Foundational Step of Regrouping - In Addition and Subtraction, Work Separately With Tens and Ones: Except When Regrouping.
We will now take all the separate strategies of the previous steps and combine them to move into our next step of adding and subtracting 2 digit numbers. Within this step, however, are five smaller steps we will be taking to arrive at the larger goal of successfully regrouping in addition and subtraction problems. As with the previous foundational steps within each of these sub-steps students will be allowed the time needed for mastery of the concept and we will continue to use the concrete – pictorial – symbolic models. The work we have been doing with word problems, number lines, base 10 blocks, place value and place value cards, problem solving skills and the depth of knowledge students have achieved is all utilized in this step.
Each of these sub-steps are simple to understand from an instructional point of view and can easily be rushed through and addressed in only a few lessons rather than taught to mastery. Student will build their mathematical foundation deeply and solidly as they progress assisting them in mastery of regrouping skills, and preparing them to move onto more difficult mathematical concepts. Suggested time allotment will be minimum of a week and a half per sub-step.
Addition and Subtraction using multiples of 10.
Within this sub-step students will only be working with the tens digit. In the first foundational step students were given time and models to develop a deep understanding of the ones place and how numbers work within the ones place. Now they are given that same opportunity to do so with multiples of 10. This is an important step and should not be overlooked. Students are continuing to work with word problems, place value and problem solving skills will have the opportunity to fully explore multiples of 10 before moving on to multiples of 100. This will also show students the steps necessary to break down any number within its place value for faster and effective problem solving. Students will work with a variety of problems in all three of the model areas. For example, with the concrete model students can use the base 10 rods to manipulate and work through word problems, then transition to the use of place value cards and finally, in the symbolic model students should practice solving problems of the following type:
20 + 30 =___, 50 – 30 =___, 50 + ___ = 70, and 50 - ___ = 80.
Addition and Subtraction using a 2 digit number and ones.
In this sub-step students will work with problems that have a digit in both the tens and the ones place. This is a subtle difference that can easily throw student off. It is important here to remember that this sub-step does not involve regrouping. For comparison's sake, a problem that would involve regrouping that we will at this stage avoid would be 12+9= (10+2) +9 = 10 + (2+9) = 10+11 = 10+(10+1) = (10+10)+1 = 20+1 = 21. To avoid confusing problems like this it will be necessary to keep the digits in the ones column in agreement with this idea, which means the sum can not exceed 9. For example, problems modeled like the following should be used:
25+4, 17+2, or 41+5, 49-3, 99-9, or 33+? =38,
and so on. Once again this allows students to work with one place value at a time even through one of the numbers is a two digit number and one is a single digit number. For example with 25 + 4, 25 would be broken into two tens and five ones, next the five ones and the four ones from the original problem will be combined to make nine, then the two tens and the nine ones will be combined to a sum of twenty-nine. The Associative Property (the ability to regroup in addition is any way desired) is further developed in this stage as well.
25 + 4 = (20 + 5) + 4 = 20 + (5 + 4) = 20 + 9 = 29
Addition and Subtraction 2 digit numbers and multiples of tens.
In this sub-step, students will move on to add and subtract a general 2-digit number with a multiple of ten, chosen so that regrouping is not required. This means limiting problems to those involving numbers whose tens values will not exceed a sum of 9. For example, 23+90 would involve regrouping within the tens place and should be avoided. I will continue to use the base 10 blocks and word problems as the first concrete model of this sub-step, where students will see and develop a deeper understanding of how the tens place works separately from the ones. In the earlier sub-step students only manipulated numbers with in the ones, while now they will build up or take apart the tens while also keeping track of the ones. Again this is a subtle but important step. This is a balancing act of sorts where student are one piece at a time learning to balance the whole of what will grow into a more complicated problem. Once they are able to master these small steps they will have the capabilities to move on to adding and subtracting the more difficult 3, 4 and multi-digit numbers as well as multiplication and division. The Associative Property and Commutative Property (the ability to add numbers in any order) are further developed in this sub-step as illustrated in the following example:
47 + 30 = (40 + 7) + 30 = (40 + 30) + 7 = 70 + 7 = 77
Addition and Subtraction using general 2 digit numbers with no regrouping.
In this fourth sub-step students will now work with both the tens and ones places, however they will still work with only one place at a time. Again, regrouping should be avoided in problems such as 98 + 25 (which requires regrouping in both the tens and ones place). Students will continue to gain a deep and lasting understanding of how the tens and ones are separate places and work independently. As students learn to add or subtract using columns, the understanding of each place value being different is important. Consider the following examples, which illustrate two ways students might add multi-digit numbers.
The differences in the above problems are subtle but of course one method (namely the right-most) is fundamentally incorrect. With a strong sense of place value this mistake can be addressed (as well as in later problems containing decimals involving money, for example). While this might seem like a long and drawn out way to teach, one should remember the benefits of long lasting mathematical foundations. After all, we spend weeks working on a given letter and its sound in Kindergarten and weeks on digraphs and diphthongs in 2 nd grade. We need to do the same for numbers and place value. The following is an example that demonstrates how utilizing place value, the Associative Property, and the Commutative Property work together allowing students to correctly combine like terms and find the correct solution:
36 + 42 = (30 + 6) + (40 + 2) = (30 + 40) + (6 +2) = 70 + 8 = 78
Addition and Subtraction using 2 digit numbers with regrouping.
Finally at this stage we will address addition and subtraction that involves regrouping. Although it has taken some time to get here the journey we have taken will pay off in the end. Normally I begin working with my students in the 2 nd nine weeks of the year on regrouping. They learn the algorithm or short cut so to speak to do this but often never really understand why or how it works. With the journey we have just taken students should now know the why and the how of addition and subtraction. This last sub-step of regrouping is now just a small hurdle to overcome. I also want to point out that by this stage student will have worked with their basic facts long enough and deep enough that they have them memorized and may be beginning, if they have not already, to use various mental math strategies on their own. This is just another benefit of students having a deep lasting understanding of the concept and not just knowing how to compute using short cuts. In this step student will continue to utilize the Associative and Distributive Properties as well as the "Break Apart to Make a Ten" method. Once again we are bringing all the skills we have previously mastered and are using them together. I should note that this is not the stage to regroup up to 100. That will come in the next step and just as before should be built up to. Here is an example of an addition problem with two 2-digit numbers that involves regrouping. Note the repeated use of the Associative Property and the Commutative Property.
46 + 37 = (40 + 6) + (30 + 7) = (40 + 30) + (6 + 7) = (40 + 30) + (6 + 4 +3) =
(40 + 30) + (10 + 3) = (40 + 30 + 10) + 3 = 80 + 3 = 83.
I will have students write out their work at each sub-step to ensure accuracy and the correct understanding of each step of the problem. Short cuts are the luxury of the expert. Once student have mastered a given sub-step they will be allowed to simply show the answer. However, as we progress to the next model they will be required to show all their work until they can show mastery. When they miss a problem I will also require them to go back and redo their work using the expanded form.
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