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:
Task
 Arrange the fractions in order from least to greatest. Explain your answer with a picture.
 $\frac{1}{5}, \frac{1}{7}, \frac{1}{3} $
 $\frac{2}{5}, \frac{2}{7}, \frac{2}{3} $
 $\frac{5}{6}, \frac{3}{6}, \frac{1}{6} $
 $\frac{5}{12}, \frac{8}{12}, \frac{4}{12} $

Concept/skill: 
Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
 Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.

Answer: 
Yes. By arranging the fractions in this problem, students learn to compare fractions by reasoning about their size.

Example 3:
Problem/activity:
Problem 1
Solve using the standard algorithm. If you get stuck, use an area model and/or the partial products algorithm to help. Then assess the reasonableness of your answer.
a.
$$5,803\times46$$
b.
$$63\times2,116$$
Problem 2
Dunkin' introduced a croissant doughnut in 2014. In the first three months of its debut, they sold 7,870 dozen of them per day. How many croissant doughnuts were sold in the first three months?


Concept/skill: 
Fluently multiply multi-digit whole numbers using the standard algorithm.

Answer: 
Yes. This set of problems involves multi-digit whole numbers because the numbers in each problem are four and two digit integers, and the mathematical calculations involved in both problems focus on multiplication. The first problem explicitly asks the student to solve its subproblems using the standard algorithm.

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_pos_0

K

im_task_000157_pos_0

3

fl_problem_000674_pos_0

5
standard_description: Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3^(2) × 3^(–5) = 3^(–3) = 1/3^(3) = 1/27.
problem_activity: Activity
Students extend their understanding of exponents to include negative exponents and explain patterns in the placements of the decimal point when a decimal is multiplied by 10 or
\(\frac{1}{10}\)
. Students use repeated reasoning to recognize that negative powers of 10 represent repeated multiplication by
\(\frac{1}{10}\)
and generalize to the rule
\(10^{\text- n} = \frac{1}{10^n}\)
(MP8).
A table is used to show different representations of decimals, fractions, and exponents. The table is horizontal to mimic the structure of decimals. As students work, notice the strategies they use to go between the different representations of the given powers of 10. Ask students who use contrasting strategies to share later.
Launch
Tell students to complete the table one row at a time to see the patterns most clearly. Ask a student to read the first question aloud. Select a student to explain the idea of a “multiplier” in this context. Give students 5–7 minutes to work. Follow with a whole-class discussion.
Student Facing
Complete the table to explore what negative exponents mean.
[IMG]
As you move toward the left, each number is being multiplied by 10. What is the multiplier as you move right?
How does a multiplier of 10 affect the placement of the decimal in the product?  How does the other multiplier affect the placement of the decimal in the product?
Use the patterns you found in the table to write
\(10^{\text -7}\)
as a fraction.
Use the patterns you found in the table to write
\(10^{\text -5}\)
as a decimal.
Write
\(\frac{1}{100,000,000}\)
using a single exponent.
Use the patterns in the table to write
\(10^{\text -n}\)
as a fraction.
Student Response
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for free access to Student Response.
Anticipated Misconceptions
Some students may think that, for example,
\(\frac{1}{1,000,000} = 10^{\text-7}\)
because the number 1,000,000 has 7 digits. Ask these students if it is true that
\(\frac{1}{10} = 10^{\text-2}\)
.
Activity Synthesis
One important idea is that multiplying by 10 increases the exponent, thus multiplying by
\(\frac{1}{10}\)
decreases the exponent. So negative exponents can be thought of as repeated multiplication by
\(\frac{1}{10}\)
, whereas positive exponents can be thought of as repeated multiplication by
\(10\)
. Another key point is the effect that multiplying by 10 or
\(\frac{1}{10}\)
has on the placement of the decimal.
Ask students to share how they converted between fractions, decimals, and exponents. Record their reasoning for all to see. Here are some possible questions to consider for whole-class discussion:
“Do you agree or disagree? Why?”
“Did anyone think of this a different way?”
“In your own words, what does
\(10^{\text- 7}\)
mean? How is it different from
\(10^7\)
?”
Introduce the visual display for
\(10^{\text- n} = \frac{1}{10^n}\)
and display it for all to see throughout the unit. For an example that illustrates the rule, consider displaying
\(10^{\text- 3} = \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10} = \frac{1}{10^3}\)
.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections. For example, annotate the patterns students notice to help synthesize how negative exponents can be thought of as repeated multiplication by
\(\frac{1}{10}\)
, whereas positive exponents can be thought of as repeated multiplication by
\(10\)
.
Supports accessibility for: Visual-spatial processing
Conversing, Representing, Writing: MLR2 Collect and Display.
During the whole-class discussion, capture the vocabulary and phrases students use to describe their strategies for converting between fractions, decimals, and exponents. Listen for students who make connections between repeated multiplication and the placement of the decimal point. Record this language onto a visual display that can be referenced in future discussions. This routine provides feedback to students that supports sense-making and increases meta-awareness of mathematical language.
Design Principle(s): Support sense-making; Maximize meta-awareness


aligns: yes