Measurement
Table of Contents
1. Introduction
My goal is to make this class as hands-on as possible. I could just stand at the front of the class and lecture every day, but I would find that boring and I think you would too. So, I intend to construct the class in such a way that we spend at least two class period every week doing lab activities. (That, incidentally, is a promise to you, and one you should feel free to hold me to).
In order to be able to work independently in lab, there are a lot of skills you need to develop first. In particular, you need to know how to make measurements of length, mass, and volume, you need to be able to record and present this information in an organized way, and you need to have some idea of how accurate your data is.
In this unit, we will devote a lot of time to developing these basic skills. It may seem strange to be spending three or four weeks just learning how to measure things, but the time we invest here will pay off immensely in the rest of the year when you find that you are able to comfortably deal with a wide variety of tasks you encounter in lab.2. Measurement Systems
2.1. I can measure an amount using a variety of different units
Understand that you can measure something using various different units.
"Units of liquid measure"
- Go over do-now
- Ranking these measures
- Measuring with different units
10/20/08 (Monday)
Class notes: Measurement Introduction (pdf)
Detailed aims
By the end of this class, students will know what is meant by unit and measure when used in the scientific sense. These two are defined as follows:
I measure something by seeing how many times some standard amount fits into it. That standard amount is called a unit. A measurement is written as some number of a particular unit.
Students should also be able to explain why a number without a unit attached is not a measurement - that is, why saying that "I am 6.25 tall" is a meaningless statement.
Students will also recognize that there are many different units I can use to measure any given thing, and that I will end up with a different number depending on what unit is used. Thus, there are many different measurements that can represent the same amount. For example, "12 cups" or "6 pints" or "3 quarts" are all ways of representing the same amount.
Once an amount has been measured in one unit, students will be able to predict whether the number will come out larger or smaller when it is measured in a different unit. Doing this requires, of course, that you know which unit is larger. The general rule is that a bigger unit can measure the same thing in a smaller number of units. This thought process will be practiced repeatedly in future classes as a check on conversions.
Students will recognize measurement as a quotition, dividing some amount into pieces of a standard size.
Description of instruction
The goal in going over the do-now is to "activate prior knowledge" about these various liquid measures. We will do this by talking about various situations in which those measures are used. For example, everyone will be familiar with a gallon as the unit that milk usually comes in. For the other units, I might tell a story like:
"When my family makes buttermilk pancakes, we get a quart of buttermilk from the store. We also get a pint of whipping cream and whip it up to put on top. I might put about a cup of syrup on my pancakes, if I really like syrup, and I might drink a cup of orange juice poured from a quart container. If we want some fruit on top, we might open up a pint can of peaches..."
This story - or whatever else students can remember - would be accompanied by holding up the exemplars of the various units that I have with me.
By the time we're done with that, it should be really obvious to everyone what the correct size order is, but if we don't explicitly mention the order in talking about what the units are used for, we can still be able to give students a chance to fix up their order before we share it.
Unfortunately, this is science class, so it's not enough to be convinced of the order by looking at it; we have to find a way to prove that that is the right order. ("I have a really bad sense of volume - whenever I'm putting away leftovers I end up getting out a tupperware that is either just a bit too small to hold everything, or quite a bit too large. So to me, it's just not obvious by looking that a gallon is larger than a quart - you have to prove it to me somehow.")
Once we have proved that our order is right, we need to define some terms before we move on: measure and unit. Once we are clear on what is meant by that, we will try measuring something using a variety of different units. Ideally, I'd like to have something with a volume around two quarts (a 2 liter soda bottle?), and let various lab groups (or individuals, since we won't be in partner desk configuration) measure it out using a gallon, quart, pint, or cup. We could then compare the measurements that they produce. This would give fairly interesting results because of the fact that two liters is not an exact number in any of these systems.
There are two things I want students to get out of this demo. First, they should notice that we have four different measurements (½ gallon, 2 quarts, 4 pints, 8½ cups) that are all ways of measuring exactly the same amount of water. It is hard for students at first to accept the fact that all these measurements can mean the same thing. Once they understand that, the more interesting thing to note is that the larger the unit used, the smaller the number of that unit that I end up with. This is something that we will need to discuss in order to understand why, mathematically, that is the case.
Once students grasp that idea - that a bigger unit will produce a smaller number when measuring the same thing - they will be able to try several different types of problems, which will be given on the practice that we do at the end of the class and for homework:- Predict whether the number will become smaller or larger when I measure come amount in a smaller or larger unit.
- Tell me which of two units is larger by comparing the number part of two measurements of the same thing.
- Given two measurements, either tell me that one is definitely the larger (because it is a larger number of a larger unit) or tell me that we don't know enough yet to compare the two (because one is a larger unit but a smaller number)
This last problem type will lead directly into what we will do tomorrow. It is the most challenging because it requires students to recognize when they don't actually know enough to answer a problem.2.2. When I measure with a smaller unit I get a bigger number
Understand that with two measurements of the same amount, the one using a smaller unit will have a larger number.
"Bigger unit, smaller number"
- Go over do-now
- Measuring with unusual units
- Comparing measurements
10/21/08 (Tuesday)
Class notes: Number and Units (pdf)
Detailed aims
The goal of this class is to reinforce what we did yesterday. Specifically, many students could not figure out on the homework and in class how it would be possible to compare two units without needing to look up somewhere an authoritative explanation of the size of those units. So, we will repeat all the sort of problems we were thinking about yesterday, but using units that are clearly not part of any kind of standard.
Students will be able explain how to compare the size of two containers by filling one and pouring into the other. They will be able to explain that if A fills B with water left over, A is bigger; that if A goes into B with space left, B is bigger, and that if A exactly fills B with nothing left over, they are the same size. They will be aware that there are also other, less certain ways to compare container size, such as timing how long it takes to fill each one or comparing their weights when full.
Students will understand that if I have already measured a container using one unit, I know that I will get a larger number when remeasuring it with a smaller unit, and a smaller number when remeasuring it with a larger unit. They will be able to explain this in ordinary language: "More of these fit in, because they're smaller" or "because this one is bigger, it doesn't take as many to fill it up."
Students will be able to rank several unknown units in order of size, if they are told how some amount measures in each. In doing this, they will make use of and refer back to for proof the idea that if I can fit more of a particular unit into something, it must be smaller.
Given two measurements in different units, where it is known which unit is larger, students will be able to say whether the two might possibly be equal. This is simply another extension of the same ideas; the two measures can only be equal if the bigger unit is with the smaller number. Hence, we have looked at the statement, "If two measures are equal, the one using a bigger unit should have a smaller number" from all three possible directions: we have compared the size of the units to predict the relationship in the number, have compared the numbers to predict the relationship of the units, and have looked at the unit and number relationships to establish whether equality of the two measures is possible. This last skill acts as a useful check on any result of conversion.
In the case where two measures are definitely not equal, students will be able to explain which is larger and which smaller, because one has a larger number of a bigger unit and the other has a lesser number of a smaller unit.
Description of instruction
I have brought three tupperwares from home. The clear ones hold roughly 2 cups, the green ones roughly three cups, and the blue ones roughly four cups. It should be clear by observation that the blue is larger than the clear, but we will prove this by pouring water and describe in ordinary English how we established that proof.
A soda bottle measures approximately 3¾ clears, and thus we can predict that it will measure a smaller number of blues. The actual measurement is 2¼ blues. We use the green unit somewhat differently, first measuring that the soda bottle is exactly 3 greens, and then using htat to establish that the green is intermediate in size, larger than the clear but smaller than the blue. Time permitting, we can prove this by pouring water.
The next section of class is meant to reinforce the part of the homework that most students seemed to have struggled with: the part where we look at two measurements and try to use the relation between the units and the number to determine if the two measurements may be equal or if one is definitely larger. This is a much more difficult and abstract procedure to get used to, because instead of using one comparison to predict another comparison, knowing that the measures are equal, we are using the agreement of two comparisons to say whether equality is possible or not.2.3. I can figure out how many of one unit go into another
Understand that you can measure something using various different units.
"Coin conversions"
- Go over do-now
- Figuring out conversion rates
- Converting liquid measures
10/22/08 (Wednesday)
Class notes: Conversion (pdf)
Cards: liquidMeasureCards.pdf (This smaller version might be easier to use at home)
Detailed aims
Students will be able to determine the conversion rates between units by measuring out one unit with another, and will understand why doing this once allows us to be able to convert, without measuring, any measurement in one of those units into another. We will define what it means to convert:
To convert a measurement into different units means to figure out how big that amount would be in the other units, without actually measuring anything, simply by knowing how many of one unit is equal to the other.
Students will know to check their work by making sure that a larger number always goes with a smaller unit. Students will at this point be thinking of conversion as a process of trading in a fixed number of one unit for a fixed number of another. They will not necessarily see this as division, but that is what tomorrow is about.
There are two specific procedures that I expect all students should be comfortable doing at this point:- Converting a mixture of measurements in various units down into the smallest unit so that the total can be written as one measurement
- Converting a measurement into larger units, when it comes out to a whole number of those units.
- Starting with a large number of a small unit and converting up into larger units as much as possible, so that the total number of units used is at a minimum. This is analogous to determining digits in a place value system.
- Figure out the conversion rate between two units given measurements of the same amount in two different units.
- Use a conversion rate given for unfamiliar units to perform conversions.
Note what is absent from this: students are not yet being required to perform conversion that give a fractional answer. So, for example, they might be asked to convert six cups, one pint, and two quarts into gallons, but not three cups and a quart.
Description of instruction
The point in going over the do-now is not to figure out what the right answers are, but to notice what we need to know in order to get that answer. We can convert easily between different types of coins because we know how many of one coin are equal to another coin.
In talking over the do-now, we will define the term conversion, and point out that if you know how many of one thing adds up to another, you can translate a measurement into different units without having to redo the measurement.
Our goal today is to learn how to convert with the liquid measures that we saw yesterday. In order to do this, our first step is to figure out the conversion rates. We will do this in lab groups, in partner desk configuration. Starting with a full gallon of water, I will have five lab groups work to determine the conversion from gallons into quarts, while the other groups watch. Then we will pass off the last full quart to another set of five lab groups and have them measure it out in pints. Finally, the last set of five groups will take the pints and measure them out into cups. We will end up knowing how many quarts I can trade in a gallon for, how many pints I can trade in a quart for, and how many cups I can trade in a pint for.
Once we have these exchange rates worked out, the "wet" part of the lab is over. We'll continue working in partners, but now we'll be using the cards, instead of actual measures. We need to make sure the desks are dry, then pass out one set of cards to each lab group.
We'll start with a "real" problem in which one group measures something out in cups, and we work together to convert that up into a mixture of quarts, pints, and cups; we can then have someone measure it out in that way to verify our result. This verifies that our way of converting does in fact work properly. Another quick conversion we could do for verification would be to predict how many pints are in a gallon, and verify that by demonstration. Then, groups can quickly work through individually a series of problems that require them to perform conversions by trading cards for each other.2.4. Mathematical language is used to describe converting measurements
Understand what is meant when I write that some combination of measurements is equal to another.
"Converting liquid measures"
- Go over do-now
- Mathematical language
10/23/08 (Thursday)
Class notes: Measurement and Conversion (pdf)
Cards: liquidMeasureCards.pdf (This smaller version might be easier to use at home)
Detailed aims
In yesterday's class, I recorded some of the conversions that we had done in mathematical language, like this:
1 gallon = 4 quarts = 8 pints = 16 cups
It was clear in class, and even more clear on the homework, that students were not understanding what a sentence like this means.
Someone comfortable with mathematical language reads this as, "One gallon is the same amount as four quarts, which is the same amount as eight pints, which is the same amount as sixteen cups. This person probably has a picture in their head, and can explain in words, that if the gallon were originally full of water, that same water could be poured off into exactly four quarts, and then each of those poured off into two pints, and so on.
For a student coming out of elementary school, none of this is obvious. In elementary school, math is a matter of doing calculations, not expressing relationships. So, students are familiar with equations only in the sense of "7 + 4 = 11", and as a result they tend to assume that the "=" sign means "the answer is" or "comes out to." At some point in middle school, students make the transition to being able to understand what the "=" really means, and making this transition is thought by many researchers to be one of the most important steps a clid makes in going from arithmetic to algebra.
By the end of this class, students will have been exposed to the idea that a sentence written in math language need not be a "problem" to fill in, but can simply be a statement of a fact we have discovered. This, of course, is a fact that we will need to reinforce over and over again, partly by using mathematical language whenever possible to record things we have found out, and then checking understanding. So, for example, when in the coming days we find that a tupperware holds 11 cups, we ought to write "Blue tupperware = 11 cups" and then check that students understand that this means that they hold the same amount of water, that eleven full cups of water could be poured into the tupperware and would exactly fill it.
Students will be able to translate a sentence in mathematical language into English. In particular, if the sentence involves equality between several combinations of measurements, students will be able to list off the measurements present in each group and connect the two with "is the same amount as."
Students will understand that, in the particular case of liquid measures, an "=" sign means that if all the units on one side were full of water, they could be poured into the units on the other side and would exactly fit.
Description of instruction
I'm afraid this class was rather boring because I was trying to emphasize an important point, but had not thought of any way other than lecture to do it. I had originally thought that we could quickly get through the language stuff and move on to the card game, but it became clear in class that the language was a real challenge, so I ended up taking the whole class for that.
The main thing that I emphasized was going over and over each problem, showing it as a sentence in English, as an expression in math, and as an operation done with liquid measures, in each emphasizing that the things on either side of the "=" were to be grouped together, and emphasizing that the "=" was establishing equality between those two groups.2.5. Converting by trading units for others
Be able to convert among liquid measures and record mathematically what you did.
"Converting liquid measures"
- Go over do-now
- Conversion card game
10/27/08 (Monday)
Class notes: Recording Conversion (pdf)
Cards: liquidMeasureCards.pdf (This smaller version might be easier to use at home)
Detailed aims
Students will be able to convert from any collection of units into other unit by using the idea of "trading" some number of one for another. For today, they will be expected to be able to do this only when they have a handy visual representation of the units, such as the cards we will use in class and the units to color in that we have on the homework.
Students will be able to record the process of conversion using mathematical language. In particular, when writing out a mathematical sentence expressing the conversion, they will be aware that the "=" signs separate that sentence into sections, with each section listing out the combination of units present at a particular step.
They will also understand that by putting an "=" between two of these sections they are saying that the collection of units in one section holds the same amount as the collection of units in the other section. They may choose to make this clearer to themselves by drawing a box around each section of the sentence. They will recognize that each new section is produced simply by pouring the water present in the previous section into different containers.
Students will not be expected, for today, to know how to figure out what order to trade units in in order to make a more complex conversion. They will either be explicitly instructed in what to trade, or will be given problems (as in the do-now) that only require one step.
The goal of playing the units card game is for students to reach the point where they think it is too slow and they would rather just work things out mathematically and write them down on paper. However, I am asking all students to practice using the cards, at least for the next two days, so that they will have the image of trading in units for others firmly embedded in their heads and will be able to use that idea intuitively as they are doing conversions "on paper."
Description of instruction
There is not really any new instruction going on today; this is all practice of what we learned last week. I'll be using the document camera so that it is very clear exactly what I expect to be written on the worksheet and where, and exactly how the manipulation of cards is to be done. When we are doing the card game, we will do one step at a time together as a class, then check to make sure everyone has it right; tomorrow I will be asking students to take on a bit more responsibility and actually go through an entire problem on their own.2.6. Planning what units to trade first
Be able to plan what order to convert things in.
Explaining the steps of conversion
- Go over do-now
- Conversion card game
10/28/08 (Tuesday)
Class notes: Planning Conversion (pdf)
Cards: liquidMeasureCards.pdf (This smaller version might be easier to use at home)
Detailed aims
Students will be able to read a problem calling for a conversion into particular units, and plan out what steps to take to accomplish that. The general rule is to start by converting whatever units, among what you have, are farthest away in size from what you are looking for.
The big change from yesterday is that instead of telling students what to trade in at each step, I am now expecting them to take that initiative on their own. I am also expecting them to be able to work through a whole problem without direction, instead of going one step at a time.
Description of instruction
The do-now proved to be a real challenge because we haven't done exactly that task before. I did the first problem together as a class, using the markup we have developed for math sentences - that is, I put a box around each side of the equation, then circled the unit that was being traded in, and wrote under it what it was exchanged for. Most students found that once they saw the process, it was easy to duplicate.
In going over the do-now, the big point to discover was that whoever did that conversion wasted a lot of effort. They should have converted the gallons into quarts first, so that they could convert the quarts all together; instead, they ended up having to trade in quarts in two separate steps. In general, when planning what to convert first, you want to start with the unit that is furthest away, in size, from the units you want to end up with.
In doing the problems, the challenge for students was not getting the answer (in most cases), but having the discipline to correctly write out each step of the process. We have been using a particular pattern for doing this:
You start by writing down what you are told you have at the beginning. Then, in each step, you first circle the units you want to trade in, then write under them what they turn into. This makes it fairly straightforward to then write an equal sign, followed by whatever units you have now, combining the units you just traded for with the other units that were already there. We are still using a box around each part of the equation to show that those units all belong together and that the = sign is talking about two whole sets of units rather than just what is immediately left and right of it.2.7. Practicing our liquid measure conversions
Become very comfortable converting liquid measures.
Writing in Math Language
- Go over do-now
- Conversion card game
10/29/08 (Wednesday)
Class notes: Practicing Conversion (pdf)
Handout: Conversion Card Game (pdf)
Cards: liquidMeasureCards.pdf (This smaller version might be easier to use at home)
Detailed aims
Most students are able to do any conversion problem at this point, if they are careful and determined. The hope for today is just to practice doing this so many times that converting liquid measures will be one thing students can be absolutely confident that they will get right on the test. For those students who are already really good at this, the activity also provides one section at a higher level of difficulty, one that forces even me to have to think carefully.
Description of instruction
The do-now proved fairly simple for most students, although some are still confused by being faced with a unit (blue tupperwares) for which they don't know how big it is, and some students are still trying to turn it into a math problem - adding up the numbers somehow - instead of just directly translating the English sentence into math. However, once they saw that all they needed to write was "2 blue tupperwares = 1 gallon + 1 quart + 1 pint + 1 cup," most students seemed to see why that was the answer. Maybe the problem was just too simple, so they were trying instinctively to find a way to make it harder.2.8. Applying our measuring skills to area
Apply what we know about units and measurement to area.
Measuring area
- Go over do-now
- Area units
- Mixing units
10/30/08 (Thursday)
Class notes: Area Measures (pdf)
Handout: areaMeasures.pdf
Manipulatives: smallArea.pdf midArea.pdf largeArea.pdf
Detailed aims
Our goal is to review two "big ideas" from the very beginning of the unit:- I measure something by counting how many of some unit fit into it.
- If I measure something with two different units, the smaller unit will go in a bigger number of times.
By applying these rules in a completely unfamiliar context (area, rather than liquid volume), I hope to make students' understanding of these rules more connected and flexible.
Description of instruction
2.9. Applying our conversion skills to area
Apply what we know about conversion to area.
Figuring out the conversion rules
- Go over do-now
- Combining into larger units
- Breaking down into reds
10/31/08 (Friday)
Class notes: Area Conversion (pdf)
Handout: areaMeasurables.pdf
Manipulatives: smallArea.pdf midArea.pdf largeArea.pdf
Detailed aims
As in class yesterday, the goal for today is to reinforce basic ideas about measurement by applying them to a new sort of situation. We know already that we can measure area by seeing how many area units fit into something, and that we can expect that when measuring with a smaller unit we will fit in a bigger number of them.
Students will recall that in order to figure out the conversion rule between two units, I need to measure the larger unit with the other. Having discovered the conversion rules for our area units, they will be able to apply those rules to convert some mixture of units either into as many larger units as possible, or into all the smallest unit.
The goal, again, is not to develop specific proficiency with this particular measurement system - after all, these units are ones they will never see again - but to practice the process fo working out what the conversion rules are and then applying them.
Students will also practice using math sentences to show the process they are going through in doign these conversions.
Description of instruction
2.10. Applying our measuring and conversion skills to length
Apply what we know about units, measurement, and conversion to length.
???
11/3/08 (Monday)
Class notes: Length Measures (pdf)
Manipulatives: smallLength.pdf midLength.pdf largeLength.pdf lengthMeasurables.pdf
Detailed aims
Today we will try to do, all in one day, working independently as much as possible, the whole process of figuring out a measurement system. Given a collection of units, students will be able to measure with those units, either purely with one unit or with a combination of units. They will be able to relate the size of a unit to the number of times that it goes into something, so that they have an intuitive check that says that a larger number is expected when converting or remeasuring in a smaller unit.
Students will also be able to independently discover and apply the conversion rules. This process will require them to know to measure one unit with another, to record what they discovered, and then to be able to use that fact to convert a mixture of units into either all the smallest ones or as many larger ones as possible.
Description of instruction
2.11. Reviewing what we have learned about measurement
Review what we have learned about measurement.
???
11/4/08 (Tuesday)
Class notes: Measurement Review (pdf)
Manipulatives: isometricAreaSmall.pdf isometricAreaMedium.pdf isometricAreaLarge.pdf isometricAreaHuge.pdf isometricAreaExamples.pdf
Detailed aims
Description of instruction
3. Scraps left over from planning
3.1. Sometimes converting requires us to use fractions
-Establish that a cup is a half pint, a pint is a half quart, a quart is a quarter gallon
-Express amounts such as three quarts as a fraction: "three quarter gallons"
-Problem type: Convert generically from any unit to any other using fractions to express the answer when necessary
-Work problems using cards and then demo using water; for example, 23 cups is 1 gallon, 1 quart, 1 pint, 1 cup, and is also 1 7/16 gallons; 7/16 is nearly half.
-Problem type: Convert a measure with fractions in it to another measure. This is a lot trickier because you need to either know how to multiply fractions, or you need to be able to look at the denominator and recognize what unit the numerator is.
-Work problem with cards and then demo using water: How can I measure out 1 5/8 gallon using the least number of units?3.2. What we did for volume may also be done with length
The goal of this class will be to quickly walk through the same process we just did for volume - ranking in order of size, determining conversion rates, and converting, possibly with fractions - for a completely different sort of measure: linear size.
This is the "Y'all do" part of the release of responsibility model; the goal is to have the next section (area) be done independently.3.3. Extending to different measures: Area
To really reinforce what we have been doing with volume, we could also repeat the same process with area. This is interesting as a problem because depending on the area measures used, you would probably have no choice but to measure some more complicated shape with a mix of measures, and yet you can still, in the end, express it using any one of those measures alone, using fractions.
3.4. Various scraps left over from planning
By the end of this unit, all students will...- Understand that when measuring something (length, volume, etc.) there are a number of different units that I can measure it in. A measurement is reported as some number of that unit.
- Understand that I can measure out the same amount using different units and get different numbers.
- Discuss which unit is bigger, and what that means about which unit will yield a bigger count.
- Determine how many of one unit is equivalent to another, and express that relationship in words and as an equation.
- Given words or an equation describing the relationship between two units, identify which unit is bigger and perform conversions.
- Check your work in conversions using what you know about which unit is bigger.
- Given many related units, express some quantity using a minimum total number of units. (In other words, no more than one cup, one pint, three quarts)
- Talk about what fraction one unit is of another (only unit fractions)
- Convert correctly even when a fraction is produced.
- Refine a measurement system by breaking a unit into halves, fourths, etc.
- Express verbally and mathematically the relationships among fractions: a quarter is half of a half, two eighths equal one quarter (treating fractions as separate units)
- Relate the process of generating fractions to the relationships among them.
- Measure in inches with appropriate fractions
- Add power-of-two fractions by converting as you go
3.5. More scraps: Conversions with Liquid Measures
Understand that measurements can be expressed in many different units.
What do you know about these four measures of liquid?
Quart, Gallon, Pint, Cup
- Go over do-now
- Find relationships between measures
- Converting Game
10/20/08 (Monday)
In doing the do-now, I expect students, at the very least, to try to rank the four measures from least to greatest. Hopefully they can also mention some examples of things that come in a unit that large (gallon or quart of milk, cups of various things in baking). In going over the do-now, I will show them what each of those measures looks like, and we will all agree together on which is the biggest and which the smallest. Then, various lab groups will try to answer the question of how many quarts in a gallon, how many pints in a quart, and how many cups in a pint. By the end of class, we should be able to do a bit of converting using the cards, solving problems (for now) that just convert everything down into cups.
In the next day, we would address a few other problems:- How do I convert straight from gallons to pints or cups, or from quarts to cups? We can build up these conversion factors based on the conversions we already know.
- How do I express a number of cups as an amount of gallons (or similar problems). This is introducing the use of fractions in this context.
- How can I resolve a number of cups, or a mix of units, into a collection with the least total number of measures? This is the problem of getting a number with a proper digit in each place. What are the rules for when I'm done? (no more than one of cups and pints, no more than three quarts...)
- Our final goal would be to be able to look at a mixture of different measures and express it as a single fraction of a gallon. This would mean choosing which denominator to use and converting all others. Good practice for adding fractions later on.
- Introduce the idea of using several different units together and converting between them, in the context of american liquid measure units (cup, pint, quart, gallon)
- Introduce the idea of using fractions in measurements. Since there are four quarts in a gallon, three quarts can be written as 3/4 gallon.
- (insert a day here to practice conversions and binary fractions?)
- Introduce length measures. Have students measure lengths using strips cut from a sheet of paper. These don't correspond to a "real" measure. The goal would be to arrive at the idea that the strip may be broken down into halves, quarters, and so on to improve accuracy.
- Reinforce what we just did by a lab measuring volumes of rice in plastic cups. How do we subdivide the cup?
- Introduce the typical form of inch ruler, and learn to name all the fractions on it.
- Practice measuring things in inches
- Review all this binary measurement systems stuff
- Test or quiz
4. Fractions in measurement
4.1. Using fractions in measurement
Be able to use fractions in writing a measurement.
Liquid measure fractions
- Go over do-now
- Fractions of a gallon
- Converting with fractions
- Combining fractions
Thursday 11/6/08
Class notes: Fractions in Measurement (pdf)
Detailed aims
Students will be able to figure out what fraction one unit is of another. So, for example, recalling that there are four quarts in a gallon, they will be able to express that same idea by saying that a quart is one fourth of a gallon. They should be able to have a picture in their minds that this means that when a gallon is divided into four equal pieces, each is a quart.
Once they know what unit fraction of a gallon is represented by a given unit, students will also be able to recognize that some number of that unit can also be written as a fraction. So, for example, three quarts is three fourths of a gallon, since each quart is one fourth of a gallon.
Description of instructions
The big surprise in teaching this lesson was how difficult it was for students to understand what a fraction means at all. "Three fourths of a gallon," to most students, does not evoke the picture of a gallon divided into fourths, of which three are reserved. We need to do a lot more work, then, on simply understanding fractions, before we try to do anything fancy with them.4.2. Unit fractions
Be able to read and understand measurements with fractions.
Figuring out the fractions
- Go over do-now
- Unit fractions of a rod
- Writing fraction measurements
- Writing mixed number measurements
Friday 11/7/08
Class notes: Unit Fractions (pdf)
Detailed aims
Students will be able to use and manipulate fractions in simple ways using ordinary language. So, for example, they will be able to generate sentences like "A pint is an eighth of a gallon. So, if I have five pints of water, that is five eighths of a gallon."
Students will be able to use a sequence of relationships among units to express all the units as a fraction of the largest one. The picture I expect them to use to do this is one of breaking that unit down into smaller units, which are broken down into smaller still, keeping track of how many pieces are present at each step.
So, for example, I get quarts by breaking down a gallon into four pieces, and I get pints by breaking each of those four pieces in half. Since this leaves me with eight pieces, a pint must be an eighth of a gallon.
Students will be able to express a measurement in smaller units as a fraction of a larger unit, by first naming that measurement in words, then substituting for the name of the unit the fraction it is of the larger unit, and then writing that in math symbols. So, a student presented with "five pints" would rewrite that as "five eighths of a gallon" and then translate that into math as "5/8 gallon".
Students will be able to combine a smaller unit with some number of a larger unit to produce a mixed number of the larger unit. So, for example, two gallons and three pints is 2 3/8 gallon. The mixed number has three different "numbers" in it, but it needs to be understood as a single quantity.
Students will also practice reading in English phrases, not only fractional measurements, but also mixed number measurements. This is an essential skill for two reasons: firstly, because we need to be able to use words to communicate the math symbols to each other in class, and secondly, because embedded in the syntax of the English phrase describing a mixed number is all the information you need to know in order to figure out what that mixed number looks like.4.3. Constructing fractional lengths
Break down units into pieces to represent a particular mixed number or fraction.
Fractions in measurement review
- Go over do-now
- The parts of a mixed number
- Constructing fractions
Thursday 11/14/08
Class notes: Constructing Fractions (pdf)
Detailed Aims
So far, we have focused on just naming with fractions an amount that is laid out in known units. So, for example, since a cup is a sixteenth of a gallon, another way to say three cups is 3/16 gallon. This kind of connection is easy to make because when I am looking at that measurement 3/16 I have in my mind the idea that the 3 is how many cups I have, and the 16 comes because a cup is a sixteenth of a gallon.
The first step toward building a more robust understanding of fractions - one that does not depend on being able to name each possible fraction as a particular, smaller unit - is to be able to take a unit and break it up into arbitrary fractions. I have chosen to focus on halves, quarters, and eighths since these fractions can be constructed easily and accurately by eye.
In reading a mixed number measurement (W N/D unit), a student should be able to explain that this means you have W whole units, and then N pieces out of a unit broken into D pieces. They should be able to draw this measurement on a pictorial representation of the units by following a three-step process:- Fill in the first W whole units
- Draw lines splitting the next unit into D pieces
- Fill in the first N of those pieces
When faced with a string of boxes representing units, or which some full boxes and some fraction of a box are filled in, a student should be able to write that amount as a mixed number by reversing this process; that is:- Write down as the "big number" W the number of completely filled-in units
- Count the total number of pieces the next unit is broken up into, and write that down as D
- Count how many of those D pieces are filled in, and write that down as N
(Originally, I had intended to do this lesson with paper strips instead of coloring in pictures. The paper strips are much easier to break up accurately into fractions, but the first two classes proved that the frustration of trying to get a whole class of 30 to handle paper and scissors in a controlled way while following instructions was just not worth it. In the end, I'm really glad we switched over to a picture way of showing what we are doing, since it makes it a lot easier to record how you did a problem and refer back to it to discuss it in class.
Class notes: Constructing Fractions (pdf)4.4. Halves, Quarters, and Eighths
Be able to read and understand measurements with fractions and mixed numbers.
Halves, quarters, and eighths
- Go over do-now
- Name that measurement
- Which measurement is largest?
- Odd one out
Friday 11/14/08
Class notes: Halves, Quarters, and Eighths (pdf)
Detailed aims
The goal of this class is just to practice what we have learned yesterday about constructing fractions. To make this more fun, I have arranged the classwork into a number of "games" in which there is some further step required after simply identifying or drawing a measurement.4.5. Crazy Fractions
Stretch your understanding of what a mixed number is.
Bend your mind
Tuesday 11/18/08
Class notes: Crazy Fractions (pdf)
Detailed Aims
The purpose of introducing "mixed mixed numbers" is twofold:- Often carrying a procedure one step further along the same pattern is a good way to help students finally grasp the underlying concept and structure, rather than just relying on being able to do the procedure by rote.
- This is a use of mixed numbers that looks cool and complicated, but which all students can grasp. Thus, it provides more of a sense of challenge in a class that has been largely review of very basic math concepts.
By the end of class, students will be able to draw a mixed number in which the numerator of the fraction part is another mixed numbers. Most students will be able to carry this one step farther, figuring out what would happen if the mixed number in the numerator had a mixed-number numerator of its own.
The underlying concept here is that since a mixed number is constructed out of three numbers, and is itself regarded as a single number, I can put a mixed number into either the numerator or denominator of a fraction and still have a meaningful number. The goal is to stretch both the concept of "number" and the understanding of the procedure in which we fill in a whole number of units, split the next unit into parts, and fill in some number of those parts.4.6. Measuring with Fractions
Use fractions to measure a length more accurately.
Review of mixed numbers
- Go over do-now
- Measuring dimensions
- Finding equivalent lengths
Thursday 11/20/08
Class notes: Measuring with Fractions (pdf)
Detailed aims
I inserted this day here for two reasons. First, I wanted to put in more review of the basics - what a mixed number is - before we move on to converting mixed numbers and fractions. Secondly, I was afraid that we might have gotten so focused on drawing and reading fractions in the particular form of a filled in portion of a bar that we might have forgotten that the goal is to show how long that measurement is.
The new skill for today is that of continuing to break down a unit until you reach the point where the edge of one of the fractional pieces lines up with whatever you are trying to measure. Here, we are still restricting ourselves to halves, quarters, and eighths, since it gives us a simple procedure:- Fill in all the full units, and stop once you get to the unit within which the line to be measured ends. Write how many full units that is.
- Cut that unit in half. If the half line lines up with the edge, stop and write 1/2
- Cut each half into quarters. If either the 1/4 or 3/4 line lines up with the edge, stop and write that.
- Cut the half into eighths, and write your best guess of which eighth line the edge lines up with.
Apart from this new skill, students will also hopefully come out of class remembering that our purpose in drawing fractions has been to show what length a particular mixed number measurement has.4.7. Equivalent Fractions
Understand how fractions and mixed numbers may be converted into equivalent forms.
Finding equivalent measurements
- Go over do-now
- Equivalent fractions
- Mixed numbers and improper fractions
- Changing the denominator
Thursday 11/20/08
Class notes: Equivalent Fractions (pdf)
Detailed aims
Most of our students are familiar with the idea that a fraction can be converted to another denominator, or that improper fractions can be converted into mixed numbers and vice versa. But many of them, even the top students, make mistakes in doing this that betray a lack of understanding of the fundamentals of what a fraction is.
The goal in this unit has been to give students a concrete, consistent picture of what a fraction is, so that they will be ready to use fractions in making real-world measurements. Today, we will be using that picture to explain how converting a fraction to a different form works. This will both strengthen our understanding of this model of fractions, and reinforce what students know about converting fractions.
Students will leave class considering it intuitively obvious that I can produce an equivalent fraction by multiplying the numerator and denominator by the same amount. Visually, this just means taking each of the pieces into which the unit has been broken to show a fraction, and breaking them down into more pieces. If I have a fraction bar showing 5/6, I can split each piece in half, producing 12 pieces of which 10 are colored: 10/12. If instead I were to break each piece in thirds, I would have 18 pieces of which 15 are colored. In both cases, the numerator and denominator are multiplied by the same number, because if I break all the pieces in thirds, that of course means that the colored pieces are broken into thirds.
Students will also find it intuitively obvious now what I need to do to convert a mixed number into an improper fraction or vice versa. Converting a mixed number into an improper fraction means splitting up the whole pieces each into the same number of pieces specified by the denominator. So, if I have 2 2/3, I split the two whole units into thirds (6 pieces) and thus have a total of eight thirds filled in.4.8. Fractions of an inch
Convert between equivalent combinations of fractions.
Working out equivalences
- Go over do-now
- Fractions of an inch
- Writing the fraction in a different way
- Measuring with an inch ruler
Friday 11/21/08
Class notes: Fractions of an Inch (pdf)
Detailed aims
- Recognize that the marks on an inch break it up into halves, quarters, eighths, and sixteenths
- Measure the length of a line in inches
- Measure out and draw a line of a given length
4.9. Adding fractions
Tuesday 11/25/08
Class notes: Adding Fractions (pdf)4.10. Finding a common denominator
Wednesday 11/26/08
Class notes: Common Denominator (pdf)4.11. Fractions of a cup
Work with fraction-of-a-cup units.
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Wednesday 11/12/08
Class notes: Fractions in Measurement (pdf)
Detailed aims
Description of instructions
5. The metric measurement system
5.1. Metric measurements
Develop an understanding of the size of meters, decimeters, centimeters, and millimeters.
- The metric system
- Constructing the metric units
- Measuring with metric units
5.1.1. The metric system
You are probably used to using a ruler that is marked off in inches and feet. In science, however, we more often use a system of measurement called the metric system. The name comes from the fact that all the various units of measurement are based on a single length called a meter, which is a little longer than three feet.
The system of feet and inches that we use is actually, when you think about it, rather difficult to work with. There are three feet in a yard, twelve inches in a foot, and inches are usually broken up into sixteenths; this means that converting between the different sorts of measurement takes a lot of math.
In contrast, the metric system was designed so that each new unit is made up of ten of the next smaller unit. A meter is broken up into ten decimeters, and each decimeter is broken up into ten centimeters. A centimeter can also be broken down into ten parts; a tenth of a centimeter is called a millimeter. This works very well with our place value system, which also has each place worth ten times as much as the next smaller.
So, for example, if something is 4.82 meters long, this means 4 meters plus 8 decimeters plus 2 centimeters. Each place refers to a particular unit. This also makes it easy to convert from one type of units to another; if I want to say how many centimeters are in 4.82 meters, I just move the centimeters over into the ones place; 4.82 meters = 482 centimeters.5.1.2. Constructing the metric units
I've brought in to class a large number of strips of paper that are all exactly one meter long. Your job, working with your lab partner, is to figure out how to break up a one meter strip into decimeters. Remember, you will have to split it up into ten equal parts - some thinking about fractions should help you come up with a strategy for that.
Once you've constructed your decimeters, you can try to construct centimeters as well. In order to do that, you will have to break up a decimeter into ten equal parts. (This should be easier to do now that you have a strategy that worked for breaking up the meter) You might even be able to split up one of your centimeters into millimeters.
You can label your decimeters, centimeters, and millimeters with the abbreviations dm, cm, and mm. The abbreviation for meters is just m.5.1.3. Measuring with metric units
Now that you have created your own set of metric units, your ticket to leave will be to use them to successfully measure some things. Remember, you have a choice of which units to use; for smaller things you might want to use centimeters, whereas for larger things you might want to use meters.
Measure for me:- The width and height of an ordinary sheet of paper
- The width and height of your desk
- The length of your pen or pencil
- The width and height of your DEAR book
5.2. Using metric units
Build intuition for the size of different metric measurements.
- Combining metric units
- Interpreting metric measurements
- Switching the type of metric units
5.2.1. Combining metric units
Suppose that I start off measuring something in decimeters, and I find that it is 8 dm long, with a bit left over. Then I decide that I want to switch into centimeters to be more specific. Do I need to start over and measure out the whole thing with just centimeters?
As you probably realized, I don't have to remeasure the 8 dm at all; I know that it is equal to 80 cm because each decimeter is ten centimeters. If I find that I have to add three more centimeters to get to the actual length of the object, then I have a total length of 83 cm: 80 centimeters in the form of 8 decimeters, plus 3 more individual centimeters. The tens place of my centimeters measurement should really be called the "decimeters" place, since ten centimeters is a decimeter.
In general, I could make any metric measurement at all using just nine of each type of unit: meters, decimeters, and centimeters. There are other units for the other places as well:- 10 m = 1 decameter (dam)
- 10 decameters = 1 hectometer (hm)
- 10 hectometers = 1 kilometer (km)
- .1 meter = 1 decimeter (dm)
- .1 decimeter = 1 centimeter (cm)
- .1 centimeter = 1 millimeter (mm)
So, for example, the length of the main hallway is:
2 decameters + 9 meters + 2 decimeters + 6 centimeters
How many centimeters, in total, would that be?
One way to solve this would be to think to yourself, "2 decameters is 20 meters, so that is a total of 29 meters. But 29 meters is 290 decimeters..."5.2.2. Interpreting metric measurements
The cool thing about metric measurements is that every place value in the number can be translated directly into a particular measurement. When you see a number like "2926 cm" or "29.26 m", you can line it up like this:
| km | hm | dam | m | dm | cm | mm |
2926 cm = | | | 2 | 9 | 2 | 6 | |
29.26 m = | | | 2 | 9 | 2 | 6 | |
0.038124 km = | 0 | 0 | 3 | 8 | 1 | 2 | 4 |
Notice how, in the first example, it is measured in centimeters, and the 6 is in the ones place, so I put the 6 in the centimeters bin; the second example is measured in meters, and the 9 is in the one's place (note the decimal point!), so I put the 9 in the meters bin. It turns out the two numbers are measuring exactly the same length!
The third example looks like a complicated decimal number, but when I lay it out in bins, remembering that the one's place is kilometers, it becomes a lot clearer. If I wanted to, I could measure it in meters instead, moving the decimal place to after the meters bin; it would be just 38.124 m.
Even without the table above, you should be able to quickly look at any given metric number and make sense of it. So, for example:- My pencil is 0.0041 hm long. Is that a rather long pencil, or a rather short one?
The 4 here is in the decimeters place, so the pencil is about 41 cm long. That's long for a pencil, but not unreasonably so.
- I can run 12,493 mm in just three seconds. Is that very impressive, or not?
The 2 is in the meters place, so this is equal to 12 m. This is comparable to the example length of the hallway that I gave you; I am saying that I can run a little less than half the length of the hallway in just three seconds. Again, this is really not all that impressive; it just sounded fast when I expressed it in millimeters.
5.2.3. Switching the type of metric units
As we saw above, the fact that metric units are all labeled by place value also makes it very easy to convert from one type of metric units to another; it is just a matter of moving the decimal place. I can do this very quickly, and I don't even need to put the number in bins first. Suppose that I want to convert 309.2 cm into meters:
The trick here is that if you first identify what unit the ones place is, then identify the places around it using the relationships between the metric units, then you can easily convert into any other sort of units just by moving the decimal point to after the proper place.
The one possible point of confusion is what to do if the place value you want doesn't show up in the number you are trying to convert. You can fix this by simply adding zeros to the number. For example:
The classroom is 5.73 m long. How long is that in millimeters?
If the 5 is the meters place, then the smallest place I have is centimeters (where the 3 is). That's no problem; I'll just add an extra zero to get 5.730 m, with the 0 in the mm place, and then move the decimal point to get 5.730 m = 5730 mm.
I can add zeros in the other direction as well:
The classroom is 5.73 m long. How long is that in hectometers?
If the 5 is the meters place, then I need to add two zeros to the left of it in order to have one in the hectometers place. My number is then 005.73 m. I move the decimal place to after the 0 in the hectometers place, and then I have 0.0573 hm.
5.3. Metric Units Checkpoint
Check that everyone has a sense for the size of metric measurements and can interpret and convert measurements using place value.
- What is a "checkpoint?"
- Metric Checkpoint
In order to move on to the next point in this unit, we need to make sure that everyone:- Has a good sense of how big centimeters, decimeters, and meters are
- Can measure an edge of something in metric by lining up units of different types, and express that measurement as a single number with the correct label
- Can lay out units to show any length, given as a number
- Can identify which place in a number is which metric unit, based on what the number as a whole is labeled as
- Can convert a number from one set of metric units to another by using their understanding of place values
In order to give everyone a chance to practice this and get immediate feedback on how you're doing, we'll be working on computers. So, as soon as you are done with the do-now, pack up your metric measures and get ready to head out to the computer lab.
6. How accurately can you measure?
Our goal for the last section of this unit was to learn about the metric measurement system, and to understand how place value makes it very easy to work with metric measurements. The measures that you made were useful as a way to understand how measurement works, but of course most of the time when you are measuring, you will use a ruler. The ruler-reading skills that we will develop in this section will, as you will soon discover, be useful for any sort of measurement device you may have cause to use.6.1. Estimating to a tenth
Estimate accurately to within a tenth of what is marked on a ruler.
- The difficulties of measuring
- Estimating to a tenth
6.1.1. The difficulties of measurement
The first thing that we did in class was to make a table of the perimeter and area that different people found for the do-now. We then tried to figure out, based on that table, exactly how many different shapes of do-nows we had in the classroom. The interesting thing here is that often two people would have numbers that were very close, but not exactly the same, and so we had to decide: did those two people really have two different shapes, or did they just measure a little bit differently?
The rulers I gave you were marked out in centimeters, with no more accurate markings in between. One of the dimensions of your rectangle was very close to being exactly on one of the centimeter lines, but the other dimension was somewhere in between two lines. That left you with a difficult choice. If the length is about halfway between 12 and 13, can I write down that it is 12.5, even though there is no 12.5 mark on the ruler? Or should I instead round it to 12 or 13, whichever seems closer?
Because different people made different choices about whether to round, or estimated differently when a number was in between, we had to decide that measurements that were fairly close to each other were probably actually the same. Or, perhaps a better way of saying this is that, given the amount of inaccuracy in how we measured, if two numbers were close enough together we could not with any confidence say that they were actually measurements of different things. We decided that there were three groupings of area measurements that were far enough away from each other to be distinguishable, and three groupings of perimeter measurements that fit the same criteria. Then, by looking at the different combinations of area and perimeter measurements, we decided that there were actually only four distinct shapes.
This activity demonstrates some of the complications that arise as soon as you ned to measure something in the real world. We've seen that even in a very simple task like finding the dimensions of a rectangle, we have to struggle with the question of how to estimate when a measurement doesn't exactly line up with a mark on your ruler. We had to be aware of the fact that there are always little inaccuracies when someone makes a measurement, and that therefore it is not unreasonable for two people to measure different lengths for the same thing.
Because we will be doing a lot of lab activities that require you to measure things, we really need to figure out good solutions to these two problems: how to estimate consistently, and how to deal with inaccuracy in measurement. We'll tackle the first issue today, and come back to the other issue later on.6.1.2. Estimating to a tenth
With a few tricks of the eye, and a lot of practice, it should be possible to estimate the value of any measurement to within a tenth of what is actually marked on the ruler. So, for example, once you learn these tricks, you will be able to confidently report a length of 3.6 cm or 4.3 cm using the rulers you have today, even though the marks go from 3 to 4 to 5 with no marks in between.
Suppose that you are trying to identify the point marked with an arrow in the picture to the right. The first step in doing this is to picture where the midpoint is in that segment. I have shown this by drawing a small blue line in the middle. Dividing a line exactly in two is something that your eyes are particularly good at doing, since you can compare visually to see if one side is bigger than the other.
Once you can picture where the middle of the space is, it turns out to be fairly easy to decide the exact decimal point value of the mark. If the mark is on one of the lines, or at the midpoint, it is clear that it is at .0 or .5. If it is close to the ends or the midpoint, it will be one of .1, .4, .6, or .9. If it is out in the middle of either the first half or the second half, ask yourself: is it slightly closer to the middle, or to the end? That will tell you if it is .2, .3, .7, or .8.
The practice that we did to learn how to interpret the different place values of metric measurements will come in handy to you here. For example, if I am trying to estimate to within a tenth for a centimeter measurement, then I am estimating millimeters.
6.2. Place value and measurement
Big idea: I can measure one place beyond what is marked, no matter whether that is the 1's or the 1000's or the .01's.
Computer activity for most of class.6.3. Measurement and accuracy
Big idea: Using +/- notation; error is .1 of mark size7. Using a ruler with any mark spacing
At this point, we know how to read any sort of ruler, provided that it is marked off in some power of ten. Unfortunately, in the real world we can't always count on that being the case.7.1. Using a ruler with any mark spacing
This takes quite a lot of care to teach and learn. The idea is to always be saying, "This is 4.3 marks past the 20, and each mark is 4, so..."7.2. Mark spacing practice
Computer activity takes up most of class7.3. Reporting accuracy of any ruler
Some sort of pop quiz or other activity to serve as a checkpoint for the rulers unit
The new type of interesting question we can introduce at this point is whether two measurements agree.7.4. Adding and averaging uncertain numbers
8. Using common lab measurement devices
8.1. Graduated cylinders
8.2. Conservation of volume
8.3. Volume of sand and gravel
8.4. Data tables
8.5. Three-beam balance
8.6. Conservation of mass
8.7. Experimental evidence of mass conservation
8.8. Assessing the accuracy of a measurement device
Lab design activity: give each group a stopwatch, and ask them to design an experiment to decide what kind of +/- ought to be used when measuring times with that stopwatch.9. Review