The Farther Away You Are, The Faster Away You Move

We believe our Universe is expanding in size. This concept may seem familiar and commonplace now, but try to imagine what it was like in 1929 when Edwin Hubble first published his data and came to this conclusion. Our Universe was expanding, literally and figuratively.

Lesson Summary and Pre-Requisites

If your math class needs a fresh way to for students to explore linear relationships or if your science class needs to develop an understanding of the expanding Universe, then this lesson is for you. The lesson consists of 3 Labs, only 2 of which are required. Lab C is an enrichment for higher level classes.

Lab A will ask students to take distance measures of sequins on elastic and plot the measures. From these plots, students will discover a linear relationship that will lead to the conclusion that "the further away you are, the further you travel when stretched."

Lab B will have students take the activity in the first lab one step further. Students will measure the velocity of each sequin and plot it (just as Hubble did with galaxies more than 70 years ago). Students will take time and distance measures to calculate the velocity. This provides a nice extension to Lab A, and demonstrates a connection to other subjects (i.e., astronomy).

For higher level students, Lab C can be substituted for Lab B, or done in addition to it. It is based on the fact that the relationship investigated here is not based on absolute values, but one that is concerned with the relative movement of the sequins. Absolute time is not important, only that we measure the locations of the sequins at different times (albeit the same times) for all of the sequins. In other words, "snapshots" in time work just fine and no measurement of time is needed. If your students perform all 3 Labs, have them compare and discuss the 3 results.

Student Labsheets

• Lab A Worksheet
• Lab B Worksheet
• Lab C Worksheet

Teacher Notes

Background Information

If you look at the data Hubble (1929), and 2 years later, Hubble and Humason (1931), published, you were led to the following inevitable conclusion: the further away from us you were, the faster away from us you were moving. While this may seem to be an odd thing, it is a consequence of the expansion of space. In this lesson, we will explore this concept; you will discover for yourself that when space expands, the further away you are, the faster away you move.

In the 1920s, American astronomer Edwin Hubble changed the way we thought of our Universe. First, he showed us that the fuzzy patches of light seen through the telescope were in fact other galaxies, far outside of our own Milky Way. Then, he spread the light of those galaxies into a spectrum. From this, he could see what each was made of and how fast it was moving. The latter was determined by the red shift, or Doppler Shift. You can read about the Doppler Shift in the next section. He could then plot on a graph velocity versus the distance for each galaxy he measured. He had determined the distance by the apparent brightness of the galaxy. When he made his plot, he found that the galaxies seemed to lay on a straight line -- in other words, their velocities were proportional to their distances! This discovery led to what is now called Hubble's Law. This rate of expansion is called Hubble's constant (H_o). It is written as

v = H_o * d

Based on these observations, an amazing conclusion was reached by Edwin Hubble and his colleagues: our Universe was expanding.

Important Note: Most galaxies are moving away from us. Not all, but most. Our nearest spiral neighbor, Andromeda, is in fact moving toward us and will one day collide with our Milky Way. Andromeda is moving toward us for the same reason galaxies themselves are not expanding -- gravity. It is important to distinguish this. Space itself is expanding, not the sizes of objects in the space. Gravity stops them from getting any larger than they are.

Doppler Shift

When you spread out the light from a stationary galaxy, you see dark lines (called spectral lines) at particular wavelengths due to the elements in that galaxy. This fact, coupled with the existence of the Doppler Shift due to the movement of sources of waves, provides us with an important tool in understanding the expansion of the Universe. Specifically,

• If the galaxy is moving toward you, the light waves are compressed. Its spectral lines move toward the shorter, or bluer, wavelengths.
• If the galaxy is moving away from you, the light waves are stretched. Its spectral lines move toward the longer, or redder, wavelengths.
• The faster you move toward or away, the larger the shift to the blue or red.

Think About It

If each 1 cm of elastic stretches to be 1.5 cm in length, then an object 1 cm away will move to 1.5 cm away. However, an object 10 cm away will move to 15 cm -- a full 5 cm from where it started. This is because each cm of the 10 cm will increase by 0.5 cm, leading to a cumulative total change of 5 cm. So an object 100 cm away will move to 150 cm! In other words, the farther away you were to start with, the farther away you move with each stretch. Thanks to the linear expansion of the elastic.

This is an analogy to what is happening in the Universe. The galaxies aren't moving, but they appear to be moving away from us because the space between us and them is expanding. And we now understand that the farther away you started, the more away you move with each expansion. Since space continually expands, the expansion is a function of time, and we measure your movement away from us by your velocity. The result is that since space is expanding, the farther away you are, the faster away you move!

Notes on Lab A

• To begin, read through the whole procedure and lab worksheet in order to familiarize yourself with this lab. In sum, students will be using elastic with sewed-on sequins to model an expanding universe. The elastic represents space; the sequins /buttons represent galaxies.
• Sewing is involved. If you would like to save time, have a strip of elastic and 5 sequins or buttons sewed on (for each group) prepared in advance. This is not as hard as it sounds, since this need only be done once. The elastic can be used over and over again for every class to which you'd teach this lesson. You can glue on the sequins, but be warned that this tends not to hold up very well under student wear. Also, it is important that you do not draw dots on the elastic and use them as the objects for measurement. As you stretch the elastic, the dots will expand in size. This teaches an incorrect concept about the Universe. As space expands, the sizes of objects in space do not expand.
• Before students begin the lab, you should let students know that, ultimately, they are about to examine what happens in a 1-dimensional Universe as it expands. An excellent closing activity may be to ask them how their experiments relate to our spatially 3-dimensional Universe.
• Students should work in groups of 3. Group members should decide on roles. One group member should be the "Zero Mark", another member should be the "Stretcher" and the last member should be the "Recorder".
• Students should stretch the elastic a short length before making the first measurement. Do not measure the location of the sequins when the band is unstretched.
• Here are some sample data (measured in centimeters) for step 4.

Galaxy #	Initial Distance (cm)	Final Distance (cm)	Change in Distance (cm)
1	22.0	24.0	2.0
2	34.6	38.7	4.1
3	49.9	56.8	6.9
4	63.5	72.8	9.3
5	77	88.5	11.5

• Students should notice in step 5 that galaxy 1 moved much less than any other galaxy. In our sample data, for instance, galaxy 1 only moved 2.0 cm whereas galaxy 5 moved 11.5 cm.
• Below is a plot of the sample data showing Initial Distance as the independent variable and the change of distance as the dependent variable. All students should get a linear relationship in their plots.



• Students should see in their plots that the relationship shows that the farther away you started, the more away you moved. Get them to think about how this can be so. Perhaps lead them through the thinking discussed in the "Think About It".

Notes on Lab B

• In this Lab, rather than having the final result being a plot of change in distance versus distance, we want to examine velocity versus distance. Students will perform essentially the same activity as in Lab B, only this time they will measure how many seconds they take to stretch the elastic. They will then generate velocity (distance stretched over time) and plot that data set.
• In step 6, students should see that the farther away the galaxy is, the greater its velocity is away from you. This is what Edwin Hubble saw; it was our first observational evidence that the Universe was, and is, expanding.

Notes on Lab C

• For each galaxy, the students will plot the following coordinate pairs (Initial Distance, to) and (Final Distance, t1), where to and t1 have undefined values, but t1 to. The times to and t1 represent any two measurement times that you observe the distance or the change in distance to the galaxies. You will then calculate the slope of the line which connects these two points. Remind students that in this case, the slope is the galaxy's velocity: velocity = distance / time. Discuss with the students that it does not really matter what the values of to and t1 are, and why this is so.
• The slope for each galaxy is calculated from the sample data as follows:
(#1) 2 / (t1-t0)
(#2) 4.1 / (t1-t0)
(#3) 6.9 / (t1-t0)
(#4) 9.3 / (t1-t0)
(#5) 11.5 / (t1-t0)

• When the students plot velocity versus Initial Distance, they will get essentially the same plot they got before. Since the times of observation for the galaxies on the elastic strip are the same, dividing each distance value by same time (even if it is unknown) just scales the results. It does not change the trend.

National Standards

Science

Origin and Evolution of the Universe: In grades 9 - 12 all students should:

"Early in the history of the universe, matter, primarily in the form of hydrogen and helium, clumped together by gravitational attraction to form countless trillions of stars. Billions of galaxies, each of which is a gravitationally bound cluster of billions of stars, now form most of the visible mass of the universe; Stars produce energy from nuclear reactions, primarily the fusion of hydrogen to form helium. These and other processes in stars have led to the formation of all the other elements."

Mathematics

Algebra: In grades 9 - 12 all students should:

• Understand patterns, relations, and functions
• understand relations and functions and select, convert flexibly among, and use various representations for them;
• analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior;
• Represent and analyze mathematical situations and structures using algebraic symbols
• understand the meaning of equivalent forms of expressions, equations, inequalities, and relations;
• write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency -- mentally or with paper and pencil in simple cases and using technology in all cases;
• use symbolic algebra to represent and explain mathematical relationships;
• judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.
• Use mathematical models to represent and understand quantitative relationships
• identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships;
• use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts;
• draw reasonable conclusions about a situation being modeled.
• Analyze change in various contexts
• approximate and interpret rates of change from graphical and numerical data.

Measurement: In grades 9 - 12 all students should:

• Understand measurable attributes of objects and the units, systems, and processes of measurement
• make decisions about units and scales that are appropriate for problem situations involving measurement.
• Apply appropriate techniques, tools, and formulas to determine measurements
• analyze precision, accuracy, and approximate error in measurement situations;
• use unit analysis to check measurement computations.