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: 
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

Answer: 
No. Though this task involves addition, it does not involve any word problems.

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: 
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.

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

shot_info: im_task_001477_neg_0

K

fl_problem_001097_pos_0

7

fl_problem_001894_neg_0

HS_0
standard_description: Order three objects by length; compare the lengths of two objects indirectly by using a third object.
problem_activity: Narrative
The purpose of this activity is for students to notice structure in a series of diagrams and the expressions that represent them. They investigate how these expressions vary as the number of rows and columns in the diagram change. Students see how the diagram represents the multiplication expression and also how the diagram helps to find the value of the expression (MP7). Through repeated reasoning they also begin to see how to find the value of a product of any two unit fractions (MP8).
Launch
Groups of 2
Display the images from the task.
“What is different about these diagrams?” (The number of rows increases by 1. The blue shaded piece gets smaller.)
“What is the same?” (The size of the big square. There are always 4 columns. Only one piece is shaded.)
1 minute: quiet think time
Share and record responses.
Activity
1–2 minutes: independent work time to complete the first problem
1–2 minutes: partner discussion
“Now, complete the rest of the problems with your partner.”
5 minutes: partner work time
Monitor for students who:
choose different diagrams to represent with multiplication expressions
represent the same diagram with different multiplication expression, for example,
\(\frac{1}{2}\times\frac{1}{4}\)
and
\(\frac{1}{4}\times\frac{1}{2}\)
Student Facing
A
[IMG]
B
[IMG]
C
[IMG]
D
[IMG]
Choose one of the diagrams and write a multiplication expression to represent the shaded region. How much of the whole square is shaded? Explain or show your thinking.
If the pattern continues, draw what you think the next diagram will look like. Be prepared to explain your thinking.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Advancing Student Thinking
If students do not write correct expressions to represent the diagrams, write the correct expressions and ask, “How do the expressions represent the area of the shaded piece of the diagram?”
Activity Synthesis
Display the diagrams from the student workbook.
Select previously identified students to share.
As students explain where they see the multiplication expression in each diagram, record the expressions under the diagram for all to see.
Refer to the diagram that shows
\(\frac{1}{2}\times\frac{1}{4}\)
or
\(\frac{1}{4}\times\frac {1}{2}\)
.
Display both expressions.
“How does this diagram represent both of these expressions?” (It shows one half of one fourth shaded in and it also shows one fourth of one half shaded in.)
Represent student explanation on the diagrams.
“Why is the area of the shaded region getting smaller in each diagram?” (Because we are shading a smaller piece of
\(\frac{1}{4}\)
each time.)
Display:
\(\frac{1}{2} \times \frac{1}{4}= \frac{1}{2\times 4}=\frac{1}{8}\)
\(\frac{1}{3} \times \frac{1}{4}= \frac{1}{3\times 4}=\frac{1}{12}\)
\(\frac{1}{4} \times \frac{1}{4}= \frac{1}{4\times 4}=\frac{1}{16}\)
\(\frac{1}{5} \times \frac{1}{4}= \frac{1}{5\times 4}=\frac{1}{20}\)
“These equations represent the diagrams. What patterns do you notice?” (They all have
\(\frac {1}{4}\)
in them. The denominators in the first fractions go 2, 3, 4, 5. The denominators in the middle fractions are multiplication expressions, the denominators in the middle are all multiplied by 4, the denominator in the fraction that shows the value of the shaded piece goes up by 4 each time.)
“How do the diagrams represent
\(\frac{1}{2\times 4}\)
,
\(\frac{1}{3\times4}\)
,
\(\frac{1}{4\times4}\)
,
\(\frac{1}{5\times4}\)
?” (The numerator tells us that there is 1 piece shaded and the denominator tells us the size of the piece. The denominator also tells us the rows and columns that the whole is divided into.)


aligns: no