Models & Strategies for Two-Digit Addition & Subtraction
understand numbers, ways of representing numbers, relation- ships among standing and procedural understanding of addition and subtraction. Building on multiple representations and the reasoning required for the mean- ingful use and addition and subtraction, such as the number line, a hundreds chart, and base-. In a similar manner, we can specify numbers in other "bases" (besides 10), using the given base) that must be added together to obtain the value of our number. . In each step above, we are just dividing by 5 and looking at both the quotient. Examining the relationships between addition and subtraction and seeing .. Many features of multidigit procedures (e.g., the base elements and how they .
Consider the remainders seen upon division of the following numbers by 5: In each step above, we are just dividing by 5 and looking at both the quotient and remainder -- no knowledge of higher powers of 5 is necessary!
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Wonderfully, this technique works in any base. Can you explain why? So, for example, if we wanted to find the binary base 2 representation ofwe simply calculate the following: To see the similarities, let's count to 41 in base 10 and base 3 as shown in the table below.
Pay particular attention to how "2" in base 3 plays the same role as "9" in base It represents the last digit you can use before increasing the digit to the immediate left. It was uphill both ways, through the snow and blazing heat.
Enter the Romans In Roman numerals, two was one, twice. Three was one, thrice: And of course, there are many more symbols L, C, M, etc. The key point is that V and lllll are two ways of encoding the number 5.
Give each number a name Another breakthrough was realizing that each number can be its own distinct concept. Rather than represent three as a series of ones, give it its own symbol: Do this from one to nine, and you get the symbols: In our number system, we use position in a similar way. We always add and never subtract.
And each position is 10 more than the one before it. Our choice of base 10 Why did we choose to multiply by 10 each time?
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Most likely because we have 10 fingers. More formally, subtracting 3 is the inverse of adding 3. It is similar with division and multiplication. Just as people sometimes want to form sets of the same size into one larger set, they sometimes want to break up a large set into equal-sized pieces.
Thus division by 3 undoes implicit multiplication by 3 and leaves you with the original amount. It is the same no matter what amount you start with: More formally, dividing by 3 is the inverse of multiplying by 3. Two interpretations of division deserve particular mention here. If I have 20 cookies, and want to sort them into 5 bags, how many go in each bag? This is the so-called sharing model of division because I know in how many ways the cookies are to be shared.
- Models & Strategies for Two-Digit Addition & Subtraction
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- Number Systems and Bases
I can find the answer by picturing the 20 cookies arranged in 5 groups of 4 cookies, which will be the contents of 1 bag. If the cookies originally came out of 5 bags of 4 each, when I put them back into those bags, I will again have 4 in each.
Thus, division by 5 undoes multiplication by 5, or division by 5 is the inverse of multiplication by 5. The picture below shows the sharing model for this situation. If I have 20 cookies that are to be packaged in bags of 5 each, how many bags will I get? In the sharing model also called the partitioning model or partitive divisionyou know the number of groups and seek the number in a group.
In the measurement model also called quotative divisionyou know the size of the groups and seek the number of groups. The circled numbers in the figures above and below illustrate a crucial difference between the two models: Note that because multiplication is commutative, 5 bags of 4 cookies each is the same total number of cookies as 4 bags of 5 cookies each. Eventually students come to see the two kinds of division as interchangeable and use whichever model helps them with a particular division problem.
Subtraction and the integers We might summarize the story so far by saying that there are two pairs of operations—addition and subtraction, and multiplication and division—and these are inversely related in the sense described above. However, this summary would not quite be correct.
In fact, subtraction is not actually an operation on whole numbers in the same sense that addition is.
Numbers in Different Bases
You can add any pair of whole numbers together, and the result is again a whole number. Sometimes, however, you cannot subtract one whole number from another. This situation can be described by using negative numbers: