Patterns in Addition and Subtraction

Est. Class Sessions: 2–3

Developing the Lesson

Part 3: Addition and Subtraction Patterns on Rule Machines

Use the display of the Rule Machine Master with the rule
"Add 3." See Materials Preparation and Figure 2. Remind students that when they saw the Rule Machine before they had to guess the rule. Now they will be using the rule to add and subtract.

For the first input, number 8, ask students to determine the output number. Have connecting cubes, ten frames, number lines, and 100 Charts readily available.

Ask:

What is the output? (11)
Explain how you solved the problem. (Possible response: I counted on. I put 8 in my head and then counted on 9, 10, 11.)
What is your number sentence? (8 + 3 = 11)

Fill in the Output and the Number Sentence columns on the Rule Machine Master. For the next problem, ask a student to be the "teacher." He or she calls on other students to say the output number, explain the strategy they used, and give the number sentence. The "teacher" writes the answer in the Output column and determines whether the answer is correct or not.

For the input number 22, try to elicit the strategy of using the known fact 2 + 3 = 5 to find 22 + 3 = 25. Use a display of the Ten Frames page in the Student Activity Book Reference section to demonstrate the problem.

Ask:

Show 22 on the ten frames. (2 full ten frames, 2 in a third frame)
Do you know an addition fact that could help you add 22 + 3? (2 + 3)
How does that math fact help? (Adding 3 to the 2 in the third ten frame makes 5 in that frame; the two full ten frames do not change.)
Let's try to use math facts to help solve the problems in the Rule Machine.

Continue to demonstrate how to use known math facts as a strategy for solving problems on the Rule Machine display. For example, knowing that 7 + 3 = 10 can help solve 27 + 3: add
7 + 3 to make 10, and then add two more tens, 20, and 30.

Allow a few more students to take a turn at being the "teacher." Try other rules, including some subtraction rules, but keep the same numbers in the Input column.

After a few rounds, have students work in pairs to complete the Add and Subtract with Rule Machines pages in the Student Activity Book. They can take turns with their partners being the "teacher."

Upon completion, use the display of Add and Subtract with Rule Machines to discuss patterns in the Rule Machines, particularly in Questions 1, 2, and 4.

Ask:

Look at the input and output numbers on the Rule Machine. Do you see a pattern?
Were you able to use the pattern to help you solve the rest of the problems?
Did you use the 100 Chart?

Use Sample Dialog 2 to guide your discussion of these questions.

Use Sample Dialog 2 to guide your discussion of Questions 1–4 on the Add and Subtract with Rule Machines pages.

Teacher: Look at the input and output numbers on the Rule Machine for Question 1. Do you see a pattern?

Tanya: For each input, both the input and output numbers end the same. The tens digit is bigger by 1 ten.

Teacher: Why do you think so?

Tanya: I think when you add 10 to a number, the ones digit always stays the same. We just add 1 ten to the tens digit. On the 100 Chart, if you go straight down each column, all the numbers end the same.

Teacher: Good thinking, Tanya! Now, look at the input and output numbers on the Rule Machine for Question 2. Do you see a pattern?

Jerome: All the numbers in the Output column end with 1, but all numbers in the Input column end with 6.

Teacher: Why do you think that happened?

Jerome: I think because we're not adding 10. Ten has a zero for the ones digit, so none of the ones digits would change for the output. But for this Rule Machine, we're adding 5, and 5 + 6 = 11. So all the ones digits for the output are 1. When you start at 6 and you count 5 more on the 100 Chart, you land at 11. When you start at 16, you land at 21. For all the input numbers we went down one row to the number with 1 for the ones digit.

Teacher: Did you use that pattern to find the outputs for the rest of the numbers?

Jerome: Yes, it was easy when we saw the pattern.

Teacher: That's great! Let's look at Question 3. Is there a pattern in the answers?

Brenda: There is no pattern. All the numbers in the Output column are different.

Teacher: Why do you think that happened?

Brenda: I think because we had all different ones digits in the Input column. Even though we subtracted 2 for each number, we started in different columns on the 100 Chart.

Teacher: Great answer, Brenda! There wasn't really a pattern to help us find all the outputs. Now, let's look at Question 4. Is there a pattern for this question?

Daniel: All the numbers in the Input column end with 6 and all the numbers in the Output column end with 2.

Teacher: Why do you think all the ones digits are the same in the Output column?

Daniel: I think because all the ones digits are the same in the Input column. For each input number we knew that
6 − 4 = 2. We kept all the tens digits the same and just changed the ones digit to 2.

Teacher: Wonderful, Daniel! So we learned that sometimes we can use a math fact such as 6 − 4 = 2 to help us solve problems with larger numbers.

Assign the Find Patterns in Rule Machines page in the Student Activity Book to assess students' understanding of Rule Machines. Have 100 Charts available for students to use if they choose.

Use the Find Patterns in Rule Machines page with the Feedback Box in the Student Activity Book to assess students' abilities to use mental math strategies to solve addition and subtraction problems with larger numbers (e.g.,
2 digit + 1 digit) [E4]; represent addition problems using number sentences [E1]; and to identify and describe patterns in addition problems [E6].

For targeted practice, place prepared copies of the Rule Machine Master in a learning center. The copies will have different rules and input numbers on the Rule Machines. Student pairs can take turns being "student" and "teacher." The "student" gives the output and describes his or her strategy and the "teacher" writes it in the Output column and determines if the answer is correct or not.