Students are introduced to Pascal's law, Archimedes' principle and Bernoulli's principle. Fundamental …
Students are introduced to Pascal's law, Archimedes' principle and Bernoulli's principle. Fundamental definitions, equations, practice problems and engineering applications are supplied. A PowerPoint® presentation, practice problems and grading rubric are provided.
The famous story of Archimedes running through the streets of Syracuse (in …
The famous story of Archimedes running through the streets of Syracuse (in Sicily during the third century bc) shouting ''Eureka!!!'' (I have found it) reportedly occurred after he solved this problem. The problem combines the ideas of ratio and proportion within the context of density of matter.
This learning video deals with a question of geometrical probability. A key …
This learning video deals with a question of geometrical probability. A key idea presented is the fact that a linear equation in three dimensions produces a plane. The video focuses on random triangles that are defined by their three respective angles. These angles are chosen randomly subject to a constraint that they must sum to 180 degrees. An example of the types of in-class activities for between segments of the video is: Ask six students for numbers and make those numbers the coordinates x,y of three points. Then have the class try to figure out how to decide if the triangle with those corners is acute or obtuse.
This learning video deals with a question of geometrical probability. A key …
This learning video deals with a question of geometrical probability. A key idea presented is the fact that a linear equation in three dimensions produces a plane. The video focuses on random triangles that are defined by their three respective angles. These angles are chosen randomly subject to a constraint that they must sum to 180 degrees. An example of the types of in-class activities for between segments of the video is: Ask six students for numbers and make those numbers the coordinates x,y of three points. Then have the class try to figure out how to decide if the triangle with those corners is acute or obtuse.
In this problem, students are given a picture of two triangles that …
In this problem, students are given a picture of two triangles that appear to be similar, but whose similarity cannot be proven without further information. Asking students to provide a sequence of similarity transformations that maps one triangle to the other focuses them on the work of standard G-SRT.2, using the definition of similarity in terms of similarity transformations.
This Nrich activity is particularly good in a number of mathematical aspects …
This Nrich activity is particularly good in a number of mathematical aspects of learning: Using mathematical ideas and methods to solve "real life" problems Using and understanding vocabulary and notation related to money Organizing and using data Choosing and using appropriate number operations and calculation strategies Explaining methods and reasoning Making and investigating general statements
Create your own shapes using colorful blocks and explore the relationship between …
Create your own shapes using colorful blocks and explore the relationship between perimeter and area. Compare the area and perimeter of two shapes side-by-side. Challenge yourself in the game screen to build shapes or find the area of funky figures. Try to collect lots of stars!
This short video and interactive assessment activity is designed to give fourth …
This short video and interactive assessment activity is designed to give fourth graders an overview of composite figures composed of squares and rectangles.
An interactive applet and associated web page that demonstrate the area of …
An interactive applet and associated web page that demonstrate the area of a circle. A circle is shown with a point on the circumference that can be dragged to resize the circle. As the circle is resized, the radius and the area computation is shown changing in real time. The radius and formula can be hidden for class discussion. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.
Unit: Area and Surface Area Lesson 6: Area of Parallelograms This lesson …
Unit: Area and Surface Area Lesson 6: Area of Parallelograms
This lesson allows students to practice using the formula for the area of parallelograms, and to choose the measurements to use as a base and a corresponding height. Through repeated reasoning, they see that some measurements are more helpful than others. For example, if a parallelogram on a grid has a vertical side or horizontal side, both the base and height can be more easily determined if the vertical or horizontal side is used as a base.
Along the way, students see that parallelograms with the same base and the same height have the same area because the products of those two numbers are equal, even if the parallelograms look very different. This gives us a way to use given dimensions to find others.
Reasoning to Find Area: Lesson 4 Students were introduced to parallel lines …
Reasoning to Find Area: Lesson 4
Students were introduced to parallel lines in grade 4. While the standards do not explicitly state that students must work with parallelograms in grades 3–5, the geometry standards in those grades invite students to learn about and explore quadrilaterals of all kinds. The K–6 Geometry Progression gives examples of the kinds of work that students can do in this domain, including work with parallelograms.
In this lesson, students analyze the defining attributes of parallelograms, observe other properties that follow from that definition, and use reasoning strategies from previous lessons to find the areas of parallelograms.
By decomposing and rearranging parallelograms into rectangles, and by enclosing a parallelogram in a rectangle and then subtracting the area of the extra regions, students begin to see that parallelograms have related rectangles that can be used to find the area.
Throughout the lesson, students encounter various parallelograms that, because of their shape, encourage the use of certain strategies. For example, some can be easily decomposed and rearranged into a rectangle. Others—such as ones that are narrow and stretched out—may encourage students to enclose them in rectangles and subtract the areas of the extra pieces (two right triangles).
After working with a series of parallelograms, students attempt to generalize (informally) the process of finding the area of any parallelogram (MP8).
Images of rectangle on coordinate plane, one with easily identifiable area, other …
Images of rectangle on coordinate plane, one with easily identifiable area, other image with same rectangle rotated. Distance formula shown to calculate same area.
Unit: Area and Surface Area Lesson 8: Area of Triangles This lesson …
Unit: Area and Surface Area Lesson 8: Area of Triangles
This lesson builds on students’ earlier work decomposing and rearranging regions to find area. It leads students to see that, in addition to using area-reasoning methods from previous lessons, they can use what they know to be true about parallelograms (i.e. that the area of a parallelogram is ) to reason about the area of triangles.
Students begin to see that the area of a triangle is half of the area of the parallelogram of the same height, or that it is the same as the area of a parallelogram that is half its height. They build this intuition in several ways:
by recalling that two copies of a triangle can be composed into a parallelogram; by recognizing that a triangle can be recomposed into a parallelogram that is half the triangle’s height; or by reasoning indirectly, using one or more rectangles with the same height as the triangle. They apply this insight to find the area of triangles both on and off the grid.
An interactive applet and associated web page that deals with the area …
An interactive applet and associated web page that deals with the area of a kite, (a quadrilateral with two distinct pairs of equal adjacent sides). The applet shows a kite and the user can reshape it by dragging any vertex. The other vertices move automatically to ensure it always remains a kite. As the vertices are dragged, the area is continuously recalculated and displayed. The kite is filled with a grid of unit squares so that the students can estimate the area. The on-screen calculation can be hidden until the estimates are done. The web page lists two different ways to compute the area of a kite. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.
A web page and interactive applet showing the ways to calculate the …
A web page and interactive applet showing the ways to calculate the area of a parallelogram. The user can drag the vertices of the parallelogram and the other points change automatically to ensure it remains a parallelogram. A grid inside the shape allows students to estimate the area visually, then check against the actual computed area, which is continuously recomputed and displayed. The text on the page gives three different ways to calculate the area with a formula for each. The applet uses one of the methods to compute the area in real time, so it changes as the rhombus is reshaped with the mouse. A companion page is http://www.mathopenref.com/parallelogram.html showing the definition and properties of a parallelogram Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.
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