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: Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
problem_activity: Narrative
In this activity, students continue to use the idea of partitioning a number line into smaller increments to reason about and generate equivalent fractions. Through repeated reasoning, students begin to see regularity in how the process of decomposing parts on a number line produces the numbers in the equivalent fractions (MP8). The task encourages students to think of the relationship between one denominator and the other in terms of factors or multiples (even if they don’t use those terms which connect to work in a previous unit).
Partitioning a number line into smaller parts becomes increasingly inconvenient when the denominator gets larger. As students begin to think about the relationship between tenths and hundredths, they see some practical limitations to using a number line to find equivalent fractions and are prompted to generalize the process of partitioning. (Students are not expected to draw a full number line with 100 parts.)
Launch
Groups of 2
Read the first problem as a class. Ask students to think quietly for a moment about whether what Priya wants to do can be done.
30 seconds: quiet think time
1 minute: partner discussion
Activity
“Take about 7–8 quiet minutes to work on the task. Afterwards, discuss your responses with your partner.”
7–8 minutes: independent work time
2–3 minutes: partner discussion
For the first problem, monitor for students who partition the lines by:
guessing and checking
reasoning multiplicatively (3 times what number gives 9, 10, or 12?) or in terms of multiples (Is 9, 10, or 12 a multiple of 3?)
reasoning in terms of division (9 divided by 3 is what number?) or in terms of factors (Is 3 a factor of 9, 10, or 12?)
For the second and third problem, monitor for students who find equivalent fractions for
\(\frac{1}{10}\)
by:
partitioning the number lines into smaller increments quantifying the new number of parts
finding multiples of 1 and 10, and using this strategy to write an equivalent fraction with denominator 100
Student Facing
Priya wants to find fractions that are equivalent to
\(\frac{2}{3}\)
, other than
\(\frac{4}{6}\)
. She wonders if she can find equivalent fractions with denominator 9, 10, and 12.
\(\frac{\phantom{000}}{9} \qquad \frac{\phantom{000}}{10} \qquad \frac{\phantom{000}}{12}\)
Can it be done? Use number lines to show your reasoning.
[IMG]
[IMG]
[IMG]
[IMG]
Represent
\(\frac{1}{10}\)
on a number line. Then, find two fractions that are equivalent to
\(\frac{1}{10}\)
. How would you use the number line to show that they are equivalent to
\(\frac{1}{10}\)
?
[IMG]
[IMG]
Can you find an equivalent fraction for
\(\frac{1}{10}\)
with 100 for the denominator? Explain or show your reasoning.
Student Response
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Advancing Student Thinking
If students conclude that the answer to the last question is “no, it can’t be done” because they find it impractical to partition each tenth on the number line into so many parts, ask them to visualize the line and think about how they’d partition it. Consider asking: “Into how many parts should each one-tenth section on the line be split to get hundredths? How do you know?”
Activity Synthesis
Invite previously identified students to share their strategies for answering the first two problems.
If not done by students in their explanations, consider asking students to revoice their reasoning in terms of factors and multiples.
See lesson synthesis.


correct_option_index: 0