Addition and Subtraction Situations
10 min
Narrative
The purpose of this Warm-up is to elicit observations about patterns in addition tables containing sums of two-digit addends that are multiples of 10. Each table is partially completed to show certain behaviors of the sums and highlight some properties of operations. For example, the sums in the first table can illustrate the commutative property, and the sums in the second table can help students intuit the associative property. (Students are not expected to generate equations as shown here.)
Commutative property: 10+30=40 and 30+10=40
Associative property: 50+10=40+10+10=40+10+10=40+20
While students may notice and wonder many things about the addition tables, focus the discussion on the patterns in the tables and possible explanations for them. When students make sense of patterns in sums and try to explain them in terms of the features of the addends and how they are added, they look for and make use of structure (MP7)
Launch
- Groups of 2
- Display the tables.
- “What do you notice? What do you wonder?”
- 1 minute: quiet think time
Teacher Instructions
- “Discuss your thinking with your partner.”
- 1 minute: partner discussion
- Share and record responses.
Student Task
What do you notice? What do you wonder?
Solution Steps (6)
- 1Examine Table 1 structure→Addition table with headers 10, 20, 30, 40, 50 - middle row and column are filled
- 2Notice pattern in Table 1→Sums increase by 10 going left-to-right and top-to-bottom (30+10=40, 30+20=50, etc.)
- 3Examine Table 2 structure→Same headers, but diagonals are filled instead
- 4Notice diagonal pattern→Main diagonal: 10+10=20, 20+20=40, 30+30=?, 40+40=80, 50+50=100 (doubles)
- 5Notice anti-diagonal pattern→10+50=60, 20+40=60, 40+20=60, 50+10=60 - all equal 60
- 6Find the missing value→30+30=60 (the ? in both tables)
Sample Response
Students may notice:
- The two tables have the same numbers across the top and along the left side.
- All the numbers end with 0 and are tens (or groups of 10).
- Each table has a question mark.
- In the first table:
- Sums in the middle row and the middle column are the same set of numbers.
- Sums increase by 10 each time from left to right and from top to bottom.
- In the second table:
- Cells that run diagonally are filled with numbers.
- Numbers from the upper left to the lower right go up by 20 each time.
- Numbers from the lower left to the upper right are all 60.
- A 100 is in the bottom right corner.
Students may wonder:
- Why do the numbers in the first table go up by 10 from left to right and from top to bottom?
- In the first table, why are the two sets of numbers the same?
- In the second table, why does one set of numbers go up by 20, from top to bottom, while the other set shows the same number, 60?
- How do the tables work? What numbers go in the blank cells?
- What numbers do the question marks represent?
Activity Synthesis (Teacher Notes)
- “How do you think the tables work? How do we know what numbers go in the cells?” (Each number in the row at the top is added to each number in the first column on the left.)
- Ask questions to invite students to explain some of the patterns they noticed in the table. For each question, give students a minute of quiet think time. Record their responses, with equations, if possible.
- If students do not notice the following patterns, consider asking:
- “In the first table, why are the sums in the middle row and the middle column the same set of numbers?” (The same pairs of numbers are added. The first number in the middle row and in the middle column are 40 because both are the sum of 10 and 30, just added in different orders: 10 + 30 and 30 + 10.)
- “In the second table, why are the sums from the lower left corner to the upper right corner all 60?” (Each time, the first number being added goes up by 10 and the second number goes down by 10, so the sum stays the same.)
Standards
- 3.OA.9·Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. <em>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.</em>
- 3.OA.D.9·Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. <span>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.</span>
25 min
10 min
Knowledge Components
All skills for this lesson
Addition and Subtraction Situations
10 min
Narrative
The purpose of this Warm-up is to elicit observations about patterns in addition tables containing sums of two-digit addends that are multiples of 10. Each table is partially completed to show certain behaviors of the sums and highlight some properties of operations. For example, the sums in the first table can illustrate the commutative property, and the sums in the second table can help students intuit the associative property. (Students are not expected to generate equations as shown here.)
Commutative property: 10+30=40 and 30+10=40
Associative property: 50+10=40+10+10=40+10+10=40+20
While students may notice and wonder many things about the addition tables, focus the discussion on the patterns in the tables and possible explanations for them. When students make sense of patterns in sums and try to explain them in terms of the features of the addends and how they are added, they look for and make use of structure (MP7)
Launch
- Groups of 2
- Display the tables.
- “What do you notice? What do you wonder?”
- 1 minute: quiet think time
Teacher Instructions
- “Discuss your thinking with your partner.”
- 1 minute: partner discussion
- Share and record responses.
Student Task
What do you notice? What do you wonder?
Solution Steps (6)
- 1Examine Table 1 structure→Addition table with headers 10, 20, 30, 40, 50 - middle row and column are filled
- 2Notice pattern in Table 1→Sums increase by 10 going left-to-right and top-to-bottom (30+10=40, 30+20=50, etc.)
- 3Examine Table 2 structure→Same headers, but diagonals are filled instead
- 4Notice diagonal pattern→Main diagonal: 10+10=20, 20+20=40, 30+30=?, 40+40=80, 50+50=100 (doubles)
- 5Notice anti-diagonal pattern→10+50=60, 20+40=60, 40+20=60, 50+10=60 - all equal 60
- 6Find the missing value→30+30=60 (the ? in both tables)
Sample Response
Students may notice:
- The two tables have the same numbers across the top and along the left side.
- All the numbers end with 0 and are tens (or groups of 10).
- Each table has a question mark.
- In the first table:
- Sums in the middle row and the middle column are the same set of numbers.
- Sums increase by 10 each time from left to right and from top to bottom.
- In the second table:
- Cells that run diagonally are filled with numbers.
- Numbers from the upper left to the lower right go up by 20 each time.
- Numbers from the lower left to the upper right are all 60.
- A 100 is in the bottom right corner.
Students may wonder:
- Why do the numbers in the first table go up by 10 from left to right and from top to bottom?
- In the first table, why are the two sets of numbers the same?
- In the second table, why does one set of numbers go up by 20, from top to bottom, while the other set shows the same number, 60?
- How do the tables work? What numbers go in the blank cells?
- What numbers do the question marks represent?
Activity Synthesis (Teacher Notes)
- “How do you think the tables work? How do we know what numbers go in the cells?” (Each number in the row at the top is added to each number in the first column on the left.)
- Ask questions to invite students to explain some of the patterns they noticed in the table. For each question, give students a minute of quiet think time. Record their responses, with equations, if possible.
- If students do not notice the following patterns, consider asking:
- “In the first table, why are the sums in the middle row and the middle column the same set of numbers?” (The same pairs of numbers are added. The first number in the middle row and in the middle column are 40 because both are the sum of 10 and 30, just added in different orders: 10 + 30 and 30 + 10.)
- “In the second table, why are the sums from the lower left corner to the upper right corner all 60?” (Each time, the first number being added goes up by 10 and the second number goes down by 10, so the sum stays the same.)
Standards
- 3.OA.9·Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. <em>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.</em>
- 3.OA.D.9·Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. <span>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.</span>