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 to multiply a fraction or whole number by a fraction.

Answer: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

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 problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: im_task_001477_pos_0

K

fl_problem_001097_neg_0

7

fl_problem_001894_pos_0

HS_0
standard_description: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
problem_activity: Optional activity
In the previous activity, students generalized that the diagonals of squares are related to the side length of the square by a factor of about 1.4, or exactly
\(\sqrt2\)
.
In this activity, students apply their generalization about the diagonals of squares to isosceles right triangles. To find the lengths of the unlabeled sides in the second figure, students will need to generalize that a right triangle with one 45 degree angle is isosceles (because it’s half a square, or because both base angles are congruent, or because it’s similar to the isosceles triangle in the first figure).
To find the unknown values in the third figure, students will have to use the ratio of diagonal length to side length to find unknown side lengths. Students may use the approximate ratio
\(1.4:1\)
(or
\(1:0.7\)
) to find the side lengths, or they might use the exact ratio
\(\sqrt2:1 \text{ or } 1:\frac{1}{\sqrt2}\)
. Students are not expected or encouraged to rewrite answers of
\(\frac{24}{\sqrt2}\)
in another form.
Launch
Action and Expression: Internalize Executive Functions.
Provide students with a three-column table to organize their work. Use the column headings: the side adjacent to the 45 degree angle, the side opposite to the 45 degree angle, and the hypotenuse. The table will provide a visual support for students to identify the type of triangle and pattern.
Supports accessibility for: Language; Organization
Student Facing
[IMG]
[IMG]
[IMG]
Calculate the lengths of the 5 unlabeled sides.
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?
Square
\(ABCD\)
has a diagonal length of
\(x\)
and side length of
\(s\)
. Rhombus
\(EFGH\)
has side length
\(s\)
.
How do the diagonals of
\(EFGH\)
compare to the diagonals of
\(ABCD\)
?
What is the maximum possible length of a diagonal of a rhombus of side length
\(s\)
?
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
If students are struggling, encourage them to analyze the three triangles, look for patterns, and identify the triangles as isosceles right triangles. Students can then use the patterns from the previous activity.
Activity Synthesis
Make sure all students understand that the three triangles are isosceles right triangles, and represent half of a square, and so are prepared to connect their reasoning from earlier activities to this activity.
Display this list of solutions for triangle
\(PQR\)
:
\(QR=14, RP=\sqrt{392}\)
\(QR=14, RP=14\sqrt{2}\)
\(QR=14, RP = 19.8\)
\(QR=14, RP = 1.4 \boldcdot 14 \)
or 19.6
Invite students to explain the reasoning behind each. Encourage students to think about which answers are most accurate, and which are most efficient. (The first two methods are equally accurate, but the first probably took more calculation, while the second could be found using scale factors quite easily. The third answer was obtained by using a calculator to evaluate either of the first two answers, which actually loses both efficiency and accuracy. The fourth answer is very efficient, as it was found using scale factors, but is a bit less accurate. For some applications, the fourth answer may be best, and it’s great for estimating.)


aligns: yes