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- 1. Cesar Chavez Academy 6th Grade Math Matrix (Tri 1) PLC Skills Academic Vocabulary Decimals - Place Value (100ths) Addition: - Rounding sum, increase, more than, plus, total, - Comparing/Ordering altogether, gain, deposit - Key Operation Words - Adding/Subtracting Subtraction: - Multiplying/Dividing difference, decrease, less than, minus, loss, withdraw Multiplication: double, product, times, triple, twice, of Division: each, quotient, divisible Order of - Exponents base, power Operations - Order of Operations Integers - # line absolute value, opposite, # line, integer, - compare compare, order - order - absolute value - opposites - adding - subtracting - multiplying - dividing Algebraic - Expressions w/ phrases variable, algebraic expression, evaluate, Expressions - Expressions w/ word problems substitution, constant, coefﬁcient - Evaluating Expressions Equations - Equations w/ word problems variable, inverse operation, coefﬁcient, - Solving Equations (single step w/ constant, expression, isolate equations) + integers Triangles & - classify triangles Triangle Sides: Quadrilaterals - solve for missing angle scalene, isosceles, equilateral - classify quads - solve for missing angle (interior angle Triangle Angles: bisected) acute, obtuse, right, corresponding - congruency marks Quads: rhombus, square, rectangle, trapezoid, rectangle© 2012 Greg Kozak & Megan Murphy
- 2. PLC Skills Academic Vocabulary Angle Pairs - angle pairs vertical, complementary, supplementary, (Missing Angles) - multi-step missing angles exterior & interior angles, straight & right angle© 2012 Greg Kozak & Megan Murphy
- 3. Place ValueVideos: Place Value 1, Place Value 2Practice: Place Value, Place Value w/ DecimalsThe value given to the space a digit holds because of its place in a numeral. These values arenamed according to the spot each takes up.Example:RoundingVideos: Rounding Whole #’s 1, Rounding Whole #’s 2, Rounding DecimalsPractice: Rounding NumbersRewriting a number as its nearest multiple of 10, 100, 1,000, and so on.If the circled number to theright of the place you are rounding to is 0–4, the number stays the same. If it is 5–9, the numberrounds up to the next greater number. The rest of the numbers to the right of the original numberbecome zeros.Example 1:Round 3,294 to the nearest thousand.Look at the number in the place you are rounding; underline it and circle the number right after it.3,294The circled number to the right of the 3 is 2, so the 3 will stay the same. The rest of the numbers willchange to zeros. Therefore, 3,294 rounds to 3,000 when rounding to the nearest thousand.© 2012 Greg Kozak & Megan Murphy
- 4. Example 2:Round to the nearest 10 566 –––> 570 562 –––> 560 3,049 –––> 3,050Round to the nearest 100 566 –––> 600 536 –––> 500 Examples: 76,824 –––> 76,800Round to the nearest 10,000 1) .7 = 7 tenths 2) 3.5 = 3 and 5 tenths 64,340 –––> 60,000 3) .78 = 78 hundredths 868,473 –––> 870,000 4) 9.34 = 9 and 34 hundredths 406,299 –––> 410,000 5) .983 = 983 thousandths 6) 23.802 = 23 and 802 thousandthsComparing DecimalsVideos: Comparing Decimals, Using a Number Line to Compare DecimalsTo determine which decimal is larger or smaller, place value must be compared. 1. Line up the decimal points. 2. Compare tenths, then hundredths, and then thousandths.1 Line up decimal points 0.41 0.2752 Look at place value: 4 tenths is larger than 2 tenths, so 0.41 is larger than 0.275Key Operation WordsAddition: sum, increase, more than, plus, total, altogether, gain, depositSubtraction: difference, decrease, less than, minus, loss, withdrawMultiplication: double, product, times, triple, twice, ofDivision: each, quotient, divisible© 2012 Greg Kozak & Megan Murphy
- 5. Adding DecimalsVideos: Adding Decimals 1, Adding Decimals 2Practice: Adding Decimals 1. Line up the decimal points so that the place value is in line. 2. Change all whole numbers into decimals (7 = 7.00, 23 = 23.00) 3. Add as normal. Zeros can be added after the decimal point in order to make adding easier, and it won’t change the answer. 4. Bring the decimal point straight down into the answer.Examples:.75 + 2 + 1.674 = 3.5 + .46 = .4 + 3 + .27 = .750 3.50 .40 2.000 + .46 3.00 + 1.674 3.96 + .27 4.424 3.67Subtracting DecimalsVideos: Subtracting Decimals 1, Subtracting Decimals 2Practice: Subtracting Decimals, Adding & Subtracting Decimals with Word Problems 1. Line up the decimal points so that the place value is in line. 2. Change all whole numbers into decimals (7 = 7.00, 23 = 23.00) 3. Subtract as normal. Zeros can be added after the decimal point in order to make subtracting easier, and it won’t change the answer. 4. Bring the decimal point straight down into the answer.© 2012 Greg Kozak & Megan Murphy
- 6. Multiplying DecimalsVideos: Multiplying Decimals 1, Multiplying Decimals 2Practice: Multiplying Decimals 1. Write and solve the problem as normal. 2. Count the total number of places after each decimal in the numbers being multiplied. 3. Position the decimal the same number of places in the answer.Dividing DecimalsVideos: Dividing Decimals 1, Dividing Decimals 2, Dividing Decimals 3, Dividing Decimals 4Practice: Dividing DecimalsDivision Terms:1) Rational Number: Any number that can be written as a ratio, decimal, and fraction (Ex: 3, 3.00, or 3/1).2) Terminating Number: A number that stops dividing after a number of decimal points. It does not go on forever (Ex: .34, 3.45)3) Irrational Number: A number that cannot be written as a ratio, decimal, or fraction since it goes on forever and never terminates (Ex: π = 3.1415926535897932384626433832795 (and more...)4) Non Terminating Number: A number that will not terminate or stop dividing and go on forever.5) Quotient: The answer to a division problem6) Dividend & Divisor: The dividend is the number that is to be divided while the divisor is the number the dividend is being divided by.© 2012 Greg Kozak & Megan Murphy
- 7. Decimal in the dividend only: 1. Write the problem as normal. 2. Bring the decimal point straight up. 3. Divide as normal. 4. No remainders. You must add a zero in the dividend, drop it, and continue dividing.Example:Decimal in the dividend and divisor:1. Move the decimal in the divisor all the way to the right.2. Move the decimal in the dividend the same number of times as was done in the divisor.3. Bring the decimal in the dividend straight up.4. Divided as normal.5. No remainders. You must add a zero in the dividend, drop it, and continue dividing.Example:© 2012 Greg Kozak & Megan Murphy
- 8. ExponentVideos: Exponents 1, Exponents 2Practice: ExponentsA small symbol placed above and to the right of a number or letter that shows how many times thebase is to be multiplied by itself. Example: 64 = 6 x 6 x 6 x 6, so 64 = 1,296 b3 = b x b x b 57 = 5 x 5 x 5 x 5 x 5 x 5 x 5, so 57 = 78,125 72 = 7 x 7, so 72 = 49Order of OperationsVideos: Operations 1, Operations 2, Operations 3, Operations 4Practice: Order of OperationsA set of rules agreed upon by mathematicians that outline the steps to take when solving multi-operational problems. These rules help reduce confusion when solving problems and ensure thatthe same answer can be reached every time.Rules:1) Complete the work in parentheses (or brackets) first.2) Simplify exponents.3) Multiply or divide from left to right.4) Add or subtract from left to right.Example:62 + 15 x (1 + 3) ÷ 5 =62 + 15 x 4 ÷ 5 =36 + 15 x 4 ÷ 5 =36 + 60 ÷ 5 =36 + 12 = 48© 2012 Greg Kozak & Megan Murphy
- 9. Sometimes problems may not have all of the components listed above (parentheses, exponents,multiplication/division, and addition/ subtraction). If so, continue down the list of steps until youreach a step that fits the problem.Example: (There are no parenthesis or exponents, so start on rule 3)9–3+2x6=9 – 3 + 12 =6 + 12 = 18Number LineVideos: Number Line 1, Decimals on a Number LinePractice: Number Line 1, Number Line 2, Number Line 3, Decimals on a Number Line 1, Decimals ona number line 2© 2012 Greg Kozak & Megan Murphy
- 10. Comparing Integers (Integers are counting numbers, their opposites, and zero)Videos: Comparing Integers 11) Create a number line2) Negative integers are always smaller than positive integers3) Puts dots on the number line for each integer4) The integer that is on the left is always the smaller integer of the set of integersAbsolute ValueVideos: Absolute Value 1, Absolute Value 2, Absolute Value 3, Absolute Value 4Practice: Finding Absolute Value, Comparing Absolute ValuesThe absolute value of a number is its distance from zero.Example:|+4| = 4 and |-4| = 4© 2012 Greg Kozak & Megan Murphy
- 11. OppositesOpposites are numbers that are the same distance from zero. Therefore, -6 and 6 are opposites.Example:-(6) means the opposite of 6 which is -66) means the opposite of -6 which is 6Adding IntegersVideos: Adding Integers w/ Same Signs, Adding Integers w/ Diff Signs 1, Adding Integers w/ DiffSigns 2Practice: Adding & Subtracting IntegersWhen the signs are the same:1) Positive + Positive = Positive (+5) + (+10) = (+15)2) Negative + Negative = Negative (-3) + (-4) = (-7)When the signs are different:If the signs are different (one is positive and one negative) you must subtract. The larger numbergoes on top and use the sign of the larger number in your answer.(+9) + (-12) = (-3)(-10) + (+4) = (+6)© 2012 Greg Kozak & Megan Murphy
- 12. Using a number line to add integers:1) If the second integer is positive then you move right on the number line2) If the second integer is negative then you move to the left on the number lineSubtracting Integers (Add the second integer’s opposite)Videos: Subtracting IntegersPractice: Adding & Subtracting Integers Subtracting Integers 1) Change the subtraction sign into a plus sign 2) Change the sign of the 2nd integer 3) use addition rules to add Example: (-10) - (+3) = (-10) + (-3) = -13 Example: (+7) – (+2) = (+7) + (-2) = +5 Area of a rectangle or square Area = Length x Width (A=lw) (area is always squared)©Example: Kozak & Megan Murphy 2012 Greg
- 13. larger #. Example: (-10) + (+3) = -7Multiplying Integers Example: (+15) + (-6) = +9Videos: Multiplying Integers 1 Area of a rectangle or square Multiply & Dividing IntegersPractice: Multiplying & Dividing Integers Area = Length x Width (A=lw) 1) If the signs of the integers are the same then (area is always squared) your answer is always positive (+) Example: Example: (+5) x (+4) = +20 Adding Integers Subtracting Integers Example: (-12) ÷ (-3) = +4 1) If the integers have the same sign (either both 1) Change the subtraction sign into a plus 2) If theor negative) then you just add positive signs of the integers are different then 2) Change the sign of the 2nd integer your answer is always= +9 Example: (+5) + (+4) negative (-) 3) use addition rules to add Example: (-10)+x(-3) ==-7 Example: (-4) (+3) -30 Example: (-10)x- Width Area = Length (+3) = Example: (+54) ÷ (-6) = -9 Area = 14 x 7 + (-3) = -13 (-10) 2) If the signs of the integers are different (one is Area = 98cm² positive and one negative) then you must subtract. Example: (+7) – (+2) =Area of a triangle on top and use the sign of the The larger # goes (+7) + (-2) = +5 Area of a parallelogram larger #.Dividing IntegersArea = 1/2(-10) + (+3) height (A= 1/2bh) Example: x base x = -7Videos: Dividingsquared) = +9(area is always Integers 1 Example: (+15) + (-6) Area = base x height (A= bh)Practice: Multiplying & Dividing Integers (area is always squared) Area of a rectangle or square Multiply & Dividing Integers Area = Length x Width (A=lw) 1) If the signs of the integers are the same then (area is always squared) your answer is always positive (+) Example: Example: (+5) x (+4) = +20 Example: (-12) ÷ (-3) = +4Example: Example: 2) If the signs of the integers are different then your answer is always negative (-) Example: (-10) x (+3) = -30 Area = Length x Width Example: (+54) ÷ (-6) = -9 Area = 14 x 7 Area = 98cm²Area = 1/2 triangle heightArea of a x base x Area of a parallelogramArea = 1/2 x 12 x 15 Area = base x heightArea = 180/2 x base x height (A= 1/2bh)Area = 1/2 Area = 15 x 5Area is always(area = 90m² squared) Area = 75cm²x height (A= bh) Area = base (area is always squared)© 2012 Greg Kozak & Megan Murphy
- 14. Expressions with PhrasesPractice: Writing Expressions, Writing Expressions 2Terms:1) Algebraic Expression - a mathematical phrase that has at least one variable and one operation (+, -, x, ÷) Example: 6n (multiplication), n + 6 (addition), n - 6 (subtraction), n/6 or 6/n (division)2) Evaluate - to determine the value or amount3) Variable (unknown quantity) - a quantity that can be change or vary and is often represented bya letter. Example: 5n + 4 (n is the variable or unknown quantity)4) Coefficient - The numerical part of an algebraic expressionExamples:3x2 3 is the coefficient.2y 2 is the coefficient.5(a + b) 5 is the coefficient.5) Constant - a number that is on its own, or a fixed numberExample: 5n + 4 (4 is the constant)© 2012 Greg Kozak & Megan Murphy
- 15. Phrases: Word Phrase Operation Algebraic Expression 3 more than a number n addition n+3 6 less than a number n subtraction n-6 ﬁve times a number n multiplication 5n a number divided by 4 division n/4 4 divided by a number n division 4/n ﬁve less than three times a multiplication & division 3n - 5 number n two less than a number n division & subtraction n/4 - 3R A D E CA L I F O R N I A S TA N DA R D S T E S T divided by 4 6 Math Released Test Questions Expressions with Word Problemsᮀ46 1) Substitute company charges $0.05 per minute A telephone any number for the variable for local calls and $0.12 per minute for long- ᮀ 48 The steps Quentin took to evaluate the expression 3m − 3 ÷ 3 when m = 8 are 2) Determine which operation would be used to solve the problem once the variable is substituted distance calls. Which expression gives the total shown below. cost in dollars for x minutes of local calls and 3) minutes ofkey operation words if possible y Use your long-distance calls? 3m – 3 ÷ 3 when m = 8 3 ¥ problem 4) write0the+ 0.12 y expression to match the operation to be used to solve the 8 = 24 A 0. 5 x correct 24 – 3 = 21 B 0.05 x − 0.12 y 21 ÷ 3 = 7 C 0.17( x + y) What should Quentin have done differently D 0.17xy in order to evaluate the expression? CSM01299 A divided (24 − 3) by (24 × 3) Example: B divided (24 − 3) by (24 − 3)ᮀ47 Rita is moving a pile of 120 rocks by hand to build a rock wall. If h represents the number C subtracted (3 ÷ 3) from 24 of rocks that she can carry in one load, which D subtracted 3 from (24 ÷ 3) expression represents the total number of loads CSM10804 needed to move the entire pile of rocks? 120 + h A 4 rocks in one load. 4, ᮀ 1) Substitute a number for the variable h. Let’s use 49 so that will represent8that + 2 =can move 8 + ÷ 2 Rita A 4 2) No key operation words can help us here. Since Rita needs to move 120 rocks and can only B 8 B move 4 at a time, the only way to ﬁnd out the total amount of loads needed is to divide 120h 3) So our expression will be 120/4 which is 120/n whenC 10 we re-substitute the variable. © 2012 Greg Kozak & Megan Murphy D 14 C 120 − h CSM02058
- 16. Evaluating ExpressionsVideos: Expressions 1, Expressions 2, Expressions w/ 2 VariablesPractice: Evaluating Expressions w/ 1 Variable, Evaluating Expressions w/ 2 Variables1) Determine the operation to be used based on the algebraic expression2) Substitute the number given for the variable3) Solve or evaluate using order of operationsExample: Evaluate the expression 3n + 2 when n = 51) This expression includes two operations (multiplication and addition)2) Next substitute 5 for the variable n3) Your problem will now look like this: 3 x 5 + 24) Based on the order of operations, we must first multiply then add5) Our problem will be solve in the following manner: 3 x 5 + 2 (multiply first) 15 + 2 (now add) 17 (our answer)Equations with Word Problems1) Determine the operation that would be used to solve the problem normally (try using the key operation words)2) Write the equation that would allow you to perform the operation to solve the problem3) The equation is always the opposite or inverse of the operation used to solve the problem. For example if you need to add to solve the problem then you would write a subtraction equation and if you need to multiply to solve the problem then you would need to write a division equation to solve the problem.4) Your variable will be the unknown quantity that is to be found.Example: Lindsi had 8 apples on Monday and then lost 2 on Wednesday, how many applesdoes she have left.1) First determine the operation to solve the problem. The key words “have left and lost” indicate that subtraction is the operation to use.2) The correctly solve this problem we need an equation that will give us 8 - 2.3) Since we need to subtract, we will write an equation with the opposite or inverse operation which would be addition.4) Our variable is the unknown quantity this is to be found.5) The equation would be x + 2 = 86) When worked out, the equation will give us the operation we need of 8 - 2.© 2012 Greg Kozak & Megan Murphy
- 17. Solving Single Step Equations (equations with only one operation)Videos: Single Step 1, Single Step 2, Single Step 3, Single Step 4Practice: Single Step EquationsTerms:1) Equation - a number sentence that uses an equal sign. Everything on one side an equal sign has to equal everything on the other side. Equations normally have a variable or unknown quantitythat must be found to make the number sentence true.2) Inverse Operation - Operations that are opposite and undo each other. Addition and subtraction are inverses of each other as well as multiplication and division.3) Variable (unknown quantity) - a quantity that can be change or vary and is often represented bya letter. Example: 5n + 4 (n is the variable or unknown quantity)4) Coefficient - The numerical part of an algebraic expressionExamples:3x2 3 is the coefficient.2y 2 is the coefficient.5(a + b) 5 is the coefficient.5) Constant - a number that is on its own, or a fixed numberExample: 5n + 4 (4 is the constant)© 2012 Greg Kozak & Megan Murphy
- 18. Solving One Step Equations:1) Perform the inverse or opposite operation to both sides of the equations in order to isolate or get the variable all by itself.2) Evaluate or solve each side of the equation.3) Check your answer by substituting your answer for the variable. Both sides of the equal sign should equal each other if your answer is correct.Examples:© 2012 Greg Kozak & Megan Murphy
- 19. Classifying TrianglesPractice: Classifying TrianglesTriangle - A polygon with three segments as sides and three angles. The sum of the three angles ofa triangle equals 180 degrees.Classifying Triangles by Sides:1) Scalene Triangle - A triangle with no sides congruent or the same length. All of its angles will be different sizes as well.2) Isosceles Triangle - A triangle with two opposite sides congruent or the same length. The base angles opposite the equal sides will also be equal.3) Equilateral Triangle - A triangle with all sides congruent or the same length. The angles are also all congruent and will always be 60 degrees.© 2012 Greg Kozak & Megan Murphy
- 20. Classifying Triangles by Angles1) Acute Triangle - A triangle with all acute angle (less than 90 degrees).2) Obtuse Triangle - A triangle with one obtuse angle (more than 90 degrees).3) Right Triangle - A triangle with one right angle (exactly 90 degrees)© 2012 Greg Kozak & Megan Murphy
- 21. Finding Missing Angles of TrianglesVideo: Triangle Missing AnglesPractice: Triangle Missing Angles1) Add the two angles that are present together2) Subtract by 180 (all three angles in a triangle must add up to 180 degrees)Example: (find the measure of ∠ABC)1) add the two given angles together (39 + 55 = 94)2) subtract 94 from 180 (180 - 94 = 86)3) ∠ABC = 86°For Isosceles Triangles:1) Subtract the given angle from 1802) divide your answer by two (both angles opposite the congruent sides are equal which is why you divide by 2)Example: (find the measure of ∠ACB and ∠BAC)1) Subtract 120 from 180 (180 - 120 = 60)2) Divide your answer (60) by two since the base angles are congruent (60 ÷ 2 = 30)3) So both angles are 30°© 2012 Greg Kozak & Megan Murphy
- 22. Classifying Quadrilaterals (A Polygon with Four Sides. All angles add up to 360°)Videos: QuadrilateralsPractice: Classifying Quadrilaterals1) Square - a quadrilateral with four right angles and all sides congruent or the same length. A square can also be a rectangle, a rhombus, and a parallelogram.2) Rectangle - a quadrilateral with four right angles. A rectangle is also a parallelogram.3) Parallelogram - a quadrilateral with two pairs of opposite sides parallel. The opposite sides and angles congruent.Rhombus - a quadrilateral with all sides congruent. A rhombus is also a parallelogram and asquare.© 2012 Greg Kozak & Megan Murphy
- 23. 5) Trapezoid - a quadrilateral with one pair of opposite sides parallel. An Isosceles trapezoid is one with both base angles congruent.Finding Missing Angles of QuadrilateralsPractice: Missing Angles of Quadrilaterals1) Add the given angles together2) subtract your answer from 360 (all 4 angles in a quadrilateral add up to 360°)Example: (Find the measure of ∠PQR)1) Add the given angles together (120 + 60 + 70 = 250)2) Now subtract your answer (250) from 360 since all four angles in a quadrilateral add up to 360° (360 - 250 = 110)3) So ∠PQR is 110°Example: (When Missing Angle is Bisected) Find the measure of ∠MNO1) Add all of the given angles together (55 + 125 + 55 = 235)2) Subtract your answer (235) from 360 (360 - 235 = 125)3) The measure of ∠MNO is 125°© 2012 Greg Kozak & Megan Murphy
- 24. Congruency Marks (≅ means congruent or the same size and shape)Congruency Marks - The corresponding congruent sides of triangles are marked with small straightline segments called hash marks. The corresponding congruent angles are marked with arcs.Examples:1) ∠ABC ≅ ∠DFE2) Line CB ≅ Line FEAngle PairsVideos: Complementary & Supplementary, Complementary & Supplementary 2, Measuring AnglesPractice: Angle Types, Corresponding & Congruent Angles, Supplementary, Complementary,Vertical, Complementary & Supplementary, Measuring AnglesTerms:1) Angle - a figure formed by two rays, called sides, that share a common endpoint called a vertex.© 2012 Greg Kozak & Megan Murphy
- 25. Angle Pairs1) Complementary Angles - Two angles whose measures add up to 90 degrees.2) Supplementary Angles - Two angles whose measures add up to 180 degrees3) Vertical Angles - A pair of angles that are formed by intersecting lines and have no side in common.4) Corresponding Angles - When two lines are crossed by another line (called a transversal) thematching corner angles are corresponding angles and are congruent (same measure).© 2012 Greg Kozak & Megan Murphy
- 26. Multi-Step Missing AnglesVideos: Missing Angles 1, Missing Angles 2, Missing Angles 3, Missing Angles 4Practice: Multi-Step Missing Angles, Multi-Step Missing Angles 21) Highlight the missing angle that is to be found2) Decide what other angles need to be found in order to find the angle that you need3) Use your knowledge of triangles (all angles add up to 180), quadrilaterals (all angles add up to 360), supplementary angles (add up to 180) complementary angles (add up to 90), and vertical angles (opposite angles that are congruent) to help find any angles that you need.Example 1: (Find ∠JGI)Step 1: Highlight the angle that is to be foundStep 2: Use supplementary angles. Notice that ∠EGJ and ∠JGI are supplementary angles andthus should together add up to 180 degrees.© 2012 Greg Kozak & Megan Murphy
- 27. Step 3: Add ∠EGK and ∠KGJ (46 + 83 = 149). This gives you ∠EGJStep 4: Now subtract 149 from 180 (180 - 149 = 31). This will give you ∠JGIStep 5: ∠JGI = 31°Example 2: (Find ∠EFB)Step 1: Highlight the angle that is to be found 48°© 2012 Greg Kozak & Megan Murphy
- 28. Step 2: See what other angles you need to help you find your answer• Notice that ∠EFB is in a triangle. You know that all 3 angles of a triangle add up to 180 degrees.• If you knew ∠EBF then you could add that to ∠BEF and then subtract it from 180 to get your answer. We need to find what ∠EBF is.Step 3: Find ∠BEF• Notice that ∠EBF is vertical to 48°. Remember that vertical angles are opposite angles that are congruent which means that ∠EBF is also 48°.Step 4: Finding ∠EFB• Now add ∠EBF (48°) and ∠BEF (66°) together. 48 + 66 = 114°• Next subtract 114 from 180 since all of the angles in a triangle add up to 180• 180 - 114 = 66°• ∠EFB = 66°© 2012 Greg Kozak & Megan Murphy

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