This activity prompts students to generate a numerical pattern that follows the rule “start with 9, keep adding 9.” They then notice and explain other features of the pattern that may include:

• Patterns in even and odd numbers or digits
• Patterns in the digits in the ones place or tens place
• Ways the pattern can be represented with multiplication
• Ways they can use familiar multiples of 10 to predict terms in the “add 9” pattern

Students use what they know about the place value and operations to explain the patterns they notice (MP7). For instance, students may reason that, because 9 is 1 less than 10, to find the value of is to find the value of and then subtract 1 group of 12 (or ) from the product. Encourage students to use drawings, diagrams, or equations as needed to explain their thinking.

• Invite 3–4 previously selected students to share and explain what they observed about Andre’s pattern.
• Invite 2–3 students to share how they predicted the 25th number.
• If no students mentioned that the list of numbers are multiples of 9, encourage them to make this observation.
• If no students reason that adding 9 can be thought of as adding 10 and subtracting 1 each time, bring this understanding to their attention. Consider recording Andre’s pattern next to the count by 10 table from the Warm-up.
• In other words, adding 9 once means , which is 9. Adding 9 again means adding another to 9, or , which is 18.  Adding 9 a third time means , which is 27. And so on.
• “Adding 9 eight times is the same as Adding 10 eight times and subtracting 1 eight times, or , which is or 72.”
• As needed, “How could we use this to predict the 12th number in Andre’s pattern? The 25th?”

In this activity, students continue to generate and analyze a numerical pattern. This time, they generate a pattern that has an addition rule that they are less likely to apply with fluency.  The intent is to encourage all students to use what they know about place value or properties of operations to identify features of the pattern that are not explicit in the rule as they both generate and analyze the given pattern (MP7). Although the use of the distributive property over subtraction is not expected or made explicit, the work in both activities in this lesson develops students’ intuition for seeing, for instance, that .

This activity uses MLR3 Critique, Correct, Clarify. Advances: reading, writing, representing

• Display the response for students to consider: “Add 99 five times is the same as adding 100 five times. You just need to subtract some.”
• Read the explanation aloud.
• “What parts of this response are unclear, incorrect, or incomplete?” (Adding 99 isn't the same as adding 100. It's not clear how much you would need to subtract. It's not clear how this would work for adding 99 a different number of times.)
• 1 minute: quiet think time
• 2 minute: partner discussion
• Invite 2–3 groups share what they discussed. Record for all to see.
• “With your partner, work together to write a revised explanation.”
• 3–4 minutes: partner work time
• Select 1–2 groups to read their revised draft aloud and record for all to see. Scribe as each group shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.
• “How can we use our revised explanations to find the 20th multiple of 99?” (Find and subtract from it.)