Lesson 1

The purpose of this How Many Do You See is for students to subitize or use grouping strategies to describe the images they see. Students may see equal groups in the rows or the columns of the array. Recording the equations for each way of seeing the groups is an opportunity to reinforce the commutative property.

When students use different ways to group dots within the same array to find the total number of dots they look for and make use of structure (MP7).

• Groups of 2
• “How many do you see? How do you see them?”  
• Flash the image.
• 30 seconds: quiet think time

• Display the image.
• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.
• Repeat for each image.

• “How did the arrays allow us to find the number of dots in different ways?” (We were able to look at the number of dots in each row and think about how many rows there were. I counted the number in each column and multiplied it by the number of columns.)
• As students share, record equations to match their thinking.
• Consider asking: “How do the equations change if we think about the rows as the groups or the columns as the groups?” (For the image with 12 dots, if we think about the rows as the groups, we write \(4 \times 3 = 12\), but if we think about the columns as the groups, we write \(3 \times 4 = 12\). The order of the factors is reversed, but the product stays the same.)

The purpose of this activity is for students to compare shapes to decide which is larger. Given their prior experiences with length, students may initially use length to reason about what it means for a shape to be larger than another shape. The synthesis should bring out the idea that length alone is not enough to compare two-dimensional shapes. Ideas around how much space the shapes cover should be emphasized. If students disagree about which shape is larger, encourage them to share their reasoning so that the class can consider multiple ideas and come to a resolution together (MP3).

• Groups of 2
• Display or sketch the two triangles in the first problem.
• “Which triangle do you think is larger? Be prepared to explain your reasoning.”
• 1 minute: quiet think time
• Share and record responses.
• “How could you decide for sure which shape is larger?” (I could think about putting one shape on top of the other. I could measure which is longer. I could cut one out to see if it fits inside the other.)
• 1 minute: quiet think time
• 1 minute: partner discussion
• Share and record responses.
• Give each group scissors.

• “Now you are going to decide which shape in each pair is larger. Consider trying some of the strategies we discussed.”
• 3 minutes: independent work time
• “Tell your partner which shape in each pair you thought was larger. Explain how you decided.”
• 3 minutes: partner discussion
• Monitor for justifications about how much space each shape covers and disagreements students discuss.

• “Which shapes did you change your mind about as you discussed your ideas with your partner?” (In the last problem, I thought that the skinny rectangle was larger because it was taller. My partner showed me how to cut the rectangles to compare them.)
• “Are there shapes you still disagree about?”
• Have students share their justifications for any lingering disagreements.
• As students share, bring out the idea that the larger shape covers more space.  
• Consider asking:
  • “What questions do you have?”
  • “Do you agree with their reasoning?”
  • “Did you justify your choice in a different way?”

The purpose of this activity is for students to compare shapes by covering them with pattern blocks. Students experience tiling as a way to see which shape covers the most space. There are several ways to tile the shapes, but it may prove most useful to use the same units, such as triangles. The rectangle can only be fully tiled with square pattern blocks. To compare shapes B and C, students will need to notice that the rectangle and parallelogram can be made the same length, but the square pattern blocks used to tile the rectangle are taller than the blocks used to tile the parallelogram. 

The work here prepares students to tile figures with square tiles in the next lesson and to think of area in terms of square units.

• Each group of 2 needs at least 2 hexagons and trapezoids,  4 squares and rhombuses, and 8 triangles.

• Groups of 2
• Give each student a copy of the blackline master.
• Give each group pattern blocks.
• Display the first two shapes (the hexagon and the heart-shaped octagon).
• “What do you notice? What do you wonder?” (Students may notice: Both shapes could be made with pattern blocks. The second shape looks like the first shape, but it’s been bent. Students may wonder: Which shape covers more space? Which shape is larger?)
• 1 minute: quiet think time
• Share and record responses.
• “Which shape covers more space? How do you know?” (We can cover them with the same blocks, so they cover the same amount of space.)
• 2 minutes: partner discussion
• Share and record responses.

• “Let's continue comparing shapes. Work with your partner to decide which of these shapes covers the most space. Use pattern blocks if they are helpful.”
• 5 minutes: partner work time
• Monitor for students who:
  • use pattern blocks to determine which shape covers the most space
  • notice that shape B and C would be the same length if one of the ends of B was moved over, but C covers more space because it is taller

• Ask selected students to share strategies for tiling each shape with pattern blocks and how the tiling helped them decide which shape covers the most space.
• “When we decide which shape covers the most space, we are talking about the area of a shape. We can think about the area as the amount of space covered by a shape.”
• “Which of these shapes has the greatest area?” (C takes up the most space, so it has the greatest area.)