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.
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im_task_000157

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fl_problem_000674

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options: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
problem_activity: Activity
This activity formally introduces students to the absolute value function as the function that takes an input value and gives its distance from the origin as the output. Students see two different ways to represent this function algebraically: using the absolute value notation and using the cases notation:
\(A(x)=|x|\)
\(A(x)=\begin{cases} x,& x \geq 0 \\ \text{-}x, & x < 0 \end{cases}\)
The equation in cases notation focuses on what has to happen algebraically to an input value to give its distance from the origin. Different rules apply to different intervals of the domain.
As students work, look for those who plot points for the graph of function
\(A\)
and those who sketch two lines. Ask students who sketch different types of graphs to share during discussion.
Launch
Arrange students in groups of 2. Give students a few minutes of quiet work time, than time to share their response with their partner. Follow with a whole-class discussion.
Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve their verbal and written responses to the last question. Give students time to meet with 2–3 partners to share and get feedback on their responses. The listener should press for detail and precision in language by asking “Can you give a few examples?”, “How do you know?”, or “Can you demonstrate your thinking using the graph?” Invite students to go back and revise or refine their written explanation based on the feedback from peers. This will help students understand different ways to represent an absolute value function.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Student Facing
The function
\(A\)
gives the distance of
\(x\)
from 0 on the number line.
Complete the table and sketch a graph of function
\(A\)
.
|       \(x\)       | \(A(x)\) |
|-------------------|----------|
|         8         |          |
|                   |   5.6    |
|      \(\pi\)      |          |
|    \(\frac12\)    |          |
|                   |    1     |
|         0         |          |
| \(\text-\frac12\) |          |
|        -1         |          |
|       -5.6        |          |
|                   |    8     |
[IMG]
Andre and Elena are trying to write a rule for this function.
Andre writes:
\(A(x)=\begin{cases} x,& x \geq 0 \\ \text-x, & x < 0 \\ \end{cases}\)
Elena writes:
\(A(x) = |x|\)
Explain why both equations correctly represent the function
\(A\)
.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Students who interpret the “
\(\text-x\)
” in Andre's rule to mean "negative
\(x\)
" (rather than "the opposite of
\(x\)
") may be unsure how to use that information. Ask students to evaluate the function for specific values of
\(x\)
and to write down each step. For instance, when
\(x\)
is -2,
\(A(\text{-}2)\)
is
\(\text{-}(\text{-}2)\)
, which is 2.
Activity Synthesis
Select previously identified students to display their graph of function
\(A\)
. Ask students who plotted only the ordered pairs in the table whether the pairs that are not in the table, if plotted, would also fall on the same two lines. Emphasize that
\(A\)
can be shown with two lines.
Next, ask students to share their response to the last question. Students may find Elena's rule easier to explain because of their work with absolute value in recent activities and struggle to explain Andre's rule.
We can reason about Andre's equation a couple of ways:
By thinking about rules: The equation is that of a piecewise function because different rules are applied to different parts of the domain to give the "distance from 0" as the output:
When the input
\(x\)
is positive or 0, its distance from 0 is that same number,
\(x\)
.
When the input
\(x\)
is negative, its distance from 0 is the opposite of the number,
\(\text-x\)
.
By using the graph: The two halves of the graph are lines with different slopes.
If we cover up the left side of the vertical axis and see only positive values of
\(x\)
, we see a line with a slope of 1 and that represents
\(y=x\)
.
If we cover up the right side and see only negative values of
\(x\)
, we see a line with a slope of -1 and that represents
\(y=\text{-}x\)
Explain to students that:
Function
\(A\)
is the
absolute value function
. It gives the distance of an input value from a certain value, the origin, in this case.
The graph of function
\(A\)
is a V shape with the two lines of points converging at
\((0,0)\)
, which is the minimum of the graph. We call this point the
vertex
of the graph. It is the point where the graph changes direction.
The absolute value function is also a piecewise function because different rules are applied to different parts of the domain to get the output.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of: absolute value function, vertex, cases notation.
Supports accessibility for: Conceptual processing; Language


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