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


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Answer: 
Yes. An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation.

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

shot_info: im_task_001477_pos_0

K

fl_problem_001097_pos_0

7

fl_problem_001894_pos_0

HS_0
standard_description: Understand ordering and absolute value of rational numbers.
problem_activity: Activity
Students reason abstractly and quantitatively about temperatures over time graphed on coordinate axes (MP2). The goal of this activity is for students to use inequalities to describe the location of points on a coordinate grid in one direction. This activity also introduces the idea of vertical difference on the coordinate plane using a familiar context. Students may use previous strategies such as counting squares, but are not expected to explicitly add or subtract using negative numbers.
Launch
Arrange students in groups of 2. Allow students 3–4 minutes of quiet work time and 1–2 minutes to check results with their partner. Follow with a whole-class discussion.
Engagement: Develop Effort and Persistence.
Connect a new concept to one with which students have experienced success. For example, remind students about the sea levels and elevation number line representations.
Supports accessibility for: Social-emotional skills; Conceptual processing
Student Facing
The coordinate plane shows the high and low temperatures in Nome, Alaska over a period of 8 days. The axis labeled
\(T\)
represents temperatures in degrees Fahrenheit. The axis labeled
\(d\)
represents the day.
[IMG]
What was the warmest high temperature?
Write an inequality to describe the high temperatures,
\(H\)
, over the 8-day period.
What was the coldest low temperature?
Write an inequality to describe the low temperatures,
\(L\)
, over the 8-day period.
On which day(s) did the
largest
difference between the high and low temperatures occur? Write down this difference.
On which day(s) did the
smallest
difference between the high and low temperatures occur? Write down this difference.
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
Before doing this problem, do the problem about taxicab distance in an earlier lesson.
The point
\((0,3)\)
is 4 taxicab units away from
\((\text{-}4,3)\)
and 4 taxicab units away from
\((2,1)\)
.
Find as many other points as you can that are 4 taxicab units away from
both
\((\text{-}4,3)\)
and
\((2,1)\)
.
Are there any points that are 3 taxicab units away from both points?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Because the high temperature on day 7 was positive and the low temperature was negative, some students may not notice that the difference was 10 degrees on this day as well. Consider prompting them to use tracing paper to compare the temperature differences on days 6 and 7.
Activity Synthesis
This discussion should lead to two key takeaways. First, students express the range of values for the low and high temperatures using inequalities. Second, students share strategies for finding a difference between two values on the coordinate plane.
Ask students to share their inequalities for
\(H\)
and
\(L\)
. It is expected that students have inequalities that describe the maximum high temperature for
\(H\)
and the minimum low temperature for
\(L\)
, but the discussion should bring out that each variable has 4 statements that capture its possible values:
\(L>\text-3\)
,
\(L=\text-3\)
,
\(L < 26\)
, and
\(L=26\)
for the variable
\(L\)
and
\(H>2\)
,
\(H=2\)
,
\(H < 28\)
, and
\(H=28\)
for the variable
\(H\)
.
Ask students to share their strategies for finding the vertical distance between points. Push them to explain how they took the scale of the vertical axis into account. Invite students to explain how they used the context to make sense of their answers.
Listening, Speaking: MLR8 Discussion Supports.
To support students in producing statements about details in the inequalities, provide sentence frames for students to use when they explain their reasoning to their peers, such as: “This inequality should/should not also be equal to because _____”, “A situation uses an equal sign when _____”, “I know that the inequality _____ is correct because _____.”
Design Principle(s): Support sense-making


aligns: no