Question

With two minutes left in a basketball game, your team is 12 points behind. What are two different numbers of 2 -point and 3-point shots your team could score to earn at least 12 points? Write a verbal model for the situation. Assign labels to each part of the verbal model and write an inequality.

Short answer

Using the points system, the two different combinations to score at least 12 points can be: a) 6 two-point shots (x=6 and y=0) or b) 4 three-point shots (x=0 and y=4). The inequality representing this situation is 2x + 3y ≥ 12.

Step-by-step solution

Identifying Points System

The problem states that in basketball, shots can be either 2-point or 3-point. Label the number of 2-point shots as 'x' and the number of 3-point shots as 'y'.

First Combination of Shots

To make up 12 points, one can score 6 two-point shots. Hence, in this case, x=6 and y=0.

Second Combination of Shots

Alternatively, one can score 4 three-point shots. In this scenario, x=0 and y=4.

Formulate the Inequality

The objective of the game is to score at least 12 points. Therefore, the inequality will be 2x + 3y ≥ 12, where x is the number of 2-point shots and y is the number of 3-point shots.

Key concepts

Inequality

Understanding inequalities in algebra is essential for solving problems that require a range of solutions. An inequality is similar to an equation, but instead of the equals sign, it uses symbols to indicate that one side is greater than, less than, greater than or equal to, or less than or equal to the other side. In the context of a basketball game, scoring at least 12 points to catch up can be represented by an inequality.

Let's take the basketball problem as an example. By using the inequality \(2x + 3y \geq 12\), we state that the team needs to score a minimum of 12 points to tie or beat the opponent's lead. Here, \(x\) represents the number of 2-point shots, and \(y\) represents the number of 3-point shots. This inequality helps us find various combinations of \(x\) and \(y\) that satisfy the condition of scoring at least 12 points.

As students approach problem-solving, they should remember that an inequality signifies a set of possible answers rather than a single solution. They must check for all possible pairs of \(x\) (2-point shots) and \(y\) (3-point shots) that, when plugged into the inequality, make the statement true.

Verbal Model

A verbal model uses words to describe a real-world situation mathematically. Creating a verbal model involves identifying what we need to find, what we know, and how to use what we know to find what we need. It acts as a bridge between a real-world scenario and its mathematical representation.

In our basketball scenario, a verbal model assists us in formulating the algebraic inequality. The verbal model might be: 'The team needs a combination of 2-point and 3-point shots that total at least 12 points to tie the game.' From there, we label each part of the model: '2-point shot' as \(x\) and '3-point shot' as \(y\). We then use these labels to construct the inequality \(2x + 3y \geq 12\), representing the verbal model in mathematical terms.

Training in verbal models equips students with the skill to translate everyday situations into mathematical language, which is often the first step in solving algebraic problems. By mastering the construction of a verbal model, students gain the ability to approach complex problems by breaking them down into manageable algebraic expressions.

2-Point and 3-Point Shots

In basketball scoring, 2-point and 3-point shots are the main ways teams earn points during a game. A 2-point shot is scored when a player shoots the ball through the hoop from inside the three-point line, while a 3-point shot is awarded for successful attempts from beyond the three-point line.

When solving algebra problems involving basketball scoring, the difference between these shot types is crucial. As we established in our example problem, each 2-point shot is denoted by \(x\) and each 3-point shot by \(y\). This distinction allows us to explore different scoring strategies.

For example, if a team prioritizes 2-point shots, they may choose a strategy to maximize \(x\) while minimizing \(y\), as in \(x=6\), \(y=0\). Conversely, if they are confident in their 3-point shooting, they might minimize \(x\) and maximize \(y\), with \(x=0\), \(y=4\). It's essential to recognize that basketball scoring problems are often open-ended, with multiple solutions, each combination of \(x\) and \(y\) presenting a different strategic approach to the game.

Students should also keep in mind the real-world implications of focusing on either type of shot, such as the time it takes to set up a 3-point shot versus a 2-pointer, and the respective likelihood of scoring based on skill and defensive pressure.