At a Glance...

B2.4 represent and solve problems involving the addition and subtraction of whole numbers and decimal numbers, using estimation and 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 must be more oriented towards the understanding of operations, the exploration of mental calculation and the use of various strategies to carry out operations. It is in this sense that the mathematics framework program stipulates the expectation for students in the junior cycle, namely that the student must be able to solve problems related to the operations studied by using various strategies or personal algorithms.

In order to meet this expectation, students must be put in a problem-solving situation. This will allow them to develop and explore various strategies or personal algorithms.

To develop effective strategies in students, it is important to offer students various types of problems in such a way as to allow them to grasp the multiple meanings of operations. A well-chosen problem and the application of a thoughtful strategy are more profitable than a series of mechanically completed exercises. It is therefore necessary to allocate the necessary time that will allow the students to understand and consolidate the strategies.

The exploration of strategies (including flexible algorithms) is essential as these are tangible examples of the number sense and operations sense that students have acquired. These strategies and flexible algorithms indicate how they "play" with numbers and operations. These strategies that are put on paper have the potential to translate into mental math strategies. For example, students who have the opportunity to write their reasoning on paper or who use a number chart to perform a calculation such as \(36\; + \;52\) may later follow a similar reasoning mentally.

In addition, it is essential to animate mathematical exchanges relating, for example, to these strategies and to personal algorithms. These exchanges promote the sharing of strategies and the recognition of the links between them. Each student's personal strategies are clarified, perfected and become more effective as they establish links between them. Thus, "teachers guide the discussion by using strategies that students have used to initiate understanding of specific mathematical concepts and to direct students' progress towards effective methods" (Ministry of Education of the Ontario, 2004a, page 18).

The teacher's role is to help students organize their work. For example, students can be led to use semi-concrete representations, pointing out that a student who has combined objects shows the same reasoning as the student who wrote the number as a combination.

Similarly, when the student physically takes 10 groups of 10 units and realizes that it is a group of 100 units called a hundred, we can mention that the student is using the concept of grouping and that it is the same grouping which is symbolized by the writing of the number “1” above the position of the hundreds in the usual algorithm.

Scaffolding by teachers helps students understand the underlying concepts associated with various operations (for example, exchange in subtraction). In addition, the mathematical exchange makes it possible to present new strategies. Teachers and students can also model strategies by making sure to verbalize the reasoning behind them. Later, the usual algorithm can be presented, ensuring that the students understand the underlying concepts and the reasons for the actions taken. Common algorithms should be seen by students as just another way of performing operations.

Throughout the junior division, it is important to present a variety of problem-solving situations, even if students have mastered several strategies for performing the various operations. This allows them to build a network of representations, skills and connections and to develop flexibility in the use of operations. Students reflect on the types of calculations to be performed (for example, estimation, exact calculation), the operations to be performed, and the effective strategies to be used depending on the situation (for example, mental calculation, usual algorithm, personal strategy).

Source: A Guide to Effective Instruction in Mathematics, Grades 4-6, p 75-78.

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 simply 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 due to their context. However, for students, each situation represents a particular problem. It is by mastering these various types of problems that students acquire a mastery of addition and subtraction.

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.

• 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?
  

• Addition : Unknown added value - 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 $1 500. She gives $500 to her brother. How many dollars does she have left now?
  

• Separating : Change unknown - Nadia has $1 500. She gives it to her brother. She now has $1 000 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 $1 000 left. How much money did Nadia have at the beginning?
  

Meeting Problems

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

• Combining : Part of the whole unknown - The class has 800 coloured 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 coloured 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 comparison problems. In this example, we use the Cuisenaire rulers 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 on top of the number line starting at 0. I take another lime green rod and place it below the number line starting at 0 and I adds another lime green rod since Judith has $300 more than Jane. I replace the two lime green rods with the dark green rod that represents 6 (hundreds). Judith therefore has $600.

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

• 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.

\(600\; - \;?\; = \;300\)

Source: Adapted from A Guide to Effective Instruction in K-6 Mathematics^{, p 7-10.}

Adding and subtracting problems are seen by students as active situations, easier to model and “see” as the initial quantity increases or decreases. Meeting problems, however, assume a static situation because no action or change occurs, which makes them more abstract and harder to understand. Comparison problems, on the other hand, deal with the relationship between two quantities by opposing them: there is therefore no action, but a comparison of one quantity with 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 in which the variable is the initial quantity, the quantity added or the quantity removed. These problems help develop a more solid understanding of the operations of addition and subtraction and the relationships between the operations. For example, in addition problems where the unknown value 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 or counting 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 links between quantities in relation to this operation. But when students are learning, there is no need to impose a strategy.

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. 83-84.

Relationships Between Operations

The fundamental operations of addition, subtraction, multiplication and division are closely related despite their apparent differences. The more opportunities students have to work with the operations, the more they can notice and understand the connections between them.

Addition and subtraction are inverse operations. However, when they are learning, students often have difficulty solving equations such as \(17 + \Delta = 31\). Many teachers encourage their students to use the inverse operation, namely subtraction. However, it may be learning a trick, unless students understand why subtraction is a possible strategy. They must first grasp the relationship of the whole and its parts as well as the meaning of a difference. For example, a number can be represented as follows:

This way of representing the relationship between a number and its parts allows us to see that subtraction is the inverse operation of addition. Thus, since \(17+ 14; = 31) and \(14 + 17; = 31), then \(31 - 17; = 14) and \(31 - 14; = 17).

Additionally, students can see why addition is commutative (\(14 + 17\; = 17 + 14\)) and why subtraction is not (\(31 - 17 \ne 17 - 31\) ). Those who have acquired good number sense and are able to break down and regroup numbers can use their knowledge to more effectively solve equations such as \(17\; + \;?\; = \;31\ ) understanding that we are looking for the difference between 17 and 31.

Compensation Principle/Strategy

In addition to inverse operations, students can use the principle of compensation that follows from the equality relation. This involves changing the terms of an operation without changing the result. The principle of compensation involves the conservation of the equality of quantities in operations between numbers.

According to this principle, we can change the parts of an addition without changing the sum. For example, to calculate 143 + 218, we can intervene on the number 143 in order to facilitate the calculation. To do this, we subtract 3 from the first term and add 3 to the second term. In concrete terms, if we consider that we have 143 objects in a pile and 218 objects in a second pile, we move 3 objects from the first pile to the second pile. The total number has not changed. The numerical expression becomes 140 + 221. Here is the number sentence that illustrates what happens: \(143\; + \;218\; = \;143\; - \;3\; + \;218\; + \;3\). Therefore, \(143\; + \;218\; = \;140\; + \;221\), for a sum of 361.

Compensation is used to make an expression easier to evaluate. It can be used to add a quantity to get an easier number to work with, such as a multiple of 10 or 25, and subtract the same quantity at the end. In the following example, we add 10 to the number 390 to obtain the number 400, which is easy to add. Then subtract 10 from the answer.

\(\begin{array}{l}268\; + \;390\; = \;?\\268\; + \;400\; = \;668\\668\; - \;10\; = \;658\\\end{array}\)

Therefore, 268 + 390 = 658

We can visualize the situation using a vertical numerical line. We can see that in this case, a number greater than that of the expression (\( + \;400\)) has been added. The desired result has thus been exceeded. We must therefore subtract the excess (\( - \;10\)).

Compensation also applies to subtraction. Since we are looking for the difference between the terms, we modify the two terms in the same way to keep the same difference. You can add the same quantity to both terms or you can subtract the same quantity from both terms. For example, to calculate \(72\; - \;37\), one can add 3 to 72 to get a more familiar number, 75, while adding 3 to 37 to maintain the same difference.

\(\begin{align}72\; - \;37\; &= \;\left( {72\; + \;3} \right)\; - \;\left( {37\; + \; 3} \right)\\ &= \;75\; - \;40\\ &= \;35\end{align}\)

For this same expression, we could also compensate by subtracting 2 from each term.

\(\begin{align}72\; - \;37\; &= \;\left( {72\; - \;2} \right)\; - \;\left( {37\; - \;2} \right)\ &= \;70\; - \;35\ &= \;35\end{align}\)

We can also use compensation to subtract a larger number than is required in the mathematical expression and then add to the difference. In the following example, we subtract 53 (3 more than needed) to make the subtraction easier. We then add 3 to the result.

\(\begin{array}{l}173\; - \;50\; = \;?\\173\; - \;53\; = \;120\\120\; + \;3\; = \;123 \end{array}\)

Therefore, \(173 - 50 = 123\)

Compensation in subtraction uses the principle of constant difference, i.e. the difference between two numbers is the same if we add to them or if we remove the same quantity from them (for example, the difference between 645 and 185 is the same as that between 650 and 190 or that between 640 and 180).

The concept of constant difference can be used to perform operations such as subtraction with zeros (for example, \(1;000 - \;354 = \;999 - \;353)\).

Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p 97-101.

Computational Strategies

Personal or invented strategies offer several advantages over the traditional teaching of usual algorithms, starting with the pride and self-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 ten number system, on which most calculation 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 size disparities between personal algorithms and usual algorithms. Personal algorithms are usually oriented to the direction of the digits, depending on 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 digits regardless of 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 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 perform addition, subtraction, multiplication or division. These students can present these methods to the class, which can provide new strategies.

The usual algorithms are often presented as the main strategy for calculation. Although they are effective, they are not always appropriate. When instruction focuses on the usual 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: \(300\; - \;10\; = \;290\), \(290\; - \;5\; = \;285\). In addition, the usual algorithm is not the best method to use where estimation is sufficient. Therefore, it is suggested that the usual algorithm be considered as one of many computational strategies.

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 reason about other ways of doing things.

Source: A Guide to Effective Instruction in Mathematics^{, Grades 4-6, p 118-119.}

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, etcetera.

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.

Source: A Guide to Effective Instruction in Kindergarten to ^{Grade 6 Mathematics, p. 56.}

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:

Source: A Guide to Effective Instruction in Mathematics, ^{Grades 4 to 6, p. 122.}

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.

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.

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 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: Adapted from A Guide to Effective Instruction in Mathematics, Grades 4 to 6, 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.

Source: A Guide to Effective Instruction in Mathematics, ^{Grades 4 to 6, p. 127.}

The base ten material allows students to perform subtraction using an indentation. For example:

If they use the place value mat for subtraction, some students tend to represent both terms. The subtraction is then performed by comparison. For example :

Source: A Guide to Effective Instruction in Mathematics, ^{Grades 4 to 6, p. 128.}

Subtraction With Regrouping

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 \(325\; - \;118\) problem, students model the first number (325) with base ten materials on the top portion of the place value mat. Unable to withdraw 8 units since there are only 5, the students exchange a ten for 10 units.

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: A Guide to Effective Instruction in Kindergarten to Grade 6 Mathematics^{, 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 expressions, 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: A Guide to Effective Instruction in Mathematics, Grades 4 to 6^{, 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 shrinkage can be expressed by bars on the drawing. For \(1\;369 - 821\), we must subtract 8 hundreds, 2 tens and 1 unit from the 13 hundreds, 6 tens and 9 units.

Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6^{, p. 129.}

Link between a personal algorithm and a common algorithm

Giving students the opportunity to model addition and subtraction with regrouping on multi-digit numbers allows them to visualize the steps of the corresponding common algorithm. The steps of an algorithm are not out of context when conceptual understanding is present. Students who have forgotten a step in the usual algorithm can still find a solution to the problem since they have the ability to use a personal algorithm.

With practice modeling, students can move on to writing down the usual algorithm using blank tables to keep track of the steps modeled on the place value mat.

Addition

Subtraction

Source: Math: A Little, a lot, a lot, a lot," Teacher's Guide, Revised Edition, Numeration and Number Sense, ^{Grade 4, Module 1, Series 3, p 235-238.}

Skill: Estimating the Outcome of an Operation

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Estimating the result of an operation means determining the approximate value of the operation in writing or mentally. Students often have difficulty with estimation because they have difficulty recognizing that more than one answer may be acceptable for a given situation. For example, they may calculate the correct answer for an estimate and then round it off.

Estimation is extremely useful in everyday situations. It may even be the only possible or desired answer. For example, when shopping, you want to know the total cost of your purchases to be able to stick to your budget. In this case, we want an approximate idea of the result, not the exact total cost.

Estimation is often used to provide a preview of the expected result or to check the reasonableness of the solution. Students need to develop the ability to make a quick mental estimation as soon as a calculation is to be performed in order to get a general idea of the result. Unfortunately, many students do not make the connection between the estimation and the correct result; the estimation is then seen as another calculation that must be performed and not as an informally performed strategy that confirms the reasonableness of the calculation.

Teachers need to use a variety of teaching strategies to make students aware of the purpose of estimation. For example, students do not see the relevance of estimation if the exact result always accompanies the estimation. Teachers should present situations in which the solution to the problem is an estimate or problems that do not require a precise answer (for example, determining each citizen's share of the national debt or shopping for school supplies for next year).

Since on a daily basis, estimations are often the result of informal mental calculations, it is appropriate that students practice and learn to estimate in this context. Teachers should therefore ask them to estimate the result of an operation (without using paper and pencil) and to communicate this approximate result (for example, \(346 + 516\), it is close to 850).

You can also present a series of operations and ask students to perform only those that meet a certain condition (for example, perform those whose result is greater than 300). This teaching strategy can be integrated with problem solving. In the following problem, students use their estimation skills to determine which calculations need to be done with precision:

In a warehouse there are several bags of bales that contain either 1 256, 542, 368, 1 856, 325, 1 379 or 730 balls. In order to facilitate transport, the bags are put in boxes that can contain between 2 000 and 3 000 balls. On each box, you must clearly indicate the exact number of balls it contains. Determine 10 different combinations of bags that a box can contain.

Because the result of an estimation represents an approximate rather than a precise quantity, it should not be communicated exclusively using a number (for example, instead of stating that it is 350, one can say that it is about 350). In addition, the result of an estimation can help indicate the order of magnitude or the scale of the answer (for example, it will be at least , it will be more than, the answer must be betweenand, or the answer will be greater than).

The ability to estimate the result of an operation is a characteristic of operational sense. It demonstrates an ability to use numbers and operations in a variety of ways. To estimate, students use a variety of strategies based on their number sense and operational sense such as rounding, using benchmarks, applying properties of operations, decomposition, or compensation.

The degree of accuracy in estimating the result of an operation depends on the situation and the meaning of the number. It would be important at the end of the cycle to discuss acceptable estimation ranges in various contexts. For example, the estimate of the difference between 315 and 185 could be a number between 100 and 200.

Estimating the difference between 10 853 and 9 445 would rather be a number between 1 000 and 2 000. Estimating the outcome of an operation requires a good understanding of the effect of operations. Rounding is often used to estimate a result and it is important to be aware of its impact on the result of the estimate. For example, rounding to the nearest hundred generally creates a larger discrepancy between the actual number and the rounded value than rounding to the nearest ten.

Example

\(353\; + \;129\)

Exact result : 482

Estimate by rounding the numbers to the nearest hundred: \(400\; + \;100\; = \;500\)

Estimate by rounding the numbers to the nearest 10: \(350\; + \;130\; = \;480\)

Rounding numbers has little influence on the result of an addition or subtraction since these operations produce little effect on the numbers. Rounding can still help to refine the estimate. For example, the result of \(387\; + \;295\) is approximately 700. However, it can be said that the result is less than 700, since both numbers have been rounded up (\( 400\;+\;300\)). Similarly, \(1\;300 - 1\;170\) is approximately equal to 100. However, rounding up from 1170 to 1200 allows us to specify that the difference is more than 100.

Source: A Guide to Effective Instruction in Mathematics, ^{Grades 4 to 6, p 92-95.}

Skill: Representing and Solving Problems Involving the Addition of Decimal Numbers to Hundredths, Using Appropriate Tools, Strategies, and Algorithms

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To add decimals effectively, students need to understand the place value of the digits that make up each number and factor this into their calculations. They must also recognize that the decimal point is a marker that identifies the place value of digits. During an addition, to ensure the correspondence of the place values, one can align the decimal points. For students who have a good sense of addition and place value, decimal alignment is not a rule to memorize, 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 one can add 3 hundreds to 8 hundreds to form 11 hundreds, one can add 3 tenths to 8 tenths to form 11 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.

Example

{\(8\;{\rm{tenth}}\; + \;3\;{\rm{tenth}}\; = \;11\;{\rm{tenth}}\)

(The tab represents the unit)

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

The quantity "11 tenths" is written 1.1.

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, hundredths are added together with hundredths. When students work with base-ten 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 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.

Based on their knowledge of whole number addition strategies and their understanding of decimal numbers, students can add them using base ten materials, a number line, a personal algorithm, or the standard algorithm. It is important that teachers engage students in making connections between these strategies in order to consolidate addition of decimal numbers.

Source: A Guide to Effective Mathematics Teaching, Grades 4 to 6^{, p. 98-100.}

It is also important that the student estimate the answer to the problem before solving it. Estimation is often used to provide a preview of the expected outcome or to check the reasonableness of the solution. Students need to develop the reflex of mentally making a quick estimate as soon as a calculation is to be performed so that they can get a general idea of the outcome. Unfortunately, many students do not make the connection between the estimate and the correct result; the estimate is then seen as another calculation that must be performed and not as an informally performed strategy that confirms the likelihood of the calculation result.

Teachers need to use a variety of teaching strategies to make students aware of the purpose of estimation. For example, students do not see the relevance of estimation if the exact result always accompanies the estimation. Teachers should present situations in which the solution to the problem is an estimate or problems that do not require an exact answer.

Example:

Determine each citizen's share of the national debt or purchase school supplies for next year.

Since estimation in everyday life is often the result of informal mental calculation, it is only natural that students should practice and learn to estimate in this context.

Teachers should therefore ask them to estimate the result of an operation (without using paper and pencil) and communicate this approximate result

Source: A Guide to Effective Instruction in Mathematics, ^{Grades 4 to 6, p 92-93.}

Example:

\(1,572\; + \;2,724\)

In order to estimate the sum, it is possible to reason as follows: \1.572; + 2.724 is about 2; + 3 is about 5.

Students can then confirm their estimate using algorithms. Here are different strategies for adding two decimal numbers.

Addition using base ten materials

To add the two quantities, each of the two numbers is represented with base ten material on a place value mat. By putting the material together, we obtain 3 units, 12 tenths, 9 hundredths and 6 thousandths. We gather 10 tenths which we exchange for 1 unit. We then have 4 units, 2 tenths, 9 hundredths and 6 thousandths, that is 4.296.

Addition Using a Number Line

Addition Using a Student-Generated Algorithm

Numbers are decomposed according to place values.

Example:

Addition Using a Standard Algorithm

Example:

Source: Adapted from A Guide to Effective Instruction in Mathematics^{, Grades 4 to 6, p 100-101.}

Skill: Representing and Solving Problems Involving Subtraction of Decimal Numbers up to Tenths, Using Appropriate Tools, Strategies and Slgorithms

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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 estimating the subtraction of decimal numbers and for subtracting decimal numbers are essentially the same as those used for subtracting whole numbers.

Example:

\(2,724\; - \;1,572\)

In order to estimate the difference, it is possible to reason as follows: \(2.724\; - \;1.572\) is approximately \(3\; - \;2\), so approximately 1.

Students can then confirm their estimate using algorithms. Here are different strategies for subtracting two decimal numbers.

Subtraction Using Base Ten Materials

When using concrete materials to represent subtraction, students can actually manipulate quantities. To determine a difference, they 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.

Example of Comparison

\(3,465\; - \;1,214\)

Represent 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 3.465 (in blue). Thus, we get \(3,465\; - \;1,214\; = \;2,251\).

Example with Separating

\(3,405\; - \;2,1\)

The number 3.405 is represented using base ten material. Next, the equivalent of the number 2.1 is removed. The difference between the two numbers, 1.305, remains on the mat.

Example of Regrouping

\(2,423\; - \;1,26\)

The number 2.423 is represented using base ten materials. When we try to use the withdrawal strategy to perform the subtraction, we realize that there are only 2 hundredths on the mat when we should withdraw 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, 1.163, remains on the mat.

Subtraction Using a Number Line

Subtraction Using a Student-Generated 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.

Source: A Guide to Effective Instruction in Mathematics, Grades 4-6, p 101-104.

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, & Findell, 2001, p. 103)

Source: A Guide to Effective Instruction in Mathematics, Grades 4-6, p. 75.

Traditionally, algorithms (standardized computational steps) were designed at a time when an elite group of "human calculators" did not have calculators (Ma, 2004). Algorithms were not designed to foster the level of understanding that we expect of students today. (Ontario Ministry of Education, 2004a, p. 13)

Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p. 118.

Knowledge: Usual Algorithm

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Standardized way to perform an operation, for example:

Knowledge: Student-Generated Algorithm

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Strategy, generally developed by the student, to carry out an operation, for example:

It is a strategy found intuitively by the student or his peers, allowing him to perform an operation.

Not all students invent algorithms. However, all students benefit from sharing personal strategies.

Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p. 76.

Example:

\(299\; + \;299\)

It is more efficient to perform this addition by thinking of \(300; + \;300; - \;2\), rather than using all the steps of the usual algorithm.

Source : L’@telier - Ressources pédagogiques en ligne (atelier.on.ca).

Knowledge: Estimation

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Estimation is a process by which visual and mental information is used to assess the magnitude of a quantity. Estimating the result of an operation is determining its approximate value in writing or mentally. Estimates play an important role in our daily communications by giving us approximate quantities. It may even be the only possible or desired answer.

Examples

Through the media, nearly 10 000 people were at the rally.

In our own exchanges, grocery shopping costs about $200 per week.

Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p 36-92.

Estimation is an important skill associated with multi-digit number operations. This skill helps students develop their number sense and use it to understand each step of standard algorithms.

The goal of estimation is not to arrive at an exact answer, but a logical approximation. Questions such as "Is the answer less than 25?" and "Is the answer more than 10?" help students recognize possibilities and assess the reasonableness of their answers. The table below outlines some of the strategies teachers can use to help students develop the skill of estimating. These strategies should not be taught as terms and procedures to be memorized or used constantly. Teachers can use them to help students better understand what they are doing and why they are doing it, and they can be introduced to students only when the opportunity arises. Therefore, students should not be expected to use all of these strategies, at least not until the later grades.

In Grade 6, students can use the following estimation strategies:

Clustering is useful when numbers are easier to calculate. Clustering allows for repeated addition.

Compatiball numbers: the compatible number strategy is to use numbers that are easy to handle. In addition and subtraction, students look for numbers whose sum or difference is near 10.

Estimation from the left: In this type of estimation, the operation is performed using the highest place value number, with an adjustment of the remaining values.

Rounding: rounding is a more complex estimation method than left-handed estimation. It requires two steps: first rounding each number and then calculating the estimate.

Many students do not understand the importance and relevance of estimation. They believe that estimating is a way of arriving at the correct answer, so they find it necessary to change their estimation after making an accurate calculation. By providing students with many opportunities to practice estimating each day, teachers can help them improve this skill and thus develop a sense of estimation and its usefulness in everyday life.

An estimate provides students with a guide to determine if their solution is likely.

Students who have multiple opportunities to estimate are more likely to understand the importance of estimating and reasoning logically when working with large numbers.

Students' conceptual understanding of fundamental operations and their ability to perform mathematical operations with single- and multi-digit numbers efficiently and effectively are developed in problem-solving contexts. Students' self-image as mathematicians has a significant effect on their confidence and willingness to learn.

Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 6^e année, p. 68-70.