Video: Investigating Patterns in the Length and Area of Shapes

Isabella is making patterns of rectangles by using the rule add 2 cm to the length. What is the length of Rectangle 4? The lengths make the pattern 3, 5, 7, 9, .... She notices that the areas of the rectangles also make a pattern. Which of these is the pattern of the areas of the rectangles? [A] 6, 12, 24, ... [B] 6, 14, 18, ... [C] 6, 10, 14, ... [D] 6, 14, 22, ... [E] 6, 24, 28, .... What is the rule for the pattern of the areas?

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Video Transcript

Isabella is making patterns of rectangles by using the rule add two centimeters to the length. What is the length of rectangle four? The lengths make the pattern three, five, seven, nine. She notices that the areas of the rectangles also make a pattern. Which of these is the pattern of the areas of the rectangles? Six, 12, 24; six, 14, 18; six, 10, 14; six, 14, 22; or six, 24, 28. What is the rule for the pattern of the areas?

There’s lots to do in this problem and lots of text explaining it. So let’s take our time and go through the problem slowly, step by step. To begin with, we’re introduced to Isabella, who we’re told is making patterns of rectangles. And she has a rule to help her do this. And the rule, we’re told, is to add two centimeters to the length. We can see the first three rectangles in Isabella’s pattern. The first thing that we can tell by looking at the rectangles is that Isabella doesn’t change the height of her rectangle at all.

They’re all the same height; they’re all two centimeters tall. But we can see that the length of each rectangle is changing. Of course, we know this because Isabella’s rule means that the length increases each time. Her first rectangle has a length of three centimeters. Then she adds two centimeters to this. Three plus two equals five centimeters. So the length of Isabella’s second rectangle is five centimeters. To find the length of rectangle three, she adds another two centimeters to the length. Five plus two equals seven centimeters. So the length of rectangle three is seven centimeters. We’re now asked, what is the length of rectangle four?

Now, we can’t see rectangle four. But we know how Isabella could draw it. She’d add two centimeters to the length of rectangle three. And rectangle three is seven centimeters long. Seven plus two equals nine. So the length of rectangle four must be nine centimeters. Let’s carry on reading through the question. The lengths make the pattern three, five, seven, nine. Our answer to the last part must have been correct because we’re told it here. Next, we’re told that Isabella notices that the areas of the rectangles also make a pattern. And we’re given five possible patterns of the area of the rectangles. Which one is correct?

Now, to find the area of a rectangle, we need to multiply the length by its width. In other words, we multiply the two dimensions by each other. Now so far, we’ve been calling the dimensions in these rectangles’ length and height. But it doesn’t matter. These are still the two dimensions of each rectangle. And we need to multiply them together to find the area. For example, the first rectangle has a length of three centimeters and a height of two centimeters. Three times two is six. Ad so the area of rectangle one is six square centimeters. Now, if we look at our five possible answers, we can see that they all begin with the number six. So we can’t narrow it down yet.

Let’s work out the area of rectangle two. Remember, the height of this rectangle is still the same. It’s still two centimeters. So to find the area of rectangle two, we just need to multiply five by two. Five centimeters multiplied by two centimeters equals an area of 10 square centimeters. Now, only one of our possible answers has 10 as the second number. And it has 14 is the third number. Let’s check whether the area of rectangle three is 14. The length of rectangle three is seven centimeters. And as we know that the height is two centimeters, all we have to do is double seven. And double seven is 14. The area of rectangle three is 14 square centimeters. And so the pattern of the areas of the rectangles is six, 10, 14, and so on.

Now remember, Isabella noticed that the area’s made a pattern. It’s not just a string of numbers. The final part of the problem asks us what the rule for this pattern is. The areas go from six to 10 to 14. How do they change? Six plus four equals 10, and 10 plus four equals 14. We’re adding four each time. Let’s go through our three answers to the problem. The length of rectangle four is nine centimeters. The pattern for the areas of the rectangles is six, 10, 14, and so on. And finally, the rule for the pattern of the areas is add four.

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