Part 1: Justifications for each step

Let's break down each step and identify the property of whole numbers used.

Step (a):

$43 + 33 = (4 \times 10 + 3) + (3 \times 10 + 3)$

• Property: This is expanded notation, and no property is directly applied here, just a rewriting of the numbers.

Step (b):

$(4 \times 10 + 3) + (3 \times 10 + 3) = 4 \times 10 + [3 + (3 \times 10 + 3)]$

• Property: Associative Property of Addition. The parentheses are rearranged, but the numbers inside the parentheses remain the same. This property allows the grouping of terms to be changed without affecting the sum.

Step (c):

[ 4 \times 10 + [3 + (3 \times 10)] + 3 = 4 \times 10 + [ (3 + 3 \times 10) + 3 ] ]

• Property: Commutative Property of Addition. The terms $3 + 3 \times 10$ are reordered in the parentheses without changing their sum.

Step (d):

[ 4 \times 10 + [ (3 \times 10 + 3) + 3 ] = 4 \times 10 + [ 3 \times 10 + (3 + 3) ] ]

• Property: Commutative Property of Addition. Again, the order of addition is rearranged within the parentheses.

Step (e):

$4 \times 10 + [ 3 \times 10 + (3 + 3) ] = (4 \times 10 + 3 \times 10) + (3 + 3)$

• Property: Distributive Property of Multiplication. The multiplication is distributed across addition, combining the multiples of 10.

Step (f):

$(4 \times 10 + 3 \times 10) + (3 + 3) = (4 + 3) \times 10 + (3 + 3)$

• Property: Associative Property of Multiplication. The numbers in the multiplication expression are grouped together, simplifying $4 \times 10 + 3 \times 10$ as $(4 + 3) \times 10$.

Step (g):

$(4 + 3) \times 10 + (3 + 3) = 7 \times 10 + 6$

• Property: Simplification. This is simply simplifying the expression $4 + 3 = 7$ and $3 + 3 = 6$.

Step (h):

$7 \times 10 + 6 = 76$

• Property: Simplification. The final step just involves multiplying and adding to get the result.

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Summary of Properties:

• (a) No property applied (just expanded notation)
• (b) Associative Property of Addition
• (c) Commutative Property of Addition
• (d) Commutative Property of Addition
• (e) Distributive Property of Multiplication
• (f) Associative Property of Multiplication
• (g) Simplification
• (h) Simplification

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Part 2: 5 Related Questions for Further Understanding

1. Can you explain how the Distributive Property works with different operations (like subtraction or division)?
2. How does the Commutative Property apply to multiplication?
3. What is an example where the Associative Property of Multiplication helps simplify a problem?
4. How does the Additive Identity Property of zero apply in algebraic equations?
5. Can you demonstrate the Distributive Property in a word problem involving money or shopping?

Tip:

When you rearrange numbers or change how they are grouped in addition or multiplication, always check if the property you're using changes the result (it shouldn't!).