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 problem or activity shown below 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: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x^(2 )+ y^(2) = 3.
problem_activity: Activity
The purpose of this task is to give students experience working with exponential expressions and to promote making use of structure (MP7) to compare exponential expressions. To this end, encourage students to rewrite expressions in a different form rather than evaluate them to a single number.
For students who are accustomed to viewing the equal sign as a directive that means “perform an operation,” tasks like these are essential to shifting their understanding of the meaning of the equal sign to one that supports work in algebra, namely, “The expressions on either side have the same value.”
Launch
Before students start working, it may be helpful to demonstrate how someone would figure out whether or not an equation is true without evaluating each expression. For example:
Is
\(4^2=2^3\)
true? Well, let's see. We can rewrite each side like this:
\(\displaystyle 4 \boldcdot 4 = 2 \boldcdot 2 \boldcdot 2\)
Then we can replace one of those
\(2 \boldcdot 2\)
's with a 4, like this:
\(\displaystyle 4 \boldcdot 4 = 4 \boldcdot 2\)
Now we can tell this equation is not true.
These problems can also be worked by directly evaluating expressions, which is fine, as it serves as practice evaluating exponential expressions.
Representation: Internalize Comprehension.
Begin with the demonstration as described in the launch to support connections between new situations involving evaluating exponential expressions and prior understandings. Use color or annotations to highlight what changes and what stays the same at each step.
Supports accessibility for: Conceptual processing; Visual-spatial processing
Student Facing
Decide whether each equation is true or false, and explain how you know.
\(2^4=2 \boldcdot 4\)
\(3+3+3+3+3=3^5\)
\(5^3=5 \boldcdot 5 \boldcdot 5\)
\(2^3=3^2\)
\(16^1=8^2\)
\(\frac12 \boldcdot  \frac12 \boldcdot  \frac12 \boldcdot  \frac12 = 4 \boldcdot \frac12\)
\(\left( \frac12 \right)^4=\frac{1}{8}\)
\(8^2 = 4^3\)
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Invite students who evaluated the expressions and students who used structure or the meaning of exponents and operations to present their work. Compare and connect the strategies by noting where the use of structure might prove more efficient, where evaluating might be simpler, where thinking about the meaning of exponents and operations can make the true or false determination simpler.
Some guiding questions to highlight the meaning of exponents:
“Can we switch the order with exponents like we can with addition and multiplication—specifically, are
\(a^b\)
and
\(b^a\)
equivalent? How do you know?” (No, you can try different values of
\(a\)
and
\(b\)
or use the meaning of exponents to see that
\(a\)
multiplied
\(b\)
times is not always the same as
\(b\)
multiplied
\(a\)
times.)
“What change can we make to the equation
\(3+3+3+3+3=3^5\)
to make it true?” (Change addition to multiplication on the left, or change the exponent to multiplication on the right.)
“Your friend claims the equation
\(\frac12 \boldcdot  \frac12 \boldcdot  \frac12 \boldcdot  \frac12 = 4 \boldcdot \frac12\)
is true. What do you think they are misunderstanding? How can you convince them it is false?” (Possibly by saying that the left side shows multiplying
\(\frac12\)
4 times, but the right side shows multiplying 4 by
\(\frac12\)
, which means
adding
4 copies of
\(\frac12\)
, not multiplying them. You can show similar examples with other numbers, but the best way to convince them is to talk about what exponents and multiplication mean.)
“Can we show that
\(8^2 = 4^3\)
is true without evaluating both sides? What understanding about the meaning of exponents and operations can help us?” (
\(8^2\)
means
\(8\boldcdot 8\)
or
\(4\boldcdot 2\boldcdot 4\boldcdot 2\)
, which also equals
\(4\boldcdot 4\boldcdot 2\boldcdot 2\)
or
\(4\boldcdot 4\boldcdot 4\)
. Another way to write
\(4\boldcdot 4\boldcdot 4\)
is
\(4^3\)
. We are using the understanding that we can multiply in any order, or the commutative and associative properties of multiplication.)
Speaking, Writing: MLR8 Discussion Supports.
Revoice language and push for clarity in reasoning when students discuss their strategies for determining whether the equations are true or false. Provide a sentence frame such as “The equation is true (or false) because ______.” This will strengthen students’ mathematical language use and reasoning when discussing the meaning of exponents and operations that can make the equivalence of expressions true or false.
Design Principle(s): Maximize meta-awareness


aligns: no