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: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.  For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
problem_activity: Optional activity
This activity is optional. It gives students an opportunity to reason about and to graph the rules of piecewise functions without a context.
Students are given the equations that define two piecewise functions, along with strips of paper, each containing a part of a graph and a portion of the horizontal axis (no scale is shown). Their job is to arrange the strips, apply a scale on each axis, and add open and closed circles to the graph to accurately represent the function values at each interval of input.
Unlike the piecewise functions students saw earlier in the lesson, each function here contains intervals defined by non-constant linear expressions. Support students as needed in reasoning about those intervals. If desired, this activity could be done over two class periods. (Students could piece together the graph of the first function in one class period and do the same for the second function at another time.)
To create an accurate graph, students need to make sense of the values, expressions, and inequality symbols in each equation and persevere in discerning their connections to the graph (MP1).
Here are images of the graphs on the blackline master for reference and planning. Cut each graph along the dashed lines and the vertical axis before giving it to students.
Function
\(f\)
[IMG]
Function
\(g\)
[IMG]
Launch
Arrange students in groups of 2–3. Give each group the strips for function
\(f\)
(pre-cut from the blackline master), glue or glue stick, and a sheet of paper on which to adhere the pieces.
Tell students that they are to arrange the pieces according to the rules that define
\(f\)
, as shown in the activity statement, and to add a label and a scale to each axis after the pieces are arranged. Explain that each piece contains a part of the horizontal axis, and the gridlines are 1 unit apart. Encourage students to double check their graph before gluing the pieces on another sheet of paper.
Consider discussing students' graphs of function
\(f\)
before giving students the pieces for function
\(g\)
and asking them to do another round.
Student Facing
Your teacher will give your group strips of paper with parts of a graph of a function. Gridlines are 1 unit apart.
Arrange the strips of paper to create a graph for each of the following functions.
\(\displaystyle f(x)=\begin{cases} \begin{align} \text-5,&\qquad \text-10< x< \text-5 \\ x,&\qquad \text-5\leq x< 0 \\ 1, &\qquad 0\leq x< 3\\ x-2, &\qquad 3\leq x< 8\\6, &\qquad 8\leq x < 1 0\ \end{align} \end{cases}\)
\(\displaystyle g(x)=\begin{cases} \begin{align} 5.5,&\qquad \text-10< x\leq \text-8 \\ 4,&\qquad \text-8< x\leq \text-3 \\ \text-x, &\qquad \text-3< x\leq 2\\ \text-3.5, &\qquad 2< x\leq 5 \\ x-5, &\qquad 5< x\leq 10 \end{align} \end{cases}\)
To accurately represent each function, be sure to include a scale on each axis and add open and closed circles on the graph where appropriate.
Student Response
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Activity Synthesis
Invite students to display their completed graphs and their reasoning. In particular, discuss how students determined the appearance of the intervals defined by non-constant linear expressions such as
\(x\)
,
\(x-4\)
, or
\(\text-x\)
.
If different groups created different graphs for the same function, ask them to analyze one another's work and try to reach an agreement.
If time permits, ask students to identify the domain and range of each function.
Speaking, Representing: MLR8 Discussion Supports.
Make sure students can explain their visual display and how they reasoned about the rules for different intervals. If needed, give them additional time to prepare their explanation. Invite groups to rehearse what they will say when they share with the whole class. Rehearsing provides students with additional opportunities to speak and clarify their thinking, and will improve the quality of explanations shared during the whole-class discussion.
Design Principle(s): Support sense-making; Cultivate conversation


correct_option_index: 1