At a Glance...

B2.4 Represent and solve problems involving the addition and subtraction of whole numbers that add up to no more than 100 000, and of decimal numbers up to hundredths, using appropriate tools, strategies, and algorithms.

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

---

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.

In contrast to the traditional approach where students learn to apply standard algorithms, operations learning should be more oriented towards understanding operations, exploring mental math and using various strategies to perform operations. It is in this sense that the mathematics curriculum includes the expectation for students in the junior grades to be able to solve problems related to the operations being studied using a variety of strategies or personal algorithms.

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 75-76.

Written Problems : Addition and Subtraction

There are four types of problems in addition and subtraction : joining, separating, part-part-whole and comparing problems. 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. The following table presents a variety of addition and subtraction problems.

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 unique problem. It is through mastery of these different types of problems that students gain mastery of addition and subtraction.

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

Joining Problems

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

• Joining: Result Unknown

Jamil has a bag of 600 candies. He buys 500 more pieces of candy. How many pieces of candy does Jamil have now?

• Joining : Start Unknown

Jamil has a lot of candy. He buys 500 more. He now has 1100. How many pieces of candy did Jamil have at the beginning?

• Joining : Change Unknown

Jamil has a bag of 600 candies. He buys many more. He now has 1100 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 seperation of a quantity.

• Separating: Result Unknown

Nadia has $1500. She gives $500 to her brother. How many dollars does she have left?

• Separating: Change Unknown

Nadia has $1500. She gives some money to her brother. She now has $1000 left. How much did Nadia give her brother?

• Separating: Start Unknown

Nadia gave $500 to her brother. She has $1000 left. How much money did Nadia have at the beginning?

Part-Part-Whole Problems

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

• Part-Part-Whole: Part Unknown

The class has 800 crayons. 300 of these crayons are red. The remaining crayons are blue. How many blue crayons does the class have?

• Part-Part-Whole: Whole Unknown

The class has lots of crayons. 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 Cuisenaire rods and the double number line.

• Comparing: Difference Unknown

Judith has $600 and Jane has $300. How many more dollars does Judith have than Jane? OR Judith has $600 and Jane has $300. How much less does Jane have than Judith?

I know that the dark green rod represents 6 (hundreds), so I place it at the top of the number line at 0. I know that the lime green rod represents 3 (hundreds), so I place it below the number line at 0. I compare the 2 rods and see that the lime green rod represents 3 (hundreds) less than the dark green rod. I find the difference or gap between the 2 quantities. There is a difference of $300. Judith has $300 more than Jeanne or Jeanne has $300 less than Judith.


• Comparing: Larger Quantity Unknown

Judith has $300 more than Jane. Jane has $300. How much money does Judith have? OR Jane has $300 less than Judith. Jane has $300. How much money does Judith have?

I know that the lime green rod represents 3 (hundreds), so I place it at the top of the number line at 0. I take 1 more lime green rod and place it under the number line at 0 and add 1 more lime green rod since Judith has $300 more than Jane. I replace the 2 lime green rods with the dark green rod which represents 6 (hundreds). Judith has $600.



\(?\; - \;300\; = \;300\)

• Comparing: Smaller Quantity Unknown

Judith has $600 and Jane has $300 less than Judith. How much money does Jane have? OR Jane has $300 less than Judith. Judith has $600. How much money does Jane have?

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

Source: Guide d'enseignement efficace des mathématiques de la maternelle à la 6e année, p. 9-10.

Joining and separating problems are perceived by students as active situations, easier to model and "see" as the initial quantity increases or decreases. Part-part-whole 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 2 quantities by contrasting them: so there is no action, but a comparison of one quantity to another.

Since students are regularly exposed to problems whose final quantity is sought, they solve them more easily. However, they have more difficulty solving problems where the unknown is the initial quantity, the quantity joining, or the quantity being separated. These problems help develop a more solid understanding of the operations of addition and subtraction and the relationships between the operations. For example, in the case of joining problems where the unknown is the initial quantity, students more easily see the advantages of addition (for example, \(?\; + \;12\; = \;37 \)) which makes it possible to respect the order in which the action takes place in the problem. This allows them to use a strategy (for example, counting up or down) to determine the initial 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 relationships between quantities in relation to this operation. However, a strategy should not be imposed on students while they are learning.

Requiring students to subtract will not help students who do not see the relevance of this strategy. However, if they are regularly exposed to a variety of problems and participate in the mathematical exchanges that follow, they are able to see the connections between various strategies and to 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

Personal or invented strategies offer several advantages over traditional instruction of standard algorithms, starting with the pride and confidence they provide. Students who use personal algorithms make fewer mistakes because they understand what they are doing. In addition, they improve their knowledge and understanding of the base-10 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 strategies perform as well as or better than others on standardized tests.

There are major differences between personal algorithms and standard algorithms. Personal algorithms are usually oriented towards the meaning of the digits, according to their position (for example, in the addition 323 + 20, I add 2 tens to 323, which gives 343) whereas standard algorithms tend to use the digits without taking their position into account (for example, in the addition \(323 + \;20\), we do : \(3\; + \;0\), that's 3; \(2\; + \;2\), that's 4…).

Standard algorithms usually start on the right, whereas in their personal algorithms, students often start on the left, enabling them to maintain a sense of the magnitude of the quantities involved. Since a personal algorithm is the fruit of each student's imagination and understanding, it remains highly 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 add, subtract, multiply or divide. Students can then present these methods to the class, which can provide students with additional 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. Students should be given opportunities to reason other ways.

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, etc.

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 changing groups of 10 ones into tens, groups of 10 tens into hundreds, and so on. They need the support of visual representations of groupings to develop a conceptual understanding of the algorithm.

Source: Guide d'enseignement efficace des mathématiques de la maternelle à la 6e année, p. 57.

Base 10 materials helps some students visualize the operation more clearly. Here is how base 10 materials 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 numeric expression (\(186\; + \;156\)) can be represented using drawings. In this way, students demonstrate some level of abstraction since a drawing is used to 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 separation, although it is not very effective when dealing with large numbers.

Base 10 materials allow students to perform subtraction using separation.

Example 1

If students use the place value mat for subtraction, some students are inclined to represent the 2 terms. The subtraction is then done by comparison.

Example 2

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, 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 10 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 through modelling.

In the case of the calculation \(325\; - \;118\), students represent the 1st number (325) with base 10 materials on the top portion of the place value mat. They cannot remove 8 ones since there are only 5, so students regroup to get 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.

Source: Guide d'enseignement efficace des mathématiques de la maternelle à la 6e 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 \(263\; + \;2\; = \;265\), we can perform \(265\; - \;45\; = \;220\) 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 the 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):

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

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

Skill: Representing and Solving Problems Involving Addition and Subtraction of Decimal Numbers to Hundredths

---

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 correspond, decimal 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, just as we can add 3 hundreds to 8 hundreds to form 11 hundreds, we can add 3 tenths to 8 tenths to form 11 tenths. However, since the decimal system does not allow 2 digits to be placed in the same position, students need to understand the concept of regrouping, as illustrated in the table below.

Base 10 materials and the place value mat are a great help. With these materials, parts that have the same value can be physically combined, for example, hundredths are added together with hundredths. When students work with base-10 materials, they use their knowledge of place value and, as a result, extend the concept of grouping, transferring the concept they applied to whole numbers to situations involving decimal numbers. Students recognize that regardless of place value, whenever 10 elements are in one position, they are replaced by 1 group of 10 that is placed in the position to its left.

Using these types of materials increases students' understanding and introduces them to algorithms for adding decimal numbers. Based on their knowledge of addition strategies for whole number and their understanding of decimal numbers, students can add them using base 10 materials, a number line, a personal algorithm, or the standard algorithm. It is important for teachers to engage students in making connections between these strategies in order to consolidate addition of decimal numbers.

Here are different strategies for adding 2 decimal numbers:

Example

In order to estimate the sum, it is possible to reason as follows: \(1.57\; + \;2.72\) is about \(2\; + \;3\), so about 5.

Addition using base 10 materials

To add the 2 quantities, represent each of the 2 numbers using base ten materials on a place value mat. Putting all the material together, we get 3 ones, 12 tenths and 9 hundredths. We combine 10 tenths that we exchange for 1 one. We then have 4 ones, 2 tenths and 9 hundredths, or 4.29.

Addition using a number line

Addition using a personal algorithm

Numbers are decomposed according to place values.

Example

Student reasoning

Addition using a standard algorithm

Example

Student reasoning

During subtraction, it is important, as it was with 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 10 materials

When students use 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 recognize through the use of these materials that sometimes it is necessary to regroup in order to more easily determine the difference between quantities.

• Example of comparison

\(3.46\; - \;1.21\)

Model each number using base ten materials and match like quantities (in red) in each position. The difference is represented by the quantities that remain in the number 3.46 (in blue). Thus, we get \(3.46\; - \;1.21\; = \;2.25\).

• Example with separating

\(3.40\; - \;2.1\)

The number 3.40 is represented using base 10 materials. Then the equivalent of the number 2.1 is removed. The difference between the two numbers, 1.3, remains on the mat.

• Example of regrouping

\(2.42\; - \;1.26\)

The number 2.42 is represented using base 10 materials. When we try to use the separating strategy to perform the subtraction, we realize that there are only 2 hundredths on the mat when we should separate 6 hundredths. In this case, we exchange 1 tenth for 10 hundredths. We then remove the equivalent of the number 1.26.

The difference between the two numbers, 1.16, remains on the mat.

Subtraction using a number line

Subtraction using a personal algorithm

Subtraction using a standard algorithm

We can use a standard algorithm for subtraction with decimal numbers. However, you have to make sure to match the place values.

Student reasoning

Source: Guide d'enseignement des mathématiques de la 4^e à la 6^e année, p. 98-104

Knowledge: Algorithm

---

Algorithms are sets of rules and ordered actions required to solve an addition, subtraction, multiplication or division. In simple terms, an algorithm is the "recipe" for an operation. (Kilpartick, 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: Personal Algorithm

---

A strategy for solving a problem or performing a calculation, usually developed by the student.

Example

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