few_shots: 
Example 1:
Concept/skill: 
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

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: 
Yes. This problem solves a real-world problem involving mixing ingredients for cookies. A student doing this problem would need to reason about equivalent ratios of butter amounts to sugar amounts.

Example 2:
Concept/skill: 
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

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: 
Yes. This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).

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_pos_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.
problem_activity: Narrative
The purpose of this activity is for students to apply conservation of length by measuring with a tool that does not start at zero. Until now, students have focused on measuring by lining up the edge of an object with zero. In this activity, students consider how the units on a ruler, and in later lessons, a number line, can help us solve problems and measure even when we are not starting at zero. It is important for students to recognize that the tick marks show a consistent distance between numbers, so any part of the ruler can show a specific number of units. The synthesis focuses on how addition (counting on from the starting number) and subtraction can be used to find the length of an object. The teacher records students’ thinking with an equation with a question mark for the unknown.
It is recommended that students use rulers to measure the length of smaller objects (2
–
10 inches) in this activity. This smaller number range helps students connect what they notice about measuring starting at different numbers on the ruler to familiar addition and subtraction equations within 20. However, this activity can be adapted to use measuring tapes or yardsticks to measure longer objects if they are available.
When students measure objects starting at different non-zero spots on the ruler and observe what stays the same (the difference) and what changes (the starting and ending spots) they observe a structure which repeats itself no matter which spot they on the ruler they begin the measurement (MP7, MP8).
Action and Expression: Develop Expression and Communication
. Give students access to inch tiles to double check their measurement. Reiterate how the measurement is the same regardless of the whole inch they start at on the ruler.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing
Required Materials
Materials to Gather
Objects of various lengths
Rulers (inches)
Required Preparation
Each group of 2 needs access to several objects between 2
–10 inches long. Consider using classroom objects such as markers, colored pencils, paperclips, or other objects or images used in previous activities.
Launch
Groups of 2
Give each student a ruler.
Display the image.
Activity
“Han measured the same notebook as Jada. Jada’s measuring tape was torn, but she got the same answer. How did that happen?”
1 minute: quiet think time
“Work with a partner to figure out how Jada got the same answer as Han. Be prepared to explain your thinking.”
3–4 minutes: partner discussion
Monitor for a student who explains the problem as
\(10 - 2 = {?}\)
and one who explains the problem as
\(2 + {?} = 10\)
.
Share responses.
“Now, using your ruler, you and your partner should find 1 thing in the classroom to measure without starting at zero. Each of you should start at a different number and compare your measurements.”
“Don’t forget to include the unit with your measurement and to use a question mark for the unknown in your equation.”
5 minutes: partner work time
Monitor for students who discuss:
that the length must be the same because the object doesn’t change
the equations they write have the same value or show the same relationship
the relationship between the numbers in their different expressions (for example, “the numbers I subtracted are two more than yours because I moved the object 2 inches over on the ruler. The difference is the same.”)
Student Facing
Jada and Han used an inch ruler to measure the short side of a notebook.
[IMG]
Han says 8 inches.
[IMG]
Jada says 8 inches.
How did Han and Jada get the same measurement?
Write an equation that could show Jada’s thinking.
Measure an object using Jada’s method.
I measured a
________________________________
.
I started with the number
____________
.
I ended with the number
____________
.
Equation: ________________________________.
The length of my object is
____________
.
What do you notice about you and your partner’s measurements?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Advancing Student Thinking
Students may find the correct length when counting length units on the measuring tool, but identify the length as the total number if they represent their measurement with an addition equation. (For example, they count on from 2 inches to 10 inches on the ruler and write
\(2 + 8 = 10\)
, but say the length of the object is 10). Consider asking:
“How did you measure the ___?”
“What do the numbers in your equation represent?”
Activity Synthesis
Invite selected students to share their measurements and their reasoning.
Write equations that correspond to the students’ thinking using a question mark to represent the unknown value.
“Why do you think you get the same measurement each time, even if you measure starting from a different place?” (The object didn't change, so the length has to stay the same. The total length in inches is still the same between the number that you start with and end with.)


aligns: no