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: Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
problem_activity: Optional activity
This activity examines how measurement errors behave when they are added together. In other words, if I have a measurement
\(m\)
with a maximum error of 1% and a measurement
\(n\)
with a maximum error of 1%, what percent error can
\(m + n\)
have? In addition to examining accuracy of measurements carefully (MP6), students work through examples and look for patterns (MP8) in order to hypothesize, and eventually show, how percent error behaves when measurements with error are added to one another.
Monitor for students who look for patterns, recognize the usefulness of the distributive property, or formulate the problem abstractly with variables.
Launch
Read the problem out loud and ask students what information they would need to know to be able to solve the problem. Students may say that they need to know what length the boards are supposed to be, because it is likely that they haven't realized that they can solve the problem without this information. Explain that floor boards come in many possible lengths, that 18-inch and 36-inch lengths are both common, but the boards can be anywhere between 12 and 84 inches. Ask students to pick values for two actual lengths and figure out the error in that case. Then they can pick two different examples, make the calculations again, and look for patterns.
Provide access to calculators.
Representation: Internalize Comprehension.
Represent the same information through different modalities by using diagrams. If students are unsure where to begin, suggest that they draw a diagram to help illustrate the information provided.
Supports accessibility for: Conceptual processing; Visual-spatial processing
Student Facing
A wood floor is made by laying multiple boards end to end. Each board is measured with a maximum percent error of 5%. What is the maximum percent error for the total length of the floor?
Student Response
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Anticipated Misconceptions
Some students may pick some example lengths but then struggle with knowing what to do with them. Ask them “what would be the maximum measured lengths? The minimum? What would be the error if both measurements were maximum? What if they were both minimum?”
Some students may pick numbers that make the calculations more complicated, leading to arithmetic errors. Suggest that they choose simple, round numbers for lengths, like 50 inches or 100 centimeters.
Activity Synthesis
The goal of this discussion is for students to generalize from their specific examples of measurements to understand the general pattern and express it algebraically.
Poll the class on the measurements they tried and the maximum percent error they calculated. Invite students to share any patterns they noticed, especially students who recognized the usefulness of the distributive property for making sense of the general pattern.
Guide students to use variables to talk about the patterns more generally.
If a board is supposed to have length
\(x\)
with a maximum percent error of 5%, then the shortest it could be is
\(0.95x\)
and the longest it could be is
\(1.05x\)
.
If another board is supposed to have length
\(y\)
, it could be between
\(0.95y\)
and
\(1.05y\)
.
When the boards are laid end-to-end, the shortest the total length could be is
\(0.95x + 0.95y\)
, which is equivalent to
\(0.95(x + y)\)
.
The longest the total length could be is
\(1.05x + 1.05y\)
, or
\(1.05(x + y)\)
.
Because of the distributive property, we can see that the maximum percent error is still 5% after the board lengths are added together.
One interesting point to make, if students have also done the previous activity about measuring pencils, is that you could
measure
the sum of the board lengths with a lower percent error than you could measure each individual board (assuming your tape measure is long enough), just like an error of 1 mm was a smaller percentage of the length of the longer pencil.


correct_option_index: 0

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