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.


Concept/skill: 
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.

Answer: 
Yes. This task involves representing addition with objects such as number cards and dice.

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
$$-\frac{1}{4}$$


Concept/skill: 
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

Answer: 
Yes. 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).

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


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Answer: 
Yes. 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.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: im_task_001477_pos_0

K

fl_problem_001097_pos_0

7

fl_problem_001894_pos_0

HS_0
standard_description: 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).
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?” (
\(100 + 40 + 3\)
)


aligns: yes