Price Per Item Calculations

Yesterday I introduced the subject of money in the elementary math class. Talking about buying things is a fairly popular part of math class, although our conversations should lean toward thrift, not wild abandon.

We have to be careful in the curriculum not to use any brand names or illustrations that might imply we are endorsing a certain product. And lately, we have had to remove all references to sugary foods.

Here's a typical lesson on unit pricing. This requires higher order problem solving skills. [click on the image to enlarge it]

We have co-mingled prices given "by the ounce" and "by the pound" to give kids practice in unit conversions. They should understand the relationship of ounces to pounds. They must know the number of ounces in a pound. They need to do a division problem to learn the cents per ounce.

In Clarissa's birthday roller skating problem, we provided the beginning of a story and kids are asked to finish the story, then create several of their own problems. We have provided teachers with a sample problem and how to find the answer.

This is a relatively complex problem where data is given in the narrative portion of the story. One tricky part is for the student to remember that Clarissa must pay for admission too, in addition to her 14 friends.

As the roller rink is giving a discount of one dollar per person, you must remember to deduct that from the total of the entrance fees. In this example, we took the dollar off of the two prices ($10 becomes $9; $15 becomes $14) but we could just as easily have waited until the end and subtracted $15 ($1 x 15 admissions).

In most comparison shopping situations, you are in a store, faced with choices, and must make a decision relatively quickly. The ability to calculate these types of prices in your head is a useful skill.

Once in a while you have time to analyze things more carefully. This earlier blog on the cost of a cup of tea is an example of a complex cost-per-unit problem solved over the course of an evening or two.

This challenge interested me, because the tea was from the same source and of the same type. Creative study of the purchasing options indicated we could cut the price from 58 cents to 16 cents per cup, for the same tea!

The price of everything?

In Excel Math elementary math curriculum we help kids explore the concept of careful shopping. In fact, schools are encouraged to teach thrift (not greed or covetousness) as a desirable character trait and perspective.

Thrift is:

• The quality of using money and other resources carefully and not wastefully.
• Wise economy in the management of money and other resources
• Frugality
• An association or organization that helps people save money (credit union, bank, etc.)
• From Old Norse, from thrífa "to grasp, get hold of"

In the process of teaching thrift we investigate a variety of concepts:

• price comparisons
• determining unit prices (if N items cost $X.XX, how much is one item?)
• reverse unit pricing (if one item costs $X.XX, how many items do you get for $Y.YY)
• the razor and the blade pricing (printer is free but toner is expensive)
• quantity discounts and bulk buying
• reading advertising and sales materials
• common discounting techniques (percentage off the list price, 2 for 1, etc.)
• calculating sales tax
• how interest is charged on loans
• how interest is paid on deposits
• not spending more than you have in your bank account (no deficit spending)
• additional, after-purchase costs (energy consumed, maintenance, repairs, etc.)
• tipping or gratuities

We mention profit and loss too, even though 3rd grade math class is not designed to teach economics. We avoid things like subscriptions, airline tickets, rental cars, used items, negotiation strategies, etc. These limitations are not necessarily based on the youthfulness and inexperience of our students. We just don't have enough time. In reality, there are SO MANY THINGS we might buy that NO ONE can possibly understand all the options.

Here are some numbers shared in a recent TED speech by economists Tim Harford and Cesar Hidalgo.

Tim suggested that for many centuries, our brains needed to compare 300+ items, yet now we are faced with 10,000,000,000 potential choices. Whether this number is accurate or not, we generally only have time to study for a purchase just before we buy that item. We cram. And sometimes we later end up with buyer's remorse.

Once our students understand a bit about the items they want and their prices, they have to complete an exchange with the merchant. The price is usually X plus taxes Y which becomes total Z with possibly an additional tip A.

Depending on their funds, students can use one or more payment methods:

• coins
• bills
• check
• cash card
• debit card
• credit card
• gift certificate
  
• purchase order 
• traveler's check
• put it on account

We don't have the time or ability to get into all these items in detail but we provide an introduction for students. I'll give some examples tomorrow.

Who decides which goes first? Part IV

I have a few more observations to make about the mathematical implications on which is the first day in a week? (our customer's question), and how we teach kids about weeks and calendars in the Excel Math curriculum.

In kindergarten, we introduce words that describe adjacent days. Then we connect them with the days of the week.

Days of the week and "adjacent day words" today, tomorrow, yesterday

Finally we introduce the numbers for days, weeks, months and years. We show the "rotation" as the months of the year go by, and we hint at the seasons.

The calendar, and the words and numbers for days, weeks, months and years

Week: a period of 7 days; 168 hours [man-made]

Fortnight: 14 days, two weeks, or half of a moon cycle (14.7 days) [lunar]

Month: a time unit based on the motion of the moon; 29.5 days [lunar]

Year: a time unit based on the rotation of the earth around the sun; consisting of 12 months, 52 (non-ISO 8601) weeks, and 365.25 days. [solar]

Decade: 10 years [solar + man-made]

Century: 100 years [solar + man-made]

The week is the most familiar time-keeping interval to be set at an arbitrary length, by people in different religious traditions and cultures. This Wikipedia article describes 3-day through 10-day weeks used in various times and places. If you consider the options, the length of a "week" can't be too short (it must be longer than a day) or too long (shouldn't be be too close to a month).  By definition a week is shorter than a fortnight.

The fortnight is commonly used in the UK and other Commonwealth countries, and is a typical interval for paychecks and other social benefit payments. Since it's a two-week time period, its first day is the same as the first day of the week that country has chosen to observe.

Nobody worries much today about the first day of the month except when they don't get a benefits check. In previous times, the first day could be very significant. Some Jewish holy men were given relief from traveling restrictions on the Sabbath so they could go to the eastern edge of the mountains and see the sunset or moonrise, marking holy days and months.

Although we have been debating which day is the proper first day of the week, the actual first day of the week has little consequence to most of us, other than affecting how we read calendars. I will make sure the curriculum conveys that sense to kids.

 

For an easy conclusion of this series of blog posts, I'll portray a couple watches that tell you the sequence of days of the week.

The watch above indicates day of the week with an orange spot that moves alongside the hour markers. You can set the 7th day to be either Saturday or Sunday, as you choose. During the evening of the 7th day it speeds up, racing past 8, 9, 10, 11 and 12 to land next to the 1 at midnight.

 

 

This colorful watch is showing the time as 10:10:38 on a Saturday.  Alain Silberstein created the Smileday graphical image watches to convey a changing mood for each day of the week. 

Who decides which goes first? Part III

Today in the Excel Math blog we are tackling how the weeks fit into a year. It started with the question: Why do we say Sunday is the first day when everyone knows Monday is the first day of the week?

Providing an answer is turning into a week-long project. Luckily I started it on Friday, and not on Sunday or Monday!

In our normal calendar, weeks (a set of 7 days) fall across the boundaries of the months and years. Here's how we would normally display them in the USA, if we start the week with Monday, or Sunday [click the graphic if you want a larger version]:

We could also display the weeks using the ISO 8601 standard we talked about yesterday. Here is our year indicating the numbered weeks in rows, and an alternative with numbered weeks in columns [click the graphic if you want a larger version]:

One advantage to numbering the weeks is that you can choose certain dates in advance - for example, the company annual report deadline can be Monday of week 15 in every year. Or paychecks will be issued on Fridays of evenly-numbered weeks.

All companies in your country can use the same systems - for example, a food producer's system can say a packaged meal was prepared in 2011 during Week 25 on Day 3  (the actual ISO 8601 code is 2011W253) plus "use within 15 weeks" and a retailer's system could easily calculate the date that food will be stale.

Are there disadvantages? Here are some potential issues:

• This system has leap weeks - some years have 52 weeks and some have 53 weeks
• Some days in a year may be assigned to weeks from the previous or future year
• There are no set relationships between weeks and months

Users (people, companies, countries) can disregard these voluntary standards if they want to. Across the world there are still many calendars and many ways of dividing up a year. Here's one programmer's attempt to list them all.  Here's another programmer's site. Here is a site that will tell you the ISO 8601 week and day numbers for today.

So far (just in case you were worried) we do not teach the ISO 8601 system to elementary school kids. But as an adult, you wanted to know this, right?

Who decides which goes first? Part II

My last blog post introduced this question:

Which day of the week is considered the first day: is it Sunday or Monday? 

This is a math issue, as we teach calendars, time-keeping, etc. in math class and curriculum. After beating around the bush for quite a while describing the context of the question, I am now ready to provide an answer.  

It depends.  It depends on you, your country, and your employer. Not very satisfactory, is it?

The ISO 8601 standard recommends that everyone adopt these points when discussing dates:

• The first day of the week is Monday
• The first week of the year is the week containing the first Thursday in the year
• Dates should be presented as YYYY-MM-DD
• Dates and times are shown as YYYY-MM-DDThh:mm:ss

A minor major problem with this ISO 8601 scheme is that people tend not to use it. We have our own methods. Rather than using Year, Month, Day, we tend to say Month, Day, Year. People in other countries outside the US tend to format the date as Day, Month, Year.

• July 25, 2011 is today's date, written American style
• 25 July 2011 is how it would be written in the UK

Why does it matter?

Some decades ago I lived in the UK and received a new credit card from my bank in the US. The card said valid 5-8-90  which I believed was May 8, 1990. When I tried to use it in the UK, the merchants said it was not valid until 5 August 1990 and refused to accept it. I can confirm that this was a great nuisance.
 
After many years of confusion and completely scrambled up entries, the US Customs and Border Protection agency finally switched the blanks on its landing cards into the date format expected by foreign travelers (rather than expected by its US clerks).  Notice that they did not use the ISO 8601 format! It's Day Month Year instead.

Imagine the confusion with applications, orders, flight reservations, etc. when people try to communicate across borders with the assumption that the people on the other side use the same time and date system!

So if we (in the US or EU or elsewhere) don't follow the recommended ISO method for writing dates, why would we (in the US) adopt the recommendation that Monday is the first day of the week?

Convention / tradition / habit often overrules a politician / mathematician / scientist / programmer's desire for order and efficiency. 

PS - Did you notice the bullet point stating the first week is the week containing the first Thursday?

The weeks of the year system involves another recommendation that we Americans don't often use. Technically, it's called a "leap week calendar system." I will try to explain it tomorrow.

Notice this special "leap week calendar"watch is showing 10:14 on Monday, the 28th day of August (how do I know that?) in the 35th week of the year.

Who decides which goes first?

One of the most common discussions on the playground is "which one goes first." That's an occasional discussion among adults too - especially on the road when two lanes are converging into one. There general rules for this sort of thing but often we ignore the rules and do what we feel like doing.

It happens in the math world too. Someone has to decide who goes first. For example, someone decided that addition and multiplication are performed BEFORE subtraction and division.

In previous blogs I've discussed times and dates and calendars. ( Here's one on time zones. Here's another on leap years. And another on setting your clocks for daylight savings. )

Today's blog was prompted by a customer's visit this week, asking "why is Sunday the first day of the week in Excel Math, when everyone knows it's the 7th day of the week?"

My answer was, Not everyone knows or thinks that Monday is first. 

A DAY Seems Simple, but it's not! Bear with me now as I get into this. We need to know when a day starts in order to know when to start the week (a collection of 7 days).

WHEN DOES A DAY START?

Hebrew and Islamic traditions begin their days at sunset. Or to be more precise, some said, from the moment that the sun's disk stands distant from the horizon by the length of its own diameter.

WHAT ARE DAWN and SUNRISE and DUSK and SUNSET?

The US Naval Institute of Standards says:

• Sunrise and sunset are the exact times when the upper edge of the disc of the Sun is at the horizon. 
• Dawn occurs when the geometric center of the Sun is 18° below the horizon in the morning. 
• Dusk occurs when the geometric center of the Sun is 18° below the horizon in the evening.
• Twilight refers to periods between dawn and sunrise or sunset and dusk.

Twilight's hazy light is an effect caused by the scattering of sunlight in Earth's upper atmosphere. The brightness of twilight is subjective, depending upon on your location and elevation, the time of year and local weather conditions. Twilight is divided into 3 categories based on the angle of the Sun below the horizon.

• Astronomical twilight is the period when the Sun is between 18° and 12° below the horizon. 
• Nautical twilight is the period when the Sun is between 12° and 6° below the horizon.
• Civil twilight is the period when the Sun is between 6° below the horizon until sunrise. 

The same designations are used for periods of evening twilight.

Roman and Western European medieval monastic days began at 6:00 am. In the Western world today, we normally end and start the day at midnight (the middle of the night).

WHEN IS MIDNIGHT?

• Halfway between dusk and dawn? Not exactly. That depends on where you are.
• How about 12 hours or opposite from noon? Ok, but when's noon?
• Noon used to be when the sun was directly over your head, but that definition changed with Time Zones.
• Noon is when the sun crosses the meridian, or is at its highest elevation in the sky. Solar noon varies depending on your longitude and the date.
  

WHAT DO WE CALL MIDNIGHT?

The US government style manual recommends: use 12 a.m. for midnight and 12 p.m. for noon. Time up to midnight (but not including it) is PM and time after midnight (but not including it) and before noon (but not including it) is AM. 

The US Naval Institute of Standards says: 12 a.m. and 12 p.m. are ambiguous and should not be used.  The shortest measurable duration after noon should be designated as PM. For example, a digital clock changing from 11:59:59 a.m. to 12:00:00 should indicate PM as soon as it the 12:00 appears, and not delay the display of PM for a minute, or even a second.

Depending on your digital clock, midnight could be 12:00 or 00:00  

WHICH DAY GOES FIRST?

Traditionally, countries whose primary religion was based on the Bible (Judaism, Christianity and Islam) have generally considered the first day of the week to be Sunday.

Some countries do not use the naming structure of Sunday Monday Tuesday (Domingo, Lunes, Martes, etc.) but instead say First Day, Second Day, etc. These might use Monday as the first day.

In China, Sunday means "week day".  Monday is "first day of the (seven-day) week cycle", Tuesday is "second day of the (seven-day) week cycle", etc. When China adopted the Western calendar Sunday was at the beginning of the calendar week, but now Monday is preferred.

In Swahili a traditional day begins at sunrise, rather than sunset. It is twelve hours before Arabic and Hebrew calendars, so some countries may consider Saturday as the first day of the week.  

WHAT IS THE GENERAL RULE?

In the USA for the last 10 years or so Sunday been considered the last day of the week, as our more secular view now places Monday at the top of the pyramid of importance. When Janice Raymond created Excel Math (1978), Sunday was definitely the first day of the week in the US school systems.  As I began revising the curriculum in 2000, we reduced the places where kids were asked to count days of the week and solve math problems with those numbers.

It all comes down to how you want to display your weeks on a calendar. Most of the calendars in my house start with Sunday and end on Saturday. How about yours? There is no right answer.

The international standards organisation (ISO) decided describing the time of day was an important issue, and created Standard 8601 to answer all our questions. Here's a paraphrase of the introduction:

Although ISO standards have been available since 1971, various ways to represent dates and times are used in different countries. Where this happens across national borders, misinterpretation can occur, resulting in confusion, errors and economic loss. The purpose of this International Standard is to reduce the risk of misinterpretation. 

We include specifications for numerically showing the date and time. In order to achieve similar formats for calendar dates, ordinal dates, week-number dates, time intervals, recurring time intervals, combined date and time of day, and local time or Coordinated Universal Time, it is necessary to use numeric characters, single alphabetic characters and/or one or more other graphic characters.

The Standard provides a unique representation of any date and/or time. It retains commonly-used expressions for date and time of day, and provides unique representations for some new expressions. It can be used with computers and should eliminate errors and misinterpretation. It will facilitate interchange across international boundaries, improve the portability of software, and ease  communication within and between organizations.

Stay tuned for the next blog! It comes After Midnight.