Lesson 6

Different Square Units (Part 1)

Warm-up: Notice and Wonder: Squares, Squares (10 minutes)

Narrative

This warm-up elicits the idea that squares of different sizes can be used to measure area, which will be useful when students encounter different square units later. While students may notice and wonder many things, focus the discussion on how different-size squares could be used to tile the rectangle.

Launch

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

Activity

• 1 minute: partner discussion
• Share and record responses.

Student Facing

What do you notice? What do you wonder?

Student Response

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Activity Synthesis

• “We spent some time thinking about how rectangles could be tiled with different-size squares. Keep this in mind as we move on to the next activity.”

Activity 1: Same Rectangle, Different Units (15 minutes)

Narrative

The purpose of this activity is for students to see that there are different types of square units that can be used to measure area and that an area with the same number of square units can be larger or smaller depending on the unit that is used. To facilitate comparison, one partner works on inch grid paper and one works on centimeter grid paper. In the synthesis, students are introduced to square inches and square centimeters.

MLR8 Discussion Supports. Synthesis: For each observation that is shared, listen for the appropriate use of the word area, and invite students to turn to a partner and restate what they heard using precise mathematical language.

Required Materials

Materials to Gather

Materials to Copy

• Same Rectangle, Different Units

Launch

• Groups of 2
• Give each group 1 copy of the blackline master.
• “Work with your partner to create a rectangle for each expression. Partner 1 will use grid 1 and partner 2 will use grid 2. Think about how the rectangles might be different.”
• 30 seconds: quiet think time

Activity

• 3–5 minutes: partner work time
• “Discuss how the rectangles are the same and how they are different with your partner.”
• 2–3 minutes: partner discussion

Student Facing

Your teacher will give you two kinds of grid paper. Use them to create a rectangle for each expression.

Partner 1: Use grid 1.

Partner 2: Use grid 2.

1. $$2 \times 5$$
2. $$7 \times 3$$
3. $$6 \times 8$$

Student Response

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If students say that rectangles with the same number of squares have the same area, even if the squares are different sized, consider asking:

• “What are we measuring when we measure area?”
• “If there’s the same amount of squares, how does changing the size of the squares affect the area that the squares cover?”

Activity Synthesis

• “How are the rectangles the same and different?” (The rectangles have the same number of squares, but they do not have the same area because they do not cover the same amount of space.)
• Give students rulers.
• “Work with your partner to investigate the squares on both grids. Be prepared to share what you find.” (On the grid with the large squares, each side is 1 inch. On the grid with the small squares, each side is 1 centimeter.)
• “These are 2 types of squares we can use to measure area. The square inch has a side length of 1 inch. The square centimeter has a side length of 1 centimeter.”
• “Grid 1 is inch grid paper. Inch grid paper has squares with side lengths of 1 inch. Grid 2 is centimeter grid paper. Centimeter grid paper has squares with side lengths of 1 centimeter.”

Activity 2: What’s the Area? (20 minutes)

Narrative

The purpose of this activity is for students to estimate and then find the area of a square using different standard units of area (square inches and square centimeters). Students discuss why using square inches and square centimeters gives different measurements for the same area.

To measure the area of the square, students trace the square or cut it out, place the traced or cut square on the grids from the previous activity, and determine the number of squares covered on each grid. The measurement in square centimeters is not exactly 25 square centimeters because the square is 2 inches by 2 inches. The main idea is that it takes more of a smaller unit than a larger unit to cover the same area. When students reference units in their area measurements they are attending to precision in language (MP6).

Action and Expression: Develop Expression and Communication. Give students access to a square centimeter cube and a square inch tile to support their estimation.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

Required Materials

Materials to Gather

Materials to Copy

• Same Rectangle, Different Units

Required Preparation

• Prepare additional copies of the grids from Same Rectangles, Different Units so students can have a fresh copy to measure the area of the square.
• Have patty paper available, in case requested.

Launch

• Groups of 2
• “Estimate how many square inches and how many square centimeters you think it will take to tile this square. Record your estimates.”
• 1 minute: independent work time
• 1–2 minutes: partner discussion

Activity

• “Now you’re going to measure the area of the square using your inch grid paper and your centimeter grid paper.”
• “You can trace the square or cut it out if you want to.”
• “Then, you can place the square over the grids from Activity 1 to find the area in square inches and square centimeters.”
• “Finally, write a multiplication expression to match the area you found.”
• 5 minutes: partner work time

MLR2 Collect and Display

• Circulate, listen for, and collect the language students use to discuss square inches and square centimeters. Listen for: square inch, square centimeter, 1 inch on each side, 1 centimeter on each side, takes up less space than square inches, takes up more space than square centimeters, takes more square centimeters than square inches to cover the same space, or takes fewer square inches than square centimeters to cover the same space.
• Record students’ words and phrases on a visual display and update it throughout the lesson.

Student Facing

Estimate how many square centimeters and inches it will take to tile this square.

• square inches (estimate) _________
• square centimeters (estimate) _________

1. Use the inch grid and centimeter grid to find the area of the square

• square inches ____________________
• square centimeters ____________________
2. Write a multiplication expression that describes the rows and columns in the square and can tell us the area in each unit.

• square inches ____________________
• square centimeters ____________________

Student Response

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Activity Synthesis

• “Are there any other words or phrases that are important to include on our display?”
• As students share responses, update the display, by adding (or replacing) language, diagrams, or annotations.
• Remind students to borrow language from the display as needed.
• “How did measuring with different square units affect the measurement of the area of the square?” (It took more of the smaller squares to tile the same area. It took fewer of the larger squares to tile the same area.)

Lesson Synthesis

Lesson Synthesis

Display a square inch and a square centimeter.

“Why is it useful to have different types of square units to use when measuring area?” (We could use square inches to measure something larger and square centimeters for something a little smaller. If we didn’t want to use as many squares, we could use the larger square.)

Cool-down: Which Square? (5 minutes)

Cool-Down

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