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.


Mathematical concept/skill options: 
A. Write and interpret numerical expressions.
B. Generate and analyze patterns.
C. Analyze patterns and relationships.
D. Use properties of operations to generate equivalent expressions.
E. Use the four operations with whole numbers to solve problems.
F. Work with radicals and integer exponents.
G. Solve systems of equations.
H. Represent and solve problems involving multiplication and division.
I. Perform arithmetic operations on polynomials.
J. Understand the relationship between zeros and factors of polynomials.
K. Understand properties of multiplication and the relationship between multiplication and division.
L. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
M. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
N. Use polynomial identities to solve problems.
O. Apply and extend previous understandings of arithmetic to algebraic expressions.
P. Rewrite rational expressions.
Q. Interpret the structure of expressions.
R. Add and subtract within 20.
S. Analyze and solve linear equations and pairs of simultaneous linear equations.
T. Represent and analyze quantitative relationships between dependent and independent variables.
U. Solve equations and inequalities in one variable.
V. Create equations that describe numbers or relationships
W. Reason about and solve one-variable equations and inequalities.
X. Represent and solve equations and inequalities graphically.
Y. Understand solving equations as a process of reasoning and explain the reasoning.
Z. Understand and apply properties of operations and the relationship between addition and subtraction.
AA. Gain familiarity with factors and multiples.
AB. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
AC. Work with addition and subtraction equations.
AD. Multiply and divide within 100.
AE. Work with equal groups of objects to gain foundations for multiplication.
AF. Represent and solve problems involving addition and subtraction.
AG. Write expressions in equivalent forms to solve problems.
AH. Understand the connections between proportional relationships, lines, and linear equations.


Thought: This task teaches students to understand addition as putting together counters, or adding to already placed counters.
Answer: AB

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} $

Mathematical concept/skill options: 
A. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
B. Extend understanding of fraction equivalence and ordering.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
C. Use equivalent fractions as a strategy to add and subtract fractions.
D. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
E. Understand decimal notation for fractions, and compare decimal fractions.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
F. Develop Understanding of fractions as numbers.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.


Thought: This problem involves understanding fractions as numbers by comparing their quantities.
Answer: F

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?


Mathematical concept/skill options: 
A. Use place value understanding and properties of operations to perform multi-digit arithmetic. (A range of algorithms may be used.)
B. Understand place value.
C. Understand the place value system.
D. Extend the counting sequence.
E. Work with numbers 11–19 to gain foundations for place value.
F. Use place value understanding and properties of operations to perform multi-digit arithmetic.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.
G. Perform operations with multi-digit whole numbers and with decimals to hundredths.
H. Generalize place value understanding for multi-digit whole numbers.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.
I. Use place value understanding and properties of operations to add and subtract.


Thought: This set of problems asks students to perform multiplication, which is an operation, on multi-digit whole numbers.
Answer: G

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_001477

K

im_task_000157

3

fl_problem_000674

5
options: Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Perform operations on matrices and use matrices in applications.

Perform operations on vectors.

Represent and model with vector quantities.

Apply and extend previous understandings of numbers to the system of rational numbers.

Compute fluently with multi-digit numbers and find common factors and multiples.

Represent complex numbers and their operations on the complex plane.

Perform arithmetic operations with complex numbers.

Use properties of rational and irrational numbers.

Use complex numbers in polynomial identities and equations.

Extend the properties of exponents to rational exponents.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Reason quantitatively and use units to solve problems.
problem_activity: Activity
The purpose of this activity is for students to learn and practice a strategy for rewriting rational numbers with decimal representations that repeat eventually into their fraction representations. Students begin by arranging cards in order that show how the strategy was used to show
\(0.4\overline{85}=\frac{481}{990}\)
. Next, students use the strategy to calculate the fraction representations of two other values.
The extension problem asks students to repeat the task for decimal expansions with a single repeating digit, such as
\(0.\overline{3}\)
and
\(0.\overline{9}\)
.  While simpler to process algebraically, it may be counter-intuitive for students to conclude that
\(0.\overline{9}=1\)
. The fact that one number can have two different decimal expansions is often surprising, and hints at the fact that the number line has some quite subtle aspects to it.  Students might insist that
\(0.\overline{9}\)
must be strictly less than 1, which can invite an interesting discussion as to the nature of infinite decimal expansions.  If such a discussion arises, invite students to try to make their argument precise, and to try to explain where they feel there is a flaw in their argument that
\(0.\overline{9}=1\)
.  Discussion points to help resolve lingering dissonance include:
Asking whether any numbers on the number line could be
between
\(0.\overline{9}\)
and
\(1\)
.
Whether expressions like
\(0.\overline{9}5\)
make any sense.  (They do not, since the notation
\(0.\overline{9}\)
notation means the pattern continues
forever.
)
A second argument for those who accept that
\(\frac{1}{3}=0.\overline{3}\)
is to multiply both sides by 3.
Launch
Demonstrate the algorithm with an example such as converting
\(0.\overline{12} = \frac{12}{99}\)
.
Arrange students in groups of 2. Do not provide access to calculators. Distribute a set of the slips cut out from the blackline master to each group. Tell students that once they have the cards arranged, they should work on the second problem individually and then compare their work with their partner. Finish with a whole class discussion.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner, detailing the steps they took to calculate the fraction representation of the repeating decimals. Display sentence frames to support student conversation such as: “First, I _____ because . . .”, “Next I . . .”, and “Finally, in order to solve, I _____ because . . . .”
Supports accessibility for: Language; Social-emotional skills
Writing, Speaking: MLR1 Stronger and Clearer Each Time.
Use this routine to give students a structured opportunity to revise and refine their response to “Use Noah’s method to calculate the fraction representation of 0.1
\(\overline{86}\)
and 0.7
\(\overline{88}\)
”. Ask each student to meet with 2–3 other partners for feedback. Display prompts for feedback that will help students strengthen their ideas and clarify their language. For example, “Can you explain how...?” and “You should expand on....”, “Can you say that another way?”, etc. Give students 1–2 minutes to revise their writing based on the feedback they received.​
Design Principle(s): Optimize output (for generalization)
Student Facing
Your teacher will give your group a set of cards. Each card will have a calculations side and an explanation side.
The cards show Noah’s work calculating the fraction representation of
\(0.4\overline{85}\)
. Arrange these in order to see how he figured out that
\(0.4\overline{85} = \frac{481}{990}\)
without needing a calculator.
Use Noah’s method to calculate the fraction representation of:
\(0.1\overline{86}\)
\(0.7\overline{88}\)
Student Response
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Student Facing
Are you ready for more?
Use this technique to find fractional representations for
\(0.\overline{3}\)
and
\(0.\overline{9}\)
.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
The goal of this discussion is to help students build a more robust understanding of how the strategy works. Begin the discussion by selecting 2–3 students to share their work for the second problem, displaying each step for all to see. Ask if anyone completed the problem in a different way and, if so, have those students also share.
If no students notice it, point out that when rewriting
\(0.7\overline{88}\)
, we can multiply by 100, but multiplying by 10 also works since the part that repeats is only 1 digit long.
End the discussion by asking students to rewrite
\(0.3\overline{0}\)
using this strategy. This is like using a sledgehammer for a nail, but it works and is reflective of the work they did in an earlier activity.


correct_option_index: 1