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.


Standard options: 
A. 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.
B. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
C. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
D. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
E. Fluently add and subtract within 5.


Thought: This task involves representing addition with objects such as number cards and dice.
Answer: A

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} $

Standard options: 
A. Understand a fraction as a number on the number line; represent fractions on a number line diagram.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
B. 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.
C. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.


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

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?


Standard options: 
A. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
B. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
C. Fluently multiply multi-digit whole numbers using the standard algorithm.


Thought: 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.
Answer: C

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: im_task_001477

K

im_task_000157

3

fl_problem_000674

5
options: Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
problem_activity: Activity
In the previous activity students looked at 9 different equations. In this activity, each student focuses on just one of those equations to learn about the real-world situation it represents. They interpret the variables and values in terms of the situation (MP7). Adding to the table they saw in the previous activity gives students an opportunity to practice both evaluating expressions and solving equations.
This activity also asks students to create a graph of their assigned relationship on graph paper. In previous activities, students were given a grid with the axes already labeled and numbered whenever they were asked to create a graph of a relationship. Here they must decide for themselves how to scale each axis.
Launch
Decide whether you want to assign the equations or allow students to select which equation they will work with. Make sure there is at least one student working with each equation.
Give students 5–10 minutes of quiet work time followed by partner and whole-class discussion. If there is more than one student working with the same equation, consider letting them share their answers and reasoning with each other, after they have had some time to work on their own.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts. Check in with students within the first 2-3 minutes of work time. Look for students who need additional support deciding how to scale each axis for table. Invite students to share strategies they have used so far, as well as any questions they have before continuing.
Supports accessibility for: Memory; Organization
Student Facing
The equations in the previous activity represent situations.
\(S - 2 = T\)
where
\(S\)
is the number of sides on a polygon and
\(T\)
is the number of triangles you can draw inside it (from one vertex to the others, without overlapping)
\(G = J + 13\)
where
\(G\)
is a day in the Gregorian calendar and
\(J\)
is the same day in the Julian calendar
\(P = I - 47.50\)
where
\(I\)
is the amount of income and
\(P\)
is the profit after
$
47.50 in expenses
\(C + 273.15 = K\)
where
\(C\)
is a temperature in degrees Celsius and
\(K\)
is the same temperature in Kelvin
\(e = 6s\)
where
\(e\)
is the total edge length of a regular tetrahedron and
\(s\)
is the length of one side
\(m = 8.96V\)
where
\(V\)
is the volume of a piece of copper and
\(m\)
is its mass
\(y = \frac{1}{12}x\)
where
\(x\)
is the number of eggs and
\(y\)
is how many dozens that makes
\(t = \frac{d}{2.5}\)
where
\(t\)
is the amount of time it takes in seconds to jog a distance of
\(d\)
meters at a constant speed of 2.5 meters per second
\(g = 28.35z\)
where
\(g\)
is the mass in grams and
\(z\)
is the same amount in ounces
Your teacher will assign you one of these equations to examine more closely.
Rewrite your equation using words. Use words like product, sum, difference, quotient, and term.
In the previous activity, you matched equations and tables. Copy the values from the table that matched your assigned equation into the first 3 rows of this table. Make sure to label what each column represents.
| independent variable:                             ________________________ | dependent variable:                             ________________________ |
|----------------------------------------------------------------------------|--------------------------------------------------------------------------|
|                                                                            |                                                                          |
|                                                                            |                                                                          |
|                                                                            |                                                                          |
|                                     60                                     |                                                                          |
|                                                                            |                                   300                                    |
Select one of the first 3 rows of the table and explain what those values mean in this situation.
Use your equation to find the values that complete the last 2 rows of the table. Explain your reasoning.
On graph paper, create a graph that represents this relationship. Make sure to label your axes.
Student Response
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Anticipated Misconceptions
Some students may struggle to understand what each step is asking them to do. Consider preparing an example to share with them, such as this for the equation
\(2.54i = c\)
, and point out which part of the example corresponds to the question they are on.
The number of centimeters is the product of the number of inches and 2.54.
| independent variable:                       length (inches) | dependent variable:                       length (centimeters) |
|-------------------------------------------------------------|----------------------------------------------------------------|
|                              5                              |                              12.7                              |
|                             36                              |                             91.44                              |
|                             75                              |                             190.5                              |
|                             60                              |                                                                |
|                                                             |                              300                               |
If something is 36 inches long, this is equivalent to 91.44 when measured in centimeters.
In the table above, fill in 152.4 in the right column, next to the 60, because
\(2.54 \boldcdot 60 = 152.4\)
. Fill in 118.11 in the left column, next to the 300, because
\(2.54 \boldcdot 118.11 \approx 300\)
A graph with “length (inches)” on the horizontal axis, “length (centimeters)” on the vertical axis, and points at
\((5, 12.7), (36, 91.44), (75, 190.5), (60, 152.4)\)
, and
\((118.11, 300)\)
Some students may struggle with setting up their graph from scratch on graph paper. Prompt them to think about the maximum value they want to represent on each axis and what number they could count by to get to that maximum value in the amount of space they have.
Activity Synthesis
Continue to the next activity. The discussion of these relationships occurs after students have created their visual displays.


correct_option_index: 1