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: Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
problem_activity: Narrative
The purpose of this activity is for students draw and shade rectangles with a unit fraction side length and a whole number side length. Then they find the areas of the shaded regions. The tactile experience of drawing and shading encourages students to count the number of shaded parts and then either reason about their size or think about moving them to make full unit squares. They also consider a diagram where not all of the unit squares are shown. Students estimate how many of the unit squares are hidden. This helps to highlight that finding the total area can be done with multiplication where one factor is the area of each shaded part and the other factor is the total number of shaded parts (MP7).
Required Materials
Materials to Copy
Grid Paper 5
Launch
Groups of 2
Give students grid paper.
Activity
5–7 minutes: independent work time
5 minutes: partner work time
Monitor for students who:
use the grid structure on the paper to draw their rectangles
count the number of unit squares that might be hidden under the yellow rectangle
Student Facing
Represent each rectangle on grid paper:
\(\frac{1}{2}\)
unit by 1 unit
\(\frac{1}{2}\)
unit by 2 units
\(\frac{1}{2}\)
unit by 3 units
\(\frac{1}{2}\)
unit by 4 units
Find the area of each rectangle that you drew.
What information do you need to find the area of the shaded region?
[IMG]
What might the area of the shaded region be? Explain or show your reasoning.
Student Response
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click here to register or sign in
for free access to Student Response.
Advancing Student Thinking
If students do not draw the rectangles correctly, show them the shaded region from the previous activity with side lengths 6 units and
\(\frac{1}{2}\)
unit and ask “How could you adapt this diagram to show a rectangle that is 4 units by
\(\frac{1}{2}\)
unit?”
Activity Synthesis
Display a student generated image of the
\(\frac{1}{2}\)
unit by 4 unit rectangle.
Ask previously selected students to describe how they drew they 4 by
\(\frac{1}{2}\)
rectangle.
Display image from student workbook of the rectangle that is partly covered.
“What do you need to know to determine the area of the shaded region?” (We need to know how many unit squares are under the yellow rectangle.)
“How many unit squares do you think make up the rectangle?” (6 or 7)
“How did you use the number of these unit squares to make an estimate for the shaded region?” (I know that each shaded blue rectangle is
\(\frac{1}{2}\)
of a square so if there are 6 or 7 of those, that would be
\(\frac{6}{2}\)
or
\(\frac{7}{2}\)
square units.)


aligns: no