One of the most important skills teachers can equip their students with is the ability to apply a general math concept to problem-solving. However, students often learn the properties of arithmetic operations without appreciation of their real importance and application. That’s why Happy Numbers pays a lot of attention to clear, step-by-step guidance for learning the conceptual foundation as rules and properties and then using these to solve a variety of problems.
In this post, you’ll find an overview of Happy Numbers’ pedagogy relating to multiplication properties. See how introductory tasks explain and help students formulate a property. Explore exercises that not only develop procedural fluency but also demonstrate their power and attraction through diverse visual models and scenarios.
1. The Commutative Property
The commutative property is the simplest of multiplication properties. It has an easily understandable rationale and impressive immediate application: it reduces the number of independent basic multiplication facts to be memorized. For example, due to the property, it’s sufficient to know that the product 4 × 6 equals 24 to also know the product 6 × 4.
Happy Numbers leads students through a series of interactive steps that results in discovery of the commutative property:
Prior to choosing the correct sign, students have figured out the results of the two multiplications.
The main point of the exercise is achieved through animation that rotates the array, transforming rows into columns and vice versa, without changing the number of objects. To see the animation and full exercise, follow this link.
In the next exercises, students solidify the new skill by filling in the blanks in tasks like these:
Students who need to correct an error get a visual hint showing the corresponding array in its original and turned configurations.
Completing these tasks prepares students for the articulation of the commutative property:
Тo see the full exercise, follow this link.
The above wording perfectly suits the main application of the commutative property — given a multiplication fact, compose the related fact using swapped factors. To solidify understanding and application of the commutative property, students solve a number of tasks. For example:
Watch the video below to see the support students receive this time: it’s a textual hint recalling the commutative property.
The possibility of swapping factors without changing the product means a major development in multiplication of single-digit numbers. About half of all products in the multiplication table below can be found without calculation: by filling in a missing product in any cell with the number in its correlating cell (with respect to diagonal symmetry).
Students use this strategy in two types of exercises that demonstrate the power of the commutative property and help students master basic multiplication facts.
Exercises of the first type are preparatory. Students locate pairs of cells that correspond according to multiplication that differs only in the order of the factors. For example:
… and also find the corresponding (common) product:
In exercises of the second type, students use the acquired skill to fill in certain cells by finding in the table the (equal) product of the swapped factors. Watch the video below to see in particular the support students get in case of an incorrect answer.
2. The Associative Property
The associative property can be interpreted similarly to the commutative property by calculating the number of objects in an original and rotated array. The substantial difference is that in the case of the commutative property, arrays are two-dimensional. For example:
while the associative property corresponds to three-dimensional arrays. For example, the three-dimensional array:
is made up of 3 copies of a two-dimensional array including 4 × 2 objects, so there are (4 × 2) × 3 objects in total.
The number of objects in the rotated array is the same and can be calculated as 4 × (2 × 3), and the two calculations results in an equation illustrating the associative property:
Since interpreting an image of a three-dimensional array is hardly an appropriate task for third-grades, Happy Numbers adopts a model that separates the “layers” of a 3D array and puts them side by side:
The screenshot shows the number of berries on each plate, number of plates on each table, and number of tables, all already identified by students. Next, students calculate the total number of berries two different ways:
– Starting with the number on berries on each plate: 4 × 2, the total is (4 × 2) × 3 = 24
– Starting with the number of plates on each table: 2 × 3, the total is 4 × (2 × 3) = 24
This results in an equation expressing the associative property (4 × 2) × 3 = 4 × (2 × 3) for the given factors.
To see the full exercise, follow this link.
To get accustomed to the associative property, students work on pairs of multiplication expressions that only differ in the grouping of the factors, for example:
Operations are numbered in each expression to help students evaluate the expression and reflect on their common value, similarity, and difference.
Based on their experience, students complete the wording of the associative property:
Students are now ready to start applying the associative property to simplify some calculations. This simplification is particularly effective when the associative property is applied together with the commutative or distributive property. To prepare students for such calculations, Happy Numbers first provides a number of simpler tasks in which the associative property is applied on its own. In these tasks, students simplify multiplication of a two-digit by single-digit number:
Applying the associative property reduces the calculation to 7 × (2 × 3), which uses two basic facts of single-digit numbers mutiplication. Watch the video below to see the slide parentheses technique Happy Numbers provides to help students master regrouping the factors.
A slightly more advanced task of this type starts with completing decomposition of the two-digit number, given one of its factors:
There is a particular case of this exercise that deserves special attention: when the two-digit factor is 20, 30, … 90. Experience with multiplying the tens is important since it initiates the mental math skill of multiplying 70 × 4, 300 × 8, 6,000 × 5, etc. The first step of the calculation is decomposing the tens. For example:
To see the full exercise, follow this link.
3. Juggling the Factors || Commutative and Associative Properties Combined
In numerical expressions that only include multiplication, for example 25 × 7 × 8 × 31, rearranging factors in any way does not change the product, due to the commutative and associative properties. The benefit of rearranging factors can be substantial. For example, it can reduce the calculation to mental math.
The Happy Numbers grade 5 curriculum provides interactive tasks that
– Briefly review both properties
– Lead students through all steps of applying the properties to simplify the calculation in a representative example only involving multiplication
– Help students grasp that in such cases rearranging the factors in any way does not change the product
This is reflected in the following screen:
To see the full exercise, follow this link.
“Juggling the factors” is important in particular for multiplication of multi-digit numbers, including calculations with the standard algorithm.
The property applies in particular to calculating products like 70 × 4, 300 × 8, 6,000 × 5, etc. (a step in the standard algorithm). This multiplication can be performed mentally, and Happy Numbers provides a sequence of tasks to develop this skill. Prerequisites of this skill are:
– Rearranging the factors
– Factoring out the largest possible power of 10, for example 6,000 = 6 × 1,000
This is in addition to basic skills of multiplication of single-digit numbers and multiplication of powers of tens, like 100 × 10 = 1,000.
Students apply a step-by-step strategy that starts with factoring out powers of 10. The following screenshot shows, for example, the first step of multiplying 30 × 400:
The first factor is already decomposed as required; decomposition of the second factor is pending.
The next step — recording the multiplication with the decomposed factors — can be easily visualized through animation.
Juggling the factors in the next step is the heart of the strategy. It separates the multiplication of single-digit numbers from multiplication of powers of 10:
The separated multiplication expressions are pretty simple:
…and the remaining operation — multiplication by a power of 10, is simple, too.
In case of an error, students receive all needed support. Тo see the full exercise, follow this link.
It’s worth mentioning that the new skill immediately gets further development. First of all, Happy Numbers provides training to transform it into a mental math skill. Understandably, the practice problems are similar to the ones above, they just require the answer without any intermediate steps. However, if students need to correct an answer, they are provided the step-by-step process. Another direction to develop this skill is applying the same strategy to slightly more complicated multiplication problems, like 430 × 20 (one of the factors has two non-zero digits).
4. The Distributive Property of Multiplication over Addition
The two most important points about the distributive property are:
– It involves two different operations– in this context, multiplication and addition, in contrast to the commutative and associative properties.
– It is the basis of multiplication methods.
The First Touch
Happy Numbers first touches the distributive property as soon as the ×2 and ×3 tables and the commutative and associative properties are introduced. Since the distributive property is more complicated than the other two, its presentation proceeds step-by-step, carefully adding each new zone of proximal development.
Introduction of the distributive property is based on the array illustration of multiplication, and the very first step does not even include multiplication itself. Students just count the number of rows in each of two given parts of the array, and then find their total number:
Notice that in the second task the array is screened, so students cannot count the rows. Instead, they have to solve the task focusing on the numbers and adding them.
In the next step, composing expressions is added to the tasks:
Here, students have already composed the multiplication expressions 4 × 2 and 2 × 2, with visual support of the array. Step-by-step calculation of 6 × 2 is pending.
When students complete the calculation, it is verified by revealing the array:
Students solve a set of such tasks for various sizes of arrays and their parts. To see the full set of tasks using ×3, follow this link.
Happy Numbers then provides tasks that help students to reflect on the newly introduced strategy:
– Break one of the factors into two numbers
– Multiply each of the two by the second factor
– Add the two partial products
These tasks no longer include an array and focus on mathematical statements:
To avoid monotony and increase mathematical flexibility, the task is also presented in a slightly different form, for example:
After composing the statements, students use them to find the corresponding products:
At this point, calculations just follow hints.
These exercises provide simple examples of applying the distributive property. More sophisticated and practically important applications are discussed below.
Parentheses and Equations of the Distributive Property
Deeper understanding and effective application of the distributive property is connected touse of parentheses.
The Happy Numbers grade 3 curriculum includes a number of exercises that build on equations like:
representing the distributive property. These exercises employ the “break apart and distribute” strategy:
To see steps of the calculation and full exercise, follow this link.
It’s worth mentioning that the distributive property is not identical to the break apart and distribute strategy. For example, the property can be applied in reverse, as in the following calculation:
Distributive Property and the Area Model
In the Happy Numbers grade 4 curriculum, students find many examples where applying the distributive property is essential. They start with finding the area of a rectangle using a strategy based on the distributive property:
Here, one of the factors is broken apart specifically to the expanded form: in our 10-based number system, this simplifies further calculation — reducing it to multiplying single-digit numbers, multiplying numbers by powers of 10, and addition.
Then students apply the distributive property:
… and complete the calculations:
To see the full exercise, follow this link.
This exercise is not only about finding the area of a rectangle: it is the first example illustrating the distribution property with the area model. This modelling includes splitting one or two (adjacent) sides of a rectangle and the corresponding splitting of the rectangle and its area:
Here, the geometrical and numerical approaches reinforce each other in finding area and multiplying numbers. Applying the distributive property immediately results in the partial product method. This will be discussed in more detail in another article, along with the standard algorithm for multiplication.
Through step-by-step exercises and engaging interactive animation, Happy Numbers helps students easily master complex abstract math skills. A student always has the ability to make a mistake, because according to the latest research, qualitative mathematical thinking is built on the basis of personal mistakes, to a large extent. When students do make a mistake, Happy Numbers provides hints, which are worded in a neutral or positive tone to establish a productive learning environment. On-screen highlighting also helps students quickly identify the area to focus on.
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