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: Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.
problem_activity: Activity
This activity revisits a polynomial students first encountered in an earlier lesson in this unit, relating the polynomials to the integers. The purpose of this activity is to make clear that technology allows us to identify solutions to equations that can be tedious to do by hand. The last question reinforces this point by asking students to write their own polynomial and find the
\(x\)
value that corresponds to a certain output, since they will need to estimate the solution from a graph.
The question about possible factors is meant to preview the work of the next several lessons in which students will identify factors of polynomial expressions written in standard form and eventually show that for a polynomial
\(p(x)\)
, if
\(p(a)=0\)
, then
\((x-a)\)
is a factor of
\(p(x)\)
.
Launch
Display functions
\(p\)
and
\(q\)
for all to see. Ask, “Without using a graphing calculator, what is the most points of intersection the two functions could have? The least?” Encourage students to make sketches to justify that 3 is the most and 1 is the least.
Provide access to devices that can run Desmos or other graphing technology.
Speaking, Reading: MLR5 Co-Craft Questions.
Use this routine to spark students’ curiosity about a polynomial written in standard form. Display only the first sentence of this problem (“Consider the functions . . . .”) and ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the activity’s questions. Listen for and amplify any questions involving visualizing the graph, points of intersection, or possible factors. This will help students create the language of mathematical questions before feeling pressure to produce solutions.
Design Principle(s): Maximize meta-awareness; Support sense-making
Engagement: Develop Effort and Persistence.
Connect a new concept to one with which students have experienced success. For example, to help students see how questions 1 and 2 are related, ask them to find the solutions for an equation such as
\(x^2-4=12\)
. Then, invite them to briefly reflect on the steps used. Next, ask them what they would do differently if they were told to find the
\(x\)
-values where two functions
\(f(x)= x^2-4\)
and
\(g(x)=12\)
intersect. Ask students to notice what elements were similar in each process.
Supports accessibility for: Social-emotional skills; Conceptual processing
Student Facing
Consider the functions
\(p(x) = 5x^3+6x^2+4x\)
and
\(q(x) = 5640\)
.
Use graphing technology to find a value of
\(x\)
that makes
\(p(x)=q(x)\)
true.
For the
\(x\)
-value at the point of intersection, what can you say about the value of
\(5x^3+6x^2+4x-5640\)
?
What does your answer suggest is a possible factor of
\(5x^3+6x^2+4x-5640\)
?
Write your own polynomial
\(m(x)\)
of degree 3 or higher.
Use graphing technology to estimate the values of
\(x\)
that make
\(m(x)= q(x)\)
true.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
If students have trouble using the value of
\(5x^3+6x^2+4x-5640\)
at the
\(x\)
-value where the functions intersect to find a possible factor of
\(5x^3+6x^2+4x-5640\)
, encourage them to think about the relationship between factors of an expression and
\(x\)
-values for which the value of the expression is 0.
Activity Synthesis
The goal of this discussion is for students to see how to use technology to answer an algebraic question involving a higher degree polynomial. Invite students to share the polynomial
\(m(x)\)
that they wrote. If possible, display several versions of
\(m(x)\)
for all to see, and use the graph to estimate the
\(x\)
-values where
\(m(x)=q(x)\)
. If this is not possible, ask students to share the solutions they estimated. Students should understand that not all polynomials can be factored by hand, and that technology is a valuable tool for helping us interpret such polynomials.
Then ask students to suggest possible factors of
\(5x^3+6x^2+4x-5640\)
. After a few suggestions, ask students how they could prove something is a factor. It is important that students understand that for now, the only way to prove something like
\((x-10)\)
is a factor of
\(5x^3+6x^2+4x-5640\)
is to find the quadratic function that satisfies
\(5x^3+6x^2+4x-5640=(x-10)(ax^2+bx+c)\)
, where
\(a\)
,
\(b\)
, and
\(c\)
are real numbers. Tell students that the next lessons are about how to do this.


aligns: no