Question

properties and order of operations diagnostic assessment
for each situation, two expressions are set equal to one another. determine if the
situation is true or false.

\\(3(6 + 4) + 5 = 4(5 + 3) + 6\\)

ask students: is it true or false? how do you know?

\\(1 + (23 + 24) = (1 + 23) + 24\\)

is it true or false? how do you know?

\\((4)(6) + 6 = 6(6 + 4)\\)

is it true or false? how do you know?

looking at these three situations, what do you notice? what do you wonder?

Explanation:

Step1: Evaluate left side of first equation

Calculate parentheses first, then multiply, then add:
$3(6+4)+5 = 3(10)+5 = 30+5 = 35$

Step2: Evaluate right side of first equation

Calculate parentheses first, then multiply, then add:
$4(5+3)+6 = 4(8)+6 = 32+6 = 38$

Step3: Compare first equation results

$35
eq 38$, so the first statement is false.

Step4: Evaluate left side of second equation

Calculate inner parentheses first, then add:
$1+(23+24) = 1+47 = 48$

Step5: Evaluate right side of second equation

Calculate inner parentheses first, then add:
$(1+23)+24 = 24+24 = 48$

Step6: Compare second equation results

$48 = 48$, so the second statement is true.

Step7: Evaluate left side of third equation

Multiply first, then add:
$(4)(6)+6 = 24+6 = 30$

Step8: Evaluate right side of third equation

Calculate parentheses first, then multiply:
$6(6+4) = 6(10) = 60$

Step9: Compare third equation results

$30
eq 60$, so the third statement is false.

Step10: Analyze patterns/observations

The second equation follows the associative property of addition, which guarantees equality. The other equations do not follow valid arithmetic properties, leading to unequal results.

Answer:

1. $3(6 + 4) + 5 = 4(5 + 3) + 6$: False, since the left side equals 35 and the right side equals 38.
2. $1 + (23 + 24) = (1 + 23) + 24$: True, since both sides equal 48 (this follows the associative property of addition, which states $(a+b)+c = a+(b+c)$).
3. $(4)(6) + 6 = 6(6 + 4)$: False, since the left side equals 30 and the right side equals 60.

Observations:

• Equations that follow a valid arithmetic property (like the associative property of addition) are true, while those that do not apply proper properties or order of operations are false.
• The order of operations (parentheses first, then multiplication/division, then addition/subtraction) is critical to correctly evaluating each side of the equations.

Questions to wonder:

• How can we rearrange the false equations to make them true?
• What other arithmetic properties can we use to quickly verify if similar equations are true or false?