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: 
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_pos_0

K

fl_problem_001097_neg_0

7

fl_problem_001894_neg_0

HS_0
standard_description: Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
problem_activity: Activity
The goal of this activity is to help students flexibly transition between different notations for powers of 10 and introduce the property of multiplication of values with the same base. Students observe that
\(10^n \boldcdot 10^m = 10^{n+m}\)
for values of
\(n\)
and
\(m\)
that are positive integers. The second question hints at the reasoning that will extend exponent rules to include zero exponents, but students will investigate that more deeply in a later lesson.
Notice students who need help writing the general rule in in terms of
\(n\)
and
\(m\)
. You might ask, “What patterns did you notice with the exponents in the table? So if the exponents are
\(n\)
and
\(m\)
, how do you write what you did with the exponents?”
Launch
Give students 1 minute of quiet think time to complete the first unfinished row in the table before asking 1–2 students to share and explain their answers. When it is clear that students know how to complete the table, explain to them that they can skip one entry in the table, but they have to be able to explain why they skipped it. Give students 7–8 minutes to work before a brief whole-class discussion.
Student Facing
Complete the table to explore patterns in the exponents when multiplying powers of 10. You may skip a single box in the table, but if you do, be prepared to explain why you skipped it.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "expression"}, {"_attributes": {"scope": "col"}, "_value": "expanded"}, {"_attributes": {"scope": "col"}, "_value": "single power of 10"}]}]}], "tbody": [{"tr": [{"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "4541"}, "_value": "\\(10^2 \\boldcdot 10^3\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "40881"}, "_value": "\\((10 \\boldcdot 10)(10\\boldcdot 10 \\boldcdot 10)\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "237"}, "_value": "\\(10^5\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "38079"}, "_value": "\\(10^4 \\boldcdot 10^3\\)"}]}, {}, {}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "40919"}, "_value": "\\(10^4 \\boldcdot 10^4\\)"}]}, {}, {}]}, {"td": [{}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "40960"}, "_value": "\\((10 \\boldcdot 10 \\boldcdot 10)(10 \\boldcdot 10 \\boldcdot 10 \\boldcdot 10 \\boldcdot 10)\\)"}]}, {}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "41006"}, "_value": "\\(10^{18} \\boldcdot 10^{23}\\)"}]}, {}, {}]}]}]}]}
If you chose to skip one entry in the table, which entry did you skip? Why?
Use the patterns you found in the table to rewrite
\(10^n \boldcdot 10^m\)
as an equivalent expression with a single exponent, like
\(10^{\boxed{\phantom{3}}}\)
.
Use your rule to write
\(10^4 \boldcdot 10^0\)
with a single exponent.  What does this tell you about the value of
\(10^0\)
?
The state of Georgia has roughly
\(10^7\)
human residents. Each human has roughly
\(10^{13}\)
bacteria cells in his or her digestive tract. How many bacteria cells are there in the digestive tracts of all the humans in Georgia?
Student Response
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Student Facing
Are you ready for more?
There are four ways to make
\(10^4\)
by multiplying powers of 10 with smaller, positive exponents.
\(\displaystyle 10^1 \boldcdot 10^1 \boldcdot 10^1 \boldcdot 10^1\)
\(\displaystyle 10^1 \boldcdot 10^1 \boldcdot 10^2\)
\(\displaystyle 10^1 \boldcdot 10^3\)
\(\displaystyle 10^2 \boldcdot 10^2\)
(This list is complete if you don't pay attention to the order you write them in. For example, we are only counting
\(10^1 \boldcdot 10^3\)
and
\(10^3 \boldcdot 10^1\)
once.)
How many ways are there to make
\(10^6\)
by multiplying smaller powers of 10 together?
How about
\(10^7\)
?
\(10^8\)
?
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Create and post visual displays showing the exponent rules for reference throughout the unit, with one visual display for each rule. The visual display could include an example to illustrate how the rule works, along with visual aids and use of color. Here is a sample visual display:
[IMG]
Explain the visual display to students and display it for all to see throughout the unit.
Writing: MLR3 Clarify, Critique, Correct.
Present a hypothetical student statement that represents a misunderstanding of the exponent rule, such as “
\(10^2 ⋅ 10^3 = 10^6 \)
because
\(2⋅3 = 6\)
.” Prompt discussion by asking, “Do you agree with the statement? Why or why not?” Then, ask students to individually write an improved statement. Improved responses should include connections between the representations of a single expression, expanded form, and the single power of 10. This will help students evaluate, and improve on, the written mathematical arguments of others and highlight why
\(10^n\boldsymbol\cdot 10^m=10^{n+m}\)
.
Design Principle(s): Maximize meta-awareness.


aligns: no