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 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_pos_0

7

fl_problem_001894_pos_0

HS_0
standard_description: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
problem_activity: Activity
The goal of this activity is for students to understand that the way an equation is written to define a function depends on how the domain of the function is interpreted. In the previous activity, starting the sequence
\(C\)
with
\(C(0)\)
made sense since
\(n\)
was defined as the number of cuts while
\(C(n)\)
is the area of a paper square after the cuts. In this activity, students consider two equations written for a sequence represented by a visual pattern. One of the equations assumes the sequence starts at Step 0 and the other assumes the sequence starts at Step 1.
To help all students understand that both equations generate the same sequence and correctly represent the visual pattern, during the activity and whole-class discussion students should be encouraged to use precise language as they explain why an equation is valid (MP6). Confusion is likely to arise unless students are clear about the definition for
\(n\)
each equation is assuming. Some students may find the idea that both equations are equally valid challenging, so it is important to take time during the discussion for these students to understand that part of mathematics is recognizing what assumptions you are making and addressing them appropriately.
Monitor for students with clear definitions to select to share during the whole-class discussion. In particular, students that create different representations, such as tables, to see that the two equations result in the same list of numbers as their recursive definitions should be highlighted.
Making a spreadsheet available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Arrange students in groups of 2.
Student Facing
A Sierpinski triangle can be created by starting with an equilateral triangle, breaking the triangle into 4 congruent equilateral triangles, and then removing the middle triangle. Starting from a single black equilateral triangle:
[IMG]
Let
\(S\)
be the number of black triangles in Step
\(n\)
. Define
\(S(n)\)
recursively.
Andre and Lin are asked to write an equation for
\(S\)
that isn't recursive. Andre writes
\(S(n)=3^n\)
for
\(n\ge0\)
while Lin writes
\(S(n)=3^{n-1}\)
for
\(n\ge 1\)
. Whose equation do you think is correct? Explain or show your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Here is a geometric sequence. Find the missing terms.
3, ____, 6, ____, 12, ____, 24
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
Students may assume that at least one person has to be wrong because their equations aren't equivalent. Encourage them to make a table for each equation that starts with the first value for
\(n\)
according to the definitions.
Activity Synthesis
The goal of this discussion is for students to understand why Andre and Lin's equations are both representations of the visual pattern due to the interpretation of
\(n\)
. Display the visual pattern for all to see. Invite previously selected students to share their reasoning about why either Andre or Lin is correct, recording student reasoning for all to see. If students still think starting with 0 is unusual after the share out, give some more time for students to reason about Andre's point of view, and then select students to share. (For example, Andre could be thinking about
\(n\)
as the breaking apart phase, so for the solid black triangle that has happened 0 times.)
Conclude the discussion by telling students that identifying an appropriate domain for a function is partly dependent on the situation and partly dependent on how they see the relationship. There are often many correct equations that represent a function. An important takeaway for students is that when they write an equation to represent a situation, they need to be clear what domain they have identified so other people can correctly interpret what they've done.
Writing, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to prepare students for the whole-class discussion by providing them with multiple opportunities to clarify their explanations through conversation. Before the whole-class discussion begins, give students time to meet with 1–2 other groups to share their response to the question, “Whose equation do you think is correct?” Invite listeners to ask questions, and to press for details and mathematical language.
Design Principle(s): Optimize output (for explanation); Cultivate conversation


aligns: no