A pinch of salt, a dollop of daisy – Nonstandard Measurement

cook

For Further Discussion…

  • As adults we use standard measurements almost without thought. Nonetheless, nonstandard measurements also play a role in speech and behavior. We ask for a “pinch” of salt or a small slice of dessert. We promise to be somewhere shortly or to serve a “dollop” of whipped cream. In some situations we might even prefer nonstandard measurements – for example, in cooking, building, or decorating. Do you or someone you know use nonstandard measurements on a regular basis or even instead of a standard system? Give some examples and discuss why a nonstandard measurement might be preferred in some cases.

The only example I can think of when using a nonstandard measurement might be preferred is while cooking.  Ever since I was little, I’ve been trying to duplicate my mom’s recipes and it’s just impossible!  My sister was even determined to formally write up all of my mom’s recipes one time.  It was so funny to watch!  My mom would say “just add a little bit of this”, and before she could throw it in the pot my sister would catch it from my mom’s hand and formally measure it before adding it to the recipe.  But while she was measuring that, my mom would sneak a “bit of this” and a “tad of that” into the pot behind her back.  It was hysterical to watch!  My mom would always tell my sister, “Instead of writing so much, you should be tasting it!”.  Now that I’m older and cook everyday, I find that I “cook by taste” as well, but I still can’t duplicate my mom’s recipes!!  I guess things just taste better when mom makes it!

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Case Studies – Length

The last question on the case studies assignment really got me thinking.  I truly felt like although we expect students to progress throughout the grade levels by becoming more efficient, accurate, and precise in their measuring skills, I wouldn’t say that there’s any ideas that no longer warrant discussion at a certain grade level.  Here’s what I wrote down, but I’m curious to see what other people responded with…

By comparing the cases from second, third, fourth, and seventh grades to Barbara’s kindergarten (case 12), can we identify ideas that, by the older grades, are understood by the children and no longer warrant discussion. What are some issues that still lie ahead for Barbara’s students to sort out? 

It’s hard for me to say which issues warrant further discussion, and which don’t, because as you can see in the cases, every individual child is different.  Overall, I would like to say that students understand a more efficient, accurate, and precise way to measure objects the further they advance in grade levels, but I don’t think that means that sometimes a “refresher course” wouldn’t benefit the class as well.  For example in case 16, Lisa (a fourth grader) incorrectly measured the chalkboard when she made her marks at the end of the meterstick (approximately 39 ¼ inches), but labeled them 39 inches.  In case 14, Les (a third grader), took extra time to realize that the smaller the hand used to measure the object, the bigger the number of measurement became because it would take “more hands” to measure the object.  This concept of the larger the unit, the fewer the number of units needed to cover a length, is something that I would still expect some fourth, fifth, sixth, (and so on), graders to still struggle with.  As an aspiring teacher, I think it’s important to realize that you really can’t say anything no longer warrants discussion.  As Josie proved in case 16, your students can always surprise you.  And we’ve seen in many other cases this semester, and speaking from my personal experience as well, that some students might appear that they know the information being taught, but then when you revisit the concept, it seems that you’re right back to square one with them.  Hence why it’s called a learning process!

 

TCM Article – Measure Up

measure up

Hmmm…. I don’t remember seeing this article on our list from our class syllabus.  I opted not to purchase the NCTM membership this summer, and instead went to the library at the beginning of the course and made copies of the listed articles.  But I was able to find a free copy of Measure Up available online: http://web41.its.hawaii.edu/manoa.hawaii.edu/crdg/wp-content/uploads/measureup-2007-05b.pdf

One of the main ideas I took from this article was that “Measurement is often thought of as the determination of size, amount, or degree of something by using an instrument or device marked in standard units.  Measure Up, however, uses a broader definition of measurement that includes (1) comparing something with an object of known size; (2) estimating or assessing the extent, quality, value or effect of something; and (3) judging something by comparing it with a certain standard.  These three aspects of measurement allow students to explore mathematical structures and develop an understanding of qualitative relationships that offers access to more sophisticated mathematical ideas at earlier grades” (page 452).

I found it interesting that the article said that children often say an object is larger or smaller, but rarely specify how it is larger or smaller.  This is something I brought up in my last post, about the Ordering Rectangles activity we just completed.  I spoke about how rectangle C had one of the largest perimeters, but also the smallest area out of the set.  I loved this, because I thought it would lead to a great class discussion if we should refer to it as a small or large rectangle.

Overall, I was really surprised by the first graders’ work highlighted in this article!  I think it’s just amazing that they’re able to make so many different relationships between two different shapes.  I also really liked that they were being taught to use letters in math, a concept that I’ve seen even adults still struggle with!  Not only did I think this was a great lesson on measurement, but I think it gave students a chance to work on some skills that will benefit them in mathematics as a whole- For instance, being able to answer WHY their solution is thought to be correct, being able to establish relationships, and being able to explain their answers orally and through their written work.  I just can’t believe this was a first grade class!  Must be something in the waters in Hawaii 🙂

Ordering Rectangles Activity

1. Take the seven rectangles and lay them out in front of you. Look at their perimeters. Do not do any measuring; just look. What are your first hunches? Which rectangle do you think has the smallest perimeter? The largest perimeter? Move the rectangles around until you have ordered them from the one with the smallest perimeter to the one with the largest perimeter. Record your order.

Here’s my hunch, from smallest to largest perimeter:

C

E

D

B

F

A & G (tied)

 

2. Now look at the rectangles and consider their areas. What are your first hunches? Which rectangle has the smallest area? The largest area? Again, without doing any measuring, order the rectangles from the one with the smallest area to the one with the largest area. Record your order.

Here’s my hunch, from the smallest to the largest area:

C

D

B

F

E

A

G

 

3. Now, by comparing directly or using any available materials (color tiles are always useful), order the rectangles by perimeter. How did your estimated order compare with the actual order? What strategy did you use to compare perimeters?

I used the color tiles as my nonstandard unit of measurement, and recorded the length and width of each rectangle.  Here’s what I came up with:

A: length = 5, width = 3

B: length = 6, width = 2

C: length = 8, width = 1

D: length = 2, width = 5

E: length = 3, width = 4

F: length = 7, width = 2

G: length = 4, width = 4

To calculate the perimeter, I used the equation 2 * (length+width):

A = 2 * (5 + 3) = 16

B = 2 * (6 + 2) = 16

C = 2 * (8 + 1) = 18

D = 2 * (2 + 5) = 14

E = 2 * (3 + 4) = 14

F = 2 * (7 + 2) = 18

G = 2 * (4 + 4) = 16

So the actual order of the rectangles, from the smallest to largest perimeter would be:

D & E

A, B, & G

C & F

Okay, so my prediction was way off!  The main one being that I thought C had the smallest perimeter, and it actually was tied with F for having one of the largest!  I guess I found the “slenderness” of the rectangle really deceiving!

 

4. By comparing directly or using any available materials (again…color tiles), order the rectangles by area. How did your estimated order compare with the actual order? What strategy did you use to compare areas?

To calculate the areas of the rectangle, I started with some previous knowledge I had already obtained from calculating the perimeters (each rectangle’s length and width, using the color tiles):

A: length = 5, width = 3

B: length = 6, width = 2

C: length = 8, width = 1

D: length = 2, width = 5

E: length = 3, width = 4

F: length = 7, width = 2

G: length = 4, width = 4

To calculate the area, I used the equation length times width:

A = 5 * 3 = 15

B = 6 * 2 = 12

C = 8 * 1 = 8

D = 2 * 5 = 10

E = 3 * 4 = 12

F = 7 * 2 = 14

G = 4 * 4 = 16

So the actual order of the rectangles, from the smallest to largest area would be:

C

D

B & E

F

A

G

I did much better at predicting the area, rather than the perimeter.  In fact, I only made two mistakes: not realizing that E’s area is smaller than F’s, and that E’s is also equivalent to B’s area.  Again, the slenderness of F compared to the chunckiness of E made me believe that E had a larger area.

 

5. What ideas about perimeter, about area, or about measuring did these activities help you to see? What questions arose as you did this work? What have you figured out? What are you still wondering about? Share your responses to all the questions in 1-5 on your blog. Spend some time discussing your answers with your blog partner.

 

This activity helped me see that certain first impressions of a shape can be misleading to what it’s perimeter or area truly is.  For example, I kept thinking that the slender rectangles would have less of an area compared to the other chunkier rectangles.  But I guess it’s understanding that it just depends on the how long (the length of) the slender rectangle is, compared to how wide (the width of) the chunky rectangle is.

 

Another thing I noticed is that rectangle C had the one of the largest perimeters, but also the smallest area out of the set.  I think this would be important to discuss with my students if I were doing this activity in the classroom, because it brings up the question: How do we measure what’s larger?  One might argue that rectangle C is the largest rectangle because it has one of the largest perimeters, but another might argue that it’s the smallest because it has the smallest area.  In the end, I think it all depends on content.  For instance, think if these rectangles as swimming pools, and I had to swim around the wall of them.  I would say that C is one of the largest swimming pools because its perimeter is one of the largest, and would take me the most time to swim around.  But on the other hand, what if I had to fill each swimming pool with as much water as it can hold?  Then I would say C is the smallest swimming pool because it would take me the least amount of time to fill it up, since it holds the least amount of water, having the smallest area out of all of the swimming pools.  Hopefully this discussion would lead my class to think about the different perspectives of measurement, and the content and why they’re measuring something.  It could also lead students to become more precise with their solutions and explanations, stating “C has the largest perimeter in the set”, rather than “C is the largest rectangle”.

 

Annenberg – What does it mean to measure?

math rocks

My notes from the Annenberg lesson on measurement…

Questions to ask yourself, to determine whether an attribute is measureable…

1. Can this attribute be quantified?

2. If I combine this object with another similar object, will the attribute increase?  (If yes, than the attribute is measureable.)

 

Surface Area – the area enclosing a three-dimensional or solid object, the units for surface area will be square (exps: square inches, square centimeters, square millimeters, etc.)

 

Volume – one method for determining the volume of irregular objects (i.e. rocks) uses a technique called displacement: the volume of the immersed object will be exactly equal to the volume of the displaced fluid

 

Weight

  • Using a two-pan balance, you can determine a measure (i.e., an approximation) for the weight of your rock relative to other known weights, but not the exact weight of the rock.
  • The three-arm balance works on the lever principle, in which moving a weight farther from a balance point produces a greater force on that side of the balance. (This is the same principle used in balancing a seesaw.) We can determine an approximation of the rock’s weight using this type of scale, but not the exact weight.
  • Mass is a measure of the amount of material making up an object (specifically, its molecules). All objects have mass, but not all have weight, which is the effect of a gravitational field on a body that has mass. For example, a U.S. flag placed on the Moon has the same mass as one placed on the Earth, but it weighs less as a result of the Moon’s gravitational pull. Objects can be weightless, but they can never be without mass.
  • The precision is based on how fine the measuring instrument is. In a two-pan balance, precision is based on the values of the pan weights being used. The smaller the value of the unit, the more precise the measurement. For example, measurements made using milligrams are more precise than those using grams or kilograms.

 

What is measurement?

Measurement is the process of quantifying the properties of an object by expressing them in terms of a standard unit. Measurements are made to answer such questions as, How heavy is my parcel? How tall is my daughter? How much chlorine is in this water?

 

How do we measure?

The process of measuring consists of three main steps. First, you need to select an attribute of the thing you wish to measure. Second, you need to choose an appropriate unit of measurement for that attribute. Third, you need to determine the number of units.

 

What procedures are used to determine the number of units?

Some measurements require only simple procedures and little equipment — measuring the length of a table with a meter stick, for example. Others — for example, scientific measurements — can require elaborate equipment and complicated techniques.

 

Is it possible to measure objects without using standard units?

Yes. Nonstandard units (i.e., units that are not agreed upon by large numbers of people) can be used to make comparisons and order objects. But because the units are nonstandard, there is limited value in using them to convey information.

 

How precise are measurements?

Measurement, by its very nature, is approximate. The precision of the measuring device tells us how finely a particular measurement was made. Measurements made using small units, such as square millimeters, are more precise than measurements made using larger units, such as square centimeters. The accuracy of a measure is determined by how correctly a measurement has been made. Accuracy can be affected by the person making the measurement and/or by the measurement tool. Precision and accuracy, and how to determine them, will be covered in later sessions.

 

Okay, then — how large is my rock?

It all depends on how you define the word large. Your answer will be based on the attributes you decide to consider, such as weight, volume, surface area, and height.

 

Measurement Introduction PowerPoint

Just a few notes from the Measurement PowerPoint that I thought were important enough for repeating.  I know some of these notes are taken verbatim from the presentation, but I find it helpful to use my blog as a study guide, with everything in one place…

  • Measurement of time, measuring quantities, measuring in determining capacity, measuring distance… measurement pervades in almost all aspects of our lives

Three Distinct Agendas

  1. Help children understand what measurement is and how to go about doing it.
    • Understanding of the attribute (length, width, weight, etc…) that is being measured.
    • How units (inches, pounds, etc…) are used in measuring.
    • Understanding of how measuring tools work.
  2. Measurement Sense – A familiarity of the most commonly used standard units, ability to estimate, and flexibility with related units.
  3. Development and use of a few standard formulas.

Measurement Process

1. Select an attribute of something that you wish to measure.  Length is one of the first attributes a student typically learns about.

I wonder how long my desk is?

2. Select a unit that has an attribute (whether standard or non-standard).

I can measure the length of my desk with a meter stick.

3. Compare the units, by filling, covering, matching or using some other method, with the attributes of the object being measured.  The number of units required to match the object is the measure.

My desk is 65 centimeters long.

Prerequisite Concepts and Skills

  • Conservation –  objects maintain their same size and shape when measured
  • Transitivity – two objects can be compared in terms of a measureable quality using a third object
  • Units – the type of units used to measure an object, depending on the attributes being measured
  • Unit Iteration – the idea that the units must be repeated or iterated in order to determine the measure of an object

Estimation in Measurement

  • By asking children to estimate before they measure, they will come to realize that all measurements are approximate.
  • Estimation helps students focus on and understand the attribute that is being measured.
  • Estimation provides intrinsic motivation. It adds interest and challenge to classroom estimation activities.
  • The use of a benchmark promotes multiplicative reasoning.