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: Develop Understanding of fractions as numbers.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.

Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

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

Understand decimal notation for fractions, and compare decimal fractions.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Extend understanding of fraction equivalence and ordering.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Use equivalent fractions as a strategy to add and subtract fractions.
problem_activity: Narrative
This activity prompts students to use number lines to illustrate the decomposition of a fraction into sums of other fractions, reinforcing their work from an earlier lesson. Along the way, students recognize that one way to decompose a fraction greater than 1 is to write it as a sum of a whole number and a fraction less than 1. This insight prepares students to interpret and write mixed numbers in later activities.
Action and Expression: Develop Expression and Communication
. Provide access to colored pencils. Invite students to trace or draw each “jump” with a different color. Then invite students to circle or write out each addend in the corresponding color.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing, Organization
Launch
Display or draw this number line:
[IMG]
“What number does the point describe?” (8)
“What do you think the ‘jumps’ represent?” (The numbers that are put together to go from 0 to 8)
“What equations can we write to represent the combination of jumps?” (
\(6 + 2 =8\)
or
\(2 + 6 = 8\)
)
“Let’s look at the jumps on some other number lines and see what they may represent.”
Activity
Groups of 2
“Work independently on the activity for a few minutes. Afterwards, share your responses with your partner.”
5–7 minutes: independent work time
2 minutes: partner discussion
Student Facing
On each number line, draw two “jumps” to show how to use sixths to make a sum of
\(\frac{8}{6}\)
. Then, write an equation to represent each combination of jumps.
[IMG]
[IMG]
Noah draws the following diagram and writes:
\(\frac{8}{6} = \frac{6}{6} + \frac{2}{6}\)
and
\(\frac{8}{6} = 1 + \frac{2}{6}\)
. Which equation is correct? Explain your reasoning.
[IMG]
On each number line, draw “jumps” to show how to use thirds to make a sum of
\(\frac{7}{3}\)
. Then, write an equation to represent each combination of jumps.
[IMG]
[IMG]
Write
\(\frac{7}{3}\)
as a sum of a whole number and a fraction.
Student Response
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Advancing Student Thinking
Students may not make a connection between the fractions in the problems and the number lines because the latter show no fractional labels. Consider asking students to label the number line to show fractions or asking, “Where do you think
\(\frac{1}{6}\)
would be labeled on this number line?”
Activity Synthesis
Invite students to share their equations. Record them for all to see.
Focus the discussion on part b: writing
\(\frac{7}{3}\)
as a sum of a whole number and a fraction. Students are likely to write
\(1 + \frac{4}{3}\)
and
\(2 + \frac{1}{3}\)
.
Explain that
\(2 + \frac{1}{3}\)
can be written as
\(2\frac{1}{3}\)
, which we call a
mixed number
. This number is equivalent to
\(\frac{7}{3}\)
.
“Why might it be called a mixed number?” (It is a mix of a whole number and a fraction.)


correct_option_index: 1