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.


Your response:
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.


Your response:
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.


Your response:
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: Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:
- 100 can be thought of as a bundle of ten tens — called a “hundred.”
- The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Count within 1000; skip-count by 5s, 10s, and 100s.
problem_activity: Narrative
The purpose of this activity is for students to write three-digit numbers as the sum of the value of each digit,
expanded form
. Students connect the order and values of the addends in expanded form to the order and value of each place in a three-digit number. Use expanded form and its definition interchangeably throughout the activity so that students feel comfortable with the new vocabulary. When students represent numbers as sums by place value, they interpret the three-digit numbers in terms of its digits and the operation of addition (MP7).
MLR8 Discussion Supports.
Provide all students with an opportunity to produce mathematical language by inviting students to chorally repeat “expanded form” in unison 1–2 times.
Advances: Speaking
Required Materials
Materials to Gather
Base-ten blocks
Launch
Groups of 2
Display Andre, Tyler, and Mai’s situation and the image of their blocks.
“What would the expression look like?”
1 minute: independent work time
1 minutes: partner discussion
Share responses.
Display 357 and
\(300 + 50 + 7\)
.
“We can represent the value of the blocks by writing a three-digit number.”
“A number can also be represented as a sum of the value of each of its digits. This is called
expanded form
.”
“Like a three-digit number, expanded form shows the sum starting with the place that has the greatest value on the left to the place with the least value on the right.”
As needed, discuss reasons why any expressions generated in the launch would or would not be examples of expanded form.
Activity
“Now you will practice writing the value of the base-ten diagrams in expanded form and as a three-digit number.”
7 minutes: partner work time
Student Facing
Andre has 3 hundreds. Tyler has 5 tens. Mai has 7 ones. They want to represent the amount they have using an equation.
[IMG]
Write an expression to represent the sum of their values.
__________ + __________ + __________
Write the total value as a three-digit number:
_______________
Write each number as the sum of hundreds, tens, and ones, and as a three-digit number.
[IMG]
Expanded form: _________________________
Three-digit number: _________________________
[IMG]
Expanded form: _________________________
Three-digit number: _________________________
[IMG]
Expanded form: _________________________
Three-digit number: _________________________
[IMG]
Expanded form: _________________________
Three-digit number: _________________________
Student Response
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Activity Synthesis
“What is the value of
\(40 + 100 + 3\)
? Explain how you know.” (143. I saw there was 1 hundred, 4 tens, and 3 ones. I just rearranged it in my head like expanded form.)
“We know we can rearrange the addends to add in any order when we find the value of a sum. When a number is written in expanded form the values are written in place value order.”
“How would we write 143 in expanded form?” (
\(100 + 40 + 3\)
)


correct_option_index: 0

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