Guidance: Original Problem Mr. Smith Is Thinking Of A

Guidanceïƒ Original Problem: Mr. Smith is thinking of a

The assignment involves multiple math related problems focusing on solving equations, understanding and graphing inequalities, and applying problem-solving strategies. The specific tasks include solving a basic algebraic problem, explaining the steps and strategies used, creating similar practice problems, understanding how to graph inequalities, and demonstrating the solution and graph for given inequalities.

Firstly, the core task is to determine the number Mr. Smith is thinking of, based on the equation "three times the number minus 12 equals 36". Additionally, the student is asked to explain the "Guess, Check, and Revise" problem-solving strategy, list its seven steps, and create two similar word problems with solutions. They also need to clarify how to interpret open and closed circles on a graph of inequalities, what shading signifies, and then to graph and solve two inequalities: 9 – x > 4 and –6t

Paper For Above instruction

Solving algebraic equations and understanding inequalities are foundational components of mathematics that bridge abstract concepts with real-world applications. The problem involving Mr. Smith provides a straightforward example of solving a linear equation, which is essential in developing algebraic reasoning skills. Furthermore, understanding how to graph inequalities enhances spatial reasoning and supports the visualization of solutions, facilitating a deeper comprehension of relationships between variables.

Solving the Equation: Mr. Smith’s Number

The problem states: "Mr. Smith is thinking of a number. Three times the number minus 12 equals 36." Mathematically, this is written as:

3x - 12 = 36

To find x, the number Mr. Smith is thinking of, we follow algebraic steps. First, add 12 to both sides of the equation to isolate the term with x:

3x - 12 + 12 = 36 + 12

which simplifies to:

3x = 48

Next, divide both sides by 3 to solve for x:

x = 48 ÷ 3 = 16

Therefore, the number Mr. Smith is thinking of is 16.

Explanation of the Guess, Check, and Revise Problem Solving Strategy

The Guess, Check, and Revise strategy is an iterative problem-solving process often used in mathematics to approach complex problems systematically. The seven steps typically include:

1. Understand the problem: Read carefully to grasp what is being asked.
2. Make an initial guess: Think of a reasonable estimate based on the problem context.
3. Calculate or check the guess: Substitute the guess into the problem or equation to see if it satisfies all conditions.
4. Analyze the result: Determine whether the guess was too high, too low, or correct.
5. Revise the guess: Adjust the estimate accordingly.
6. Repeat the process: Continue guessing, checking, and revising until reaching an acceptable solution.
7. Verify the solution: Confirm that the final answer satisfies the original problem.

This strategy encourages patience and systematic reasoning, especially useful when solutions are not immediately apparent or when working with complex problems.

Create Two Similar Word Problems for Practice

Problem 1: Sarah is thinking of a number. Four times the number plus 8 equals 24. What is the number?

Answer: Let the number be x. Thus, 4x + 8 = 24. Subtract 8 from both sides: 4x = 16. Divide both sides by 4: x = 4. The number is 4.

Problem 2: A bakery sold a total of 60 cookies in a day. If they sold twice as many chocolate chip cookies as oatmeal cookies, how many of each type did they sell?

Answer: Let o = number of oatmeal cookies, c = number of chocolate chip cookies. Then, c = 2o, and c + o = 60. Substitute c: 2o + o = 60 → 3o = 60 → o = 20. Therefore, c = 2(20) = 40. They sold 20 oatmeal cookies and 40 chocolate chip cookies.

Understanding Graphing Inequalities: Open/Closed Circles and Shading

When graphing inequalities, the type of circle—open or closed—indicates whether the solution includes the boundary point. A closed circle signifies that the number is included in the solution set (e.g., ≥ or ≤), whereas an open circle indicates that the boundary point is excluded (e.g., > or

Shading on the number line shows the set of solutions that satisfy the inequality. If the inequality is greater than or equal to or less than or equal to, shading includes the boundary point, which corresponds to a closed circle. For strict inequalities, shading is to the right (for > or ≥) or to the left (for or <.>

The shading represents all possible values of the variable that satisfy the inequality, providing a visual way to interpret the solutions.

Graphing and Solving the Inequalities

a. 9 – x > 4

To solve for x:

1. Subtract 9 from both sides: 9 – x – 9 > 4 – 9
2. -x > -5
3. Multiply both sides by -1 (remember to flip the inequality sign): x

This inequality states that x is less than 5. When graphing, draw an open circle at 5 and shade to the left, indicating all values less than 5 satisfy the inequality.

b. –6t

To solve for t:

1. Divide both sides by -6 (and flip the inequality sign): t > -2

In graphing, draw an open circle at -2 and shade to the right, representing all t-values greater than -2.

This visual representation helps understand the solution sets and their relation to the boundary points, considering whether the boundary is included or excluded based on the inequality symbol.

Conclusion

Mastering algebraic equations and inequalities requires comprehension of key concepts such as solving equations, graphing inequalities, and applying strategic problem-solving methods like Guess, Check, and Revise. Clear understanding of how to interpret graph features—like open or closed circles and shading—is essential for visualizing solutions. Practice through creating similar problems reinforces understanding, while systematic methods ensure solutions are accurate and reliable. As these skills develop, students enhance their quantitative reasoning, which is vital beyond the classroom into everyday problem-solving and advanced mathematics.

References

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• Blitzer, R. (2018). Algebra and Trigonometry. Pearson.
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• National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. NCTM.
• Larson, R., Hostetler, R., & Edwards, B. (2012). Elementary and Intermediate Algebra. Brooks Cole.
• Mathematics Assessment Resource Service. (2002). Developing Algebra. University of Wisconsin-Madison.
• Khan Academy. (n.d.). Solving inequalities. https://www.khanacademy.org/math/algebra/inequalities
• Paul, D. (2017). The Art of Problem Solving. MathCounts Foundation.
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• Merrill, P. (2014). Creative Approaches to Teaching Mathematics. National Council of Teachers of Mathematics.