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: Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
problem_activity: Activity
This activity requires students to make sense of negative powers of 10 as repeated multiplication by
\(\frac{1}{10}\)
in order to distinguish between equivalent exponential expressions. If students have time, instruct them to write the other expressions in each table as a power of 10 with a single exponent as well. Look for students who have productive debate regarding the interpretation of, for example,
\((10^2)^{\text-3}\)
versus
\((10^{\text-2})^3\)
and ask them to share their reasoning later (MP3).
Launch
Arrange students in groups of 2. Give students 15 minutes of partner work time followed by a whole-class discussion. Ask students to explain their reasoning to their partner as they work. If there is disagreement, tell students to work to reach an agreement.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time.
Supports accessibility for: Organization; Attention
Student Facing
Match each exponential expression with an equivalent multiplication expression:
\(\left(10^2\right)^3\)
\(\left(10^2\right)^{\text -3}\)
\(\left(10^{\text -2}\right)^3\)
\(\left(10^{\text -2}\right)^{\text-3}\)
{"table": [{"_attributes": {"align": "center", "border": "1"}, "tbody": [{"tr": [{"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "43590"}, "_value": "\\(\\frac{1}{(10 \\boldcdot 10)} \\boldcdot \\frac{1}{(10 \\boldcdot 10)} \\boldcdot \\frac{1}{(10 \\boldcdot 10)}\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "43636"}, "_value": "\\(\\left(\\frac{1}{10} \\boldcdot \\frac{1}{10}\\right)\\left(\\frac{1}{10} \\boldcdot \\frac{1}{10}\\right)\\left(\\frac{1}{10} \\boldcdot \\frac{1}{10}\\right)\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "43680"}, "_value": "\\(\\frac{1}{ \\frac{1}{10} \\boldcdot \\frac{1}{10} }\\boldcdot \\frac{1}{ \\frac{1}{10} \\boldcdot \\frac{1}{10} } \\boldcdot \\frac{1}{ \\frac{1}{10} \\boldcdot \\frac{1}{10} }\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "43719"}, "_value": "\\((10 \\boldcdot 10)(10 \\boldcdot 10)(10 \\boldcdot 10)\\)"}]}]}]}]}]}
Write
\((10^2)^{\text-3}\)
as a power of 10 with a single exponent. Be prepared to explain your reasoning.
Match each exponential expression with an equivalent multiplication expression:
\(\frac{10^2}{10^5}\)
\(\frac{10^2}{10^{\text -5}}\)
\(\frac{10^{\text -2}}{10^5}\)
\(\frac{10^{\text -2}}{10^{\text -5}}\)
{"table": [{"_attributes": {"align": "center", "border": "1"}, "tbody": [{"tr": [{"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "43945"}, "_value": "\\(\\frac{ \\frac{1}{10} \\boldcdot \\frac{1}{10} }{ \\frac{1}{10} \\boldcdot \\frac{1}{10} \\boldcdot \\frac{1}{10}\\boldcdot \\frac{1}{10}\\boldcdot \\frac{1}{10} }\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "43989"}, "_value": "\\(\\frac{10 \\boldcdot 10}{10 \\boldcdot 10 \\boldcdot 10 \\boldcdot 10 \\boldcdot 10}\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "44030"}, "_value": "\\(\\frac{ \\frac{1}{10} \\boldcdot \\frac{1}{10} }{ 10 \\boldcdot 10\\boldcdot 10\\boldcdot 10\\boldcdot 10 }\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "44069"}, "_value": "\\(\\frac{ 10 \\boldcdot 10 }{ \\frac{1}{10} \\boldcdot \\frac{1}{10} \\boldcdot \\frac{1}{10}\\boldcdot \\frac{1}{10}\\boldcdot \\frac{1}{10}}\\)"}]}]}]}]}]}
Write
\(\frac{10^{\text -2}}{10^{\text -5}}\)
as a power of 10 with a single exponent. Be prepared to explain your reasoning.
Match each exponential expression with an equivalent multiplication expression:
\(10^4 \boldcdot 10^3\)
\(10^4 \boldcdot 10^{\text -3}\)
\(10^{\text -4} \boldcdot 10^3\)
\(10^{\text -4} \boldcdot 10^{\text -3}\)
{"table": [{"_attributes": {"align": "center", "border": "1"}, "tbody": [{"tr": [{"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "44231"}, "_value": "\\((10 \\boldcdot 10 \\boldcdot 10 \\boldcdot 10) \\boldcdot ( \\frac{1}{10} \\boldcdot \\frac{1}{10}\\boldcdot \\frac{1}{10})\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "44275"}, "_value": "\\(\\left(\\frac{1}{10} \\boldcdot \\frac{1}{10} \\boldcdot \\frac{1}{10} \\boldcdot \\frac{1}{10}\\right) \\boldcdot \\left( \\frac{1}{10} \\boldcdot \\frac{1}{10} \\boldcdot \\frac{1}{10}\\right)\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "44316"}, "_value": "\\(\\left(\\frac{1}{10}\\boldcdot \\frac{1}{10} \\boldcdot \\frac{1}{10} \\boldcdot \\frac{1}{10}\\right) \\boldcdot \\left(10 \\boldcdot 10 \\boldcdot 10\\right)\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "44356"}, "_value": "\\((10 \\boldcdot 10 \\boldcdot 10 \\boldcdot 10) \\boldcdot (10 \\boldcdot 10 \\boldcdot 10)\\)"}]}]}]}]}]}
Write
\(10^{\text-4} \boldcdot 10^3\)
as a power of 10 with a single exponent. Be prepared to explain your reasoning.
Student Response
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Student Facing
Are you ready for more?
Priya, Jada, Han, and Diego stand in a circle and take turns playing a game.
Priya says, SAFE. Jada, standing to Priya's left, says, OUT and leaves the circle. Han is next: he says, SAFE. Then Diego says, OUT and leaves the circle. At this point, only Priya and Han are left. They continue to alternate. Priya says, SAFE. Han says, OUT and leaves the circle. Priya is the only person left, so she is the winner.
Priya says, “I knew I’d be the only one left, since I went first.”
Record this game on paper a few times with different numbers of players. Does the person who starts always win?
Try to find as many numbers as you can where the person who starts always wins. What patterns do you notice?
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Some students may struggle to distinguish between
\(\left(10^{\text-2}\right)^3\)
and
\(\left(10^2\right)^{\text-3}\)
. Breaking down each expression into parts that emphasize repeated multiplication will help to illustrate the difference. For the first expression,
\(10^{\text-2} = \frac{1}{10} \boldcdot \frac{1}{10}\)
. The outer exponent of 3 means that
\(\frac{1}{10} \boldcdot \frac{1}{10}\)
is multiplied repeatedly 3 times. Similarly for the second expression,
\(10^2 = 10 \boldcdot 10\)
and the outer exponent of
\(\text- 3\)
means that the
reciprocal
of
\(10 \boldcdot 10\)
is multiplied repeatedly 3 times. In the end, both expressions are equal to
\(10^{\text-6}\)
.
Activity Synthesis
Select students who had disagreements during the activity to share what they disagreed about and how they came to an agreement. Consider asking:
“How is
\((10^{\text-2})^3\)
different from
\((10^2)^{\text-3}\)
? How are they the same?”
“Do you agree or disagree? Why?”
“Could you restate __'s reasoning in a different way?”
It is important for students to understand that the exponent rules work even with negative exponents. To see why, the whole class discussion must make a clear connection between the exponent rules and the process of multiplying repeated factors that are 10 and
\(\frac{1}{10}\)
. Contrast the expanded version of
\(\left(10^{\text-2}\right)^3\)
and
\(\left(10^2\right)^{\text-3}\)
. For
\((10^{\text-2})^3\)
, there are 3 factors that are
\(10^{\text-2}\)
, where
\(10^{\text-2}\)
is two factors that are
\(\frac{1}{10}\)
, so
\(\displaystyle \left(10^{\text-2}\right)^3 = \left( \frac{1}{10} \boldcdot \frac{1}{10} \right)\left(\frac{1}{10} \boldcdot \frac{1}{10}\right)\left(\frac{1}{10} \boldcdot \frac{1}{10}\right) = \frac{1}{10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10} = 10^{\text-6}.\)
For
\((10^2)^{\text-3}\)
, there are 3 factors that are
\(\frac{1}{10^2}\)
, so
\(\displaystyle (10^2)^{\text-3} = \left( \frac{1}{10 \boldcdot 10} \right)\left(\frac{1}{10 \boldcdot 10}\right)\left(\frac{1}{10 \boldcdot 10}\right) = \frac{1}{10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10} = 10^{\text-6}.\)
Both
\(\left(10^{\text-2}\right)^3\)
and
\(\left(10^2\right)^{\text-3}\)
are equal to
\(10^{\text-6}\)
in the same way that
\(\text-2 \boldcdot 3 = 2 \boldcdot \text-3 = \text-6\)
.
Listening, Conversing: MLR8 Discussion Supports.
As students share the disagreements they had with their partner, press for details in students’ statements by requesting that students elaborate on an idea or give an example of their process of reconciliation. For example, ask students, “How did you explain your reasoning to your partner in a way that convinced them?” In this discussion, it may be useful to revoice student ideas to model mathematical language use by restating a statement in order to clarify, apply appropriate language, or involve more students.
Design Principle(s): Maximize meta-awareness


aligns: no