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

FitBit Math

Personal Reflection:

I have a FitBit.  This is a Fitbit Zip, the smallest and most basic of the systems.  It clips on to your waistband, pocket, bra or shirt and tracks your steps.  I assume, based on the info you provide when you set up your online account, it then translates that movement into a distance traveled and a number of calories burned.  (This post will give your students a chance to explore whether or not this is true!)

Anyway, we've all heard that a goal of 10,000 steps a day is a great way to increase movement and to stay healthy.  How far is that?  How many calories does that burn?  Is that consistent?

As you know, I do a lot of thinking about "how much I have left" when I'm working out.  This is similar to my treadmill post, but slightly different.


Grade Level: 6-7

Course: Pre-Algebra, 6th and 7th grade math

Standards:  6.RP.1, 6.RP.2, 6.RP.3, 7.RP.1, 7.RP.2, 7.RP.3
SMP:  SMP1, SMP2, SMP3, SMP4

Skills: Ratios, Proportions, Unit Rates, Problem solving, Real world problems

How to use this as a mad minute:
You have 60 seconds. Estimate the number of steps someone takes in one mile.

How to use this as a warm up:
You could ask the students to consider one of the following:
1.  What is the relationship between steps and distance?
2.  What is the relationship between steps and calories?
3.  What is the relationship between calories and distance?
4.  Estimate the number of steps you would take in 10 miles.
5.  Estimate how far you would have to walk (either in steps or in distance) to burn off a large McDonald's French Fries.  (500 calories.)

How to use this as a mini-lesson:

0:00--I'd like you to take 60 seconds to brainstorm everything you know about ratios or proportions.
1:00--Partner up and share your ideas with a partner.  Make sure to add anything you forgot to your list!
2:00--Can we make a list of properties of ratios and proportions and define them?
4:00--I'm going to show you three photos.  (Link here.) When you look at them, don't talk to your friends, but take a minute to write down your immediate "math wonders" about the photos.
5:00--Take a second to reflect.  Are your questions mathematical?  Are you focused on applications of math and not off-topic?  If so, please share them with your partner.  When both have shared, select two questions you feel are your "best" and write them on the board.
7:00--Here you will want to zoom in on the most relevant and appropriate questions.  I suspect that several will be able to be answered through solving proportions.  Feel free to "prime" groups as you observe to encourage them to think proportionally.  I will post "pretend" questions for the remainder of the lesson based on what I would expect kids to "wonder" about.  It looks like we have a lot of questions about the distance and the steps!  Do you know what a "unit rate" is?  Think about this, if you can buy 4 candy bars for $1, how much is each candy bar?  (Allow time.)  Finding the cost for one candy bar is the UNIT rate, how much for 1 of that thing.  I think we are wondering how many steps for 1 mile.  Can you set up a proportion that shows steps compared to distance? 
9:00--How could you change that to find out how many steps are in 1 mile?  (This assumes previous knowledge of solving proportions.  If I were introducing the skill for the first time, I would have spent at least a class period on setting up equivalent fractions and observing/discovering the property of cross products being equal.  Kids should know how to set up a proportion with a missing value.)  
10:00--Please solve your proportion and determine how many decimal places you need.  When are done, discuss with a partner and come to an agreement. 
12:00--(Discuss the answers as a class.  The exact answer, rounded to the nearest hundredth is 2148.22.  I rounded here because the next two places are zeros.  However, I would round to the nearest whole step, or even to the nearest ten.)  How did you solve?  Why?  How can you be sure that makes sense?  How did you round?  Why?  How can you be sure that is reasonable?  (Choose two students who rounded differently and ask them to explain.  I hope someone would note that 2150 is much easier to use in long term estimating than 2148 or 2149.22, I also hope someone would note that 2148 is easier than 2148.22 and there is no such thing as .22 of a step.  Finally, I hope someone would note that 2148.22 is the exact value and that the extra decimals are negligible.) 
14:00--Can you repeat this magic?  Can you tell me how many calories I'll burn in an hour?  Or how many steps it takes to burn 100 calories? 
16:00--Are both of the questions I asked Unit Rates?  Why or why not?  Be prepared to back up your answer!
17:00--Who thinks they are?  Who thinks they are not?  (Hold a mini debate, or allow students to change sides of the room.  Revisit the definitions you established if there is still a question at the end of your "debate".)
19:00--Can anyone, after looking at these examples, think of a time when they might solve a proportion to answer a real life question?

How to use this as a full lesson?
I always recommend extending the mini-lesson into a full lesson with further exploration.  First, I wondered if the FitBit readings and unit rates would differ for a different person.  I had to enter my height and weight in the online program when I registered, so I asked another user to share a few screen shots of HER FitBit.  Adding this second set of data opens another opportunity for exploration and extension.  For example, are the unit rates the same?  If not, what can you tell about each person's rates?  Can you graph the data?  Can you compare the two sets on a single graph?  Will one person "go farther" with the same number of steps or burn more calories with the same distance?  This is a great introduction to slope!  Slope is a rate of change, or a relationship between two numbers, just like a proportion!  Even if you don't calculate the unit rates for the second FitBit, you could definitely have students graph the data (assuming both started at 0,0,0) and talk about what the slope represents.  What does a steeper line mean in this real life situation?

I included both sets of FitBit data on FitBit Worksheet 2.

How to use this as an assessment?
If your students are proficient with unit rates, it would be perfectly reasonable to provide the FitBitMath1 or FitBitMath2 worksheets and ask them to calculate unit rates and explain the meaning of their answers.  Short, simple, effective.


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 2013