Number Sense Return to TopPlace Value, Expanded and Standard Notation
(Note:Not on
Standards for 9th grade—
assessed at lower RIT
band)
Learning Continuum
Teacher Observable
DMAECAISAT
For Learning:Have students write their own “logic number problem “and then have them trade with a partner and solve. The problems should have several clues that have to do with place value.
Of Learning:RIT 231-240
NS#1
Give students a problem from Logic Number Problems. After students solve the problem, have them explain which clues they used to get to their answer. See sample problem.
Logic Number Problems Dale Seymour Publications(problems deal with place value)
Logic Algebra Problems Dale Seymour Publications
Estimation and Computation Return to Top
347.01.aWhole
Numbers, Decimals, Fractions
Learning Continuum
Learning Continuum
Learning Continuum
For Learning:Give students word problems involving decimals and have them solve without using a calculator. See sample problems.
Give students word problems involving fractions and have them solve without using a calculator. See sample problems. EC#1-3
Use the worksheets from Mental Math for the Junior High to show students different strategies for adding and subtracting whole numbers without using a calculator. After teaching each method,
Mental Math for the Junior High by Hope, Reys, and Reys(Dale Seymour Publications)
Have students describe the relationship between the size of the multiplier/divisor and size of the product/quotient.
Of Learning:RIT 231-240 (Decimals--Computation)
RIT 231-240 (Fractions--Computation)RIT 241-250 (Fractions--Computation)
RIT 231-240 (Whole Numbers)
RIT 231-240 (Money)
RIT 231-240 (Decimals—Number Sense)
RIT 231-240 (Fractions—Number Sense)
have students practice with the skill worksheets
Have students investigate the relationship between the size of the multiplier/divisor and size of the product/quotient.
Give students problems with decimals and have them choose the best estimate for the answers.See sample problems.
Give students several pairs of fractions and have them practice getting common denominators for each pair. Also, discuss which operations with fractions require a common denominator and which ones don’t.
Dale Seymour
(Great for warm-up problems)
347.01.aPercents
Learning Continuum
Teacher Observable
DMAECAISAT
For Learning:Have students write equations and solve story problems involving percents.
Of Learning:RIT 231-240
EC#4
Have students solve percent problems using the percent equation:Part is percent of wholeOR New is percent of old
Give students percent of change problems. (See problems)
Give them three different types of problems, one with percent missing, one with is number missing, one with of number missing. Discuss
Mental Math for the Junior High by Hope, Reys, and Reys(Dale Seymour Publications)
Estimation Books A and B Published by Dale Seymour
how the same proportion (equation) works for all three kinds.
347.01.aIntegers
Learning Continuum
Learning Continuum
Teacher Observable
DMAECAISAT
For Learning:Have students write equations and solve word problems involving integers.
Have students draw a Venn diagram that shows how the set of whole numbers relates to natural numbers, integers, rational numbers and real numbers.
Of Learning:RIT 231-240RIT 241-250RIT 251-260
EC#5
Have students compare sign rules for adding integers with sign rules for multiplying integers.
Have students use color counters (or poker chips) to model integer problems
Mental Math for the Junior High by Hope, Reys, and Reys(Dale Seymour Publications)
Estimation Books A and B Published by Dale Seymour
347.01.cRoots
Learning Continuum
Learning Continuum
Teacher Observable
DMAECAISAT
For Learning:Have students plot roots on a number line.
Of Learning:RIT 231-240RIT 241-250
EC#6
Write a list of numbers on the board. Have students decide which ones are perfect squares and which are not. Discuss the difference between “square” and “square root.”
Tell students: If you list all of the factors of a number in numerical order, the square root of the number either is the middle number or lies between the two middle numbers.
Then have students list the factors for certain numbers. Have them estimate the square root for each number.
Have students use rules of exponents to simplify expressions.
When solving problems with negative exponents, have the students do one whole step where all they are allowed to do is move any term with a negative exponent to make all exponents positive. Then let them combine terms.
347.01.aScientific Notation
Teacher Observable
DMAECAISAT
For Learning:Show students problems that look like they are in scientific notation but don’t fit the requirement of having the first number fall between 1 and 10. Ask what they can do to fix the problems so they are in scientific notation. Make sure they understand when to increase the exponent and when to decrease the exponent. (Examples)
EC#8Show students definition of scientific notation. Practice with them the skill of moving the decimal until the resulting number falls into the required boundaries of 1 and 10. The whole class can participate while you visually move your finger across the number written on the board until they tell you to stop.
347.01.bProperties
Of RealNumbers
Learning Continuum
Teacher Observable
DMAECAISAT
For Learning:Have students make “Property Posters”. Write out the names of the properties on slips of paper and have the students randomly select one. They have to illustrate the property (commutative, associative, whatever) with pictures and math symbols but no numbers or letters. Then have students show their posters
EC#9
Compare the commutative property with the symmetric property. Show that in both the order changes but the commutative property only has one equation while the symmetric property
the class and have the class decide which property is being illustrated.
has two. Discuss the similarities and differences of other properties as well.
347.01.dNumber Theory
Teacher Observable
DMAECAISAT
For Learning:Give students a number or list of numbers and have them list all of the numbers 2-11 that will divide evenly into the number(s). Make sure they use divisibility rules and not calculators.
EC#10Show students how to tell if a number is divisible by numbers 2-11 using the divisibility rules. (NO CALCULATORS ALLOWED)
To investigate factors, have students determine the different rectangular configurations for a given number of square tiles.
Logic Number Problems Dale Seymour Publications(problems deal with divisibility)
Logic Algebra Problems Dale Seymour Publications
347.02.aOrder of
OperationsTeacher
ObservableDMAECAISAT
For Learning:Have students write out the order of operations. Make sure they include what to do if there is more than one operation from each level in the problem. (Ex. With multiplication and division, if there is more than one, work left to right). Have them check their lists with another student.
Given an equation with grouping symbols “missing” have students insert symbols to make the equation true.
EC#11
Write a problem with several operations on the board. Have students tell you step by step what to do first, second, etc. using the order of operations.
Given an equation with grouping symbols “missing” have students insert symbols to make the equation true.
Have students write equations, remove grouping symbols and exchange with other students to solve (as above).
Have students solve simple matrix equations involving addition and subtraction of matrices. example
347.02.bSequences Teacher
ObservableDMAECAISAT
For Learning:See sample problems. EC#14
Have students come up with their own sequences (either arithmetic or geometric) and then trade with a partner and figure out the next 3 terms for each sequence.
Have students write a rule and initial value to generate a sequence.
347.03.aDaily Life & Real WorldSituations
Teacher Observable
DMAECA
ISAT
For Learning:See sample problems.
For Learning:See sample warm-up problem.
PS#1 Teach students the formulas for solving uniform motion problems, dry and liquid mixture problems, and interest problems. Also, show them how to set up charts to organize the information and solve the problems.Teach students how to estimate the tip on a restaurant check using 15% as the tip. Tell them
to find 10% first and then add half of that to get the total amount of the tip. Have them practice using several different amounts.
Mathematical Reasoning and Problem Solving Return to Top 348.01.aProblem Solving Strategies
Learning Continuum
Teacher Observable
DMAECAISAT
For Learning:Given the dimensions of a room, including door(s) and window(s) have the students determine amount and cost of wallpaper.
Of Learning:RIT 231-240RIT 241-250RIT 251-260
PS#1 Have students solve a variety of everyday problems (commission, ratios, percents, area, etc.) using multiple strategies.
Have students discuss and compare strategies.
348.02Reasoning and Logic Teacher
ObservableDMAECA
ISAT
For Learning:Have students work in pairs to measure their height and the distance around their head. Then have them make a conjecture about the relationship between the two measurements. Share each pair’s conjecture with the class. They should be finding that a person’s height is approximately three times the distance around his or her head.
Have students defend/justify their solution(s) to problem(s).
See sample problems.
Have students work in groups to solve logic problems. Emphasize that the explanation of HOW they arrived at their conclusions is as important as the actual conclusions.
PS#2After teaching the definitions of both inductive and deductive reasoning, give students examples and have them decide between deductive and inductive for each one.
Given a number pattern/sequence ask students to justify/explain how to determine the next term.
Show students how solving an algebra problem can be set up in proof form where they have to justify each step with a property from Algebra.
“She’s a witch” scene from Monty Python and the Holy Grail
Discuss the following problem solving strategies with students and demonstrate how these strategies can aid in solving logic problems.
*List the possibilities*Process of elimination* Draw a diagram* Identify subgoals* Work backward* Look for a pattern* Use a graph or table* Guess and check* Make a model* Write an equation
348.03.aUsingTechnology Teacher
ObservableDMAECA
ISAT
For Learning:Have students use graphing calculators to solve systems of equations. Start with linear equations and then expand the lesson to include quadratic and possibly cubic equations also.
Have students describe the similarities and differences between the graphs a) and b) and c) and
PS#5
Have students use graphing calculators to explore the effect of changing a parameter in a linear equation. sample
348.03.bMathematicalModels Teacher
ObservableDMA
For Learning:See sample problem.
PS#6 Given a problem situation, model the problem using a table, equation, and graph.
PS#7 Discuss the different ways of communicating mathematical information (charts, graphs, written explanations, pictures, equations, scatter plots, etc).
Concepts and Principles of Measurement Return to Top349.01.a349.02.a349.03.a349.04.aUnits in
Real WorldSituations
Learning Continuum (area, perim,
circum)
Learning Continuum (Leng, Weig,
Vol)
Learning Continuum (time, temp)
Learning Continuum
(Units)
Teacher Observable
DMAECAISAT
For Learning:Have students solve distance-rate-time problems where the units have to be converted before solving. See sample problem.
RIT 231-240 (Length, Weight, Volume)RIT 251-260 (Length, Weight, Volume)
RIT 231-240 (Time, Temperature)
RIT 251-260 (Angle Ident/Measure)
IM#1
Have students match appropriate labels to different situations. For example, with a distance problem, they should choose hours for time, miles for distance and mph for rate.
349.05.a,bLearning
Continuum For Learning: M#2Have students use various approximations for
(1.4, 1.41, 1.412,…) and evaluate the impact when working with large numbers. How do you determine which, if any, approximation to use? (Significant digits in Science…)
Algebra, Functions, and Mathematical Models Return to Top
353.01.aSolve
ProblemsInvolving Functions
Teacher Observable
DMAECAISAT
For Learning:See sample problems and questions.
Of Learning:RIT 231-240 (Pattern, Sequence, Function)RIT 241-250(Pattern, Sequence, Function)
A#1Have students solve problems like: Plumber A charges $75 for a house call plus $50/hr. Plumber B charges $125 for a house call plus $45/hr. Compare each plumber rate.
A#2Create sets of 3x5 cards (3 per set – table, equation, graph) Shuffle cards, have the students match up the cards by set.
353.03.aReal WorldProblemsInvolvingFunctions
Learning Continuum
Teacher Observable
DMAECAISATECA
For Learning:See sample problem.
A#3Show students how to make sketches for functions that deal with real life situations. Then give them a sketch and have them write an explanation of what the graph could be representing.See sample ideas.
For Learning:Have students solve word problems by factoring. See sample problems.
A#4 Have students factor polynomials, using the GCF.
Have students factor quadratic equations.
Have students solve quadratic equations by factoring.
350.03 bCompleting the Square
Teacher Observable
DMAECAISAT
For Learning:Have students solve word problems by completing the square.
A#5 Have students solve equations by completing the square.
350.03 bQuadratic
Formula and Equations
Learning Continuum
Teacher Observable
DMAECAISAT
For Learning:Have students solve word problems that involve using the quadratic formula. See sample problems.
Of Learning:RIT 251-260
A#6 Teach students how to use the quadratic formula.
350.01.aVariables,Algebraic
Expressions
Teacher Observable
DMAECAISAT
For Learning:Have students translate phrases into algebraic expressions. See sample problems.
A#7Make a chart on the board that has four columns. Put the following symbols as column headings: + - X ÷ Have students brainstorm words that would fit in each column.
350.02.aEvaluatingAlgebraic
Teacher Observable
For Learning:Write an algebraic expression for each verbal expression. Then, evaluate the expression if a = 3, b = ½, and c = 0.
A#8Review order of operations before doing examples.
1) twice the sum of a and b2) twice the product of a and b3) the square of b increased by c4) the cube of a decreased by b
350.03.aSolvingLinear
Equations and
Inequalities
Learning Continuum
Teacher Observable
DMAECAISAT
For Learning:Have students solve consecutive integer problems by setting up variables, writing an equation, and showing the correct steps to solve the equation. See sample problems.
Of Learning:RIT 231-240RIT 241-250RIT 251-260
A#9Teach students that solving linear equation is using inverse operations.
Have student solve word problems by translating the English into an equation and solving it.
Show students how to define variables for consecutive integers. X = 1st integer X + 1 = 2nd integer X + 2 = 3rd integerEtc.Then show them the difference when the problem says “consecutive even” or “consecutive odd”. X = 1st integer X + 2 = 2nd integer X + 4 = 3rd integer.
350.04.aSystems
Of EquationsAnd
Inequalities
Teacher Observable
DMAECAISAT
For Learning:
Have students solve word problems involving systems of equations.See sample problems.
Use systems of inequalities to find a maximum profit.
A#10Have students make connections between what graphs look like for one solution, no solution, and infinitely many solutions.
Have students chose a method to solve a system of equations (graphing, substitution, elimination)
Concepts and Principles of Geometry Return to Top351.01.a
Congruency and Similarity
Learning Continuum
Teacher Observable
DMAECAISAT
For Learning:
Give students several pictures with similar triangles. Have them solve for missing measures.
Of Learning:RIT 241-250RIT 251-260
G#1 Give students similar triangles and have them find missing measurements.
Have students determine whether a pair of figures is congruent, similar or neither and explain their reasoning.
Geometry: Plane and Simple by Creative Publications
351.01.aSymmetry
and Transform-
ations
Learning Continuum
Teacher Observable
DMAECAISAT
For Learning:Have students create a Name Inversion. Use the book Inversions by Scott Kim for examples and teaching instructions.
Of Learning:RIT 231-240RIT 241-250RIT 251-260
G#2 Write the alphabet on the board. Have students decide which letters have a horizontal axis of symmetry and which have a vertical axis of symmetry. Then have them create words that have horizontal and vertical symmetry like DECODE and TOMATO.
Inversions by Scott Kim
351.01.bScale
DrawingsAnd
Models
Teacher Observable
DMAECAISAT
For Learning:Have students bring a cartoon to class. Students should draw a square grid over the entire cartoon and then enlarge the cartoon, square by square, onto larger paper. Advanced students could also do a cartoon distortion where the new shape is changed to a rectangle instead of a square. This will distort the original cartoon either lengthwise or widthwise. Another idea is to change the shape to triangles or trapezoids and see what kind of distortion results.
G#3Give students problems that deal with scale drawings (like blueprints of a house) and converting into actual measurements.
For Learning:Have students create a dot-to-dot picture with a list of instructions (coordinates of points and when to lift pencil and when to shade, etc.) on a separate paper. Have students trade instruction pages and draw each other’s pictures.
G#5Teach the vocabulary of quadrants, origin, ordered pairs, coordinates, x-axis, y-axis and have students practice plotting points on a coordinate plane.
351.02.aBasic
Trigono-metryRatios
Learning Continuum
Teacher Observable
DMAECAISAT
For Learning:Give students several triangles that have two measures given and ask students which trigonometric ratios they could set up to solve for the unknown measures.
Of Learning:RIT 251-260
G#7Use “soh-cah-toa” to teach the three basic trig ratios.Or use “some old hags can’t always hide their old age.”
Sine= opposite/hyp. Cosine= adjacent/hyp.
Tangent = opp/adj
Have students solve problems that require trig. See sample problems.
351.02.bProblemSolving
With Trig.Ratios
Teacher Observable
DMAECAISAT
For Learning:Have students draw pictures to solve story problems that deal with trig. Ratios. See sample problems.
G#9 Teach students the difference between angles of elevation and angles of depression.
351.02.cPythagorean
Theorem
Learning Continuum
Teacher Observable
DMAECA
For Learning:Give students several Pythagorean Triples and ask them to construct triangles using those lengths. They should be creating right triangles each time. Discuss how the converse of the Pythagorean
G#8 Give students measures of two sides of a right triangle and have them solve for the third side using the Pythagorean Theorem. Give whole number measures and
ISAT Theorem is true and how their constructions should show that.
Of Learning:RIT 231-240 (Properties/Terminology)RIT 241-250 (Properties/Terminology)RIT 251-260 (Properties/Terminology)
well as measures that are in radical form.
Teach students the distance formula, using the coordinate plane to demonstrate the derivation.
Data Analysis, Probability, and Statistics Return to Top
352.01.aData Analysis
Learning Continuum
Teacher Observable
DMAECAISAT
For Learning:ST#1 Give students data in
various formats (bar graphs, circle graphs, stem-and-leaf plots, line graphs, scatter plots, etc.) and ask them questions about each one.
352.02.aCollect,
Organize,And Display
Data
Teacher Observable
DMAECAISAT
For Learning:Have students work in groups to conduct an experiment (like how long does it take a popcorn kernel to pop for different brands of popcorn). Have each group create two different kinds of graphs to display the results of their experiment and have them explain their posters to the class.
Of Learning:RIT 231-240 (Graphing)RIT 241-250 (Graphing)RIT 251-260 (Graphing)
ST#2 Have students bring in various advertisements that have statistical data in them. Discuss the different ways the information is presented.
352.04.aProbability
Learning Continuum
Teacher Observable
DMA
For Learning:
Of Learning:RIT 241-250 (Probability & Predictions)RIT 251-260 (Probability & Predictions)
ST#3 Have students play the game of PIG as a class. See instructions.
Teach students the difference between probability and odds.
Give them problems that include both. See sample problems.
352.04.bIndependentAnd DependentEvents
Teacher Observable
DMAECAISAT
For Learning:All of the letters to the word “probability” are written on pieces of paper and placed in a hat. Ask students to find the probability of drawing an ‘a’, ‘b’,’i’,etc. in two consecutive drawings if the letters are NOT replaced between drawings. Have them do the problems assuming the letters ARE replaced between each drawing.
ST#4 Explain to students the difference between independent and dependent events. Independent: 1st event has no effect on 2nd event.Dependent: results of 2nd event depend on results of 1st event.Example:Find the probability of drawing 2 red blocks out of a bag. If the blocks are replaced after each drawing, this is independent. If the blocks are not replaced, it is dependent.
352.04.cRandomnessAnd Chance Teacher
ObservableDMAECAISAT
For Learning:ST#5 Explain randomness
a) one event has no effect on a second: flipping a coin b) having 2 possibilities does not guarantee that each occurs half the time
Explain chance as being the probability or odds of an event occurring.
352.05.aPredictions
Teacher
For Learning:See sample problem.
ST#6 Show students how to use “best fit lines” to make predictions given certain data.
For Learning:In the Land of Prob, anyone who insults the Queen is sentenced to death. The only way to escape punishment is to successfully tie six strings together as follows: The Queen holds out 6 strings in her hand with the ends extending on either side. The condemned person must tie the strings together in pairs on each side of her hand. If the strings form one closed ring when the Queen opens her hand, the lucky insulter goes free. What is the probability that one could escape punishment?
Have students simulate this situation to come up with the probability. Should be 8/15.
ST#7 Use The Monty Hall Problem.
Have students roll 2 die and list outcomes … find the probability of getting a specific sum.
Make up several paper sacks of colored markers (skittles or something of the same size) example:
12 red6 blue5 green2 yellow
make all the sacks identical. Have students in groups decide how many experiments (draws) the will do with replacement to determine how many of each color are in the sack. Compare results with all students.
The outcome will be that the students will find several different possible combinations. Then have them look in the sack and discuss why they did not find the correct number of
ST#8 Using spinners is a good way to show the bias of data as they are rarely unbiased.
Combinations and
Permutations
(Note: not on State
Standards)
Learning Continuum
Teacher Observable
DMAECAISAT
For Learning:A lottery which requires guessing 6 numbers out of 50 in any order sells $1 tickets. A group of people decides to buy 12,000,000 tickets. Can they ensure that they win? Justify your answer.Answer: No, the number of combinations of 50 things taken 6 at a time is 15,890,700. Their chance of winning is about 75%.
Of Learning:RIT 241-250
ST#9Show students that with combinations, order does not matter, but with permutations, order does matter.
352.03.aStatistics
Learning Continuum
Teacher Observable
DMAECAISAT
For Learning:Have students keep a record of the high temperatures for their city for a couple of weeks. Then have them calculate the mean, median, mode and range of the set of numbers.Of Learning:RIT 231-240
ST#10 Show students how to calculate the mean, median, and mode, as well as the range for different sets of numbers.
346. MATHEMATICS STANDARDS – GRADES 9 THROUGH 12, SECTIONS 347 THROUGH 353.The samples associated with the content standards are meant to illustrate meaning and to represent possible areas of application. They are not intended to be an exhaustive list, but are samples of applications that would demonstrate learning.
347. BASIC ARITHMETIC, ESTIMATION, AND ACCURATE COMPUTATIONS.Rationale: An understanding of numbers and how they are used is necessary in the everyday world. Computational skills and procedures should be developed in context so the learner perceives them as tools for solving problems.
Standard - The student will:
Content Knowledge and Skills: Samples of Applications:
01. Understand and use numbers.
Return to EC: Whole Numbers, Decimals, Fractions
Return to Percents
Return to Integers
a. Understand and use positive and negative numbers, fractions, decimals, percentages, and scientific notation.
Return to Scientific Notation
i. Use positive and negative numbers (credits and debits) in accounting.
ii. Use fractions when mixing solutions, in measuring with a ruler, or in recipes.
iii. Use decimals in computing gas mileage or in measuring with a micrometer.
iv. Use percentages when computing sales tax, tips, or commissions.
v. Use scientific notation when working with very large or small numbers, such as distance in outer space or microscopic scales.
Return to Properties of Real Numbers
b. Understand properties of the real number system.
i. Analyze real number relationships based on the position of numbers on a number line (e.g., using relative magnitude, absolute value).
ii. Explain why the set of even numbers is closed underaddition and the set of odd numbers is not.
Return to Roots
Return to Exponents
c. Understand properties of roots, exponents, and logarithms.
i. Plot exponential growth using log scale graph paper.
Return to Number Theoryd. Use number theory concepts
(divisibility rules, factors, multiples, primes) to solve problems.
i. Find the largest size square tile that you could use to tile two rooms of different dimensions using only whole tile.
02. Perform computations accurately.
a. Use the proper order of operations. Perform operations with real numbers.
Return to Order of Operations
i. Use mental math to determine correct change. Balance a checkbook.
ii. Find the average of a set of data.
Return to Matrices
Return to Sequences
b. Use graphs, matrices, and sequences to represent and solve problems.
i. Compute compound interest from the number of compounding periods, principal, and annual interestrate.
ii. Use a matrix to adjust a recipe for six to serve crowds of 20, 30, and 40.
03. Estimate and judge reasonableness of
a. Apply number sense to every day situations.
i. Estimate how much lumber you need to build a deck.
ii. Estimate how much you can afford to borrow on a loan given the interest rate.
iii. Estimate driving time to a given destination.
348. MATHEMATICAL REASONING AND PROBLEM SOLVING.Rationale: These processes are essential to all mathematics and must be incorporated in all other mathematics standards.
Standard - The student will:
Content Knowledge and Skills: Samples of Applications:
01. Understand and use a variety of problem-solving skills.
Return
a. Use a variety of methods, including common mathematical formulas, to solve problems drawn from daily life.
i. Determine the payment required on a loan.
ii. Compute the amount of wallpaper, paint, or curtains needed in your kitchen, bath or bedroom.
iii. Determine the amount of paint needed to paint a house.
02. Use reasoning skillsto recognize problems and express them mathematically.
Return to Reason / Logic
a. Use inductive and deductive reasoning to set up a problem.
i. Write a paragraph explaining a solution to a problem.
b. Use logic to make mathematical proofs.
i. Prove a corner is square using the Pythagorean Theorem.
c. Make and evaluate logical arguments.
i. Explain why it is not possible to divide by zero.
03. Apply appropriate technology and models to find solutions to problems.
a. Understand the purpose and capabilities of appropriate technology.
Return
i. Use graphing calculators to fit curves to data.
ii. Use computers for manufacturing process control.
b. Understand the nature and use of mathematical models.
R Return
i. Set up a spreadsheet to model financial or statistical
ii. problems.
04. Communicate results using appropriate terminology and methods.
a. Select the appropriate means to communicate mathematical information.
Return
i. Create charts or graphs to represent demographic data.
ii. Plot the graph of a function based on experimental data.
iii. Use a control chart to determine whether a change is needed in a manufacturing process.
iv. Determine whether a table, pie chart, or bar graph best communicates a set of data.
349. CONCEPTS AND PRINCIPLES OF MEASUREMENT.Rationale: The first step in scientific investigation is to understand the measurable attributes of objects.
Standard - The student will:
Content Knowledge and Skills: Samples of Applications:
01. Understand and use U.S. customary and metric measurements.
a. Determine length, area, capacity, weight, time, and temperature, with appropriate units.
Return
i. Measure the length of a board to the nearest sixteenth of an inch.
ii. Determine the volume of an aquarium in liters.
02. Apply concepts of rates and other derived or indirect measurements.
Return
a. Understand equivalent units, comparable units, and conversions.
i. Compute speed such as kilometers per hour.
ii. Compute gas consumption in miles per gallon.
iii. Calculate snow load on a roof in pounds per square foot.
iv. Compute the percentage of body fat.03. Apply the concepts
of ratios and proportions.
Return
a. Understand and use proportions, ratios, and scaling.
i. Build and use scale models.ii. Determine distance from map scale.iii. Determine the mechanical advantage
of levers or gears.iv. Calculate size limitations based on
strength of materials.v. Calculate amounts of concentrated
ingredients needed for a specified mixture.
04. Apply dimensional analysis.
Return
a. Understand units and their relationship to one another and to real world applications.
i. Check reasonableness of a calculation based on the resulting units.
ii. Convert miles per hour to seconds per mile.
05. Perform error analysis.
Return
a. Understand tolerance, precision, and their applications.
i. Explain how the error in computing the area of a rectangle depends on the errors in measuring its length and width.
ii. Calculate error introduced by uncalibrated laboratory equipment.
b. Understand that error accumulates in a computation when there is rounding at intermediate steps.
i. Determine the amount of money lost by a resaurant if portions are too large.
350. CONCEPTS AND LANGUAGE OF ALGEBRA.Rationale: Algebra is the language of mathematics and science. Through the use of variables and operations, algebra allows students to form abstract models from contextual information.
Standard - The student will:
Content Knowledge and Skills: Samples of Applications:
01. Use algebraic symbolism as a tool to represent mathematical relationships.
a. Understand and use variables, expressions, equations and inequalities.
Return
i. Represent FICA as a percentage of gross salary.
ii. Model voltage, current, and resistance in electrical circuits.
iii. Express the formula for determining the amount of concentrate needed to make a diluted solution.
a. Understand and use procedures for operating on algebraic expressions.
i. Calculate gas mileage from distance traveled and gallons used.
ii. Determine sales tax plus tip to compute the final bill in a restaurant.
iii. Determine the monthly payment on a loan given the amount borrowed, term, and interest rate.
iv. Find the value of an annuity given the payment, interest rate, and number of years.
03. Solve algebraic equations and inequalities.
Return to Solve Linear Equations
a. Understand and use appropriate procedures to solve linear equations and inequalities such as
3x - 4 = 2or
3x - 4 > 2.
i. Convert temperatures between Fahrenheit and Celsius.
ii. Determine the rate per kilowatt-hour in a utility bill given the amount charged and kWh used for two data points.
iii. Solve time-rate-distance problems.
Return to Factoring
Return to Completing the Square
Return to Quadratic Formula
b. Use appropriate procedures to simplify and solve polynomial equations and inequalities such as
x2 + 3x = 7or
x2 + 3x 7.
i. Solve braking and acceleration problems for automobiles.
ii. Find trajectories for falling objects such as baseballs or arrows.
iii. Determine how area scales in relation to side length.
04. Solve simple linear systems of equations or inequalities.
Return
a. Understand and use appropriate procedures to solve simple linear systems of equations and inequalities such as
x + y = 72x + 3y = 21
orx + y < 7
2x + 3y 21.
i. Do break-even analysis given linear supply and demand. Solve mixture problems.
ii. Use linear programming to find feasible regions for manufacturing processes.
351. CONCEPTS AND PRINCIPLES OF GEOMETRY.Rationale: The study of geometry helps students represent and make sense of the world by discovering relationships and developing spatial sense.
Standard - The student will:
Content Knowledge and Skills: Samples of Applications:
01. Apply concepts of size, shape, and spatial relationships.
Return to Congruency / Similarity
Return to Symmetry / Transformations
a. Understand congruence and similarity as they apply to reflection, rotation, and translation.
i. Determine the amount of carpet needed for an irregularly shaped room.
ii. Determine the amount of concrete needed for a foundation or driveway.
iii. Determine how to lay out the pieces when making a quilt.
Returnb. Understand scaling as it relates to
size variations in one, two, and three-dimensional objects, while shape is maintained.
i. Create scale models of buildings. ii. Create a drawing to scale of a
a. Understand the basic concepts of right triangle trigonometry (basic trigonometry ratios such as sine, cosine, and tangent).
i. Find the center of a circle using inscribed right triangles.
ii. Define the tangent, sine, and cosine ratios for an acute angle in a right triangle.
Returnb. Use trigonometric ratio methods to
solve problems.i. Determine the pitch of a roof from its
width and the length of the rafters.ii. Measure the height of a tree from the
length of its shadow using elementary trigonometry.
Returnc. Know and apply the Pythagorean
Theorem to solve real world problems.
i. Determine whether a corner is square using the “3-4-5” right triangle.
03. Apply graphing in two dimensions.
Return
a. Understand concepts of the Cartesian Coordinate System.
i. Graph linear equations and inequalities.
ii. Rotate solutions to inequalities using a graphing calculator.
iii. Represent experimental data with graphs.
iv. Use computer assisted drafting and design.
b. Understand the characteristics and uses of vectors.
i. Determine the relative velocity of boat moving upstream in a river.
ii. Determine the force necessary to prevent a barrel from rolling down an incline.
352. DATA ANALYSIS, PROBABILITY, AND STATISTICS. Rationale: With society’s expanding use of data for prediction and decision-making, it is important that students develop an understanding of the concepts and processes used in analyzing data.
Standard - The student will:
Content Knowledge and Skills: Samples of Applications:
01. Understand data analysis.
Return to Data Analysis
a. Read and interpret tables, charts, and graphs (scatter plots, line graphs, three-dimensional graphs, and pie charts).
i. Analyze and interpret bar graphs and pie charts in magazines or newspapers.
ii. Use topographical maps.iii. Choose the line of best fit from a
scatter plot of heights and weights.02. Collect, organize,
and display data.
Return
a. Collect and organize data, and display the data in tables, charts, and graphs (scatter diagrams, frequency tables, bar graphs, or pie charts).
i. Conduct a survey of product preferences.
ii. Collect data for surveying land.iii. Represent demographic data
graphically.iv. Collect and display data for a
financial audit.03. Apply simple
statistical measurements.
Return
a. Understand basic statistical concepts including mean (average), median, mode, range, and standard deviation.
i. Compute average grade in a class.ii. Explain the differences between the
a. Understand experimental and theoretical probability.
Return to Probability
i. Analyze card games, dice games, and lotteries as chance events.
Returnb. Distinguish between independent
and dependent events.i. Determine the probability of having
an accident using past accident data.
Returnc. Know that probability ranges from
0% to 100%. Understand randomness and chance.
i. Explain what the meteorologist means by “35% chance of rain tomorrow.”
05. Make predictions or decisions based on data.
Return
a. Use appropriate technology to employ simulation techniques, curve fitting, correlation, and graphical models to make predictions or decisions based on data.
i. Predict population trends using mathematical models.
ii. Predict economic trends from economic models.
Returnb. Design, conduct, and interpret results
of statistical experiments.i. Predict a team's final win/loss record
at midseason. ii. Use data from a manufacturing
process to determine whether the process is out of control.
Returnc. Analyze the effect of biased data on
statistical predictions.i. Select a non-random sample from a
student population and examine inherent bias.
353. FUNCTIONS AND MATHEMATICAL MODELS.Rationale: One of the cnetral themes of mathematics is the study of patterns, relationships, and functions. Exploring patterns helps students develop mathematical power.
Standard - The student will:
Content Knowledge and Skills: Samples of Applications:
01. Understand the concept of functions.
Return
a. Solve problems that involve varying quantities with variables, expressions, equations, inequalities, and absolute values.
i. Explain how a power bill depends on the amount of electricity used.
ii. Explain how the amount of the sales tax depends on the cost of an item.
iii. Identify domain and range in an income tax table.
02. Represent equations, inequalities and functions in a variety of formats.
Return
a. Represent a set of data in a table, a graph, and as a mathematical relationship.
i. Use function formulas with paper and pencil, program function formulas into graphing calculators, or input function formulas into spreadsheets.
ii. Display function graphs on graph paper, graphing calculators, or computer displays.
iii. Display experimental data in a table.03. Apply functions to a
variety of problems.
Return
a. Model real-world phenomena using polynomial, rational, and basic exponential functions, noting restricted domains.
i. Represent revenue as a function of items sold.
ii. Model crop yield as a function of fertilizer used.
iii. Illustrate velocity as a function of time.
iv. Model gross income as a function of years of education.
v. Investigate the growth of a population versus the birthrate.
Approved by the State Board of Education March 6, 2003
ADVANCED: Exceeds Standards Back to Top
The student demonstrates thorough knowledge and mastery of skills that allows him/her to function independently above their current educational level.
The student demonstrates a comprehensive understanding of all relevant information relevant to the topic at level.
The student demonstrates comprehension and understanding of knowledge and skills above his/her grade level.
The student can perform skills or processes independently without any significant errors.
PROFICIENT: Meets Standards Back to Top
The student demonstrates mastery of knowledge and skills that allow them to function independently on all major concepts and skills related to their educational level.
The student demonstrates a comprehensive understanding of all information relevant to the topic, at level.
The student can perform skills or processes independently without any significant errors.
The student demonstrates basic knowledge and skills usage but cannot operate independently on concepts and skills related to his/her educational level. Requires remediation and assistance to complete tasks without significant errors.
The student has an incomplete knowledge of the topic and/or misconceptions about some information.
The student requires assistance and coaching to complete tasks without errors.
BELOW BASIC: Critically Below Standards Back to Top
The student demonstrates significant lack of skills and knowledge and is unable to complete basic skills or knowledge sets without significant remediation.
The student has critical deficiencies of relevant knowledge of topic and/or misconceptions about some information.
The student cannot complete any skill set without significant assistance and coaching.
RIT Below 150-160 RIT 161-170 RIT 171-180 RIT 181-190
Algebraic Concepts
Return to Top
o € for answer to equation (151-160)
o ( ) used in equations o <, > symbols signo – used as negative
RIT 191-200 RIT 201-210 RIT 211-220 RIT 221-230
if-then, product, sumo variables, exponents
(signs / symbols)
input, output, table, associative, equation
o ,
commutative, solve, square numbers, input, output, number sequence
absolute value, evaluate, quotient
o square root symbol, %, absolute value symbol
RIT 231-240 RIT 241-250 RIT 251-260 RIT 261-300
inequality, if-then statement, term in sequence, square numbers, negative coefficient
o I = p x r x t
regression equation, varies inversely as the square, slope of parallel lines, solution to system, factor (used with equations)
parabola, intercepts, number of solutions, simultaneous equations, counterexample, solution set, sin A, cosine, tangent, discriminant, imaginary solutions, difference of two squares, read solution set from graph, factor completely
transformation, hypotenuse, edge, isosceles, scalene triangle, similar trapezoids, alternate interior angles, sum of angles in triangle
symmetrical halves, diameter, radius, angle bisector, tangent, corresponding parts of congruent triangles, Pythagorean theorem, corresponding angles, complementary angles, construction
o sign for parallel lines
slope, non-vertical and non-horizontal, midpoint, endpoint, rotational symmetry, reflected, perpendicular bisector, similar triangles, similar trapezoids, similar cylinders, conditional, counterexample, conclusion of in-then statement, interior of angle, exterior angle of regular polygon, interior angle of regular polygon, sum of angles in triangle equals 180 degrees, congruent angles, adjacent
o ” = inches, yd = yards, b = base, h = height, r = radius, s = side, angle symbol, - for negative, + for positive, d = distance, tsp = teaspoon, pt = pint, gal = gallon, qt = quart, c = cup
missing number, pentagon, simplest fraction, biggest, hundredths, one less, decimal numeral, squared, divisible, inequality, expanded numeral, equivalent, point, standard numeral, sequence
o + used as positive symbol, - used as negative symbol
counting number, portion, fractional part, lowest terms, reciprocal, order, scientific notation, prime factorization, not prime, factor tree, common multiples, greatest common denominator, equivalent, nearest dollar
o square root symbol
always true, proportion, multiply, between, tens digit, standard numeral, ratio
RIT 231-240 RIT 241-250 RIT 251-260 RIT 261-300
Base ten, prime factors infinite non-repeating decimal (261-270)
New Vocabulary (l) and Symbols (¢)ISAT “Sub-Goal” RIT Below 150-160 RIT 161-170 RIT 171-180 RIT 181-190
Reasoning and Problem Solving
Return to Top
how many, not, in all, altogether
product, number sentence, problem, how many more, multiples, greater than, left, how much, money, answer, about, outside, inside, shape, calendar, picture
add, subtract, multiply, divide, rate, kilometers, hours, between, equals, cost, time, less than, closest to, equation, weigh, minutes, pieces, dozen, cups, inches, pounds, miles, sum, dime, nickel, pennies, quarter
o tally marks, =, - as an integer sign, %, °C, € for answer
equation, product, increased, number line, information not needed, division, estimate, odd, prime, cube, pattern, geometric patterns, extra information
Use the correct order of operations to evaluate numeric and algebraic expressions
Simplify and evaluate expressions that include positive and negative integral components
Simplify polynomials by combining like terms
Use the rules of exponents to multiply and divide monomials
Solve simple linear equations and inequalities over the rational numbers
Create a table of (x, y) values for the given linear equation and graph the function
Students solve equations and inequalities involving absolute values
Solve a system of two linear equations in two variables algebraically and interpret the answer graphically
Graph a linear function in two variables using the slope-intercept method and identify intercepts
Solve a system of two linear inequalities in two variables and identify the solution set
Understand the concepts of parallel lines and perpendicular lines and how those slopes are related
Add, subtract, multiply, and divide monomials and polynomials
Apply basic factoring techniques to second- and simple third-degree polynomials, including finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials
Add, subtract, multiply, and divide rational expressions and functions
Find the difference of two squares
Simplify embedded expressions before solving linear equations and inequalities in one variable
Solve problems that use variables in expressions describing geometric quantities by solving for one variable
Solve equations with variables as exponents
Use the rules of exponents to multiply polynomials by monomials
Derive linear equations by using the point-slope function
Find the slope given two points on the line of a given graph
Write the equation of a line when given the graph of the line, two points on the line, or the slope of the line and a point on the line
Simplify monomials containing integer powers and roots
Find the solution set for inequalities that include absolute values
Identify the equation of a parabola Simplify expressions containing cube
roots Solve expressions containing factorials Find the number of possible solutions
for a system of equations Cube a binomial
Patterns, Sequences, Functions
Return to Guide
Using whole numbers, complete a function table based on a given rule
Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is constant
Students analyze a given set of data for the existence of a pattern and represent the pattern algebraically and graphically
Determine whether a relation is defined by a graph, a set of
ISAT “Sub-Goal” RIT 231-240 RIT 241-250 RIT 251-260number or a fraction
Multiply three factors using fractions, mixed numbers, and whole numbers in any combination, answer in lowest terms Divide a whole number, fraction, or
mixed number by a mixed numberIntegers
Return to Guide
Add integers with unlike signs Add several integers Multiply integers with like signs Divide integers with like signs
Subtract integers, subtracting a negative
Use the additive inverse property to solve problems Use order of operations rules
Logarithms
Return to Guide
Solve logarithmic functions
Matrices
Return to Guide
Multiplication of matrices
Percents
Return to Guide
Find the percent one number is of another (8 is what % of 20?)
Find a number from a percent (20 is 45% of what number?)
Roots
Return to Guide
Simplify an expression containing imaginary roots
Whole Numbers
Return to Guide
Divide a 2-digit number or a 3-digit number by a 1-digit number with remainder
Divisibility Rules: return to curriculum guide2: A number is divisible by two if the ones digit is an even number. (0,2,4,6,8)3: A number is divisible by three if the sum of the digits is divisible by three.
( 936: 9+3+6 = 18, 1+8 = 9 which is divisible by three so 936 is divisible by 3)4: A number is divisible by four if the last two digits are divisible by 4.
(19,824: 24 is divisible by four so 19,824 is divisible by 4)5: A number is divisible by five if the ones digit is a 0 or 5.6: A number is divisible by 6 if it is divisible by BOTH two and three.7: To test if a number is divisible by 7: take the ones digit and double it, then subtract this result from the original number with the ones digit crossed out. Repeat the process until left with a one or two digit number. If that number is divisible by seven, then the original problem is divisible by 7.
Original number: 66,3746637 (original number without ones digit)- 8 (result of doubling the original ones digit)
6629 (repeat process with this number)
662 (number without ones digit) -18 (ones digit times two) 644 (repeat process with this number)
64 (number without ones digit) -8 (ones digit times two) 56 Since this number is divisible by 7, so is the original number 66374.
8: A number is divisible by eight if the last three digits are divisible by eight.(335,696: 696 is divisible by 8 so 335,696 is also divisible by eight)
9: A number is divisible by 9 if the sum of the digits is divisible by nine.(231,147: 2+3+1+1+4+7 = 18 is divisible by nine so 231,147 is divisible by 9)
10: A number is divisible by 10 if the ones digit is a zero.11: To test if a number is divisible by 11: add every other digit, go back and add remaining digits, subtract the smaller result from the bigger result. If the answer is divisible by 11, so is the original problem.
Estimation and Computation with Real World Situations: return to curriculum guide
Put a list of items and their costs on the board or on a worksheet. Then give students problems that list different combinations of the items and tell them how much money they have to spend. Ask them to estimate if they have enough money to cover the cost of the items on the list. This would be a good warm-up activity for the first few minutes of class.
List of items:(would be better to have actual pictures with the prices next to them…)pair of socks: $5.25three-ring binder: $6.18pack of bubblegum: $.88ball-point pen: $.45package of notebook paper: $1.29
Questions:1. You have $20. You want to buy two pairs of socks, a three-ring binder, and a pack of gum. Do you have enough money?
2. You have $30. You want to buy three pens, a package of notebook paper, two binders and four packs of gum. Do you have enough money?
Etc.
Evaluate and name the property used in each step: return to curriculum guide
Function Problems and Questions: Return to curriculum guide
1) When students are reviewing the definition of what a function is, ask:Can the domain values repeat?Can the range values repeat?
2) Ask students to recognize which chart represents an inverse variation.
Example:
A 10 15
B 3 2
This chart shows an inverse variation and students can easily recognize this if they understand that they should get the same answer when they multiply the two numbers in each column. (10)(3) = 30 and (15)(2) = 30
A 2 4
B 1 8
This second chart does not show an inverse variation and students can check quickly by multiplying the numbers in each column. (2)(1) = 2 and (4)(8) = 32
3) Give students some information about a function in chart form and have them write an algebraic equation for the function.
Negative Exponent Problem: return to curriculum guide
1. a-2 b3 c- 4 d-1
________________
a3 b5 c-5 d-7
b3 c5 d7
________________ ** step to make all exponents positive
a3 a2 b5 c4 d1
c d6
_______ ** step to combine exponents
a5 b2
Order of Operations: return to curriculum guide
1) Simplify the expressions inside grouping symbols, such as parentheses, brackets, and braces, and as indicated by fraction bars. (With more than one set of grouping symbols, work inside to outside).2) Evaluate all powers.3) Do all multiplications and divisions from left to right.4) Do all additions and subtractions from left to right.
Patterns and Sequences Sample Problems: (From Algebra I, Glencoe, pages 23, 104, 110)return to curriculum guide
1.
X 1 10 50 100
a) Copy and complete the chart above.b) Describe the pattern that is emerging.c) What would the approximate value of the expression be if x were a very large number?Solution:a)
X 1 10 50 1000
b) As the value of x gets larger, the value of the expression approaches 1.c) 1
2. An arithmetic sequence is a sequence in which the difference between any two consecutive terms is the same.a) Find the next three terms in the arithmetic sequence 4.53, 5.65, 6.77, 7.89…b) What is the common difference?c) Write the first five terms of an arithmetic sequence in which the first term is -2 and the common
difference is . Find the sum of the first five terms.
Solution:a) 9.01,10.13,11.25b) 1.12
c) -2, , , , 1 Sum: -2
3. A geometric sequence is a sequence in which the ratio of any term divided by the term before it is the same for any two terms.
a) Write the next three terms in the geometric sequence 9, 3, 1, , …
b) What is the common ratio?c) Write the first five terms in a geometric sequence in which the first term is -6 and the common ratio is 0.5. Then find the sum of the five terms.Solution:
1. In a certain mythical community, politicians always lie, and nonpoliticians always tell the truth. A stranger meets three natives, and asks the first of them if he is a politician. The first native answers the question. The second native then reports that the first native denied being a politician. Then the third native asserts that the first native is really a politician. How many of these three natives are politicians?
2. The members of a small loan company are Mr. Black, Mr. White, Mrs. Coffee, Miss Ambroce, Mr. Kelly, and Miss Earnshaw. The positions they occupy are manager, assistant manager, cashier, stenographer, teller, and clerk, though not necessarily in that order. The assistant manager is the manager’s grandson; the cashier is the stenographer’s son-in-law; Mr. Black is a bachelor. Mr. White is 22 years old; Miss Ambroce is the teller’s step-sister; and Mr. Kelly is the manager’s neighbor. Who holds each position?
3. Of three prisoners in a certain jail, one had normal vision, the second had only one eye, and the third was totally blind. All were of at least average intelligence. The jailer told the prisoners that from three white hats and two red hats he would select three and put them on the prisoners’ heads. Each was prevented from seeing what color hat was placed on his own head. They were brought together and the jailer offered freedom to the prisoner with normal vision if he could tell what color hat was on his head. The prisoner confessed that he couldn’t tell. Next, the jailer offered freedom to the prisoner with only one eye if he could tell what color hat was on his head. The second prisoner confessed that he couldn’t tell. The jailer did not bother making the offer to the blind prisoner, but agreed to extend the same terms to him when he made the request. The blind prisoner then smiled broadly and said:
“I do not need to have my sight;From what my friends with eyes have said,I clearly see my hat is _______!”
Scientific Notation Problems: return to curriculum guide
1. Is .8975 X 10 5 in scientific notation? No, the number in front is too small. What has to be changed to make it scientific notation?
Solution: 8.975 X 104
(If first number has to be made bigger, then exponent will be made smaller)
2. Is 546 X 103 in scientific notation? No, the number is not between one and ten. What has to be changed to make it scientific notation?
Solution: 5.46 X 10 5
(If first number has to be made smaller, the exponent will become bigger)
Logic Number Problem (taken from book of same title, Dale Seymour Publications) Return to curriculum guide
Find the number that works for all of the following clues:1) It is a three-digit whole number.2) It is divisible by 5.3) It is an even number.4) Each of its digits is different.5) Its tens digit is greater than its ones digit.6) Its hundreds digit is greater than its tens digit.7) It is less than 400.8) It is divisible by 3.9) It has only one odd digit10) Its tens digit is 1.
** Answer: 210**Note: this is a really easy one. They get harder towards the end of the book.
Just For Fun Definitions (Geometry) **don’t know the source** return to curriculum guide
Match the following geometrical terms and their “Just For Fun” definitions.1. a broken angle a. line2. place where people are sent b. pi for committing crimes c. chord3. a beast d. prism4. a clever angle e. tangent5. an angle that is never wrong f. geometry6. used to tie up packages g. acute angle7. what girls want to find at the beach h. polygon8. They voted “yes” on tractors for Cuba. i. rectangle
9. printer’s dessert j. protractor10. a sharp weapon k. coincide11. what little acorns say l. right angle when they grow up m. inverse12. the one in charge n. sphere13. what a person should do o. ruler when in rains14. a dead parrot15. the way the poet wrote his love letters
Vectors return to curriculum guide
A boat is traveling north at 20 miles per hour. A wind from the west is blowing the boat eastward at 5 miles per hour. Find the speed of the boat and the direction in which it is moving.
Solution:
Magnitude: Use the Pythagorean Theorem to find the length of the hypotenuse. (20.62 mi/hour)
Direction: Use Inverse Tangent of 5/20 to find the measure of 76 degrees.
Ratio Problems with Unit Conversion return to curriculum guide
Express both measurements in the same unit. Then give their ratio as a fraction in simplest form.1. 20 sec to 2 min2. 3 h to 40 min3. 75 cents to 2 dollars4. 60 cm to 4 m5. 1 kg to 250 g
Real World Problem Involving Units return to curriculum guide
At 7:00 A.M., Brooke leaves home to go on a business trip driving 35 miles per hour. Fifteen minutes later, Bart discovers that Brooke forgot her presentation materials. He drives 50 miles per hour to catch up with her. If Bart is delayed 30 minutes with a flat tire, when will he catch up with Brooke?*** Remind students that time needs to be in terms of HOURS so they will have to change minutes to hours.
Percent Story Problems return to curriculum guide
1. After taking his girlfriend out to dinner, Trent decides to leave a tip of 15% of the bill. If the bill is $38, how much should the tip be?2. Mandy got 10 out of 16 problems on an algebra test correct. What percent were correct?3. A team won 13 games, lost 15 games, and tied 2 games. What percent of its games did the team win?4. In a magazine drive a school keeps 40% of all sales dollars. How many dollars worth of magazines must be sold for the school to earn $5000?5. Dustin made 54 out of 80 free throws. What percent did he miss?
Integer Story Problems return to curriculum guide
1. The Eagles football team made the following gains on four plays: 14 yards, -32 yards, 3 yards, and -19 yards. What was the net change in position of the Eagles as a result of the four plays?
2. Jake had a balance of $345.00 in his checking account. During the week he wrote checks for $65.00, $24.00, and $118.00. He then made a deposit of $56.00. What was his balance after the deposit?
3. At its first stop, a bus picked up 17 people. At the next stop, 12 people got on and 7 got off. At the third stop, 21 people got on and 13 got off. At the fourth stop, 5 people got on and 18 got off. How many passengers were then on the bus?
Algebraic Expression Problems return to curriculum guide
Translate each phrase into an algebraic expression.
1. 3 times a number2. 3 more than a number3. 3 decreased by a number4. 3 less than a number5. one third of a number6. 8 more than 3 times a number7. 8 less than 3 times a number
Story Problems Involving the Quadratic Formula return to curriculum guide
1. The diagonal measure of a movie screen is 6 m. The length of the screen is 2 m greater than the height. Find the dimensions of the screen.
2. A rectangular pond measures 3 m by 5 m. A concrete walk of uniform width is constructed around the pond. If the walk and pond together cover an area of 39 square meters, how wide is the walk?
3. A flower garden is in the shape of a right triangle. The longest side of the triangle measures 13 m. One of the shorter sides is 7 m longer than the other. Find the length of the shortest side.
Angles of Elevation and Depression return to curriculum guide
1. A surveyor is 130 feet from a tower. The tower is 86 feet high. The surveyor’s instrument is 4.75 feet above the ground. Find the angle of elevation.
2. A plane P is 3 miles above the ground. The pilot sights the airport A at an angle of depression of 15 degrees. He sights his house H at a angle of depression of 32 degrees. What is the ground distance d between the pilot’s house and the airport?
Systems of Inequalities return to curriculum guide
A pottery shop makes 2 kinds of birdbaths: glazed and unglazed. The glazed birdbath takes 1 hour on the potter’s wheel and 6 hours in the kiln. The unglazed birdbaths take ½ hour on the potter’s wheel and 1 hour in the kiln. The potter’s wheel is available for at most 8 hours a day and the kiln is available for at most 20 hours a day. The shop’s profit on each unglazed birdbath is $10 while the profit per glazed birdbath is $15. How many of each kind of birdbath should be produced in order to maximize the profit? (see solution below)
Solution:x = # of glazed y = # unglazed
Profit: P = 15x + 10y (this is what you want to maximize)
Constraints: 1x + .5y ≤ 8 (for the potter’s wheel)6x + 1y ≤ 20 (for the kiln)x ≥0 y ≥ 0 (can’t have a negative number of birdbaths)
Final answer: Graph all 4 inequalities and find the shaded region. The vertices of this region are where the profit will be minimized or maximized. Take each ordered pair of the vertices and plug them into the profit statement (P = 15x + 10y) to see which ordered pair gives you the maximum value for P. This tells you how many glazed and unglazed birdbaths you should make to maximize your profit. Answer: 0 glazed and 16 unglazed for a profit of $160.
Problem Solving – Every Day Situations return to curriculum guide
1. Two cyclists begin traveling in the same direction on the same bike path. One travels at 20 miles per hour, and the other at 14 miles per hour. After how many hours will they be 15 miles apart?
2. The Nut House sells walnuts for $4.00 a pound and cashews for $7.00 a pound. How many pounds of cashews should be mixed with 10 pounds of walnuts to obtain a mixture that sells for $5.50 a pound?
3. Melanie invested $5000 for one year, part at 9% annual interest and the rest at 12% annual interest. The interest from the investment at 9% was $198 more than the interest from the investment at 12%. How much money did she invest at 9%?
(taken from The University of Chicago School Mathematics Project book Functions, Statistics and Trigonometry)
Work with a partner on this problem. You will need three playing cards, two of one color and a third of a different color.
Monty Hall was the host of the TV game show called Let’s Make a Deal! Contestants were told that behind each of three doors was a prize, not all of which were considered desirable. The contestant was asked to pick a door, and then Monty Hall would offer money to the contestant, who could take it or choose what was behind the door, sight unseen.
Here is a variation of that game. Suppose that behind two of the doors is a goat and behind the third door is a new car. You are the contestant and you pick one door, say Door Number 2. The host, who knows what is behind each door, then opens one of the other two doors and shows you a goat. The host now asks you,” Do you want to stay with the door you chose or do you wish to choose the other unopened door?” The problem is to determine whether it is to your advantage to switch.
1) Decide who will play the role of the host, and who will play the contestant. Let the two cards of the same color represent the goats, and the card with the unique color represent the car.2) The host should shuffle the cards, place them face down without the other person’s watching, and remember which card represents the car.3) The contestant will then choose one of the cards. The host will not turn over that card, but the host will turn over a remaining card which represents a goat.4) The contestant will then decide to keep the first choice or to switch. Then the host will turn over the final choice to see if the contestant won the game (got the car) or lost the game (got a goat). 5) Play the game (that is, repeat steps 2-4) 15 times. Keep a tally on a chart. Then reverse roles and play the game 15 more times, and record your results on the same tally chart.6) On the basis of your data, what should the strategy of the contestant be?7) Under the direction of the teacher, incorporate your data with that of the rest of the class. On the basis of the class data, what should be the strategy of the contestant?
Here is the theoretical solution to the problem. The probability of winning if you do not switch is 1/3. Your probability of winning if you switch is 2/3.
Rationale: Suppose Door Number 1 has the car, Doors 2 and 3 have goats. If you pick Door 1, the host will show you door 2 or 3. Either way, if you switch, you will lose. If you pick Door 2, the host will show you door 3. If you switch, you win. If you pick Door 3, the host will show you Door 2, and if you switch, you win. Hence, when you switch, you win 2 out of 3 times.
Decimal Estimation Sample Problems return to curriculum guide
1. A U.S. penny weighs 0.1 oz. The smallest hummingbird on record was 2.24 inches long and weighed 0.056 oz. How much less than a penny did the hummingbird weigh?
2.A team of 4 students is running in a 400-meter relay race. The first 3 runners had times of 14.7 s, 16.2 s, and 15.3 s. What time must the 4th runner beat in order for the total to be less than 60 seconds?
3. A pack of construction paper is 0.5 cm thick. If there are 20 sheets of paper in the pack, how thick is each sheet?
4. A window is made using 2 panes of glass separated by an insulating air space. The glass is 0.25 cm thick, and the separation between panes is also 0.25 cm. How thick is the window?
Fraction Problems return to curriculum guide
1. Julia studied math for 3 1/3 hours during the 4 days before her last math test. What was the average amount of time she studied each day?2. Sean used ¾ cup of sugar to make a dozen brownies. How much sugar is in each brownie?3. A store sells trail mix in 10-ounce packages. Half the weight is peanuts. There are also 2 oz of almonds, 1 oz of cashews, and 3 oz of raisins. The rest is chocolate chips. What fraction of the mix is chocolate chips?
4. A backpacking club can average 2 ½ miles per hour. At that rate, how long will it take for a hike of 8 ¾ miles?
Instructions for PIG (Probability) return to curriculum guide
Have students put the letters P-I-G across the top of their paper. Each letter represents a different round of the game. During each round, two dice will be rolled. Students can accumulate points by choosing to stand up while the dice are being rolled. If a 4 and 6 are rolled, the students would record a 10 under the first column on their paper. In between each roll, students can choose to stay standing or sit down. If they stay standing, they can add to their total as long as a one is NOT rolled. As soon as a one is rolled, everyone who is standing loses all of their points for that round and the round is over. The students who sat down before a one was rolled get to keep their points. The same procedure is followed for each of the three rounds. The only other rule is that if a double one is rolled, students lose all of their points for ALL ROUNDS if they are standing up instead of just the current round.
After playing, have students figure out the probability of a one being rolled. Of a double one.
Graph Problems return to curriculum guide
1. Lisa is concerned about our natural resources. She has made a study about the use of water in the United States. Use the information below to make a circle graph that would help Lisa illustrate the use of water to her classmates. Water Use in the U.S. : Agriculture 36%, Public Water 8%, Utilities 33%, Industry 23%.
2. In a sample of fish caught in a net, 112 were cod, 259 were pollack, 45 were halibut, and 6 were sharks. Draw a graph to show the amount of each fish in the catch.
Capacity and Weight Problems return to curriculum guide
1. A large can of frozen orange juice contains 354 ml. To make orange juice, you add 3 full cans of water. How many liters of orange juice does this make?
2. ABC Corporation is mailing a report to stockholders. The report includes a cover and 100 sheets of paper. If the cover weighs 0.2 kg and each sheet of paper weighs 5 g, how much does the report weigh?
3. A gasoline can was filled with 17.5 L of gasoline. A lawnmower fuel tank that holds 1.4 L has been filled 8 times from the gasoline can. How much gasoline is left in the can?
Add or subtract. Simplify if possible.1. 2 h 30 min + 3 h 45 min2. 5 min 40 s + 8 min 50 s3. 8 min 10 s – 3 min 30 s4. 12 h 25 min - 5 h 45 min5. 9 min - 5 min 32 s
Thinking With Mathematical Models return to curriculum guide(Taken from Thinking With Mathematical Models Dale Seymour Publications)The astronomy club at King Middle School is planning a field trip to the science center to see a new 3-D science film. Renting a bus for the trip will cost $125. Admission to the film is $2.50 per person.
a. Write an equation to show the relationship between the number of students who go on the tip, n, and the cost of the trip, c.
b. Make a table showing the cost of the trip for 0 students, 5 students, 10 students, and so on, up to 50 students.
c. Make a graph of the equation on grid paper or by using a graphing calculator.d. What do the slope and the y-intercept of the graph tell you about the cost and the number of
Table and Graph Problem return to curriculum guide
(Taken from Thinking With Mathematical Models Dale Seymour Publications)
Mandy is saving money to pay for the first year of her car insurance, which will cost about $1200. She already has $500 in her savings account. She gets a job working at a fast-food restaurant and plans to save $45 per week.
a. Make a table to show the number of weeks and how much money Mandy will have in her account after each week. For example, on Week 1, she will have $545. Graph the results.
b. When will she have enough money to pay her insurance premium?
Testing Paper Bridges (Predictions) return to curriculum guide
Materials needed: several 11-inch-by-4-inch strips of paper, two books of the same thickness, a small paper cup, and about 50 pennies.
Directions:Make a paper “bridge” by folding up 1 inch on each long side of one of the paper strips. Suspend the bridge between the two books. The bridge should overlap each book by about 1 inch. Place the paper cup in the center of the bridge. Put pennies into the cup, one at a time, until the bridge crumples. Record the number of pennies you added to the cup. This number is the breaking weight of the bridge. Put two strips together to make a bridge of double thickness. Find the breaking weight for this bridge. Repeat this experiment to find breaking weights for bridges made from three, four, and five strips of paper.
Problems:1. Do the experiment described above to find breaking weights for bridges 1,2,3,4, and 5 layers
thick.2. Make a table and a graph of your data.3. Describe the pattern of change in the data. Then, use the pattern to predict the breaking
weights for bridges 6 and 7 layers thick.4. Suppose you could use half-layers of paper to build the bridges. What breaking weights
would you predict for bridges 2.5 layers thick and 3.5 layers thick?
Probability and Odds return to curriculum guide
1. If the probability that an event will occur is 2/3, what are the odds that the event will occur?
2. If the odds that an event does not occur are 9:14, what is the probability that the event does occur?
3. A shipment of 100 clocks was just received by The Clocke Shoppe. There is a 4% probability that one of the clocks was damaged during shipment, even though the package does not give any indication that the clock is damaged. If Alison buys one of these clocks, what are the odds that she is buying a damaged clock?
4. A reception is scheduled for an opening area of Spring Grove Park, and a tent has been ordered in case of rain. Weather reports indicate a 50% chance of rain on the day of the reception. Whether or not it rains, there is a 50% chance the tent will not arrive in time for the reception due to traffic problems. What is the probability the people at the reception will get wet?
Communication Mathematical Information return to curriculum guide(Taken from Thinking With Mathematical Models Dale Seymour Publications)
Suppose you and some friends are planning a 20-mile bike trip to a park campsite.a) At what average speed would you have to ride to complete the trip in 4 hours? In two hours? In 1.5
hours? In 1 hour?b) Write an equation for the relationship between riding time, t, in hours and average speed, s, in miles
per hour for a 20-mile trip.c) Is the relationship between t and s linear? Use a table and a graph to support your answer.d) Which of the following causes the greater change in required average speed: increasing the riding
time from 1 hour to 2 hours or increasing the riding time from 2 hours to 3 hours? Explain.
Sketching Functions return to curriculum guide
Make a sketch for each function described below. Use your knowledge of the relationships described.1. At a fixed price per ounce, the cost of buying gold is a function of the number of ounces you
buy.2. The height of your head above the ground as you ride a Ferris wheel is a function of the time
since you got on.3. The total cost of operating a lemonade stand is a function of the amount of lemonade sold.4. The amount of water in a pan on a burner that is turned on “high” is a function of the time since
the burner was turned on.5. The height of a ball that is dropped from a height of 10 feet is a function of the time since it was
Real World Problems – Functions return to curriculum guide
(Taken from Thinking With Mathematical Models Dale Seymour Publications)
Spartan Publishing Company’s delivery truck has a broken fuel gauge. Luckily, the driver always keeps track of mileage and gas consumption. She uses her data to write the equation G = 25 – (1/15)M for the relationship between the number of gallons of gasoline in the tank, G, and the number of miles driven since the last fill-up, M.
a) The driver has just filled the tank and is about to start the engine. What is the value of M for this situation? Use the equation to figure out the value of G for this situation. What does the result tell you about the gas tank?
b) If the driver travels 50 miles after filling the tank, how much gas will be left?c) After filling the tank, how many miles can the driver travel before 5 gallons remain?d) Solve the equation 10 = 25 –(1/15)M. What does the solution tell you about the amount of gas left
and then number of miles driven since the last fill-up?e) Use the equation to figure out how many miles the driver would have to travel to use 1 gallon of
gas. Explain how you got your answer.f) In the equation G = 25 – (1/15)M, what do the numbers 25 and 1/15 tell you about the situation?g) What patterns would you expect to see in the table and the graph of the relationship between miles
driven and gallons of gas remaining? How would the numbers 25 and 1/15 from the equation be represented in the table and the graph?
Consecutive Integer Problems return to curriculum guide
1. Find two consecutive integers whose sum is 45.2. Find two consecutive even integers whose sum is 66.3. Find three consecutive odd integers such that the sum of the smallest and 4 time the largest is 61.
