Thumbnail: A two-dimensional perspective projection of a sphere (CC BY-3.0; Geek3 via Wikipedia).

## 6 Math Concepts Explained by Knitting and Crochet

Using yarn and two pointy needles (knitting) or one narrow hook (crochet), pretty much anyone can stitch up a piece of fabric. Or, you can take the whole yarncraft thing light-years further to illustrate a slew of mathematical principles.

In the last several years, there’s been a lot of interesting discussion around the calming effects of needlecraft. But back in 1966, Richard Feynman, in a talk he gave to the National Science Teachers’ Association, remarked on the suitability of knitting for explaining math:

I listened to a conversation between two girls, and one was explaining that if you want to make a straight line…you go over a certain number to the right for each row you go up, that is, if you go over each time the same amount when you go up a row, you make a straight line. A deep principle of analytic geometry!

Both mathematicians and yarn enthusiasts have been following Feynman’s (accidental) lead ever since, using needlecraft to demonstrate everything from torus inversions to Brunnian links to binary systems. There’s even an annual conference devoted to math and art, with an accompanying needlecraft-inclusive exhibit. Below are six mathematical ideas that show knitting and crochet in their best light—and vice versa.

#### 1. HYPERBOLIC PLANE

A hyperbolic plane is a surface that has a constant negative curvature—think lettuce leaf, or one of those gelatinous wood ear mushrooms you find floating in your cup of hot and sour soup. For years, math professors attempting to help students visualize its ruffled properties taped together paper models … which promptly fell apart. In the late ‘90s, Cornell math professor Daina Taimina came up with a better way: crochet, which provided a model that was durable enough to be handled. There’s no analytic formula for a hyperbolic plane, but Taimina and her husband, David Henderson, also a math professor at Cornell, worked out an algorithm for it: if 1^x = 1 (a plane with zero curvature, made by crocheting with no increase in stitches), then (3/2)^x means increasing every other stitch to get a tightly crenellated plane.

#### 2. LORENZ MANIFOLD

In 2004, inspired by Taimina and Henderson’s work with hyperbolic planes, Hinke Osinga and Bernd Krauskopf, both of whom were math professors at the University of Bristol in the UK at the time, used crochet to illustrate the twisted-ribbon structure of the Lorenz manifold. This is a complicated surface that arises from the equations in a paper about chaotic weather systems, published in 1963, by meteorologist Edward Lorenz and widely considered to be the start of chaos theory. Osinga and Krauskopf’s original 25,510-stitch model of a Lorenz manifold gives insight, they write, “into how chaos arises and is organised in systems as diverse as chemical reactions, biological networks and even your kitchen blender.”

#### 3. CYCLIC GROUPS

You can knit a tube with knitting needles. Or you can knit a tube with a little handheld device called a Knitting Nancy. This doohickey looks something like a wooden spool with a hole drilled through its center, with some pegs stuck in the top of it. When Ken Levasseur, chair of the math department at the University of Massachusetts Lowell, wanted to demonstrate the patterns that could emerge in a cyclic group—that is, a system of movement that’s generated by one element, then follows a prescribed path back to the starting point and repeats—he hit on the idea of using a computer-generated Knitting Nancy, with varying numbers of pegs. “Most people seem to agree that the patterns look nice,” says Levasseur. But the patterns also illustrate applications of cyclic groups that are used, for example, in the RSA encryption system that forms the basis of much online security.

#### 4. MULTIPLICATION

There’s a lot of discussion about elementary students who struggle with basic math concepts. There are very few truly imaginative solutions for how to engage these kids. The afghans knit by now-retired British math teachers Pat Ashforth and Steve Plummer, and the curricula [PDF] they developed around them over several decades, are a significant exception. Even for the “simple” function of multiplication, they found that making a large, knitted chart using colors rather than numerals could help certain students instantaneously visualize ideas that had previously eluded them. “It also provokes discussion about how particular patterns arise, why some columns are more colorful than others, and how this can lead to the study of prime numbers,” they wrote. Students who considered themselves to be hopeless at math discovered that they were anything but.

#### 5. NUMERICAL PROGRESSION

Computer technician Alasdair Post-Quinn has been using a pattern he calls Parallax to explore what can happen to a grid of metapixels that expands beyond a pixel’s usual dimensional constraint of a 1x1. “What if a pixel could be 1x2, or 5x3?” he asks. “A 9x9 pixel grid could become a 40x40 metapixel grid, if the pixels had varying widths and heights.” The catch: metapixels have both X and Y dimensions, and when you place one of them on a grid, it forces all the metapixels in the X direction (width) to match its Y direction (height), and the other way around. To take advantage of this, Post-Quinn charts a numerical progression that’s identical on both axes—like 1,1,2,2,3,3,4,5,4,3,3,2,2,1,1—to achieve results like the ones you see here. He’s also in the process of writing a computer program that will help him plot these boggling patterns out.

#### 6. MÖBIUS BAND

A Möbius band or strip, also known as a twisted cylinder, is a one-sided surface invented by German mathematician August Ferdinand Möbius in 1858. If you wanted to make one of these bands out of a strip of paper, you’d give an end a half-twist before attaching the two ends to each other. Or, you could knit one, like Cat Bordhi has been doing for over a decade. It ain’t so simple to work out the trick of it, though, and accomplishing it requires understanding some underlying functions of knitting and knitting tools—starting with how, and with what kind of needles, you cast on your stitches, a trick that Bordhi invented. She keeps coming back to it because, she says, it can be “distorted into endlessly compelling shapes,” like the basket pictured here, and two Möbii intersecting at their equators—an event that turns Möbius on its ear by giving it a continuous “right side.”

## How to help at home

There are lots of everyday ways you can help your child to understand geometry. Here are just a few ideas.

### 1. Explore shapes around us

Find everyday opportunities to talk to your child about common 2D shapes like rectangles, triangles, and pentagons. Try to use the language of polygons and non-polygons:

- A polygon is a 2D shape that has three or more straight sides and angles (for example, a rectangle).
- A non-polygon is a 2D shape with sides that are not all straight (for example, a semicircle).

When talking about 2D shapes, encourage your child to use language like *straight*, *curved*, *side,* and *corner* (or *vertex/vertices*) to describe them. This will help them to understand the properties of 2D shapes and to know how they differ from 3D shapes.

Your child should know that 2D shapes are completely flat shapes. See how many different 2D shapes your child can spot in everyday life, such as on buildings, cushions, clothes, curtains, or in picture books!

### 2. Talk about 3D shapes

Talk to your child about common 3D shapes such as cuboids and prisms (a prism is a 3D shape that has two identical parallel bases, such as a cylinder).

Point out 3D shapes in different orientations and sizes when out and about with your child. See if they can describe the shapes to you using precise language like *face, vertex (vertices),* and *edge*.

Seeing lots of real-world examples will help your child realise that shapes like cuboids and pyramids are not always similar to each other. For instance, they could identify pyramids with different bases, such as triangular or square-based pyramids.

### 3. Guess the shape

Try this game with your child:

- Imagine a 2D or 3D shape.
- Encourage your child to ask you a series of questions to work out what shape you are thinking about.
- See how many questions they need before they guess correctly.

Alternatively, you could draw a shape without your child seeing. Describe the shape to your child and see if your child can draw it from your description. Compare the shapes to see how close they are.

Remember that it is important for children to see shapes in different orientations. For example:

You could also try challenging your child to find certain shapes – a 3D shape with a square base, a 2D shape that has three sides, and so on. This helps them understand exactly what all these maths words mean.

### 4. Compare, order, and sort shapes

Your child needs to be able to compare shapes using the correct mathematical language.

Show your child two shapes, 2D or 3D (for example, building blocks, cereal boxes, cans). Can your child name the shapes? Ask them to describe to you what is the same about them and what is different. For example:

You may choose a cube and a cuboid. Your child may tell you the shapes are

similarin that they are both 3D shapes with 8 vertices, 12 edges and 6 faces. They aredifferentin that a cube has 6 square faces whereas the cuboid may have 6 rectangular faces (which could be a mixture of squares and rectangles).

To practise sorting 2D shapes, your child could put the shapes into groups according to the number of sides, whether they are regular or irregular, and so on. When working with 3D shapes, they could sort the shapes according to the number of edges, vertices, faces, whether they can roll or not, and so on.

You could ask your child to sort their shapes in any way they like and decide on the headings for the groups. This helps them to figure out properties of shapes independently.

## (h). Earth-Sun Geometry

In this equation, **L** is the latitude of the location in degrees and **D** is the declination. The equation is simplified to **A = 90 - L** if Sun angle determinations are being made for either equinox date. If the Sun angle determination is for a solstice date, declination ( **D** ) is added to latitude ( **L** ) if the location is experiencing summer (northern latitudes = June solstice southern latitudes = December solstice) and subtracted from latitude ( **L** ) if the location is experiencing winter (northern latitudes = December solstice southern latitudes = June solstice). All answers from this equation are given relative to **True North** for southern latitudes and **True South** for northern latitudes. For our purposes only the declinations of the two solstices and two equinoxes are important. These values are: June solstice **D** =23.5, December solstice **D** =-23.5, March equinox **D** =0, and September equinox **D** =0. When using the above equation in tropical latitudes, Sun altitude values greater than 90 degrees may occur for some calculations. When this occurs, the noonday Sun is actually behind you when looking towards the equator. Under these circumstances, Sun altitude should be recalculated as follows:

Get outside and explore geometry (and other math) all around you. A math trail is a walk with various stops where you look at math in the world around you, and ask questions about it.

KaBOOM!

Find a playground in your neighborhood, complete with reviews and photos. You can also upload photos of your own.

National Math Trails

Discover math in the world around you.

Voronoi Toy

This open-source program lets users play with adding points to a Voronoi diagram.

Geometry Games

A number of downloadable games that let you explore topology, polygons, tilings, and more.

Geometry Playground

This is a free ruler and compass application for multiple geometries. (Not related to the exhibition.)

subblue

Tom Beddard writes programs&mdashsome interactive, some downloadable&mdashthat make beautiful geometric designs.

Cinderella

Interactive geometry software. The current version is not free, but the older version is. Visit the &ldquoDownload&rdquo page to find it.

SketchUp

Free 3D modeling software from Google.

Then go here for polyhedron models to use in SketchUp.

## Grade 6 Math

Looking for lessons, videos, games, activities and worksheets that are suitable for 6th Grade Math? Find loads of resources here.

These compilations of lessons cover measurements, integers, number properties, fractions & mixed numbers, scientific notation, estimation & rounding, decimals, algebra, exponents & square roots, geometry, coordinate geometry, ratios, proportions & percents, probability and statistics.

### Measurements

### Integers

Integer Games

Integer Number Line, Comparing Integers, Add, Subtract, Multiply & Divide Integers

Orbit Integers

Add integers. Click the correct answer to power your spaceship. Up to 4 players.

Integer Warp

Multiply Integers. Click the correct answer to power your spaceship. Up to 4 players.

Integer Practice

Add integers, subtract integers, multiply integers, & divide integers One or two players.

Integers Jeopardy Game

This game has 4 categories: add integers, subtract integers, multiply integers, & divide integers. You can play it alone or in teams.

## 5 Fighter Jets Crashed Because of the Angle of the Runway

You don't have to be a pilot to guess that landing on an aircraft carrier is really fucking hard. It's a tiny little landing strip crowded with other planes, bobbing up and down in the waves. Keep in mind, this is with a whole host of instruments, computers and signals to help guide planes in. The early planes didn't even have that.

But there was another problem .

The Laughably Simple Flaw:

Here's what the earlier carriers looked like. Couldn't be simpler, right?

It's a floating runway. How else would you design it?

Well, that design was kind of a suicide factory. As you can see, planes waiting to take off sit at the other end of the runway you're trying to land on. If you don't get stopped in time, you're going to create one hell of a fireball. And getting stopped in time was no small thing -- catching the arresting wire (the thing that stopped the plane) was a tricky business. Eventually carriers went with the cartoon-logic solution and installed barrier nets to stop planes if they missed all the wires. However, it wasn't all that uncommon for aircraft to bounce over the barrier.

So what was the brilliant innovation that allowed them to make landings that much safer?

They angled the landing strip about nine degrees.

Don't laugh -- it took years to come up with it. While some of the greatest technological advances in history, including space flight and splitting the goddamn atom, came from developments during World War II, we didn't think of angling the flight deck until 1952. Prior to that, every landing was a potential rear-end collision.

By angling the deck, a plane that missed the wires could go to full throttle, take off again and come around for another pass. Planes waiting to take off are near the bow, out of harm's way.

Angling the deck also allowed for the tactical advantage of being able to launch and recover aircraft simultaneously, whereas in WWII, launching had to be postponed while landings were occurring, and vice versa. Who knows how many lives could have been saved if someone had thought of doing this about 10 years sooner.

### Related: Spain Has An Abandoned Jumbo Jet Problem

## Geometry Math Games

Are you looking for free geometry math games? Check out the activities provided on this website!

The following geometry games are suitable for elementary and middle school students.

2D Shapes (Jeopardy Game)

Play this fun jeopardy-style math game alone, with another friend, or even in teams.

Geometry Math Vocabulary Game

Discover important math terms based on given properties or definitions.

Classifying Angles Game

Classify angles as acute, right, obtuse, or straight when you play this interactive Tic Tac Toe game against the computer.

Classifying Triangles Game

In this interactive game, kids will practice classifying triangles as as acute, right, or obtuse by dragging and dropping different images in the correct basket in less than two minutes.

2D Shapes Game (Concentration)

In this game students click on two cards to match the figure of a two-dimensional shape with its name. If there is a match, the problems remain on the page if not, the cards are turned over.

Classifying Geometric Figures Game

Classify geometrical figures as 2-dimensional or 3-dimensional by playing this fun and interactive geometry game.

Polygon or Not?

Do you know if a given geometric figure is a polygon or not? Play this fun game to demonstrate your skills! How many points can you score in one and a half minutes?

3D Shapes Game (Concentration)

Have fun matching pictures of three-dimensional shapes with the correct words. If there is a match, the problems remain on the page if not, the cards are turned over.

Types of Polygons

In this game you have to quickly name different types of polygons based on given clues. For each question you will have only 30 seconds to write your answer.

Angles Jeopardy Game

This game is a fun way to assess your knowledge about measuring and classifying angles. The game has a single-player mode and a multi-player feature.

3d-Shapes Game

Discover the names of the most important 3d shapes.

Pythagorean Theorem Game

Find the legs or the hypotenuse of a right triangle, and solve word problems by applying the Pythagorean Theorem.

Polygon Game

Learn how to clasify different polygons based on their characteristics.

## Primary 6 PSLE Math Syllabus

The PSLE Math syllabus consist of topics learnt in Primary 5 and Primary 6 as PSLE is a 2-year course for students.

The 2021 Primary 6 PSLE Math Syllabus can be divided into 3 main branches:

### P6 PSLE Math: Numbers and Algebra

The Primary 6 (P6) topics that are covered under Number and Algebra are Algebra, Whole Number, Fraction, Decimal, Percentage, Ratio and Speed, Rate and Time. Refer below for the breakdown of the skills in each topic.

### Whole Numbers*

- Notate & Represent Place Values Up to 10 000
- Read and Write Numbers in Words Up to 1 Mil
- Apply Order of Operations
- Divide by 10, 100, 1000 & their Multiples
- Multiply with 10, 100, 1000 & their Multiples
- Solve Problem Sums Involving 4 Operations

### Algebra

- Notate & Interpret Simple Algebraic Expressions
- Solve Simple Linear Equations
- Solve Simple Linear Expressions by Substitution
- Form & Solve Simple Linear Equations in Problem Sums

### Fraction

- Divide Whole Number by Proper Fraction
- Divide Proper Fraction by Proper Fraction
- Divide Proper Fraction by Whole Number
- Solve Fraction Problem Sums

### Ratio

- Understand the Relationship Between Fractions & Ratio
- Find Ratio of 2 Quantities in Direct Proportion
- Solve Problem Sums Involving 3 Quantities
- Solve Problem Sums that Involves Changing Ratio

### Percentage

- Find the Whole given a Part of the Percentage
- Find Percentage Increase/Decrease
- Solve Problem Sums Involving Percentages

### Decimals*

- Add and Subtract Decimals
- Round Off Decimals to the Nearest Whole Number, 1 Decimal or 2 Decimal Places
- Multiply Decimals up to 2-digit Whole Number
- Divide Decimals up to 2-digit Whole Number
- Multiply Decimals by 10, 100 1000 and their Multiples
- Divide Decimals by 10, 100 1000 and their Multiples
- Convert Measurements of Length, Mass & Volume
- Solve Problem Sums Involving 4 Operations

### Speed

- Write Speed in Different Units
- Find Time, given Distance and Speed
- Find Speed, given Time and Distance
- Find Distance, given Time and Speed
- Differentiate between Speed & Average Speed

### P6 PSLE Math: Measurement and Geometry

The Primary 6 (P6) topics that are covered under Measurement and Geometry are Area and Perimeter, Volume, Angles, Shapes and Properties, 2D/3D Representation and Nets. Refer to the following for the breakdown of the skills in each topic.

## Problem Solving

**Problem 564:Exploring the Stars in Orion - Light Year Madness**

Students explore the light year and its relationship to light travel time for observing events in different parts of space. When would colonists at different locations observe the star Betelgeuse become a supernova? [Grade: 6-8 | Topics: time lines time interval calculations time = distance/speed ] [Click here]

**Problem 507: Exploring the Launch of the Falcon 9**

Students use data from the launch of the Falcon 9 booster to determine its speed and acceleration. [Grade: 6-8 | Topics: speed=distance/time Time calculations] [Click here]

**Problem 505: SDO Sees Coronal Rain - Estimating Plasma Speeds**

Students estimate the speed of plasma streamers near the solar surface using images from a Solar Dynamics Observatory. [Grade: 6-8 | Topics: scale models speed=distance/time proportions] [Click here]

**Problem 488: RBSP and the Location of Dawn Chorus - II**

Students use hypothetical information from the twin RBSP spacecraft to triangulate the location of the Chorus signal near Earth using angle measurements, graphing and protractors to identify the intersection point of the CHorus signals. [Grade: 6-8 | Topics: Angles graphing protractors ] [Click here]

**Problem 452: The Closest Approach of Asteroid 2005YU55 - I**

Students work with a scaled drawing of the orbit of the moon and the asteroid trajectory to predict where the asteroid will be relative to earth and the orbit of the moon. [Grade: 6-8 | Topics: time=distance/speed scale models metric math] [Click here]

**Problem 451: The Spectacular Cat's Eye Planetary Nebula**

Students measure the diameter of the nebula and use speed information to estimate the age of the nebula [Grade: 6-8 | Topics: time=distance/speed scale models metric math] [Click here]

**Problem 445: LRO - The relative ages of lunar surfaces**

Students examine two Apollo landing areas using images from the LRO spacecraft to estimate the relative ages of the two regions using crater counting. [Grade: 6-8 | Topics: scale histogramming] [Click here]

**Problem 438: The Last Flight of the Space Shuttle Endeavor**

Students use tabular data and graphing to determine the launch speed and acceleration of the Space Shuttle from the launch pad. [Grade: 6-8 | Topics: tabular data, graphing, metric measurement, speed=distance/time] [Click here]

**Problem 437: Saturn V Rocket Launch Speed and Height**

Students use tabular data to determine the launch speed of the Saturn V rocket from the launch pad. [Grade: 6-8 | Topics: tabular data, graphing, metric measurement, speed=distance/time] [Click here]

**Problem 436: Space Shuttle Challenger Deploys the INSAT-1B Satellite**

Students use a sequence of images to determine the launch speed of the satellite from the Space Shuttle cargo bay. [Grade: 6-8 | Topics: scale, metric measurement, speed=distance/time] [Click here]

**Problem 435: Apollo-17 Launch from Lunar Surface**

Students use a sequence of images to determine the speed of ascent of the Apollo-17 capsule from the lunar surface. [Grade: 6-8 | Topics: scale, metric measurement, speed=distance/time] [Click here]

**Problem 434: Dawn Spacecraft Sees Asteroid Vesta Up-Close!**

Students use an image of the asteroid to determine the diameters of craters and mountains using a millimeter ruler and the scale of the image in meters per millimeter. [Grade: 6-8 | Topics: scale, metric measurement] [Click here]

**Problem 433: Space Shuttle Atlantis - Plume Speed**

Students use a sequence of images from a video of the launch to determine speed from the time interval between the images, and the scale of each image. [Grade: 6-8 | Topics: scale, metric measurement, speed=distance/time] [Click here]

**Problem 432: Space Shuttle Atlantis - Exhaust Speed**

Students use a sequence of images from a video of the launch to determine speed from the time interval between the images, and the scale of each image. [Grade: 6-8 | Topics: scale, metric measurement, speed=distance/time] [Click here]

**Problem 431: Space Shuttle Atlantis - Launch Speed**

Students use a sequence of images from a video of the launch to determine speed from the time interval between the images, and the scale of each image. [Grade: 6-8 | Topics: scale, metric measurement, speed=distance/time] [Click here]

**Problem 430: Space Shuttle Atlantis - Ascent to Orbit**

Students use a sequence of images from a video of the launch to determine speed from the time interval between the images, and the scale of each image. [Grade: 6-8 | Topics: scale, metric measurement, speed=distance/time] [Click here]

**Problem 429: Tracking a Sea Turtle from Space**

The latitude, longitude, elapsed time and distance traveled are provided in a table. Students use the data to determine the daily and hourly speed of a leatherback turtle as it travels from New Zealand to California across the Pacific Ocean. [Grade: 4-6| Topics: scale, metric measurement, speed=distance/time] [Click here]

**Problem 404: STEREO Spacecraft give 360-degree Solar View** Students use STEREO satellite images to determine which features can be seen from Earth and which cannot. They learn about the locations and changing positions of the satellites with respect to Earth's orbit. [Grade: 6-8 | Topics: angular measure, extrapolation distance = speed x time] [Click here]

**Problem 267: Identifying Materials by their Reflectivity** The reflectivity of a material can be used to identify it. This is important when surveying the lunar surface for minerals, and also in creating 'green' living environments on Earth. [Grade: 6-8 | Topics: percentage, interpreting tabular data, area ] [Click here]

**Problem 237: The Martian Dust Devils** Students determine the speed and acceleration of a Martian dust devil from time laps images and information about the scale of the image. [Grade: 6-8 | Topics: scales Determining speed from sequential images V = D/T] [Click here]

**Problem 247: Space Mobile Puzzle** Students calculate the missing masses and lengths in a mobile using the basic balance equation m1 x r1 = m2 x r2 for a solar system mobile. [Grade: 6-8 | Topics: metric measure, algebra 1, geometry] [Click here]

**Problem 245: Solid Rocket Boosters** Students learn how SRBs actually create thrust, and study the Ares-V booster to estimate its thrust. [Grade: 6-8 | Topics: volume, area, unit conversions] [Click here]

**Problem 238: Satellite Drag and the Hubble Space Telescope** Satellite experience drag with the atmosphere, which eventually causes them to burn up in the atmosphere. Students study various forecasts of the altitude of the Hubble Space Telescope to estimate its re-entry year. [Grade: 6-8 | Topics: interpreting graphical data predicting trends] [Click here]

**Problem 211: Where Did All the Stars Go?-** Students learn why NASA photos often don't show stars because of the way that cameras take pictures of bright and faint objects. [Grade: 6-8| Topics: multiplication division decimal numbers.] [Click here]

**Problem 209: How to make faint things stand out in a bright world!-** Students learn that adding images together often enhances faint things not seen in only one image the power of averaging data. [Grade: 6-8| Topics: multiplication division decimal numbers.] [Click here]

**Problem 148 Exploring a Dying Star** Students use data from the Spitzer satellite to calculate the mass of a planetary nebula from a dying star. [Grade: 9 - 11 | Topics:Scientific Notation unit conversions volume of a sphere ] [Click here]

**Problem 141 Exploring a Dusty Young Star** Students use Spitzer satellite data to learn about how dust emits infrared light and calculate the mass of dust grains from a young star in the nebula NGC-7129. [Grade: 4 - 7 | Topics: Algebra I multiplication, division scientific notation] [Click here]

**Problem 134 The Last Total Solar Eclipse--Ever!** Students explore the geometry required for a total solar eclipse, and estimate how many years into the future the last total solar eclipse will occur as the moon slowly recedes from Earth by 3 centimeters/year. [Grade: 7 - 10 | Topics:Simple linear equations] [Click here]

**Problem 124 The Moon's Atmosphere** Students learn about the moon's very thin atmosphere by calculating its total mass in kilograms using the volume of a spherical shell and the measured density. [Grade: 8-10 | Topics:volume of sphere, shell density-mass-volume unit conversions] [Click here]

**Problem 115 A Mathematical Model of the Sun** Students will use the formula for a sphere and a shell to calculate the mass of the sun for various choices of its density. The goal is to reproduce the measured mass and radius of the sun by a careful selection of its density in a core region and a shell region. Students will manipulate the values for density and shell size to achieve the correct total mass. This can be done by hand, or by programming an Excel spreadsheet. [Grade: 8-10 | Topics: scientific notation volume of a sphere and a spherical shell density, mass and volume.] [Click here]

**Problem 95 A Study on Astronaut Radiation Dosages in SPace -** Students will examine a graph of the astronaut radiation dosages for Space Shuttle flights, and estimate the total dosages for astronauts working on the International Space Station. [Grade level: 9-11 | Topics:Graph analysis, interpolation, unit conversion] [Click here]

**Problem 83 Luner Meteorite Impact Risks -** In 2006, scientists identified 12 flashes of light on the moon that were probably meteorite impacts. They estimated that these meteorites were probably about the size of a grapefruit. How long would lunar colonists have to wait before seeing such a flash within their horizon? Students will use an area and probability calculation to discover the average waiting time. [Grade level: 8-10 | Topics: arithmetic unit conversions surface area of a sphere ] [Click here]

**Problem 74 A Hot Time on Mars -** The NASA Mars Radiation Environment (MARIE) experiment has created a map of the surface of mars, and measured the ground-level radiation background that astronauts would be exposed to. This math problem lets students examine the total radiation dosage that these explorers would receive on a series of 1000 km journeys across the martian surface. The students will compare this dosage to typical background conditions on earth and in the International Space Station to get a sense of perspective [Grade level: 6-8 | Topics: decimals, unit conversion, graphing and analysis ] [Click here]

**Problem 71 Are the Van Allen Belts Really Deadly? -** This problem explores the radiation dosages that astronauts would receive as they travel through the van Allen Belts enroute to the Moon. Students will use data to calculate the duration of the trip through the belts, and the total received dosage, and compare this to a lethal dosage to confront a misconception that Apollo astronauts would have instantly died on their trip to the Moon. [Grade level: 8-10 | Topics: decimals, area of rectangle, graph analysis] [Click here]

**Problem 68 An Introduction to Space Radiation -** Read about your natural background radiation dosages, learn about Rems and Rads, and the difference between low-level dosages and high-level dosages. Students use basic math operations to calculate total dosages from dosage rates, and calculating cancer risks. [Grade level: 6-8 | Topics: Reading to be Informed decimals, fractions, square-roots] [Click here]

**Problem 66 Background Radiation and Lifestyles -** Living on Earth, you will be subjected to many different radiation environments. This problem follows one person through four different possible futures, and compares the cumulative lifetime dosages. [Grade level: 6-8 | Topics: fractions, decimals, unit conversions] [Click here]

**Problem 54 Exploring Distant Galaxies -** Astronomers determine the redshifts of distant galaxies by using spectra and measuring the wavelength shifts for familiar atomic lines. The larger the redshift, denoted by the letter Z, the more distant the galaxy. In this activity, students will use an actual image of a distant corner of the universe, with the redshifts of galaxies identified. After histogramming the redshift distribution, they will use an on-line cosmology calculator to determine the 'look-back' times for the galaxies and find the one that is the most ancient galaxy in the field. Can students find a galaxy formed only 500 million years after the Big Bang? [Grade level: 6-8 | Topics: Decimal math using an online calculator Histogramming data] [Click here]

**Problem 49 A Spiral Galaxy Up Close. -** Astronomers can learn a lot from studying photographs of galaxies. In this activity, students will compute the image scale (light years per millimeter) in a photograph of a nearby spiral galaxy, and explore the sizes of the features found in the image. They will also use the internet or other resources to fill-in the missing background information about this galaxy. [Grade level: 6-8 | Topics: Online research Finding the scale of an image metric measurement decimal math] [Click here]

**Problem 41 Solar Energy in Space** Students will calculate the area of a satellite's surface being used for solar cells from an actual photo of the IMAGE satellite. They will calculate the electrical power provided by this one panel. Students will have to calculate the area of an irregular region using nested rectangles. [Grade level: 7-10 | Topics: Area of an irregular polygon decimal math] [Click here]

**Problem 36 The Space Station Orbit Decay and Space Weather**Students will learn about the continued decay of the orbit of the International Space Station by studying a graph of the Station's altitude versus time. They will calculate the orbit decay rates, and investigate why this might be happening. [Grade: 5 - 8 | Topics: Interpreting graphical data decimal math] [Click here]

**Problem 31 Airline Travel and Space Weather** Students will read an excerpt from the space weather book 'The 23rd Cycle' by Dr. Sten Odenwald, and answer questions about airline travel during solar storms. They will learn about the natural background radiation they are exposed to every day, and compare this to radiation dosages during jet travel. [Grade: 6 - 8 | Topics: Reading to be informed decimal math] [Click here]

**Problem 10 The Life Cycle of an Aurora** Students examine two eye-witness descriptions of an aurora and identify the common elements so that they can extract a common pattern of changes. [Grade: 4 - 6 | Topics: Creating a timeline from narrative ordering events by date and time] [Click here]

NASA’s STEAM Innovation Lab is a think tank with an emphasis on space science content applications.