few_shots: 
Example 1:
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?


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.

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:
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
$$-\frac{1}{4}$$


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

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:
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?


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.

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, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_000972_pos_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
- An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.
- An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
problem_activity: Narrative
The purpose of this activity is for students to compare shapes by covering them with pattern blocks. Students experience tiling as a way to see which shape covers the most space. There are several ways to tile the shapes, but it may prove most useful to use the same units, such as triangles. The rectangle can only be fully tiled with square pattern blocks. To compare shapes B and C, students will need to notice that the rectangle and parallelogram can be made the same length, but the square pattern blocks used to tile the rectangle are taller than the blocks used to tile the parallelogram.
[IMG]
[IMG]
The work here prepares students to tile figures with square tiles in the next lesson and to think of area in terms of square units.
MLR8 Discussion Supports.
Synthesis: Create a visual display of shapes A, B, and C. As students discuss their comparisons, annotate the display with their observations. For example, when comparing shape B to C, write “same length, but taller.”
Advances: Speaking, Listening
Action and Expression: Develop Expression and Communication.
Synthesis: Identify connections between strategies that result in the same outcomes but use differing approaches.
Supports accessibility for: Conceptual Processing
Required Materials
Materials to Gather
Pattern blocks
Scissors
Materials to Copy
Pattern Blocks to Compare Shapes
Required Preparation
Each group of 2 needs at least 2 hexagons and trapezoids,  4 squares and rhombuses, and 8 triangles.
Launch
Groups of 2
Give each student a copy of the blackline master.
Give each group pattern blocks.
Display the first two shapes (the hexagon and the heart-shaped octagon).
“What do you notice? What do you wonder?” (Students may notice: Both shapes could be made with pattern blocks. The second shape looks like the first shape, but it’s been bent. Students may wonder: Which shape covers more space? Which shape is larger?)
1 minute: quiet think time
Share and record responses.
“Which shape covers more space? How do you know?” (We can cover them with the same blocks, so they cover the same amount of space.)
2 minutes: partner discussion
Share and record responses.
Activity
“Let's continue comparing shapes. Work with your partner to decide which of these shapes covers the most space. Use pattern blocks if they are helpful.”
5 minutes: partner work time
Monitor for students who:
use pattern blocks to determine which shape covers the most space
notice that shape B and C would be the same length if one of the ends of B was moved over, but C covers more space because it is taller
Student Facing
Your teacher will give you handouts with some shapes on them.
Use pattern blocks to decide which shape covers the most space. Be ready to explain your reasoning.
[IMG]
Student Response
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click here to register or sign in
for free access to Student Response.
Activity Synthesis
Ask selected students to share strategies for tiling each shape with pattern blocks and how the tiling helped them decide which shape covers the most space.
“When we decide which shape covers the most space, we are talking about the area of a shape. We can think about the
area as the amount of space covered by a shape.”
“Which of these shapes has the greatest area?” (C takes up the most space, so it has the greatest area.)


aligns: no