Fibonacci Numbers in Plants
Exponent challenge question
At OAME I learned several new problems which I have been sharing with my classes, colleagues, and friends. One that’s been particularly fun with grade 9s has been an exponent question.
The question is: how many digits are in the answer to this question.
We quickly notice that calculators can’t handle this problem (phones can, so I’ll need to change it for the future). It’s fun to tackle problems that are too big for calculators, but our brains can figure them out.
This question illustrated so many misconceptions my students have about exponents. It was good to see them exposed so we could tackle them together and build more number sense.
The first misconception was that the base and the exponent would both be changed when raised to a new power. This makes me think we’ve done too much practice with algebraic bases where we’ve ignored the base because it’s a letter. They knew that the power of a power multiplies the exponents together, but they missed the idea that it’s that many 2s or 5s being multiplied together.
The next misconception is that when multiplying, we add the exponents and multiply the different bases. They are used to adding exponents together when the bases are the same, and maybe thought that you do that all the time. They are working on remembering rules, and not understanding the concepts.
We brought it down to a smaller scale to see if that would help. We tried 2 squared x 5 squared etc,
Students were making connections more quickly now, seeing that all the answers were powers of 10. We’d just done scientific notation so this was a nice connection to see.
Some spent a lot of time exploring various powers of 2 and 5 and seeing how they grew. We practiced how to calculate this (where the exponent buttons are) and how to interpret scientific notation on our calculators.
While consolidating the lesson we had time to talk about many aspects of the math that we’ve learned. We reviewed what exponents mean, and what the power of a power does, and why the base stays the same.
We reviewed 2 different ways to multiply 16×625, using area model or by doubling 4 times.
We looked at powers of 10 as well and made the connection to the scientific notation that we’ve practiced.
We looked at how a power of a product is the product of the power as well.
All in all we have enjoyed working with this problem in several classes now. It helps to show what skills are solid, and where we need some more practice.
Fractions
I’ve been working with some grade 8 students, supporting their understanding of fractions. We have been using various representations and manipulatives to help. Today I brought Cuisinaire rods, and we used concept circles.
We used 12 as our “1” and represented a variety of fractions based on that. I wrote the fractions and the students needed to place the blocks.
The second time we cleared the values in each segment, put different blocks in each section, and then the students needed to write down the fraction.
Finally we tried it again using the 10 rod as our “1” which pushed their thinking. By changing what the “whole” is, each block takes on a different meaning. When the whole is 12, each white block is 1/12 since it takes 12 white blocks ro build the whole. When the whole is 10 each white block is 1/10, since it takes 10 white blocks to build the whole.
Students are starting to engage more, and respond well to the manipulatives. We are working on conceptual understanding and filling in some gaps in both skills and confidence. It’s been a fun challenge!
How Tall is the Novelis Sign?
We have a very tall sign visible from the park beside our school. I’ve always wondered how tall it is, so decided we should try as a class to figure it out. We can’t actually walk to the sign, so we are trying to calculate the height using 2 different angles of elevation separated by a known distance.
We have clinometers that we are learning to use, and we tried our best to get good measurements, but it was surprisingly windy on the field, so even with classmates becoming windbreaks it was hard to get accurate measurements. We calculated the height to be anywhere from 6m (which is very much too small) to 40m which seems too large.
It was a lovely day to go outside, and get our hands on the measuring tools and build some sense around how we might apply our trigonometry skills to solve real problems. Next time I’ll check for wind in the forecast as well as for rain!
Cookie Towers
In grade 9 we built cookie towers to generate some single variable data to use for our statistics and box and whisker plots.
Each group made 12 towers and kept track of how many cookies were in each. From there, they calculated the mean, median, mode and range of their data.
Then each group made box and whisker plots for their data. We learned about how each quartile will hold the same number of data points, and the graph will give us lots of information about how the data is spread out. Students used the idea of density to talk about which quartiles had data that was more dense and less dense.
I apparently forgot to capture pictures of their box and whisker plots! It’s been a busy time.
this activity was inspired by “math equals love” blog years ago, but I can’t seem to find the post.
What’s been happening in Grade 11
Lately grade 11s have been working on understanding periodic and sinusoidal functions. After our linguine math day we started off by making tables of values for both f(x)=sinx and f(x)=cosx.
We made connections between the CAST rule and the quadrants we’ve become familiar with, and these graphs. We learned a lot of vocabulary like amplitude, period, cycle, and axis.
Next we started to do some modelling, thinking about ferris wheels, or pebbles stuck in tires, and the height over time, or height over distance in each of those situations. Next we got into some transformations, a good chance to review mapping statements once more.
Today we explored some fun problems like where does sinx=cosx? We saw tables of values, graphs, quadrants and the cast rule and special triangles, and also algebra where we divided both sides by cosx and found ourself solving for where tanx=1. (I should have taken pictures, it was really neat work).
further to that, we looked at graphing tanx, a bit of an extension, but touches on periodic functions and reviews rational functions and identities which is always good to revisit. Again, there were so many interesting approaches, and strategies explored to end up with the graph for f(x)=tanx.
We’re going to work backwards next, starting with graphs and making equations. New challenges to explore tomorrow!
How Tall Is The Atrium?
Today it’s raining, which put a dent into plans to go outside to use shadows and similar triangles to measure heights of things near the school, so we used a good rainy day method instead.
To start off, we did some review of similarity, starting with some similar rectangles, and some work on scaling dimensions proportionally. We then looked at some similar right angle triangles, and finally some nested right triangles which will look like what our experiment will use. We got good at figuring out the scale factor, then using it to solve for an unknown side. Once we had those skills back at the top of our minds, we went downstairs and worked on getting some measurements to use for our calculations.
We held a meter stick vertically in our outstretched arm, then lined up the top of the meter stick with the edge of the ceiling and the wall of the atrium.
We measured arm length, and the height of the ruler above our hand, and then the distance from us to the wall, and the height of our eyes, and used that to make similar triangles and calculate the height of the atrium.
We had 3 groups all working at a different distance from the wall, so each group would have different information, and we were curious to see how close the answers would be.
We got back to class and did our calculations. We had a lot of variability in our answers, from 6 meters to 14 meters. We used some estimates of how tall one classroom is, and then extended that to 2 floors, and saw that the real answer is likely somewhere in between those values.
Fraction Work and Building Pyramids
Today we did a bit of numeracy work at the start of class to work on giving students a way to visualize fractions and solve problems. Here’s the sequence we did.
we looked at solving a problem like 4 is 1/2 of a number, find the number. This was an easy enough entry point to show the model: we made a rectangle, split it into 2 halves. One half is 4, so the other half is also 4, and the whole number is 8.
We levelled up a bit with the next: 12 is 1/4 of a number, what is the number. We drew our rectangle, split it into 4, and wrote 12 in one quarter. Each other quarter is also 12 so the whole number is 48.
next we did 5 is 2/3 of a number, what’s the number. We made a rectangle, split it into 3, and since 2/3 is 5, each third is 2.5, so the whole number is 7.5
Next we looked at a very fractionny example where 1/2 is 1/4 of a number, what’s the number. We figured it is 2.
the last one we looked at was 1/2 is 3/4 of a number, what’s the number. We made a rectangle, split it into 4 pieces, 3 of the pieces totalled 1/2, we used our fraction magnets to test the theory that each piece is 1/6. We also can use ratios 1/2=3/? And then calculate that the ? is 6. Next we determine the whole number is 4/6 or 2/3.
After that progression we switched back to working on volume and today our challenge was to build a square based pyramid with a volume of 300cm^3. We’ve build rectangular prisms with the same volume, and we know the volume relationship between prisms and pyramids so this was a good combination challenge.
Many groups decided on dimensions of 10×10 with a height of 9. Groups got to the end of their fabrication and it was time to check for accuracy. Most groups have a frustrated “ahaa” moment at this point when they realize their pyramid is too short. For most groups, they made the slant height of the side triangles equal to 9, so when inclined, the triangles form a pyramid that is 8cm tall.
Several groups thought about this before constructing and added an extra cm to be safe. We all had a moment to realize that the pythagorean theorem is hiding in these questions, and we can make use of it to help us calculate the exact slant height needed.
We then calculated the surface area of our pyramid and we put them all in our gallery of pyramids on the windowsill.
Volume Relationships
Today in grade 9, we took a few minutes break from our review to explore the volume relationships between prisms and the pyramids with the same base and height.
These transparent solids are great for modelling the relationship. We can pour water using the pyramid and try to fill up the prism. The class took some guesses and most landed on 2, or maybe 1.5 as the guess for how many times the pyramid will be filled and dumped into the prism to fill it.
Here’s the picture after 1 time, 2 times, and after 3 times. We noticed that for each prism it takes 3 of the pyramid to fill it up. That means the volume of the pyramid is 1/3 of the volume of the prism. We’ll be using this on our next cycle after tomorrow’s test.
Linguine Math
My grade 11s are starting to learn periodic functions today, and we used the activity 8.5.2 Freddy the Frog rides the Mill Wheel, listed here
We started off by drawing 15 degree increments around the wheel, and then we measured the height of each wedge using linguine, which we snapped to length and glued onto the graph paper.
Students were able to make predictions about the repetitive nature of the graph, and felt that they didn’t need to snap and glue more pasta. The top curve of the graph shows Freddy’s height above the water, and the bottom curve of the graph shows Freddy’s height below the surface.
We consolidated with an introduction of new words like period, and amplitude, and axis, crest and trough. We looked at how we can do a sinusoidal regression on desmos that will give us a periodic graph. We also made connections to the quadrants and CAST rule, where we know that from 0-90 degrees, quadrant 1 sine is positive (just like on our graph) same for quadrant 2, from 90-180 degrees sine is positive, but from 180-270 quadrant 3 sine is negative, and also from 270-360 quadrant 4 sine is negative.
We’ll be looking at cosine graphs and sine graphs in more detail tomorrow.
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