few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Concept/skill: 
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Answer: 
No. This problem does not directly teach how side ratios in right triangles lead to definitions of trigonometric ratios. Instead, it involves solving a trigonometric function.

Example 2:
Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


Concept/skill: 
Solve quadratic equations in one variable.

Answer: 
Yes. This task shows one quadratic equation with one variable, x.

Example 3:
Problem/activity:
Task
Suppose $A$ is an angle with vertex $P$, as pictured below:
[IMG]
Draw a circle with center $P$ and with  radius $r \gt 0$.  Explain why the circle
meets each ray of angle $A$ in a single point.  Label these points $Q$ and $R$ respectively.
Draw circles with centers $Q$ and $R$ respectively and radius $r$.  These circles meet at $P$ and a second point to be labelled $B$.
Show that ray $\overrightarrow{PB}$  bisects angle $P$.


Concept/skill: 
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.

Answer: 
Yes. This task asks students to draw a ray that bisects an angle. The act of drawing circles and rays may involve compass and straightedge or dynamic geometric software.

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

shot_info: fl_problem_001894_neg_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
problem_activity: Activity
In the previous activity, students interpreted given tape diagrams and explained how they represented a story. Here, they have a chance to draw tape diagrams to represent a story. The first story is a bit more scaffolded because it specifies what
\(x\)
represents. In the other two stories, students need to decide which quantity to represent with a variable and choose a letter to use. As with all activities in this lesson, students are
not
expected to write and solve an equation. This preliminary work supports the understanding needed to be able to represent such situations with equations.
Digital
Launch
Keep students in the same groups. You might have
each
student draw
all three
diagrams and compare them with their groups, working together to resolve any discrepancies. Or if time is short, you might assign
each
student in the group
a different story—
ask each student to explain their diagram to their group to see if their group members agree with their interpretation.
For classrooms using the digital version of the materials, take a minute to demonstrate how the controls work in the applet. Some students may prefer to draw the diagrams in their notebooks or on scratch paper.
Action and Expression: Internalize Executive Functions.
Provide students with a blank template of a tape diagram to represent each story.
Supports accessibility for: Language; Organization
Student Facing
Here are three more stories. Draw a tape diagram to represent each story. Then describe how you would find any unknown amounts in the stories.
Noah and his sister are making gift bags for a birthday party. Noah puts 3 pencil erasers in each bag. His sister puts
\(x\)
stickers in each bag. After filling 4 bags, they have used a total of 44 items.
Noah’s family also wants to blow up a total of 60 balloons for the party. Yesterday they blew up 24 balloons. Today they want to split the remaining balloons equally between four family members.
Noah’s family bought some fruit bars to put in the gift bags. They bought one box each of four flavors: apple, strawberry, blueberry, and peach. The boxes all had the same number of bars. Noah wanted to taste the flavors and ate one bar from each box. There were 28 bars left for the gift bags.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Keep students in the same groups. You might have
each
student draw
all three
diagrams and compare them with their groups, working together to resolve any discrepancies. Or if time is short, you might assign
each
student in the group
a different story—
ask each student to explain their diagram to their group to see if their group members agree with their interpretation.
For classrooms using the digital version of the materials, take a minute to demonstrate how the controls work in the applet. Some students may prefer to draw the diagrams in their notebooks or on scratch paper.
Action and Expression: Internalize Executive Functions.
Provide students with a blank template of a tape diagram to represent each story.
Supports accessibility for: Language; Organization
Student Facing
Here are three more stories. Draw a tape diagram to represent each story. Then describe how you would find any unknown amounts in the stories.
Noah and his sister are making gift bags for a birthday party. Noah puts 3 pencil erasers in each bag. His sister puts
\(x\)
stickers in each bag. After filling 4 bags, they have used a total of 44 items.
Noah’s family also wants to blow up a total of 60 balloons for the party. Yesterday they blew up 24 balloons. Today they want to split the remaining balloons equally between four family members.
Noah’s family bought some fruit bars to put in the gift bags. They bought one box each of four flavors: apple, strawberry, blueberry, and peach. The boxes all had the same number of bars. Noah wanted to taste the flavors and ate one bar from each box. There were 28 bars left for the gift bags.
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?
Design a tiling that uses a repeating pattern consisting of 2 kinds of shapes (e.g., 1 hexagon with 3 triangles forming a triangle). How many times did you repeat the pattern in your picture? How many individual shapes did you use?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Much of the discussion will take place in groups. Here are some ideas for synthesizing students’ learning about creating tape diagrams:
Ask students if they had any disagreements in their groups and how they resolved them.
Ask students how they decided which unknown quantity to find in the story. The first story specifies
\(x\)
stickers, but the other stories do not define a variable.
Display one diagram for each story and ask students to explain how they are alike and how they are different.


aligns: yes