Explain the relationship between multiplication and addition.
Multiplication and addition has a unique relationship; multiplication is a repeated addition. The result of multiplication is the product or the total number, obtained by simply multiplying the multiplicand and the multiplier, respectively. For example, 5 times 3 is equal to 15; number 5 is the multiplicand, number 3 is the multiplier, and number 15 is the product. The result obtained in the repeated addition is the same; this is to combine the groups with the objects. For example, 3+3+3+3+3=15 will obtain the same value by doing the multiplication 5×3=15.
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)?
For a thorough understanding for the students to know how multiplication extends the addition concepts, it is better to illustrate a real world problem. For example, you are to buy 8 yogurts from a grocery store, and per yogurt is 3 dollars. At the counter, the sales associate or cashier scans each yogurt. In figure, it shows 3 + 3 + 3 + 3 + 3 + 3 +3 + 3 = 24; that 3 happening 8 times. Alternatively, to make a fast transaction, the cashier may type 8, press the symbolx, and scan 1 yogurt. In figure, it shows 8×3=24. Therefore, if you will buy 8 yogurts of 3 dollars each will cost you 24 dollars.
Explain the commutative, associative, and distributive properties using examples.
Commutative Property of Addition and Multiplication
It states that the order of the two addends or two factors or the variables does not affect the sum or the product . The root word of the word ‘commutative’ is commute; ‘commute’ means to interchange. In this manner, the order of the two addends or the two factors reversed without changing the sum or product. Say, a+b=b+a (in addition), and a×b=b×a (in multiplication), respectively. For example, 8+3=3+8, in addition, and 8×3=3×8, in multiplication. It shows that only the order of numbers changes, however, the results are the same.
Associative Property of Addition and Multiplication
It states that the three or more addends, factors, or variables grouped before adding or multiplying, and it does not affect the sum and the product . This type of property usually simplifies calculations when adding, multiplying, or used to simplify the calculations. Say, a+b+c=a+b+c in addition, and a×b×c=a×b×c in multiplication . For example, 8+3+2=8+3+2, in addition, and 8×3×2=8×3×2, in multiplication. It shows that even the numbers, grouped according before applying the operation and after the calculation it obtained the same results.
Distributive Property
It states that when multiplying the two addends by a certain factor, the result is the same as multiplying each addend by the factor, and then adds the products . Say, a×b+c=a×b+a×c . For example, 5×3+6=5×3+5×6=45. In a simplified solution,5×3+6=5×9=45, while 5×3+5×6=15+30=45. In words, multiply a number or variable by a group of numbers or variables added together, or perform each, multiply separately then add.
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).
Complex illustrations made the thinking strategies of students improved that include signed numbers or algebraic expressions. Each property becomes critical and the students should integrate deep analysis and understand the significance of the properties. The students should synthesize both the addition and multiplication operations. For example, in distributive property, 53-2+6+3-5+3-4=5×7+3×-6=35+-18=35-18=17. Alternatively, 53-2+6+3-5+3-4=5×3+5×-2+5×6+3×-5+3×3+3×-4=15-10+30+-15+9-12=35+-18=17.
Provide specific examples of at least two common conceptual errors.
Describe an instructional strategy that would serve to correct and/or avoid each of these conceptual errors (1 strategy per conceptual error for at least 2 strategies).
In Numbers and Operations, specifically in counting numbers, the two instructional strategies among the many strategies available that helped the students in preK or fifth grade perform the activities, develop ideas, and avoid conceptual errors. These are to match the counting aloud with the concrete objects and using a circle game to assist the students with the difficult numbers.
Explain how each of the instructional strategies serves to correct and/or avoid its associated conceptual error.
For example, to much the counting aloud with the concrete objects, the strategy is to count the number of students in class through tapping a student on the shoulder while saying the word. This strategy supports the understanding of the students of the one-one or stable-order principles or concepts.
In using a circle game to assist, the students with the difficult numbers simply arrange students in a circle with its designated number, for example, twelve. The students start to count by ones, the student who says the number twelve sits down automatically. Then, the next student starts to count from number one until it says the number twelve again, and the student sits down, so on and so forth. This strategy supports the alertness and accuracy of the students.
Reference
Bamberger, H. J., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math Misconceptions Pre K-Grade
5: From Misunderstanding to Deep Understanding. Portsmouth, NH. Retrieved from http://www.heinemann.com/shared/onlineresources/e02613/BambergerSample1.pdf.
Kinder, C. N. (2004). Using Basic Properties to Solve Problems in Math. Retrieved from
http://yale.edu/ynhti/curriculum/units/2004/5/04.05.07.x.html.
Math is Fun. (2014). Commutative, Associative and Distributive Laws. Retrieved from
http://www.mathsisfun.com/associative-commutative-distributive.html.
Math Worksheet Center. (2014). How Are Addition and Multiplication Related? Retrieved from
http://www.mathworksheetscenter.com/mathtips/multiplicationandaddition.html.
