This task provides an opportunity to model a concrete situation with mathematics. Once a representative picture of the situation described in the problem is drawn (the teacher may provide guidance here as necessary), the solution of the task requires an understanding of the definition of the sine function. When the task is complete, new insight is shed on the ``Seven Circles I'' problem which initiated this investigation as is noted at the end of the solution.

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: The geometry of the earth-sun interaction plays a very prominent role in many aspects of our lives that we take for granted, like the variable length o...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: In the July 2013 issue of United Airlines' Hemisphere Magazine the following article appeared: Write down an equation that describes Captain Bowers' me...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important aspects of the task and its potential use.

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important aspects of the task and its potential use.

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Below is a picture of $\triangle ABC$: Draw a triangle $DEF$ which is similar (but not congruent) to $\triangle ABC$. How do $\frac{|DE|}{|DF|}$ and $\...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: In rectangle $ABCD$, $|AB|=6$, $|AD|=30$, and $G$ is the midpoint of $\overline{AD}$. Segment $AB$ is extended 2 units beyond $B$ to point $E$, and $F$...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Suppose we take a square piece of paper and fold it in half vertically and diagonally, leaving the creases shown below: Next a fold is made joining the...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Milong and her friends are at the beach looking out onto the ocean on a clear day and they wonder how far away the horizon is. About how far can Milong...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Below is a picture of a right triangle $ABC$ with right angle $C$ along with the point $D$ so that $\overleftrightarrow{CD}$ is perpendicular to $\over...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important aspects of the task and its potential use.

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important aspects of the task and its potential use.

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: In the two triangles pictured below $m(\angle A) = m(\angle D)$ and $m(\angle B) = m(\angle E)$: Using a sequence of translations, rotations, reflectio...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: In the picture below, points $A$ and $B$ are the centers of two circles and they are collinear with point $C$. Also $D$ and $E$ lie on the two respecti...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important aspects of the task and its potential use.

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important aspects of the task and its potential use.

Game Over Gopher is an exciting tower defense game that guides students in plotting coordinate pairs, differentiating negative coordinates from positive coordinates, and identifying the four quadrants. Hungry space gophers are marching towards a prize carrot, and to defend it players place tools around the coordinate grid to “feed” gophers and make them lose interest. Ruby mines (which must be placed at designated x, y coordinates) yield currency that players spend to strategically place carrot launchers, garlic rays, corn silos, and beet traps – gopher feeding tools. To introduce students to the coordinate grid, Game Over Gopher introduces new skills via interactive tutorials and uses consistent visual clues (e.g., x red, y blue) to guide players in plotting coordinates. This means it’s not necessary to teach students coordinate plotting ahead of time. As the levels progress, the number of tools available increases, the level of math vocabulary increases, the scale of the grid changes, and players are asked to expand their mastery of the grid by reflecting points across axes. The game lowers intimidation about the coordinate grid, helps students understand how positive and negative numbers reflect each other across the axes, and helps students get comfortable with the four quadrants.

While students need to be able to write sentences describing ratio relationships, they also need to see and use the appropriate symbolic notation for ratios. If this is used as a teaching problem, the teacher could ask for the sentences as shown, and then segue into teaching the notation. It is a good idea to ask students to write it both ways (as shown in the solution) at some point as well.

In this activity, students study gas laws at a molecular level. They vary the volume of a container at constant temperature to see how pressure changes (Boyle's Law), change the temperature of a container at constant pressure to see how the volume changes with temperature (Charles’s Law), and experiment with heating a gas in a closed container to discover how pressure changes with temperature (Gay Lussac's Law). They also discover the relationship between the number of gas molecules and gas volume (Avogadro's Law). Finally, students use their knowledge of gas laws to model a heated soda can collapsing as it is plunged into ice water.

"Shadows are corrupting the land. Restore the balance of nature by exploring place value. Gate guides students in: lowering intimidation about large numbers and
decimals, understanding the meaning of place value, and realizing that the same mathematical concepts that apply to the ""easy"" integers apply to every order of magnitude."