few_shots: 
Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
[IMG]
Counters, 10 per pair of students
Decks of number cards 0-5 (with 4 of each number) or two 0-5 dice
Setup
This activity is done in pairs. The students sit side by side with 10-frame and deck of cards or dice. If they have cards, the cards are shuffled and placed face down in front of them.
Action
Player 1 flips a card or rolls the die. They place that many counters on the 10-frame. Player 2 flips a card or rolls the die. They place that many more counters on the 10-frame next to the counters already on the ten frame. The students count the total together.
A recording sheet can be added when the students have been introduced to equations and number sentences. For those students who can handle higher numbers, a deck of 0-10 cards and two 10-frames can be used.
This game can also be played alone.


Concept/skill: 
Represent addition and subtraction with objects, fingers, mental images, drawings (Drawings need not show details, but should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)), sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

Answer: 
Yes. This task involves representing addition with objects such as number cards and dice.

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 to multiply a fraction or whole number by a fraction.

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

Example 3:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Concept/skill: 
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Answer: 
No. This problem does not directly teach how side ratios in right triangles lead to definitions of trigonometric ratios. Instead, it involves solving a trigonometric function.

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

shot_info: im_task_001477_pos_0

K

fl_problem_001097_neg_0

7

fl_problem_001894_neg_0

HS_0
standard_description: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
problem_activity: Optional activity
In this activity, students solve a variety of systems of equations, some involving fractions, some involving substitution, and some involving inspection. This gives students a chance to practice using the methods they have learned in this section for solving systems of equations to solidify that learning. Some of the systems listed are ones students could have used in an earlier activity in this lesson, to describe the situations there. In the discussion, students compare the systems here to the ones they wrote in that activity and interpret the answer in that context.
Launch
Keep students in groups of 2. Allow students 5–7 minutes of partner work time followed by a whole-class discussion.
Representation: Internalize Comprehension
. Activate or supply background knowledge. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Here are a lot of systems of equations:
\(\begin{cases} y=\text-2x+6 \\ y=x-3 \end{cases}\)
\(\begin{cases} y=5x-4 \\ y=4x+12 \end{cases}\)
\(\begin{cases} y=\frac23x-4 \\ y=\text-\frac43x+9 \end{cases}\)
\(\begin{cases} 4y + 7x = 6 \\ 4y+7x = \text-5 \end{cases}\)
\(\begin{cases} y=x - 6\\ x=6 + y \end{cases}\)
\(\begin{cases} y=0.24x\\ y=0.18x+0.9 \end{cases}\)
\(\begin{cases} y=4.5x+15 \\ y=5x+12.5 \end{cases}\)
\(\begin{cases} y=3x \\ x+y=52 \end{cases}\)
Without solving, identify 3 systems that you think would be the least difficult for you to solve and 3 systems you think would be the most difficult. Be prepared to explain your reasoning.
Choose 4 systems to solve. At least one should be from your "least difficult" list and one should be from your "most difficult" list.
Student Response
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Activity Synthesis
There are two key takeaways from this discussion. The first is to reinforce that some systems can be solved by reasoning whether its possible for a solution to one equation to also be a solution to another. The second takeaway is that there are some systems that students will only be able to solve after learning techniques in future grades.
For each problem, ask students to indicate if they identified the system as least or most difficult. Record the responses for all to see.
Bring students' attention to this system:
\(\begin{cases} 4y + 7x = 6 \\ 4y+7x = \text-5 \end{cases}\)
Ask students what the two equations in the system have in common to each other and to think about whether a solution to the first equation could also be a solution to the second. One can reason there is no solution because
\(4y+7x\)
cannot be equal to both 6 and -5.
Ask students to return to the earlier activity and see if they can find any of those systems in these problems. (Lin's family ride is the sixth system. Noah's kayaking trip is the seventh system. Diego's baking is the eighth system.) After students notice the connection, invite students who chose those systems to solve to interpret the numerical solutions in the contexts from the earlier activity.
Students may be tempted to develop the false impression that all systems where both equations are given as linear combinations can be solved by inspection. Conclude the discussion by displaying this system that students defined in the last activity about sugar and flour:
\(\begin{cases} 0.8x+0.5y = 9 \\ x+y = 15 \end{cases}\)
Tell students that this system has one solution and they will learn more sophisticated techniques for solving systems of equations like this in future grades.
Speaking, Listening, Conversing: MLR8 Discussion Supports.
Before the whole-class discussion, use this routine to give students an opportunity to discuss their reasons for labeling systems of equations as “least difficult” or “most difficult.” Provide the following sentence frame: “_____ is the least/most difficult to solve because _____.” Encourage listeners to press for detail by asking questions such as “What would make that problem easier to solve?”; “What do your least/most difficult problems have in common?” This will help students make connections between the structures of given systems of equations, and possible strategies they can use to solve them.
Design Principle(s): Cultivate conversation; Maximize meta-awareness


aligns: no