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: 
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

Answer: 
No. Though this task involves addition, it does not involve any word problems.

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_neg_0

K

fl_problem_001097_pos_0

7

fl_problem_001894_pos_0

HS_0
standard_description: Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
problem_activity: Activity
Students have seen quadratic expressions in both standard form and factored form since the beginning of the unit. In this activity, they learn to distinguish the expressions by their forms and to refer to each form by its formal name. Refining their language about the different forms prepares students to be more precise in their thinking about the graphs of quadratic functions in later lessons (MP6).
Launch
Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.
Writing, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve their writing, by providing them with multiple opportunities to clarify their explanations through conversation. Give students time to meet with 1–2 partners to share their response to the final question, “Why do you think that form is called factored form?” Provide listeners with prompts for feedback that will help their partner add detail to strengthen and clarify their ideas. For example, students can ask their partner, “Can you say more about what each expression means?” or “I understand _____, but can you clarify . . .” Next, provide students with 2–3 minutes to revise their initial draft based on feedback from their peers.
Design Principle(s): Optimize output (for explanation)
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer that can be used to brainstorm the activity. Include one column for standard form and one column for factored form. Include a space at the bottom of the organizer for student-generated notes or reminders about standard form and factored form.
Supports accessibility for: Language; Organization
Student Facing
The quadratic expression
\(x^2 + 4x + 3\)
is written in
standard form
.
Here are some other quadratic expressions. The expressions on the left are written in standard form and the expressions on the right are not.
Written in standard form:
\(x^2 – 1\)
\( x^2 + 9x\)
\(\frac12 x^2\)
\(4x^2 – 2x + 5\)
\(\text-3x^2 – x + 6\)
\(1 - x^2\)
Not written in standard form:
\((2x + 3)x\)
\((x+1)(x-1)\)
\(3(x-2)^2 +1\)
\( \text-4(x^2 + x) +7\)
\( (x+8)(\text{-}x+5)\)
What are some characteristics of expressions in standard form?
\((x+1)(x-1)\)
and
\((2x + 3)x\)
in the right column are quadratic expressions written in
factored form
. Why do you think that form is called factored form?
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?
What quadratic expression can be described as being both standard form and factored form? Explain how you know.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Display the two columns of expressions for all to see. Solicit students’ ideas on the features of each form. Record their responses for all to see. Invite other students to express agreement or disagreement or to clarify their fellow students’ responses.
Define a quadratic expression in
standard form
explicitly as
\(ax^2 + bx + c\)
. Explain that we refer to
\(a\)
as the coefficient of the squared term
\(x^2\)
,
\(b\)
as the coefficient of the linear term
\(x\)
, and
\(c\)
as the constant term.
Ask students:
“How would you write
\((2x + 3)x\)
in standard form?” (
\(2x^2 + 3x\)
)
“The expression
\(2x^2 + 3x\)
only has two terms. Is it still in standard form?” (Yes, there is no constant term, which means
\(c\)
is 0.)
“How would you write
\(\text {-4}(x^2 + \frac14x) +7\)
in standard form?” (
\(\text {-4}x^2 - x + 7\)
)
“What are the values of the coefficients
\(a\)
and
\(b\)
?” (-4 and -1) “What about the constant term?” (7)
Then, clarify that a quadratic expression in
factored form
is a product of two factors that are each a linear expression. For example,
\((x+1)(x-1)\)
,
\((2x + 3)x\)
, and
\(x(4x)\)
all have two linear expressions for their factors. An expression with two factors that are linear expressions and a third factor that is a constant, for example:
\(2(x+2)(x-1)\)
, is also in factored form.
Tell students that each form gives us interesting or useful information about the function that the expression represents, and they will learn more about this in future lessons.


aligns: no