Lesson 1

The purpose of this warm-up is for students to describe the fraction of macaroni and cheese that is left in the pan. While students may notice and wonder many things about this image, the amount of macaroni and cheese in the pan is the important discussion point.

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “The picture shows a pan of macaroni and cheese. What other food is baked in pans like this one?” (lasagna, casseroles, cakes)
• “About how much macaroni and cheese is left in the pan?” (It’s less than \(\frac{1}{2}\) and more than \(\frac{1}{3}\). It looks like it is about \(\frac{2}{5}\).)

The goal of this activity is for students to draw diagrams that represent a unit fraction multiplied by another unit fraction in context. The macaroni and cheese context was introduced in the warm-up to motivate students to draw a diagram to represent the pan. The focus in this activity is on the different diagrams students draw and how they represent the same situation (MP2). Some students may identify that Lin ate \(\frac{1}{6}\) of the pan. Invite these students to share their observation at the end of the synthesis when they think about the diagrams in relation to the fraction of the whole pan of macaroni and cheese Lin ate.

• “We are going to solve problems about a pan of macaroni and cheese that was served at a big family dinner. Lin is excited that her aunt made her famous baked macaroni and cheese. Tell your partner a story about a dish that you love to eat for dinner.”
• 1–2 minutes: partner discussion

• 3–5 minutes: individual work time
• As students work, consider asking:
  • “How does your diagram show \(\frac{1}{2}\)?”
  • “How does your diagram show \(\frac{1}{3}\) of \(\frac{1}{2}\)?”
  • “How did you decide how to partition the rectangle?”
• Monitor for students who draw different diagrams to show \(\frac{1}{3}\) of \(\frac{1}{2}\) such as those shown in the student solutions. 

If students do not draw a diagram that represents the situation, suggest they draw a diagram to show how much of the pan of macaroni and cheese is left. Then ask: “How can you adapt your diagram to show that one third of one half of the pan was eaten?”

• Ask selected students to display their responses side by side for all to see or use the images provided in the student solutions.
• For each diagram ask: “How does the diagram represent \(\frac{1}{3}\) of \(\frac{1}{2}\) of the pan?” (First, the rectangle or pan is divided in half and then a third of one half is shaded.)
• “How are the diagrams the same?” (They all show the full pan cut in half. Then they show a half cut into 3 equal pieces and one of those pieces is shaded.)
• “How are the diagrams different?” (One diagram cuts the pan in half horizontally and the other two cut it in half vertically. The other cuts into 3 equal pieces are also sometimes horizontal and sometimes vertical.)
• “How much of the whole pan did Lin eat?” (Students may say \(\frac{1}{6}\) or other fractions.)
• Record all responses and revisit this in the lesson synthesis.

Continuing the macaroni and cheese context from the previous activity, the purpose of this activity is for students to interpret diagrams showing a fraction of a fraction of the pan. Then students address what fraction of the whole pan the shaded piece of the diagram represents. Because the whole pan is not subdivided, students may need to add the extra divisions or think carefully to identify the fraction of the whole pan represented by the diagrams. To identify that the shaded pieces in the two diagrams have equal area students may

• cut out and compare the shaded pieces explicitly
• reason that they are each \(\frac{1}{4}\) or \(\frac{1}{2}\) of the same amount
• reason that they are each \(\frac{1}{8}\) of the whole

• Groups of 2

• 1–2 minutes: quiet think time
• 5–8 minutes: partner discussion
• Monitor for students who:
  • extend the dashed lines in diagram A to determine that \(\frac{1}{8}\) of the whole square is darkly shaded
  • partition the rest of the square in diagram B to determine that \(\frac{1}{8}\) of the whole square is darkly shaded

If students do not explain why each diagram represents \(\frac{1}{4}\) of \(\frac{1}{2}\), suggest they draw their own diagram to represent \(\frac{1}{4}\) of \(\frac{1}{2}\). Ask: “What is the same about your diagrams and the ones in the tasks? What is different?”

• Ask previously selected students to share in the given order.
• “How does each diagram represent \(\frac {1}{4}\) of \(\frac {1}{2}\)?” (They each show \(\frac{1}{2}\) shaded in the lighter blue and then \(\frac{1}{4}\) of that half is shaded darker.)
• “How do we know the darkly shaded pieces are the same size?” (I cut them out to check. They are both \(\frac{1}{4}\) of \(\frac{1}{2}\). They both represent \(\frac{1}{8}\) of the whole pan.)
• If not already mentioned by students, ask: “How can we figure out how much of the whole pan of macaroni cheese the dark shaded piece represents?” (We can extend the lines in diagram A and we can partition the rest of the square in diagram B.)
• “\(\frac {1}{4}\) of \(\frac {1}{2}\) is equal to how much of the whole pan of macaroni and cheese?” (\(\frac{1}{8}\) of the whole pan.)