Illustrative Mathematics
Illustrative Mathematics: G C, G Srt Neglecting the Curvature of the Earth
Jerry and Ashley are trying to find out if it is possible to see the lowest point in the USA from the highest point in the USA. It turns out that the highest point in the United States, the peak of Mount Whitney, is only 85 miles from...
Illustrative Mathematics
Illustrative Mathematics: G Gpe, G Co, G Srt Unit Squares and Triangles
In this task, young scholars are shown three unit squares and two line segments connecting two pairs of vertices. They are asked to find the area of a triangle formed by intersecting lines. They can solve it using coordinate geometry or...
Illustrative Mathematics
Illustrative Mathematics: G C Right Triangles Inscribed in Circles I
This task provides a good opportunity to use isosceles triangles and their properties to show an interesting and important result about triangles inscribed in a circle with one side of the triangle a diameter. The fact that these...
Illustrative Mathematics
Illustrative Mathematics: G C Tangent to a Circle From a Point
This task builds on a previous one called ''Right triangles inscribed in circles I.'' Here students are asked to construct a tangent line to a circle from a point outside of the circle. Aligns with G-C.A.4.
Illustrative Mathematics
Illustrative Mathematics: G C Right Triangles Inscribed in Circles Ii
In this task, students analyze the characteristics of a right triangle that has two sides labeled with midpoints and show that the triangle is inscribed in a circle with a diameter the length of its hypotenuse. Aligns with G-C.A.2.
Illustrative Mathematics
Illustrative Mathematics: G Gpe.5 Triangles Inscribed in Circle
In this task, students use ideas about linear functions to determine when certain angles are right angles. They use the fact that two lines (if not vertical or horizontal) are perpendicular when their slopes are inverse reciprocals, as...
Illustrative Mathematics
Illustrative Mathematics: G C Mutually Tangent Circles
Three circles are mutually tangent. Young scholars are asked to find the total area of the circles plus the area inside where they meet. Students will need to know how to calculate the area of a circle, a sector of a circle, and an...
Illustrative Mathematics
Illustrative Mathematics: G C, G Srt Setting Up Sprinklers
In this task, students must decide where to place sprinklers on a lawn so that the maximum area of the lawn will be watered. To solve it, they must find areas of sectors of circles and use trigonometric ratios to solve right triangles....
Illustrative Mathematics
Illustrative Mathematics: G C Placing a Fire Hydrant
You have been asked to place a fire hydrant so that it is an equal distance from three locations on a map. Students must fold a paper so that this point is where two folds intersect, and they must explain why their construction works and...
Illustrative Mathematics
Illustrative Mathematics: G C Inscribing a Triangle in a Circle
This problem introduces the circumcenter of a triangle and shows how it can be used to inscribe the triangle in a circle. It also shows that there cannot be more than one circumcenter. Aligns with G-C.A.3.
Illustrative Mathematics
Illustrative Mathematics: G C Circumcenter of a Triangle
The goal of this task is to construct the circumcenter of a triangle, that is, the point which is the same distance from each of the triangle's three vertices. This task shows that the three perpendicular bisectors of the sides of a...
Illustrative Mathematics
Illustrative Mathematics: G C Similar Circles
In this task, students are asked to show that all circles are similar using two different methods, a visual one and an algebraic one. Includes a GeoGebra worksheet that can be used by the instructor to demonstrate the material. Aligns...
Illustrative Mathematics
Illustrative Mathematics: G Srt Shortest Line Segment From a Point P to a Line L
In this task, students develop an understanding that the shortest path from a point to a line always meets the line at a right angle. Aligns with G-SRT.C.8.
Illustrative Mathematics
Illustrative Mathematics: G Srt Tangent Line to Two Circles
The purpose of this task is to use similar triangles and setting up a proportion in order to calculate a side length. Aligns with G-SRT.B.5.
Illustrative Mathematics
Illustrative Mathematics: G Srt, G Mg Coins in a Circular Pattern
In this task, students investigate how many coins will fit around a central coin when they are all of the same denomination, and if there is room left over. The teacher is given a chart of four coin denominations and their diameters, but...
Illustrative Mathematics
Illustrative Mathematics: G Srt, G Mg Seven Circles Iii
In this task, students are shown circle formations where an inner circle is surrounded by a set of circles and all circles are touching. In the first situation, all circles have the same diameter. In the second, the inner circle is...
Illustrative Mathematics
Illustrative Mathematics: G Srt Pythagorean Theorem
The purpose of this task is to prove the Pythagorean Theorem using similar triangles. Aligns with G-SRT.B.4.
Illustrative Mathematics
Illustrative Mathematics: G Srt, G Mg How Far Is the Horizon?
Some friends are at the beach looking out onto the ocean on a clear day and they wonder how far away the horizon is. A second problem asks how far the horizon would be from the top of a mountain. Students must also answer a question...
Illustrative Mathematics
Illustrative Mathematics: G Srt Similar Triangles
This task works toward establishing the AA criterion for similarity of triangles. Students are asked to provide a detailed sequence of transformations that moves one of the given triangles onto the other. Aligns with G-SRT.A.3.
Illustrative Mathematics
Illustrative Mathematics: G Srt Ask the Pilot
In the July 2013 issue of United Airlines' Hemisphere Magazine, a pilot responded to a question about how far in the distance he could see at different altitudes. The solution involved right triangles, lines of tangency to a circle, and...
Illustrative Mathematics
Illustrative Mathematics: G Srt Similar Triangles
The goal of this task is to examine similarity of triangles from the point of view of proportional sides, showing that this is sufficient to conclude that two triangles are similar. Aligns with HSG-SRT.A.2.
Illustrative Mathematics
Illustrative Mathematics: G Srt Folding a Square Into Thirds
The goal of this task is to apply knowledge about similar triangles in order to understand an origami construction: trisection of the side of a square. Aligns with G-SRT.B.5.
Illustrative Mathematics
Illustrative Mathematics: G Srt Bank Shot
This task asks students to use similarity to solve a pool table problem. Students must use the fact from physics that when an object bounces off of a wall, the angle of incidence equals the angle of reflection. Aligns with G-SRT.B.5.
Illustrative Mathematics
Illustrative Mathematics: G Srt Are They Similar?
In this task, students are shown two triangles that appear to be similar, but whose similarity cannot be proven without further information. Students are asked to provide a sequence of similarity transformations that maps one triangle to...