Math Toolkit

Système Internationale d'Unités (SI) is an internationally agreed upon system of measurements. A base unit is a defined unit in a system of measurement that is based on an object or event in the physical world, and is independent of other units. SECTION 2.1 Units and Measurements SI Units

SECTION 2.1 Units and Measurements SECTION 2.1 Units and Measurements Units (cont.)

SECTION 2.1 Units and Measurements Units (cont.)

megakilo hecto deca Basic deci centi milli micro nano Unit (M)(K) (h) (da) gram(g) (d) (c)(m) (n) 1,000,0001, liter (l) meter(m)

The SI base unit of time is the second (s), based on the frequency of radiation given off by a cesium-133 atom. The SI base unit for length is the meter (m), the distance light travels in a vacuum in 1/299,792,458th of a second. The SI base unit of mass is the kilogram (kg), about 2.2 pounds SECTION 2.1 Units and Measurements Units (cont.)

The SI base unit of temperature is the kelvin (K). Zero kelvin is the point where there is virtually no particle motion or kinetic energy, also known as absolute zero. Two other temperature scales are Celsius and Fahrenheit. SECTION 2.1 Units and Measurements Units (cont.)

The Difference between Mass and Weight Mass is a measurement of how much matter is in an object eg bag of sugar has a mass of 1kg. Weight is the measure of the amount of matter but also depends on how much gravity is acting on you eg Moon vs earth.

Not all quantities can be measured with SI base units. A unit that is defined by a combination of base units is called a derived unit. SECTION 2.1 Units and Measurements Derived Units

Volume is measured in cubic meters (m 3 ), but this is very large. A more convenient measure is the liter, or one cubic decimeter (dm 3 ). SECTION 2.1 Units and Measurements Derived Units (cont.)

Density is a derived unit, g/cm 3, the amount of mass per unit volume. The density equation is density = mass/volume Units are g/cm 3 or g/ml SECTION 2.1 Units and Measurements Derived Units (cont.)

Measuring Density Regular Shape Solid Measure mass (g) and volume (cm 3 ) directly Units: g/cm 3 Irregular Shape or Liquid Measure mass (g) directly and volume (ml) by the water displacement method. Units: g/ml

Water Displacement Method Steps 1) Add water to a graduated cylinder and record the initial volume (ml) 2) Add the solid and record the final volume (ml) Volume (ml) = final volume (ml) – initial volume (ml)

What is the density of an unknown metallic element that has a volume = 5.0 cm 3 and a mass = 13.5 g?

When a piece of Aluminum is placed in a 25 ml measuring cylinder that contains 10.5ml of water, the water level rises to 13.5ml. What is the mass of the Aluminum?

Which of the following is a derived unit? A.yard B.second C.liter D.kilogram Section Check SECTION 2.1

What is the relationship between mass and volume called? A.density B.space C.matter D.weight Section Check SECTION 2.1

Scientific Notation & Dimensional Analysis

scientific notation dimensional analysis conversion factor Scientists often express numbers in scientific notation and solve problems using dimensional analysis. SECTION 2.2 Scientific Notation and Dimensional Analysis

Scientific notation can be used to express any number as a number between 1 and 10 (a, known as the coefficient) multiplied by 10 raised to a power (n, known as the exponent). a x 10 n Eg Carbon atoms in the Hope Diamond = 4.6 x is the coefficient and 23 is the exponent. SECTION 2.2 Scientific Notation and Dimensional Analysis Scientific Notation

800 = 8.0  = 3.43  10 –5 The number of places moved = n. When the decimal moves to the left, n is positive and when the decimal moves to the right, n is negative. Count the number of places the decimal point must be moved to give a coefficient between 1 and 10. SECTION 2.2 Scientific Notation and Dimensional Analysis Scientific Notation (cont.)

Addition and subtraction –Exponents must be the same. –Rewrite values to make exponents the same. –Ex x x 10 17, you must rewrite one of these numbers so their exponents are the same. Remember that moving the decimal to the right or left changes the exponent x x –Add or subtract coefficients. –Ex x x = 3.2 x SECTION 2.2 Scientific Notation and Dimensional Analysis Scientific Notation (cont.)

Multiplication and division –To multiply, multiply the coefficients, then add the exponents. Ex. (4.6 x )(2 x ) = 9.2 x 10 0 –To divide, divide the coefficients, then subtract the exponent of the divisor from the exponent of the dividend. Ex. (9 x 10 7 ) ÷ (3 x ) = 3 x Note: Any number raised to a power of 0 is equal to 1: thus, 9.2 x 10 0 is equal to 9.2. SECTION 2.2 Scientific Notation and Dimensional Analysis Scientific Notation (cont.)

Dimensional analysis is a systematic approach to problem solving that uses conversion factors to move, or convert, from one unit to another. A conversion factor is a ratio of equivalent values having different units. SECTION 2.2 Scientific Notation and Dimensional Analysis Dimensional Analysis

Writing conversion factors –Conversion factors are derived from equality relationships, such as 1 dozen eggs = 12 eggs. –Percentages can also be used as conversion factors. They relate the number of parts of one component to 100 total parts. SECTION 2.2 Scientific Notation and Dimensional Analysis Dimensional Analysis (cont.)

Using conversion factors –A conversion factor must cancel one unit and introduce a new one. SECTION 2.2 Scientific Notation and Dimensional Analysis Dimensional Analysis (cont.)

Dimensional Analysis Steps: 1. Write the given number and unit. (55mm) 2. Set up a conversion factor (55mm = __?__ m) Place the given unit as a denominator of conversion factor. Place desired unit as the numerator. Place a 1 in front of the larger unit. Determine the number of smaller units needed to make “1' of the larger unit. 3. Cancel units. Solve the problem.

What is a systematic approach to problem solving that converts from one unit to another? A.conversion ratio B.conversion factor C.scientific notation D.dimensional analysis SECTION 2.2 Section Check

Which of the following expresses 9,640,000 in the correct scientific notation? A.9.64  10 4 B.9.64  10 5 C.9.64 × 10 6 D.9.64  6 10 SECTION 2.2 Section Check

Accuracy and Precision

accuracy precision error Measurements contain uncertainties that affect how a result is presented. percent error significant figures SECTION 2.3 Uncertainty in Data

Accuracy refers to how close a measured value is to an accepted value. Precision refers to how close a series of measurements are to one another. SECTION 2.3 Uncertainty in Data Accuracy and Precision

Error is defined as the difference between an experimental value and an accepted value. SECTION 2.3 Uncertainty in Data Accuracy and Precision (cont.)

The error equation is error = experimental value – accepted value. Percent error expresses error as a percentage of the accepted value. SECTION 2.3 Uncertainty in Data Accuracy and Precision (cont.)

Often, precision is limited by the tools available. Significant figures include all known digits plus one estimated digit. SECTION 2.3 Uncertainty in Data Significant Figures

Rules for significant figures –Rule 1: Nonzero numbers are always significant. –Rule 2: Zeros between nonzero numbers are always significant. –Rule 3: All final zeros to the right of the decimal are significant. –Rule 4: Placeholder zeros are not significant. To remove placeholder zeros, rewrite the number in scientific notation. –Rule 5: Counting numbers and defined constants have an infinite number of significant figures. SECTION 2.3 Uncertainty in Data Significant Figures (cont.)

Calculators are not aware of significant figures. Answers should not have more significant figures than the original data with the fewest figures, and should be rounded. SECTION 2.3 Uncertainty in Data Rounding Numbers

Rules for rounding –Rule 1: If the digit to the right of the last significant figure is less than 5, do not change the last significant figure. –Rule 2: If the digit to the right of the last significant figure is greater than 5, round up the last significant figure. –Rule 3: If the digits to the right of the last significant figure are a 5 followed by a nonzero digit, round up the last significant figure. SECTION 2.3 Uncertainty in Data Rounding Numbers (cont.)

Rules for rounding (cont.) –Rule 4: If the digits to the right of the last significant figure are a 5 followed by a 0 or no other number at all, look at the last significant figure. SECTION 2.3 Uncertainty in Data Rounding Numbers (cont.)

Addition and subtraction –Round the answer to the same number of decimal places as the original measurement with the fewest decimal places. Multiplication and division –Round the answer to the same number of significant figures as the original measurement with the fewest significant figures. SECTION 2.3 Uncertainty in Data Rounding Numbers (cont.)

Determine the number of significant figures in the following: 8,200, 723.0, and A.4, 4, and 3 B.4, 3, and 3 C.2, 3, and 1 D.2, 4, and 1 SECTION 2.3 Section Check

A substance has an accepted density of 2.00 g/L. You measured the density as 1.80 g/L. What is the percent error? A.20% B.–20% C.10% D.90% SECTION 2.3 Section Check

Representing Data

Create graphics to reveal patterns in data. independent variable: the variable that is changed during an experiment graph Interpret graphs. Graphs visually depict data, making it easier to see patterns and trends. SECTION 2.4 Representing Data

A graph is a visual display of data that makes trends easier to see than in a table. SECTION 2.4 Representing Data Graphing

A circle graph, or pie chart, has wedges that visually represent percentages of a fixed whole. SECTION 2.4 Representing Data Graphing (cont.)

SECTION 2.4 Representing Data Graphing (cont.) Bar graphs are often used to show how a quantity varies across categories.

On line graphs, independent variables are plotted on the x-axis and dependent variables are plotted on the y-axis. SECTION 2.4 Representing Data Graphing (cont.)

If a line through the points is straight, the relationship is linear and can be analyzed further by examining the slope. SECTION 2.4 Representing Data Graphing (cont.)

Interpolation is reading and estimating values falling between points on the graph. Extrapolation is estimating values outside the points by extending the line. SECTION 2.4 Representing Data Interpreting Graphs

This graph shows important ozone measurements and helps the viewer visualize a trend from two different time periods. SECTION 2.4 Representing Data Interpreting Graphs (cont.)

____ variables are plotted on the ____-axis in a line graph. A.independent, x B.independent, y C.dependent, x D.dependent, z Section Check SECTION 2.4

What kind of graph shows how quantities vary across categories? A.pie charts B.line graphs C.Venn diagrams D.bar graphs Section Check SECTION 2.4

Chemistry Online Study Guide Chapter Assessment Standardized Test Practice CHAPTER 2 Analyzing Data Resources

Key Concepts SI measurement units allow scientists to report data to other scientists. Adding prefixes to SI units extends the range of possible measurements. To convert to Kelvin temperature, add 273 to the Celsius temperature. K = °C Volume and density have derived units. Density, which is a ratio of mass to volume, can be used to identify an unknown sample of matter. SECTION 2.1 Units and Measurements Study Guide

Key Concepts A number expressed in scientific notation is written as a coefficient between 1 and 10 multiplied by 10 raised to a power. To add or subtract numbers in scientific notation, the numbers must have the same exponent. To multiply or divide numbers in scientific notation, multiply or divide the coefficients and then add or subtract the exponents, respectively. Dimensional analysis uses conversion factors to solve problems. SECTION 2.2 Scientific Notation and Dimensional Analysis Study Guide

Key Concepts An accurate measurement is close to the accepted value. A set of precise measurements shows little variation. The measurement device determines the degree of precision possible. Error is the difference between the measured value and the accepted value. Percent error gives the percent deviation from the accepted value. error = experimental value – accepted value SECTION 2.3 Uncertainty in Data Study Guide

Key Concepts The number of significant figures reflects the precision of reported data. Calculations should be rounded to the correct number of significant figures. SECTION 2.3 Uncertainty in Data Study Guide

Key Concepts Circle graphs show parts of a whole. Bar graphs show how a factor varies with time, location, or temperature. Independent (x-axis) variables and dependent (y-axis) variables can be related in a linear or a nonlinear manner. The slope of a straight line is defined as rise/run, or ∆y/∆x. Because line graph data are considered continuous, you can interpolate between data points or extrapolate beyond them. Study Guide SECTION 2.4 Representing Data

Which of the following is the SI derived unit of volume? A.gallon B.quart C.m 3 D.kilogram CHAPTER 2 Analyzing Data Chapter Assessment

Which prefix means 1/10 th ? A.deci- B.hemi- C.kilo- D.centi- CHAPTER 2 Analyzing Data Chapter Assessment

Divide 6.0  10 9 by 1.5  A.4.0  10 6 B.4.5  10 3 C.4.0  10 3 D.4.5  10 6 CHAPTER 2 Analyzing Data Chapter Assessment

Round to 3 significant figures. A.2.35 B C.2.34 D.2.40 CHAPTER 2 Analyzing Data Chapter Assessment

The rise divided by the run on a line graph is the ____. A.x-axis B.slope C.y-axis D.y-intercept CHAPTER 2 Analyzing Data Chapter Assessment

Which is NOT an SI base unit? A.meter B.second C.liter D.kelvin CHAPTER 2 Analyzing Data Chapter Assessment

Which value is NOT equivalent to the others? A.800 m B.0.8 km C.80 dm D.8.0 x 10 4 cm CHAPTER 2 Analyzing Data Standardized Test Practice

Find the solution with the correct number of significant figures: 25  0.25 A.6.25 B.6.2 C.6.3 D CHAPTER 2 Analyzing Data Standardized Test Practice

How many significant figures are there in meters? A.4 B.5 C.6 D.11 CHAPTER 2 Analyzing Data Standardized Test Practice

Which is NOT a quantitative measurement of a liquid? A.color B.volume C.mass D.density CHAPTER 2 Analyzing Data Standardized Test Practice

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