Integer Addition Blueprint Notes Sheet Step-by-Step Reference Notes
Integer Addition Blueprint
Part 1: Multi-Digit Column Addition & Carrying (Regrouping)
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The Golden Rule of Column Addition:
Stack numbers strictly by place value (Tens under Tens, Ones under Ones). Always add from Right to Left (Ones first!). Each column can only hold one digit (0–9) in the answer line.
1 The 3 Steps to Add in Columns
Step 1: Align Columns
Push to the Right
Write the numbers vertically. Keep the ones column lined up against the right wall.
Step 2: Add the Ones
Sum the Right Column
Add the digits in the ones place. If the total is 9 or less, write it down and move to the tens.
Step 3: Carry (Regroup)
Total ≥ 10?
Write the ones digit in the answer, and carry the 1 ten into the carry box up top!
Worked Example: 2-Digit Column Carrying Step-by-Step Problem: 38 + 47
T O
1 —
3 8
8 5
Total = 85
Step A (Ones Column): 8 + 7 = 15
15 has 1 ten and 5 ones. Write 5 in the ones answer spot. Carry the 1 to the carry box above the Tens column.
Step B (Tens Column): (carried 1) + 3 + 4 = 8
Add all three numbers in the tens column: 1 + 3 + 4 = 8 tens. Write 8. Final answer is 85.
Don't Miss the Carry!
Write the carried "1" right away in the carry box up top so you don't forget to add it.
Lined Paper Trick
Turn your lined paper sideways! The blue lines become instant columns for Tens and Ones.
P1 Foundation Checkpoint: Add with Carry Boxes Fill in the carry box at top, then sum each column
Problem 1: 46 + 28 2-Digit Carrying
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4 6
Problem 2: 357 + 286 Double Carry
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3 5 7
Mastery Guide
Adding Signed Integers in Columns
Part 2: Applying Single-Digit Integer Addition to Multi-Digit Columns (Pure Addition!)
Page 2 of 2
You already know single-digit integer addition! We use those exact same facts inside each column!
6 + (−4) = +2 | (−4) + (−8) = −12
A Case 1: Same Signs (Both Negative) → Add & Keep the Negative! (−24) + (−38)
1 —
−2 4
−6 2
Answer: −62
1. Ones Column: 4 + 8 = 12. Write 2 below, carry 1 to the tens.
2. Tens Column: Carried 1 + 2 + 3 = 6 tens.
3. Final Sign: Both numbers were on Team Negative, so the answer stays negative: −62.
B Case 2: Opposite Signs → Column Addition (No Subtraction Needed!) Stack Positive on Top
Model 1: (−24) + 36 Commutative: 36 + (−24)
T O
3 6
+1 2
• Ones Column:
• Tens Column:
Final Answer: +12
Model 2: 90 + (−42) Regroup 1 Ten → 10 Ones
(8) (10)
9 0
+4 8
• Regroup 90:
• Ones: 10 + (−2) = +8
• Tens: 8 + (−4) = +4
Final Answer: +48
P2 Student Practice: Solve Using Column Addition Show vertical columns & add column by column
1. (−35) + (−27)
Both negative (carry 1 ten)
Work Space
Answer: ____________
2. (−32) + 56
Stack: 56 + (−32)
Work Space
Answer: ____________
3. 80 + (−34)
Regroup 80 → 7 tens, 10 ones
Work Space
Answer: ____________
Column Addition Checklist: □ Numbers aligned in Tens & Ones □ Added single-digit integers per column □ Applied the final sign (+ or −)
Integer Multiplication Notes Sheet Visual Reference & Guided Notes
Integer Multiplication Leap
Part 1: Connecting Multiplication to Repeated Addition & Skip Counting
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The Big Secret: Multiplication is Just Shortcut Addition!
When you see 4 × 3, read it aloud as: "4 groups of 3". You are simply adding 3 four times: 3 + 3 + 3 + 3 = 12!
The Skip-Counting Number Ladder (Your Multiplication Helper) No memorization needed — just count the jumps!
Skip by 2s:
2, 4, 6, 8, 10, 12, 14, 16, 18, 20
Skip by 3s:
3, 6, 9, 12, 15, 18, 21, 24, 27, 30
Skip by 4s:
4, 8, 12, 16, 20, 24, 28, 32, 36, 40
Skip by 5s:
5, 10, 15, 20, 25, 30, 35, 40, 45, 50
Skip by 6s:
6, 12, 18, 24, 30, 36, 42, 48, 54, 60
Skip by 7s:
7, 14, 21, 28, 35, 42, 49, 56, 63, 70
Skip by 8s:
8, 16, 24, 32, 40, 48, 56, 64, 72, 80
Skip by 9s:
9, 18, 27, 36, 45, 54, 63, 72, 81, 90
What Happens With a Negative? Positive × Negative Example: 3 × (−4)
Translate to "Groups Of":
3 × (−4) means 3 groups of −4.
(−4) + (−4) + (−4) = −12
Think: You owe $4 to 3 friends. You are in debt by $12!
Number Line Hops (Left = Negative):
−12 −8 −4 0
← Hop 3 (−4) ← Hop 2 (−4) ← Hop 1 (−4)
Start at 0, take 3 hops of 4 to the left → land at −12.
P1 Guided Checkpoint: Translate Repeated Addition to Multiplication Use the skip-counting ladder above if needed!
Problem 1:
5 + 5 + 5 + 5
How many groups? _____
Multiplication: ___ × ___
Skip-Count Result: _______
Problem 2:
(−6) + (−6) + (−6)
How many groups? _____
Multiplication: ___ × (−6)
Skip-Count Result: _______
Problem 3:
4 × (−7)
Repeated Addition:
(___) + (___) + (___) + (___)
Final Product: _______
Key takeaway: A positive times a negative is always a NEGATIVE answer! □ Checked with skip counting
Mastery Guide
Signed Integer Multiplication
Part 2: The Negative × Negative Mystery & The Two-Step System
Page 2 of 2
The Countdown Pattern: Why (−) × (−) = (+) Watch the Pattern Add 4!
3 × (−4) = −12
Integer Multiplication Slides Math Foundations Algebra Readiness
Integer Multiplication Leap
Connecting Repeated Addition, Skip Counting, and Positive/Negative Signs
No memorization required • Use skip-counting tools • Master the signs
Concept 1: The Core Meaning
Multiplication IS Repeated Addition
When you see:
4 × 3
Read it as: "4 groups of 3"
You are adding 3 four times:
3 + 3 + 3 + 3 = 12
Multiplication is just a faster shortcut for addition!
Key takeaway: If you can add, you can multiply! Slide 2 / 6
Tool: No Memorization Needed
Skip Counting is Your Superpower
Example: Find 5 × 6 Skip count by 6s five times
Hop 1 6
Hop 2 12
Hop 3 18
Hop 4 24
Hop 5 30
Use your fingers or your notes ladder: count the hops to reach the total!
Every multiplication table is just a skip-counting pattern. Slide 3 / 6
Concept 2: Adding Negatives
Positive × Negative = Negative
Problem: 3 × (−5)
Read as: "3 groups of −5"
(−5) + (−5) + (−5) = −15
Real-World Thinking:
You buy 3 snacks that each cost $5.
Your wallet is down by $15 (−15)!
Repeated negative hops always go further left on the number line.
Rule: Different signs always multiply to a NEGATIVE product. Slide 4 / 6
The Big Reveal
Why Negative × Negative = Positive
3 × (−4) = −12
2 × (−4) = −8
1 × (−4) = −4
0 × (−4) = 0
(−1) × (−4) = +4
(−2) × (−4) = +8
Look at the Pattern:
Each step down, the answer grows by +4.
Mathematical Certainty:
Past zero, the numbers MUST become positive to keep the pattern intact!
Rule: Same signs always multiply to a POSITIVE product. Slide 5 / 6
Summary Blueprint
The 2-Step Multiplication System
1
Find the Number
Ignore the signs for a second. Use skip counting to find the numerical product.
Example: 6 × 7 → 7, 14, 21, 28, 35, 42
2
Pick the Sign
Same Signs: (+ · +) or (− · −) + POSITIVE
Diff Signs: (+ · −) or (− · +) − NEGATIVE
Ready! Now transition to your guided notes sheet practice! Slide 6 / 6
