Factor Finder Worksheet
Delta • Lesson 1 Foundations of Division
Factor Finder Worksheet
Discovering division as finding the missing factor & mastering division rules
Name:
Date:
1. The Missing Factor: Area & Rectangles
In multiplication: Side \(\times\) Side = Area. In division, you know the total Area and one Side. Find the unknown side!
Rectangle 1: Total Area = 8
\(2 \times \underline{\hspace{1.2cm}} = 8\)
\(8 \div 2 = \)
2 by ?
Rectangle 2: Total Area = 14
\(2 \times \underline{\hspace{1.2cm}} = 14\)
\(14 \div 2 = \)
2 by ?
2. Anatomy & Three Ways to Write Division
Inline Symbol
\(12 \div 2 = 6\)
"12 divided by 2"
Division Bracket
\(2 \overline{) 12} = 6\)
"2 into 12"
Fraction Bar
\(\frac{12}{2} = 6\)
"12 over 2"
Rewrite each problem in the other two formats and solve:
Given: \(16 \div 2 = \)
Bracket: Fraction:
Given: \(\frac{10}{2} = \)
Inline: Bracket:
3. Rules of Division: Dividing by 1 & Itself
Rule 1: Dividing by 1 \((n \div 1 = n)\)
A number divided by 1 stays unchanged.
Rule 2: Dividing by Itself \((n \div n = 1)\)
Any non-zero number divided by itself equals 1.
\(5 \div 1 =\)
\(6 \div 6 =\)
\(1 \overline{) 9} =\)
\(8 \overline{) 8} =\)
\(\frac{12}{1} =\)
\(\frac{15}{15} =\)
\(25 \div 25 =\)
\(100 \div 1 =\)
4. Fact Family Foundations
Write 2 multiplication and 2 division facts for each number trio:
Trio: 2, 6, 12 Family 1
\(\underline{\hspace{0.4cm}} \times \underline{\hspace{0.4cm}} = 12\)
\(12 \div \underline{\hspace{0.4cm}} = \underline{\hspace{0.4cm}}\)
\(\underline{\hspace{0.4cm}} \times \underline{\hspace{0.4cm}} = 12\)
\(12 \div \underline{\hspace{0.4cm}} = \underline{\hspace{0.4cm}}\)
Trio: 2, 8, 16 Family 2
\(\underline{\hspace{0.4cm}} \times \underline{\hspace{0.4cm}} = 16\)
\(16 \div \underline{\hspace{0.4cm}} = \underline{\hspace{0.4cm}}\)
\(\underline{\hspace{0.4cm}} \times \underline{\hspace{0.4cm}} = 16\)
\(16 \div \underline{\hspace{0.4cm}} = \underline{\hspace{0.4cm}}\)
Math-U-See Delta • Lesson 1: Introduction to Division Page 1 of 2
Delta • Lesson 1
Factor Finder: Fluency & Application
Name:
Page 2
5. Dividing by 2: Equal Shares & Halves
Think: "What times 2 equals this number?"
Solve each division fact. Notice how dividing an even number by 2 finds its exact half:
Problem 1
\(4 \div 2 =\)
Problem 2
\(6 \div 2 =\)
Problem 3
\(2 \overline{) 8}\)
Problem 4
\(2 \overline{) 10}\)
Problem 5
\(\frac{12}{2} =\)
Problem 6
\(\frac{14}{2} =\)
6. Mixed Fluency Practice
Apply your knowledge of dividing by 1, dividing by itself, and dividing by 2:
1. \(18 \div 2 =\)
2. \(7 \div 1 =\)
3. \(9 \div 9 =\)
4. \(20 \div 2 =\)
5. \(2 \overline{) 16} =\)
6. \(1 \overline{) 14} =\)
7. \(13 \overline{) 13} =\)
8. \(2 \overline{) 2} =\)
7. Real-World Story Problems
Write the division equation, show your thinking, and write your answer:
Problem A: Sharing Math Blocks
Maya has 16 blue unit blocks. She divides them equally between herself and her brother. How many blocks does each get?
Show work / equation:
Blocks each:
Problem B: The Garden Rectangle
Leo builds a rectangular garden bed with a total area of 18 square feet. If the width is 2 feet, what is the length?
Show work / equation:
Garden length:
8. Deep Thinker: Why Can't We Divide by Zero?
Think about our rectangle model. If someone asks for \(6 \div 0 = \underline{\hspace{0.5cm}}\), they are asking: "What number times 0 equals 6?"
Explain why no number can solve this problem:
Math-U-See Delta • Lesson 1: Introduction to Division Page 2 of 2
Decade Decoders Worksheet
Delta • Lesson 2 Enrichment Lab Advanced Base-10 Analysis
Decade Decoders Worksheet
Base-10 digit shifts, algebraic unknowns, and algorithmic mental math
Investigator:
Date:
1. The Place-Value Shift Engine
Decomposition: \((A + B) \div 10 = (A \div 10) + (B \div 10)\)
Dividing by 10 shifts every digit exactly one place value to the right. Observe how expanded form proves this rule: \(360 \div 10 = (300 \div 10) + (60 \div 10) = 30 + 6 = 36\).
Case A: \(580 \div 10\)
Expand: \((500 \div 10) + (80 \div 10)\)
= + =
Case B: \(1,420 \div 10\)
Expand: \((1400 \div 10) + (20 \div 10)\)
= + =
Case C: Multiple Shifts
Compute double decade shift:
\((7,400 \div 10) \div 10 = \)
2. Remainder Forensics: Non-Multiples of 10
Because our number system is base-10, dividing any integer by 10 produces a quotient formed by all higher digits and a remainder equal to the ones digit!
a. \(87 \div 10\)
Quotient:
Remainder:
b. \(349 \div 10\)
Quotient:
Remainder:
c. \(1,206 \div 10\)
Quotient:
Remainder:
d. Grand Challenge
Remainder of:
\(84,937,251 \div 10\)
R =
3. The Decade Hack: Rapid Mental Division
Mathematicians use 10 as a stepping stone. To divide by 5, divide by 10 first, then double the result (\(N \div 5 = (N \div 10) \times 2\)). To divide by 20, divide by 10 first, then halve the result!
Strategy A: Divide by 5 via Decade Double
Problem: \(140 \div 5\) \((140 \div 10) \times 2 = 14 \times 2 = \mathbf{28}\)
1. \(230 \div 5 =\) \((230 \div 10) \times 2 =\)
2. \(410 \div 5 =\) \((410 \div 10) \times 2 =\)
Strategy B: Divide by 20 via Decade Halving
Problem: \(160 \div 20\) \((160 \div 10) \div 2 = 16 \div 2 = \mathbf{8}\)
1. \(280 \div 20 =\) \((280 \div 10) \div 2 =\)
2. \(340 \div 20 =\) \((340 \div 10) \div 2 =\)
4. Algebraic Balances: Solving for Unknowns
Determine the value of each variable using inverse operations and balance logic:
Equation 1
\(10 \times z = 240 - 60\)
\(z =\)
Equation 2
