Study Guide. Summer Packet 06/03/2014 Area of Triangle - B

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1 Study Guide Summer Packet 06/03/2014 Area of Triangle - B This skill requires the student to find the area of a triangle, which is one half the area of a rectangle that has the same base and height. The area of a rectangle can be found by multiplying its base by its height. If a rectangle is rectangle. A regular triangle regular triangle is The formula for divided in half along its diagonal, each triangle formed is half the area of the has three 60-degree angles, and sides that are all the same length. The area of a also one half the area of a rectangle with the same base and height. the area of a triangle is: Step 1: Write the formula for the area of a triangle. Step 2: Substitute 13 ft for the base and 10 ft for the height in the formula. Step 3: Find the product of 13 ft and 10 ft. Step 4: Multiply 130 by 1/2 (or divide 130 by 2). Answer: 65 ft 2 Example 2: The side of the lengths of the side of a hay storage building is in the shape of a triangle. What is the area of the Remem building if its base is 10 ft, its height is 12 ft, and the other two sides are 13 ft? ber: Units for area are always squared. Examples: ft 2, in. 2, m 2

2 Step 1: Write the formula for the area of a triangle. As you can see from the formula, the lengths of the other two sides, besides the base, are not needed to calculate the area. Step 2: Substitute 10 ft for the base and 12 ft for the height in the formula. Step 3: Find the product of 10 ft and 12 ft. Step 4: Multiply 120 by 1/2. Answer: 60 ft 2 An activity that can help reinforce the concept of area of a triangle is to show students several examples of triangles with the height and lengths of all three sides given. Ask them to write down the equations that would allow them to find the area of each triangle, reminding them that the only two dimensions they need are the height and base. Accuracy - B Accuracy and precision in measurement can be extremely important. There are two systems of measurement that are commonly used: the metric system and the U.S. customary (or standard) system. The Metric System: The meter is the basis of length measurements in the metric system. Here is a basic breakdown of the metric system of length. 1,000 millimeters (mm) = 1 meter (m)100centimeters (cm) = 1 meter (m) 10 decimeters (dm) = 1 meter (m)1 dekameter (dam) = 10 meters (m)1 hectometer (hm) = 100 meters (m) 1 kilometer (km) = 1,000 meters (m) The gram is the basis of weight measurements in the metric system. Here is a basic breakdown of the metric system of weight. 1,000 milligrams (mg) = 1 gram (g)100centigrams (cg) = 1 gram (g) 10 decigrams (dg) = 1 gram (g)1 dekagram (dag) = 10 grams (g)1 hectogram (hg) = 100 grams (g) 1 kilogram (kg) = 1,000 grams (g) The liter is the basis of capacity measurements in the metric system. Here is a basic breakdown of the metric system of capacity. 1,000 milliliters (ml) = 1 liter (l)100centiliters (cl) = 1 liter (l)

3 The U.S. Customary System: 10 deciliters (dl) = 1 liter (l)1 dekaliter (dal) = 10 liters (l)1 hectoliter (hl) = 100 liters (l) 1 kiloliter (kl) = 1,000 liters (l) The foot is the basis of length measurements of the U.S. customary system. Here is a basic breakdown of the U.S. customary system of length. 12 inches (in) = 1 foot (ft)1 yard (yd) = 3 feet (ft) The pound is the basis of weight measurements of the U.S. customary system. Here is a basic breakdown of the U.S. customary system of weight. 16 ounces (oz) = 1 pound (lb)1 ton (ton) = 2,000 pounds (lb) The gallon is the basis of capacity measurements of the U.S. customary system. Here is a basic breakdown of the U.S. customary system of capacity. 16 cups (C) = 1 gallon (gal)8 pints (pt) = 1 gallon (gal)4 quarts (qt) = 1 gallon (gal) Comparing the Two Systems: Length: Weight: Capacity: 1 inch (in) = 2.54 centimeters (cm)1 foot (ft) = meters (m)1 yard (yd) = meters (m) 1 ounce (oz) = grams (g)1 pound (lb) = grams (g) 1 cup (C) = liters (l)1 pint (pt) = liters (l)1 quart (qt) = liters (l)1 gallon (gal) = liters (l) Example 1: Choose the measure that is most precise. A cm B. 72 mm C m D. They are all of equal precision. Step 1: It will be easiest to compare the measurements if they are all converted to the same unit.

4 Convert 7.22 cm into meters by dividing by 100, since there are 100 cm in 1 meter cm 100 = m Convert 72 mm into meters by dividing by 1,000, since there are 1,000 mm in 1 meter. 72 mm 1,000 = m Step 2: Compare the three measures cm = m 72 mm = m m Solution: Since 7.22 cm is carried out to more decimal places when the measurements are converted to the same unit, it is the most precise. Answer: A Example 2: Choose the best estimate for the capacity of a cereal bowl. Solution: A. 2 l B. 2 oz C. 2 C D. 2 gal 2 liters is 1 large bottle of soda, so it is too large to be the capacity of a cereal bowl. 2 ounces are less than one cup, so it is much too small to be the capacity of a cereal bowl. 2 cups is less than two liters, but more than 2 ounces, so it is a possible capacity of a cereal bowl. 2 gallons is even larger than 2 liters, so it is definitely too large to be the capacity of a cereal bowl. Answer: C Order Numbers: Fractions/Dec./Percents In this grade level, students will make comparisons between fractions, decimals, and percents. They will then determine the order of least to greatest or greatest to least. It may be necessary to review fractions, decimals, and percents with the student. As the student practices comparisons, remind him or her that fractions, decimals, and percents all represent specific portions or parts. For example, the shaded portion in the diagram below can be represented by the fraction 3/4, the percent 75%, or the decimal To compare fractions, decimals, and percents, each number must be converted to the same form. For each problem, there are several conversions that can be made to compare one number type to the other.

5 However, converting to decimals first is almost always the easiest method. Converting Fractions or Percents into Decimals: To convert a fraction into a decimal, simply divide the numerator (top number) by the denominator (bottom number). In the example 4/5, divide 4 by 5. The answer is 0.8, the decimal equivalent to 4/5. To convert a percent into a decimal, we must recall that percent means "per one hundred." Therefore, 54% is equivalent to 54/100. To convert this fraction into a decimal, divide 54 by 100. Hence, 54% = (Dividing by 100 is the same as moving the decimal point two places to the left. Since the decimal point in 54 is to the right of the 4 (54.0), moving the decimal point two places to the left would also produce 0.54.) Converting Decimals or Percents into Fractions: When converting a decimal into a fraction, remember that decimals are parts of a whole having place values in powers of ten. For example, 3.23 expresses 3 wholes and 23 hundredths of a whole. Therefore, to change 3.23 into a fraction, rewrite it as 3 23/100, or 323/100. Reduce the fraction, if possible. When converting a percent into a fraction, write the percent over 100, and if possible, reduce the fraction. Hence, 54% = 54/100, which can be divided by their common factor of 2, yielding 27/50. Converting Fractions or Decimals into Percents: When converting a fraction into a percent, first change the fraction into a decimal by dividing the numerator by the denominator. In the example above, 4/5 became 0.8. To change a decimal into a percent, multiply it by 100, or move the decimal point two places to the right. Thus, 0.8 becomes 80%. Hence 4/5 = 80%. Example 1: Put the following numbers in order from least to greatest. Step 1: Determine a good method to compare the given numbers. In most problems, including this example, it is relatively simple to convert all of the given numbers into decimals. Step 2: Convert 4/5 into a decimal by dividing 4 by 5, yielding 0.8. Step 3: Convert 75% into a decimal by dividing it by 100, or moving the decimal point two places to the left. 75% = Step 4: 0.9 is given in the form of a decimal already. Step 5: Order these numbers from least to greatest, 0.75, 0.8, and 0.9. Therefore, the answer is 75%, 4/5, and 0.9. Example 2: Put the following numbers in order from greatest to least. Step 1: Determine a good method to compare the given numbers. In most problems, including this example, it is relatively simple to convert all of the given numbers into decimals. Step 2: Convert 270% into a decimal by recalling that percent means "per one hundred". Therefore, 270% = 270/100, which is the same as moving the decimal point two places to the left. 270% = 2.70 = 2.7. Step 3: Convert 8/3 into a decimal by dividing 8 by 3. The result is approximately Step 4: Order these numbers from greatest to least, 2.8, 2.7, Therefore, the answer is 2.8,

6 270%, and 8/3. A good activity to reinforce this skill could involve a bag of snack food or dollar bill (broken into change). The student could be asked if they would rather have 37%, 3/8, or 0.34 of the contents of the bag or the dollar. Whatever answer they decide on is how much they get, as long as they provide reasoning for why their choice is the best choice. Conversion: Variable Expressions/Words Learning to convert variable expressions into words or words into variable expressions is an important problem solving skill. In this tutorial, actual solutions to the problems will not be determined. It is important for the student to understand the difference between an expression and an equation. Expressions are variables or combinations of variables, numbers, and symbols that represent a mathematical relationship. Expressions do not have equal signs, but can be evaluated or simplified. Example: y - 6 Equations are expressions that contain equal signs. They can be solved, but not evaluated. Example: n + 5 = 9 Writing Variable Expressions To Represent Word Phrases: To represent a word phrase or story problem using algebraic symbols, there are three steps that the student should follow. Step 1: Choose a variable to represent the unknown quantity, or use the variable provided. Some problems will tell the student which variable to use. Step 2: Look for key words in the phrase or story problem that indicate the use of a particular operation. The chart below shows several key words with their corresponding operations. Step 3: Use the chosen words to set up the Example 1: Translate In August, a realtor sold sold in January. How h = the number of homes sold in January. variable, relevant information, and operation key variable expression. the following story problem into an expression. 6 less than 4 times the amount of homes she many homes did the realtor sell in August? Let (1) h = the number of homes sold in January Step 1: Identify the variable. Step 2: Look for key words and use the chart above to determine the necessary operations. Less than implies subtraction and times implies multiplication. Step 3: Set up the variable expression. Be sure to double check that all parts of the word phrase have been represented. Also, it is important to recognize that subtraction, although usually stated first in the word phrase, is normally placed at the end of the expression. This can be tricky for many students since they are taught to read from left to right.

7 Answer: 4h - 6 Example 2: Translate the following word phrase into an expression. nine less than ten times the sum of a number, y, and three (1) y = the number Step 1: Identify the variable. Step 2: Look for key words and use the chart above to determine the necessary operations. Less than implies subtraction, times implies multiplication, and sum of implies addition (usually as a quantity written in parentheses). Step 3: Set up the variable expression. Be sure to double check that all parts of the word phrase have been represented. Also, it is important to recognize that subtraction, although usually stated first in the word phrase, is normally placed at the end of the expression. Writing Word Phrases To Represent Variable Expressions: A similar process is used to write word phrases from variable expressions. In most cases, the student will be provided with a choice of word phrases as opposed to actually having to create one. This is because a variety of different phrases could accurately represent one expression. The student has two options. Option 1: The student can review the answer choices and apply the same three steps listed above to determine the corresponding variable expression. Option 2: The student can review the variable expression determining the operations used, values, and order in which it is set up. This information will guide the student to the correct word phrase. Example 3: Translate the following expression into words. (A) the difference of half the peanuts, p, and four (B) four plus twice the number of peanuts, p (C) the sum of one-half the number of peanuts, p, and four (D) one-half the sum of the number of peanuts, p, and four Solution: The student should rule out option (A) because it mentions difference which implies subtraction. This description would represent the expression?(p) - 4.

8 The student should rule out option (B) because it mentions twice which implies multiplying by 2 instead of?. This description would represent the expression 2p + 4. The student should rule out option (D) because it represents taking one half of the sum, which means you have to add the number, p, and four first, and then multiply by one-half. This description would represent the expression?(p + 4). Answer: The correct answer is option (C). Area of Parallelogram - B A parallelogram is a quadrilateral (a four-sided figure) with two pairs of parallel and congruent sides. Area is the measure, in square units, of the interior region of a two-dimensional figure. To find the area of a parallelogram, multiply the length of the base (b) by the height (h). The base is one of the sides of the parallelogram. The height is the length of the segment going from the base at a right angle (or perpendicular) to the opposite side. Here is the formula: Example 1: Find the area of the parallelogram. Solution: The Area = base formula for the area of a parallelogram is height. that the height Therefore, the area of the parallelogram is 3.5 cm 8 cm = 28 cm 2 Answer: 28 cm 2. The height of this parallelogram is 3.5 cm and the base is the length of the side is perpendicular to, in this case, 8 cm. One way to help the student reinforce the concept of finding the area of parallelograms is to use a ruler to draw a few parallelograms. Have the student measure the base and height of the parallelograms and then calculate the area using the formula given above. Also, try to find parallelograms in real world figures (such as those in designs) that can be measured so the area can be computed. Equivalent Fractions - B A fraction is made up of two parts: a numerator and a denominator. The numerator is the number on the top of the fraction and the denominator is the number on the bottom of the fraction. For example, in the fraction 4/5, 4 is the numerator and 5 is the denominator. A mixed number is a combination of a whole number and a fraction. An improper fraction is a fraction in which the numerator is larger than or equal to the denominator. An improper fraction can be rewritten as a mixed number or as a whole number.

9 Example 1: Find the missing number. Solution: Convert 4 2/3 into an improper fraction by multiplying the whole number by the denominator (4 x 3 = 12), then adding that product to the numerator ( = 14). 4 2/3 can be written as 14/3. The missing number is 14. Example 2: Find the missing number. Step 1: Replace the question mark with the variable, 'N'. Then cross multiply to begin the process of determining the missing number. Multiply the denominator of the first fraction by the numerator of the second fraction (N x 36 = 36N). Next multiply the denominator of the second fraction by the numerator of the first fraction (12 x 12 = 144). Step 2: Place an equal sign between the two products. Divide both sides of the equation by = 4 Answer: N = 4 Coordinate Geometry - C A coordinate graph is used to name the position of points. The x-coordinate (horizontal) is listed first and the y-coordinate (vertical) is listed second. For example, the coordinate pair (3, 2) is at the horizontal position 3 and the vertical position 2. It may be helpful to use graph paper to develop a coordinate graph. Help the student plot points on the graph and determine the coordinate pair. Example 1: What is the ordered pair for point J? Answer: (2, -2) because the point J is 2 units over and 2 units down. Equivalent Forms: Dec./Fract./Percent Fractions can be written as decimals and percents. For example, 1/4 is 0.25 or 25%. The numerator of a

10 fraction is the number on the top of the fraction and the denominator of a fraction is the number on the bottom of the fraction. Develop a series of fractions and decimals and help the student find the equivalent forms. The table below will help get you started. Example 1: Write 2/5 as a decimal and as a percent. Step 1: Every fraction can also be written as a division problem by dividing the numerator by the denominator. Step 2: Complete the division problem to write 2/5 as a decimal. Step 3: To write a decimal as a percent, multiply the decimal number by 100. This involves moving the decimal point two places to the right. Answers: Decimal 0.4 and Percent 40% Example 2: Write 8.2% as a decimal and as a fraction. Step 1: To change a percent into a decimal, divide the percent by 100. This involves moving the decimal point two places to the left. Step 2: The decimal is read "eighty-two thousandths," so it can be written as the fraction 82/1000. Step 3: Since 82 and 1000 can both be divided by 2, the fraction can be reduced to 41/500. Answers: Decimal and Fraction 41/500. Divisibility/Multiples/Factors - B Divisibility occurs when one number is divided by another and the remainder is zero. For example, 15 divided by 3 is 5. Factors are numbers that when combined in a multiplication equation give the product. The factor of a number is a whole number that divides it exactly. For example, 1, 2, 4, and 8 are factors of 8. A common factor of two or more numbers is a factor of all the numbers. The greatest common factor (GCF) is the greatest number in a list of common factors of the numbers. A multiple is a number that is the product of a given number and a whole number. For example, the multiples of 3 are 3, 6, 9, 12, etc. The common multiple of two numbers is any number that is a multiple of both numbers. The least common multiple (LCM) of two or more numbers is the least number in the list of their common multiples.

11 It may be helpful to create a divisibility game. On small pieces of paper, write the numbers 1 to 100. Put the 100 pieces of paper in a bag or a hat. Have the student draw two pieces of paper from the hat. Help the student determine if one number is divisible by the other number. A strong knowledge of the multiplication tables will be helpful. A similar game can be created for factors and multiples. On small pieces of paper, write the numbers 1 to 100. Put the 100 pieces of paper in a bag or hat. Have the student draw one piece of paper from the hat. Help the student determine the factors and multiples of the number. It may be helpful to note that there are an infinite amount of multiples for a specific number, but there are only a certain amount of factors for the same number. Example 1: Find the least common multiple (LCM) of 4 and 6. Step 1: Determine the multiples of 4 and 6. (Multiply 4 times 1, 2, 3, 4, etc. and the same with the number 6) multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36,.... multiples of 6: 6, 12, 18, 24, 30, 36, Step 2: List the common multiples. common multiples of 4 and 6: 12, 24, 36,.... Step 3: The least common multiple is the least number in the list of common multiples, which is the number 12. Answer: The LCM of 4 and 6 is 12. Example 2: Find the greatest common factor (GCF) of 24 and 56. Step 1: Determine the factors of 24 and 56. factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24 factors of 56 are: 1, 2, 4, 7, 8, 14, 28, and 56 Step 2: List the common factors of 24 and 56. Common factors of 24 and 56 are: 1, 2, 4, and 8 Step 3: The greatest common factor is the greatest value in the list, which is the number 8. Answer: The GCF of 24 and 56 is 8. Compare Whole Number Equations - B Comparing whole number equations involves determining which side of the equation is larger in value than the other. Knowledge of the greater than (>), less than (<), and equal to (=) signs is needed for this skill.

12 It may be helpful to review the ordering symbols with the student. Example 1: 11, ? 70 x 2 (1) 11, = 105 (2) 70 x 2 = 140 (3) 105? 140 Step 1: Simplify the expression on the left. Step 2: Simplify the expression on the right. Step 3: Rewrite the mathematical sentence with the new numbers and determine which symbol to place between the two numbers. The answer is: 11, < 70 x 2. Example 2: 200,000 20? 20,000 2 (1) 200, = 10,000 (2) 20,000 2 = 10,000 (3) 10,000? 10,000 Step 1: Simplify the expression on the left. Step 2: Simplify the expression on the right. Step 3: Rewrite the mathematical sentence with the new numbers and determine which symbol to place between the two numbers. The answer is: 200, = 20, Multiple Operations: Whole Numbers - B Multiplication skills at this level include operations with multi-digit numbers (237 x 56) which require regrouping (carrying, borrowing, renaming) when the product of the numbers in the ones or tens (hundreds, thousands, etc.) position are equal to or greater than ten. Necessary division skills include long division, remainders, and two-digit quotients. When performing multiple operations on expressions, it is important to remember that operations inside parentheses are completed first. If there are two sets of parentheses, then the set that comes first when reading from left to right is completed first. Example 1: (237 x 56) 12 =? (1) 237 x 56 = 13,272 (2) 13, =? (3) 13, = 1,106

13 Step 1: Multiply the numbers inside the parentheses. (237 x 56) Step 2: Rewrite the problem with the new value in place of the parentheses. Step 3: Divide 13,272 by 12. Answer: 1,106 Subtract Decimals: Hundred Thousandths Subtracting decimal numbers with more than one decimal position (columns of numbers) is very similar to subtracting whole numbers. Subtracting decimal numbers requires the ability to regroup (carry, borrow, or rename) when the number being subtracted is greater than the other number. Example: = Step 1: Write the problem vertically. Remember to line up the decimal points and place a zero at the end of to hold the hundred thousandths place. Step 2: Subtract the numbers in the hundred thousandths column (5-0 = 5). Place the 5 in the hundred thousandths place. Step 3: Before the numbers in the ten thousandths column can be subtracted we must "borrow" or "trade" from the thousandths column. Cross out the 9 and make it an 8, then make the 2 in the ten thousandths column a 12. Now, subtract the numbers in the ten thousandths column (12-7 = 5). Place the 5 in the ten thousandths place. Step 4: Subtract the numbers in the thousandths column (8-1 = 7). Place the 7 in the thousandths place. Step 5: Subtract the numbers in the hundredths column (6-2 = 4). Place the 4 in the hundredths place. Step 6: Before the numbers in the tenths column can be subtracted, we must "borrow" or "trade" from the ones column. Cross out the 7 and make it a 6, then make the 3 in the tenths column a 13. Now, subtract the numbers in the tenths column (13-8 = 5). Place the 5 in the tenths place. Step 7: Subtract the numbers in the ones column (6-4 = 2). Place the 2 in the ones place. Answer: = Divide Decimals by Whole Number Dividing a decimal number by a whole numberis very similar to dividing whole numbers. The decimals point must remain in the same position in the answer. Example: Solve 18.9 divided by 9 =? Step 1: Write the problem in long division format. Step 2: Division follows the same format as with whole numbers. 9 goes into 18 two times because 9 x 2 = 18. Place 2 in the ones position. Subtract 18 from 18 resulting in 0. Bring down the 9. Step 3: Place the decimal point.

14 Step 4: 9 goes into 9 one time because 9 x 1 = 9. Place 1 in the tenths position. Subtract 9 from 9 resulting in 0. The answer is 2.1. Multiplying Integers Integers are the set of positive and negative whole numbers, including zero. Students should understand how integers appear on a number line. Numbers to the right of 0 on a number line are positive and numbers to the left of 0 are negative. The number -3 is a negative integer and the number 3 is a positive integer. The integer 0 is neutral. To multiply and divide integers, follow these rules. The product of two positive integers is positive (Example: 9 x 4 = 36). The product of a positive integer and a negative integer is negative (Example: 9 x -4 = -36). The product of two negative integers is positive (Example: -9 x -4 = 36). Notice from the above examples that when you multiply integers with the same sign, the answer is positive. When multiplying integers with different signs, the answer is negative. The order of operations must be followed when working with grouping symbols and/or multiple-step operations. The order of operations are as follows: 1. Work inside grouping symbols 2. Multiply and divide from left to right 3. Add and subtract from left to right Example 1: (3 x -5) + 7 =? (1) 3 x -5 = -15 (2) = -8 Step 1: Using the order of operations, calculate within grouping symbols. 3 x -5 = -15, so we can replace the parentheses with the value of -15. Step 2: Perform the addition. Answer: -8 Example 2: 8(-3 x 4) x (-5 + 7) =? (1) -3 x 4 = -12 and = 2 (2) 8(-12) x 2 =? (3) 8(-12) = -96 (4) -96 x 2 = -192

15 Step 1: Work inside grouping symbols. Step 2: Rewrite the problem with the new numbers. Step 3: Multiply from left to right, 8 x -12 = -96 Step 4: Multiply -96 by 2 to get Answer is: -192 Story Problems Integers Integers are the set of positive and negative whole numbers, including zero. In integer story problems, students must determine the elements of the story that make up an integer equation, decipher the correct operation, and solve the problem. To begin, the student should be familiar with how to add, subtract, multiply, and divide integers. Use the following definitions and examples to review adding and subtracting integers. When adding two integers with the same sign, add their absolute values. Then give the sum (answer) the sign of the integers. Answer: =? =? 3 +2 = 5, then make the result negative. When adding integers with different signs, first find their absolute values. Then subtract the lesser absolute value from the greater absolute value, and give the result the sign of the integer with the greater absolute value =? -7 = 7 and 3 = 3 (find the absolute values) 7-3 =? (subtract the lesser from the greater) 7-3 = = -4 (The result is given the sign of the greater integer.) Subtracting integers is the same as adding the opposite =? = 10 (add the opposite) = 10 Use the following definitions and examples to review multiplying and dividing integers. The product of two positive integers is positive. Example: 9 x 4 = 36. The product of a positive integer and a negative integer is negative. Example: 9 x -4 = -36. The product of two negative integers is positive. Example: -9 x -4 = 36. The quotient of two positive integers is positive. Example: 9 3 = 3. The quotient of a positive integer and a negative integer is negative. Example: 9-3 = -3. The quotient of two negative integers is positive. Example: -9-3 = 3.

16 Once the student is comfortable with these skills, help him or her apply these skills in a real life scenario. Example 1: The temperature on Sunday was -9 degrees, and the temperature on Monday was -3 degrees. How much warmer was it on Monday than on Sunday? (1) Sunday: -9 degreesmonday: -3 degrees (2) -3 - (-9) =? (3) =? (4) = 6 Step 1: Identify the relevant information in the story problem. Step 2: Determine the necessary operation. The story requires the difference to be found. Step 3: Subtracting a negative number makes a positive. Step 4: Complete the addition problem. If the student has difficulty with the above step, draw a number line and label the points -9 and -3 on the number line. Count the difference. Answer: Monday was 6 degrees warmer than Sunday. The student should review finding averages with integers and be able to apply this skill to story problem situations. Example 2: During a diving course for diving certification, John had to make 5 dives. His first dive was 20 feet down, his second was 30 feet down, his third was 55 feet down, his fourth was 75 feet down and his last was 100 feet below the surface. What was the average depth of all 5 dives? Step 1: Identify the relevant information. Below the surface represents negative integers so -20, -30, -55, -75, -100 Step 2: Determine the necessary operations. Find the sum of the numbers and divide by the total amount of numbers. Step 3: (-20) + (-30) + (-55) + (-75) + (-100) = -280 Step 4: = -56 Answer: 56 feet below the surface (because -56 represents 56 feet below the surface) Percent One Number is of Another Percent means "per one hundred." When you state that 70% of the students ate tacos, then you are stating that 70 out of 100 students ate tacos. If 35 out of 50 students ate tacos, then 70% ate tacos because 35 out of 50 (35/50) is equal to 70 out of 100 (70/100). To understand percents, students must draw upon knowledge of decimal numbers, ratios, and fractions. The following example should help the student better understand the steps needed for determining the percent one number is of another.

17 Example: What percent of 45 is 36? Step 1: Rewrite the equation. Since the percent is what we want to find, we can represent it with the variable n%. Note that the word "of" means multiply and the word "is" means equal. Step 2: Change n% to a fraction and multiply it by 45. n% times 45 should equal 36. Write the new equation. Step 3: Multiply both sides of the equation by 100 to begin to isolate the n on one side of the equal sign. Step 4: Rewrite the equation with the new numbers. Step 5: Divide both sides of the equation by 45. Step 6: = 80. Answer: 80% Expressions: Addition Expressions are number sentences which do not have equal signs, but need to be evaluated or simplified. Example: y - 6 An equation is a number sentence that does have an equal sign. Example: y - 6 = 14 Example 1: Evaluate the expression x + 23, when x = Solution: Substitute the value 5 in place of x in the expression. Answer: 28 Example 2: For x = -7, find 2x (1) 2(-7) (2) (3) -26 Step 1: Substitute -7 in for the value of x. Step 2: Multiply 2 by -7 and rewrite the expression with the new value. Step 3: Add -14 and -12. Answer: -26 Example 3: Write a mathematical expression to represent the following: The sum of a number and 23. Solution: Remember that "sum" is the answer to an addition problem, so the expression is x + 23.

18 Answer: x + 23 Expressions: Multiplication pressions are number sentences which do not have equal signs, but need to be evaluated or simplified. Example: y - 6 An equation is a number sentence that does have an equal sign. Example: y - 6 = 14 Example 1: Evaluate the expression below for x = x (1) 5 + 3(18) (2) (3) 59 Step 1: Replace x with 18. Step 2: Multiply 3 by 18 to get 54. Step 3: Add 5 and 54. Answer: 59 Example 2: Write a mathematical expression that represents the following word expression. four times a number less 6 (1) four times x less 6 (2) 4x less 6 (3) 4x - 6 Step 1: Replace the words "a number" with a variable (x was chosen) Step 2: "four times x" can be written as 4x. Make this replacement. Step 3: "less 6" can be written as "- 6." Make this replacement and the expression is complete. Answer: 4x - 6 Expressions: Story Problems Story problems for equations (word problems) require students to read passages, determine variables, write equations, and solve. Expressions are number sentences which do not have equal signs, but need to be evaluated or simplified. Example: y - 6

19 Equations are number sentences which contain equal signs. Eample: n + 5 = 9 Example: The length of John's model airplane is 10 inches more than twice the width. The width is 15 inches. What is the length of John's model airplane? (1) L = 2W + 10 (2) L = 2(15) + 10 (3) L = (4) L = 40 Step 1: The length is equal to two times the width plus 10. Develop a formula with W = Width and L = Length. Step 2: Replace W with 15. Step 3: Calculate the right side of the equation by adding 30 and 10. Answer: The length of John's model airplane is 40 inches. Distance/Rate/Time The formula for solving distance (D), rate (R), and time (T) is: D = R x T Students should be able to read story problems, decipher two elements of the distance formula, plug the elements in the distance formula, and solve. The following are examples of problems requiring the D = R x T formula. Example 1: If Carson drove 55 miles per hour on the freeway for 495 miles, how long did he drive? Step 1: Determine the distance, rate, and time values and substitute them into the D = R x T formula. D = 495, R = 55, and T =? Step 2: Solve for T (Time) by dividing both sides of the equation by 55. Answer: Carson drove for 9 hours. Example 2: Stan ran for hours. He ran a total of 2500 miles. What was the rate of speed that Stan ran? (Round miles per hour to the nearest hundredth). Step 1: Determine the distance, rate, and time values and substitute them into the D = R x T formula. D = 2500, R =? and T = Step 2: Solve for R (Rate) by dividing both sides of the equation by Step 3: Round the answer to the nearest hundredth.

20 Answer: Stan ran at a rate of miles per hour. Exponential Notation - C An exponent is a number that represents how many times the base is used as a factor. The base number 5 to the 3rd power translates to 5 x 5 x 5 which equals to the 3rd power is not 5 x 3. To perform operations with exponents, all exponential properties must be understood. Have the student find the equivalent whole number forms of these exponential numbers: Scientific Notation: Scientific notation is based upon exponential properties and is used to communicate very large or very small numbers. Scientific notation deals with significant digits. The most significant digit in a number is the first non-zero digit in the number (reading from left to right). To write a number using scientific notation, place the decimal point to the right of the most significant digit and count the digits between the new placement of the decimal point and the old placement of the decimal point. The number of places that the decimal point moved will be represented by a power of 10. Example 1: Write 123,000,000 using scientific notation. Step 1: Determine where the decimal point is in the number to be written using scientific notation. Step 2: Place the decimal point to the right of the most significant digit and count the number of places the decimal point was moved. Step 3: Write all of the significant digits (with the decimal in the new postion) and multiply by 10 to a power. The power on the ten is the number of places that the decimal point was moved. The power is positive because the decimal point was moved from the right to the left. Answer: To take a number out of scientific notation, move the decimal point the same number of places as the exponent in the power of ten. Example 2: Answer: 62,900,000 (move the decimal 7 places to the right) Example 3: Find the equivalent form.

21 Step 1: Write the number in the first set of parentheses in standard form and rewrite the expression. Step 2: Write the number in the second set of parentheses in standard form and rewrite the expression. Step 3: Add 6,000 and 200 to get 6,200. Answer: 6,200 Scientific Notation Scientific notation is a condensed way to write very large or small numbers without including each digit. Scientific notation is a number written as the product of a number between 1 and 10 and a power of 10. To write a large number using scientific notation, count the digits (from right to left) to be represented by a power of ,000,000 can be written in scientific notation as 1.23 x 10 to the 8th power. To write a small number, count the digits from left to right. To undo scientific notation, move the decimal point the same number of places as the exponent in the power of ten. Example 1: Answer: 62,900,000 (move the decimal 7 places to the right) Example 2: Answer: The missing exponent would be 4. Place Value: Whole Numbers - C Each digit of a number is given a value for the position it is located. For example, in the number 56 the place value of the 6 is ones. The place value of the 5 is tens. It may be helpful to develop a place value chart like the one below to help the student visualize place value. Example 1: What does the digit 5 mean in 7,235,689? Solution: Help the student fill in the place value chart with the correct digits (as seen in the place value chart below). Then use the chart to determine the meaning of the indicated number.

22 Since the 5 is in the thousands column, the 5 in 7,235,689 means 5 thousands. Example 2: In which place is the underlined number? 123,458,079 Solution: Help the student fill in the place value chart with the correct digits. Then use the chart to determine the meaning of the indicated number. The 3 is in the millions place. Radicals and Roots Mastering roots and radicals is an essential step toward learning advanced mathematics concepts. A radical sign looks like a check mark with a line attached to the top. The radical sign is used to communicate square roots. The following rules are required to perform operations with roots and radicals. 1. If x multiplied by x equals y, then x is a square root of y. For example, 6 multiplied by 6 is 36, so 6 is a square root of 36. In fact, 36 is called a perfect square because its square root, 6, is a whole number. Most algebra text books contain a table of perfect squares and 3 are both square roots of 9 because -3 x -3 = 9 and 3 x 3 = 9. 3 is referred to as the principal square root because it is the positive square root of To find the simplest radical form of a radical expression, factor the number under the radical sign (the radicand). The square root of 45 could be factored to be the square root of 9 multiplied by the square root of 5. The square root of 9 multiplied by the square root of 5 can be simplified further by finding the square root of 9. The result is 3 (the square root of 9) multiplied by the square root of 5. Example 1: Find the equivalent form. Solution: Multiply the numbers under the radical symbols.

23 Example 2: What symbol would best replace the? in the given statement? A. = B. < C. > There are two methods that can be used to solve this problem. Each method is shown and explained below. Solution Method 1: Step 1: Simplify the square root of 12 by making it the square root of 6 x 2. Step 2: Further simplify the square root of 6 x 2 by making it the square root of 3 x 2 x 2. (If you multiply 3 x 2 x 2, you will get 12.) Step 3: The square root of 3 x 2 x 2 becomes 2 times the square root of 3, because the square root of 2 x 2 is 2 and the 3 must remain under the square root symbol. Step 4: Two times the square root of 3 can also be written as the square root of 3 plus the square root of 3. Step 5: Now, we can make a comparison. We know that the larger a number is, the larger that number's square root will be. We can determine that the square root of 5 plus the square root of 7 will be greater than the square root of 3 plus the square root of 3 because 5 and 7 are both larger than 3. Answer: C Solution Method 2: Step 1: Estimate the square root of 5, the square root of 7, and the square root of 12. This estimation can be done using a calculator. Step 2: Add together the 2.24 and the 2.65 to get Step 3: Replace the question mark with the > symbol because 4.89 is greater than Answer: C Example 3: Solve for the value of x. Step 1: The square root of 36 is 6 because 6 x 6 = 36. Step 2: Subtract 6 from each side of the equation to isolate the square root of x.

24 Step 3: The square root of x is equal to 8. We can replace the x with 64, since the square root of 64 is 8. Step 4: Since the square root of 64 equals 8, the value of x is 64. Answer: x = 64 Properties - D Students must be able to solve for a missing value in a given equation. Understanding properties such as the order of operations is the key to correctly solving these problems. Please review the following rules with the student: 1. Multiplication by 0: the product of any integer and 0 equals x 0 = 0 3 x 0 = 0 2. Associative Property of Addition: (a + b) + c = a + (b + c). (1 + 2) + 3 = 1 + (2 + 3) 3. Associative Property of Multiplication: (a x b) x c = a x (b x c). (1 x 2) x 3 = 1 x (2 x 3) 4. Reciprocals: two numbers are reciprocals if their product equals Commutative Property of Addition: a + b = b + a = Commutative Property of Multiplication: a x b = b x a 1 x 2 = 2 x 1 7. Order of Operations: A. When calculations for a given expression or equation require both addition and multiplication, the rule is to multiply first and add second. (3)(2) + 3 =? =? = 9 B. The number outside the parentheses is multiplied with each number within the parentheses: x(y + z) = xy + xz. (1) 3(x + y) (2) 3(x) + 3(y) (3) 3x + 3y

25 C. If a given expression contains both parentheses and brackets, calculations should be completed working from the innermost parentheses or bracket outward. The following are sample questions using the above properties. Example 1: Which answer best completes the number sentence? 5 = (5 x 4) + (5 x 6) A. x (4 + 6) B. + (4 + 6) C. x ( ) D. + (4 x 6) Answer: A (because of rule 7B) Example 2: Which one of the following best completes the number sentence? ( ) x 5.6 =? A. 3.5 x 5.6 B. 7.2 x 5.6 C. 9.1 x 5.6 D. 5.4 x 5.6 Answer: D (because of rule 7A) Example 3: What is the value of n in the following statement? 13 x (3.4 x 0) = n A B. 0 C. 13 D. 3.4 Answer: B (because of rule 1) Inequalities - A An inequality is a number sentence that uses "is greater than", "is less than", or "is not equal to" symbols. For example, 6n > 4 is a number sentence with an inequality symbol. It may be useful to review the inequality symbols.

26 Example 1: Solve for y: 8y > 40 Get the variable being solved for (y) on one side of the inequality and the whole number on the other. To do this, divide both sides by 8. The correct answer is that y is greater than 5. Inequalities can be represented as a value on a number line. The following number line represents the inequality Example 2: Which inequality represents the value shown on the number line below? A. n < 3 B. n > 3 C. n = 3 The answer is A. n < 3 because the dot on the number line is open. Multiple-step Story Problems - E These problems are designed to test a student's ability to interpret data from story (word) problems. Answers are found by solving equations with multiple operations. It may be helpful to develop a series of multiple-step word problems that relate to the student's activities, such as allowance. The following is a step-by-step example of a multiple-step story problem. Example 1: On Saturday, Stella earned $3.50 for each hour of work. She earned $3.25 for each hour of work on Sunday. She worked 5 hours each day. How much money did she earn for both days? (1) $3.50 x 5 =? $3.25 x 5 =? (2) $3.50 x 5 = $17.50 $3.25 x 5 = $16.25 (3) $ $16.25 = $33.75 Step 1: Develop 2 separate equations. One to find the earnings on Saturday, and one to find the earnings on Sunday. Step 2: Find the products of the two equations. Step 3: Add the two products together.

27 Answer: Stella earned $ Example 2: Saman ate 3 times as many cookies as Alli. Alli ate 5 cookies less than Josh. Josh ate 10 cookies. How many cookies did Saman eat? (1) 10-5 = 5 (2) 5 x 3 = 15 Step 1: Since Alli ate 5 cookies less than Josh, subtract 5 from 10 to determine the number of cookies she ate. Step 2: Now that we know how many cookies Alli ate, we can determine the number of cookies Saman ate by multiplying 5 by 3. Answer: Saman ate 15 cookies. Perimeter - C Perimeter is the measurement of the distance around a figure. To calculate the perimeter of a figure, add the lengths of all the sides of the figure. Example 1: What is the perimeter of a figure that has four sides measuring 3 inches, 7 inches, 3 inches and 7 inches? P = = 20 Answer: 20 inches Example 2: What is the perimeter of the figure? Step 1: Add the length of each side together. Step 2: Since the length of each side includes a fraction, it may be easier to add the numbers vertically. Find the common denominator. The common denominator is 12 because 2 x 3 x 4 = 12. Rewrite the fractions so they all have the common denominator (12). Step 3: 30/12 is an improper fraction because the numerator (top number) is larger than the denominator (bottom number). Twelve will divide into 30 two times with 6 left over. Add the 2 to the 35 and now the fraction is 37 6/12. Six and 12 can both be divided by six, so the final answer is 37 1/2.

28 Answer: 37 1/2 Area of Trapezoid The area of a trapezoid is the number of square units needed to cover the surface of the figure. The following is the formula needed for calculating the area of a trapezoid: Example 1: Solve for height equal to 4 meters. Step 1: Apply the within the parentheses. the answer. Answer: 28 square the area of a trapezoid with bases equal to 6 meters and 8 meters, and amounts given in the problem to the formula. Step 2: Add the numbers Step 3: Multiply the whole numbers. Step 4: Perform calculations to find meters Example 2: Find x if the area of the trapeziod is 73.5 centimeters squared. Step 1: Apply the given values to the formula for the area of a trapeziod. (NOTE: This time you are given the area of the trapeziod.) Step 2: Add the numbers within the parentheses. Step 3: Perform the multiplications on the right side of the equation. Step 4: Multiply both sides of the equation by 2. Simplify. Step 5: Divide both sides of the equation by 21.

29 Answer: x = 7 Area of Circle The area of a circle is the number of square units needed to cover the surface of the figure. The following is the formula needed for calculating the area of a circle: Pi is approximately equal to The symbol for Pi is Example 1: Solve for the area of a circle with a radius (1) Area = 3.14 x (4 x 4) (2) Area = 3.14 x 16 (3) Area = Step 1: Apply the amounts given in the problem to the the parentheses. Step 3: Perform calculations to find the The area of the Angles - B An angle is created by two rays with the same endpoint. That The radius is the length from the center of the circle to the outside edge. The diameter equal to 4 meters. formula. Step 2: Multiply the numbers within answer. is the line A semicircle is half of a circle. The area of a semicircle is segment exactly half of the area of a circle with the same radius. that connects Example 2: What is the area of the following semicircle? two Round your answer to the nearest hundredth. points on the Step 1: The diameter of the semicircle is 13 inches, so the radius is outside 13 inches divided by 2 (6.5 inches). Step 2: Determine the area of a circle with radius 6.5 inches. Step 3: Divide the area of the circle by 2 to find the area of the semicircle with radius 6.5 inches. Step 4: Round to the nearest hundredth. edge of the circle and passes through the center of the circle. The length of the diameter is twice the length of the radius. semicircle is square inches. endpoint is called the vertex.

30 An interesting method for improving the student's understanding of angles is to have him or her draw the various types of angles. Then, develop a series of flash cards. On one side of the card, draw the figure. On the other side of the card, write the name. The following are definitions to help get you started: Obtuse Angle - an angle with a measure greater than 90 degrees and less than 180 degrees Right Angle - an angle with a measure equal to 90 degrees Acute Angle - an angle with a measure greater than 0 degrees and less than 90 degrees Adjacent angles - two angles with a common vertex and a common side Complementary angles - two angles whose measures have a sum of 90 degrees Supplementary angles - two angles whose measures have a sum of 180 degrees Triangles - A A triangle is a polygon with three sides. The following are common triangles used at this level: Scalene Triangle - a triangle with three unequal sides Isosceles Triangle - a triangle with at least two equal sides Equilateral Triangle - a triangle with three equal sides

31 Right Triangle - a triangle with one right (90º ) angle It is also important for the student to understand that the sum all of the angles in a triangle equals 180º. Example 1: If one angle of a triangle is 10º and a second angle is 45º, what is the measure of the third angle? (1) 10º + 45º = 55º (2) 180º - 55º = 125º Step 1: Add the measures of the two known angles. Step 2: Since the measures of the three angles of a triangle add up to 180º, subtract 55º from 180º. Answer: The measure of the third angle is 125º. An alternate method for determining the measure of the third angle of a triangle is to set up an equation. Example 2: A triangle has two angles, each measuring 47º. What is the measure of the third angle? (1) 180º = 47º + 47º + x (2) 180º = 94º + x (3) 180º - 94º = 94º + x - 94º (4) 86º = x Step 1: Since the sum of the angles of a triangle equals 180º, let 180º = 47º + 47º + x where x represents the measure of the third angle. Step 2: Combine like terms (47º +47º = 94º ) Step 3: To isolate x, subtract 94º from both sides of the equation. Step 4: Simplify both sides of the equation. Answer: The measure of the third angle is 86º. Lines/Segments/Points/Rays At this level, the lines skill involves the following concepts: lines, rays, points, and segments. It is important for the student to understand the following definitions: Point - a specific location on a figure, usually on a line or plane Line - a straight path extending in both directions with no endpoints. A line AB is denoted as Segment - a part of a line that ranges from one point to another. A line segment AB is denoted as Skew lines - lines that are not intersecting and are not parallel.