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

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fl_problem_000674

5
options: 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.

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

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
Students continue to compare quadratic and exponential functions in this activity. This time, they decide how to compare the functions.
Monitor for different strategies students may use to compare the functions. Identify students who:
Create one or more tables of values and compare the values of
\(p(x)\)
and
\(q(x)\)
at increasingly large values of
\(x\)
.
Graph the functions defined by
\(p(x) = 6x^2\)
and
\(q(x) = 3^x\)
and compare the graphs.
Create one or more tables of values, calculate the growth factors at equal intervals of input, and then compare the growth factors.
If students choose graphing or spreadsheet tools strategically to perform comparisons, they practice MP5.
Launch
Ask students to observe the equations representing the two functions and determine which function is exponential and which is quadratic. Invite students to share how they know. Make sure students recognize that
\(p\)
is quadratic and
\(q\)
is exponential before they begin the activity.
Provide access to graphing technology and spreadsheet tools. This may be a good opportunity for students to experiment with the graphing window. If the horizontal dimension is very small (for example,
\(0<x<5\)
) or the vertical dimension is very large (for example,
\(0<y<3,\!000\)
), the two graphs will be hard to distinguish. As needed, remind students to think about adjusting the graphing window to make the graphs more informative.
Action and Expression: Provide Access for Physical Action.
Provide access to tools and assistive technologies such as a graphing calculator or graphing software. Some students may benefit from a checklist or list of steps to be able to adjust the graphing window to experiment with the horizontal and vertical dimensions.
Supports accessibility for: Organization; Conceptual processing; Attention
Student Facing
Here are two functions:
\(p(x) = 6x^2\)
and
\(q(x) = 3^x\)
.
Investigate the output of
\(p\)
and
\(q\)
for different values of
\(x\)
. For large enough values of
\(x\)
, one function will have a greater value than the other. Which function will have a greater value as
\(x\)
increases?
Support your answer with tables, graphs, or other representations.
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?
Jada says that some exponential functions grow more slowly than the quadratic function
\(f(x)=x^2\)
as
\(x\)
increases. Do you agree with Jada? Explain your reasoning.
Let
\(f(x) = x^2\)
. Could you have an exponential function
\(g(x)=b^x\)
so that
\(g(x) < f(x)\)
for all values of
\(x\)
?
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
Select students to present their strategies in the sequence listed in the Activity Narrative. If no students chose to graph the functions, consider displaying the graphs for all to see.  By presenting strategies that involved calculation of growth factors last, students will deepen their understanding of why exponential values will eventually overtake the quadratic values.
Again, one way to make sense of why exponential functions eventually grow faster than quadratic functions is by thinking of the growth factors. The output of an exponential function always increases by the same factor when the input increases by 1 (for example, 3 for the exponential function
\(q\)
studied here). The output of a quadratic function, on the other hand, increases by smaller and smaller amounts when the input increases by 1. So even though a quadratic function may take larger values than an exponential function for many inputs, the values of the exponential function will eventually overtake the quadratic.
Conversing: MLR2 Collect and Display.
As students share their analysis with the class, listen for and collect the language students use to identify and describe how the output of the exponential function eventually outgrows that of the quadratic function. Write the students’ words and phrases on a visual display and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students read and use mathematical language during their partner and whole-group discussions.
Design Principle(s): Maximize meta-awareness


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