Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
* Shared factors: 1, 2, 3, 6. Greatest factor: 6.
Prime Factor Listing Method:
Write out the prime factors of 24 and 36. Circle the common prime factors. Multiply them together to find the GCF.
Shared: 2, 2, and 3. GCF = 2 × 2 × 3 = 12.
Problem 1 Answer GCF of 15 and 35
Factors of 15: 1, 3, 5, 15
Factors of 35: 1, 5, 7, 35
Or prime: 15=3×5; 35=5×7.
GCF Answer: GCF = 5
Problem 2 Answer GCF of 24 and 60
24 = 2 × 2 × 2 × 3
60 = 2 × 2 × 3 × 5
Shared: 2, 2, 3
GCF Answer: GCF = 12
Find the GCF of 32 and 48:
32 = 2 × 2 × 2 × 2 × 2
48 = 2 × 2 × 2 × 2 × 3
Shared: 2 × 2 × 2 × 2 = 16
GCF Answer:
16
Teacher Answer Key • Week 1
IAS 6.NS.4
Teacher Resource
Misconception Alert
Students often make addition errors when skip-counting. Encourage them to verify skip-counting lists using basic multiplication facts (e.g., 6 × 1, 6 × 2, 6 × 3).
Pacing Guide
2m: Warm-Up • 8m: Walk through skip-counting (listing) vs prime factors for LCM • 10m: Ind. Practice • 5m: Exit Ticket.
Multiples of 6: 6, 12, 18, 24, 30, 36
Multiples of 8: 8, 16, 24, 32, 40, 48
* LCM of 6 and 8 is 24.
Explain the difference: factors go inside a number, multiples count up.
For GCF of 4 and 5, it is 1. But for LCM of 4 and 5, we skip-count until they meet. Since 4 and 5 share no prime factors, their LCM is simply their product: 4 × 5 = 20.
Problem 1 Answer LCM of 4 and 10
Multiples of 4: 4, 8, 12, 16, 20, 24
Multiples of 10: 10, 20, 30
LCM Answer: LCM = 20
Problem 2 Answer LCM of 9 and 12
Multiples of 9: 9, 18, 27, 36, 45
Multiples of 12: 12, 24, 36, 48
LCM Answer: LCM = 36
Find the LCM of 8 and 12. Show your workspace by listing multiples!
8: 8, 16, 24, 32, 40
12: 12, 24, 36, 48
LCM Answer:
24
Teacher Answer Key • Week 1
IAS 6.NS.4
Teacher Resource
Key Clues
Help students identify: "Greatest number of groups", "equal bags", "no leftovers" point to GCF. "Next time", "happens again", "intervals" point to LCM.
Pacing Guide
2m: Warm-Up matching • 8m: Analyze sample pretzel/chocolate scenario • 10m: Ind. Practice • 5m: Exit Ticket.
Scenario Clues:
A: "Divide items into equal bags"
B: "Events happen at the same time"
Match Answers:
Clue A matches with: GCF
Clue B matches with: LCM
Walk through Sienna's bag packing scenario: 24 pretzels, 36 chocolates. We want the *greatest* number of equal bags, so we divide. GCF of 24 and 36 is 12 bags. Show that each bag contains 2 pretzels (24 ÷ 12) and 3 chocolates (36 ÷ 12).
Problem 1 Key (LCM)
Juice boxes (6) and cookies (8). Least number to match sets:
LCM(6, 8) = 24 items.
24 ÷ 6 = 4 boxes of juice.
24 ÷ 8 = 3 packs of cookies.
LCM Answer: 24 of each
Problem 2 Key (GCF)
Row A (18 chairs) and Row B (27 chairs). Align columns:
GCF(18, 27):
18 = 2 × 3 × 3
27 = 3 × 3 × 3
GCF = 3 × 3 = 9 columns.
GCF Answer: 9 columns
How many seconds until they flash together again? (12s and 15s)
Find LCM(12, 15):
12: 12, 24, 36, 48, 60
15: 15, 30, 45, 60
Seconds:
60 sec
Teacher Answer Key • Week 1
IAS 6.NS.4
Teacher Resource
Scoring Rubric:
Problems 1-2: Factor trees. 1.5 pts each (1 pt tree, 0.5 pt exp expression).
Problems 3-4: GCF/LCM. 2 pts each (1 pt work/lists, 1 pt final answer).
Problem 5: Word problem. 3 pts (1.5 pts identify GCF & solve, 1.5 pts final).
Part 1: Prime Factorization Answers 3 Points
1. Factor tree for 60:
60 -> 2 × 30 -> 2 × 2 × 15 -> 2 × 2 × 3 × 5
Prime Factorization: 22 × 3 × 5
2. Factor tree for 72:
72 -> 2 × 36 -> 2 × 2 × 18 -> 2 × 2 × 2 × 9 -> 23 × 32
Prime Factorization: 23 × 32
Part 2: GCF and LCM Skills Answers 4 Points
3. GCF of 45 and 60:
45 = 3 × 3 × 5
60 = 2 × 2 × 3 × 5. Common: 3, 5.
GCF = 15
4. LCM of 12 and 15:
12: 12, 24, 36, 48, 60, 72
15: 15, 30, 45, 60, 75
LCM = 60
Part 3: Word Problems Answers 3 Points
5. Solving a Real-World Challenge:
Marcus has 16 oatmeal bars and 24 fudge brownies. Identical boxes with no leftovers:
We find the GCF of 16 and 24.
16 = 2 × 2 × 2 × 2; 24 = 2 × 2 × 2 × 3.
GCF = 2 × 2 × 2 = 8 boxes.
Greatest Boxes:
8 boxes
Intervention Action Plan Students scoring < 7 should focus on skip-counting visual matrices next week.
Standard-Met: 7+ / 10
If \( \frac{1}{3} \) is divided in half:
1/3
Quotient:
Problem 2 Solve: \( \frac{1}{4} \div 3 \)
If \( \frac{1}{4} \) is divided in thirds:
1/4
Quotient:
A strip of length \( \frac{1}{2} \) yard is cut into 4 equal segments. How long is each segment?
Length:
Week 2: Fraction Division
IAS 6.NS.1
25-Min Intervention
Name: Date: Score: / 10
Write the reciprocal (flipped form) of each fraction below.
Recip of: \( \frac{2}{3} \rightarrow \frac{?}{?} \)
Recip of: \( \frac{1}{5} \rightarrow \frac{?}{?} \)
Recip of: \( \frac{4}{7} \rightarrow \frac{?}{?} \)
Recip of: \( 6 \rightarrow \frac{?}{?} \)
The Standard Algorithm: Keep, Change, Flip (KCF)
K - KEEP
Keep 1st fraction
\( \frac{1}{2} \)
C - CHANGE
Change ÷ to ×
×
F - FLIP
Flip 2nd fraction
\( \frac{4}{3} \)
\( \frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3} \)
Problem 1 Solve: \( \frac{2}{5} \div \frac{1}{2} \)
Quotient:
Problem 2 Solve: \( \frac{3}{4} \div \frac{2}{5} \)
Quotient:
Apply Keep, Change, Flip to solve: \( \frac{3}{5} \div \frac{2}{3} \)
Quotient:
Week 2: Fraction Division
IAS 6.NS.1
25-Min Intervention
Name: Date: Score: / 10
Multiply the fractions. Always simplify your final answers if possible.
\( \frac{2}{3} \times \frac{3}{4} = ? \) Ans: ___
\( \frac{1}{5} \times \frac{5}{8} = ? \) Ans: ___
\( \frac{4}{5} \times \frac{2}{3} = ? \) Ans: ___
Simplifying quotients:
Sometimes division results in an improper fraction (numerator is bigger). We can convert it to a mixed number!
IMPROPER QUOTIENT:
\( \frac{7}{4} \)
MIXED NUMBER:
How many 4s fit inside 7?
1 whole with 3 leftover.
Answer: 1 \( \frac{3}{4} \)
Problem 1 \( \frac{5}{6} \div \frac{2}{3} \)
Mixed Number:
Problem 2 \( \frac{3}{8} \div \frac{3}{4} \)
Simplified Ans:
Divide and write your quotient in simplest form: \( \frac{2}{3} \div \frac{4}{9} \)
Final Answer:
Week 2: Fraction Division
IAS 6.NS.1
25-Min Assessment
Name: Date: Score: / 10
Part 1: Visual Models 3 Points
1. Model \( 3 \div \frac{1}{3} \)
Quotient: _________
2. Model \( \frac{1}{2} \div 2 \)
Quotient: _________
Part 2: Division Mechanics 4 Points
3. Solve: \( \frac{4}{5} \div \frac{1}{2} \)
Quotient = _________
4. Solve: \( \frac{3}{10} \div \frac{2}{5} \)
Quotient = _________
Part 3: Word Problem 3 Points
5. Fractional Sharing Challenge:
An instructor has \( \frac{3}{4} \) of a gallon of paint. She needs to share it equally among 6 student project groups. How much paint does each group receive?
Paint per Group:
Weekly Reflection How do you feel about dividing fractions, modeling problems, and using Keep-Change-Flip?
🌱 Just Starting 🏃 Growing ⭐ Mastered
5 = \( \frac{?}{1} \) ? = 5
8 = \( \frac{8}{?} \) ? = 1
10 = \( \frac{?}{1} \) ? = 10
Use a visual area model on a grid. Shade half of the rectangle to represent \( \frac{1}{2} \) of a lasagna pan. Draw horizontal lines to split the entire pan into 3 rows (sharing with 3 people). Highlight that 1 shared piece is exactly \( \frac{1}{6} \) of the entire pan. Show equation: \( \frac{1}{2} \div 3 = \frac{1}{6} \).
Problem 1 Answer \( \frac{1}{3} \div 2 \)
Split \( \frac{1}{3} \) in half.
Makes 2 equal smaller pieces.
Each piece is \( \frac{1}{3 \times 2} = \frac{1}{6} \)
Quotient: 1/6
Problem 2 Answer \( \frac{1}{4} \div 3 \)
Split \( \frac{1}{4} \) into 3 parts.
Total segments in 1 whole = 12.
Each part is \( \frac{1}{4 \times 3} = \frac{1}{12} \)
Quotient: 1/12
Segment length of \( \frac{1}{2} \) yard cut into 4 segments:
\( \frac{1}{2} \div 4 = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8} \)
Each piece is \( \frac{1}{8} \) yard long.
Length:
1/8 yd
Teacher Answer Key • Week 2
IAS 6.NS.1
Teacher Resource
Misconception Alert
Students often flip the *first* fraction instead of the *second* (divisor) fraction. Remind them: "The leader stays standing (Keep), the sign transforms (Change), and only the follow-up dancer flips (Flip)!"
Pacing Guide
2m: Warm-Up reciprocals • 8m: Connect visual division to multiplication algorithm • 10m: KCF independent practice • 5m: Exit Ticket.
Recip of: \( \frac{2}{3} \rightarrow \frac{3}{2} \)
Recip of: \( \frac{1}{5} \rightarrow \frac{5}{1} \) (5)
Recip of: \( \frac{4}{7} \rightarrow \frac{7}{4} \)
Recip of: \( 6 \rightarrow \frac{1}{6} \)
Show how dividing by \( \frac{3}{4} \) is mathematically equivalent to multiplying by its reciprocal \( \frac{4}{3} \). Write on the board: Keep \( \frac{1}{2} \) → Change ÷ to × → Flip \( \frac{3}{4} \) to \( \frac{4}{3} \). Multiply straight across: \( 1 \times 4 = 4 \) and \( 2 \times 3 = 6 \). Simplify \( \frac{4}{6} \) to \( \frac{2}{3} \).
Problem 1 Answer \( \frac{2}{5} \div \frac image{1}{2} \)
Keep: \( \frac{2}{5} \)
Change: ×
Flip: \( \frac{2}{1} \)
\( \frac{2}{5} \times \frac{2}{1} = \frac{4}{5} \)
Quotient: 4/5
Problem 2 Answer \( \frac{3}{4} \div \frac{2}{5} \)
Keep: \( \frac{3}{4} \)
Change: ×
Flip: \( \frac{5}{2} \)
\( \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} \)
Quotient: 15/8 or 1 7/8
Solve \( \frac{3}{5} \div \frac{2}{3} \) using Keep, Change, Flip:
\( \frac{3}{5} \times \frac{3}{2} = \frac{9}{10} \)
Quotient:
9/10
Teacher Answer Key • Week 2
IAS 6.NS.1
Teacher Resource
Misconception Alert
Students leave final quotients as improper fractions. Week 2 goal is fluency with converting values (e.g., \( \frac{5}{4} \rightarrow 1 \frac{1}{4} \)) and simplifying.
Pacing Guide
2m: Warm-Up multiplication review • 8m: Modeling how to simplify and convert improper fractions • 10m: Ind. Practice • 5m: Exit Ticket.
\( \frac{2}{3} \times \frac{3}{4} = \frac{6}{12} \) Ans: 1/2
\( \frac{1}{5} \times \frac{5}{8} = \frac{5}{40} \) Ans: 1/8
\( \frac{4}{5} \times \frac{2}{3} = \frac{8}{15} \) Ans: 8/15
Explain how to change improper quotients to mixed numbers: If the numerator is bigger, divide the top by the bottom. The whole number quotient becomes the whole number of the mixed value. The remainder becomes the new numerator. Keep the denominator!
Problem 1 Answer \( \frac{5}{6} \div \frac{2}{3} \)
\( \frac{5}{6} \times \frac{3}{2} = \frac{15}{12} \)
Simplify \( \frac{15}{12} = \frac{5}{4} \)
Convert: \( 5 \div 4 = 1 \) remainder 1
Mixed: \( 1 \frac{1}{4} \)
Mixed Number: 1 1/4
Problem 2 Answer \( \frac{3}{8} \div \frac{3}{4} \)
\( \frac{3}{8} \times \frac{4}{3} = \frac{12}{24} \)
Simplify \( \frac{12}{24} \)
Divide numerator and denominator by 12:
\( \frac{12 \div 12}{24 \div 12} = \frac{1}{2} \)
Simplified Ans: 1/2
Divide and simplify \( \frac{2}{3} \div \frac{4}{9} \):
\( \frac{2}{3} \times \frac{9}{4} = \frac{18}{12} \)
Simplify: \( \frac{18 \div 6}{12 \div 6} = \frac{3}{2} \)
Mixed Number: \( 1 \frac{1}{2} \)
Final Answer:
1 1/2
Teacher Answer Key • Week 2
IAS 6.NS.1
Teacher Resource
Scoring Guidelines:
Part 1 (Visuals): 1.5 pts each (1 pt for accurate division partitions, 0.5 pt for correct quotient).
Part 2 (Mechanics): 2 pts each (1 pt for KCF work setup, 1 pt for simplified quotient).
Part 3 (Word problem): 3 pts (1 pt for setting up equation, 2 pts for solution 1/8).
Part 1: Visual Models Answers 3 Points
1. Model \( 3 \div \frac{1}{3} \):
Draw 3 squares. Split each in thirds → 9 pieces.
\( 3 \times 3 = 9 \)
Quotient Answer: 9
2. Model \( \frac{1}{2} \div 2 \):
Shade half a bar. Slice that half into 2 equal parts.
Each part is \( \frac{1}{4} \) of the whole bar.
Quotient Answer: 1/4
Part 2: Division Mechanics Answers 4 Points
3. Solve: \( \frac{4}{5} \div \frac{1}{2} \)
\( \frac{4}{5} \times \frac{2}{1} = \frac{8}{5} = 1 \frac{3}{5} \)
Quotient: 1 3/5 (or 8/5)
4. Solve: \( \frac{3}{10} \div \frac{2}{5} \)
\( \frac{3}{10} \times \frac{5}{2} = \frac{15}{20} = \frac{3}{4} \)
Quotient: 3/4
Part 3: Word Problem Answer 3 Points
5. Fractional Sharing Challenge:
\( \frac{3}{4} \) gallon shared equally among 6 student groups:
Equation: \( \frac{3}{4} \div 6 \)
Algorithm: \( \frac{3}{4} \times \frac{1}{6} = \frac{3}{24} = \frac{1}{8} \) gallon.
Paint per Group:
1/8 gal
Next Steps Prepare students for Week 3 by introducing mixed number-to-improper conversion.
IAS 6.NS.1
