In earlier lessons, students found sums and differences of fractions (including mixed numbers) with the same denominator. In this activity, they reason about problems that involve combining or removing fractional amounts with different denominators—2, 4, and 8—in the context of stacking playing bricks. Because the denominators are familiar and are multiples or factors of one another, students can rely on what they know about the relationships of halves, fourths, and eighths to compare amounts (for example, how much greater than or less than) or to combine them.

The last question asks students to reason about the height of a tower of bricks created by combining three shorter stacks. Students may arrive at two different answers, depending on their familiarity with playing bricks and attention to precision. Some students may notice that each playing brick has studs that disappear into the bottom of another brick when stacked, so the combined height of two stacks will be less than the sum of the heights of individual stacks (MP6). Both answers are acceptable as long as they are supported. 

• Select students to share their responses and their reasoning.
• Focus the discussion on how students found the fractional differences between the measurements. Highlight explanations about the relative sizes of halves, fourths, and eighths, and about equivalence. (See Student Responses.)

Previously, students used their knowledge of equivalence to reason about the sums and differences of fractions with denominators 2, 4, or 8. In this activity, they do the same with fractions with denominators 2, 3, and 6. As before, students are not expected to write addition expressions in which the fractions are written with a common denominator (though some students may choose to do so). Instead, they rely on what they know about the relationship between and , and between and , to solve the problems. Students may choose to use visual representations to support their reasoning. When students create and compare their own representations for the context, they develop ways to model the mathematics of a situation and strategies for making sense of and persevering to solve problems (MP1, MP4).

The measurements in the task—, , and —are given in feet. Because each of them has a whole-number equivalent in inches, some students may choose to reason entirely in inches, which is a valid strategy. Ask these students to think about how they’d approach the problems if the given unit is unfamiliar or doesn’t convert handily to whole numbers in another unit.

• Invite students to share their strategies for determining whether or how certain combinations of blocks would make a specified height. Record and display their reasoning.
• To highlight different ways to combine different-size fractional parts, consider sketching or displaying diagrams as shown:
  • Making 1 foot with all blocks:

    

  • Making foot with and blocks:

    

  • Making foot with and blocks: