IM 6.4.13 Lesson: Rectangles with Fractional Side Lengths

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[/img][br]What do you notice about the areas of the squares?[br]

Kiran says “A square with side lengths of [math]\frac{1}{3}[/math] inch has an area of [math]\frac{1}{3}[/math] square inches.” Do you agree? Explain or show your reasoning.

In the app, draw a square with side lengths of 1 inch. Inside this square, draw another square with side lengths of ¼ inch.

Use your drawing to answer the questions.[br]How many squares with side lengths of [math]\frac{1}{4}[/math] inch can fit in a square with side lengths of 1 inch?[br]

Use your drawing to answer the questions.[br]What is the area of a square with side lengths of [math]\frac{1}{4}[/math] inch? Explain or show your reasoning.[br]

On the graph paper, draw a rectangle that is 3 ½ inches by 2 ¼ inches.

Use your drawing to answer the questions. Write a division expression and then find the answer.[br][br][list=1][/list]How many [math]\frac{1}{4}[/math]-inch segments are in a length of [math]3\frac{1}{2}[/math] inches?[br]

Use your drawing to answer the questions. Write a division expression and then find the answer.[br][br][list=1][/list]How many [math]\frac{1}{4}[/math]-inch segments are in a length of [math]2\frac{1}{4}[/math]inches?[br]

Each of these multiplication expressions represents the area of a rectangle. All regions shaded in light blue have the same area. Match each diagram to the expression that you think represents its area.

Explain your reasoning for the matches above.

Use the diagram that matches 2 ½ · 4 ¾ to show that the value of 2 ½ · 4 ¾ is equal to 11 ⅞.

The following rectangles are composed of squares, and each rectangle is constructed using the previous rectangle. The side length of the first square is 1 unit. Draw the next four rectangles that are constructed in the same way.

Then complete the table with the side lengths of the rectangle and the fraction of the longer side over the shorter side.

Describe the values of the fraction of the longer side over the shorter side. What happens to the fraction as the pattern continues?

[size=150]Noah would like to cover a rectangular tray with rectangular tiles. The tray has a width of [math]11\frac{1}{4}[/math] inches and an area of [math]50\frac{5}{8}[/math] square inches.[/size][br][br]Find the length of the tray in inches.

If the tiles are [math]\frac{3}{4}[/math] inch by [math]\frac{9}{16}[/math] inch, how many would Noah need to cover the tray completely, without gaps or overlaps? Explain or show your reasoning in the applet below. [br][br]Draw a diagram to show how Noah could lay the tiles. Your diagram should show how many tiles would be needed to cover the length and width of the tray, but does not need to show every tile.[br]