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: Write a function that describes a relationship between two quantities.

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
problem_activity: Activity
The pattern in this activity creates the Fibonacci sequence. One purpose of this task is for students to understand that sometimes it is straightforward to define a sequence recursively and difficult to determine a definition for the
\(n^{\text{th}}\)
term. This task also challenges students to expand their notion of recursive definitions since the current term in a Fibonacci sequence depends on the previous two terms rather than just the previous term.
This sequence was chosen to provide a contrast to the cake activity. With the cake, students had a situation whose domain does not reasonably extend above a certain value given physical constraints. Since this sequence is a mathematical construct, the domain has no upper bound restrictions on it.
Launch
Distribute 1 piece of graph paper to each student. Ask them to complete the first question and pause. Ensure that they are interpreting the description of the pattern correctly before they proceed with the rest of the activity.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time.
Supports accessibility for: Organization; Attention
Student Facing
On graph paper, draw a square of side length 1. Draw another square of side length 1 that shares a side with the first square. Next, add a 2-by-2 square, with one side along the sides of
both
of the first two squares. Next, add a square with one side that goes along the sides of the previous two squares you created. Next, do it again.
Pause here for your teacher to check your work.
Write a sequence that lists the side lengths of the squares you drew.
Predict the next two terms in the sequence and draw the corresponding squares to check your predictions.
Describe how each square’s side length depends on previous side lengths.
Let
\(F(n)\)
be the side length of the
\(n\)
th
square. So
\(F(1)=1\)
and
\(F(2)=1\)
. Write a recursive definition for
\(F\)
.
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?
Is the Fibonacci sequence arithmetic, geometric, or neither? Explain how you know.
Look at quotients
\(\frac{F(1)}{F(0)}, \frac{F(2)}{F(1)}, \frac{F(3)}{F(2)}, \frac{F(4)}{F(3)}, \frac{F(5)}{F(4)}\)
. What do you notice about this sequence of numbers?
The 15th through 19th Fibonacci numbers are 610, 987, 1597, 2584, 4181. What do you notice about the quotients
\(\frac{F(16)}{F(15)}, \frac{F(17)}{F(16)}, \frac{F(18)}{F(17)}, \frac{F(19)}{F(18)}\)
?
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
Some students may not realize that it is okay to use both
\(F(n-1)\)
and
\(F(n-2)\)
when defining a function recursively. Encourage these students to think about how the “previous side lengths” for term
\(F(n)\)
can be expressed using function notation.
Activity Synthesis
Display the sequence 1, 1, 2, 3, 5, 8, 13 and the recursive definition
\(F(1)=1, F(2)=1, \)
\(F(n)=F(n-1)+F(n-2)\)
,
\(n\ge3\)
for all to see and check their answer against. Either give students a minute to think about how to write a definition for the
\(n^{\text{th}}\)
term of the sequence, or just tell them that it is more complicated to derive than class time allows. An important modeling point to make is that the goal is to create a model that suits the needs of what you are trying to do. If we wanted to know the value of
\(F(100)\)
, then a spreadsheet and a recursive definition is likely going to be faster than working out a definition for the
\(n^{\text{th}}\)
term.
Ask students, "What is the domain of this function? Does it have any restrictions?" (The integers
\(n\)
where
\(n\ge1\)
or equivalent. We can always add another square onto the image, even if only in our imaginations once the squares get too big.) Select students to share their responses.
If time allows, tell students that we could define the Fibonacci sequence as
\(F(0)=1, F(1)=1, \)
\(F(n)=F(n-1)+F(n-2)\)
,
\(n\ge2\)
and we would still get the same list of numbers.


correct_option_index: 1