31. Communicate and Justify Which is greater, \( \frac{4}{7} \times \frac{1}{4} \) or \( \frac{4}{7} \times \frac{1}{6} \) ? Explain,

To compare \( \frac{4}{7} \times \frac{1}{4} \) and \( \frac{4}{7} \times \frac{1}{6} \), we can look at the fractions involved. Since \( \frac{4}{7} \) is common to both expressions, we only need to compare \( \frac{1}{4} \) and \( \frac{1}{6} \). Because \( \frac{1}{4} \) is greater than \( \frac{1}{6} \) (the smaller the denominator, the greater the fraction), we conclude that \( \frac{4}{7} \times \frac{1}{4} \) is greater than \( \frac{4}{7} \times \frac{1}{6} \). Thus, \( \frac{4}{7} \times \frac{1}{4} > \frac{4}{7} \times \frac{1}{6} \). For additional clarity, think of these fractions as pieces of pizza! If you have a pizza cut into 4 slices, each slice is larger than if it’s cut into 6 slices. So the pizza slice you get from a 4-slice cut is definitely bigger! Finally, when performing these kinds of comparisons, it's important to simplify first if possible. Students might mistakenly multiply the fractions before making comparisons, which complicates things unnecessarily—keep it simple and focus on the fractions directly!