Education ∪ Math ∪ Technology

# Tag: realmath(page 1 of 3)

Act One

Show this video. Ask students what questions they have.

Act Two

Give the students these photos. From these they should be able to figure out the distances travelled by the car, but you may wan to point out those handy reference points in the background (hint: the parking spots).

Act 3

I don’t know exactly what Act 3 should look like. Would a graph make the most sense? A function in standard form? Would an image overlay of the car in all four positions work the best (I don’t have the tools to make this on my current computer)?

This article originally appeared in my school’s monthly magazine, and was co-written with Chelsea Todd, a 4th grade teacher at my school. Note that the lesson idea presented below is Chelsea’s idea.

The 4th grade math classes were recently completely engaged with a challenging but slightly unusual mathematical task. Students had been studying area and perimeter with the guiding question, “How do area and perimeter help us to construct things?”. For their summative project, students designed and constructed a model building design, starting first with the architectural blueprints and completing their project with the structure. They had the choice to build their world virtually using Minecraft or to use Lego blocks. During the two classes, it looked like every student was engaged in the task for the entire class time, and in fact sometimes their concentration on their work during the task made it difficult to interrupt them to ask questions!

In the first class, students designed blueprints on graph paper, being sure to include accurate labels of the values and calculations of each shape’s area and perimeter. They spent the entire time busy at work, trying to create complicated designs, knowing that they would be asked to recreate these designs the following day. They used their knowledge of area and perimeter to accurately label their work, often re-checking their values to ensure that their blueprints accurately captured their intentions.

In the second class, students recreated their blueprints, either using Minecraft or Lego blocks. They brought in their own computers if they were using Minecraft, and at one point we noticed that there were seven different kinds of computers in use. If they had Lego at home, the students brought that in as well, and extra Lego and building boards were provided for all other students.

The students had to use their blueprints and their models to calculate the area and perimeter of their designs, with some students also calculating the total volume, and explain their solutions. This task therefore assessed the students’ ability to create and read a blueprint, calculate perimeter and area, and communicate their understanding of mathematics. It also required them to demonstrate their visualization skills to transfer their 2-D plans to a 3-D structure, an area of mathematics that often falls second to calculations.

During the lesson, students were observed problem-solving using mathematics and applying mathematics skills they had previously learned. The students had to continue to develop their numeracy skills while at the same time starting some pre-algebraic reasoning, as they tried to get the scale and dimensions of their blueprint to match their constructed diagrams.

The context of the problem was realistic, so that students should have seen that mathematics is something that one can use in one’s life. The context helped students draw connections that they may have otherwise not understood; for example, the relationship between multiplying two numbers and the area of a rectangle. We hope this will keep students, and their teachers, inspired about mathematics!

Probably the best sign that this is a project worth doing actually came from the fifth grade students. When they heard that this year’s fourth grade class was breaking out the Lego, they said, “Oh, I remember doing that project. That was fun!” The fifth graders did a project that involved practicing skills, learning mathematics in context, and they not only remember doing it, they found it fun. That sounds like a success to me.

One of my jobs at my current school is work on mathematics enrichment projects with three 4th grade students. For the past couple of weeks, in between their camps, my camp, and other end of year events, we have been working on looking into a mathematics modelling task, specifically, a fork catapult. The 4th grade boys and I completed this project over a series of 3 lessons with me, and 2 more lessons working on it independently between our sessions.

I built the fork catapult fairly quickly for the boys, mostly because we lacked a lot of time to work on this particular project, and partially because I wanted them to focus on the collection of data, not specifically the design of their catapult (As it turned out, the boys found time to create three more of their own designs anyway).

I set up the catapult, starting opening up the clip on the clipboard, and right away one of the boys asked, "I wonder how far it will go?"

So, we collected data. What we did was open up the clip part of the clipboard so that it was 1cm, 1.5cm, 2cm, 2.5cm, and 3cm open, and shot a mini-whiteboard eraser by releasing the clip, 10 times for each position of the clip. We then measured the distance (as accurately as we could) the eraser flew across the room. The initial prediction from the boys was that the wider the clip was open, the farther the eraser would fly, which the boys continued to believe, even once we had actually collected all of the data.

I showed the boys how to plot 1 or 2 points, and then I then asked them to carefully graph the rest of the data. This is what they produced (the red dots were added later).

Once the boys had their data graphed (do you see the small error in the graph above?), we looked at together to see what it meant. From the graph, the boys decided that it was clear that their initial hypothesis (they didn’t use this language, I did) was incorrect and that there was a maximum distance that the eraser could be launched. One of the boys had the insight that the problem was that when the clip was "open too wide" that too much of the energy went into throwing the clip up, and not enough went into catapulting the clip forward.

The boys also reasoned that the eraser would be shot the furthest when the clip was between 2cm and 2.5cm open. Our next step was to test their new hypothesis. As you can see from the red dots graphed above, their hypothesis was probably right, although in order for the boys to see that, we first had a discussion about finding average values from the graph.

At the end of the second class, we started cleaning up, and the boys asked me to photocopy their graph and data, and let them take home their other catapult designs. One of them said to the other boys, "Let’s keep working on this next year!"

There are a lot of ways one could go with this project, but one thing I really liked about it was that we created a mathematical investigation from some pretty basic supplies, that the data that results has enough experimental error that it makes it more obvious that collecting many trials is useful, and that the graph was clear enough that students could read the results from the graph fairly easily.

I’m always on the look-out for ways of finding connections between mathematics and other areas of knowledge. Music is one of the areas of knowledge that I know has some similarities with mathematics, and so I’ve been brainstorming ways one could incorporate music into a mathematics classroom. Here are a few examples.

1. A musical scale is an example of a sequence (of notes) and could be used to show the idea the order of objects, related to the order of numbers. As each note in an ascending scale is played in sequence, students should be able to hear that the notes have a order, and then you can relate this order to the order we associate with the counting numbers.

2. Introducing students to patterns can also be done nicely with music, either with notes, or with percussion instruments. Here are two sample patterns. One simple activity to do with students here is to have them produce their own different types of patterns.

3. You can also use music to develop some conceptual understanding of skip counting. Often children are taught to count by 2s and 3s but do not necessarily understand what this means. Obviously one should use manipulatives and other techniques to develop this understanding, but here’s an example of how skip counting sounds in music. This example could also be used as an introduction to simple linear functions as well at a later grade.

4. You could introduce students to fractions by comparing relative sizes of different notes. In the example below, the music starts off with 16th notes, followed by 8th notes, quarter notes, half notes, and finally a whole note. Can you hear how obvious the difference is between the notes?

5. Music notes themselves are sound waves, which if you have an oscilloscope, you can visualize directly as you listen to a note. A pure note has a relatively simple associated wave, but notes as played on a music instrument are almost always composed of multiple harmonics (or waves of different frequences added together). This is an example of a capture from a digital oscilloscope. What do you think the seemingly random waves that appear between the notes are from?

You can also visualize the volume of the notes (by opening up an audio recording of some music being played in a program like Audacity, for example), and notice an interesting drop-off that occurs. If you measure this drop-off closely, it should match an exponential decay function.

Notice also what the volume of the notes over time looks like when we zoom in on one of them.

6. Imagine you played one note on the piano at one constant speed, and another note at a different constant speed. After how many notes would you play both notes at the same time? This is an application of the lowest common multiple (provided you express the number of notes played per unit time in lowest terms). Below is a video where one note is being played at a rate of 120 times per minute, and in a different recording, the same note is being played at a rate of 150 times per minute. Do you notice something interesting when both recordings are played simultaneously?

7. Another area where mathematics comes into play is in the ratio of the wavelengths of different notes. Karen Cheng does an excellent job of explaining how this relates to why we appreciate some music more than other music.

Hopefully these short examples give you some examples of how mathematics and music are related. In another post, I intend to look at musical instruments, and how mathematics can be used to construct them.

* Musical scores created with Noteflight. This program has a free demo one can use without signing in, but if you want to save your work, you will need to sign up for a free account.

** If you are viewing this post in your email, none of the videos will be visible, so I recommend reading it online here.

I’ve been playing with paper folding recently, and exploring the mathematics involved. I’m simply amazed by the number of mathematical ideas that can be represented by paper folding, so I thought I would share a few of my discoveries here.

Sequences

As you can see above, you can generate the sequence of numbers 1, 2, 4, 8, 16, 32 and so on, just by folding the paper in half again each time. This means that there is an exponential relationship between the number of folds you have made and the number of areas created on the paper.

Notice that if I instead fold the paper into thirds each time, the sequence changes into 1, 3, 9, 27, etc… which suggests that folding a piece of paper is a little bit like multiplication.

Fractions

First, form the fraction by folding the paper into quarters and shading three of them in.

Now fold the paper in the other direction into thirds, and shade , ideally in the other direction. Where your two shadings have overlapped is the product of your two fractions, in this case .

Symmetry

Here is an example of folding the paper around the centre to produce rotational symmetry. I worked with a student to produce snowflakes with  9 points, 12 points, and other points, after watching this interesting video by Vi Hart

Tessellations

If you fold a paper in half a bunch of times, you can create a tesselation by cutting portions of the paper out. The number of folds and the size of the repeated portion of the tessellation have an interesting relationship.

Circle geometry

If you very carefully cut a circle out of a piece of paper (which will finally give you a use for all of those CDs you have laying around you aren’t using anymore), you can prove quite a large number of the theorems from circle geometry by folding the paper in certain ways.

For example, if you fold the paper in half twice in two different directions, the intersection of the folds has a useful property.

For further resources on paper folding and mathematics, see this TED talk by Robert Lang, this book on the mathematics of paper-folding, and this useful PDF describing some geometry theorems that can be demonstrated through paper folding. See also this very interesting article on fraction flags (via @DwyerTeacher).

I am presenting in Hope, British Columbia today, on the topic of Math in the Real World. Here are my presentation slides.

You will probably notice that sections 6 and 7 of my presentation are not completely focused on the topic of “math in the real world” but I feel like they are such important concepts for mathematics teachers to understand that I needed to include them in my presentation.

(Image credit: Kumaravel)

After reading Bruce’s post this morning about finding the area of a leaf, it occurred to me that this could be solved using calculus. The basic project would be for students to collect some leaves, trace them onto graph paper, determine through modeling the equations that correspond to the edges of the leaf, and then use integration to find the area as exactly as they can. Students could then confirm their answer works by splitting the area of the leaf into smaller shapes and estimating the total area without calculus.

As an added bonus, students could all choose the same kind of leaf, pool their results, and use some statistics to determine the total area of the leaves in a park, or on their street.

This is another in a series of posts about how one could find mathematics in the world around us.

My son loves to play with train tracks. A few days ago, while playing with his train tracks, he observed, "Daddy, I can’t turn a train around." I asked him what he meant. "No matter which way I go on this track, I can’t get my train to start facing in the other direction. I’d have to pick it up, but that’s cheating." (Note: I’m paraphrasing here)

Observations like this are mathematical observations about the world. He has abstracted from his train tracks to a property of his train tracks, specifically the direction his train is able to travel. He has then attempted, and I watched him do this, to verify this statement is true by running his trains around the track in every possible comination.

My wife and I spoke about this later, and she came to the observation that in order to be able to turn around his train on the track (without "cheating" by lifting it up), he needs a closed loop with a single entrance and exit point included in his track somewhere, and this entrance and exit point has to connect to the rest of the track in a certain way. So I asked the question, does he have the right track to be able to create a closed loop? If you look at the picture above, you may be able to answer this question yourself.

The area of mathematics that deals with these kinds of issues is called graph theory, and it was invented by Euler for a very different purpose many years ago. It is unfortunately not in most school curriculums, but it is certainly an interesting area of exploration, and one which is accessible to students.

My uncle called me today, and asked me a math question. Normally, I get called and asked technology related questions, but occasionally people remember that I have a mathematics background and call me in to assist.

My aunt wants to build a raised garden bed with a very particular shape. My uncle has been tasked with building it. She wants 3 of the sides of the shape to be 4 feet long, and the 4th side to be three feet long, and the whole shape should form a trapezoid (with a line of symmetry down the middle of the trapezoid). It took a little bit of chatting on the phone to get this to be clear, and I can see how being able to send each other pictures would have been really useful. To be able to build this shape as accurately as he would like, he needs to know all of the angles of the shape, so he can cut the pieces of the wood with the angles in the right position using a miter saw.

I looked at the shape and decided that the fastest solution would be to build the shape in Geogebra, and measure the angles, which resulted in this.

Not the exact solution, but close enough that my uncle would be able to use the miter saw (which has a maximum accuracy of 1 degree, according to my uncle) and cut the wood for his shape. It took me about 3 or 4 minutes to draw the shape in Geogebra and measure the angles.

After my phone call with my uncle was over, I decided that I should double check this solution though, and verify that I knew how to solve it.

I drew an imaginary line across the shape, and labelled that side x. This allowed me to create a pair of equations using the Cosine law, and I ended up with the following equation to solve:

which simplifies to:

and finally leads to this calculation:

On my calculator, that leads to a value of the smaller angle of about 82.8° and a larger angle of 97.2°, which means that my diagram that I drew for my uncle is fairly close. Wanting to be sure that my answer was correct, I also checked it using Wolfram Alpha, and on my graphing calculator.

After I told my uncle the solution, he told me that my aunt had suggested drawing the diagram carefully on a piece of paper and measuring the angles with a protractor, but he had complained that solution wasn’t "mathematical enough." Of course, this leads to a discussion of what it means to do mathematics, anyway.

Does it matter which way I solve this problem for my uncle? Which of these techniques would you classify as "mathematics"? All of them? None of them?

This is another post in my series on mathematics in the real world.

Thanks to a colleague of mine, I rediscovered the Google Public Data explorer. Within 10 minutes, I had constructed the above graph, which shows adolescent fertility rate for 15 to 19 year olds, versus life expectancy, measured against (look at the colors) average income for all of the countries in the world. If you click play, you can see a happy trend; life expectancy is increasing across the world for almost all countries, and the fertility rate is also decreasing.

This type of graph also lends itself well to questions from your students. For example, they may ask why so many teenagers have babies in some countries. They may also why there is a relationship (and from the above graph, it looks like the relationship is reasonably strong), between births from teenage moms, and life expectancy. They may also ask about trend itself, and why that is happening. Further, they may ask, how strong is this relationship? They may also confuse correlation with causation, which in itself can lead to an interesting conversation.

A natural extension of an activity related to this graph would be to have students construct their own graphs, perhaps even collecting their own data. What kind of social data do you think would interest your students?