Whenever you multiply by a bottom tens digit, check for the "Hero Zero":
0
Is there a zero in the first spot of the second row? If not, stop and fix it!
Cross out old regrouping digits above so you don't add them to your second-row multiplication!
Precision Blueprint Strategy SLIDE 4 OF 6
Division
Our Mnemonic Division Algorithm:
D Divide: How many times divisor fits into dividend segment.
M Multiply: Multiply quotient digit by divisor.
S Subtract: Subtract that product from top segment.
C Check & Bring Down: Ensure remainder is smaller, bring down next digit.
R Repeat or Remainder: Restart cycle with new number.
1 0 8
14 ) 1 5 1 2
-1 4
1 1
-0
1 1 2
-1 1 2
0
Precision Blueprint Strategy SLIDE 5 OF 6
Showdown
3-Digit × 2-Digit
\( 384 \times 42 = \text{?} \)
First partial product: \( 384 \times 2 = 768 \)
Placeholder zero required on the second line!
4-Digit ÷ 2-Digit
\( 2,340 \div 15 = \text{?} \)
List multiples of \( 15 \): \( 15, 30, 45, 60, 75 \dots \)
Check your remainder at each step!
Ready to build your mathematical blueprint? Grab your worksheet! SLIDE 6 OF 6
1 0 8
14 ) 1 5 1 2
-1 4
1 1
-0
1 1 2
-1 1 2
0
✂️ Printable Division Companion (Cut Along Dashes)
Place answers directly above the last digit of the number you are dividing!
Always write: Divisor × 1, 2, 3, 4, 5... down the side first!
Grade 5 Precision Grid Masters Math Lab Page 2 of 2
Find the product of \( 476 \times 65 \):
4 7 6
× 6 5
PROBLEM 10 3-Digit × 2-Digit
Find the product of \( 352 \times 97 \):
3 5 2
× 9 7
Division Section Ahead!
Excellent work on multiplication. The next two pages cover 10 division problems using the division house alignment templates.
Grade 5 Precision Grid Masters Math Lab Page 2 of 4
WS-ID: PGM-WS-03
Solve each long division problem. Use the boxes above the house for the quotient digits.
Division Mnemonic Check:
Does McDonald's Sell Cheeseburgers Raw? (Divide → Multiply → Subtract → Check → Bring Down → Repeat).
PROBLEM 11 4-Digit ÷ 2-Digit
Find \( 1,584 \div 12 \):
12 ) 1 5 8 4
PROBLEM 12 4-Digit ÷ 2-Digit
Find \( 2,340 \div 15 \):
15 ) 2 3 4 0
PROBLEM 13 4-Digit ÷ 2-Digit
Find \( 4,896 \div 24 \):
24 ) 4 8 9 6
PROBLEM 14 4-Digit ÷ 2-Digit
Find \( 9,610 \div 31 \):
31 ) 9 6 1 0
PROBLEM 15 4-Digit ÷ 2-Digit
Find \( 1,872 \div 16 \):
16 ) 1 8 7 2
Grade 5 Precision Grid Masters Math Lab Page 3 of 4
WS-ID: PGM-WS-04
Complete the final 5 division problems using the visual guides.
PROBLEM 16 4-Digit ÷ 2-Digit
Find \( 3,564 \div 18 \):
18 ) 3 5 6 4
PROBLEM 17 4-Digit ÷ 2-Digit
Find \( 5,280 \div 22 \):
22 ) 5 2 8 0
PROBLEM 18 4-Digit ÷ 2-Digit
Find \( 8,436 \div 12 \):
12 ) 8 4 3 6
PROBLEM 19 4-Digit ÷ 2-Digit
Find \( 7,168 \div 32 \):
32 ) 7 1 6 8
PROBLEM 20 4-Digit ÷ 2-Digit
Find \( 6,552 \div 13 \):
13 ) 6 5 5 2
Grade 5 Precision Grid Masters Math Lab Page 4 of 4
PROBLEM 09 Solved
476 × 65 = 30,940
443
476
×65
2380
+28560
30940
PROBLEM 10 Solved
352 × 97 = 34,144
431
352
×97
2464
+31680
34144
Check placeholders
For Problem 10 (\( 352 \times 97 \)), ensure students have double regrouped correctly above. The partial products are \( 2,464 \) and \( 31,680 \).
Grade 5 Precision Grid Masters Math Lab Answer Key Page 2 of 4
TEACHER KEY • PGM-AK-03
Quotient values and algorithmic solutions for Problems 11-15.
PROBLEM 11 RESOLVED
1,584 ÷ 12 = 132
1 3 2
12 ) 1 5 8 4
-1 2
3 8 -3 6
2 4 -2 4
0
PROBLEM 12 RESOLVED
2,340 ÷ 15 = 156
1 5 6
15 ) 2 3 4 0
-1 5
8 4 -7 5
9 0 -9 0
0
PROBLEM 13 RESOLVED
4,896 ÷ 24 = 204
2 0 4
24 ) 4 8 9 6
-4 8
0 9 -0 0
9 6 -9 6
0
PROBLEM 14 RESOLVED
9,610 ÷ 31 = 310
3 1 0
31 ) 9 6 1 0
-9 3
3 1 -3 1
0 0 -0 0
0
PROBLEM 15 RESOLVED
1,872 ÷ 16 = 117
1 1 7
16 ) 1 8 7 2
-1 6
2 7 -1 6
1 1 2 -1 1 2
0
Grade 5 Precision Grid Masters Math Lab Answer Key Page 3 of 4
TEACHER KEY • PGM-AK-04
Quotient values and algorithmic solutions for Problems 16-20.
PROBLEM 16 RESOLVED
3,564 ÷ 18 = 198
1 9 8
18 ) 3 5 6 4
-1 8
1 7 6 -1 6 2
1 4 4 -1 4 4
0
PROBLEM 17 RESOLVED
5,280 ÷ 22 = 240
2 4 0
22 ) 5 2 8 0
-4 4
8 8 -8 8
0 0 -0 0
0
PROBLEM 18 RESOLVED
8,436 ÷ 12 = 703
7 0 3
12 ) 8 4 3 6
-8 4
0 3 -0 0
3 6 -3 6
0
PROBLEM 19 RESOLVED
7,168 ÷ 32 = 224
2 2 4
32 ) 7 1 6 8
-6 4
7 6 -6 4
1 2 8 -1 2 8
0
PROBLEM 20 RESOLVED
6,552 ÷ 13 = 504
5 0 4
13 ) 6 5 5 2
-6 5
0 5 -0 0
5 2 -5 2
0
Grade 5 Precision Grid Masters Math Lab Answer Key Page 4 of 4
Quotient (Simplify!): ______
Problem 09: \( \frac{4}{9} \div \frac{2}{3} = \) ______
Problem 10: \( \frac{8}{15} \div \frac{4}{5} = \) ______
Fraction Alchemy Mastery Class Page 2 of 3
LAB-ID: ALCH-6-03
Solve each real-world fractional alchemy scenario. Show your full formula transformations.
PROBLEM 11: THE GLOWING POTION Lab A
Grandmaster Alchemist Raymond has \( \frac{3}{4} \) of a liter of Glowing Liquid. He needs to pour exactly \( \frac{1}{8} \) of a liter into individual glowing potion vials. How many full vials of Glowing Potion can Raymond fill? Show your keep-change-flip formula below.
Transformation Workspace:
Grader Target Check
• Initial State: \( \frac{3}{4} \div \frac{1}{8} \)
• Transform: \( \frac{3}{4} \times \frac{8}{1} \)
Final Quotient: ___________________ vials
PROBLEM 12: ELIXIR RECIPE SCALING Lab B
A giant batch of Wisdom Elixir calls for \( 2 \) kilograms of stardust root powder. If Raymond’s measurement cup holds exactly \( \frac{2}{5} \) of a kilogram, how many cups of stardust root powder must he scoop to make the wisdom batch?
Transformation Workspace:
Grader Target Check
• Initial State: \( 2 \div \frac{2}{5} \)
• Transform: \( 2 \times \frac{5}{2} \)
Final Quotient: ___________________ scoops
Fraction Alchemy Mastery Class Page 3 of 3
\( \frac{105}{168} \) simplified: 5/8
Problem 09: \( \frac{4}{9} \times \frac{3}{2} = \) 2/3
Problem 10: \( \frac{8}{15} \times \frac{5}{4} = \) 2/3
Fraction Alchemy Mastery Class Answer Key Page 2 of 3
TEACHER KEY • ALCH-AK-03
Fully written equations, mathematical models, and context-based outcomes.
PROBLEM 11 RESOLVED: THE GLOWING POTION
Grandmaster Raymond has \( \frac{3}{4} \) liter of Glowing Liquid. He pours \( \frac{1}{8} \) liter into individual vials. How many vials can he fill?
Grader Transform Check:
\( \frac{3}{4} \div \frac{1}{8} \)
\( = \frac{3}{4} \times \frac{8}{1} \)
\( = \frac{24}{4} \)
\( = 6 \) whole vials
Pedagogical Metric
Verify that students did not leave the answer as \( \frac{24}{4} \). They must simplify to the whole number \( 6 \) to earn full credit!
Correct Answer: 6 vials
PROBLEM 12 RESOLVED: ELIXIR RECIPE SCALING
Wisdom Elixir calls for \( 2 \) kg of stardust root powder. The measurement cup holds \( \frac{2}{5} \) kg. How many scoops are needed?
Grader Transform Check:
\( 2 \div \frac{2}{5} \)
\( = \frac{2}{1} \times \frac{5}{2} \)
\( = \frac{10}{2} \)
\( = 5 \) scoops
Pedagogical Metric
Students should represent the whole number \( 2 \) as \( \frac{2}{1} \) inside their workspace before executing the reciprocal multiplication.
Correct Answer: 5 scoops
Fraction Alchemy Mastery Class Answer Key Page 3 of 3
Step 1: Draw the Dividend. We start by drawing a bar partitioned into \( 4 \) equal parts to represent fourths. We shade in exactly \( 3 \) parts to show \( \frac{3}{4} \).
Step 2: Subdivide to find common units. Our divisor is in eighths, so we must slice our fourths into eighths. How do we do that? We split every fourth in half! Now, our bar has \( 8 \) equal parts total, and our shaded section represents \( \frac{6}{8} \).
Step 3: Measure the Divisor. Our divisor is \( \frac{1}{8} \). We want to count how many blocks of \( \frac{1}{8} \) are in our shaded area of \( \frac{6}{8} \).
Let's count them together: \( 1, 2, 3, 4, 5, 6 \)! There are exactly \( 6 \) blocks of \( \frac{1}{8} \) inside \( \frac{3}{4} \). So, \( \frac{3}{4} \div \frac{1}{8} = 6 \). Our formula matches our visual map!"
The "Shrinking Quotient" Trap: Students expect division to always yield a smaller number. Explain that dividing by a number less than \( 1 \) increases the quotient value.
Common Denominator Ignorance: Attempting to divide numerators and denominators directly without reciprocal steps.
Stage 1 (Fourths): Stage 2 (Eighths): Outcome:
3/4 Shaded
1/4
1/8
1/8
1/8
1/8
1/8
1/8
There are exactly 6 units of \( \frac{1}{8} \) in \( \frac{3}{4} \)!
Fraction Alchemy Lesson Plan Page 2 of 3
DOC-ID: FAM-TG-03
Connecting Visuals to standard reciprocal mathematical algorithms
Guided Script: Keep-Change-Flip Mechanics
"Now let's translate our visual tape diagrams into a speedy math shortcut called **Keep-Change-Flip** (KCF). We will compute \( \frac{2}{5} \div \frac{3}{10} \).
1. KEEP: Keep the first fraction exactly as it is: \( \frac{2}{5} \).
2. CHANGE: Change the division sign \( (\div) \) to a multiplication sign \( (\times) \).
3. FLIP: Flip the second fraction upside down to write its **reciprocal**! \( \frac{3}{10} \) flips to become \( \frac{10}{3} \).
4. MULTIPLY: Multiply straight across. Numerator times numerator: \( 2 \times 10 = 20 \). Denominator times denominator: \( 5 \times 3 = 15 \). That gives us \( \frac{20}{15} \).
5. SIMPLIFY: Simplify by dividing by common factors. Divide both by \( 5 \) to get \( \frac{4}{3} \), which is \( 1 \frac{1}{3} \)."
1. Write \( \frac{5}{8} \div \frac{5}{6} \) on student whiteboards.
2. Have them write 'K', 'C', 'F' boxes under each part.
3. Solve together: \( \frac{5}{8} \times \frac{6}{5} = \frac{30}{40} = \frac{3}{4} \).
• "What is the mathematical definition of a reciprocal?" (Two fractions that multiply to make 1, e.g. \( 2/3 \times 3/2 = 1 \)).
• "Why can we multiply by the reciprocal to solve division?" (Because division is the inverse operation of multiplication).
Have students complete a 2-minute "Recipe Ticket" explaining why flipping the divisor works. Release students to work independently or in pairs on the **Fraction Alchemy Worksheet (ALCH-6-01)**, using their desk models as visual companion tools!
Fraction Alchemy Lesson Plan Page 3 of 3
Flip the second fraction upside down to get its **reciprocal**.
\( \frac{3}{10} \) → \( \frac{10}{3} \)
The Algorithmic Reciprocal Strategy SLIDE 4 OF 6
Formula Flow
Let's multiply straight across.
Original problem: \( \frac{2}{5} \div \frac{3}{10} \)
KCF Formula: \( \frac{2}{5} \times \frac{10}{3} \)
Calculation: \( \frac{2 \times 10}{5 \times 3} = \frac{20}{15} \)
We found the quotient is \( \frac{20}{15} \). But we are not finished! Potion recipes must be in simplest form.
\( \frac{20 \div 5}{15 \div 5} = \frac{4}{3} = 1\frac{1}{3} \)
\( \mathbf{1\frac{1}{3}} \) cups or beakers represents the exact formula solution!
The Algorithmic Reciprocal Strategy SLIDE 5 OF 6
Practice
Tape Diagram
How many blocks of \( \frac{1}{6} \) fit inside \( \frac{5}{6} \)? Draw a quick tape diagram on your scratch paper to show it.
\( \frac{5}{6} \div \frac{1}{6} = \text{?} \)
KCF Algorithm
Convert and compute: \( \frac{3}{4} \div \frac{2}{3} \). Write your final quotient as a mixed number.
\( \frac{3}{4} \times \frac{3}{2} = \text{?} \)
Ready to master the potion lab? Grab your Fraction Alchemy Worksheet! SLIDE 6 OF 6
Grade 6 Fraction Alchemy Masters Reference Page 2 of 2
