## Teacher resources

Overview

### Suggestion for teaching to count change

• People use money to buy goods and services.
• Money can be sorted by its value.
• Similar coins have similar values.
• Sort coins by value.
• Draw pictures of coins and paper money.
• Have students lay a number of coins on their desk (two quarters, two dimes and a nickel).
• Represent the coins on the board (draw a circle for each coin with the value inside: (25, 25, 10, 10, 5).
• Ask students how to count them.
• As students count them circle the first two numbers (say 25, 25) and write the sum (50).
• Draw an arrow to the next number they want to add (10) and write the sum (60).
• Draw a line with an arrow to the next (10) write the sum (70).
• Draw a line with an arrow to the last (5) and write the sum (75).
• Ask the students why they added the coins in the order that they did.
• Discuss the differences that order can make.
• Try other orders and combinations to see if they get the same value.
• Discuss which orders are easier to add and develop suggestions for adding different kinds of money.
• Create more problems and discuss as above.

Can use the counting money strategy for counting decimal numbers. Write the starting value (1.89), select a count (.01), write it, write the sum (1.90), write the next coin (.10), then the sum (2.00), and continue as before.

### Metric units

Meter is the unit for linear measurement.

• One meter is a little longer than one yard.
• One millimeter = .001 meter or one thousandth meter (diameter of a paper clip)
• One centimenter = .01 meter or one hundredth meter (width of a paper clip or little finger)
• One kilometer = 1000 meters (a bit more than half a mile)
• 1000 mm = 1 m
• 100 cm = 1 m
• 1000 m = 1 km

Liter is the unit for volume. One liter is a little more than a quart.

• milliliter is .001 liters or one thoundth liter (five ml make a teaspoon)

Gram is the unit mass. One gram is abouit the weight of a paper clip.

• kilogram 1000 grams (can of peaches about 2.2 pounds)

Most used prefixes for metric units:

• milli = one thousandth (.001)(10 -3)
• centi = one hundredth (.01) (10-2)
• kilo = one thousand times (1000)(103)
• More metric prefixes

Some prefixes on a number line:

More in History of measurement

## All math conceptsMeasurement Concepts

### Overall K-...

• Objects have properties or attributes that can be measured with different accuracies depending on the selection and use of a measurement system, units, tools, and purpose of the measurement.
• Measuring is a process that starts with the identification of something to measure - a property of an object or event, then a selection of a measuring system and appropriate unit (standard or non standard), and finally a process to iterate the unit across the unknown object or event to be measured.

### Primary K- 3

• Properties can be counted.
• Conservation of number - the number of objects does not change with the position of the objects.
• Conservation of length - length of an object does not change when its position is changed or its shape is altered by bending.
• Objects can be used to compare other objects.
• Measurement is a way of detecting change.
• Linear measurement is the distance between two points.
• Volume is the measurement of space an object occupies.
• Area measures the surface of an object.
• A standard unit of measurement helps communication.
• When the thermometer goes up the temperature is hotter.
• Measurement helps in making better observations.
• Scales measure mass and weight.
• Measuring cups measure volume
• Measurements can be compared.
• Measurement is used in everyday life (recipes, plans, designing, building)
• Time is the measurement of years divided into seconds, minutes, hours, days, weeks, months, decades, centuries.
• Mathematics can be used to solve problems with time.
• Time is communicated in standard units.
• The duration of an event from the beginning to the end is measured in time.
• Time represents past, present, and future events.

### Intermediate 4-6

• Conservation of volume - volume of an object or substance does not change when the position and shape of the object or substance change position.
• Conservation of mass - mass of an object or substance does not change when the postion or shape of the object or substance changes.
• Conservation of area - area of a surface does not change when the position of the surface changes position.
• Properties and change of properties can be quantified.
• All measurement is relative to a unit, usually a standard unit.
• Scale is proportional.
• Measurement helps in making more accurate observations.
• Quantitative estimates of familiar lengths, weights, and time intervals can be confirmed by measurement.
• Measurement helps in making better observations.
• Rulers are used to measure linear measurement.
• Scales measure mass and weight.
• Measuring cups measure volume.
• Measurements can be compared.
• Measurement is used in everyday life (recipes, plans, designing, building).
• Rate is based on time.
• Standard units include:
• Linear measurement standard units are cm, m, km, inch, foot, yard, mile
• Volume- standard units include liter, ml, cup, pint, quart, gallon
• Time - measure in units of seconds, minutes, hours, days, weeks, months, years, decades, centuries. Time is used to order events. Twenty-four hours in a day, about 30 days in a month, 365 days in a year, 52 weeks in a year, 12 months in a year. Calendar is used to measure time. Clocks are used to measure time. Clocks can be analog and digital. A day is divided into daytime and nighttime. Time is determined by Earth's movement. Time is cyclic (seasons, days of weeks, months).
• Temperature measures hot and cold. Degrees in Celsius and Fahrenheit are standard units of temperature
• Mass - standard units are g, kg, pounds, ounces, tons
• Money is represented in increments of .01, .05, .1, .25, .5, 1.00, 2.00, 5.00 10.00, 20.00, 50.00, 100.00 (beats me from here)
• All measurement has error.

### Middle 6+

• Scale is a proportional relationship of characteristics, properties, or relationships within a system as its dimensions are increased or decreased.
• Rate involves a measure of change for a part relative to a whole (birth rate as part of population growth and comparing one measured quantity to another measured quantity (km per hour).

## All math conceptsEstimation concepts K+

• Numbers can be taken apart and put together in a variety of ways.
• Estimation can be done by replacing or substituting difficult numbers with close or nice numbers and worked with mentally.
• There is no right way to estimate or compute mentally.
• Estimation is used not only for estimation but also when we use mental calculation, paper - pencil calculation, a computer, and a calculator.
• Some mental computation starts with the large digits and works toward the smaller (left to right). Paper and pencil computation is digit oriented and right handed.
• Rounding is not a property of numbers but a property of a system of numbers used to represent them.
• Rounding is an approximate estimation to a selected value (... .001, .01, .1, 0, 1, 10, 100, 1ooo, ...).

## History of Measurement

• Article I, Section 8 of the U. S. Constitution gives Congress the power to "fix the standard of weights and measures" for the nation.
• In 1790 Thomas Jefferson proposed to the Congress of the United States a decimal based measurement system. Congress took no action. Metric system was not yet developed.
• France officially adopted the metric system 1795.
• History of Franklin, Washington, and Jefferson and the metric system
• John Quincy Adams submitted to congress in 1821 a report and review of systems of weight. He did not recommend the use of the metric system as he felt it would not be accepted by the public and the many changes required to implement would be impractical at the time.
• Establishment of Standard Decimal and Division of Weights, ... Vol 66. United States Congress House committee on coinage, weights, and measures. 1920. 35 pages.
• January 1866, Congress enacted legislation authorizing, but not mandating, the use of the metric system in the United States. It was signed into law by President Andrew Johnson on July 20, 1866. This regulated the the coordintion of weights and measures between the United States and other nations.
• The United States participated in the Convention of the meter in Paris in 1875, signed the Treaty of the Metre, received prototypes of the standard meter bar and standard kilogram in 1893, which became the nation's official unit standards for length and mass, both for metric and customary weights and measures.
• In 1901 the U.S. National Bureau of Standards was establishedto serve the worlds of science and technology.
• Any system of measurement must have a basic unit for its standard. This fact has been the the basis for rejection of various systems. Historically standards have included:
• The unit of length (cubit) from the back of the elbow to the tip of the middle finger.
• The unit of length (foot) of a human foot.
• The unit of length (meter) a pendulum would travel in one full swing in one second. This unit had the practical problem of varing with temperature and altitude.
• The unit of length (meter) equal to one ten-millionth of the length of the arc from the equator to the North Pole, or a quadrant of the Earth's meridian circle.
• The unit of mass (kilogram) of water contained in a cube whose sides are one-tenth the unit of length (meter).
• The unit of volume (liter) in a cube whose sides are one-tenth the unit of length (meter).
• The unit of length (meter) constructed by a precise measurement of the arc of meridian between Dunkirk, France, and Barcelona, Spain. The length of the arc from the equator to the North Pole was then to be inferred from astronomical measurements of angle. The survey was completed in November 1798, and platinum artifact reference standard for the meter and the kilogram were constructed in June 1799. These two standards, were in the French National Archive in Paris. Known as the Meter of the Archives and the Kilogram of the Archives.
• A chronology of the SI metric system
• A chronology of the metric system
• Measurement and the presidents of the United States timeline.
• History of measurement in the United States
Metric prefixes, ymbols, & powers
Prefix Symbol Power
Quecca Q 1030
Ronna R 1027
Yotta Y

1024

Zetta Z 1021
Exa E 1018
Peta P 1015
Tera T 1012
Giga G 109
Mega M 106
kilo k 103
milli m 10-3
micro μ 10-6
nano n

10-9

pico p

10-12

femto f 10-15
atto a 10-18
zepto z 10-21
yocto y 10-24
ronto r 10-27
quecto q 10-30
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