HavenHub Math • Edition 3
Unit 3 Workbook: The Swap
The Scribe's Vocabulary Bank
Commutative Law: The rule that lets factors switch seats without changing the total ($3 \times 4 = 4 \times 3$).
Associative Law: The rule that lets you group 3 factors in different ways using parentheses ().
Rotation: Turning an array sideways to see it from a different perspective.
Triad: A family of three numbers bonded by multiplication (e.g., 2, 5, 10).
Invariant: The product, which stays the same even when the factors move.
Perspective: Choosing the view that makes the math easier for your brain.
Lesson 3.1: Turning the Chocolate Bar (Rotation)
Scribe's Hint: When you rotate an array, you aren't changing the dots! You are just changing where you are standing. 2 rows of 5 is the same as 5 rows of 2. Rotation proves the truth is constant!
Part 1: The Rotate Test
Write the new multiplication sentence after rotating the array.
1.
2 rows of 4 $\implies$ ____ rows of ____
2.
3 rows of 5 $\implies$ ____ rows of ____
3.
1 row of 10 $\implies$ ____ rows of ____
4.
4 rows of 6 $\implies$ ____ rows of ____
5.
Draw It: Draw an array for $3 \times 2$. Then rotate it and draw the new shape.
Part 2: The Constancy of the Product
Does the total change when you rotate? Write the Product for both.
6.
$2 \times 5 =$ ____ and $5 \times 2 =$ ____
7.
$3 \times 1 =$ ____ and $1 \times 3 =$ ____
8.
$4 \times 3 =$ ____ and $3 \times 4 =$ ____
9.
$2 \times 8 =$ ____ and $8 \times 2 =$ ____
10.
Master Thinking: If an array is a perfect Square ($3 \times 3$), does it look different when you rotate it?
Verification Node: The Law of Perspective
Scenario: One scribe says, "I see 2 groups of 6." Another scribe stands across the table and says, "No, I see 6 groups of 2."
The Truth: Who is lying? Can both be telling the truth at the same time? Explain how the Swap Rule brings peace to their argument.
Lesson 3.2: Switching Seats (The Commutative Law)
Scribe's Hint: In multiplication, order does not matter! $A \times B = B \times A$. Think of it like two friends switching chairs—they are still the same two friends!
Part 1: The Fast Switch
Don't solve! Just fill in the missing twin using the Swap Rule.
1.
$4 \times 5 = 5 \times$ ____
2.
$2 \times 9 = $ ____ $\times 2$
3.
$10 \times 3 = 3 \times$ ____
4.
$6 \times 7 = $ ____ $\times 6$
5.
$1 \times 100 = 100 \times$ ____
Part 2: Relationship Detective
Write the "Handshake" fact for each problem.
6.
$3 \times 8 \implies$ ____ $\times$ ____
7.
$2 \times 4 \implies$ ____ $\times$ ____
8.
$5 \times 9 \implies$ ____ $\times$ ____
9.
$10 \times 10 \implies$ ____ $\times$ ____ (Wait!)
10.
Critical Check: Can you use the Swap Rule for $10 - 2$? Why or why not?
11.
$6 \times 2 = 2 \times$ ____
12.
$1 \times 10 = 10 \times$ ____
Lesson 3.3: Half the Work (The Twin Hunt)
Scribe's Hint: Every multiplication fact has a mirror twin! If you know 6 times 2 is 12, you already know 2 times 6 is 12. You only have to learn half of the table!
Part 1: Mirror Match
Find the mirror twin for each fact and solve them both.
1.
$2 \times 7 =$ ____ $\implies 7 \times 2 =$ ____
2.
$3 \times 10 =$ ____ $\implies 10 \times 3 =$ ____
3.
$4 \times 5 =$ ____ $\implies 5 \times 4 =$ ____
4.
$2 \times 9 =$ ____ $\implies 9 \times 2 =$ ____
5.
$3 \times 8 =$ ____ $\implies 8 \times 3 =$ ____
Part 2: The Scribe's Shortcut
Which fact is easier for your brain? Circle it and solve.
6.
$2 \times 10$ OR $10 \times 2$ $\implies$ Product: ____
7.
$5 \times 2$ OR $2 \times 5$ $\implies$ Product: ____
8.
$1 \times 9$ OR $9 \times 1$ $\implies$ Product: ____
9.
$3 \times 2$ OR $2 \times 3$ $\implies$ Product: ____
10.
Mental Sovereignty: If you learn 10 new facts today, how many do you actually know because of the Swap? ____
Lesson 3.4: The Party Grouping (Associative Law)
Scribe's Hint: When you have 3 numbers, you get to choose which two "Dance" first! Put parentheses ( ) around the easy partners. It will make the whole party faster!
Part 1: The Dance Partners
Solve the numbers in the brackets first.
1.
$( 2 \times 3 ) \times 2 \implies $ ____ $\times 2 = $ ____
2.
$2 \times ( 3 \times 2 ) \implies 2 \times$ ____ $=$ ____
3.
$( 5 \times 2 ) \times 4 \implies $ ____ $\times 4 = $ ____
4.
$5 \times ( 2 \times 4 ) \implies 5 \times$ ____ $=$ ____
5.
$( 10 \times 1 ) \times 3 \implies $ ____ $\times 3 = $ ____
Part 2: Choosing Wisely
Circle the partnership that is easier to solve first.
6.
$( 2 \times 5 ) \times 7$ OR $2 \times ( 5 \times 7 )$
7.
$( 3 \times 4 ) \times 2$ OR $3 \times ( 4 \times 2 )$
8.
$( 10 \times 2 ) \times 3$ OR $10 \times ( 2 \times 3 )$
9.
Draw It: Draw 2 groups of (2 groups of 3). Count the total.
10.
Logic Check: Does the Total Product change when the partners switch?
Lesson 3.5: Multiplication Triads (Fact Families)
Scribe's Hint: These numbers are bonded for life! If you know the two kids (factors), you know the father (product). If the father and one kid are home, you can always find the missing member!
Part 1: Completing the Cord
Fill in the missing family member in each triangle house.
Part 2: The Four Stories
Write the two multiplication stories for each triad.
7.
Family: 2, 8, 16 $\implies$ ____ $\times$ ____ $=$ 16 AND ____ $\times$ ____ $=$ 16
8.
Family: 5, 3, 15 $\implies$ ____ $\times$ ____ $=$ 15 AND ____ $\times$ ____ $=$ 15
9.
Family: 10, 6, 60 $\implies$ ____ $\times$ ____ $=$ 60 AND ____ $\times$ ____ $=$ 60
10.
Final Check: If you know $4 \times 9 = 36$, what is $36 \div 4$? (Wait, that's next Edition!). What is $9 \times 4$?
Parent Mentor Guide: The Power of Perspective
The Goal: This unit introduces the Commutative Property ($A \times B = B \times A$) and the Associative Property ($(A \times B) \times C = A \times (B \times C)$). These laws allow the student to manipulate problems to find the easiest solution. This is a massive shift from Edition 1!
Key Habits to Watch:
- Strategy: Encourage the student to swap factors if one way is easier (e.g., $2 \times 9$ is easier as $9 \times 2$). This builds mental flexibility and confidence.
- Grouping: When multiplying 3 numbers, help them look for "Ten-Makers" ($2 \times 5$, $1 \times 10$) to simplify the work. This is the root of mental math sovereignty.
- Triads: Ensure they see the relationship between the 3 numbers in a family. This prevents them from seeing multiplication as random facts. If they know 3 and 4 make 12, they have found a friend for life.
- Rotation: Physically rotate the workbook or objects to show that the total headcount is an Invariant—it stays the same no matter the view.
Spiritual Connection:
The Swap is a lesson in Empathy and Perspective. Being able to see the same truth from a different side is a vital skill for a citizen of the Kingdom. Truth is constant, but our view can change! Remind them that God sees the whole array at once. He is the God of all perspectives.
The Scribe's Final Vow of Perspective
"I, ____________________________, a Navigator of the One Truth, promise to look at my numbers from every side. I will use the Swap to find the Light, honor the Family bond, and walk in the peace of the perfect Balance. Hallelujah!"
Student Navigator Signature
UNIT 3 COMPLETE!
You have mastered the Art of the Swap.
GLORIA IN EXCELSIS DEO!
Navigator's Mirror Certificate
This honor is awarded to
For mastering the Sovereignty of Perspective
Commutative Law, Associative Law, and Fact Families.
You are now ready for The Laws of 1 and 0.
Appendix: The Associative Dance Floor
Choose the best partners to dance (multiply) first. Circle them and find the Product!
| Three Factors |
Step 1 (First Dance) |
Step 2 (Final Sum) |
| $2 \times 5 \times 3$ | $10 \times 3$ | 30 |
| $3 \times 2 \times 2$ | | |
| $5 \times 2 \times 4$ | | |
| $4 \times 1 \times 10$ | | |
| $2 \times 3 \times 2$ | | |
| $10 \times 2 \times 2$ | | |
| $3 \times 3 \times 1$ | | |
| $2 \times 2 \times 5$ | | |
| $4 \times 2 \times 5$ | | |
| $1 \times 10 \times 9$ | | |
| $2 \times 3 \times 1$ | | |
| $5 \times 2 \times 5$ | | |
| $10 \times 1 \times 10$ | | |
| $3 \times 2 \times 10$ | | |
| $4 \times 2 \times 2$ | | |
| $2 \times 2 \times 2 \times 2$ | | |