Mathematics Grade 2 Addition and Subtraction 2-digit

Grade 2 Mathematics: Addition and Subtraction (2-Digit) Study Guide

1. The Driving Question

If you have 47 stickers and your friend gives you 25 more, how do you figure out how many you have without counting every single one? And if you then give away 18 stickers, how do you know how many are left—without starting over? Why can’t you just add or subtract the tens and ones separately like you did with smaller numbers?

2. The Core Idea — Built, Not Listed

Imagine you’re at a Lego store with two bins: one has 47 red bricks, and the other has 25 blue bricks. You want to know how many bricks you have total to build a castle. Instead of counting each brick one by one, you can group them by tens and ones to make it faster.

• Step 1: Count the tens. The red bin has 4 tens (40), and the blue bin has 2 tens (20). Together, that’s 6 tens (60).
• Step 2: Count the ones. The red bin has 7 ones, and the blue bin has 5 ones. Together, that’s 12 ones—but 12 ones is the same as 1 ten and 2 ones!
• Step 3: Now add the extra ten from the ones to the tens pile. 6 tens + 1 ten = 7 tens (70), plus the 2 ones left over. So, 70 + 2 = 72 bricks total.

Now, what if you give away 18 bricks to your little brother? You can break it down the same way: - Step 1: Subtract the tens. 7 tens (70) – 1 ten (10) = 6 tens (60).
- Step 2: Subtract the ones. But you only have 2 ones, and you need to give away 8 ones—that’s a problem! So, you "borrow" 1 ten (10 ones) from the tens pile, turning 6 tens (60) into 5 tens (50) and 12 ones.
- Step 3: Now subtract the ones: 12 ones – 8 ones = 4 ones. Then subtract the tens: 5 tens – 0 tens = 5 tens (50). So, 50 + 4 = 54 bricks left.

This way, you never have to count every single brick—you just group, add, and adjust.

Key Vocabulary:
- Regrouping (Borrowing/Carrying): When you have too many or too few ones (or tens), you trade 10 ones for 1 ten (or 1 ten for 10 ones) to make the math work.
- Example: If you have 35 candies and eat 7, you can’t subtract 7 from 5, so you "borrow" 1 ten (10 ones) to make it 2 tens and 15 ones, then subtract.
- Grade 2 Note: This is the first time students see that numbers aren’t "fixed"—they can be broken apart and put back together differently.

• Place Value: The idea that the position of a digit (tens or ones) tells you its value.
• Example: In 53, the 5 isn’t just "five"—it’s 5 tens (50). The 3 is 3 ones.

• Future Note: In 3rd grade, this expands to hundreds, and in 5th grade, to decimals (where the "ones place" is the middle of the number, not the end).

• Sum: The answer to an addition problem.

• Example: If you have 24 marbles and win 19 more in a game, the sum is 43 marbles.

• Future Note: Later, "sum" will apply to fractions, decimals, and even variables in algebra.

• Difference: The answer to a subtraction problem.

• Example: If you start with 60 trading cards and trade away 23, the difference is 37 cards left.
• Future Note: In algebra, "difference" can mean the result of any subtraction, even if the numbers are negative or unknown.

3. Assessment Translation

How This Appears in Classroom Assessments (Grade 2):
- Exit Tickets: Short problems like "45 + 28 = ?" or "72 – 39 = ?" with space to show work.
- Constructed Response: Problems like:

"Liam has 56 crayons. He buys 27 more. How many crayons does he have now? Show your work using tens and ones." - Word Problems: Real-life scenarios (e.g., counting toys, money, or snacks) where students must choose the operation (add or subtract) and solve.

What a "Proficient" Response Looks Like:
- Shows regrouping clearly (e.g., crossing out tens and writing the new amount).
- Labels tens and ones (even if just with a T and O above the digits).
- Writes the final answer in the correct place (not just in the work space).

What a "Developing" Response Looks Like:
- Forgets to regroup (e.g., adds 45 + 28 as 613 because they put 4+2=6 and 5+8=13 side by side).
- Miscounts the ones (e.g., says 12 ones is 20 because they forget it’s 1 ten and 2 ones).
- Writes the answer but doesn’t show work, so the teacher can’t tell if they guessed or understood.

Model Proficient Response:
Problem: 34 + 29 = ?
Student Work:

  34
+ 29
-----
  63

Explanation: 1. Add the ones: 4 + 9 = 13. Write down 3, carry over 1 ten.
2. Add the tens: 3 + 2 = 5, plus the 1 ten from the ones = 6 tens.
3. Final answer: 63.

(Note: The student doesn’t have to write the explanation—just showing the regrouping with a small "1" above the tens is enough!)

4. Mistake Taxonomy

Mistake 1: The "Side-by-Side" Addition Error
- Question: 45 + 28 = ?
- Common Wrong Answer: 613
- Why It Loses Credit: The student adds the tens (4+2=6) and the ones (5+8=13) separately, then smashes the answers together. This ignores place value and regrouping.
- Correct Approach: 1. Add the ones: 5 + 8 = 13 ones (which is 1 ten and 3 ones).
2. Write down 3 in the ones place, carry over 1 ten to the tens column.
3. Add the tens: 4 + 2 + 1 (carried) = 7 tens.
4. Final answer: 73.

Mistake 2: The "Borrowing Without Adjusting" Error
- Question: 62 – 37 = ?
- Common Wrong Answer: 35
- Why It Loses Credit: The student sees they can’t subtract 7 from 2, so they "borrow" but forget to reduce the tens place. They do 12 – 7 = 5, but then 6 – 3 = 3, ignoring that they took 1 ten away.
- Correct Approach: 1. Can’t subtract 7 from 2, so borrow 1 ten from the 6, making it 5 tens and 12 ones.
2. Subtract the ones: 12 – 7 = 5.
3. Subtract the tens: 5 – 3 = 2.
4. Final answer: 25.

Mistake 3: The "Operation Mix-Up" in Word Problems
- Question: "Emma has 48 stickers. She gives 19 to her friend. How many does she have left?" - Common Wrong Answer: 67 (student adds instead of subtracts) - Why It Loses Credit: The student misreads the problem and picks the wrong operation. Keywords like "gives away" or "left" signal subtraction, but the student might default to addition.
- Correct Approach: 1. Reread the problem: "Gives away" means subtract.
2. Set up the equation: 48 – 19 = ? 3. Regroup: Borrow 1 ten to make 18 ones, then subtract.
4. Final answer: 29.

5. Connection Layer

1. Within Math: 2-digit addition/subtraction → 3-digit addition/subtraction

2. Why it helps: Once you master regrouping with tens and ones, adding hundreds (e.g., 256 + 178) is the same process—just one more column. The "borrowing" rules don’t change, they just apply to a bigger number.

3. Across Subjects: Math (regrouping) → Science (conservation of matter)

4. Why it helps: In science, you’ll learn that matter (like water or clay) can’t be created or destroyed—it just changes form. Regrouping is like that: you’re not adding or removing bricks, you’re just rearranging them (e.g., 10 ones → 1 ten).

5. Outside School: Math (place value) → Money (making change)

6. Why it helps: When you buy a $2.75 toy with a $5 bill, the cashier doesn’t count every penny—they regroup (e.g., "2 dollars and 75 cents" becomes "4 dollars and 25 cents left" after giving you a $1 bill). The same logic applies to adding and subtracting dollars and cents.

6. The Stretch Question

If you add two 2-digit numbers and the answer is 99, what could those two numbers be? How many different pairs work?

Pointer Toward the Answer:
- The biggest 2-digit number is 99, so the two numbers you’re adding must be less than 99 (since 99 + 0 = 99, but 0 isn’t a 2-digit number).
- Start with one number, like 50. Then the other number would be 99 – 50 = 49.
- Try 60 + 39 = 99, or 75 + 24 = 99. Do you see a pattern? - Hint: The tens digits in the two numbers must add up to 9 or 8 (because of regrouping!). For example, 45 + 54 = 99 (4+5=9 in the tens place, 5+4=9 in the ones place, but with regrouping). How many pairs like this exist?