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:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Standard options: 
A. Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
B. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
C. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.


Thought: This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).
Answer: B

Example 3:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Thought: An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation. Solving this problem does not require use of the Pythagorean identity.
Answer: B

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: For exponential models, express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Distinguish between situations that can be modeled with linear functions and with exponential functions.

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
problem_activity: Activity
In this activity, students practice constructing an exponential function and evaluating it at a fractional input value as they translate between decades and years. They also think about how to express a percent increase over a fractional interval of input—specifically, how to express the annual growth rate of a population given the growth rate for one decade. They analyze and articulate why the growth rate for one year is
not
one tenth of that of one decade (MP3).
Launch
Tell students that, as of 2017, Nigeria is the seventh-most populous country in the world. It has grown at a fairly steady (and relatively high) rate since 1980.
Speaking, Reading: MLR5 Co-Craft Questions.
Use this routine to orient students to the language of exponential functions in context. Ask students to keep their books or devices closed, and display only the image of the map and the problem statement, without revealing the questions that follow. Arrange students in groups of 2, and invite them to create a list of mathematical questions that could be asked about the situation.  Listen for and amplify questions about growth and decay factors, percent increase or decrease, fractional intervals, and comparisons of growth rates. Invite 2–3 groups to share their questions with the class.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
[IMG]
In 1990, Nigeria had a population of about 95.3 million. By 2000, there were about 122.4 million people, an increase of about 28.4%. During that decade, the population can be reasonably modeled by an exponential function.
Express the population of Nigeria
\(f(d)\)
, in millions of people,
\(d\)
decades since 1990.
Write an expression to represent the population of Nigeria in 1996.
A student said, “The population of Nigeria grew at a rate of 2.84% every year.”
Explain or show why the student’s statement is incorrect.
Find the correct annual growth rate. Explain or show your reasoning.
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
If students struggle to identify the input to their function where
\(d\)
is not an integer, remind them that there are 10 years in a decade. Then ask them how many decades is 6 years? What about one year?
Activity Synthesis
Focus the discussion on why the annual growth rate is not
\(\frac1{10}\)
of 28.4%. Consider asking students:
“Why might someone think the annual population growth rate is 2.84%?” (Because 28.4 divided by 10 is 2.84, and they assume that population was growing linearly, or by the same amount each year, rather than by the same factor each year.)
“How can we determine the growth factor for one year?” (We know that when the annual growth factor, say
\(b\)
, is raised to an exponent of 10, it must equal 1.284, or
\(b^{10} = 1.284\)
, so
\(b\)
must be
\(1.284^\frac{1}{10}\)
or
\(\sqrt[10]{1.284}\)
because raising that number to an exponent of 10 gives 1.284.)
“How do we express the factor as a growth rate? (
\(1.284^\frac{1}{10} \approx 1.0258\)
, so the growth rate is 2.58%.)
Representation: Internalize Comprehension.
Use color and annotations to illustrate student thinking. As students share their reasoning about the last question, scribe their thinking on a visible display. To illustrate why the growth factor for one year is not
\(\frac{1}{10}\)
of that of a decade, it may be helpful to return to a diagram that is configured like a double number line (especially if used in the last lesson). Use the double number line of years and decades to annotate and show the growth factor. When students explain that the growth factor for one decade is 1.284, draw an arrow across the decade line and label it
\(\boldcdot 1.284\)
. Then draw small arrows for the years labeled with
\(\boldcdot b\)
for each segment. Label the decade line with the corresponding equation. Encourage students to refer to the diagram when explaining why
\(b\)
is not one-tenth the growth rate of a decade. When students first explain the use of the fractional exponent to calculate the yearly population growth, label the line representing years with the corresponding equation.
Supports accessibility for: Visual-spatial processing; Conceptual processing


correct_option_index: 1