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.


Standard options: 
A. 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.
B. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
C. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
D. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
E. Fluently add and subtract within 5.


Your response:
Thought: This task involves representing addition with objects such as number cards and dice.
Answer: A

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
$$-\frac{1}{4}$$


Standard options: 
A. Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
B. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
C. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.


Your response:
Thought: 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).
Answer: B

Example 3:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Your response:
Thought: 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. Solving this problem does not require use of the Pythagorean identity.
Answer: B

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
problem_activity: Activity
The purpose of this activity is to extend what students know about constructing a perpendicular line through a point on the given line to a new situation in which the constructed perpendicular line goes through a point
not
on the given line.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Digital
Launch
Arrange students in groups of 2. Display the image for all to see:
[IMG]
Remind students that in the previous lesson, they used straightedge and compass moves to create a line perpendicular to
\(\ell\)
that goes through point
\(B\)
, which was on line
\(\ell\)
. Now display an image of the construction of a perpendicular line through
\(B\)
for all to see throughout the activity:
[IMG]
Display a list of constructions students already know. This display should be posted in the classroom for the remaining lessons within this unit. It should look something like (only include the first 6 constructions now):
circles of a certain radius
lines and line segments through two points
regular hexagons
equilateral triangles
a perpendicular bisector of a given segment
a perpendicular line through a point on the given line
a perpendicular line through a point
not
on the given line (added in this lesson)
a parallel line through a point
not
on the given line (added in this lesson)
After quiet work time, ask students to compare their responses to their partner’s and decide whether they are both correct, even if they are different. Follow with a whole-class discussion.
Action and Expression: Provide Access for Physical Action.
Provide access to tools and assistive technologies such as dynamic geometry software.
Supports accessibility for: Visual-spatial processing;Conceptual processing; Organization
Student Facing
Here is a line
\(m\)
and a point
\(C\)
not
on the line. Use straightedge and compass tools to construct a line perpendicular to line
\(m\)
that goes through point
\(C\)
. Be prepared to share 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?
The line segment
\(AB\)
has a length of 1 unit. Construct its perpendicular bisector and draw the point where this line intersects our original segment
\(AB\)
. How far is this new point from
\(A\)
?
We now have 3 points drawn. Use a pair of points to construct a new perpendicular bisector that has not been drawn yet and label its intersection with segment
\(AB\)
. How far is this new point from
\(A\)
?
If you repeat this process of drawing new perpendicular bisectors and considering how far your point is from A, what can you say about all the distances?
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Print
Launch
Arrange students in groups of 2. Display the image for all to see:
[IMG]
Remind students that in the previous lesson, they used straightedge and compass moves to create a line perpendicular to
\(\ell\)
that goes through point
\(B\)
, which was on line
\(\ell\)
. Now display an image of the construction of a perpendicular line through
\(B\)
for all to see throughout the activity:
[IMG]
Display a list of constructions students already know. This display should be posted in the classroom for the remaining lessons within this unit. It should look something like (only include the first 6 constructions now):
circles of a certain radius
lines and line segments through two points
regular hexagons
equilateral triangles
a perpendicular bisector of a given segment
a perpendicular line through a point on the given line
a perpendicular line through a point
not
on the given line (added in this lesson)
a parallel line through a point
not
on the given line (added in this lesson)
After quiet work time, ask students to compare their responses to their partner’s and decide whether they are both correct, even if they are different. Follow with a whole-class discussion.
Action and Expression: Provide Access for Physical Action.
Provide access to tools and assistive technologies such as dynamic geometry software.
Supports accessibility for: Visual-spatial processing;Conceptual processing; Organization
Student Facing
Here is a line
\(m\)
and a point
\(C\)
not
on the line. Use straightedge and compass moves to construct a line perpendicular to line
\(m\)
that goes through point
\(C\)
. Be prepared to share your reasoning.
[IMG]
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?
The line segment
\(AB\)
has a length of 1 unit. Construct its perpendicular bisector and draw the point where this line intersects our original segment
\(AB\)
. How far is this new point from
\(A\)
?
We now have 3 points drawn. Use a pair of points to construct a new perpendicular bisector that has not been drawn yet and label its intersection with segment
\(AB\)
. How far is this new point from
\(A\)
?
If you repeat this process of drawing new perpendicular bisectors and considering how far your point is from A, what can you say about all the distances?
[IMG]
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
Some students may struggle more than is productive. Ask these students to draw a line segment and construct the perpendicular bisector of it. In that construction, the perpendicular bisector will go through an intersection point of two circles. Ask, “What happens if you create a circle centered at that intersection point that goes through an endpoint of the segment? Why does that happen? How can you use this idea in this new activity?”
Activity Synthesis
Focus on the process of using a previous construction to generate new constructions. Here are some questions for discussion:
“How was this construction different from perpendicular line constructions you have done before? How did thinking about the differences help you plan what to do?” (I didn't have a segment to bisect, because the point wasn't on the line. I realized I could still make a segment using the new point.)
“How does knowing some constructions help you do other, more complicated constructions?” (I can use the same moves, but just change them a little bit.)
Writing, Speaking, Conversing: MLR 1 Stronger and Clearer Each Time
. Use this routine to give students a structured opportunity to revise and refine the reasoning used to create their construction. Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help individuals strengthen their ideas and clarify their language. For example, “Can you explain how…?”, “Why did you choose to construct a circle there?”, “What do you know about radii that helps here?”, or “What do you mean by…?”. Students can borrow ideas and language from each partner to strengthen their final explanation.
Design Principle(s): Optimize output (for justification); Support sense-making


correct_option_index: 0