Mateo is working with the fraction $\frac{4}{6}$ and is using a number line to find equivalent fractions with different denominators. Let's break this problem down step by step:

1. Understanding Equivalent Fractions: Fractions are equivalent if they represent the same portion of a whole. For $\frac{4}{6}$, equivalent fractions can be found by multiplying or dividing both the numerator and denominator by the same number.

2. Finding an Equivalent Fraction with a Denominator Greater than 6:

   • To find an equivalent fraction with a denominator greater than 6, multiply both the numerator and denominator by the same number. For example: $\frac{4}{6} \times \frac{2}{2} = \frac{8}{12}.$ Thus, $\frac{8}{12}$ is equivalent to $\frac{4}{6}$, and its denominator is greater than 6.

3. Finding an Equivalent Fraction with a Denominator Less than 6:

   • To find an equivalent fraction with a smaller denominator, divide both the numerator and denominator by their greatest common factor (GCF). The GCF of 4 and 6 is 2, so: $\frac{4}{6} \div \frac{2}{2} = \frac{2}{3}.$ Thus, $\frac{2}{3}$ is equivalent to $\frac{4}{6}$, and its denominator is less than 6.

4. Using the Number Line:

   • The number line shows fractions divided into sixths (denominator = 6). Mateo can relabel the number line by dividing each interval into more parts (for a greater denominator) or by grouping intervals together (for a smaller denominator) to find these equivalent fractions.

Would you like further explanation of equivalent fractions or help visualizing the relabeling process?

5 Related Questions:

1. How can we verify that two fractions are equivalent?
2. What happens to a fraction when both the numerator and denominator are multiplied by the same number?
3. Can you find more equivalent fractions for $\frac{4}{6}$?
4. What does it mean to simplify a fraction, and how is it related to equivalent fractions?
5. How would the number line look if divided into twelfths instead of sixths?

Tip:

Always simplify fractions to their lowest terms by dividing both the numerator and denominator by their GCF to make comparisons easier!