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.


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.


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.


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: Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
problem_activity: Warm-up
In this activity, students practice determining the measurement of unknown angles using their relationship to angles with known angle measure and what students already know about the measurement of right angles and straight angles. After they have worked on the activity and shared their solutions, they are introduced to the vocabulary terms complementary and supplementary. Complementary describes angles whose measures sum to 90 degrees and supplementary describes angles whose measures sum to 180 degrees.
Some students may wish to use protractors, either to double check work or to investigate the different angle measures. This is an appropriate use of technology (MP5), but ask these students what other methods they could use instead.
Launch
Provide access to geometry toolkits.
Student Facing
Find the measure of angle
\(JGH\)
. Explain or show your reasoning.
[IMG]
Find and label a second
\(30^\circ\)
angle in the diagram. Find and label an angle congruent to angle
\(JGH\)
.
Angle
\(BAC\)
is a right angle. Find the measure of angle
\(CAD\)
.
[IMG]
​​​​​
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 needed, remind students that right angles have a measure of
\(90^\circ\)
and straight angles have a measure of
\(180^\circ\)
.
Activity Synthesis
The goal of this discussion is to introduce students to the terms complementary and supplementary for describing relationships between pairs of angles. Display the images for all to see and select students to briefly share their response to each question, recording any calculations used.
Explain to students that the term
complementary
describes a pair of angles whose measures sum to 90 degrees, and the term
supplementary
describes a pair of angles whose measures sum to 180 degrees. This is true even for pairs of angles that are not adjacent. Conclude the discussion by inviting students to identify pairs of angles that are complementary (
\(CAD\)
and
\(DAB\)
), supplementary (
\(FGJ\)
and
\(JGH\)
,
\(JGH\)
and
\(HGI\)
,
\(HGI\)
and
\(IGF\)
, or
\(IGF\)
and
\(FGJ\)
), or vertical (
\(FGJ\)
and
\(HGI\)
or
\(JGH\)
and
\(IGF\)
) from the activity.


correct_option_index: 1