few_shots: 
Example:
Concept/skill: 
Fluently multiply multi-digit whole numbers using the standard algorithm.

Problem/activity:
Problem 1
Solve using the standard algorithm. If you get stuck, use an area model and/or the partial products algorithm to help. Then assess the reasonableness of your answer.
a.
$$5,803\times46$$
b.
$$63\times2,116$$
Problem 2
Dunkin' introduced a croissant doughnut in 2014. In the first three months of its debut, they sold 7,870 dozen of them per day. How many croissant doughnuts were sold in the first three months?


Answer: 
Yes. This set of problems involves multi-digit whole numbers because the numbers in each problem are four and two digit integers, and the mathematical calculations involved in both problems focus on multiplication. The first problem explicitly asks the student to solve its subproblems using the standard algorithm.

Now, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: fl_problem_000674_pos_0

5
standard_description: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
problem_activity: Activity
Students learn that we can still organize our work with the distributive property in a familiar way, even with negative numbers where thinking in terms of area breaks down.
Launch
Display the image and ask students to write an expression for the area of the big rectangle in at least 3 different ways.
[IMG]
Collect responses. If students simply say “16,” ask them to explain how they calculated 16 and record these processes for all to see. Remind students that thinking about area gives us a way to understand the distributive property. This diagram can be used to show that
\(2\boldcdot 5+2\boldcdot 3=2(5+3)\)
. Be sure that students see you write the partial products in the diagram, and that they see every piece of the associated identity
\(2\boldcdot 5+2\boldcdot 3=2(5+3)\)
.
[IMG]
Tell students that when we are working with negative numbers, thinking about area doesn’t work so well, but the distributive property still holds when there are negative numbers. The expressions involved still have the same structure, and we can still organize our work the same way.
Representation: Internalize Comprehension.
Differentiate the degree of difficulty or complexity by beginning with an example with more accessible values, such as the one given. Highlight connections between representations by recording the calculations in the boxes. In addition, consider creating a display showing a general example using only variables and keeping it as a reference throughout the remainder of the unit.
Supports accessibility for: Conceptual processing
Student Facing
Write two expressions for the area of the big rectangle.
[IMG]
Use the distributive property to write an expression that is equivalent to
\(\frac12(8y + \text-x + \text-12)\)
. The boxes can help you organize your work.
[IMG]
Use the distributive property to write an expression that is equivalent to
\(\frac12(8y - x - 12)\)
.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Here is a calendar for April 2017.
[IMG]
Let's choose a date: the 10th. Look at the numbers above, below, and to either side of the 10th: 3, 17, 9, 11.
Average these four numbers. What do you notice?
Choose a different date that is in a location where it has a date above, below, and to either side. Average these four numbers. What do you notice?
Explain why the same thing will happen for any date in a location where it has a date above, below, and to either side.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Solicit responses to the second question and demonstrate thinking about one product at a time:
[IMG]
Then ask students to share how they approached the last question. Highlight responses where students noticed that
\(\frac12(8y - x - 12)\)
can be rewritten like
\(\frac12(8y + -x + -12)\)
(because of what they talked about in the warm-up). So the two questions have the same answer.
Speaking, Representing: MLR7 Compare and Connect.
Use this routine when students present their expressions. Ask students “What is the same and what is different?” about the approaches and representations involving subtraction with the distributive property. Help students connect how the expressions that have a subtraction operation are equivalent to expressions that add its additive inverse. These exchanges strengthen students’ mathematical language use and reasoning with the distributive property and the subtraction operation.
Design Principle(s): Maximize meta-awareness


aligns: no