The purpose of this Choral Count is to invite students to practice counting by 10 and notice patterns in the count. These understandings help students develop fluency and will be helpful later in this lesson when students need to be able to, given a rule, generate numerical patterns and explain features of the patterns that are not explicit in the rule. Keep the record of this count displayed for students to reference in the lesson activities.
Launch
“Count by 10, starting at 10.”
Record as students count.
Stop counting and recording at 150.
Activity
“What patterns do you see?”
1–2 minutes: quiet think time
Record responses.
Student Task Statement
None
Student Response
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Advancing Student Thinking
Activity Synthesis
“How is describing what you notice about the numbers in our count the same as what you have been doing with shape patterns? How is it different?” (Same: Even though we were counting by 10, we noticed lots of other things that are also true. “Count by 10” is like a rule for creating a pattern. Different: It’s only numbers, there aren’t any shapes or diagrams. I’m not sure if there’s a rule.)
“Today we are going to generate and describe numerical patterns that follow a rule. Numerical patterns are like what we do in the Choral Count. To create one, you need to know what to start with and what to do to get the next number.”
“In a Choral Count, we count on by a given number. Numerical patterns can have rules like start with 0 and keep adding 2 or start with 1 and keep multiplying by 3.”
Activity 1
20 mins
Keep Adding 9
Standards Alignment
Building On
3.OA.D.9
Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.
Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.
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.
MLR8 Discussion Supports. Use multimodal examples to compare the keep adding 9 pattern and counting by 10. Use verbal descriptions along with gestures, drawings, or concrete objects to show the connection between the multiples of 9 and 10. Advances: Listening, Representing
Representation: Access for Perception. Synthesis: Use pictures of the long rectangle base-ten blocks to help students visualize the patterns. For example, display a picture of eight long rectangle base-ten blocks. Count by 10 while pointing to each block. Then cross out one unit in each block and discuss how this shows that counting by 9 is like multiplying by 10 and subtracting. Supports accessibility for: Conceptual Processing, Visual Spatial Processing
Launch
Groups of 2
Read the opening paragraph as a class.
“How is Andre’s rule the same as our Choral Count by 10? How is it different?” (Same: We start with a number and then count by the same number. Adding a number over and over is like counting by a number. Different: They are different numbers. Counting by 10 is easier for me.)
30 seconds: partner discussion
“Let’s create Andre’s pattern and talk about what we notice about the numbers in the pattern.”
Activity
“Take a few quiet minutes to work on the first problem. Then share your thinking and complete the rest of the activity with your partner.”
5–6 minutes: independent work time
5–6 minutes: partner discussion
Monitor for students who explain:
Patterns with even and odd numbers
Patterns in the digits
Why the terms are the multiples of 9
How they can use what they know about multiplying by 10 to predict terms
Student Task Statement
Andre’s rule for a pattern is “start with 9, keep adding 9.”
Use what you’ve noticed about Andre’s pattern to make some predictions.
Complete the table with the first 10 numbers in Andre’s pattern.
What do you notice about the numbers in Andre’s pattern? Make at least 2 observations to share with your partner.
keep adding 9
9
Choose one observation you or your partner made. Explain or show why you think it happens.
What is the 12th number in Andre’s pattern?
What is the 15th number?
What is the 25th number?
Activity Synthesis
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?”
Activity 2
15 mins
Add 99
Standards Alignment
Building On
Addressing
4.OA.C.5
Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
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 .
“We completed a pattern that followed the rule “start with 9, keep adding 9” and explained other patterns we noticed in the numbers. Now, let’s see what we notice about a pattern that follows a different rule.”
Activity
“Work with your partner to complete the activity.”
6–8 minutes: partner work time
Monitor for students who:
Identify different patterns in the numbers.
Reason about the numbers in the “counting by 99” column (multiples of 99), by reasoning about multiples of 100.
Student Task Statement
Elena’s rule for a pattern is “start with 99, keep adding 99.”
Complete the table with the first 5 numbers in Elena’s pattern.
Look closely at the list of numbers. Make at least 3 observations about the numbers in the pattern.
keep adding 99
99
Complete the table with the next 5 numbers in Elena’s pattern.
Activity Synthesis
MLR3 Critique, Correct, Clarify
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.)
Lesson Synthesis
“Today we generated numerical patterns that follow a rule. Just like we did with shape patterns we noticed other features in the pattern that weren’t in the rule.”
“Even though the rule for Andre’s pattern was simple - “start with 9, keep adding 9” - what else did we notice?” (even and odd patterns, the ways the digits in the numbers changed each time, that all the numbers were multiples of 9)
“How did you explain what you noticed in the patterns today? What did others do that helped you understand what they noticed?” (I used equations to show decomposing numbers to make a ten. I used equations to show how each number was a multiple of 9. It helped to have drawings and diagrams.)
Have feedback on the curriculum?
Help us improve by sharing suggestions or reporting issues.
Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.
