few_shots: 
Example 1:
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.

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.


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

Example 2:
Concept/skill: 
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.

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} $

Answer: 
No. Completing this task does not involve representing fractions on a number line diagram.

Example 3:
Concept/skill: 
Fluently multiply multi-digit whole numbers using the standard algorithm.

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?


Answer: 
Yes. 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.

Now, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

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im_task_000157_neg_0

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fl_problem_000674_pos_0

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standard_description: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
problem_activity: Activity
The goal of this activity is for students to reason about what horizontal intercepts a graph of a polynomial must have based on the structure of the factored form of the polynomial (MP7). Students begin by considering a polynomial equation and graph and reasoning about why the graph window is insufficient since it only shows two out of three intercepts. Students then consider the horizontal intercepts of a different polynomial through the lens of selecting an appropriate viewing window before checking with graphing technology.
Launch
Provide access to devices that can run Desmos or other graphing technology.
Reading, Conversing, Writing: MLR5 Co-Craft Questions.
To help students consider the graph window of a polynomial, start by only displaying Mai’s graph and comment. Give students 1–2 minutes to write their own mathematical questions about the situation. Invite students to compare their questions before revealing the activity’s questions. Listen for and amplify any questions involving the terms “parabola” or “quadratic,” along with any questions about the scale on the axes. This will help students create questions about Mai’s statement and graph as they explore an appropriate graph window.
Design Principle(s): Maximize meta-awareness; Cultivate conversation
Engagement: Develop Effort and Persistence.
Break the class into small discussion groups and then invite a representative from each group to report back to the whole class. Ask that each discussion group member is equally prepared to report back to the whole group. As students work, encourage groups to use peer supports to ensure all group members are prepared to report out if selected.
Supports accessibility for: Language; Social-emotional skills; Attention
Student Facing
Mai graphs the function
\(p\)
given by
\(p(x)=(x+1)(x-2)(x+15)\)
and sees this graph.
[IMG]
She says, “This graph looks like a parabola, so it must be a quadratic.”
Is Mai correct? Use graphing technology to check.
Explain how you could select a viewing window before graphing an expression like
\(p(x)\)
that would show the main features of a graph.
Using your explanation, what viewing window would you choose for graphing
\(f(x)=(x+1)(x-1)(x-2)(x-28)\)
?
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?
Select some different windows for graphing the function
\(q(x) = 23(x-53)(x-18)(x+111)\)
. What is challenging about graphing this function?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
If students have trouble deciding what the constant term of
\(f(x)\)
is, remind them that it can be found quickly by multiplying the constant terms of each linear factor. Students can use the distributive property to understand why this is true.
Activity Synthesis
Select students to share how they selected a viewing window for
\(f(x)=(x+1)(x-1)(x-2)(x-28)\)
. In particular, ask students to state what structure in the factored form they focused on in order to select their window size.
If students question whether we can really be sure we’re seeing all the horizontal intercepts, tell them that in future lessons, they will learn more about what functions are doing to the far left and right of a graph and how we can be sure there are no other horizontal intercepts outside our viewing window. Later in this unit, they will see that all zeros of a function correspond to factors of the function.


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