3. Adding Base-10 Numbers
In the last module, you learned about place value. You will use this knowledge to add basenumbers having multiple digits. The addition process is illustrated using the following examples.
Adding two base-10 numbers
Example 1
Add 14 and 62.
Step 1
Line up the second number (62) below the first number (14).
Write a plus(+) sign to the left of the second number to indicate you are adding the numbers together.
Draw a line below the second number. The sum will be written below this line.
14
+ 62
----
Step 2
Add the right digit of the first number (4) to the right digit (2) of the second number. (The right digit is the ones place). Write the sum of those digits below the line.
14
+ 62
----
6
Step 3
Add the left digit of the first number (1) to the left digit (6) of the second number. (The left digit is the tens place). Write the sum of those digits below the line.
14
+ 62
----
76
Example: Add two numbers with two digits (with no carries and no missing zeros)
Example 2
432
+ 310
=====
742
- Add 2 to 0 to get 2.
- Add 3 to 1 to get 4.
- Add 4 to 3 to get 7.
Objective: Add two numbers with three digits (with no carries and no missing digits)
The less detail principle is used here to summarize the addition process that is explained in example 1.
Example 3
253
+ 6
=====
259
- In the ones place, add 3 and 6 to get 9.
- Since a digit is not present in the tens and hundreds place for 6, those missing zeros are 0.
- In the tens place, add 5 to 0 to get 5.
- In the hundreds place, add 2 to 0 to get 2.
Example: Add three digit numbers where there are missing zeros.
Example 4
243
+ 539
=====
782
- In the ones place, add 3 and 9 to get 12.
- Since 12 cannot fit in the ones place, subtract 10 from 12 to get 2. In the next step, 1 will be added to the tens place to compensate. This is called a carry.
- In the tens place, add the carry (1) to the sum of 4 and 3 to get 8.
- In the hundreds place, add 2 to 5.
Example: Add using one carry (not in the leading place).
Example 5
648
+ 298
======
946
- In the ones place, add 8 and 8 to 16. Since the result is greater than 9, subtract 10 from the result to get 6.
- In the tens place, add a carry of 1 to 4 and 9 to get 14. Since the result is greater than 9, subtract 10 from the result to get 4.
- In the hundreds place, add a carry of 1 to 6 and 2 to get 9.
Example: Add using two carries (not in the leading place).
Problem 1
421
+ 256
=====
Application: Add with no carries or missing zeros.
Problem 2
3102
+ 7
=====
Application: Add with missing zeros and no carries.
Problem 3
521
+ 197
=====
Application: Add with a carry (not in the leading place).
Problem 5
542
+ 289
=====
Application: Adding two numbers with multiple carries.
Problem 6
823
+ 765
=====
Inferred application: This Problem covers both of these cases, which previous examples covered separately: 1) Add missing zeros in the leading (thousands) place and 2) add with a carry in the leading place.
Problem 7
What is the maximum carry when adding two numbers? Why?
Inferred application: This is sequenced after seeing multiple carry examples both in the examples and the Problem, so the learner is more equipped to answer this question. The learner needs to see that they can try adding the maximum digits such as 999 + 999. In this case the carry will not exceed 1.
Adding three base-10 numbers
Problem 8
252
391
+ 180
======
Inferred application: The process for adding 3 numbers is the same as the process for adding 2 digit numbers.
Problem 9
262
21
+ 3
=====
Add three numbers, filling in the blank digits with zeros.
Problem 10
623
853
+ 882
======
Inferred application: Add three numbers with a carry of two in the leading position, filling in the zero in that position (This is is shown in the examples above where two numbers are added together; the learner should infer that the same applies to three digit numbers.)
Problem 11
What is the maximum carry when adding three numbers? Why?
This is an inferred application. This question was asked earlier for adding two numbers. The learner needs to see that they can try adding the maximum digits such as 999 + 999 + 999. In this case the carry will not exceed 2.
Problem 12
What are the similarities in the process of adding two numbers and adding three numbers that can be generarlized to adding any number of numbers?
This is an inferred application. The learner needs to generalize the ideas of adding two and three digit numbers to adding numbers of any length.
Problem 6
201
23
+ 5
=====
Inferred application: This Problem combines three ideas:
1) Add using consecutive carries. 2) Fill in the zero in the missing places. 3) Add using a leading place with zeros to accomodate the carry in the thousands place.
While the problem may look simpler than the others because there are fewer digits, it is actually the most complicated. It combines the examples above, that are demonstrated in the earlier examples.