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 that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
problem_activity: Narrative
The purpose of this activity is for students to apply conservation of length by measuring with a tool that does not start at zero. Until now, students have focused on measuring by lining up the edge of an object with zero. In this activity, students consider how the units on a ruler, and in later lessons, a number line, can help us solve problems and measure even when we are not starting at zero. It is important for students to recognize that the tick marks show a consistent distance between numbers, so any part of the ruler can show a specific number of units. The synthesis focuses on how addition (counting on from the starting number) and subtraction can be used to find the length of an object. The teacher records students’ thinking with an equation with a question mark for the unknown.
It is recommended that students use rulers to measure the length of smaller objects (2
–
10 inches) in this activity. This smaller number range helps students connect what they notice about measuring starting at different numbers on the ruler to familiar addition and subtraction equations within 20. However, this activity can be adapted to use measuring tapes or yardsticks to measure longer objects if they are available.
When students measure objects starting at different non-zero spots on the ruler and observe what stays the same (the difference) and what changes (the starting and ending spots) they observe a structure which repeats itself no matter which spot they on the ruler they begin the measurement (MP7, MP8).
Action and Expression: Develop Expression and Communication
. Give students access to inch tiles to double check their measurement. Reiterate how the measurement is the same regardless of the whole inch they start at on the ruler.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing
Required Materials
Materials to Gather
Objects of various lengths
Rulers (inches)
Required Preparation
Each group of 2 needs access to several objects between 2
–10 inches long. Consider using classroom objects such as markers, colored pencils, paperclips, or other objects or images used in previous activities.
Launch
Groups of 2
Give each student a ruler.
Display the image.
Activity
“Han measured the same notebook as Jada. Jada’s measuring tape was torn, but she got the same answer. How did that happen?”
1 minute: quiet think time
“Work with a partner to figure out how Jada got the same answer as Han. Be prepared to explain your thinking.”
3–4 minutes: partner discussion
Monitor for a student who explains the problem as
\(10 - 2 = {?}\)
and one who explains the problem as
\(2 + {?} = 10\)
.
Share responses.
“Now, using your ruler, you and your partner should find 1 thing in the classroom to measure without starting at zero. Each of you should start at a different number and compare your measurements.”
“Don’t forget to include the unit with your measurement and to use a question mark for the unknown in your equation.”
5 minutes: partner work time
Monitor for students who discuss:
that the length must be the same because the object doesn’t change
the equations they write have the same value or show the same relationship
the relationship between the numbers in their different expressions (for example, “the numbers I subtracted are two more than yours because I moved the object 2 inches over on the ruler. The difference is the same.”)
Student Facing
Jada and Han used an inch ruler to measure the short side of a notebook.
[IMG]
Han says 8 inches.
[IMG]
Jada says 8 inches.
How did Han and Jada get the same measurement?
Write an equation that could show Jada’s thinking.
Measure an object using Jada’s method.
I measured a
________________________________
.
I started with the number
____________
.
I ended with the number
____________
.
Equation: ________________________________.
The length of my object is
____________
.
What do you notice about you and your partner’s measurements?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Advancing Student Thinking
Students may find the correct length when counting length units on the measuring tool, but identify the length as the total number if they represent their measurement with an addition equation. (For example, they count on from 2 inches to 10 inches on the ruler and write
\(2 + 8 = 10\)
, but say the length of the object is 10). Consider asking:
“How did you measure the ___?”
“What do the numbers in your equation represent?”
Activity Synthesis
Invite selected students to share their measurements and their reasoning.
Write equations that correspond to the students’ thinking using a question mark to represent the unknown value.
“Why do you think you get the same measurement each time, even if you measure starting from a different place?” (The object didn't change, so the length has to stay the same. The total length in inches is still the same between the number that you start with and end with.)


aligns: no