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 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_pos_0

7

fl_problem_001894_pos_0

HS_0
standard_description: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
problem_activity: Optional activity
This activity reviews the work students did previously drawing shapes with given conditions. Students draw as many different triangles as they can that could be the base of the triangular prism, given two side lengths and one angle measure for the triangle.
In preparation for calculating surface area and volume in the next activity, students select one of their triangles and find its area. This will require them to draw and measure the height of the triangle. As needed, remind students that the height must be perpendicular to whichever side they are using as the base of their triangle. Also, prompt students to measure the height as precisely as possible, because it will influence the accuracy of their later calculations.
Launch
Provide access to geometry toolkits and compasses.
Student Facing
The base of a triangular prism has one side that is 7 cm long, one side that is 5.5 cm long, and one angle that measures
\(45^\circ\)
.
Draw as many different triangles as you can with these given measurements.
Select one of the triangles you have drawn. Measure and calculate to approximate its area. Explain or show 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.
Anticipated Misconceptions
Students may try to multiply two side lengths of the triangle to calculate the area. Remind them that the height must be perpendicular to the base. If necessary, demonstrate using an index card to draw in the height of the triangle.
Activity Synthesis
Ask students to share triangles they drew so that everyone has an opportunity to see all four triangles. If any of the four triangles are not presented by students, demonstrate how to construct it. Ensure the class agrees that 4 unique triangles have the given measurements. The third side length of the triangle could be 5.0 cm, 7.3 cm, 2.6 cm, or 9.7 cm.
Make sure students have calculated the area of their selected triangle correctly, because this will affect their volume and surface area calculations in the next activity.
| If the third side of the triangle is | then the area of the triangle should be about |                                                                                              possible strategies                                                                                              |
|--------------------------------------|-----------------------------------------------|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
|                5.0 cm                |          13.7 cm                   2          |                                          \(\frac12 \boldcdot 7 \boldcdot (3.9)\)                  or                    \(\frac12 \boldcdot (5.5) \boldcdot (5.0)\)                                           |
|                7.3 cm                |          18.2 cm                   2          | \(\frac12 \boldcdot 7 \boldcdot (5.2)\)                  or                   \(\frac12 \boldcdot (5.5) \boldcdot (6.6)\)                  or                    \(\frac12 \boldcdot (7.3) \boldcdot (5.0)\)  |
|                2.6 cm                |          6.3 cm                   2           | \(\frac12 \boldcdot 7 \boldcdot (1.8)\)                  or                    \(\frac12 \boldcdot (5.5) \boldcdot (2.3)\)                  or                    \(\frac12 \boldcdot (2.6) \boldcdot (5.0)\) |
|                 9.7                  |          18.9 cm                   2          | \(\frac12 \boldcdot 7 \boldcdot (5.4)\)                  or                    \(\frac12 \boldcdot (5.5) \boldcdot (6.9)\)                  or                    \(\frac12 \boldcdot (9.7) \boldcdot (3.9)\) |
Representing, Conversing: MLR7 Compare and Connect.
Invite students to prepare a visual display that shows how they measured and calculated the approximate area for their chosen triangle. Students should consider how to display their calculations so that another student can interpret them. For example, some students may wish to add notes or details to their drawings to help communicate their thinking. Arrange students in groups of 2–4. Give 2–3 minutes of quiet think time for students to read and interpret each other's calculations before they begin to discuss them. Display a list of questions that students can ask each other about their work. For example, "Can you show me how you measured the side lengths?", “How did you calculate the area of the triangle?”, etc. During the whole-class discussion, draw students’ attention to the relationship between measuring and approximating area. Emphasize the language used to make sense of the strategies used to measure and approximate area. These exchanges strengthen students’ mathematical language use and reasoning based on the relationship between measures and area.
Design Principle(s): Maximize meta-awareness


aligns: no