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.


Mathematical concept/skill options: 
A. Write and interpret numerical expressions.
B. Generate and analyze patterns.
C. Analyze patterns and relationships.
D. Use properties of operations to generate equivalent expressions.
E. Use the four operations with whole numbers to solve problems.
F. Work with radicals and integer exponents.
G. Solve systems of equations.
H. Represent and solve problems involving multiplication and division.
I. Perform arithmetic operations on polynomials.
J. Understand the relationship between zeros and factors of polynomials.
K. Understand properties of multiplication and the relationship between multiplication and division.
L. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
M. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
N. Use polynomial identities to solve problems.
O. Apply and extend previous understandings of arithmetic to algebraic expressions.
P. Rewrite rational expressions.
Q. Interpret the structure of expressions.
R. Add and subtract within 20.
S. Analyze and solve linear equations and pairs of simultaneous linear equations.
T. Represent and analyze quantitative relationships between dependent and independent variables.
U. Solve equations and inequalities in one variable.
V. Create equations that describe numbers or relationships
W. Reason about and solve one-variable equations and inequalities.
X. Represent and solve equations and inequalities graphically.
Y. Understand solving equations as a process of reasoning and explain the reasoning.
Z. Understand and apply properties of operations and the relationship between addition and subtraction.
AA. Gain familiarity with factors and multiples.
AB. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
AC. Work with addition and subtraction equations.
AD. Multiply and divide within 100.
AE. Work with equal groups of objects to gain foundations for multiplication.
AF. Represent and solve problems involving addition and subtraction.
AG. Write expressions in equivalent forms to solve problems.
AH. Understand the connections between proportional relationships, lines, and linear equations.


Thought: This task teaches students to understand addition as putting together counters, or adding to already placed counters.
Answer: AB

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


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Thought: This problem involves understanding fractions, multiplication, and division.
Answer: A

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


Mathematical concept/skill options: 
A. Prove and apply trigonometric identities.
B. Model periodic phenomena with trigonometric functions.
C. Interpret functions that arise in applications in terms of the context.
D. Build a function that models a relationship between two quantities.
E. Define, evaluate, and compare functions.
F. Use functions to model relationships between quantities.
G. Understand the concept of a function and use function notation.
H. Analyze functions using different representations.
I. Extend the domain of trigonometric functions using the unit circle.
J. Interpret expressions for functions in terms of the situation they model.
K. Build new functions from existing functions.
L. Construct and compare linear and exponential models and solve problems.


Thought: This problem applies trigonometric identities, including the angle addition formula for cosine.
Answer: A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Represent and model with vector quantities.

Apply and extend previous understandings of numbers to the system of rational numbers.

Extend the properties of exponents to rational exponents.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Perform arithmetic operations with complex numbers.

Reason quantitatively and use units to solve problems.

Use complex numbers in polynomial identities and equations.

Perform operations on vectors.

Compute fluently with multi-digit numbers and find common factors and multiples.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Represent complex numbers and their operations on the complex plane.

Perform operations on matrices and use matrices in applications.

Use properties of rational and irrational numbers.
problem_activity: Activity
This activity has two parts and serves two purposes. The first two problems aim to help students see the limits of using base-ten diagrams to add and subtract numbers and to think about choosing more efficient methods. The latter three questions prompt students to reason about addition and subtraction of numbers with more decimal places in the context of situations. Students need to determine the appropriate operations for the given situations, perform the calculations, and then relate their answers back to the contexts (MP2).
As students work, check that they remember to line up the decimal points, write zeros when needed, and label their answers with units of measurement (for the last three questions). Identify one student to present each question. For the last question, there are two likely methods for solving (as outlined in the solutions). If both methods arise, they should both be presented.
Launch
Ask students to read the first 2 problems and think about which method of reasoning might work best. Give students 5 minutes of quiet time to work on these problems. Observe which methods students choose and check their work for accuracy.
Afterwards, take a few minutes to discuss their answers and methods. Ask one or more of the following questions before moving onto the second half of the activity:
“Would base-ten diagrams work well for
\(318.8 - 94.63\)
? Why or why not?” (No, it is time consuming to draw and keep track of so many pieces. Yes, I can draw 3 large squares for 300, 1 rectangle for 10, 8 small squares for 8, etc., remove pieces in the amount of 94.63, and count what is left.)
“What challenges did you face in computing
\(0.02 - 0.0116\)
using a diagram?” (I needed to take away 116 ten-thousandths from 2 hundredths, so I had to exchange hundredths with ten-thousandths, which meant drawing or showing many smaller pieces.) “What about using vertical calculations?” (The first number is two digits shorter than the second, which meant some extra steps.)
“When might base-ten diagrams be a good choice for finding a difference (or sum)?” (When the numbers are simple or short so the diagrams do not take too long to draw.)
“When might an algorithm be a good choice for finding a difference (or sum)?” (For a difference like
\(6.739258 - 0.8536672\)
, it will be time consuming to draw pictures and label each place value. The algorithm takes some time as well but is more efficient than drawing pictures.)
After this discussion, have students work independently on the remaining problems. During this time, monitor and select individuals to share their work during the whole-class discussion.
Action and Expression: Internalize Executive Functions.
To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Check to make sure students have proper decimal point and place value unit alignment prior to calculating the value of each expression. Some students may benefit from access to graph paper to aide in proper alignment of decimals.
Supports accessibility for: Organization; Visual-spatial processing
Reading: MLR6 Three Reads.
Use this routine to support reading comprehension of the problem about Lin’s grandmother. After a shared reading of the question, ask students “what is this situation about?” (Lin’s grandmother ordered needles; the needles were different lengths). Use this read to clarify any unfamiliar language such as pharmacist or administer. After the second read, students list any quantities that can be counted or measured, without focusing on specific values (length of each type of needle, in inches). After the third read, invite students to discuss possible strategies to solve the problem. If needed, repeat this support with the remaining word problems. This helps students connect the language in the word problem and the reasoning needed to solve the problem.
Design Principle(s): Support sense-making
Student Facing
Find the value of each expression. Show your reasoning.
\(11.3 - 9.5\)
\(318.8 - 94.63\)
\(0.02 - 0.0116\)
Discuss with a partner:
Which method or methods did you use in the previous question? Why?
In what ways were your methods effective? Was there an expression for which your methods did not work as well as expected?
Lin’s grandmother ordered needles that were 0.3125 inches long to administer her medication, but the pharmacist sent her needles that were 0.6875 inches long. How much longer were these needles than the ones she ordered? Show your reasoning.
There is 0.162 liter of water in a 1-liter bottle. How much more water should be put in the bottle so it contains exactly 1 liter? Show your reasoning.
One micrometer is 1 millionth of a meter. A red blood cell is about 7.5 micrometers in diameter. A coarse grain of sand is about 70 micrometers in diameter. Find the difference between the two diameters in
meters
. Show your reasoning.
Student Response
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Anticipated Misconceptions
Students may benefit from using graph paper for calculations; they can place one digit in each box for proper decimal point placement and place value unit alignment. For the last question, students might confuse the amount of zeros in 1 millionth. It might be helpful to write 0.000001 on the board as reference.
Activity Synthesis
Discussions on the first 2 questions are outlined in the Launch section. Focus the discussion here on the last 3 questions, i.e., on how students interpreted the problems, determined which operation to perform, and found the sum or difference (including how they handled any regroupings). Select a few previously identified students to share their responses and reasoning.
If some students chose base-ten diagrams for the calculation, contrast its efficiency with that of numerical calculations. The numbers in these problems have enough decimal places that using base-ten representations (physical blocks, drawings, or digital representations) would be cumbersome, making a numerical calculation appealing.


correct_option_index: 9