few_shots: 
Example 1:
Concept/skill: 
Recognize and represent proportional relationships between quantities.

Problem/activity:
You’re mixing ingredients for cookies. The recipe says to combine 6 tablespoons, or ¾ stick, of butter with 1 cup of sugar. You accidentally mix in a whole stick of butter (8 tablespoons) with the cup of sugar. How can you fix this?


Answer: 
No. Usually problems that teach recognizing and representing proportional relationships between quantities involve deciding whether two quantities are in a proportional relationship or identifying the constant of proportionality, and then representing proportional relationships by equations or explaining proportional relationship using coordinate graphs.

Example 2:
Concept/skill: 
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Answer: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

Example 3:
Concept/skill: 
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


Answer: 
No. Completing neither of these problems requires replacing one equation by the sum of that equation and a multiple of the other equation.

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_000972_neg_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_neg_0

8
standard_description: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
problem_activity: Optional activity
This optional activity is an opportunity to practice the methods in this lesson to calculate products of decimals, and students have an opportunity to practice multiplying decimals in a real-world context. Students can choose to use area diagrams to help organize their work and support their reasoning. However, the goal of the activity is to have students practice using the multiplication algorithm on decimals.
Launch
Give students quiet think time to complete the activity and then time to share their explanation with a partner. Follow with whole-class discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge. Provide students with access to a blank area diagram to support information processing.
Supports accessibility for: Visual-spatial processing; Organization
Student Facing
Calculate each product. Show your reasoning. If you get stuck, consider drawing an area diagram to help.
\((5.6) \boldcdot (1.8)\)
\((0.008) \boldcdot (7.2)\)
A rectangular playground is 18.2 meters by 12.75 meters.
Find its area in square meters. Show your reasoning.
If 1 meter is approximately 3.28 feet, what are the approximate side lengths of the playground in feet? Show your reasoning.
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?
Write the following expressions as decimals.
\(1-0.1\)
\(1-0.1+10-0.01\)
\(1-0.1+10-0.01+100-0.001\)
Describe the decimal that results as this process continues.
What would happen to the decimal if all of the addition and subtraction symbols became multiplication symbols? Explain your reasoning.
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
Select students to share their strategies, being sure to highlight approaches using area diagrams and vertical calculations. Record the representations or strategies students share and display them for all to see.
Representing, Conversing: MLR8 Discussion Supports.
Use this routine help students prepare to share their strategies with the class. Give students time to discuss how they calculated each product for the first problem with a partner. Consider using this prompt to begin the conversation: “How did you use the multiplication algorithm to calculate the product of these two decimals? Provide a step-by-step explanation.” The listener should press for details by asking clarifying questions such as, “Why did you do that first?” and “Could you explain that a different way?” Allow each student an opportunity as the speaker and listener. This will help students build confidence in explaining the process of multiplying decimals before students are selected in the whole-class discussion.
Design Principle(s): Support sense-making; Cultivate conversation


aligns: no