standard_description: Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) ^( )Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)
problem_activity: Narrative
The purpose of this warm-up is to draw students’ attention to the first few multiples of 45, which will be helpful as students continue to work with benchmark angles and use a protractor to measure angles. Students have the skills to perform the multiplication in each equation, but computing each product may be time-consuming. Students can more efficiently tell if the equations are true or false if they consider properties of operations and look for and make use of structure.
Launch
Display one equation.
“Give me a signal when you know whether the equation is true and can explain how you know.”
1 minute: quiet think time
Activity
Share and record answers and strategy.
Repeat with each statement.
Student Facing
Decide if each statement is true or false. Be prepared to explain your reasoning.
\(2 \times 45 = 6 \times 15\)
\(4 \times 45 = 2 \times 90\)
\(3 \times 45 = 180 - 90\)
\(6 \times 45 = 45 + 90 + 135\)
Student Response
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Activity Synthesis
Some students may notice that it is handy to think in terms of
\(2 \times 45\)
because it would mean dealing with multiples of 90 rather than multiples of 45. Highlight their explanations.
If no students decomposed expressions such as
\(3 \times 45\)
,
\(4 \times 45\)
, and
\(6 \times 45\)
into sums of
\(1 \times 45\)
and
\(2 \times 45\)
, discuss how this could be done. (See sample responses.)


aligns: no