Addition & Subtraction Algorithm

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The Standard Algorithm for addition is shown below along with two step-by-step lessons that you can work through with your children to practice using it. Note that, before using the algorithmic method as illustrated below, your children should be familiar with the place value strategies that are the basis of the algorithm.

Work with your child to help him or her understand the concept of regrouping when adding numbers. The illustrated example below can be used to help explain this. Also, activities with money, such as adding cents and then exchanging groups of ten for dimes will also help master the concept of regrouping.

Review: Place Value

We use the base ten number system to write numbers with each place having a value; ones, tens, hundreds, thousands, and so on.

37 shown in tens and ones       64 shown in tens and ones

Example: Addition Algorithm with Regrouping

When we add numbers we often have to combine these groups if we end up with more than 9 in any one group. Below is an example of addition with regrouping.

28 plus 34 shown with blocks 28 plus 34 shown with regrouping of the ones 28 plus 34 showing grouping of 12 ones to give 1 ten and 2 ones 28 + 34 shown to equal 62
Start by adding the ones. We have 12 ones but only one place! We must regroup our 12 ones to give 1 ten and 2 ones Now we can put the 1 ten with the tens and write 2 in the ones place Finish by adding the tens including the 1 ten that was regrouped
2 + 3 + 1 = 6

animated example showing addition with regroupingWatch an animated example showing each of the steps from the above addition.

Addition Calculator

Follow the instructions below to enter your own numbers to add and watch the steps required to complete the calculation.

Lesson: Using the Standard Algorithm

Read the lesson below and then work through it with your children. Print off any required worksheets and gather whatever materials are required before starting.

Introduction/ Warm-up

Students normally learn best when they can associate what they are learning with what they already know. This introduction will provide your child with an opportunity to do this and help prepare him for the new content to be learned later in this lesson. This introduction (or warm-up) typically takes 10-15 minutes. Do not worry if it takes longer.

Materials Needed: About 50 buttons, or coins (ideally some sort of item that students can count with)

Problem #1

  1. Hand your child a large handful of your counting items, a piece of paper and pencil.
  2. Explain that they are going to be using them to help them with addition and subtraction.
  3. Write the following addition problem on the board (if you do not have a board write it on a large sheet of paper and show it to your child) and have him copy it on their sheet of paper.

    28 + 8 =
  4. Instruct your child to count out 28 items to help solve the problem,
  5. Once this is complete, tell your child that these 28 items represent the number 28 on the board and point to it.
  6. Now tell your child that they are going to count out 8 more beans to represent the number 8 in the problem (point to the number 8 on the board).
  7. After that is complete tell your child that they are going to count both numbers together to get the answer to the problem 28+8.
  8. Ask him to tell you his answer. Once this is complete demonstrate the answer on the board. 28 + 8 = 36

Lesson Explanation

This part should take around 20 to 30 minutes but you can take as long as necessary.

Step 1:

  • Explain to your children that they will be learning more about addition and subtraction.
  • Let your children know that what was practiced in the warm-up was a great start but that they will be learning more ways to complete harder (more complex) addition problems

Step 2:
Tell your children that you will show them how to solve addition problems by exchanging 10 ones for 1 ten.

  • Demonstrate as shown below with detailed instructions.

  1. Write the following addition problem on the board or on a sheet of paper if you do not have a board.
    34 + 9
  2. Explain that as we solve this problem we can see that 4 + 9 is 13. Discuss the amount of space there is to write 13 and that we only have enough space to write one digit underneath the 4 and 9.
  3. To solve this problem, you exchange 10 ones for 1 ten and 'carry-over' a digit into the next column. At this point, if your child does not understand place value, stop and review place value . Otherwise, complete the steps below.
  4. In order to determine which number to move over, we must know about place value. Remember, each digit in a number has a certain value within its place. For example, the digit 4 in 34 has the value of ones and represents 4 ones in the number 34. The digit 3 in the number 34 has the value of tens and represents 3 sets of ten in the number 34.
  5. When we add numbers that need to be carried over we can only carry over a digit into the right spot. For example, when we add the 4 and 9 from the problem on the board we have to carry over the answer (13) into the correct place value spots.
  6. To help solve the problem, write down the problem like this. (Draw the below information on the board or sheet of paper).
    34 + 9 with ones and tens shown
  7. Now we can easily see that when we add 4 + 9 we must leave the ones in the ones place and move the tens into the tens place. So the 3 must stay in the ones place and the 1 must move into the tens place. See this example below.
    34 + 9 partially solved
  8. To finish solving the problem add the remaining numbers in the tens column which are 1+3 to get 4 giving an answer for 34 + 9 of 43. See below.
    34 + 9 = 43

Two Important Notes

  • It is very important that your children have a proper understanding of place value before they start using algorithms. Working with objects as described above, and handling base ten blocks (ask at school for recommendations) will provide a good foundation to build this understanding.
  • When discussing place value with your child, emphasize exchanging 10 ones for 1 ten (and 10 tens for 1 hundred) as you work from right to left. Similarly, when moving to the right, talk about exchanging 1 ten for 10 ones, and so on.

 

Step 3:
Tell your children that you will now show them how to solve subtraction problems by exchanging 1 ten for 10 ones.. Explain that when students are subtracting in a problem and do not have enough in a digit to take or subtract from, they must exchange 1 ten for 10 ones.

  • Demonstrate as shown below with detailed instructions.

 

  1. Write the following subtraction problem on the board.
    57 - 29
  2. Discuss with your children that as we solve this problem we see that it is not possible to subtract 9 from 7.
  3. In order to determine which number to move over, we must know about place value. Recap how each digit in a number has a certain value within its place. For example, the digit 7 in the number 57 is in the represents 7 ones and the 5 in 57 represents 5 tens in the number.
  4. To solve this subtraction problem we must exchange 1 ten from the tens column in 57 and give it to the 7 in the ones column to make the number 17 which is larger than 9 and large enough to be subtracted from.
  5. To help use solve the problem write down the problem like this. (Draw the below information on the board).
    57 - 29 partially solved
  6. In this example (see above) we can easily see that now we must subtract 17 - 9 and 4 - 2 to get an answer of 28. See the problem solved below.
    57 - 28 = 29

Guided Practice

Step 4:

  • Together complete worksheet #1. Work through each type of problem with your children to help them master concepts from this lesson.
  • If you are working with more than one child then group them together to complete Worksheet #2.
  • If you are working one-on-one with your child, alternate turns completing problems on worksheet #2. However, make sure to let him ‘help’ you solve your problems to keep him involved in the learning.
  • Review concepts that they may not completely understand or have questions with. In Step 5 they will have a mini-assessment with worksheet #3 and that will close the lesson.

Independent Practice/ Lesson Closure

  • Use Worksheet #3 to assess your children's understanding of the lesson.
  • Have them complete the worksheet individually.
  • Grade the worksheet to determine how well they understand the material.
  • When grading the worksheets consider the following. Students who receive:
    • at least 5 out of 6 questions correct are ready to move forward.
    • 4 out of 6 may move forward, but may need additional review.
    • 3 or less correct need to repeat the lesson or have additional instruction on the concept.

Note: Remind your child that not all 2-digit addition and subtraction problems require carrying over or borrowing. e.g. 23 + 44 or 87 - 26. Set up a variety of addition and subtraction questions using this worksheet generator.

Lesson Variations

  • Children who struggle with this lesson may benefit from a review of simple addition and subtraction problems without exchanging or regrouping. They may also benefit from reviewing place value concepts.
  • If your child is advanced have him or her help you with solving the examples on the board to keep them engaged with the lesson.
  • You may also, consider giving your child a worksheet on simple addition and subtraction facts if they are significantly struggling with the materials.

Addition and Subtraction Worksheets

The 3 worksheets and worksheet generator listed above are grouped below:

And for a final challenge, you can help (if needed) your children with this 4-Page Addition/ Subtraction worksheet that promotes and demonstrates a full understanding of the addition and subtraction algorithmic methods - it gets progressively more challenging.

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