Smarties - Part II

Have you every noticed that when you open a box of Smarties, there seems to be a lot of air in there? 

What is amount of air in each box? What percentage of each box is air? What size would the box be if there was no air? Why do they not make the box that size? How many Smarties can you actually fit in a box? What do you need to figure this out? Clearly we need some boxes of Smarties to begin collecting data.

We already know how many Smarties are in each box. Now we need to know the dimensions of the box and the dimensions of a Smartie. A Smartie is an oblate spheroid:

We can approximate the volume of a Smartie by considering a cylinder with the Smarties' diameter of 12 mm and height 5 mm.

Using this, we can calculate the Smartie volume of each box:

Now we need to find the volume of each box, then the volume of air:

Next, students can calculate the percentage of air in each box:

Whoa! That's a lot of air. Did I go wrong somewhere???

Students can then design their own box that will minimize the amount of air for a particular number of Smarties. Choose a number of Smarties, determine the volume required then choose/calculate the dimensions of a rectangular prism that will have the required volume. The grand finale would be for them to build their box and see if their Smarties fit!

Smarties - Part I

There is talk in the news about Smarties boxes changing size to better reflect portion size. The new 45 g box has three compartments, each representing one portion size of 15 g. So if you have three children you only need to buy one box of Smarties which they can easily share! How many Smarties would they each get? What if you have four children? How many Smarties should they each expect to get if they share a 75 g box?

There also exists a GIANT holiday box of Smarties that weighs in at 430 g. 

What questions come to mind?
How many Smarties? How much would that cost?

Let's tackle the second question first. What do we need to know? The cost per gram or the cost per Smartie. Collect some data!

They should get something similar to this:

Graphing cost vs. mass, we get:

Drawing a line of best fit would allow students to determine a model relating cost and mass. Students then need to think about what the slope or rate of change of this line represents.

The slope would give them the cost per gram. My equation was cost = 0.0215(mass) so for a box weighing in at 430 g should cost approximately $9.25. 

Alternatively you could look at the cost per Smartie. We need to know the number of Smarties in each different size box. 

Students can count the number of Smarties in each size box and then determine the cost per Smartie with a graph or a table:

Hopefully students make the connection that the cost per Smartie is the slope of the graph.

My equation was cost = 0.0226(number of Smarties). We can use that equation to determine the approximate number of Smarties you would get for $9.25. A little linear equation solving yields 409 Smarties. Now I have to wait until December to verify my work!

MFM2P - Day 18 (Modelling Perimeter & Area of Triangles)

Today's warm up is a Daily Desmos-like challenge:

The great part about this is that, if all goes according to plan*, this is the perfect segue to what is coming next today. (*I am out of town and don't know how far the class got yesterday.) I want students to come up with the equation of the line shown (likely by finding the slope and noticing the y-intercept) and then check their equation using Desmos.

On to the main event. I wanted to tie triangles back to the linear and quadratic work we did earlier this semester so I made this handout. Students will begin by calculating the perimeter for all the triangles from family 1 (from our similar triangles work) and add a few more triangles to the list. They will look at the pattern of the results and hopefully notice that is is constant when the scale factor increases by 1 and a multiple of the previous pattern when the scale factor increases by more than one.

Next they will graph the results and determine a model which will allow them to answer some questions about larger triangles. You may notice that I am giving them a graph with the scale set up for them and the axes labeled. As we progress through the course, I will give them less and expect more.

On to area:

I predict a lot more issues coming up here. They have a column with the length of the hypotenuse, but don't use it to calculate area. All the triangles used here are right triangles which they have drawn, so they can refer back to them for help with finding the area, but will they think to do that? The pattern and pattern in the pattern (1st and 2nd differences) should work nicely if they calculate the area correctly. It will be interesting to see what their graphs look like and how many come up with an equation to represent the relationship.

On Monday I will see how it all went. I do love how this connects linear, quadratic and similar triangles - like a spiral within the first spiral : )

MFM2P - Day 17 (Similar Triangles)

I am currently in Toronto about to spend 2 days working with Apple. So this post and the next one are "what should happen", not what happened in class.

Today's warm up is this Would You Rather:

I did this one with my 2P class last semester. This may seem like a simple calculation exercise, but it was challenging for some. Many calculated how much they would make in a day, then in a week, then in a month, then in a year and they did not get the same answer as someone who went from one week to one year. There was some interesting debate about how many hours per day is "normal" - is it 8 hours or 7.5 hours? If you work on a per hour basis, does that include vacation time or sick leave? (Some of these questions are addressed on the WYR site, but I wanted my students to think about them.)

The plan for today is to learn how to find missing side lengths in similar triangles and practice that skill. I am trying to encourage them to highlight corresponding sides in matching colours and label corresponding angles with the same symbols. I find that they understand how to find the length of a missing side best by first calculating the scale factor. I suggest doing large triangle over small triangle so that the scale factor represents how many times bigger the large triangle is compared to the small triangle. (They are always welcome to use other mathematically correct methods.)

Here are a couple more examples to work through with them:

They will practice a little more (page 3 of this handout) and then they will work on this handout which includes a few shadow and mirror questions.

MFM2P - Day 16 (Similar Triangles)

We did two warms up today, as I did not think I should leave a warm up for the English teacher covering for me yesterday. This Estimation 180 was first, where they had to estimate the height of the lamppost:

They did a good job of saying that a "too low" guess was at least 6'4" as that is Mr. Stadel's height. Their reasoning for their actual estimates was solid - they said the lamppost seemed like it was 2.5 to 3 times his height so their estimates were in the 15' to 19' range. One student had said 15 m so we talked about the units they choose to use and that 15 m would be about 7 Mr. Stadel's stacked one on top of the other!

On to today's visual pattern:

I think the colours are very leading in how students saw the pattern. I would prefer to have all the blocks the same colour so that different students might see the pattern in different ways. It was great that one student thought the pattern was 3n + 2 and we talked about what that meant and why it was different from 3 + 2n.

I did show them a different way of seeing the pattern and how it was the same as what they had found.

I would like to say we took up the triangle families handout from yesterday, but they hadn't actually done it yet. So they drew triangles and measured the angles with protractors. They noticed that triangles in the same family have the same angles and there was a pattern in the sides. For the first family:

they saw this pattern as "add 3 (to the first number), add 4 (to the second number), add 5 (to the third number)". This was good, but I wanted to move toward finding the scale factor. I asked how they could get from 3-4-5 to 15-20-25. They had to think about it for a bit and came up with "multiply each number by 5". Yes! We looked at the next family and they all told me that they had to multiply by 2 this time.

We moved on to this handout which we went through a little faster than I would have liked because I will be out of town tomorrow and Friday. I had them use different colours/symbols for the corresponding angles, then I pulled out the highlighters and had them highlight corresponding sides. We measured angles and sides lengths and calculated the ratio of corresponding sides.

And that is where we had to stop.

MFM2P - Day 15 (Sum of Squares, Similar Triangles)

My 5-year old threw up yesterday afternoon so I am home with her today. In my absence my students will continue with this handout from yesterday to practice sum of squares. Then they will start looking at families of triangles, drawing each triangle and measuring the angles. Here is the handout. I hope they will notice some patterns which we will talk about tomorrow.

Now back to this cutie (who will definitely not be going skiing today!).

MFM2P - Day 14 (Sum of Squares)

We did another counting circle as a warm up today. We started at 57 and went down by 3 each time. At one point I stopped them (at student A) and asked what number we would be at when we reached <student 4 away from current student> (student B). Some counted down by 3 for each student from student A to student B, others counted the number of students to get to student B, multiplied that by 3 then subtracted the result from the number for student A. They did this well.

Back to our list of right triangles and their areas. We spent more time than I thought we would discussing how to recognize the longest side in a right triangle. This is what they were looking at:

and one student said "the bottom line" was the longest then changed his answer to "the bottom line and the diagonal line" were the same length. We talked about isosceles triangles and figured out that if both angles were 90° we would not have a triangle. They told me that the sum of the angles in a triangle was always 180° so I asked them what they knew about the two non-right angles in the triangles. They figured out that together they would be 90° so each would have to be less than 90°. Combining that with the fact that the longest side is always across from the largest angle, we found the hypotenuse!

They were now ready to use the Pythagorean theorem, known to them as the Sum of Squares, to find missing side lengths. First, they cut out and glued an example in their exercise books:

We worked through the connection between the areas as some had already forgotten and I we worked through how to find a missing side length. They practiced with 3 more examples; one had them find the length of the hypotenuse, the other two had them find the length of one of the legs (two are shown below).

Once they did those correctly (there were a lot of errors - finding the perimeter instead of area, dividing by 4 instead of taking the square root...) they got a sticker (because even 15-year olds like stickers) and started working on this handout which we will continue tomorrow.

MFM2P - Day 13 (Triangles out of Squares)

Today was our first Always-Sometimes-Never warm up:

They quickly thought of 2 + 2 = 2 * 2. I liked hearing students saying that since they found one case that worked, it couldn't be always nor never. Someone also said that it worked for 0 and 0. I asked if it could work with two numbers that were different from each other. I left them thinking about that one.

Back to our 26 squares. I first asked them to make something with their squares. I saw a rocket, a caterpillar, a cat and many towers. They didn't seem to be moving toward making triangles so I asked if they could use their squares to make a mathematical shape. It took a bit, but someone finally made a triangle so I asked others to follow suit. They got the idea of how to make a triangle and that the corners needed to match up. I asked if they could make a triangle with any three squares. Most said yes. I then chose three of their squares that could not form a triangle and asked them to make one. This was followed up by me asking how you could know whether it would work. We eventually all came to the conclusion that the sum of the two shorter sides had to be longer than the long side.

On to right triangles. We defined a right triangle and I asked them to use their squares to determine whether the triangles listed (see below) seemed to contain a 90 degree angle. Here are a couple of pictures - the first one worked, the second one didn't.

They had to come up to the SMARTboard and either cross out or place a check mark next to each triangle. Here is the list (updated to remove those that were not obvious). 

They filled in this handout with the nine cases that worked.

and I asked what they noticed about the areas. Many blank stares... I asked a student what he noticed about the triangle with areas 100, 576 and 676. They are all even numbers (true, but not all areas in the table were even numbers - what else do you notice?). The 3rd number is bigger than the other too (yes, by how much?)... Eventually we got to noticing the the third area was the sum of the other two.

On Monday we will see how we can use this information.

MFM2P - Day 12 (Quiz - continued)

Today's warm up was this Would You Rather:

I let them work on it for a minute or two then someone asked how many gallons in a litre. Perfect! I asked which holds more, 1 litre or 1 gallon? They could all picture a litre:

but a gallon is less familiar to them so I pulled up some pictures:

We all agreed that 1 gallon holds more. I wrote down the number of gallons in a litre and the number of litres in a gallon and let them work out the conversion.

We talked about checking the reasonableness of answers (which they are usually very good at). So if they are converting from gallons to litres, they should get a bigger number than the one they started with. Some have trouble understanding when to multiply and when to divide, but they can still check their answers and, if needed, recalculate.

Most decided they would rather carry 5.62 l of water because it would be easier to carry, but one student said he would rather carry the 1.85 gallons as he would have more water. So then I changed the question and asked something like "If you had to empty a big vat of water into another one 50 m away, would you rather carry the water using a 5.62 litre bucket or a 1.85 gallon bucket?" Throwing a context around the original question definitely makes a difference. I may change the question for next time...

Many students then continued their quiz. When each finished they worked on a literacy test package (all grade 10 students in Ontario must write a provincial literacy test), then on more linear and quadratic data, then on some balance benders. Some only completed the quiz. This is the challenge of giving students all the time they need to complete an assessment - you need to keep the ones who finish more quickly doing meaningful work that will not disadvantage those who need the extra time to complete the assessment.

Tomorrow we explore triangles with our 26 squares.

MFM2P - Day 11 (Quiz)

Today's visual pattern was very fitting as there is lots of snow on the ground here:

I love how many different ways there are of seeing this pattern. I will try to explain what's going on in this picture:

In brown: a student noticed that the number of snowflakes in the top row of each pattern was the same as the step number and that each row below it had one more. So step 43 would be 43 + 44 + 45. Generally this would be n + (n + 1) + (n + 2) = 3n + 3 (sadly, I do not have that written anywhere).

In pink: a student noticed that each step has 3 snowflakes added on the diagonal. Working backwards to step 0, there would be 3 snowflakes so that rule here is 3 + 3n.

In black: some looked at the number of snowflakes, not at the pattern and again worked backward to get the starting value.

In blue: same as pink, but the pattern is growing by adding one column of 3 each time.

In orange: I showed them this one - we can move one of the single snowflakes to form a rectangle. For step 1 the dimensions are 3 by 2, for step 2 the dimensions are 3 by 3, for step 3 the dimensions are 3 by 4. Generalizing to step n, we would have a rectangle with dimensions 3 by (n + 1), and 3(n + 1) = 3n + 3.

So much good stuff in here!

Next it was time for the quiz. I tried to make it as stress-free as possible. I told them they could use their exercise books and worksheets, along with the graphing calculators (and instructions sheet). I also told them they could have as much time as they needed. We got the desks into rows and they got going. There were the usual questions along the way about how to use the graphing calculator, but others too when they were stuck. I think that developing skills around using the tools available to them is really important and not something that we do very explicitly. Knowing that they are stuck on a quadratic question should lead them to open their book to look at a quadratic example - what should the graph look like? what patterns should be expected in the data? how can I find other points?, but many just sat and stared for a while. I encouraged them as I know they know far more than they had so far shown me. They need to learn to ask themselves what they can do to start or continue a question, instead of leaving it blank (not allowed in my class) or immediately asking me. That resourcefulness is something many have not practiced much so I will attempt to help them help themselves more as we progress through the semester. Confidence in math comes not only from knowing how to answer a question, but knowing strategies to use when they don't know how to answer a question.

Many did not finish the quiz today so I will have them continue tomorrow. I want to see their best work and I will not let time be a barrier to that.