few_shots: Example:
Problem/activity:
Task
Cruz and Erica were both getting ready for soccer.
Cruz ran 1 lap around the school.
Erica ran 3 laps around the playground.
Erica said,
I ran more laps, so I ran farther.
Cruz said,
4 laps around the school is 1 mile, but it takes 12 laps around the playground to go 1 mile. My laps are much longer, so I ran farther.
Who is right? Draw a picture to help you explain your answer.


Mathematical concept/skill options: 
A. 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.
B. 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.
C. Use equivalent fractions as a strategy to add and subtract fractions.
D. Develop Understanding of fractions as numbers.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
E. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
F. 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.


Your response, in json format:
{
  "explanation": "In this problem, students learn to recognize equivalent fractions. That is, three laps around the playground is 3/12 mile, which is equivalent to 1/4, which is one lap around the school.",
  "answer": "B"
}

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_000212

4
options: Analyze and solve linear equations and pairs of simultaneous linear equations.

Understand solving equations as a process of reasoning and explain the reasoning.

Use the four operations with whole numbers to solve problems.

Perform arithmetic operations on polynomials.

Work with equal groups of objects to gain foundations for multiplication.

Gain familiarity with factors and multiples.

Write and interpret numerical expressions.

Solve systems of equations.

Add and subtract within 20.

Understand properties of multiplication and the relationship between multiplication and division.

Create equations that describe numbers or relationships

Interpret the structure of expressions.

Represent and solve problems involving addition and subtraction.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Rewrite rational expressions.

Use polynomial identities to solve problems.

Use properties of operations to generate equivalent expressions.

Write expressions in equivalent forms to solve problems.

Work with radicals and integer exponents.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Analyze patterns and relationships.

Work with addition and subtraction equations.

Understand and apply properties of operations and the relationship between addition and subtraction.

Reason about and solve one-variable equations and inequalities.

Solve equations and inequalities in one variable.

Represent and solve problems involving multiplication and division.

Understand the connections between proportional relationships, lines, and linear equations.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Multiply and divide within 100.

Represent and solve equations and inequalities graphically.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Generate and analyze patterns.

Understand the relationship between zeros and factors of polynomials.

Represent and analyze quantitative relationships between dependent and independent variables.
problem_activity: Activity
In this activity, students see that taking the square root of each side of an equation doesn’t necessarily preserve solutions like other algebraic moves they have used in the past.
Launch
Arrange students in groups of 2. Encourage students to discuss their responses with a partner as they work. If they disagree with their partner, they should work to reach agreement.
Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve their written responses for Priya’s question. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “How do the graphs show that there are two solutions?”, “What is the definition of a square root?”, and “What happened when Tyler took the square root of each side?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify why the equation
\(x^2-1=3\)
must have two solutions.
Design Principle(s): Optimize output (for justification); Cultivate conversation
Student Facing
Tyler was solving this equation:
\(\displaystyle x^2 - 1 = 3\)
He said, “I can add 1 to each side of the equation and it doesn’t change the equation. I get
\(x^2 = 4\)
.”
Priya said, “It does change the equation. It just doesn’t change the solutions!” Then she showed these two graphs.
Figure A
[IMG]
Figure B
[IMG]
How can you see the solution to the equation
\(x^2 - 1 = 3\)
in Figure A?
How can you see the solution to the equation
\(x^2 = 4\)
in Figure B?
Use the graphs to explain why the equations have the same solutions.
Tyler said, “Now I can take the square root of each side to get the solution to
\(x^2 = 4\)
. The square root of
\(x^2\)
is
\(x\)
. The square root of 4 is 2.” He wrote:
\(\begin{align}x^2 &= 4\\ \sqrt{x^2} &= \sqrt{4} \\ x &= 2 \end{align}\)
Priya said, “But the graphs show that there are
two
solutions!” What went wrong?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Invite students to share their responses. Discuss the connection between the graphs and the solutions to the equations. The important idea is that the operation of taking the square root results in a positive number, but there are two square roots of every non-zero number, one positive and one negative. This means that when Tyler takes the square root of both sides of the equation, he does not get an equivalent equation. To find all possible solutions, he should remember that
\(x^2\)
has two roots,
\(x\)
and
\(\text-x\)
, and that 4 has two square roots, 2 and -2. This gives him the pair of equations
\(x=2\)
and
\(x=\text-2\)
.
Representation: Internalize Comprehension.
Use color-coding and annotations to highlight connections between representations in a problem. As students share their reasoning about the solutions to the equations, scribe their thinking on a visible display. Use the same color to highlight the connection between the graphs and the solutions to the equations.
Supports accessibility for: Visual-spatial processing; Conceptual processing


correct_option_index: 1