Lesson 13

In this warm-up, students review how to find and record the area of a square with whole-number and fractional side lengths. The first question is open-ended to encourage students to notice many things about the area of each square, the relationships between them, and other geometric ideas they might remember from earlier grades. The second question prepares students for the work in this lesson. Focus class discussion on this question.

As students discuss the second question, note those who think of the area of a \(\frac13\)-inch square in terms of:

• Tiling, i.e., determining how many squares with \(\frac13\)-inch side length cover a square with 1-inch side length and dividing the area of 1 square inch by that number
• Multiplying \(\frac13 \boldcdot \frac13\)

Arrange students in groups of 2. Display the image and the first question for all to see. Give students 1 minute of quiet time to make observations about the squares. Follow with a brief whole-class discussion.

If not mentioned by students, ask students what they notice about the following:

• The area of each square
• How to record the area in square inches
• Whether one square could tile another square completely

Then, give students 1–2 minutes to discuss the second question with their partner.

Some students may struggle in getting started with the second question. Suggest that they try marking up the given 1-inch square to show \(\frac13\)-inch squares.

Consider telling students that we can call a square with 1-inch side length “a 1-inch square.”

Ask previously identified students to share their response to the second question. Illustrate their reasoning for all to see. After each person shares, poll the class on whether they agree with the answer and the explanation. If not mentioned in students’ explanations, highlight the following ideas:

• A square with a side length of 1 inch (a 1-inch square) has an area of 1 in².
• A 2-inch square has an area of 4 in², because 4 squares with 1-inch side length are needed to cover it.
• A \(\frac12\)-inch square has an area of \(\frac14\) in² because 4 of them are needed to completely cover a 1-inch square. 
• A \(\frac13\)-inch square has a side length of \(\frac13\) inch, so it would take 9 squares to cover a 1-inch square. Its area is therefore \(\frac19\) square inch.

This activity serves two purposes:

• To review and illustrate the idea from grade 5 that the area of a rectangle with fractional side lengths can be found by multiplying the two fractions, just as in the case of whole numbers.
• To prepare students to reason about a prism with fractional edge lengths. Students connect the area of a square with fractional side length with that of a unit square. Later, they transfer this idea to find the volume of prisms with fractional edge lengths. They will then compare whole cubic units and fractional cubic units.

As students work, monitor the ways students represent and reason about the area of the rectangle with fractional side lengths in the last question. A few possibilities are shown in the Possible Responses. Select students who use different strategies to share later.

Keep students in groups of 2. Give students 7–8 minutes of quiet work time and 2–3 minutes to share their responses and drawings with their partner. Provide each student with \(\frac14\)-inch graph paper and a straightedge.

Some students may have trouble counting grid squares or using a ruler on graph paper and struggle to measure the lengths of the rectangle. Consider preparing pre-drawn copies of the rectangle for students who may benefit from them.

Focus the whole-class discussion on the last question. Invite previously selected students to share their answers and diagrams in the sequence shown in the Possible Responses. Ask students to explain how they found that \(3\frac12 \boldcdot 2\frac14\) equals \(7\frac 78 \text { in}^2\). Record their reasoning for all to see.

Consider displaying the following images and highlighting the areas of the sub-rectangles with fractional side lengths.

Compare and contrast the different strategies. Then, ask students how the area they found would compare to the product \(3\frac12 \boldcdot 2\frac14\). Ask them to calculate the product. Make sure students see that the product of the two numbers is equal to the area of the rectangle. \(\displaystyle 3\frac12 \boldcdot 2\frac14 = \frac72 \boldcdot \frac 94 = \frac{63}{8} = 7\frac78\)

This activity also revisits grade 5 work on finding the area of a rectangle with fractional side lengths. Students interpret and match numerical expressions and diagrams. Because the diagrams are unlabeled, students need to use the structure in the expressions and in the diagrams to make a match (MP7). This work reinforces their understanding of the area of rectangles and of multiplication. Specifically, it helps them see how the product of two mixed numbers (or two fractions that are greater than 1) can be found using partial products.

Give students 2–3 minutes of quiet work time. Emphasize the direction that states, “All regions shaded in light blue have the same area” before students begin working.

In answering the second question (showing that \(2\frac12 \boldcdot 4\frac34 = 11\frac78\)), some students may neglect to use the diagram and simply multiply the whole numbers in the side lengths (the 2 and 4), multiply the fractions (the \(\frac12\) and \(\frac34\)), and then add them. Allow them to pursue this path of reasoning, but later, when they recognize their answer is less than \(11\frac78\), refer them to the diagram. Ask them to identify the rectangles whose areas they have calculated and those they have not accounted for, and to think about how they could find the area of the entire rectangle.

When adding partial products with fractions in different denominators, some students may simply add the numerators and denominators. Remind them to attend to the size of the fractional parts when adding or subtracting fractions.

For each diagram, ask one or more students to share which expression they think matches. Ask students to share their reasoning for how they matched the figure to the expression. Consider displaying the four figures for all to see and recording students’ reasoning or explanations on the figures. To involve more students in the conversation, consider asking:

• “Who can restate ___’s reasoning in a different way?”
• “Does anyone want to add on to _____’s reasoning?”
• “Do you agree or disagree? Why?”

For the second question, ask students for the area of each section in Figure B. Label each section with its side lengths and its area and display for all to see. If not already articulated by students, highlight that combining all the partial areas gives us a sum of \(11\frac78\), which is the area of the entire rectangle.

This activity consolidates prior work on the area of rectangles and the current work on division of fractions. Students determine how many tiles with fractional side lengths are needed to completely cover another rectangular region that also has fractional side lengths. Besides dividing fractions, students also need to plan their approach, think about how the orientation of the tiles affects their calculation and solution, and attend carefully to the different measurements and steps in their calculation. The experience here prepares students to work with lengths and volumes in the culminating lesson (in which students determine how many small boxes with fractional edge lengths will fit into larger boxes that also have fractional edge lengths).

As students work, identify those whose diagrams or solutions show different tile orientations. Also notice students who consider both ways of laying the tiles before finding the solutions.

Keep students in groups of 2. Give students 7–8 minutes of quiet work time and 2–3 minutes to share their responses with their partner, or give 10 minutes for them to complete the activity in groups.

Students might only determine the amount of tiles needed to line the four sides of the tray. If this happens, suggest that they look at their drawing of the tray and check whether their tiles cover the entire area of the tray.

Invite students who chose different tile orientations to show their diagrams and explain their reasoning. Display these two diagrams, if needed.

Point out how in this problem, the two different tile orientations do not matter, as the length and the width of the tiles are factors of both the length and the width of the tray. This means we can fit a whole number of tiles in either direction, and can fit the same number of tiles to cover the tray regardless of orientation.

But if the side lengths of the tiles do not both fit into \(11\frac14\) and \(4\frac12\) evenly, then the orientation of the tiles does matter (i.e., we may need more or fewer tiles, or we may not be able to tile the entire tray without gaps if the tiles are oriented a certain way).

Use the opportunity to point out that a diagram does not have to show all the details (i.e., every single tile) to be useful.