Explain the relationship between multiplication and addition.
B. Describe how understanding the relationship between multiplication and addition contributes to students’ understanding of these operations. For example, how does multiplication extend addition concepts (e.g., manipulation of groups for a total product)?
C. Explain the commutative, associative, and distributive properties using examples.
D. For each property, describe how it is connected to thinking strategies students might use in performing computations (e.g., counting by two’s or five’s, groupings or “many sets of” items, adding several equal groups together).
E. Provide specific examples of at least two common conceptual errors.
1. Describe an instructional strategy that would serve to correct and/or avoid each of these conceptual errors (1 strategy per conceptual error for a total of at least 2 strategies).
2. Explain how each of the instructional strategies serves to correct and/or avoid its associated conceptual error.
The relationship between addition and multiplication is well explained by the fact that, multiplication is more so known to be just repeated addition. This statement is well explained by use of an example. Take into consideration a case where in a group of four students each holding three books. If a question is posed to calculate the total number of books the students are holding, then this calculation can be done this way; by adding 3+3+3+3 and obtain the required solution as 12 books. To explain this, since there are four students and each is holding three books, therefore adding the number of books four times gives the required answer as twelve. Similarly, if you have 5×3, you combined a total of 5 parts in 3 groups, represented as 5+5+5, to obtain 15 an equivalence of 5×3=15. This clearly explains the relationship between addition and multiplication. (Matzke, 2012).
Math uses multiplication and addition, so as to enable students to have a good understanding of the addition concepts like manipulating the groups to obtain the total products. It also helps students find it easy in finding out the quantity or even size of elements in the given groups of same size. A good instance is that where you have a total of 8 sweets in a pack and then you purchase 5 more similar packs, then how many sweets do you have in total at that instance? The solution to this can be obtained by general addition, that is to say, 8+8+8+8+8=40, more so called repeated addition that can be replaced by a multiplication statement, 8×5=40. (Matzke,2012).
Commutative property explains how a mathematical operation numbers relate commutatively. Commutative originates from the word commute which means to move around and therefore commutative property refers to that property that involves the movement of numbers or entities around in a mathematical operation. In addition, the rule can be expressed as c+b=b+c which means, if b=4 and c=7, then this can be represented as; 7+4=4+7. For multiplication, the rule is represented as, bc=cb which is equivalent to 4×7=7×4 in number form as per the stated variables.
Associative comes from the word “associate” which basically means to group. Associative property is the rule that makes its reference to the grouping technique. For the addition, the rule is expressed as; a + (b + c) = (a + b) + c which can also be given in numbers. If a=5, b=9 and c=7, then expression can be represented in number form as; 5+(9+7)=(5+9)+7. Also, associative property gives rise to (a+c)+b from the given expression. For multiplication, the rule is; a(b×c)=(a×b)c. In numbers it can be represented as; 5(9×7)=(5×9)7. In the case of associative property, you are simply required to do regrouping of the variables or the numbers if they have been stated.
Distributive property depends on the statement that multiplication distributes over addition. If an expression is represented as a (b + c), then this can be re-written as ; ab+ac. In this case, (a) has been distributed through the bracketed expression. Hence, a (b + c)= ab + bc. A distributive property done in an expression is simply taking something through the parenthesis, majorly referred to as factoring out.
The above properties can be connected to thinking strategies students might use in performing computations. In the case of commutative property, students can apply it in grouping processes. Students can take 5 pouches and place 3 pencils in each pouch or alternatively they can have 3 pouches and place 5 pencils in each pouch. This, 5×3=3×5=15 gives rise to the same product. When applying the associative property, students can make sets of and skip counting. For the distributive property, students can apply it through connecting it to adding equal groups and then making sets. For example, 3(5)6=3(5)×3(6). In this case, multiplication of 5 and 6 can be done to get 30 and then making 3 groups of 30 to obtain 90.
Conceptual errors are errors that arise from ones inability to clearly understand the given information and provide the required interpretation of the question at hand. (Spooner, M. 2002).
A conceptual error arises from inadequate mastery of facts relating to numbers.
Examples of conceptual errors:
When a student does not understand the place value system then, 150+8 becomes
A student who exhibits this error needs revision on the place value system
A student may also have conceptual errors when it comes to addition where he forgets to the `carrying` aspect. For example:
A B C D
250 528 147 545
+270 +281 +283 +365
490 799 399 899
This error can be corrected by reminding the student to add the digit carried
Another example of a conceptual error is demonstrated during multiplication of a number with zero :
This error depicts that a student has not understood the multiplication and addition process. He therefore goes ahead to perform an addition of multiplication. One way to avoid this errors is by reminding students that they are performing multiplication instead of addition and show them how the signs of multiplication and addition differ. Show the students the correct signs of both addition (+) and multiplication (×). In multiplication, showing students the relationship between addition and multiplication, will build their capability to develop a strategy that will help them discover the correct product. (Spooner, 2002).
Once an instructor has identified that there are conceptual errors, it is important that he works at correcting them immediately so that the students` grasp of mathematical concepts is not jeopardized. Most of the conceptual error will be corrected through practice. An instructor should fist make sure that the students gets a clear understanding of the concept then give the student a number of problems to test his understanding. It is important for the instructor to encourage and give positive feedback so that the student is motivated to learn (Mathematics in action, 2012).
Mathematics in action (3rd ed.). (2012). Boston: Pearson Education.
Matzke, A. H. (2012). Who’s right, addition or multiplication?. Vero Beach, Fla.: Rourke Pub..
Spooner, M. (2002). Errors and misconceptions in maths at Key Stage 2. London: David Fulton.