Mrs. Matthews splits her math class into groups for a projects. There are x groups of 4 scholars and there are 3 scholars remaining. Mr Riley's math class has double the amount of groups and no scholars that do not belong in a group. Write an expression to represent the total number of scholars in both math classes. Why does the meaning of the expression depend on the definition of the variable?

Let's break down the problem to find an expression for the total number of scholars in both Mrs. Matthews' and Mr. Riley's math classes.

### Step 1: Determine the number of scholars in Mrs. Matthews' class.
- Mrs. Matthews has \( x \) groups of 4 scholars each.
- Therefore, the number of scholars in groups is \( 4x \).
- Additionally, there are 3 scholars remaining, so the total number of scholars in Mrs. Matthews' class is:
\[
4x + 3
\]

### Step 2: Determine the number of scholars in Mr. Riley's class.
- Mr. Riley's class has double the number of groups, so he has \( 2x \) groups.
- Each group also has 4 scholars, so the total number of scholars in Mr. Riley's class is:
\[
4(2x) = 8x
\]

### Step 3: Combine the totals from both classes.
- The total number of scholars in both math classes is the sum of the scholars from Mrs. Matthews' and Mr. Riley's classes:
\[
(4x + 3) + 8x = 12x + 3
\]

### Interpretation of the Expression

The expression \( 12x + 3 \) depends on the definition of the variable \( x \). Specifically:
- \( x \) represents the number of groups in Mrs. Matthews' class.
- The meaning of the expression changes if the value of \( x \) changes. For example, if \( x \) represents a different number of groups, the total number of scholars would be different.
- The expression shows how the total number of scholars is linearly related to the number of groups in Mrs. Matthews' class. The coefficient 12 accounts for both classes, while the constant 3 represents the scholars who are not in any group in Mrs. Matthews' class.

### Conclusion
The meaning of the expression \( 12x + 3 \) depends on the variable \( x \) because \( x \) directly influences the total number of scholars calculated. Different definitions or values for \( x \) will lead to different totals, illustrating the importance of understanding what \( x \) represents.

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Do you have any questions or need further details on this? Here are some related questions to consider:

1. How would the expression change if Mrs. Matthews had 2 scholars remaining instead of 3?
2. What if Mr. Riley had three times the number of groups instead of double?
3. Can you generalize the expression for any number of remaining scholars in Mrs. Matthews' class?
4. What would the total number of scholars be if \( x = 5 \)?
5. How would the expression look if each group had 5 scholars instead of 4?
6. If \( x = 0 \), what does the expression \( 12x + 3 \) tell us about the number of scholars?
7. How does the structure of the expression \( 12x + 3 \) reflect the distribution of scholars in the groups?
8. What is the significance of the coefficient 12 in the expression?

**Tip:** Always ensure that you clearly define your variables when creating mathematical expressions to avoid confusion and ensure accurate interpretation.

Understand how to calculate the total number of scholars in Mrs. Matthews' and Mr. Riley's math classes using the expression 12x + 3. Learn about arithmetic, linear equations, and algebra in this problem-solving scenario. Suitable for students in Grades 6-8.

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