Pancake Proportions

Personal Reflection:
I'm looking down the road a few weeks to when my 7th grade class begins a long haul with Ratios and Proportions.  (Stretching and Shrinking, Comparing and Scaling for those using the CMP books!)  It makes sense that we will spend a lot of time on this unit because it's what the CCSS emphasize as the fundamental skills in 7th grade, and what we should build all 7th grade learning around.  For that reason, I suspect many of my posts in the next few weeks will be around proportional thinking.  (And I'll be going back to some of my others, such as the FitBit post, the Treadmill post, etc.)

We participate in the BIC program, Breakfast in the Classroom, so I can't make this one a hands-on activity, but I would sure LOVE to.  I'm finding that BIC (while I support it in theory) is going to force me to change my style of "bribing" kids to get engaged because of all the food I like to incorporate.  :)  Anyway, here we go!

Grade Level: 6-8

Course: Math, Pre-Algebra

Standards:   6.RP.1, 6.RP.2, 6.RP.3, 7.RP.1, 7.RP.2,

SMP: MP1, MP2, MP3, MP4, MP6
Skills:Writing ratios, analyzing ratios, analyzing proportional relationships, solving proportions, using proportions in the real world, solving for missing values using proportions


How to use this as a mad minute:
I've taken to noting that each of these not only depends on the amount of time you are willing to commit to a given activity, but also to note the proficiency level of your students.  I say this because I'm working with a population of students that is causing me to shift my thinking about what a "warm up" might look like, due to lower levels of proficiency, language challenges, etc.  For a quick check in, 60 seconds or so, I would ask:

• What is the unit rate for mix, milk and eggs for 1 pancake?

How to use this as a warm up:
The above question would also work well for a warm up, if that is the skill you've been working on.  However, I might ask the students to find the ingredients needed for a simple number of pancakes in order to highlight proportional reasoning and multiplicative relationships:

• How much mix would I need for 28 pancakes?
• How many pancakes would 6 eggs make?  What about 7 eggs?

How to use this as a mini-lesson?
As you may have discovered reading other posts, I usually find images that catch my eye because I am skeptical.  So my first thought was, does it make sense for this box of pancakes to make that many pancakes?  Is this truly in "scale" or proportion?  I would ask my students, is this in proportion? If so, how many cups does the entire box hold?

• Note, I don't think it is!  If we use the eggs as a guide, the recipe is scaled by a factor of 9, but 9 times 1 cup is 9 cups, which is not 3Q.  (A great way to work on unit conversions!  Have you seen the "big G" conversion chart?  I love it!)  (Here's one place I found the image.)
• If we also use the factor of 9, the box would contain 18 cups of mix, which I would assume is more of a "Costco" size box, not what we see here.
• Finally, a scale factor of 9 would make only 126 pancakes, not 155. 
• If we use the milk as our guide, the SF is 12.  That would mean we need 24 eggs and 24 cups of "mix".  That should also make 336 pancakes.    Hmmm.....

How to use this as a full lesson?
I don't think this could be used as a full lesson, but it depends on your students.  If you choose to use it, I would extend the warm up and mini-lesson into a full discussion AS WELL as setting aside time for students to present rebuttals and/or corrections to the "recipe."  A great interdisciplinary connection would be having the students write the company (can we tell which company this is based on the colors?  I think so.) with their discoveries.  I suspect that the company might respond with some coupons or other "swag"!!

How to use this as an assessment?
Any one of the questions listed above would be perfect to use as an exit slip, a mini-quiz or an assessment question!


Please feel free to use any of these ideas and modify them to meet your needs.  However, please acknowledge the original source of the items and my own lesson outlines.  ©NatalieRSprigg 2014

Marketing Percent Blunder

Personal Reflection:
As I stated in my previous post, I'm seeking relevant, engaging percent problems for my students.  Earlier this summer I received an email advertisement from a company that I frequent.  (Who doesn't love amazing balsamic vinegars and olive oils??)  However, my "mathematician brain" quickly targeted the 60% off claim.  I was so disappointed that my $21 bottle of olive oil was STILL $13.  Something smells fishy.  :)  I think my kids should talk about this one!

Grade Level: 6-8

Course: Math, Pre-algebra

Standards:   7.RP.3

SMP: MP1, MP2, MP3, MP6
Skills:  Solving problems with percents, finding percent increase, analyzing and interpreting mathematics.

How to use this as a mad minute:
There are a variety of questions I'd ask students to consider for 60 seconds or less:

• What is the difference between paying 60% of an item's cost and a 60% discount?
• Which of those would you use if advertising a 60% savings?
• Can you estimate the cost if you were saving 60% on the advertised bottle of olive oil?  (Note:  I use estimate because I want students to round the price and use mental math, not calculators on such an estimate!)

How to use this as a warm up:
This question feels a bit more like a warm up than a sprint.  A few more minutes to consider the phrasing, the numbers, and the claims.  If your students are proficient or nearing proficiency with the skill of percent discounts, they should be able to attack this independently.  I'd simply ask, "Do you agree with this ad?  Why or why not?"

How to use this as a mini-lesson?
I like the "Mini-lesson" feel of this ad more than anything.  I know that we'd need more than 5 minutes for this conversation, but not nearly an entire class period.  We use Connected Math at my school and this feels like a great "Launch" into other explorations of discounts.  Some questions I'd ask in a mini lesson:

• Use the warm up questions.
• Using the two prices provided, find out what percent you PAY of the original.
• Using the two prices, find the percent DISCOUNT off the original.
• What do you think of the claims made here?
• What math might the owners of this business have done to get to their conclusion?
• Can you create a more accurate ad for this business to use?

ALSO, if you have technology readily available in your classroom, I would use it to have the students "draft a response" to this advertisement email.  This will increase their literacy skills, their communication skills, and their skills at justifying their mathematical thinking.  Plus, it's a great civics lesson to work with community members to keep informed.

How to use this as a full lesson?
If this were your students' first introduction to percents, percent change, and discounts, I can see how the exploration of these relationships using this problem might last a whole hour.  My advice for such a lesson is to really scaffold the instruction and questions to help guide students to the realization that this may not be accurate.  Of course, this depends on the culture of your classroom, the instructional strategies you use, etc.

A quick outline of what I might try:

• Show the ad, explain that "mathematicians wonder mathematically" and we might wonder if this is accurate.  We are going to build our skills so that we can analyze this ad successfully.
• Start with the meaning of percent, how to find a percent given two numbers.  Do some samples.  3/5 is 60%, 2/8 is 25%, etc.  Talk about the meaning of those percents.
• When they see an ad that says 25% off, what does that mean?
• What does "off" mean mathematically?
• If we know how to find 25% of a number, how do we find 25% OFF of a number?
• If we are taking 25% OFF, what % are we paying?
• What's another way to find the cost?  (Find 75% OF the number instead of 25% OFF)
• What's the difference between PAYING 60% and SAVING 60%?
• Show the ad again, and ask them to figure out if the ad is accurate.

How to use this as an assessment?
If your students are ready for an assessment, they are ready for this.  Simply ask them if the ad is accurate and why!  :)


Please feel free to use any of these ideas and modify them to meet your needs.  However, please acknowledge the original source of the items and my own lesson outlines.  ©NatalieRSprigg 2014

Delicious Percents

Personal Reflection:
It's that time of year!  We started 7th grade off this year with a quick and dirty unit on percents.  It's my first time teaching 7th grade in 11 years, and this is a completely different population than I last taught.  The standards have changed, the expectations have drastically increased, and I'm desperately searching for ways to engage the students in real-world mathematics.  So, as I look for real world applications of percents, I found this "draft" post I started ages ago.  Perfect for my lesson this week on percent increase!

Grade Level: 6-8

Course: Math, Pre-algebra

Standards:   7.RP.3

SMP: MP1, MP2, MP3, MP6
Skills:  Solving problems with percents, finding percent increase, analyzing and interpreting mathematics.

How to use this as a mad minute:
Depending on the performance/experience level of your students, you may be able to accomplish a successful analysis of this problem in 1 minute.  If so, I would simply ask, "If a normal package contains 1 bar, and the new package contains 2 bars, is that a 200% increase?  Why or why not?"

How to use this as a warm up:
The only difference in how I would use this as a mad minute, warm up or mini-lesson is in the amount of time it would take for students to successfully analyze, interpret, and debate the reasoning in the ad.  If your students are nearly proficient with this skill, they should be able to tackle this in a 5 minute warm up.

How to use this as a mini-lesson?
As outlined above, students who are not yet proficient may need up to 20 minutes to talk through the fundamentals of percent of change, percent increase, etc.  I know that my students will need about 15 minutes to thoughtfully and successfully approach this problem.  Here is my sample "script outline" that I plan on using this week with my students.

• Mathematicians "wonder mathematically" and analyze the world around them, thinking about mathematical claims they see.  
• Here is one such example.  (Review the claim of the ad.)
• What do we already know about percent change, or percent increase?
• We know that we need to find the amount of change
• We know that we need to find the original amount
• We know that we need to divide to get a decimal.  
• We know that we need to convert our decimal to a percent.  (Alternatively we could find an equivalent fraction with a denominator of 100 in order to find a percent.)
• Knowing how we find percent increase, analyze this advertisement and prepare a short response (rebuttal, if your students can handle the vocabulary) regarding their mathematics.
• (Have students share out.)

How to use this as a full lesson?
I do not think this would warrant a full lesson in most classrooms.

How to use this as an assessment?
I would DEFINITELY incorporate this as a question on an assessment once I'd reviewed these skills with my class.  It's not too sophisticated to "mystify" students in an assessment setting.  Just remember to push your students to think this way and develop such arguments PRIOR to the assessment!  :)


Please feel free to use any of these ideas and modify them to meet your needs.  However, please acknowledge the original source of the items and my own lesson outlines.  ©NatalieRSprigg 2014

Odometer

Personal Reflection:
Like many math teachers, I have a slight obsession with really cool number patterns.  "Palindrome" numbers are some of my favorites (along with series, repetition, etc.)  I took the opportunity to take a photo (stopped, you'll notice) of my odometer when it reached 77,477 miles.  (On a side note, my new job has me doing a LOT of commuting, and I'm over 90k now, which shows how much I'm driving and how much I procrastinated on this post!)

Grade Level: 4-5

Course: 4th or 5th Grade Mathematics

Standards: 4.OA.5,  4.NBT.4, 5.OA.3, 5.NBT.5
SMP: MP2, MP5, MP6, MP8
Skills: Patterns, numeric relationships, addition, subtraction


How to use this as a mad minute:
You have 60 seconds. 
List as many palindromic numbers as you can.

How to use this as a warm up:
Tell the students what this shows, review what an odometer measures, and what a palindrome is.
1.  Why is this a palindrome?
2.  What would be the next 10 times I'd have a palindrome on my odometer?
3.  Do you see any patterns in the palindromes?

How to use this as a mini-lesson:
I would repeat the warm up above, but exclude the third question.  I would ask the students to work in groups to try to figure out how many times my odometer had "hit" a palindrome from the time I bought it (7 miles!) to the time that is shown above.  I would use the third question as a way to help kids who ares struggling trying to list ALL of the possible answers.  If they can identify patterns, they will shorten their process.

Push students to make observations about the patterns, to look for ways to generalize, and identify the final answer.  (In my own calculations, I THINK the answer is 674 times.  I am not positive, and will recheck my work.  My thinking is posted in a photo here.)


How to use this as a full lesson?
Expanding on the mini-lesson above, the focus REALLY has to be on number properties and patterns.  Students are NOT going to quickly answer the question of "How many times."  In fact, if they are not developing effective strategies and using patterns, they could easily get stuck in the listing routine for an entire period.

With a focus on selecting students who have a variety of strategies, definitely ask students to prepare to share their thinking (either on a poster, a dry erase board, on to put under a document camera).  Select a variety of approaches and sequence them in a way that will help students who are stuck make more efficient progress.

Other extensions you might offer could include:

• If this person drives 3,000 miles a month, how often can she expect to see a palindrome?
• How long until her next palindrome?
• What other "special" numbers might this person look forward to?  (I like repeated numbers "33333" and "sequences" of numbers "12345")
• How many of these such numbers would she encounter between 0 and 1,000 miles?
• How many prime number has she hit between 0 and 77,477 miles?

Car Talk, one of my favorite radio shows, had a number puzzle dealing with palindromes and odometers.  Click here.  (A very advanced explanation and solution can be found here.)

Another great challenge that deals with palindromes, as well as merging speed, is included at this site.

How to use this as an assessment?

This is not directly tied to assessed standards.  I do not feel it is appropriate as an assessment item, only as a critical thinking, perseverance, and practice problem.


Please feel free to use any of these ideas and modify them to meet your needs.  However, please acknowledge the original source of the items and my own lesson outlines.  ©NatalieRSprigg 2014  

Frozen Equations

Personal Reflection:

If you are an educator, you've probably been using Pinterest for a few years.  In fact, you probably found this post through Pinterest!  I'm fairly certain I found this image on Pinterest, but possibly on one of my other regular "fun" sites such as 9gag.com.  In an attempt to track down the original, I found this site.  It is not where I got the image, but it is a nice connection to the world of CGI and more detail about the snow effects in Frozen!

The site, linked above, has this amazing introduction, 

"Snow is a challenging natural phenomenon to visually simulate. While the graphics community has previously considered accumulation and rendering of snow, animation of snow dynamics has not been fully addressed. Additionally, existing techniques for solids and fluids have difficulty producing convincing snow results. Specifically, wet or dense snow that has both solid- and fluid-like properties is difficult to handle. Consequently, this paper presents a novel snow simulation method utilizing a usercontrollable elasto-plastic constitutive model integrated with a hybrid Eulerian/Lagrangian Material Point Method. The method is continuum based and its hybrid nature allows us to use a regular Cartesian grid to automate treatment of self-collision and fracture. It also naturally allows us to derive a grid-based semi-implicit integration scheme that has conditioning independent of the number of Lagrangian particles. We demonstrate the power of our method with a variety of snow phenomena including complex character interactions."

Wow.  That's technical.  In my own words?  "Snow is hard to animate.  While past methods worked fairly well, wet and dense snow was challenging because it acts like both a solid and a fluid.  In order to address this, engineers created a model that uses two different geometrical methods to animate snow.  They are able to use a Cartesian coordinate grid, along with programming, to simulate how snow both gathers (forms snowballs) and breaks (falls, hits, etc).  This sounds SO advanced, but I believe the analysis is totally approachable by a typical middle school student.  This is why I snagged the gif above and saved it, knowing that it would give some concrete meaning to students who are interested in the use of variables AND scientific notation!

Grade Level: 6-8

Course: Math, Pre-Algebra, Algebra

Standards:   6.EE.6, 6.EE.9, 7.EE.4, 8.EE.4

SMP: MP1, MP2, MP3, MP4, MP6, MP7, MP8
Skills: Variables, Algebraic Reasoning, Scientific Notation

How to use this as a mad minute:
You'll definitely need to preview this and explain the basics to the students.  However, after a short intro, a 1 minute number sense and reasoning check in might look like:

Compare the top two expressions carefully.  When you identify the difference in the expressions, and the subsequent snow fall, describe how you think the change in the scientific notation changes the snowfall.

How to use this as a warm up:
Again, after an intro, so students understand what this gift shows and where it comes from, I would challenge them to work with a partner to try to define what each "variable" controls, and how the change controls it.  (Hint:  It might be useful to name the snow fall quadrants A, B, C, and D in order to know which is which.  I will use A in the upper left and rotate clockwise through.)

A sample answer might be, "The equations on the right have ThetaS of 7.5 x 10^-3 and the snow is noticeably more clumpy or less-liquid than those on the left." (There are several comparisons they could make, so don't stop! Also, I would recommend you DON'T provide this example to the students prior to them working and struggling, it will be more productive that way!

How to use this as a mini-lesson?
I would start the same way as above, but prepare extension questions for students.  The exploration I outlined above, might only take 5 minutes, but a 20 minute mini-lesson could easily flow.
First, I would encourage students to share out, and convince others of their observations.  This is a great way to work on descriptive language, choosing appropriate adjectives, and talking about scientific notation.

To push students further, I would ask them to try to create their own equation that merges some of the changes.  Then challenge them to describe the resulting snowfall, and possibly even illustrate the final image.

How to use this as a full lesson?
Continuing on the trend above, I would then push students to explore more independently.

If your classroom has technology, I would visit the site:  http://www.cgmeetup.net/home/making-of-disneys-frozen-snow-simulation/ and have the students watch the video.  Another option would be for them to find a clip of the movie online and try to match which of the four quadrants a particular scene might be using.

This is another video of how artists used mathematics in creating the imagery in the movie Brave.

Depending on your focus, you may push students to analyze the scientific notation.  How big are these numbers?  What does that tell you about the size of the changes?

A great reading extension is this article from The New York Times about the Columbia University mathematicians who are working with film studios to enhance their computer graphics.

How to use this as an assessment?
I would not use this as an assessment, as it's probably a student's first exposure to this type of analysis.  However, if you've provided similar learning experiences for students, the Warm Up lesson is probably an opportunity for authentic assessment and analysis.


Please feel free to use any of these ideas and modify them to meet your needs.  However, please acknowledge the original source of the items and my own lesson outlines.  ©NatalieRSprigg 2014

How Radians Work

I can take NO credit for this, but it was powerful and I would love to have it on reference if/when I ever teach Alg II, Trig, or Pre-Calc.

I tried to find the original source and found it on Wikimedia's open source files page:
http://commons.wikimedia.org/wiki/File:Circle_radians.gif

It makes me feel better to know I'm not stealing someone's work without documenting it!

I am not writing lessons for this, as I think this is best left as an amazing resource for students and teachers when they need a visual way to explain and envision radians.

Grade Level: 10-12

Course: Algebra II, Trig, Pre-Calc

Standards: F-TF.1, F-TF.2, G-C.5,
SMP:  SMP2, SMP4

Skills: Radians, Making sense of the unit circle