3. Does Order Matter in Mathematics?
When order doesn’t matter:
Commutative Property of Multiplication
3 • 7 = 21
7 • 3 = 21
Commutative Property of Addition
3+5=8
5+3=8
Changing the
order yields a
different answer:
7 – 5 = +2
5–7=-2
10 ÷ 5 = 2
5 ÷ 10 = .5
4. The Fair
Ms. Delores is taking her children and grandchildren
to the fair. Tickets to get in cost $4 each. She paid for
5 grandsons, 2 daughters, and herself.
• How much did Ms. Delores spend on her
grandchildren’s tickets? Write an equation and
show or explain how you solved the equation.
• How much did Ms. Delores spend on adult
tickets? Write an equation and show or explain
how you solved the equation.
• Using the two equations you already wrote, write
a new equation that gives the total amount Ms.
Delores paid for tickets to go into the fair.
5. Weekend Laundry
I’m doing laundry. I discover $4 in
each of three girls’ pockets. The
next day, I find $5 in each of 2
different pockets in my husband’s
clothes.
3•4+2•5=
6. Pay-Day Shopping Splurge
15 + 2(7) + 10
Daijah-Tae went shopping. She
bought a scarf for $15, two lip
glosses for $7 each, and a cell
phone case for $10.
7. Buffet
Hot food is $7 per pound and
salads are $4.50 per pound. You
buy 5 pounds of hot food, 2
pounds from the salad bar, and
three waters that are $1 each.
5 • 7 + 2 • 4.5 + 3 • 1
Warm-UpActivity (done upon entering): With a partner, put the steps for getting to work in order.Bring everyone together. Ask for a volunteer to read the order of their steps for getting to work. “Sometimes the order is just logical.”“In this real-life example, the order in which we perform operations matters.”Sometimes activities have steps where the order does not matter. “What steps can I add in to this sequence where order would not matter?” Ideas: If I put on my seatbelt before I start the engine or I put on my seatbelt after I start the engine, the result is the same – I have my seatbelt on before I start driving. If I bring my coffee from home or I stop for coffee, I still have coffee before I get to work.(click)
“Name a time in math where order doesn’t matter.”“Name a time in math where order does matter.”If students are able to describe a situation but not able to identify the specific property, provide the proper name for the property. If students are not ready to answer this question, revisit this slide after exploring some more specific examples.
Read the situation. Allow time for students to answer the first problem. Insist that students show visually or explain how they know that their answer works. Click to show the second problem. Afterwards, discuss whether it mattered if you multiplied the cost of the tickets by the number of grandsons or the number of grandsons by the cost of the tickets. Click to show the third question that asks for an equation to show the total Ms. Delores paid for fair entrance. If students answer 5 + 3 x 4, draw their attention back to their two sets of drawings. “You’ve got 3 adults multiplied by four dollars with 5 more people added in. The cost is not figured for those five people.”Because of the two equations they just wrote, students will likely answer in this format: 4 • 5 + 4 • 3. Challenge students to write the full equation in another way. If students don’t think of it, guide them to noticing that the 4 is in both original equations and therefore using the distributive property, 4(5 + 3) could be written to express that situation. Show that these two expressions are equal. 4(5 + 3) = 4 • 5 + 4 • 3. Look back at the expression 4(5 + 3). “Do you have to start solving that with what is in the parenthesis?” Show the distributive property in action. Mention that it’s important for algebra success that they be flexible and see that there are two ways to look at this problem.
Let’s explore some other examples and decide if order matters.“What is the answer to this question?” 3 • 4 + 2 • 5 =Let students work through possibilities.“How many different answers to this question can there be? Can more than one answer be correct here?”“Let’s look at the context of this question.” (click) allow time for students to calculate a new answerDiscuss“So, did the order matter here?”6.EE.2c “Include expressions that arise from formulas used in real-world problems.”
“Write an expression that shows how to solve this. I’m not interested in the answer, but instead, I want to see how you could set up to solve this all in one expression.”“Does the order matter?”“What would you change about this shopping trip to make this an example where order does not matter?” (possible answer: buy only one lip gloss, add the cost of each lip gloss rather than multiply the cost of one by two glosses)
Read the equation contextually, “Five sets of 7 added to 2 sets of 4 and 5 tenths added to 3 sets of 1.”“Do you think that the order in which we solve this is going to matter?”(click) to give situation“Does the order matter?”
To figure out payroll costs, what would I need to do?Would order matter? In which parts would order matter?
