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 add and subtract multi-digit whole numbers using the standard algorithm.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.

Answer: 
No. This set of problems does not directly involve adding or subtracting multi-digit whole numbers. Instead, it focuses on multiplying multi-digit whole numbers.

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

im_task_000157_pos_0

3

fl_problem_000674_neg_0

5
standard_description: Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
problem_activity: Activity
This activity connects the ideas of roots, rational exponents, graphs of exponential functions, and decimal approximations. Students graph
\(y=2^x\)
for a few integer values of
\(x\)
, approximate a smooth curve passing through those points, and use that curve to estimate the value of
\(2^x\)
for various rational inputs. Students then test these estimates using their understanding of the connection between rational exponents and roots.
Launch
Arrange students in groups of 2–3. Encourage them to compare their answers with their group after each question.
Since the first problem in this activity was designed to be completed without technology, ask students to put away any devices until after they complete it.
Representing, Conversing: MLR7 Compare and Connect.
Use this routine to prepare students for the whole class discussion about the graph of
\(y=2^x\)
. After students complete the table and sketch the graph of
\(y=2^x\)
, invite them to create a visual display of their work. Then ask students to quietly circulate and observe at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their tables and graphs. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify the language students use to compare and contrast the estimates for
\(2^x\)
for various values of
\(x\)
between 0 and 2.
Design Principle(s): Cultivate conversation
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer collaboration. When students share their work with a partner, display sentence frames to support conversation such as: “First, I _____ because. . .”, “How did you get . . .?”, “That could/couldn’t be true because . . . .”
Supports accessibility for: Language; Social-emotional skills
Student Facing
Complete the table as much as you can without using a calculator. (You should be able to fill in three spaces.)
{"table": [{"_attributes": {"align": "center", "border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math"], "data-png-file-id": "50"}, "_value": "\\(x\\)"}]}], "td": [{"_value": "0"}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "14"}, "_value": "\\(\\frac13\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "781"}, "_value": "\\(\\frac23\\)"}]}, {"_value": "1"}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "4325"}, "_value": "\\(\\frac43\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "11218"}, "_value": "\\(\\frac53\\)"}]}, {"_value": "2"}]}, {"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math"], "data-png-file-id": "1139"}, "_value": "\\(2^x\\)"}], "_value": "(using exponents)"}], "td": [{"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "2323"}, "_value": "\\(2^0\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "2115"}, "_value": "\\(2^{\\frac13}\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "2132"}, "_value": "\\(2^{\\frac23}\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "1930"}, "_value": "\\(2^1\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "25624"}, "_value": "\\(2^{\\frac43}\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "25635"}, "_value": "\\(2^{\\frac53}\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "1131"}, "_value": "\\(2^2\\)"}]}]}, {"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math"], "data-png-file-id": "1139"}, "_value": "\\(2^x\\)"}], "_value": "(decimal approximation)"}], "td": [{}, {}, {}, {}, {}, {}, {}]}]}]}]}
Plot the points that you filled in.
[IMG]
Connect the points as smoothly as you can.
Use this graph of
\(y=2^x\)
to estimate the value of the other powers in the table, and write your estimates in the table.
Let’s investigate
\(2^{\frac13}\)
:
Write
\(2^{\frac13}\)
using radical notation.
What is
\(\left( 2^{\frac13}\right)^3\)
?
Raise your estimate from the table of
\(2^{\frac13}\)
to the third power. What should it be? How close did you get?
Let’s investigate
\(2^{\frac23}\)
:
Write
\(2^{\frac23}\)
using radical notation.
What is the value of
\(\left( 2^{\frac23}\right)^3\)
?
Raise your estimate from the table of
\(2^{\frac23}\)
to the third power. What should it be? How close did you get?
Student Response
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Student Facing
Are you ready for more?
Answer these questions using the fact that
\((1.26)^3 = 2.000376\)
.
Explain why
\(\sqrt[3]{2}\)
is very close to
\(1.26\)
. Is it larger or smaller than
\(1.26\)
?
Is it possible to write
\(\sqrt[3]{2}\)
exactly with a finite decimal expansion? Explain how you know.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Display several student graphs of
\(y=2^x\)
for all to see and invite students to share how close their estimates for the cubes of
\(2^{\frac13}\)
and
\(2^{\frac23}\)
were. The important takeaway from this activity is that whether we write the number as
\(2^{\frac23}\)
or
\(\sqrt[3]{2^2}\)
, the expression is just a number and its value can be approximated.


aligns: no