|Strategies for Operations|
Doubles Plus One
Doubles Plus Two
Or students may use the strategy …
Double the Skipped Number
Adding to Make Ten
Adding with Tens
Adding 8 and 9
The same strategy is used with visualizing 8 in the ten frame. Students mentally move two into the ten frame to make ten and then are able to add “10 + ___.” Adding 8 is more difficult, and second graders may not grasp it right away.
This is the understanding that any number minus itself is zero and any number minus zero is itself.
Counting up is used when the numbers are close to each other on the number line; in other words, when the difference between two numbers is three or less. Students determine the difference by counting up from the smaller number to the larger number. The answer is how many times you counted up.
Using Doubles to Subtract
Subtracting from 10
Sutracting from 9
Make Ten and Then Some
Summarization of Some Addition Strategies
More Subtraction Strategies
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Activities and teaching suggestions are provided in the Addition and Subtraction Fact Strategies booklet from the Wichita Public Schools.
Children should have multiple experiences with all the common addition and subtraction situations.
Please note that this list of strategies is not comprehensive. The examples of basic fact strategies are given as an instructional sequence that focuses on connecting multiplication to known addition facts. The list begins with the easiest multiplication fact strategies and moves towards the more difficult.
Students need to fully understand the commutative property, and "turnaround" facts should be presented together. Arrays are important tools to use as students are building a conceptual understanding of multiplication.
Doubles in multiplication connects to doubles in addition. These facts are relatively easy to learn, especially if students already mastered the doubles for addition.
These multiplication facts are usually easy for students to master because of their experience skip counting by 5s.
Zeros and Ones (x0 or x1)
Make sure to differentiate addition concepts from multiplication concepts for these facts. They are often confused. For example 6 x 0 is confused as 6 + 0.
Students can reason for these facts.
These facts are best developed using story problems to convey meaning rather than isolated rules such as "any number times 0 equals 0" or "any number times 1 equals that number."
These multiplication facts are usually easy for students to master because of their experience with standard 1.NBT.2c (The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones) and standard 2.NBT.2 (Count within 1000; skip-count by 5s, 10s, and 100s).
Multiplying by 9 is like multiplying by 10 then taking one set away.
Focus on the patterns in these facts. One pattern is the sum of the digits in the product is always equal to 9. The other pattern is the ten’s digit is always one less than the factor multiplied by 9.
For example, 9 × 6 = 54.
Notice 5 + 4 = 9 and the 5 in the product is one less 6, the number being multiplied by 9.
Another example, 8 x 9 = 72.
Again, the sum of the digits 7 and 2 is 9 and the tens digit is one less than the tens digit.
Double, Double (x4)
Because 4 is 2 doubled, multiplying by 4 is doubling doubles.
When making arrays, encourage students to make squares. Have them recall that a square has the same number of units on each side. This activity will help them visualize the products of 3 x 3, 4 x 4, 5 x 5, etc.
Use decomposition strategies, known facts, and the distributive property to solve. For instance, 7 × 6 might be rewritten as 7 x (5 + 1). This would allow a student to use the 7 × 5 fact that he knows and add that to the 7 × 1 fact to get 42.
Activities and teaching suggestions are provided in the Multiplication Fact Strategies booklet from Wichita Public Schools.
Beyond the Basic Facts
Students need to demonstrate fluency of the basic multiplication (and addition!) facts before moving into multiplication of larger numbers. And they need to develop conceptual understanding of multiplication of larger numbers before being introduced to the traditional multiplication algorithm. There are several models and strategies that help them develop conceptual understanding.
The base-model for multiplication is an extension of the base-ten models children used to build and represent numbers. Using base-ten blocks, students can model the multiplication of a one-digit by a two-digit number. The model reinforces finding the area of rectangles.
After having had experiences building base-ten models for multiplication, students can move to area models without the manipulatives. The area models are representations which simplify by drawing rectangles rather than seeing all of the units as with the base-ten model. Areas of component rectangles are found and then summed to find the product.
The area model connects to algebra and the multiplication students with perform with binomials in later grades.
The intersecton model of multiplication has its foundation in the array model. The example below shows a basic fact, but the model can be extended for multiplying larger numbers.
In multiplying larger numbers using the intersection model, students must expand the factors using place value and use the distributive property to multiply all components. Finding the product of 47 and 53 is found using the intersection model in two ways in the examples shown below.
Students can multiply two factors by using the distributive property and expanding a factor using place value.
The use of the partial products strategy provides the foundation for the traditional algorithm. Parial products requires students to use the distributive property and place value as they did with partitioning strategies, and it begins to put the work in place as students will do with the traditional algorithm.
The lattice method is also connected to the distributive property and place value, but its connection is less obvious. When students draw the "lattice" it seems a bit similar to the area model, but it differs in that it does not require students to find the areas of rectangles. Instead, students are multiplying the digits and using the lattice to keep the digits with the same place value aligned.
Mastery of multiplication facts and connections between multiplication and division are the key elements of division fact mastery.
If students have trouble with this strategy, it is best to work on the related multiplication facts and developing efficient multiplication strategies. Drill and practice of division facts without efficient strategies for related multiplication facts is ineffective.
Do not introduce premature drill and practice of division facts until students have mastered and own efficient strategies for related multiplication facts. Counting on fingers or making marks on paper can never result in quick fact recall regardless of the amount of drill. Drill without an efficient strategy present offers no benefit to students and merely reinforces inefficient practices.
Beyond the Basic Facts
Students must continually be challenged to consider the magnitude of numbers and whether or not the results they obtain are reasonable. Estimation is a skill that is useful for all computations, but it is especially useful with division. Having students predict the size of the quotient before carrying out any computations and then defend why they believe their prediction is reasonable is a good exercise in making sense of the work and in recognizing when results are out of bounds.
As we begin a discussion of division beyond the basic facts, it is also important to recognize the two types of division:
The partitive type of division allows children to share the amount into groups equally until all the amount (or as close to it as possible) is gone. An example of partitive division is illustrated below.
Partitive division can be done for smaller numbers, but when the divisor (the number of groups) or the dividend (the amount being shared equally) get very large, it becomes less efficient and more tedious. Other methods and strategies can be employed instead.
The methods and strategies discussed below should be used to develop children's conceptual understanding of division of larger numbers before introducing them to the long division algorithm.
The area model for division is closely related to the area model for multiplication. With multiplication, the product is the area of the large rectangle formed with consecutive sides the length of the factors. The factors are known and the area (product) is found. With division, the area of the rectangle (the dividend) and one of the sides (the divisor) are known, and the goal is to find teh length of the other side of the rectangle (the quotient).
Students begin by building the rectangle represented by the dividend using base-ten blocks, as in the example below.
Because 12 is the divisor, the rectangle has to have 12 as the length of one side.
The length of the other side of the rectangle is the quotient.
In this case, the length of the edge consecutive to the edge of length 12 is 23. Therefore, the quotient of 276 divided by 12 is 23.
The area model can also be used when there is a remainder. It will become obvious that there is a remainder when it is impossible to build a rectangle of the dividend having the divisor as the length of an edge. Students will have to make sense of what to do with the remainder based on the context of the problem.
Decomposing the Dividend
A strategy for developing conceptual understanding of division is decomposing the dividend. In this strategy, the student will rewrite the dividend as the sum of "friendly" numbers in relationship to the divisor. Then the division is accomplished by dividing each part. An example is shown below.
Partial quotients is a similar strategy to decomposing the dividend, and it provides the foundation for understanding the long division algorithm. As students become more and more efficient with partial quotients, they are actually performing long division by the algorithm.
Partial quotients is based on the idea of division as repeated subtraction, using powers of ten to speed up the process. An example of dividing using partial quotients is shown below.
Here, the student is not yet efficient in finding the greatest number of times 84,000 can be divided by 27 and resorts to subtracting out several multiples of 1000 instead.
When the difference gets to 3,926, the student uses a multiple of 100 to further reduce the amount.
At a difference of 1,226, the student uses a multiples of 20 to reduce further. Finally, the student uses a multiple of 5 to get a difference of 11, which is smaller than the divisor and is the remainder. The quotient is the sum of all the multiples. In this case, 84,926 divided by 27 is 3,145 with a remainder of 11.
As illustrated in this example, multiples of 10, 100, and 1000 are the easiest ones for students to use as they divide using the partial quotients strategy. Because multiplying by 2 and by 5 are relatively easy for students, they will often use multiples of 20, 200, and 2000 or 50, 500, and 5000 as well.
Children should have multiple experiences with all the common multiplication and division situations.