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: Write a function that describes a relationship between two quantities.

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
problem_activity: Activity
This activity focuses on a specific aspect of the previous activity: combining two functions to make a new function. Here, students sketch the graph of the function defined as the sum of two original functions. To help students make generalizations about the process, in each question one of the starting functions is always the same while the other is either constant, linear, or quadratic (MP8).
Monitor for students who approach the activity in different ways based on how precise they make their sketch, such as:
making a rough sketch by observation
plotting a few points by summing together the
\(y\)
-values of the two functions and then connecting them with a curve
measuring the distance of each function from the
\(x\)
-axis in a few locations and then combining those heights to determine the
\(y\)
-value of the new function at specific values of
\(x\)
making a table of values for
\(E\)
and the other function, as in the previous activity, in order to find the sums before making a sketch of the new function
Also look out for possible incorrect approaches due to students not considering the numerical values of the functions and what it means to add, such as by:
sketching a curve that is halfway between the two given graphs
letting the horizontal asymptote dictate end behavior on the right regardless of the value of the other function
Ask students who tried either of these last two approaches if they would be willing to share during the discussion. If possible, identify students who first did a sketch incorrectly and then revised their work to share at the end of the discussion.
Launch
Arrange students in groups of 2–3. Tell students that in this activity they are going to sketch graphs of functions that are the sum of two other functions. Ask students to first complete the sketch of
\(S\)
on their own and then check their work with their group before continuing on to the other graphs. Depending on time available, assign each group 3–5 of the graphs to complete.
Since this activity was designed to be completed without technology, ask students to put away any graphing devices.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, students can use different color highlighters for each graph. Students can identify and label points on each graph. Remind students that they are sketching the sum of the two functions.
Supports accessibility for: Visual-spatial processing
Student Facing
Here are the graphs of two functions,
\(E\)
and
\(L\)
. Define a new function
\(S\)
by adding
\(E\)
and
\(L\)
, so
\(S(x) = E(x) + L(x)\)
. On the same axes, sketch what you think the graph of
\(S\)
looks like.
[IMG]
Sketch the graph of the sum of
\(E\)
and each of the following functions.
[IMG]
[IMG]
[IMG]
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Here are the graphs of two functions,
\(U\)
and
\(V\)
. Define a new function
\(W\)
by multiplying
\(U\)
and
\(V\)
, so
\(W(x) = U(x) V(x)\)
. On the same axes, sketch what you think the graph of
\(W\)
looks like.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
The purpose of this discussion is for students to share how they made their sketches, focusing on how the numerical value of the new function is defined by the sum of the numerical values of the two original functions. The discussion begins with students sharing techniques that worked for them and then focuses on techniques that did not work and why, calling back to the strategies that did work in order to make sense of what went wrong.
Invite previously identified students who graphed correctly to share their process in the order listed in the Activity Narrative. Display student graphs and any accompanying work for all to see during the explanation. Since these were only sketches, students did not have to make a detailed table, but using a table illustrates clearly that adding two functions together means adding together their output values.
Next, invite previously identified students who took an incorrect approach to share in the order listed in the Activity Narrative. Where possible, identify what function the student did graph. For example, if a student sketched the curve halfway between the two starting graphs, the function they graphed is the average of the two starting graphs instead of the sum. If a student’s sketch appears to have a horizontal asymptote at
\(y=0\)
on the right side, it's possible they were thinking more of a function defined by the product of the two original functions.
If time allows, ask students to choose one of the questions and sketch the graph of a function defined by the
difference
between
\(E\)
and the other function.


correct_option_index: 0