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?


Standard options: 
A. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
B. 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.
C. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade are limited to non-complex fractions.)


Thought: 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.
Answer: B

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


Standard options: 
A. Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
B. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
C. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.


Thought: 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).
Answer: B

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?


Standard options: 
A. Solve linear equations in one variable.
B. Analyze and solve pairs of simultaneous linear equations.


Thought: The last three rows of the table in the first problem involves solving pairs of simultaneous linear equations. The second word problem can also be represented as two simultaneous linear equations.
Answer: B

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_000972

6

fl_problem_001097

7

fl_problem_001392

8
options: Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

Verify experimentally the properties of rotations, reflections, and translations:
- Lines are taken to lines, and line segments to line segments of the same length.
- Angles are taken to angles of the same measure.
- Parallel lines are taken to parallel lines.

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
problem_activity: Activity
In this activity, students calculate side lengths of similar triangles. They can use the scale factor between the similar triangles, studied in depth in previous lessons. Or they can look at internal ratios between corresponding side lengths within the triangles, introduced in the previous lesson. Students need to think strategically about which side lengths to calculate first since there are many missing values. As they discover more side lengths, this opens up more paths for finding the remaining values.
As students work, monitor for students who:
Use scale factors between triangles.
Notice that the long side is twice the short side in
\(GHI\)
and use that to find
\(c\)
,
\(d\)
, or
\(e\)
.
Notice that the long sides are equal in
\(ABC\)
and use that to find
\(h\)
,
\(d\)
, or
\(e\)
.
Select students who use different strategies to find side lengths to share during the discussion.
Launch
Tell students, “There are many ways to find the values of the unknown side lengths in similar triangles. Use what you have learned so far.” Give students 5 minutes of quiet work time followed by 5 minutes small group discussion and then a whole-class discussion.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, use the same color for corresponding side lengths.
Supports accessibility for: Visual-spatial processing
Student Facing
Triangles
\(ABC\)
,
\(EFD\)
, and
\(GHI\)
are all similar. The side lengths of the triangles all have the same units. Find the unknown side lengths.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
If students have trouble locating corresponding sides, suggest that they use tracing paper so they can rotate and or translate them. Another technique is to color corresponding side lengths the same color. For example, they could color
\(AB\)
,
\(EF\)
, and
\(GH\)
all red.
Activity Synthesis
Ask selected students to share the following strategies:
using (external) scale factors to move from one triangle to another
using quotients of corresponding side lengths within the triangles (internal scale factors)
Both methods are efficient and the method to use is guided by what information is missing and the numbers involved in the calculations. For example, if
\(h\)
is the first missing value we find, then comparing with triangle
\(ABC\)
and using internal scale factors is appropriate. To find
\(c\)
, again we can compare
\(ABC\)
and
\(GHI\)
and internal scale factors are appropriate again because
\(\frac{6}{5}\)
is half of
\(\frac{12}{5}\)
whereas comparing
\(\frac{12}{5}\)
and
\(4\)
is more involved (the scale factor is
\(\frac{5}{3}\)
from
\(\triangle GHI\)
to
\(\triangle ABC\)
).
Ask students to articulate how they knew which sides of the similar triangles correspond. Make sure to make the following reasoning pathways explicit for all:
Triangle
\(ABC\)
has two equal side lengths, so the other two triangles will as well. This insight is efficient for finding
\(h\)
.
One side of triangle
\(GHI\)
is twice the length of another side, so this will be true of the other triangles as well. This insight is helpful for finding
\(c\)
,
\(d\)
, and
\(e\)
.
Emphasize that there are many different relationships that can be used to find side lengths of similar triangles.
Speaking: MLR8 Discussion Supports:
To support students to articulate how they knew which sides of the similar triangles correspond, provide sentence frames such as, “For similar triangles _____ and _____ I know that sides _____ and _____ correspond because ________." Sentence frames invite and incentivize more student participation, conversation, and meta-awareness of language. This routine should help students reason about the ratio of corresponding sides of similar triangles and communicate their understanding.
Design Principle(s): Support sense-making, Optimize output (for comparison)


correct_option_index: 4