss at home the method their parents use to perform addition, subtraction, multiplication or division. These students can present these methods to the class, which can provide new strategies.

The standard algorithms are often presented as the main strategy for calculation. Although they are effective, they are not always appropriate. When instruction focuses on the standard algorithm, for example, to calculate \(300\; - \;15\), students tend to pull out a pencil and solve the problem in writing, with the written algorithm and its exchanges, which is a common source of error. However, it is more efficient to calculate mentally as follows: \(300\; - \;10\; = \;290,\;290\; - \;5\; = \;285\). Furthermore, the standard algorithm is not the best method to use where estimation is sufficient. Therefore, it is suggested that the standard algorithm be considered as one of several computational strategies.

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 118-119.

Students can solve written problems in a variety of ways. The following tables provide some examples of addition and subtraction algorithms. These are not the only ways to solve a problem, there are many others, so students should be given opportunities to discover other ways of solving problems.

Strategies to Facilitate the Understanding of Standard Algorithms

It is important to provide students with several activities to explore standard algorithms using manipulatives such as the place-value mat, interlocking cubes, ten frames, base ten material, number line, and so on.

Teachers need to provide students with many opportunities to create their own algorithms, to explain their strategies and the reasons for their choices. It is important to give students the opportunity and time to explore the algorithms in greater depth and to encourage discussion. It is important to encourage students to work in pairs (one student writes down the steps while the other works with the concrete representation). Understanding of the meaning of the steps in a standard algorithm develops when teachers allow students to compare it to their own algorithm in order to make connections between the two approaches, such as "add from left to right and combine.

Adding Multi-Digit Numbers Without Grouping

Adding large numbers can be represented on a number line. For example, students could perform \(435\; + \;223\) by decomposing \(223\;\left( {200\; + \;15\; + \;8} \right)\) and representing the operation as follows:

Over time, students gradually develop their sense of abstraction and can use the same strategy without using a number line, but by performing the calculation mentally.

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 122-123.

Adding Multi-Digit Numbers with Grouping

(Van de Walle and Folk, 2005, p. 191)

It is important for students to practice exchanging groups of 10 ones into tens, groups of 10 tens into hundreds, and so on. Students need the support of visual representations of groupings to develop a conceptual understanding of the algorithm.

The base ten material helps some students visualize the operation more clearly. Here is how the base ten material can be used to represent addition.

Example

Students can also use a place-value mat that organizes the material by the position of the digit in the number.

The same numerical expression \(\left( {186\; + \;156} \right)\) can be represented using pictures. In this way, students demonstrate some level of abstraction, since any drawing can represent 100, 10, or 1.

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 123-124.

Subtraction Without Regrouping

As with addition, students often use manipulatives to perform subtractions. This strategy helps them grasp the concept of separating, although it is not very effective when dealing with large numbers.

The base ten material allows students to perform subtraction by separating.

Example

If students use the place-value mat for subtraction, some students are inclined to represent both terms. Subtraction is then performed by comparison.

Example

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la6eannée, p. 127-128.

Subtraction With Regrouping

(Van de Walle and Folk, 2005, p. 193)

Exploring subtraction with regrouping promotes conceptual understanding. Teachers should encourage students to use the place-value mat and base ten materials to model subtraction with regrouping. Students can work in pairs. They can move on to the written form of the algorithm once they have developed a solid understanding of it through modeling.

In the case of the problem \(325\; - \;118\), students represent the first number (325) with base 10 material on the top portion of the place-value mat. Unable to remove 8 ones since there are only 5, students exchange one ten for 10 ones.

This gives them a group of 15 ones from which they can now remove 8 so that 7 ones remain. Students should be encouraged to group the ones on the mat to better organize their work.

Students now remove 1 ten and 1 hundred and place them outside the mat.

So, \(325\; - \;118\; = \;207\)

Source: Guide d'enseignement efficace des mathématiques de la maternelle à la 6^e année, p. 58-59.

Students can also use the number line to perform a subtraction. For example, to calculate \(263\; - \;45\), they can use compensation to work with more familiar numbers. Since \(263\; + \;2\; = \;265\), one can carry out \(265\; - \;45\; = \;220\) and then subtract 2 to compensate \((263\; + \;2\; - \;45\; - \;2)\).

Students do not have to transcribe their thinking into numerical expression, but can still use the number line to illustrate their thinking:

The number line can also be used with decomposition according to the place values of the digits of the number \((263\; - \;40\; = \;223,\;223\; - \;5\; = \;218)\) :

The open number line allows students to proceed in significant leaps \((263\; - \;3\; = \;260,\;260\; - \;40\; = \;220,\;220\; - \;2\; = \;218)\) :

Students can use drawings to quickly illustrate a numerical expression such as \(1\;369\; - \,821\). Numbers can be represented by lines, circles, dots, and so on. A separation can be expressed as bars on the drawing. For 1369, you need to remove 8 hundreds, 2 tens and 1 one from 13 hundreds, 6 tens and 9 ones.

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 127-129.

Skill: Representing and Solving Problems Involving Addition and Subtraction of Decimal Tenths

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The student will use prior knowledge of whole numbers to solve problems involving decimal numbers and will use the same types of student-generated algorithms, since they are based on the meaning of the number rather than the position of the digits. As a result, the student will be able to clearly explain the meaning of the computations performed.

Source: UPBALF Numération et sens du nombre/Mesure 6^e, module 3, série 2, p. 179

Example

Sophie is making two necklaces with beads.

She uses 1.5 m of yarn to make the first and 2.7 m of yarn to make the second. What length of yarn did she use in total?

To add decimal numbers effectively, students need to understand the place value of the digits that make up each number and factor this into their calculations. Students must also recognize that the decimal point is a marker that identifies the place value of the digits. When adding, to ensure that the place values match, points can be aligned. For students who have a good sense of addition and place value, aligning the decimal point is not a rule to be memorized, but a way to account for place values.

When adding decimal numbers, the concept of grouping is used just as when adding whole numbers. For example, 7 tenths can be added to 5 tenths to get 12 tenths. Since the decimal system does not allow two digits to be placed in the same position, students need to understand the concept of regrouping.

5 tenths + 7 tenths = 12 tenths

(The rod represents one unit)

The decimal system does not allow two digits to be in the same position.

Since 10 tenths can be grouped into 1 unit, we have 1 unit and 2 tenths.

The quantity "12 tenths" is written 1.2.

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 98-99.

The base ten material and the place-value mat are a great help. With these materials, quantities of the same value are joined together explicitly, for example, tenths are added with tenths. When students work with base-ten materials, they use their knowledge of place value and, in so doing, deepen the concept of grouping, transferring this concept from whole numbers to situations involving decimal numbers. Students then recognize that regardless of place value, whenever 10 elements are found in a position, they are replaced by 1 group of 10 that is placed in the position to its left. Using this type of material increases students' understanding and introduces them to algorithms for adding decimal numbers.

It is important for teachers to have students make connections between these strategies in order to consolidate addition of decimal numbers. The following are different strategies for solving the problem below:

\(1.5\; + \;2.7\)

In order to estimate the sum, it is possible to reason as follows: \(1.5\; + \;2.7\), is about \(2\; + \;3\), so about 5.

Addition Using Base Ten Materials

To add the two quantities, each of the two numbers is represented using base ten material on a place-value mat. By putting the material together, we obtain 3 units and 12 tenths. We gather 10 tenths which we exchange for 1 unit. We then have 4 units and 2 tenths, or 4.2.

Addition Using a Number Line

\(\begin{array}{l}1.5 + 2.7 = 1.5 + 2 + 0.7\\1.5 + 2.7 = 4.2\;\;\,\end{array}\)

Addition Using Words

\(\begin{array}{l}1{\rm{\ et \ 5 \ dixièmes}}\;{\rm{ + }}\;{\rm{2 \ et \ 7 \ dixièmes}}\\{\rm{ = }}\;{\rm{3 \ et \ 12 \ dixièmes}}\\{\rm{ = }}\;{\rm{3}}\;{\rm{ + }}\;{\rm{1 \ et \ 2 \ dixièmes}}\\{\rm{ = }}\;4{\rm{ \ et \ 2 \ dixièmes}}\\{\rm{ = }}\;{\rm{4}}{\rm{.2}}\end{array}\)

Addition Using a Student-Generated Algorithm

The numbers are decomposed according to the place values.

Addition Using a Standard Algorithm

Example of Comparison

Christophe uses 3.4 m of wire to make two beaded necklaces.

Sophie uses 1.2 m of thread to make two beaded bracelets.

How much more thread does Christopher use than Sophie?

During subtraction, it is important, as it was for addition, to consider the place value of the digits that make up the numbers. The strategies for subtracting decimal numbers are essentially the same as those used for subtracting whole numbers.

Subtraction Using Base Ten Materials

When using concrete materials to represent subtraction, students can actually manipulate quantities. To determine a difference, students can compare one quantity to another or take one quantity away from another. In addition, students discover, through the use of these materials, that sometimes exchanges are necessary to make it easier to determine the difference between quantities.

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 101

\(3.4\; - \;1.2\; = \;?\)

Each number is represented using base ten materials and similar quantities (in red) are matched in each position. The difference is represented by the remaining quantities in the number 3.4 (in blue). Thus, we obtain \(3.4 - \;1.2 = \;2.2\).

Christophe uses 2.2 m more than Sophie.

Example with Separating

Christophe uses 3.4 m of wire to make two beaded necklaces. He uses 1.2 m of wire to make the first necklace. How much wire will he use to make the second necklace?

\(3.4\; - \;1.2\; = \;?\)

Subtraction Using Base Ten Materials

The number 3.4 is represented using base ten material. Then, the equivalent of the number 1.2 is removed. The difference between the two numbers, 2.2, remains on the mat.

The wire used for the second necklace measures 2.2 m.

Example of Regrouping

Ahmed wants to make necklaces too. He had 2.4 m of wire. He uses 1.6 m for the first necklace. How much wire will he use to make the second necklace?

\[2.4 - 1.6 = ?\]

Subtraction Using Base Ten Materials

The number 2.4 is represented using base ten materials. When we try to use the separating strategy to perform the subtraction, we realize that there are only 4 tenths on the mat and we need to take away 6 tenths. In this case, we separate 1 unit into 10 tenths. We then remove the equivalent of the number 1.6.

The difference between the two numbers, 0.8, remains on the mat.

So the length of wire used to make the second necklace is 0.8 m.

Subtraction Using a Nubmer Line

Subtraction Using a Student-Generated Algorithm

\(\begin{align}2.&4\; - \;1.6\\2.84\; - \;\left( {1.4\; + \;0.2} \right)\\2.&4\; - \;1.4 = \;1\\ &1\; - \;0.2 = \;0.8\end{align}\)

So the length of wire used to make the second necklace is 0.8 m.

Subtraction Using a Standard Algorithm

The standard algorithm also allows subtraction with decimal numbers. However, you have to make sure to match the place values.

So the length of wire used to make the second necklace is 0.8 m.

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 99-104.

Particular attention should be paid to subtractions involving numbers with a 0 in the ones or tens position. Van de Walle and Folk (2005, p. 193) suggest that it is most effective to address this issue when students are working with manipulatives. It is important to give students the opportunity to discuss the different strategies used to perform subtraction with such numbers.

Source: Guide d'enseignement efficace des mathématiques de la maternelle à la 6^e année, p. 59.

Knowledge: Algorithm

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Algorithms are sets of rules and ordered actions needed to solve an addition, subtraction, multiplication or division. In simple terms, an algorithm is the "recipe" for an operation. (Kilpatrick, Swafford and Findell, 2001, p. 103)

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 75.

Knowledge: Student-Generated Algorithm

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A strategy, usually developed by the student, for carrying out a computation.

Example

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 76.