The "Be Square!" Unit.
Use this web page to find extra practice and resources that suppliment our class activities. These pages will not make much sense or seem connected without our class instruction in between.
Objectives. In this unit, students will...
Objectives. In this unit, students will...
- Achieve a basic understanding of area and perimeter of squares, triangles, and rectangles.
- Become familiar with the geometric terms "congruent" and "similar."
- Develop geometric thinking through concrete experiences.
- Connect algebra to geometry when applying a formula for perimeter or area of squares, triangles, and rectangles.
- Demonstrate persistence and the ability to try multiple approaches to solve an open-ended problem
Warm up your brain to thinking about shapes...
This worksheet opened up our unit. If you missed it, print it and give it a try to get your brain ready for our unit!
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Once you've tried flipping and moving shapes, try picturing a shape folded. Keep your brain in the game with this challenge:
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We haven't talked about angles yet, but see if you have the background knowledge to answer this question. If this arrow (the largest one below) were turned nintey degrees clockwise three times, which figure (one of the smaller four below) shows the position in which it would land?
"Area" in Everyday Language
Try taking this one-question test. See what other students thought the answer was by voting and then viewing the results.
The Area Model of MultiplicationWe tried this in class. Try it again if you'd like. Click "Rectangle Multiplication" to use the virtual manipulative from NCTM's Illuminations.
What is the name of the property of multiplication that tells us that three rows of six blocks or six columns of three blocks is the same area?
If you send me an email with the title "Rectangle Multiplication" and a good written explanation, I will bring a small reward to you in our next class period. |
Extended Practice:We used the 2x1 Multiplication sheet in class. Challenge yoursef to try 2x2 Multiplication. Get the worksheets with the button links below.
Want to see how those work? Click the video to the right for a demonstration. Learning the distributive property is useful for mentally solving multiplication, as she mentions in the video. It's also essential for solving algebraic equations, so don't pass this up just because you already know how to multiply!
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Extra Practice:In class, we illustrated 9 x6 in at least three different ways. (See left for a student's example; click to enlarge image.) For additional pratice, illustrate 8 x 6 using the area model of multiplication. (Print graph paper using link below.)
Don't forget to demonstrate how to use the distributive property (rather than the order of operations) to solve each equation.
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"How did you figure it out?"
Yes, it's a good idea to get the right answer to a math problem, but there are many ways to get to the answer. I'm not interested in your "school answer" and seeing you follow someone else's formula. I want to see what you are thinking.
People who are good at mental math are not just plugging numbers into a formula when they figure something out in their head. It's far more interesting to hear how they really manipulate the numbers. Try these extra practice sheets after we've played the Area Model of Multiplication partner game with dice and graph paper in class. Be sure to explain in complete sentences when you are asked: "How did you figure it out?"
People who are good at mental math are not just plugging numbers into a formula when they figure something out in their head. It's far more interesting to hear how they really manipulate the numbers. Try these extra practice sheets after we've played the Area Model of Multiplication partner game with dice and graph paper in class. Be sure to explain in complete sentences when you are asked: "How did you figure it out?"
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Congruency Practice
All these geometry words can get jumbled in our heads unless we use them. Let's use them! First, make sure you understand what congruent means by practicing with these worksheets. Next, send a comment that contains an example of two objects we can picture that Are Congruent. Your email address will not be posted; it is only used to notify me that you commented.
Here are the sheets I passed out during class:
Click here to read an explanation of the math word, "Congruent."
Here is more practice with congruent shapes at the basic level:
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Click either of these links for more advanced practice with congruent shapes.
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Preparing a Quote
In class, we acted as contractors and prepared a quote for retiling the elevator/kitchen. We included a sketch of the area to be retiled (where the square-foot tiles would go), markings for perimeter (where the cove base would go), the cost of materials and labor, and a description of the work to be done.
Try this at home. Use the official-looking format for a quote by opening the document below and left. Read about how to bid on a job for installing tile by clicking the button link below and right. Then, write a bid/quote to retile a room in your own house. You can add a sketch to your quote. Next, review your bid. Would you hire you?
Try this at home. Use the official-looking format for a quote by opening the document below and left. Read about how to bid on a job for installing tile by clicking the button link below and right. Then, write a bid/quote to retile a room in your own house. You can add a sketch to your quote. Next, review your bid. Would you hire you?
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Exploring a Fixed Perimeter
During class, we invstigated all the ways we could use an entire roll of cove base, 120 feet. At home, see how many ways you can design that use 200 feet of fencing using the document below and left. If you need more graph paper than is provided on the sheet, download some below and center.
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You may like using the geoboard found at this link to explore rectangles with the same perimeter.
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Doubling and Halving Rectangles
During class, we explored Sam and Frances' patios. Sam wanted to double the size of his patio, and Frances wanted to cut hers in half to make room for raised gardens. Things didn't necessarily work out how we, Sam, or Frances thought!
The worksheet below will give you an opportunity to reinforce the concepts we learned during that activity. Don't skip the "predict" portion of the questions! It will give you the best chance to reinforce what you already know and to refine what you still need to learn.
The worksheet below will give you an opportunity to reinforce the concepts we learned during that activity. Don't skip the "predict" portion of the questions! It will give you the best chance to reinforce what you already know and to refine what you still need to learn.
doubling_and_halving_rectangles.pdf | |
File Size: | 95 kb |
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That concept can be difficult to get down. Here's one more chance to make sure you've got it. Check yourself with this worksheet.
changes_to_perimeter_and_area_of_a_rectangle.docx | |
File Size: | 23 kb |
File Type: | docx |
Similar Shapes
This link is from a free demonstration and so, unfortunately, the shapes won't move and the answer boxes won't give you feedback. Still, the questions that combine congruent and similar shapes are good practice. Click the button link below to try some more challenging questions than we did in class.
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Several students asked, "Why don't we just subtract 2 units from each side of a rectangle to make a similar rectangle?" Well, we know that rectangles must be in proportion. Sometimes it is hard to tell just by looking. When we looked at the Similar Shapes questions, we found a square at was 8 units by 12 units. If you draw a 6 by 10 rectangle on the side, it looks like they might be similar. How can we tell?
To determine whether or not addition and subtraction would produce similar shapes, we grabbed a double-sided piece of graph paper with small grids. We drew a 3 x 4 rectangle on the top left of each side. On one side, we created a series of rectangles that grew by adding two to each side for each rectangle. We got as high as 17 x 18 before the figures wouldn't fit. Next, we turned the paper over and grew the rectangles by multiplying by two to each side for each rectangle. The largest we could fit was 24 by 32. If you didn't try this yet, give it a try! What do you notice? Does either side of your graph paper appear to contain similar shapes if you compare the largest rectangle to the 3 x 4 rectangle?
Sometimes it's easier to tell if shapes are similar if you use a photograph. In Microsoft Word, you can set the dimensions of a photograph with the picture tools. When you drag from a corner, you are creating a proportional shape - both dimensions are moving at the same rate. If you drag a picture from the side or the top, you will notice that the picture gets distorted. The shape is not similar. Try measuring the pictures in this document. What do you notice about the measurements that work to make similar shapes?
To determine whether or not addition and subtraction would produce similar shapes, we grabbed a double-sided piece of graph paper with small grids. We drew a 3 x 4 rectangle on the top left of each side. On one side, we created a series of rectangles that grew by adding two to each side for each rectangle. We got as high as 17 x 18 before the figures wouldn't fit. Next, we turned the paper over and grew the rectangles by multiplying by two to each side for each rectangle. The largest we could fit was 24 by 32. If you didn't try this yet, give it a try! What do you notice? Does either side of your graph paper appear to contain similar shapes if you compare the largest rectangle to the 3 x 4 rectangle?
Sometimes it's easier to tell if shapes are similar if you use a photograph. In Microsoft Word, you can set the dimensions of a photograph with the picture tools. When you drag from a corner, you are creating a proportional shape - both dimensions are moving at the same rate. If you drag a picture from the side or the top, you will notice that the picture gets distorted. The shape is not similar. Try measuring the pictures in this document. What do you notice about the measurements that work to make similar shapes?
similar_shapes_-_proportional_pictures.docx | |
File Size: | 171 kb |
File Type: | docx |
Six Rectangles
To follow up with what you learned about the connections between area and perimeter, visit this Figure This! Challenge question to determine which non-standard figure has the most chocolate. Begin by clicking the button link below. After you have determined your answer, you can click "fullscreen" on the document to look at extended questions.
Here's a problem with more than one solution and many more possible ways to figure out answers. Challenge yourself to figure one out! Test your persistence in problem solving using the button link below. There is a link for graph paper below the next documents if you'd like to print some for your problem solving.
More practice sheets to help solidify your understanding:
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STILL UNDER CONSTRUCTION FROM HERE DOWN
As we work toward these concepts in class, I will post more practice.
As we work toward these concepts in class, I will post more practice.
triangles_and_rectangles_csr.pdf | |
File Size: | 39 kb |
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typical_triangle_problems.pdf | |
File Size: | 103 kb |
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combined_rectangles.pdf | |
File Size: | 311 kb |
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800_square_feet_of_living_space.pdf | |
File Size: | 183 kb |
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The Tile Guy provides great diagrams that show how a combination of rectangles can be used to figure area.
Quilt Squares:
What percent of this quilt square is shaded purple? Explain how you figured it out.
If this pattern is repeated in 4 rows of 3 in an entire quilt, what percent is shaded purple?
What percent of this quilt square is shaded purple? Explain how you figured it out.
If this pattern is repeated in 4 rows of 3 in an entire quilt, what percent is shaded purple?
Pythagorean Theorem
Model of proof for the Pythagorean Theorem.
Here's a challenging question that can be answered without the Pythagorean theorem. You can add an additional challenge to it. Once you figure out the distance the ramp needs to be away from the steps, figure out the length of the ramp itself.
This question cannot be solved using the formula for the Pythagorean theorem, but it can be solved with the idea of the hypotenuse in mind. Even if that is not a hint for you, give this problem a try in any way that you see fit. After you solve the problem, you can click "fullscreen" on the document below to practice with an extended question.
Miles of Tiles
Before we work together on Miles of Tiles Level C, give this question a try. Be persistent! This isn't a quick-to-solve question. It is open-ended with no specific measurements given. Click the button link below prepared with lots of scrap paper and the decision to stick with it.
This is a related video that a colleague and I made for other teachers to watch. You'll hear reference to the standards that we use to help make decisions about what to teach and when. If you pause the video as intended and do the activities, you will get the most learning from the experience.
Interested in learning more about multiplication? Here is another way to look at models for multiplication that look similar to how we would use them with algebra tiles.
Here is a recording of the follow-up webinar we held:
We mention this video in the webinar:
Mystery Rectangles Answers: #1 Rectangle C (1 x 4 units); #2 Rectangle D (3 x 2); #3 Rectangle A (3 x 4 units); #4 Rectangle E (1 x 2); #5 Rectangle B (1 x 1)