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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 MATHEMATICS BENCHMARKS Sixth Grade O – means teacher should be able to observe throughout the day – possibly use anecdotal records. I – Informal Assessment—those marked ―I‖ have an assessment task attached. Uses properties to create and simplify algebraic expressions and solves linear equations and inequalities 1) I - Solves equations in one variable using addition and subtraction 2) I - Models simple addition and subtraction problems using integers on a number line 3) I - Recognizes and continues a number pattern and/or geometric representation (e.g., triangular numbers) 4) I - States a rule to explain a number pattern 5) I - Uses whole numbers to complete a function table based on a given rule 6) I - Creates and solves proportional equations using one variable Interprets, organizes, and makes predictions using appropriate probability and statistics techniques 7) I - Reads and constructs line, bar, and pictographs 8) I - Reads and interprets circle graphs using percents 9) I - Constructs and explains a frequency table 10) I - Uses probability to predict the outcome of a single event and expresses the result as a fraction or decimal 11) I - Estimates and compares data to include mean, median, and mode 12) I - Solves problems involving combinations Writes and solves problems involving standard units of measurement 13) I - Measures length to the nearest one-sixteenth inch 14) I - Identifies appropriate units for measuring length, weight, volume, and temperature in the standard (English and metric) systems 15) I - Uses appropriate mathematical tools for determining length, weight, volume, and temperature in the standard (English and metric) systems 16) I - Uses estimation to solve problems in the standard (English and metric) systems 17) I - Converts units within a measurement system (136) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Determines the relationships and properties of two and three-dimensional geometric figures and the application of properties and formulas of coordinate geometry 18) I - Locates points in all four quadrants of the coordinate plane 19) I - Draws points, lines (parallel, perpendicular, intersecting), line segments, and rays 20) I - Identifies, classifies, and measures right, acute, obtuse, and straight angles 21) I - Creates tessellations with polygons 22) I - Explores the relationships of three-dimensional figures, including vertices, faces, and edges using manipulative materials 23) I - Describes, compares, constructs, classifies, and identifies flips, slides, and, turns (reflections, translations, and rotations) 24) I - Calculates the area of parallelograms (squares and rectangles) without using calculators 25) I - Finds the circumference of a circle with and without the use of manipulative materials 26) I - Determines the area of a circle with and without the use of a calculator 27) I - Finds the volume of cubes and rectangular prisms with and without the use of calculators Uses basic concepts of number sense and performs operations involving exponents, scientific notation, and order of operations 28) I - Reads, writes, and rounds twelve-digit whole numbers 29) I - Compares and orders whole numbers using <, >, and = 30) I - Writes twelve-digit whole numbers using expanded notation 31) I - Reads, writes, and rounds decimal numbers to the nearest ten-thousandth 32) I - Compares and orders decimal numbers using <, >, and = 33) I - Writes decimal numbers through the nearest ten-thousandth using expanded notation 34) I - Uses estimation to determine accuracy of solutions 35) I - Multiplies a three-digit decimal number by a two-digit decimal number 36) I - Divides a five-digit decimal number by a two-digit decimal number 37) I - Rounds decimal quotients to the nearest whole number, tenths, and hundredth 38) I - Estimates and solves one and two-step problems involving addition, subtraction, multiplication and division of decimals, with and without calculators 39) I - Uses the rules of divisibility to determine factors and multiples of a given number 40) I - Explores the relationships among integers 41) I - Models and writes the prime factorization of a number using exponential notation 42) I - Distinguishes between prime and composite numbers with and without the use of calculators (137) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 43) I - Uses the greatest common factor (GCF) to simplify fractions 44) I - Uses the least common multiple (LCM) to find common denominators Determines relationships among real numbers to include fractions, decimals, percents, ratios, and proportions in real life problems 45) I - Converts among fractions, decimals, and percents 46) I - Finds the percent of a number 47) I - Estimates and calculates sale price and/or original price using discount rates 48) I - Compares and orders fractions as well as mixed numerals 49) I - Determines equivalent forms of fractions 50) I - Uses a variety of techniques to express a fraction in simplest form 51) I - Locates fractions, decimals, and mixed numerals on a number line 52) I - Adds and subtracts mixed numerals with and without regrouping, expressing the answer in simplest form using like and unlike denominators 53) I - Multiplies and divides proper fractions as well as mixed numerals expressing the answer in simplest form 54) I - Estimates, solves, and compares solutions to one and two-step problems involving addition, subtraction, multiplication, and division of proper fractions and mixed numerals 55) I - Demonstrates different ways to express ratios (138) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Sixth Grade Informal Assessments Guiding Questions Task Sample 1) Can the student solve equations in Provide the student a problem with one one variable using addition and number missing. Tell the student to let ―n‖ subtraction? represent the missing number. Provide counters as manipulatives so the student can model equations. n+7=9 7–n=2 3 + n = 12 n–8=3 Have the student solve for n with and/or without the manipulatives. 2) Can the student model simple Provide the students with manipulatives addition and subtraction problems and integer problems such as: using integers on a number line? -12 + -2 -5 + 8 -4 – 9 7 – (-3) -2 – (-8) 3) Can the student recognize and Provide the student with a pattern such as continue a number pattern and/or towering cans. geometric representation? * * * * * * * * * * * * * * * * * * Tower 1 Tower 2 Tower 3 Have the student draw tower 4, 5 and 6. Have the student determine how many cans are in towers 10 and 15. 4) Can the student state a rule to Provide the student with a pattern such as explain a number pattern? 8, 4, 0, -4, and -8. Have the student describe the pattern in three ways. (139) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 5) Can the student using whole Provide the student with a rule and a numbers complete a function table function such as y = x + 5. Have the based on a given rule? student complete the function chart below: Y X 0 1 2 3 4 10 100 6) Can the student create and solve Provide the student with the following proportional equations using one recipe. Create and solve proportional variable? equations using one variable. Sausage Cheese Balls (Serves 20) 1 lb. Sharp cheddar cheese, grated 1 lb. hot sausage 1 2 c Bisquick 2 Mix together. Bake at 325 degrees for 20 minutes. The student will determine the amount of ingredients needed to serve 30 people. 7) Can the student read and construct Provide the student with a graph and set line, bar, and pictographs? of questions that may be answered using the graph. Have the student determine: The number of items represented by the graph. The variable with the greatest number of items represented. The variable with the least number of items represented. The variable with a specific number of items represented. Provide the student with survey data in an organized frequency table. Have the student make a graph to display the data. (140) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 8) Can the student read and interpret Provide the student with a circle graph circle graphs using percents? (e.g., pie chart) and a set of questions that can be answered by interpreting data using percents. Have the student determine: The percentage of occurrences of a given variable (e.g., What percent represents yellow?). The number of occurrences of a given variable (e.g., If 20% were yellow, how many out of 80 were yellow?). 9) Can the student construct and Have the student construct a T-Chart explain a frequency table? (frequency table) to record responses for a given survey question. Have the student explain the data recorded in the frequency table. 10) Can the student use probability to Have the student solve problems using predict the outcome of a single event probability to predict outcomes of a single and express the result as a fraction event occurrence (e.g., When you toss a and/or decimal? coin, what is the probability of it landing on heads?) Have the student express the probability as a fraction. Have the student express the probability as a decimal. 8) Can the student estimate and Provide a given set of numbers such as compare data to include mean, {15, 19, 14, 16, 15}. Have the student median, and mode? determine: The mean of the set (an average of the numbers). The median (the center location when the numbers are ordered least to greatest). The mode (the number that occurs most often in the set). (141) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 12) Can the student solve problems Provide manipulatives like snack photos involving combinations? and drink photos. Have the student model and list the possible combinations involving one snack and one drink. 13) Can the student measure length to Provide the student with a paper ruler the nearest one-sixteenth inch? marked to sixteenths. Have a variety of objects to be measured in class like a paperclip, penny, pencil, and book. Have the student measure these to the nearest sixteenth in lowest terms. 14) Can the student identify appropriate Provide the students with two sets of units for measuring length, weight, cards. The first set names objects to be volume, and temperature in standard measured, and the second set names the (English and metric) systems? measurements. Match the appropriate cards. Example: st nd 1 Set 2 set Length of book 10 in Length of room 10 ft Length of school hallway 10 yd Example: st nd 1 set 2 set weight of paperclip 5 gm 3 volume of box 120 cm length of eraser 7 cm 15) Can the student use appropriate Provide the student with customary mathematical tools for determining measuring devices: ruler, yard stick, tape length, weight, volume, and measure, and metric rulers. Have the temperature in standard (English and student measure the following : metric) systems? Length of pencil Length and width of book Height of chalk board (142) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample Provide the student with thermometers (Celsius and Fahrenheit) and warm and cold water. Have the students find the temperatures of the water in Celsius and Fahrenheit. Provide the student with beakers (metric and English) and containers of various amounts of water. Have the student find the amount of water each container holds. Provide the student with a balance and/or scales (metric and English). Have the student find the weight/mass of: bag of sand 5 paper clips eraser 16) Can the student use estimation to Provide the student with various objects in solve problems in standard (English the classroom. Let the student estimate and metric) systems? the length in inches and centimeters, then measure using appropriate tool to find actual length. Provide the student with a piece of grid paper (1 cm). Draw the outline of their foot. Estimate the number of square centimeters then count the squares. Provide the student with a balance. Have the student estimate the mass, then mass the following items to find the actual mass. cotton balls erasers quarters Provide the student with marked beakers. Have the student estimate then measure the number of milliliters and fluid ounces in a jar or cup. (143) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 17) Can the student convert units within Provide the student with a chart of equal a measurement system? measures. Have the student convert the following measurements: yards to feet quarts to gallons 100 inches to feet Provide the student with the following line chart. kg hg dag gram dg cg mg Have the student convert the following measurements: 5g =_____ mg 7cg =____ g 90g = ___ cg 3kg = ___ g 18) Can the student locate points in all Provide the student with a coordinate four quadrants of the coordinate plane. Ask the student to plot and label plane? the following points on the graph: (1,2) (‾3, 4) (‾6, ‾6) (2, ‾5) (0,4) (‾2, 0) Determine the quadrant or axis of each point. 19) Can the student draw points, lines Given dot paper the student will draw a (parallel, perpendicular, intersecting), point, parallel lines, perpendicular lines, line segments, and rays? intersecting lines, line segments, and rays. (144) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 20) Can the student identify, classify, and Provide the student with the following measure right, acute, obtuse, and diagram: straight angles? E D C A F B Ask the student to name and/or identify a right angle, obtuse angle, acute angle, and straight angle. Provide the student with a protractor and a set of angles. Have the student measure the angles. 21) Can the student create a tessellation Provide the student with a set of shapes. using polygons? Have the student create a tessellation from the shapes. 22) Can the student identify the vertices, Display an example of a pyramid. Ask the edges, and faces of three- student to identify the vertices, edges, and dimensional figures? faces. 23) Can the student identify and Provide the student with a set of pattern construct flips, slides, and turns? blocks and have the student model a flip, slide, and turn. 24) Can the student calculate the area of Have the student build a square and a a square and a rectangle without rectangle on the geoboard and find the using a calculator? areas of each. (145) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 25) Can the student find the Give the student a can, a string, and a circumference of a circle? ruler. Using these tools, ask the student to find the circumference of the can. 26) Can the student find area of a circle Provide the student with a calculator and using a calculator? the formula to find the area of a circle (A=r2). Have the student find the area of a circle with a radius of 5. 27) Can the student find the volume of a Provide the student with a base 10 cube. cube or rectangular prism? Have the student measure and find the volume of the cube. 28) Can the student read, write, and Provide the student a card with a number round twelve-digit whole numbers? written on it. The student will read the number orally, write the number in word form, and round the number to a given place value. 29) Can the student compare and order Provide the student with a set of numbers. whole numbers using <, >, and =? Ask him or her to compare pairs of numbers using the symbols. Then place the set of numbers in order from least to greatest. 30) Can the student write twelve-digit Provide the student with a place value whole numbers using expanded chart. Have the student identify the place notation? value of digits in a given number. The student will write the original number in expanded notation. 31) Can the student read, write, and Provide the student with a place value round decimals to the nearest ten- chart. Give the student a number to read thousandth? orally, to write in word form, and to round to the ten-thousandths place. (146) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 32) Can the student compare and order Provide the student with decimal decimal numbers using <, >, and =? numbers which they have to compare and order. Example: 4.24 4.024 Place the following in ascending order: 0.141, 0.0141, 0.1043 33) Can the student write decimal Provide the student with a place value numbers through the nearest ten- chart. Have the student identify the thousandth using expanded place value of digits in a given number. notation? The student will write the number in expanded notation. 34) Can the student use estimation to Have the student estimate a problem. determine the accuracy of solutions? Then have the student solve the same problem and compare the 2 solutions to check for accuracy. 35) Can the student multiply a three-digit Provide the student with a problem such decimal number by a two-digit as: decimal number? 0.718 or 9.23 x 0.45 x 1.6 Have the student estimate, then perform the operation. 36) Can the student divide a five-digit Provide the student with problems such decimal number by a two-digit as: decimal number? 9.6075 ÷ 0.25 or 0.23193 ÷ 0.03 Have the student estimate, then perform the operation. 37) Can the student round decimal Provide the student with a division quotients to the nearest whole problem containing decimals. Have the number, tenth, and hundredth? student find the quotient rounded to the nearest whole number, tenth, or hundredth. Example: 7 65.83 or 0.48 7.1 (147) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 38) Can the student estimate and solve Provide the student with one and two- one and two-step problems involving step problems involving addition, addition, subtraction, multiplication subtraction, multiplication and division. and division of decimals with and Have the student perform the following without calculators? tasks. Example: Estimate problems with and without calculators. Solve problems with and without calculators. Luke made $4.95 per hour. He worked 4 hours Monday, 7.5 hours Tuesday, and 3.75 hours Wednesday. About how much did he make in all? 39) Can the student use the rules of Give the student a number such as 4500. divisibility to determine factors and Example: Is 4500 a multiple of 3? Have multiples of a given number? the student use the rules of divisibility to determine factors. 40) Can the student explore the Have the student draw and label a relationship among integers? number line. Have the student order the numbers from least to greatest. 4, -4, 2, 1 Write an integer for a gain of 5 yards, a loss of 10 lb., deposit of $100, and withdrawal of $8. 41) Can the student model and write the Provide the student with a number. Have prime factorization of a number using the student model and write the prime exponential notation? factorization for the number. Give the solution in exponential notation form. Example: 90 = 21 3 2 5 1 (148) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 42) Can the student distinguish between Provide the student with a variety of prime and composite numbers, with whole numbers. Have the student and without the use of calculators? determine if the numbers are prime or composite without using calculators. Check accuracy of answers using a calculator. Example: 91(Composite), 200 (Composite), 97 (Prime) 43) Can the student use the greatest Give the student a fraction. Have the common factor (GCF) to simplify student determine the greatest common fractions? factor of the numerator and the denominator. Have the student simplify the fraction using the greatest common factor. Example: 9 3 3 12 3 4 44) Can the student use the least Give the student two or more fractions common multiple (LCM) to find with unlike denominators. Have the common denominators? student find the LCM of the denominators to determine the least common denominator. 45) Can the student convert among Have the student perform the following fractions, decimals, and percents? tasks: Convert 3 to a decimal and then a 4 percent. Convert 60% to a fraction and a decimal. 46) Can the student find the percent of a Provide the student with a picture of a number? CD player that indicates the price of the CD. Have the student determine the sales tax (7%) of the CD player, which costs $129.00. (149) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 47) Can the student estimate and Provide the student with a sale page. calculate sale price and/or original Have the student estimate then find the price using discount rates? sale price of a given item using the discount rate. Then, have the student determine the original price if given the sale price and the discount. 48) Can the student compare and order Provide the student with sets of fractions fractions and mixed numbers? and mixed numbers. Have him/her compare sets with two fractions and order from least to greatest sets of three or more 1 3 1 3 1 1 3 fractions (e.g., , > , , , 2 8 2 8 2 3 8 1 3 1 ordered from least to greatest , , ). 3 8 2 49) Can the student determine equivalent Provide the student with fractions. Have forms of fractions? the student match each fraction in Set A with its equivalent fraction in Set B. 1 3 1 a) , , 2 4 8 5 9 2 b) , , 10 12 16 50) Can the student use a variety of Provide the student with a fraction. Have techniques to express a fraction in the student express the fraction in simplest form? simplest form by: Finding the greatest common factor Example: 10 2 5 . 12 2 6 Using prime factorization. Example: 10 2 5 5 12 2 23 6 51) Can the student locate fractions, Provide the student with a number line. decimals and mixed numbers on a Write fractions, decimals and mixed number line? numbers on cards. Have the students locate these places on the number line. (150) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 52) Can the student add and subtract Provide the student with addition and mixed numbers, with and without subtraction problems with mixed regrouping, expressing the answer in numbers. Example: 8 1 1 1 2 4 simplest form using like and unlike Have the student work problems without denominators? regrouping and simplify answers. Have the student work problems with regrouping and simplify answers. Example: 3 1 1 5 8 6 53) Can the student multiply and divide Provide the student with problems using proper fractions as well as mixed multiplication and division of fractions numbers expressing the answer in and mixed numbers. Have the student simplest form? solve the problems and simplify the answers. Ex. 3 x 11 4 2 1 Ex. 12 1 3 54) Can the student estimate, solve, and Provide the student with several one and compare solutions to one and two- two-step problems that involve addition, step problems involving addition, subtraction, multiplication, and division of subtraction, multiplication, and proper fractions and mixed numbers in division of proper fractions and mixed which they have to estimate, solve, and numbers? compare the solutions. For example: Sal and Felicia went fishing. Felicia caught 3 fish whose total weight was 6 1 lb. Sal caught 2 2 fish. One fish weighed 3 lb. and the other 2 4 weighed 3 1 lb. Estimate the total weight of the 2 fish Sal caught. Whose catch weighed the most? 55) Can the student demonstrate Have the student express the ratio of different ways to express ratios? boys to girls in the class in three different ways. Example: There are 5 boys and 8 girls in the class. 5:8 5 to 8 5 8 (151) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Suggested Teaching Strategies for SIXTH GRADE Uses properties to create and simplify algebraic expressions and solves linear equations and inequalities 1) Solve equations in one variable using addition and subtraction Use stories to give meaning to equations. For example there are N number of bears in a boat. Eight jump out and three were left. How many were in the boat? Have the students act out the story by pretending to be the bears. Give the student a set of numbers such as 3, 7, and 10. Student will construct a fact family. Example: 7 + 3 = 10 3 + 7 = 10 10 – 3 =7 10 – 7 = 3 Have students use Algebra tiles to model addition and subtraction equations. Example: x + 2 = ‾1 Step 1 Step 2 Step 3 Answer x + + = - x + + = - x = - x = ‾3 - - = -- - - Example: x – 4 = -6 Step 1 Step 2 x - - - - = - - - - - - x - - - - = - - - - - - ++++ ++++ Step 3 Answer x = - - x = -2 Ask the student to explain the process being used to solve each of the equations. Compare solving an equation to a balance to help students understand the property of equality. When you add or remove a quantity from one side of a balance, you must add or remove the same quantity from the other side in order to keep it balanced. The same is true for an equation. (152) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 2) Models simple addition and subtraction problems using integers on a number line Tape a number line on the floor. The students will explore integers by walking the number line. To walk the number line, always start facing the positive numbers. When adding two integers, move forward for positive numbers and backwards for negative numbers. If subtracting, the student must turn around and face the opposite direction for the subtraction sign. This strategy may be used as a game by dividing the class into two groups. Give a problem to one person in each group. Have both students walk the number line. The student that is the greatest distance from zero earns a point for the group. Repeat this process allowing other students to participate. Provide two sided counters to model addition and subtraction. Let red be negative and yellow be positive. Pull out any zeros. Zeros are one red and one yellow. Whatever is left is the answer. Example: 3 + ‾5 Y Y Y R R R R R Two reds are left; therefore, the answer is -2. Give a deck of cards to a pair of students. Let the black cards be positive and the red cards negative. Place eight cards face up on the desk. The objective of the game is to make zero using as many cards as possible. For instance, a red 5 and a black 5 represent a ‾5 and a +5 which cancel each other out and equal zero. The following cards would also equal zero: ‾4 + ‾2 + ‾10 + 10 + 6 = 0 or red 4, red 2, red 10, black 10, black 6 3) Recognizes and continues a number pattern and/or geometric representation (e.g., triangular numbers) Use unifix cubes or graph paper and have the students create the following buildings and extend the pattern. Building 1 Building 2 Building 3 Building # # of Windows 1 3 2 6 3 9 4 5 10 (153) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Students will then complete a function table for 1, 2, 3, 4, 5, and 10. Have the students extend the pattern for the 100th building and write a rule for this pattern. Students can also create and extend their own pattern using the cubes. Have the students study the pattern below: 999 x 2 = 1998 999 x 4 = 3996 999 x 3 = 2997 999 x 5 = 4995 Have the students explain the patterns they see. Have the students extend the pattern to find 999 x 9. An extension to this activity would be for the students to use the pattern to find 9,999 x 7 and 99,999 x 8. Find the product for 15 x 15, 25 x 25, 35 x 35, and 45 x 45. Have the students look at the products and the factors to determine the pattern. Have the students use the pattern to find 85 x 85, 115 x 115, and 135 x 135. Students can check answer with a calculator. 4) States a rule to explain a number pattern Have the student create a function table for the following example. The average American family wastes approximately 30,000 gallons of water a year. F is the number of families and W is the amount of water wasted in a year. Down the Drain F 1 2 3 4 5 10 100 W 30,000 60,000 90,000 120,000 ------ ------ ------ The students will find the pattern and write a rule. Before using the rule to extend the pattern, have the students check the rule with the values that they already know. An extension to this assignment would be for the students to determine how much water is wasted in their own town and in their state. The teacher will think of a rule. One student at a time will call out a number. The teacher applies the rule to the number and tells the class the new number. Repeat the process until someone determines the rule the teacher is applying. (154) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 5) Using whole numbers to complete a function table based on a given rule Give students the rule, then sing ―The Twelve Days of Christmas‖. Discuss how the rule could be used to determine the total number of gifts given. N=1st day (1 gift) Days 1 2 3 4 5 6 7 8 9 10 11 12 Gifts 1 3 6 Use a scientific calculator to complete a function for the rule 0.5(n + 8) n 1 2 5 10 15 100 0.5(n+8) 4.5 6) Creates and solves proportional equations using one variable Provide each student with several different colored cubes, such as green and red. Students will model a ratio of 1 (red) to 3 (green) using the colored cubes. Students will determine an equivalent ratio for the number of red cubes to 12 green cubes. Use the steps for solving a proportion using cross products, to solve the following: n 12 5 6 6n 60 n 10 Interprets, organizes, and makes predictions using appropriate probability and statistics techniques 7) Reads and constructs line, bar, and pictographs Have students cut out examples of line, bar and pictographs from newspapers and magazines. The students will answer questions pertaining to these graphs. Divide the class into groups. Each group will collect data on a topic decided by the group. They will decide which type of graph would best display their data. They would then construct either a line, bar, or pictograph. If computers are available, the graphs could be created using a spreadsheet. 8) Reads and interprets circle graphs using percents Have the students to predict how many hours per day they spend doing various activities. They will create a circle graph from their data. (155) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 3-D Circle Graph: Cut the margin off a piece of notebook paper. Make sure there is a space for each person in the class. Take a survey and make a Tally Chart. Example: Colors (10 people: 5 blue, 3 red, 2 yellow) B B B B B R R R Y Y At this point, when spaces are colored, tape ends of paper together and make a point on a sheet of paper for the center of the graph. Place taped paper around point and trace to a circle. Mark on the outside of the circle where each color ends. Draw a line from the center dot to each mark on the outside of the circle. Shade in appropriate colors. Change to percents. Blue 5 Yellow Red 2 3 5 3 2 = 50% = 30% = 20% 10 10 10 9) Constructs and explains a frequency table Give students post-it notes to write down the month they were born. The students will place their post-it notes by month to make a frequency table and a bar graph. Collect data on the color of students’ math notebooks (binders, folders, etc.). Have the student make a frequency table from the data. Volunteers will explain the information in the table. 10) Uses probability to predict the outcome of a single event and expresses the result as a fraction or decimal Tell students that there are 3 white marbles and 1 blue marble in the bag. The students will predict the outcome of drawing one marble from the bag 1 3 and test their theory. Example: blue = ; white = 4 4 Have each student write the type of animal they like best on an index card. Collect cards, make a frequency table, and discuss the data. Have students predict the probability of choosing their favorite animal if the cards are mixed. Express as a fraction. Use a calculator to change fractions to decimals. 11) Estimates and compares data to include mean, median, and mode Give students a small bag of candy (skittles, M&M’s). The students will count the number of candy pieces in each bag. The data will be collected and the students can find the mean, median and mode. This could be done for individual colors or for total pieces per bag. (156) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Have each student call out his/her shoe size. Make a tally sheet of sizes. From this data, find the mean, median, and mode. Discuss results. 12) Solves problems involving combinations Give students pictures of 3 different color shirts, 4 different color pants, and 2 different types of shoes. (Pictures may be from catalogs, magazines, or have students use construction paper to draw each item of clothing.) The students will manipulate the different combinations and write each combination. The class will discuss the results. Give students a list of five pizza toppings (cheese, sausage, pepperoni, bacon, hamburger) and three types of crusts (pan, thin, hand-tossed). Students will draw or use manipulatives to model the possible combinations. Writes and solves problems involving standard units of measurement 13) Measures length to the nearest one-sixteenth inch 1 Instruct students on how to count by increments. Students will simplify 16 fractions to lowest terms. Example: Provide the students with a line that 4 measures 2 in. After students place their ruler on the line, they locate the 16 1 2 3 4 2‖ mark and count by sixteen’s such as , etc. Four 16, 16, 16, 16 1 1 sixteenths will then be simplified to . The final answer is 2 . 4 4 Have students collect items such as pine needles, blades of grass, acorns, leaves, sticks, etc. and measure to the nearest one-sixteenth of an inch. 14) Identifies appropriate units for measuring length, weight, volume, and temperature in the standard (English and metric) systems Write the words length, weight, volume, and temperature on the board. Give students a list of choices like 7 feet, 90˚ F, 3 pounds, 5 ounces, 16 fluid ounces and have them sort into the correct categories. More choices could be metric like 6 cm, 58 grams, 500 milliliters. Have one half of the students give examples of length like 5 inches and the other half give an example of something that has that length or have one group name an object and the other group guess a weight, length, temperature or whatever is being emphasized. Have students bring labels of canned goods, liquid measure, lengths of objects like buttons, ribbon, etc., and categorize them according to length, weight, and volume. 15) Uses appropriate mathematical tools for determining, length, weight, volume, and temperature in the standard (English and metric) systems Bring cup, pint, quart, gallon containers, and water to class. Have students find the number of cups in a pint and so on recording their findings. Have the (157) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 students create a conversion chart. Then show containers of different sizes and let the student tell the most appropriate measure to use. Bring rulers, meter or yardsticks, and tape measures to class. Have various objects in the classroom for students to measure. Students measure, record their measurements and tell why they chose that measuring tool. 16) Uses estimation to solve problems in the standard (English and metric) system Have a Metric Olympic. a) Hand out graph paper that is marked in 1 centimeter units. Let students guess how many squares on the paper, then count the squares. b) Let students draw their hand outline on the paper, guess the number, and then count the squares. c) Have students throw a paper plate from some point and guess the distance in centimeters, then measure. d) Have students guess the mass of a handful of cotton balls or paperclips, then find the actual mass. These same activities could be done using the English measurement system. Have students estimate their heights, the heights of their desks, the length of their books, then find the actual measurements using appropriate tools. Keep a thermometer in the classroom and have someone guess the temperature, then check it. Use Celsius as well as Fahrenheit. 17) Converts units within a measurement system Provide the student with a chart of equal measures. Show students how to use cancellation to find answers. Example: Change 3 pints to cups 2c 6c 3pt 6c 1pt 1 Example: Change 500 feet to yards 1yd 500 1 500 2 500ft yd yd 166 yd 3ft 3 3 3 Example: Change 7.5 feet to inches 12 in 7.5 12 90 7.5 ft in in 90 in 1ft 1 1 Explain that whatever unit they begin with must be the unit in the denominator of the second fraction and the numerator is its equal measure. Multiply numerators. If a number is greater than 1 in the denominator, divide by that number. (158) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Provide a metric line chart for each student and one on the chalkboard. Divide students into groups and give each group enough problems for each person to have a problem. Give 2 M&M’s to each student. These are their decimal points. Place several white beans on each group’s table. These are the zeros they will add to which ever side they need. Demonstrate moving the decimal point according to the number of places to the new measurement. Metric Chart Kilo Hecto Deka M Deci Centi Milli Examples: 34 km = 34000 m 34_______ = 34000m 5 cm = 50 mm 5_______ = 50 mm 17 mm = 0.017 m 17_______ = 0.017 m The saying ―King Hector Died Monday Don’t Call Me‖ may help students remember the chart.― Convert using multiplication or division. From small unit to large unit, divide; from large unit to small unit, multiply. Example: 3 gal. = 12 qt. 120 in. = 10 ft. Multiply by 4 Divide by 12 Draw a very large capital G on the chalkboard/overhead. Explain to the students that the G stands for one gallon. Within the G write four capital letter Q’s. This stands for 4 Quarts. Inside each of the 4 Q’s, write two capital letter P’s. This stands for 2 pints in 1 quart. Within each of the P’s write two capital letter C’s. This stands for 2 cups in 1 pint. Within each of the C’s place 8 dots. These dots represent fluid ounces. There are 8 fluid ounces in 1 cup. Determines the relationships and properties of two and three-dimensional geometric figures and the application of properties and formulas of coordinate geometry 18) Locates points in all four quadrants of the coordinate plane Assign students an ordered pair. Using a large floor coordinate plane, have students locate their point by standing in its location. Given the following set of ordered pairs, plot the points on grid paper and connect the line segments between each of the coordinates given to create a picture. (-1,3) (8,3) (8, -2) (-1,3) (159) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 19) Draws points, lines (parallel, perpendicular, intersecting), line segments, and rays Using a city street map have students identify streets that are parallel, perpendicular and intersecting. Also have the students to name a point of origin and continue indefinitely North or South, East or West to represent a ray or corner to corner to represent a line segment. Ask the students where else can they find examples of parallel, perpendicular, and intersecting lines, segments and rays. 20) Identifies, classifies, and measures right, acute, obtuse, and straight angles Give students a picture of the outside of a house from a magazine. Students will identify angles in the picture and trace them on a sheet of paper. They will then classify the angles as right, acute, obtuse or straight. Then they will use a protractor and measure each angle. Use a 3-D architecture program on a computer to obtain the original picture. A B C D Name Name Name Name__________ Estimate Estimate Estimate Estimate Actual Actual Actual Actual 21) Creates tessellations with polygons Discuss tessellations. Provide examples of tessellations (tile on floor, neck ties, fabrics, quilts. Give students several geometric figures to discover which combinations of figures tessellate. (Pattern Blocks may be used) 22) Explores the relationships of three-dimensional figures including vertices, faces, and edges using manipulative materials Using polyhedrons or straw polyhedrons, have students build a net and form the following polyhedrons. Answer the following questions. Polyhedron Net (160) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 How many faces are on a cube? What is the shape of each face? How many vertices are on the cube? How many edges are on a cube? Have students build the following polyhedrons and complete the chart. Polyhedron Faces Vertices Edges Shapes of Faces Tetrahedron Octahedron Hepahedron 23) Describes, compares, constructs, classifies, and identifies flips, slides, turns (reflections, translations, and rotations) Using pattern blocks ask students to take each shape and to produce a design using slides, turns, and flips. Model for students translations (slides), rotations (turns), and reflections (flips) using pattern blocks, or other designs (tiles). slide rotation flip Be sure to model both vertical and horizontal flips. Have students create a tile design and color it. Example: Using the tile they have created, have students translate and rotate their tile design and record the pattern to form a border design. Slide: Rotate: Using another tile, have students create a flip (horizontal) border by reflections. (161) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 24) Calculates the area of parallelograms (squares and rectangles) without using a calculator Use a ruler to draw various squares and rectangles on grid paper. Students will count the number of squares used to form the figure to find the area of each figure. Have students work with a number and partners. Use snap cubes to model all possible rectangles with areas of 24 squares and 12 squares. After students have modeled with snap cubes, have the student use centimeter grid paper and draw all possible rectangles with area of 10, 16, and 36. Students complete a chart displaying the length, width, and area of the rectangles they drew and use this information to develop a formula for the area of a rectangle. 25) Finds the circumference of a circle with and without the use of manipulative materials Have students collect various size cans (Pringles, coffee, peanuts, etc.). Have students estimate the distance around each can before they measure and record. Compare the estimate to the actual results. Then using a metric tape, measure around the can and record results. Introduce students to the formula for finding the circumference of a circle x diameter or C=d, where =3.14. Have students find the diameter of the cans with the metric tape and substitute that measurement with the formula. After the students have calculated the circumference of the cans using the formula, compare the answers to those in activity 1. Example: Can Diameter measures (13cm) (3.14) = 40.8cm 26) Determines the area of a circle with and without the use of a calculator Using centimeter grid paper, draw a circle. Count the number of squares inside the circle, estimating any sections inside the circle that are not complete squares. (Student may overestimate or under estimate) Using grid paper, draw a circle with a diameter of 4 cm. Divide the circle into eight equal parts. Cut the circle into wedges. Color half of the wedges and cut the wedges apart. Fit the wedges together to form a shape that looks like a parallelogram. Example: (162) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 The height of the parallelogram is the radius of the circle. The base of the 1 parallelogram is the circumference of the circle. So, the area of the circle is 2 (height) (base) 1 r (2r) 2 1 2 r2 2 A= r2 27) Finds the volume of cubes and rectangular prisms with and without the use of calculators Investigate volume by making containers of different shapes and comparing how much each container holds. Divide the students into small groups. Using 5‖ x 8‖ cards the students will make 3 containers: one circular, one square, and one triangular. Use tape to hold cards together. Then tape the ends of each container to another card so the container has a bottom. The base of each figure has a perimeter of 8 cubes. Each figure is 5 cubes high. Estimate which container will hold the most, which will hold the least? Fill one container with filler (rice, etc.) Does the rice fill this container? Is there too much rice? Continue this activity until you can determine which container holds the most, the least. Divide the class into groups of four. Provide each group with models of prisms and cubes, centimeter cubes, and filler (rice, cereal, sand, etc.). Have students fill the rectangular prisms with centimeter cubes, counting the number needed to fill it. Have students use the appropriate formula to find the area of the base of a rectangle or square and then use the height to compute the volume. Uses basic concepts of number sense and performs operations involving exponents scientific notation, and order of operations 28) Reads, writes, and rounds twelve-digit whole numbers Hang a clothesline. Label place value through 100 billion above it. Give each student an index card with a digit, 0-9, written on it. Call students randomly to bring their number to the place value line. Students attach their card to the line with a clothespin. Students will read the number, write the word name, round it to a given number, and expand it. Give the students a map of the solar system and have students determine the planets that are within 999,999,999,999 miles from the Earth. (Make sure numbers are not in word form on the map.) Have students choose the planet that is farthest from the Earth within the range and allow them to write the number in word form. (163) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 29) Compares and orders whole numbers using <, >, and = Give students a deck of cards with face cards and the 10’s removed. The students will place a given number of cards face up in a row. They will turn up a second row of cards. 5 4 2 9 3 4 5 9 6 1 The students write a statement using the 2 numbers created. They can use all operations, compare, round, expand, read, and write the numbers. Bring 5 different brands of popcorn to class. Pop the popcorn and have the students count the number of popped or unpopped kernels from each bag and then compare and order the different brands. 30) Writes twelve-digit whole numbers using expanded notation Cut out squares from colored construction paper. Students work with a partner. The teacher will give each pair an index card with a number on it and provide each pair with a handful of cut out squares. The teacher will model the process first on the board/overhead using the number 435. The teacher will point to the digit 5 and ask in what place is this digit? 4 3 5 Ask how many ones are sitting in this place? (5). Then, draw 5 squares underneath the number 5. Do the same for 3 and 4. (See below) 4 3 5 100 10 1 100 10 1 100 10 1 100 1 1 400 + 30 + 5 = 435 You can substitute candy by assigning place values to colors. (164) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Give each student a whole number with up to twelve digits. Have the student write the number in expanded notation using parentheses. Then have students check their own or others using a scientific calculator to compare with the original number. Example: 431,175,903 (4 x 100,000,000) + (3 x 10,000,000) + (1 x 1,000,000) + (1x 100,000) + (7 x 10,000) + (5 x 1,000) + (9 x 100) + (0 + 10) x (3 x 1) Make sure students enclose numbers in parentheses on calculators, if necessary. 31) Reads, writes, and rounds decimal numbers to the nearest ten-thousandth Have students use base-ten blocks and place value charts to model decimal numbers, such as 2.24. See example below. The teacher can either put decimal numbers, one at a time, on the board for students to model or can hand each student an index card that has a decimal number written on it to model. The students will read the number to a partner, write the word name of the decimal number, and round it to the nearest __________. 2 ONES 2 TENTHS 4 HUNDREDTHS (165) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Copies of centimeter graph paper can be made, laminated, and cut out so that each student has 4 wholes (4 10-by-10 cm squares), 2 tenths (2 1-by-10 cm squares), and 20 hundredths (20 1-by-1 cm squares). Students play the ―Memory Game‖ individually, or with a partner. Each group will use a deck of 16 cards. Eight of the cards should have a decimal number written on it. The remaining 8 should have the word name match for the decimal number. The cards should be placed face down in rows of 4. Player #1 will turn up 2 cards. If a match is not made, Player #2 will turn up 2 cards. If there is a match the player continues until no match is made. The winner is the player who gets the most matches. Two and one hundred forty- 2.141 one thousandths 0.6 six tenths Cards can also be constructed for rounding. See example below. 4.24 4.2 Students use a place-value chart and blocks to do this activity. Ahead of time the teacher will take 12 index cards (1 set of 12 for each pair of students) and write a one-digit number on each of 11 cards and a decimal point on one card. These 12 cards will be kept in a baggie. One student will construct a number using all 12 cards on the place value chart. The other student will read the number, write the word names for the number and round it to the nearest tenth, hundredth, thousandth and/or ten-thousandth. Player #2 will then do the same thing. Place students in pairs. Each pair will receive 1 sheet of newspaper and 2 different colored markers. Each pair will circle as many decimals as they can find on their newspaper within 5 minutes. The pair who finds the most will place their newspaper on the bulletin board. Students will be called at random to go to the bulletin board to read one of the circled decimal numbers. Have the teacher hold up a sheet of construction paper that has a decimal number written on it. Students can write the word name for that number and round it to a given place. Set up an interactive bulletin board in which students can use as a center to practice reading, writing, and rounding decimal numbers. (166) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 32) Compares and orders decimal numbers using <, >, and = Students are provided a stack of index cards that have decimal numbers written on them. The students will place these cards in order from least to greatest. These can also be placed on a number line. Write a less than, greater than, or = statement using appropriate symbols. Play the game of ―Battle‖. Students work in groups of 2-4. Each group is provided a stack of decimal cards in which a decimal has been written on it. The stack of cards is dealt among the players. Each player places a card face-up. The player that has greatest number gets the cards. The person with the most cards at the end of the game is the winner. Prepare a paper bag containing 11 cards-one each for the numbers 0-9 plus a card with a decimal point. Each student will make an answer board by drawing a row of five boxes in which to place digit choices. (See below.) The teacher will pull one card from the bag, show it to the students, then return the card to the bag. Students will write the digit in one of their boxes. If the teacher draws the decimal point from the bag, it will go before the next digit is drawn. After the decimal point is drawn, leave it out so it cannot be drawn again. The teacher will draw from the bag until all boxes have been filled in. The student that has the largest number will draw from the sack next game. This is a good activity for small groups. 33) Writes decimal numbers through the ten-thousandths’ place using expanded notation Have students play the ―Memory Game‖ individually, or with a partner. Each group will use a deck of 16 cards. Eight of the cards should have a decimal number written on it. The remaining eight should have the expanded form written on it. (See example below.) The cards should be placed face down in rows of 4. Player #1 will turn up 2 cards. If a match is not made, Player #2 will turn up 2 cards. If there is a match the player continues until no match is made. The winner is the player who gets the most matches. 6.25 (6 1) + (2 0.1) + (5 0.01) (167) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Have students use a place-value chart and blocks to do this activity. Ahead of time the teacher will take 12 index cards (1 set of 12 for each pair of students) and write a one-digit number on each of eleven cards and a decimal on one card. These 12 cards will be kept in a baggie. One student will construct a number using 12 decimal cards on the place value chart. The other student will write the expanded form of the number. If correct, the students swap turns. The teacher will hold up a sheet of construction paper that has a decimal number written on it. Have students write the expanded form for that number. The opposite can also be done. (An expanded notation may be held up and students may write the decimal number.) 34) Uses estimation to determine accuracy of solutions Bring an apple to class and the students estimate the weight of the apple. The students discuss their conclusion. The students weigh the apple and determine the accuracy of their solution. Have the students estimate the answer to a given problem: Example: John wants to buy 3 shirts that cost $8.65 each and 2 pair of pants at $9.39 each. If he hands the cashier three $20.00 bills, how much change will he receive? First estimate the change, then solve and find actual change he will receive. Compare the answers. 35) Multiplies a three-digit decimal number by a two-digit decimal number Write several 2 and 3 digit decimal numbers on the board. Have students select one three-digit decimal number and one two-digit decimal number to multiply and solve the problem. Use a calculator to check for accuracy. Give the students several multiplication problems and allow them to use a calculator to determine the answer. (Do not allow the answer to end in zero.) The students will write down the answer. Tell the students to focus on why the decimal appears where it does. The students will discover the rule for multiplying decimals. Example: 4. 26 3 numbers behind the decimal x 5 .2 22 .152 3 numbers behind the decimal 36) Divides a five-digit decimal number by a two-digit decimal number Have students use the calculator to divide a 5-digit decimal number by a 2- digit decimal number. (168) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Give students several problems with the same numbers but with the decimal point in different places (up to five digits). These numbers will be divided by two 2-digit decimal numbers (same numbers, decimal point in different places). Possible quotients will be given. Students will match the problems with the correct quotients. Calculators can be used to check results. Example: 6.75 0.25 .675 0.25 67.5 0.25 6.75 2.5 .675 2.5 67.5 2.5 Possible Quotients: 27, 2.7, 270, 0.27, 0.027 37) Round decimal quotients to the nearest whole number, tenth, and hundredth Give the students a problem such as: Four people go to the store and purchase candy for $9.61. How much does each have to pay? Round the solution to the nearest hundredth (cent). Give students 10 proper or improper fractions with denominators, which will not terminate when changed to a decimal. Use calculators to convert each fraction to decimal form. Round each to the nearest whole number, tenth, and hundredth. 38) Estimates and solves one and two step problems involving addition, subtraction, multiplication, and division of decimals, with and without calculators Have the students create a menu for a concession stand for their school. They decide what will be sold and how much each item will cost. The students display the menu on a sheet of poster board. The teacher selects a specific amount of money and displays it on the board/overhead. Student places an order and the class will decide if the amount displayed will be enough to pay for the order or not enough to pay. The students then use a calculator to compute the exact amount needed to purchase the order to determine whether or not the estimation was adequate. Have the students work in pairs or small groups. Give each group a map and have the students plan a trip. They choose the kind of car to be used. The teacher will list several makes and models along with a list of miles per gallon of gas each car will get. The students will estimate the total number of miles they will travel and estimate the cost of the gas. The students then find the exact distance they will travel and calculate the exact cost of gas (a calculator may be used). (169) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 39) Uses the rules of divisibility to determine factors and multiples of a given number Give each student a hundreds chart and a ziploc bag containing 50 beans. (A hundreds chart is a 10 x 10 square with the numbers marked on it.) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Tell students to cover every number that 2 will divide into evenly. These numbers are divisible by 2 and are multiples of 2. Remove the beans and do the same for 3, 5, 6, 9, and 10. Discuss divisibility rules for that number each time. Provide the rules of divisibility to each student. Have each students write an example of each rule with all examples being greater than 100. Students will swap with a partner and check each other’s work. 40) Explores the relationship among integers Make a large thermometer on the board/overhead. Ask the students to locate various pairs of temperatures and write an inequality to compare them. Put students in groups and distribute scissors, rulers, three magic markers, and a foot of string to each student. Fold strings in half and color this point with one of the markers. This represents 0. Measure 1 inch on each side of this point and tie a knot. These knots represent +1 and –1. Color the right side one color (positive) and the left side another color (negative). Measure 1 inch from each of these points and tie 2 more knots. These represent +2 and –2. Color accordingly. Continue doing this as far as they can go adding a knot each time. Make a set of index cards with one number on each card using positive and negative numbers including zero. Give each student a number and have them form a line from least value to greatest value. Have a number line marked on the floor. Give each student a problem and have a student walk a problem on the number line, such as 3 + 2, 5 + -3, -6 +ˉ1, 8 – 4, ˉ7 – 2. Be sure to explain that subtract means to add its inverse. This means 8 – 4 is really 8+ ˉ4 and ˉ7 – 2 is ˉ7 +ˉ2. (170) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 41) Models and writes the prime factorization of a number using exponential notation Use a factor tree to find the prime factors of a number. Example: 110 81 11 x 10 9 x 9 11 x 5 x 2 3x3x3x3 Answers: 110 = 2 x 5 x 11 81 = 34 Make a prime factor column. Always use the smallest prime number that will go into the dividend. Factors go on the outside of the upside down division sign with the quotient on the inside of the division sign. Keep factoring with primes until the last quotient is 1. Example: 2 18 Answer: 2 x 32 = 18 3 9 3 3 1 42) Distinguishes between prime and composite numbers, with and without the use of calculators Give each student a ziploc bag of 100 cubes or tile squares. Begin with 1, then 2, 3 through 20 and arrange each number of tiles in rectangles. If the cubes can be arranged in only one way, the number is prime. If the cubes can be arranged in more than one rectangle the number is composite. Example: 3 is the same as Both are 1 x 3 so 3 is prime. 6 But 6 = 1 x 6 or 3 x 2, so 6 is composite. 3 2 (171) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Give each student a large 10 x 10 square. Have them number each square from 1 – 100. Have them cross out every second number after two, every third number after 3, every fifth number after 5, every seventh number after 7. The numbers that are left are prime. This is called the Sieve of Eratosthenes. Using a calculator, let the students find a factor of the numbers they thought were prime. 43) Use the greatest common factor (GCF) to simplify fractions Write the numerator and the denominator of a fraction as prime factors. Draw a line through the factors that are the same in both. What is left is the simplified fraction. The factors that are the same in numerator and denominator determine the GCF. Ex. 9 3x 3 3 ; GCF = 3 12 2 x 2 x3 4 Use Venn diagrams to simplify fractions. First factor the numerator and denominator into primes. The number contained by both circles is the GCF. What is not in both circles is the simplified fraction. Ex. 6 2x3 GCF = 2 16 2x2x2x2 2 3 2 3 2 2 Simplified fraction 3 2x2x2 8 44) Uses the least common multiple (LCM) to find common denominators Use factor trees to find the prime factors. Put the factors in a Venn diagram. Multiply all the factors together. Example: 4 20 12 15 12 60 7 28 15 60 3 x 4 3 x 5 3 x 2 x 2 2 3 5 2 2 3 5 = 60 2 Multiply the two denominators together. Divide by the highest number the denominators have in common. The results is the least common multiple of the two numbers. (172) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 3 and 7 4 x 12 48 4 12 4 12 2 and 7 5 x 8 40 (They have no factor in common) 5 8 9 and 7 10 x 15 150 5 30 10 15 Determines relationships among real numbers to include fractions, decimals, percents, ratios, and proportions in real life problems 45) Converts among fractions, decimals, and percents Provide students with fictitious sports data or data that has been collected from the school’s teams. Students can determine shooting percentages during basketball season and batting averages during baseball season. Example: Number Shots Made or Number of Hits Total Shots Attempted Total Number of at Bats Have students work in groups of 4. Each group is given 4 index cards. Students will write their first name on an index card and determine the number of consonants or vowels on their first name. These will be written in ratio, decimal, and percent form. These cards can be placed in order from least to greatest. Play the ―Memory Game‖. Students work with a partner. Each group is given a stack of 24 cards. Eight of the cards will have a ratio written on them, eight will have the decimal form, and the remaining eight will have the percent form. See example below. Shuffle the cards and place face down in rows of four. One player turns up 3 cards to see if he/she gets a 3-way match of a ratio, decimal, and percent. He/she continues until a 3-way match is not made. When this happens, the next player takes a turn. The student with the most matches at the end of the game is the winner. 3 4 0.75 75% Students work in groups of 2-4. Each group receives a deck of 42 cards. (These cards have fraction, decimals, and/or percents on them.) Each student is dealt 7 cards. The remainder of the cards are placed in the center of the table. Students play according to ―Gin Rummy‖ rules. Give each student an index card that has either a ratio, decimal, or percent written on it. Students are instructed to get out of their seats and locate their 3 match. For instance, if a student is holding the fraction , he/she will look for 4 the 0.75 and the 75%. (173) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Have students look in reading books or textbooks and locate 3 lines. They can do any of the following to determine fraction, decimal, and percent form: Make a line plot depicting the letters of the alphabet. For instance, X X X X X X X X X X X X A B C D E F G H I J K L M Repeat the above example and compare the number of 3-letter words vs. the number of 4-letter words. Results will be written in fraction, decimal, and percent form. Students determine the fraction, decimal, and percent form of common nouns vs. adjectives. Graphs can be constructed to display the data. Collect grocery store receipts. Try and get receipts from at least 2 different grocery stores for at least 2 weeks before this activity. Divide the class into 2 groups. One group will get the receipts from one store, while the other group gets the receipts from the 2nd store. These 2 groups can be subdivided when the activity begins. Students will predict which digit they think will appear most frequently as the last digit in a price from the receipts. Each group will construct a line plot for each digit 0-9 using the data from the receipts. See example below. 0 1 2 3 4 5 6 7 8 9 X X X X X X X X X X The results can be expressed in decimal, fraction, or percent form. Graphs can also be constructed to display data. The results can be compared as to which grocery store had the number ____ to show up the most. Students work with a partner. Each group will be given a page from the local newspaper and a transparent centimeter grid sheet to be used as an overlay. Students will estimate the total area of the newspaper page, excluding the margins, and determine the area of each category. The categories could consist of local news, national news, advertisements, entertainment, sports, photographs, weather, obituaries, etc. Students then express the area of the article to the area of the page as a fraction, decimal, and percent. The class then records all findings. The total area of each of the categories is calculated. These totals are compared to the total number of pages of the newspaper. Using this data, students decide how much of the newspaper is really news. (174) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 46) Finds the percent of a number Sample Problem: If my class has 30 students and 60% are girls, how many girls are in the class? Give the students 10 M & M’s. They must give their neighbor 20%. How many M & M’s will they give away? 47) Estimates and calculates sale price and/or original price using discount rates Set up a class store that contains pictures of items found in a department store. Have different departments set up, such as a furniture department, a clothing department, a shoe department, etc. Have prices on the items. Display a percent off sign in each department. Students rotate centers and estimate first what the sale price will be, then calculate to find the exact answer. Calculators can be used to check work. Provide students a menu, or have them make up a menu of their own. Each student orders a meal from the menu. The total cost of the meal will be determined. Students can determine the sales tax (7%) and gratuity (15%). Each student gets a flier from J.C. Penney, Wal-Mart, or K-Mart, etc. that shows the regular price and sale price, a pair scissors, glue, and a sheet of construction paper. Have students cut out 3 ads and glue to the construction paper. Students then find the discount. Swap and solve. Calculate the original price if the sale price is $60 after a 25% discount has been deducted. 48) Compares and orders fractions as well as mixed numerals Give the student a combination of two proper fractions, two mixed numbers, or one from each set. Have the student draw rectangles of similar size to illustrate each fraction. Have the student compare the two illustrations to determine if the fractions are greater than, less than, or equal to each other. 3 2 4 3 3 2 Example: > 4 3 Have students use cross-products to compare fractions. 5 4 Example: < because 5 x 7 is less than 9 x 4. 9 7 Work in groups of 5 or 6. Have the students measure the length of their shoes to the nearest 1 of an inch. Let one student record the information on 16 an index card. Have the group arrange the measures in order from least to greatest. Have students compare their data with the other groups to determine the smallest measure and the greatest measure. (175) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 49) Determines equivalent forms of fractions Have students work in pairs. Make a gameboard by using a 3 by 3 grid. Put a fraction in each of the 9 squares. Each player is given either nine 0’s or nine’s. Player 1 should put a marker over one of the fractions and call out an equivalent fraction. Player 2 will decide if the two fractions are equal. If an equivalent fraction is given, the first player claims the box. If an equivalent fraction is not given, the player must give the box to his opponent. The 1st player to get 3 in a row vertically, diagonally, or horizontally wins. On a sheet of paper, list several fractions. On the opposite side, in random order, give equivalent fractions. Have the students connect a fraction from the left to a fraction on the right. Give more fractions on the right side so that some will be left over. 50) Uses a variety of techniques to express a fraction in simplest form Give each student a fraction to simplify using two methods, such as dividing numerator and denominator by their GCF, and prime factorization of numerator and denominator. (See Strategy #43) Students will solve and swap with a partner to check. Make two lists of equivalent fractions (Set A will not be simplified. Set B will be simplified). Use a calculator that will simplify fractions to match each fraction in Set A with its equivalent form in Set B. 51) Locates fractions, decimals, and mixed numerals on a number line Have students work in groups of 4. Each group is given a roll of adding machine tape and 4 index cards with each card containing a fraction, decimal, or mixed number on it. Students will tear off a piece of tape and display their number on it. These can be put in order from least to greatest. String plastic clothesline wire across the room to serve as a number line. Clothespins can be used to clip index cards that have fraction numbers, decimal numbers, or mixed numbers on them. 52) Adds and subtracts mixed numerals, with and without regrouping, expressing the answer in simplest form using like and unlike denominators Have the students work in pairs. Give each student a stack of number cards with the digits 1 – 9 written on them. Place the cards face down and spread them out. Have the students draw 3 cards from their own pile. The students each will form one mixed number from the 3 cards. Example: If the student draws the number 2, 3, and 4 some possible 2 3 2 combinations would be 2 , 4 , 3 ). Have the students add the 2 mixed 4 3 4 numbers and simplify the answer, then have them repeat the process using subtraction. (176) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Assign each letter in the alphabet a point value (e.g., vowels are 2 1 points, 4 capitals are 2 3 ; consonants are 1 1 points). 8 2 Have the students determine the value of each of their spellings words, names, vocabulary words, etc. Have the students add and/or subtract these values. 53) Multiplies and divides proper fractions as well as mixed numerals expressing the answer in simplest form Write fractions on index cards in blue marker. On the opposite side put the reciprocal of the fraction in red marker. Give a stack of cards to each student. On another card, put a division sign on one side and a multiplication sign on the other side. Students can work alone or with a partner. Have the student draw one card and lay it face up with the blue side showing. The student will choose to multiply or divide by using the appropriate side of the operation card. The student then chooses a 2nd fraction card and lays it face up with blue showing. If the operation sign is multiplication, the student then solves the problem. If the operation sign is division, the student turns the second card over to the red side (reciprocal), exchanges the division sign for the multiplication sign, and solves the problem. Students will choose a problem from Set A (division problems) and a problem from Set B (multiplication problems) that should give the same answer. They will write reasons for their choices. Students will then exchange papers and work out each pair the other student chose. A point will be awarded for each correct match. Example: Set A Set B 1 1 1 1 2 3 2 3 1 3 3 2 2 1 1 3 3 2 2 1 1 1 2 3 2 3 (177) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 54) Estimates, solves, and compares solutions to one and two-step problems involving addition, subtraction, multiplication, and division of proper fractions and mixed numerals. Each student is given a grease pencil and a baggie. Students are to draw a picture on the baggie using the grease pencil. Students are selected at random to place their drawing on the overhead projector. Students will write a fraction word problem about the picture and solve. These can be swapped. Students work in pairs. Each pair will get a number cube that has numbers written on it. Spinners can also be used. Student #1 will draw on his/her paper the following: ─ + ─ =. Student #1 will roll the number cube or spin the spinner 4 times. The first number will be placed in the numerator spot of the fraction. The second number will be placed in the denominator of the fraction. The third number will be placed in the numerator spot of the second fraction, and the fourth number will place in the denominator of the second fraction. A word problem will be written, estimated, and solved for this problem. Player #2 will do the same. 55) Demonstrates different ways to express ratios Use circle graphs that contain percents and have students express these percents as ratios. Collect data such as birthday months, favorite ice cream, TV shows, etc. Students determine the ratio, decimal, and/or percent from the data. An extension of this would be to have students construct a graph to display this data. Have students work in groups of 4 and determine the ratio, decimal, and percent for the different colors of M&M’s that are in a small bag. Estimation can be connected to this activity as well as graphing the collected data. Draw a rectangle on the overhead/chalkboard and label the sides 6-ft. and 4- ft. Ask students to give the ratio of the length of the longer side to the length of the shorter side. Give each pair of students a baggie that contains 100 dry butter beans. One of the students will reach in the bag and pull out a handful of beans. The student will first estimate the number of beans he/she has in their hand. The student will count the beans in his/her hand. This will be written as a ratio out of 100. This ratio can also be written in decimal and percent form. Have students work with a partner. Each group will need a measuring tape. They are to measure* each other to determine each of the following listed below. Each ratio is to be expressed in simplest form. arm to foot foot to height around the neck to arm foot to arm head to foot (178) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Example: A person has a wrist measurement of 5 in. and a height of 50 in. The ratio of wrist to height is 5:50 or 1:10 in simplest form. *Measurements should be given to the nearest inch. (179) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 MATHEMATICS BENCHMARKS Seventh Grade O – means teacher should be able to observe throughout the day – possibly use anecdotal records. I – Informal Assessment—those marked ―I‖ have an assessment task attached. Uses properties to create and simplify algebraic expressions and solves linear equations and inequalities 1) I - Describes and extends patterns in sequences 2) I - Identifies and uses the commutative, associative, distributive, and identity properties 3) I - Translates between simple algebraic expressions and verbal phrases 4) I - Solves linear equations using the addition, subtraction, multiplication, and division properties of equality with integer solutions 5) I - Writes a real world situation from a given equation 6) I - Writes and solves equations that represent problem-solving situations Interprets, organizes, and makes predictions using appropriate probability and statistics techniques 7) I - Organizes data in a frequency table 8) I - Interprets and constructs histograms, line graphs, and bar graphs 9) I - Interprets and constructs circle graphs when given degrees 10) I - Interprets and constructs stem-and-leaf plots and line plots from data 11) I - Estimates and compares data including mean, median, mode, and range of a set of data 12) I - Predicts and recognizes data from statistical graphs 13) I - Determines probability of a single event 14) I - Uses simple permutations and combinations Writes and solves problems involving standard units of measurement 15) I - Converts within a standard measurement system (English and metric) 16) I - Converts temperature using the Fahrenheit and Celsius formulas 17) I - Uses standard units of measurement to solve application problems Determines the relationships and properties of two and three-dimensional geometric figures and the application of properties and formulas of coordinate geometry 18) I - Identifies polygons with up to twelve sides 19) I - Classifies and compares the properties of quadrilaterals 20) I - Classifies and measures angles 21) I - Classifies triangles by sides and angles 22) I - Finds the perimeter of polygons (180) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 23) I - Finds the area of triangles and quadrilaterals 24) I - Finds the circumference and area of a circle 25) I - Identifies congruent segments, angles, and polygons 26) I - Develops relationships of faces, vertices, and edges of three-dimensional figures 27) I - Performs transformations (rotations, reflections, translations) on plane figures using physical models and graph paper 28) I - Investigates symmetry of polygons 29) I - Develops and applies the Pythagorean Theorem to find missing sides of right triangles 30) I - Graphs ordered pairs on a coordinate plane Uses basic concepts of number sense and performs operations involving exponents, scientific notation, and order of operations 31) I - Uses powers of ten to multiply and divide decimals 32) I - Uses patterns to develop the concept of exponents 33) I - Writes numbers in standard and exponential form 34) I - Converts between standard form and scientific notation 35) I - Finds and uses prime factorization with exponents to obtain the greatest common factor (GCF) and least common multiple (LCM) 36) I - Uses patterns to develop the concept of roots of perfect squares with and without calculators 37) I - Recognizes and writes integers including opposites and absolute value 38) I - Compares and orders integers 39) I - Adds, subtracts, multiplies, and divides integers with and without calculators 40) I - Uses the order of operations to simplify and/or evaluate numerical and algebraic expressions with and without calculators Determines the relationships among real numbers to include fractions, decimals, percents, ratios, and proportions in real-life problems 41) I - Compares, orders, rounds, and estimates decimals 42) I - Adds, subtracts, multiplies, and divides decimals in real-life situations with and without calculators 43) I - Converts among decimals, fractions, and mixed numbers 44) I - Expresses ratios as fractions 45) I - Adds, subtracts, multiplies, and divides fractions and mixed numbers 46) I - Uses estimation to add, subtract, multiply, and divide fractions 47) I - Explores equivalent ratios and expresses them in simplest form 48) I - Solves problems involving proportions 49) I - Determines unit rates 50) I - Uses models to illustrate the meaning of percent 51) I - Converts among decimals, fractions, mixed numbers, and percents 52) I - Determines the percent of a number 53) I - Estimates decimals, fractions, and percents (181) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 54) I - Uses proportions and equations to solve problems with rate, base, and part with and without calculators 55) I - Finds the percent of increase and decrease 56) I - Solves problems involving sales tax, discount, and simple interest with and without calculators (182) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Seventh Grade Informal Assessments Guiding Questions Task Sample 1) Can the student describe and extend Provide the student with arithmetic (+ and a pattern in a sequence? −) and geometric (x and ÷) sequences. a) 2, 5, 8, 11, –, –, – b) 2, 2, 4, 8, 16, –, –, – Have the student extend patterns such as: c) 1, 1.25, 1.5, 1.75, 2, ___, ___, ___, ___ d) 256, 128, 64, 32, ___, ___, ___. Have the student describe the pattern and label it as arithmetic or geometric sequence. 2) Can the student identify and use the Provide the student with the following commutiative, associative, properties: distributive, and identity properties? a+b=b+a a+0=a (a b) c = a (b c) ax1=a a(b + c) = ab + ac Have the student identify the properties as the commutative, associative, distributive or identity properties. Simplify the following using the common associative or distributive properties: 25 x 168 x 4 = (25 x 4) x 168 3) Can the student translate between Read the following verbal phrases: simple algebraic expressions and three times a number verbal phrases? a number increased by five the difference of a number and six Have the student write an algebraic expression for each. (183) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 4) Can the student solve linear Provide the student with equations such equations using the addition, as the following: subtraction, multiplication, and x+4 = -6 division properties of equality with m-13 = -5 integer solutions? -3x = 21 a 24 12 The student may use manipulatives, if needed. 5) Can the student write a real world Give the student the equation: x - 5 = 17 situation from a given equation? Have the student write a word problem to represent the equation. Sample Answer: Five years ago Mary was 17. How old is she now? 6) Can the student write and solve Give the student examples of real life equations that represent problem- problems such as, ―Bob is three years solving situations? older than twice his brother’s age. If Bob is thirteen, how old is his brother?‖ Have the student write an equation and solve. Example: 2a + 3 = 13, a = 5 7) Can the student organize data in a Provide students with given data such as frequency table? temperatures for the week and have them create a frequency table. 8) Can the student interpret and Provide students with data and have them construct histograms, line graphs, construct histograms, line graphs, and bar and bar graphs? graphs. After graphs have been constructed, each group will interpret the different graphs. 9) Can the student interpret and Provide students with given degrees such construct circle graphs when given as 90, 45, 45, 75, 105. Have them degrees? construct a circle graph using a compass and a protractor. Each student must be able to interpret the circle graph. (184) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 11) Can the student estimate and Provide students with the following data: compare data including mean, 23, 85, 76, 63, 76, 94. Students are to median, mode, and range of a set of estimate and then find actual results of data? mean, median, mode, and range. Compare actual results with the estimate. 12) Can the student predict and Provide students with a statistical graph. recognize data from statistical For example, a sports graph comparing graphs? previous year’s averages to present averages. Students should predict the following year’s average after interpreting data from graph. 13) Can the student determine the Provide the student with a spinner that probability of a single event? has been divided into a specific number of sections such as eighths in which the numbers 1 through 8 will be written. Students will determine the probability of spinning a composite/prime number. 14) Can the student use simple Have students solve problem such as: permutations and combinations? Given a group of 4 people, how many different ways can they line up to go to lunch? (185) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 15) Can the student convert within a Provide the student with a chart of equal standard measurement system measures. Have the student complete the (English and metric)? following conversions and explain why one would multiply or divide. 9 gal. = ____ qt. 17 ft. = ____ in. 23 c. = ____ pt. Provide the student with the following metric system line: k h da meter d c m gram liter 7g = kg 37cm = m 5KL = L 6.9cm = mm 16) Can the student convert temperature Provide the student with the Fahrenheit using the Fahrenheit and Celsius and Celsius formulas. Have the student formulas? convert the following temperatures. 86ºF = ˚C 59º F = ˚C 35º C = ˚F 120º C= ˚F (186) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 17) Can the student use standard units of Provide the student with standard English measurements to solve application units of measure. Have the student problems? answer the following problems showing all his work. Ann needed 5 cups of milk to make a recipe. How many pints does she need to make this recipe? Tom rode his bike 2.5 miles. How many feet did he ride his bike? Janet has 3 pounds and 4 ounces of grapes to divide into 4 baskets. How many ounces are to be put into each basket? Provide the student with metric units of measure. Have the student answer the following questions: Tammie’s pencil is 13 cm long. How many millimeters is her pencil? John ran the 400-meter dash. How many kilometers did he run? 18) Can the student identify polygons Provide the student with a set of polygons. with up to twelve sides? Have the student name each polygon in the set. 19) Can the student classify and Give the student a set of quadrilaterals. compare the properties of Have the student identify which ones are quadrilaterals? rectangles, parallelograms, squares, trapezoids, and rhombi and explain why. 20) Can the student classify and Provide the student with a set of pre- measure angles? drawn angles in various positions. Have the student determine whether the angles are right, acute, obtuse, or straight and define each. Provide the student a protractor and a set of angles. Have the student measure each angle and classify each angle. (187) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 21) Can the student classify triangles Provide the student with a set of triangles. according to their sides and angles? Have the student name the triangles according to their sides and angles. 22) Can the student find the perimeter of Give the student a meter stick or trundle polygons? wheel. Have the student find the perimeter of the classroom, hallway, shapes on the floor, etc. 23) Can the student find the area of Give the student grid paper and a ruler. triangles and quadrilaterals? Have the student draw a triangle, a rectangle, and a parallelogram on the grid paper and find the area of each figure. 24) Can the student find the Give the student a circle with a diameter of circumference and area of a circle? 5 cm. Have the student find the circumference and area of the circle using the appropriate formulas. 25) Can the student identify congruent Provide the student with the following sets: segments, angles, and polygons? angles, segments, and polygons. Have the student identify the congruent shapes in each set. 26) Can the student develop Provide the student a set of three- relationships of faces, vertices, and dimensional figures. Ask the student to edges of three-dimensional figures? identify all of the faces, edges, and vertices. 27) Can the student perform Provide the student with a geoboard. Ask transformations on plane figures the student to model/demonstrate a using physical models and graph transformation of a given design. paper? (188) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 28) Can the student identify lines of Provide the student with a triangle, symmetry in polygons? square, rectangle, octagon, parallelogram, etc. Ask the student to draw lines of symmetry for each figure. 29) Can the student develop and apply Use the Pythagorean Theorem formula to the Pythogorean Theorem to find the find the longer leg of a right triangle, if the missing side of a right triangle? hypotenuse is 25 units and the shorter leg is 7 units. 30) Can the student graph ordered pairs Provide the student with a pegboard with on a coordinate plane? a coordinate plane sketched on it and a handful of golf tees. Have the student plot several ordered pairs on the pegboard from a given set of ordered pairs. 31) Can the student use powers of ten to Provide the student with a decimal multiply and divide decimals? number. Have the student: Multiply the decimal number by a power of ten. Divide the decimal number by a power of ten. Example: a) 0.125 x 10 b) 3.5 x 100 c) 152 ÷ 1000 d) 0.39 ÷ 100 32) Can the student use patterns to Give the student a number. Have the develop the concept of exponents? student find a geometric pattern using multiplication. Example: 1, 4, 9, 16 _, _, _ (12, 22, 32, 42…) (189) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 33) Can the student write numbers in Give the student a number. Have the standard and exponential form? student write the number in exponential form. For example: 154 = (1 x 102) + (5 x 10) + (4 x 1) Give the student a number written in exponential form. Have the student write the number in standard form. For example: (3 x 103) + (5 x 102) + (8 x 1) = 3, 508 34) Can the student convert between Give the student a number written in standard form and scientific notation? scientific notation and have him/her convert the number to standard form. Example: Have the student convert a number such as 6,200,000,000 to scientific notation. (6.2 x 109) 35) Can the student find and use prime Provide the student with two or more factorization with exponents to obtain whole numbers. Example: 20 = 22 x 51 the greatest common factor (GCF) 14 = 21 x 71 and least common multiple (LCM)? Have the student: Find the prime factorization of each number. Find the GCF of any 2 numbers of the given set using the factorizations. GCF = 2 1 = 2 Find the LCM of any 2 numbers of the given set using the prime factorizations. LCM = 2 2 5 1 7 1 = 140 36) Can the student use patterns to Have the student draw squares with develop the concepts of roots of lengths of 1 cm to 10 cm. Have the perfect squares with and without student calculate the area of each square. calculators? Have the student complete the following pattern: 12 = 1 1 =1 1 1 1 1 22 = 2 2 =4 4 22 2 32 = 3 3 =9 9 33 3 (190) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 37) Can the student recognize and write Provide the student a set of integers. integers including opposites and Have the student write the opposite of the absolute value? given integer. For example: Integer Opposite a) 8 -8 b) -24 24 c) 0 0 Have the student write the absolute value of the given integer. For example: Integer Absolute Value a) 4 4 b) 8 8 c) 0 0 38) Can the student compare and order Provide the student with a number line integers? and a given set of integers. Have the student: Compare sets of integers. Order from least to greatest. 39) Can the student add, subtract, Provide the student with a variety of multiply, and divide integers with and addition subtraction, multiplication, and without calculators? division problems using integers. Have the student solve the problems without a calculator and then check each problem using a calculator. For example: a) -8 + 12 a) –55 ÷ 11 b) –15 – 14 c) 4 x 16 (191) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 40) Can the student use the order of Have the student explain the correct order operations to simplify and/or evaluate in which to simplify numerical expressions. numerical and algebraic expressions Have the student simplify the following with and without calculators? numerical expressions. 15 10 5 2 6 4 2 30 5 22 2 6 4(3 2) Provide the student with a calculator to check answers. 41) Can the student compare, order, Provide the student with a set of decimals. round, and estimate decimals? Have the student: a) Compare two or more decimals b) Order the set from least to greatest, c) Round the decimal to a given value d) Estimate the decimals. Example: a) 0.56____ .05 b) 0.56, 4.5, 5.6, .05 c) Round 0.56 to the nearest tenth d) Estimate: 5.6 + 0.56 = ____ 42) Can the student add, subtract, Provide the student with several decimal multiply, and divide decimals in real- numbers that have been written on the life situations with and without board or overhead and ask them to write calculators? their own addition, subtraction, multiplication, and/or division word problems using the numbers. The student will then solve their problems. Example: 28.7 2.5 426 $1.31 7.236 Shelley’s gas tank holds 28.7 gallons of gas. If she can travel 426 miles on a tank of gas, how many miles per gallon does her car get? (192) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 43) Can the student convert among Provide the student with a chart in which decimals, fractions, and mixed they have to fill in the correct form, such numbers? as the following: Decimal Fraction Mixed Number 2.5 804 100 3 14 44) Can the student express ratios as Provide the student with 4 yellow squares fractions? and 8 brown squares that have been cut from construction paper. Have the student answer the following questions: What is the ratio of yellow squares to brown squares? What is the ratio of brown squares to yellow squares? What is the ratio of yellow squares to all squares? Answers are to be written in ratio and fraction form. 45) Can the student add, subtract, Provide the student with problems that multiply, and divide fractions and involve addition, subtraction, division, and mixed numbers? multiplication. Use denominators that are found in everyday life, such as 2,3,4,8,12,16. Example: At the family reunion Kerra saw 60 of her relatives. She remembered the names of 3 of them. How many names did she 4 remember? (193) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 46) Can the student use estimation to Have the student estimate the following: add, subtract, multiply, and divide 1 4 fractions? a) 3 8 6 5 3 7 10 5 2 b) 2 6 x 7 9 3 x 7 21 47) Can the student explore equivalent Provide the students with cards with ratios ratios and express them in simplest written on them. Have the student match form? a ratio card with another card that is an equivalent ratio in simplest forms. 48) Can the student solve problems Provide the student with proportion involving proportions? problems to solve such as the following: 9 to 15 is the same as 12 to_______ 8 12 Solve : = 12 n 49) Can the student determine unit Provide the student with an index card rates? containing information such as the following on it. Apples $1.60 5 Have the student will determine the cost of 1 apple. (194) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 50) Can the student use models to Provide the student with graph paper and illustrate the meaning of percent? a ruler in which he/she is to model percents such as the following on the graph paper. Using 10-by-10 grids, have the student model his/her answers to each of the problems. 3 a) 10 85 b) 4 100 c) 23% 51) Can the student convert among Provide the students with a chart in which decimals, fractions, mixed numbers, they have to fill in the correct form, such and percents? as the following. Decimal Fraction Mixed Number Percent 0.25 3 1 5 6 33% 10 52) Can the student determine the Provide the student with a piece of graph percent of a number? paper and have him/her model 25% of 80. The student will then calculate the answer. 53) Can the student estimate decimals , Provide the student with several word fractions, and percents? problems to estimate using decimals, fractions, and/or percents, such as the following: A Big Mac at McDonalds’s has 940 calories. Jesse 1 ate about 33 % of his. About how many calories 3 did he eat? (195) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 54) Can the student use proportions and Have the students count the number of equations to solve problems with girls and boys in the classroom. Knowing rate, base, and part with and without this information, have the students predict calculators? how many boys and how many girls are in the seventh grade class with a total of 750 students. Find the percent of the class that is boys. 55) Can the student find the percent Provide the student with several of increase and percent of decrease? basketball players’ shooting averages from last year and this year. Have the student calculate the percent of increase or percent decrease of last year’s average and this year’s average. 56) Can the student solve problems Provide the student with a restaurant involving sales tax, discount, and menu. Have the student order a meal. simple interest with and wihtout Have the student calculate the total cost of calculators? the meal, including 7% sales tax and a 15% tip. (196) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Suggested Teaching Strategies for SEVENTH GRADE Uses properties to create and simplify algebraic expressions and solves linear equations and inequalities 1) Describes and extends patterns in sequences Use pattern blocks to develop a tessellation. Use pencil and paper to use patterns to develop string art. 2) Identifies and uses the commutative, associative, distributive, and identity properties Divide the class into two groups. Give each student a card with a variable, an operation, parentheses, or an equal sign written on each card. Have the students to work cooperatively to create and identify a property using the given cards. Create one set of cards with the property names written on them and another set with an example of the property written on them. Have the students play ―Memory‖ by matching the property name with the property example. 3) Translates between simple algebraic expressions and verbal phrases Create an expression search puzzle using algebraic expressions. Give the students verbal phrases as clues. Have the student find and circle the answer in the puzzle. Example: Clue Answer A number (x) increased by five x+5 Twice a number (n) 2n Six minutes less than Bob’s time (t) t–6 3 years younger than Seth (s) s–3 Give the students a number. Have the students perform the instructed operations on that number. Example: Start with 12 12 Twice that number 12 2 = 24 Less 8 24 - 8 = 16 Divided by 4 16 ÷ 4 = 4 Increased by 2 4+2=6 Squared 62 = 36 (197) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 4) Solves linear equations using the addition, subtraction, multiplication, and division properties of equality with integer solutions Have the students solve equations using a flow chart. On the top of the flow chart create the equation and solve the equation on the bottom. Example: 3a – 4 = 11 +4 +4 3a = 15 3 3 a=5 Give the students ten equations that are already solved. Have the students mark the equations that are worked correctly and rework the equations worked incorrectly. 5) Writes a real world situation from a given equation Give the students the equations such as 3 + (2.50)x = 11.50. Have the students write a story problem based on the equation. Sample Questions: The Rollo Skating Rental charges an entrance fee of $3 and an additional $2.50 per hour. If Sam paid $11.50 to skate, how many hours did he skate? Have the students create a small business that will sell a product or service. Have the students create an advertisement for the product or service that includes a description of what they are selling and the cost. Based on this information, have the students write an equation describing the relationship. 6) Writes and solves equations that represent problem-solving situations Have the students write an equation and solve problems. Examples: The sum of a number and 8 is 21. What is the number? The sum of two consecutive numbers is 23? What are the numbers? Mike’s test scores are 75, 93, 88, and 82. What must he make on his next test to have an average of 85? John made 14 points in the basketball game on Tuesday. By the end of the second game on Friday, his two game total was 30 points. How many points did John make in the second game? Solution: Tuesday’s game + Friday’s game = total points of 14 + f = 30 Choose several occupations and write problem-solving situations for each occupation on index cards. Divide the class into groups according to the occupations chosen and distribute the cards for that occupation to the appropriate group. Each group will write and explain equations that could be (198) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 used to solve each problem. Group presentations can be made demonstrating how various occupations use the equations in the business. Example: Carpenters One sheet of 4x8 plywood will cover 32 sq. ft. How many sheets will be needed to cover 640 sq. ft.? 32 p = 640 Interprets, organizes, and makes predictions using appropriate probability and statistics techniques 7) Organizes data in a frequency table Provide the students different types of balls and collect data on how many times the balls bounce. Transfer information to a frequency table. Give students a list of their grades. Have students organize their grades in a frequency table. 8) Interprets and constructs histograms, line graphs, and bar graphs Use the computer to show different types of bar graphs and line graphs. Take the students outside and choose students to run sprints. Students will time and record data. Divide the students into groups. Assign each group a different type of graph to construct using the data collected. 9) Interprets and constructs circle graphs when given degrees Have the students use protractors to construct circle graphs when given degrees. Have the students use compasses to draw a circle. Have the students use the computer to construct circle graphs. Have the students write two questions about their graphs. 10) Interprets and constructs stem-and-leaf plots and line plots from data Time the students as they toss paper into the trash cans. Count the number of baskets each student makes. Have the students to create stem-and-leaf plots and line plots from the collected data. Give the student a set of data such as 23, 25, 40, 71, 21, 32, 66, 54, 40, 47 and 78 which could represent the number of points scored in a game. Have the students put the data in order from least to greatest. Have the student choose an appropriate stem value such as tens digits and leaf values such as one digits. Plot the data in a stem-and-leaf plot. Explain the plot giving an explanation of the stems and leaves. Be sure to give the stem-and- leaf plot a title. (199) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Examples: Number of Points Scored Stems Leaves 2 1,3,5 3 2 4 0, 0, 7 5 4 6 6 7 1, 8 List the grades made on a particular test in random order. Students will construct and interpret stem and leaf and/or line plots from this data. 11) Estimates and compares data including mean, median, mode, and range of a set of data Have students estimate the number of drops of water that will fit on a penny. Provide students with a dropper, water, and a penny. Have the student find the mean, median, mode, and range for the class data. Have the students use the mean, median, and range feature on the graphing calculator. 12) Predicts and recognizes data from statistical graphs Have the student go to a web site on the Internet dealing with the stock market page. Have students predict and recognize data. Provide students with several examples of statistical graphs from newspapers and magazines. Divide the class into groups and let each group choose a graph to use. Have the students in each group present an explanation of the data represented by their graph to the class. Then have them use their graphs to predict what might happen in the future and explain what might cause these changes. Have students collect a variety of graphs from magazines or newspapers. Have each student write questions about a selected graph. Students exchange graphs with a partner and answer the questions about the graphs. Share and discuss answers. 13) Determines probability of a single event Have the students roll a number cube to find the probability of rolling an even number, odd number, prime number, composite number, and a given number. Have the students roll two number cubes, add the numbers together, and find their sum. Have the students determine the probability of getting the sum of 9, 12, etc. (200) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 14) Uses simple permutations and combinations Announce to the students that they are going to elect a president, vice- president, and secretary for the class. Have the students nominate five students to run for the three positions. Have the students predict the possible outcomes. Examples: P (5,3) = 5 · 4· 3 = 60 *Order is important in permutations. Choose a three-member team from a group of five people. How many teams are possible? 543 C (5,3) = 10 outcomes 3 2 1 *Order is not important in combinations. Writes and solves problems involving standard units of measurement 15) Converts within a standard measurement system (English and metric) Provide the students with a chart of equal measures. Show the students how to use cancellation to convert measurements. Example: a) Convert 5 gal. to quarts. 4qt 5 gal 5 4qt 20 qt 1gal b) Change 1000 ft. to yards 1yd 1000 1yd 1000yd 1 1000ft 333 yd 3ft 3 3 3 c) Change 8.7 ft. to inches 12in 8.7 12 8.7ft in 104.4in 1ft 1 Explain that whatever unit the student starts with must be the unit in the denominator of the second fraction, and the numerator is its equal measure. Multiply numerators. Divide by denominator. Provide each student a metric line chart. Divide the students into groups of fours. Give each group a ziploc bag of cubes, a ziploc bag of beans, and two M & M’s for each student. The cubes represent all numbers except 0, the (201) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 beans represent zeroes, and the M & M’s represent decimal points. Give each group four problems. Have each student take a problem and model the conversion by moving the decimal. Example: 17 m = 17000 mm 0.45 g = 0.00045 kg Metric Chart Kilo Hecto Deka M Deci Centi Milli This saying may help students remember the chart ―King Hector Died Monday Don’t Call Me.‖ Make a floor chart of the metric system. Make a black circle for the decimal point. Have students write a conversion problem, swap problems with their neighbor, and walk the problems off on the floor chart. 16) Converts temperature using the Fahrenheit and Celsius formulas Write the formulas on the board or overhead. Demonstrate solving formulas, emphasizing that substitution is used. Have students use order of operations to convert temperatures. Example: Change 50˚ F to ˚C 5 ˚C= (˚F – 32) 9 5 C (50 - 32) 9 5 C (18) 9 C = 10C Change 0˚ to ˚F 9 F= C + 32 5 9 F= (0) + 32 5 F=0 + 32 F=32F (202) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Explain that a formula shows a relationship between quantities. Give the student the formulas for converting degrees Fahrenheit to degrees Celsius 5 and for converting degrees Celsius to degrees Fahrenheit: C = (F - 32) and 9 9 F= C + 32. 5 Give a problem such as, the temperature outside is 10C. What is the temperature measured in degrees Fahrenheit? Have the students explain which formula should be used. Then have the student write the correct formula and plug in the appropriate information. 9 Examples: F= C +32 5 9 F = (10) +32 5 F = 18 + 32 F = 50 *Note: In order for students to use this formula, they must know how to multiply fractions. Use a scientific calculator to convert between Fahrenheit Celsius temperatures. 17) Uses standard units of measurement to solve application problems Provide the students with Standard English and metric units of measure and calculators. Give a series of questions in different areas and require students to work in pairs to answer the question. Examples: 1. A beauty shop has about 66 customers per day, keeping 3 beauticians busy for most of a 9-hour day. How long does each beautician average with each customer? Give answer in minutes. 2. John said ―It is 35˚ C outside.‖ Is this a hot, a cold, or a very pleasant comfortable day? 3. Dominick is recovering from surgery and is not suppose to lift more than 25 pounds. He works in a grocery store as a stocker. A bag of flour weighs 5 pounds. Each case contains 6 sacks. Can he carry the cases to the aisle and stock the shelf? Explain. 4. John’s coach has ordered him to drink a liter of Gatorade during practice. The containers hold 1,000 ml, 500 ml, and 250 ml. Which size should he get? Explain. 5. Find the amount of carpet needed to cover the floor, the paint to paint the walls, and the border to go around a room with certain dimensions. (203) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Use plastic straws, a hole punch, and brads to construct various quadrilaterals. (D-Stix or other commercial sets may be used.) Use different colored straws for different lengths so they can be compared. Examples: 4 equal sides, angles not all equal 4 equal sides, no right angles 4 equal sides, all right angles 4 equal sides, square or rhombus Have students measure the dimensions of the classroom floor to determine the area of the floor. Determines the relationships and properties of two and three-dimensional geometric figures and the application of properties and formulas of coordinate geometry 18) Identifies polygons with up to twelve sides Students work with a partner to create shapes of polygons on a geoboard. Record on dot paper and identify each polygon by name On a geoboard construct each of the following figures with one rubber band and record each result on dot paper. a. an acute triangle b. a right triangle c. a triangle d. a isosceles triangle e. a scalene quadrilateral f. an equilateral triangle g. a rectangle with four congruent sides h. a parallelogram with four right angles i. a pentagon j. a hexagon 19) Classifies and compares properties of quadrilaterals Using a Venn diagram, compare and contrast the following quadrilaterals. Explain why they were grouped in a particular way. Provide the student with pictures of the five kinds of quadrilaterals. Have the student explain the properties of each set that are alike and the properties of each set that are different (number of sides, types of interior angles, congruent angles, congruent sides, length of sides). (204) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 20) Classifies and measures angles Cut out models of rays from tag board or card stock. Connect two rays with a paper fastener to form a model of an angle. Explore forming angles of different sizes by rotating the rays around the vertex. Sketch the angles you modeled. Next to each angle write the name of each. Measure with a protractor to validate the name of the angle. List examples around the classroom that are real-world types of angles. Let students work in small groups of 2 or 3. Give each student a sheet of white paper and a protractor. Each student will draw several angles using the protractor. Each student should draw at least one of each of the following: acute angle, obtuse angle, right angle, and straight angle. Encourage students to draw these angles opening in different directions. The students will exchange papers and use the protractor to measure and classify each angle. 21) Classifies triangles by sides and angles Use a circular geobard template to connect the points. Then measure the length of each side of the figure with a ruler. What type of triangle is represented? Correct Points Type of Triangle 1. P7 P15 P23 2. P1 P6 P11 3. P11 P15 P20 4. P20 P6 P10 Using your circular template, connect the points. Then measure each angle and classify the triangle. Correct Points Type of Triangle 1. P5 P14 P21 2. P4 P10 P16 3. P24 P16 P20 4. P5 P13 P21 (205) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Use the circular template or geoboard to investigate which of the following triangles are possible 1. an acute scalene 2. an obtuse scalene 3. an obtuse isosceles 4. an obtuse equilateral 5. an acute equilateral 6. a scalene right 7. an isosceles right 22) Finds the perimeter of polygons Give students geoboards, rubberbands, and dot paper. Students will model polygons with perimeter of 20 units each. Each polygon will be drawn on dot paper. Give students drawings of polygons and the measurement of the sides. Students will find the perimeters. 23) Finds the area of triangles and quadrilaterals Discuss the materials that are necessary to paint the interior of a house. Discuss the specific materials needed to paint each room. Determine how much of each material will be needed to paint a seven room house. Divide the student into teams. Have each team to choose an area of the school to paint a new and better color. Each team will determine perimeter and area of the floor, the walls, and ceiling. Then they are to determine the amount of paint needed for their chosen area. 24) Finds the circumference and area of a circle Give students grid paper. Trace a small and large circle on the grid paper. Have the students count the squares and approximate the area of each circle. Wrap a string around the outside of each circle. Measure the string to determine the perimeter. Use a calculator to find the area and circumference of various circles given the formulas A=r2 and C=d. Discuss the key on a calculator and the 22 common values used for are or 3.14. 7 Have several circular items in the room. Let the student choose one or more of the circular items. Have the student measure the circumference and the diameter of each item. This information should be recorded. Be sure students know that pi () is approximately 3.14. Students may use a calculator to solve the problem. 25) Identifies congruent segments, angles, and polygons Using a set of tangrams, the student will find all tangram shapes that are the same size and shape. Ask the students to explain how they know that they are the same size and shape. Tangrams that have the same size and shape (206) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 are congruent. Also have students create congruent shapes using two or more tangrams. Note: After activities read the ―Grandfather Story,‖ and play Dominoes by matching congruent segments, angles, and polygons. Examples: Define the term congruent (the same size and shape) and give the symbol used to show two items are congruent (). On a sheet of paper, draw several line segments, angles, and polygons. Be sure that some or all have a match. Have the student match the items that are congruent. 26) Develops relationship of faces, vertices, and edges of three-dimensional figures Use one or more of these strategies: Make a table, use a formula, draw a diagram or guess and check to complete the following table. N - The number of sides in the base F - The number of faces V - The number of vertices E - The number of edges Complete the table and identify the relationship between the number of vertices and edges. Figures N F V E Triangular Prism Rectangular Prism Pentagonal Prism Construct three-dimensional figures from straws or commercial sets letting each length be a different color. Describe the number of faces, the shape of each face and the number of vertices and edges. Compare with other three- dimensional figures. Example: Triangular prism and rectangular pyramid. Discuss the similarities and differences. (207) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 27) Performs transformations (rotation, reflections, translations) on plane figures using physical models and graph paper Use a geoboard to create a quadrilateral and demonstrate slides, flips, and turns. Slide Use grid paper to form a coordinate plane. Plot the following points, and then translate (slide) the point and make the new coordinates. Use colored pencils to work new points. Example: (-2, 4) Slide 4 units right (2, 4) (4, 0) Slide 3 units down (4, -3) Working in pairs, reflect the following shapes across the line of symmetry. Share your solutions with your partner. A B C D (208) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 28) Investigates symmetry of polygons Have each student take a shape (made from cardstock) and one sheet of tracing paper. Trace the cardstock shape on the tracing paper. Then have each student see if the traced shape could be folded so that the 2 halves will match exactly. If the traced shape can be folded exactly in half, then the fold line is the line of symmetry. Try to see if the traced shape can be folded another way to match exactly. There may be more than one line of symmetry. Have each student use another sheet of tracing paper and trace the same shape or different shapes. Each student then takes a mirror or mira and places it on the figure so that half the figure together with the reflection of that half forms the entire figure. The line along which the mirror is placed is the line of symmetry. Have each student use another sheet of tracing paper and trace a shape. Students will cut out the traced shape, then try to fold the shape so that each half is the same along the folded line of symmetry. *Note: Look for more then one line of symmetry in some shapes. 29) Develops and applies the Pythagorean Theorem to find missing sides of right triangles Using a geoboard construct a right triangle that has legs each 3 units + 4 units long. Form two squares using the length of each leg as the side of each square. Also form a square using the hypotenuse as one side. The area of square ―a‖ plus square ―b‖ = the area of square ―c‖. Using the Pythagorean Theorem a2 + b2 = c2, find the missing side (Use calculator if needed) Leg 1 Leg 2 Hypotenuse 3 4 5 5 12 13 8 15 17 (209) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 30) Graphs ordered pairs on a coordinate plane Use dot paper to form a coordinate grid. Give each student a grid. Call out ordered pairs, and have students plot them on the grids. Track a real or imaginary hurricane on a hurricane tracking map. Discuss latitude as the x-axis and longitude as the y-axis. Create a design by connecting points represented by x, y coordinates. (2, 1) (8,1) (4, -1) (2, 1) (6, -1) Provide the student with a coordinate plane. Discuss the x-axis and y-axis and the ordered pair of numbers which indicate the coordinates of a point (x, y). Have the student plot several coordinates on the graph. Examples: (5, 2), (3, 4), (0-03), (-4, -2). If there is a floor with square tiles, have students determine an origin and use tape to make the x-axis and the y-axis. Call out an ordered pair and have the student walk out the correct number of squares to get to the given point. Uses basic concepts of number sense and performs operations involving exponents, scientific notation and order of operations 31) Uses powers of ten to multiply and divide decimals Make a set of index cards with problems such as 2.756 x 300 on them. Make another set of index cards with the answer 826.8 on them. As students enter the classroom they are given an index card. Students are instructed to find their match. 32) Uses patterns to develop the concept of exponents Have the students use grid paper to explore the patterns of exponents. Example: 20 21 22 23 Have the students multiply or divide decimals by powers of ten using long multiplication or division. Students will then look for patterns and develop shortcuts. 33) Write numbers in standard and exponential form Have the students use dominos to write numbers in standard and exponential form. = 42 = 44 = 34 = 3333 (210) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Students find the prime factorization of a number and write the answer in standard and exponential form 810 = 2 3 3 3 3 5 = 21 34 51. 34) Converts between standard form and scientific notation Have the students use a map of the solar system to convert the distances into scientific notation. Earth to sun: 93,000,000 miles = 9.3 x 10 7 Have the students brainstorm on when they would use scientific notation. Example: Thickness of a strand of hair in inches The length of a bedroom wall in millimeters 35) Finds and uses prime factorization with exponents to obtain the greatest common factor (GCF) and least common multiple (LCM) Create a Vienn Diagram. First allow the students to create a factor tree. GCF = 12 20 2 6 2 10 22 3 list common multiples 2 2 5 12 20 3 22 5 GCF is 2 x 2 = 4 3 2 2 5 LCM LCM 3 4 5 = 60 Write a prime factorization for a set of numbers such as 18 and 30. Use common bases and lowest exponents for GCF. Use all bases and largest exponents for LCM. Example: 18 = 2 32 30 = 2 3 5 GCF = 2 3 = 6 LCM = 2 32 5 = 90 (211) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 36) Uses patterns to develop the concept of roots of perfect squares with and without calculators Provide the students with cm squares. Have the students arrange the pieces into squares and tell how many pieces it takes to make the squares Example: 1 4 9 16 1x1 2x2 3x3 4x4 Provide the students with a set of numbers. Have the students predict which number are perfect squares. Have the students use the calculator to check their answer. Number Prediction Calculator Check 1 yes 1 1 yes 10 no 10 3.2 no 18 yes 18 4.2 no 37) Recognizes and writes integers including opposites and absolute value Have the students make a human number line. Have one student walk this number line and turn every time opposite is mentioned. Opposite of –3 -3 0 3 Use this same number line to demonstrate absolute value and discuss the definition of absolute value. 38) Compares and orders integers Place students into groups of four or less. Provide the students with a set of cards with integers and >, <, =, and absolute value signs written on them. Have the students arrange the cards from least to greatest. Example: -96 -50 -31 0 30 48 101 250 Ask the students comparison questions. Have the students in each group use the cards to answer the questions. (212) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Provide the students with a number line made from a piece of rope and the rope knotted at equal intervals. Have the students go to a knot on the number line and move forward as a problem is read out loud. Have the students make comparisons. 39) Adds, subtracts, multiplies, and divides integers with and without calculators Have the students use a number line rope. Call out addition and subtraction problems. Have the students move up and down the rope and tell their new position. Have the students play ―around the world.‖ Provide each student with one integer card. Have each student stand by another student. Have the two students hold up their cards. Call out an operation. The student who answers first exchanges cards and moves around to the next person. The person who goes around the room first wins. Have the students play domino chase in pairs. Provide the students with a game board, two tokens, and a + - card. start finish Add - + -2 + 4 = 2 move forward 2 places - + -5 + 1 = -4 move backward 4 places 40) Uses the order of operations to simplify and/or evaluate numerical and algebraic expressions with and without calculators Have students simplify problems using the correct order of operations: Example: 3 + 2 – 4 3 = ‾ 7 Have the students solve problems and expressions. Have the students check their work with calculators. Students may learn the phrase ―Please Excuse My Dear Aunt Sally‖, to assist them in learning order of operations. (Parentheses, Exponents, Multiplication, and Division left to right, Addition and Subtraction left to right) (213) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Determines the relationships among real numbers to include fractions, decimals, percents, ratios, and proportions in real life problems 41) Compares, orders, rounds, and estimates decimals Provide students cards containing decimal numbers. Have the class arrange itself in order from least to greatest. Have each student remain in line and tell what his/her number is, when rounded to the nearest given place value. Have two students show their card numbers containing a decimal. Have the other students estimate the sum of these two numbers. A statement will also be written to compare the two numbers such as 42.347 > 41.38. 42) Adds, subtracts, multiplies, and divides decimals in real-life situations with and without calculators Use place value blocks and charts to model all four operations. Have students construct word problems that fit the problems modeled using the four operations. Place value blocks can be made by taking a grid sheet of 1 cm squares and dividing it into several 10-by -10 cm squares. Make and laminate enough copies for each student to have at least six 10-by-10 cm squares, 15 1-by-10 cm squares, and 15 1-by-1 cm squares. These can be stored in baggies so that each student will have his/her own squares. Write several problems with addition and subtraction of money on the overhead/board, for example, $2.96 + $0.63 and $0.73 – $0.20. Have the students solve the problems and discuss the similarity between decimals and money amounts. Explain that adding and subtracting money is similar to adding and subtracting decimals. Have the students estimate the amount their family spends on food each month. Have the students bring to class their cash register receipts for food for the month. Have the students add the cash register receipts and compare them with original estimates. 43) Converts among decimals, fractions, and mixed numbers Have students play the ―Memory Game ― individually or in groups of up to four players. Make twenty-four square-sized cards. Write different fractions on eight squares, the decimal form of the fraction on eight squares, and the percent form on eight squares. (See below) The 24 cards are placed facedown with 6 rows of 4. One player turns up 3 cards trying to get a 3-way match of a fraction, decimal, and percent card. If a correct match is not made, the next player turns up three cards. If a correct match is made, that player continues until a match is not made. The player with the most matches at the end of the game is the winner. FRACTION DECIMAL PERCENT 3 0.75 75% 4 (214) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Provide each student an index card on which either a decimal, fraction, or a 1 percent has been written, such as , 50%, or 0.5. Have the students locate 2 their matches. Have students determine what letters of their first name are vowels, and have them record this information on an index card. Convert the number to decimal and percent forms. A large piece of string can be strung across the room for a number line for students to tape their index cards. Have the students repeat procedure for their last name. Have the students develop graphs from the class data. 44) Expresses ratios as fractions Have the student write the following as a fraction. a) 3:5 b) 6 to 9 c) 10 out of 20 d) John made 4 free throws when he shot 7 times. Give the student a problem and have the student express the ratio as a fraction. Example: There are 14 girls and 17 boys. What is the ratio of girls 14 17 to boys? What is the ratio of boys to girls and boys? What is the 17 31 31 ratio of girls and boys to girls? 14 Collect data from the class such as number of boys and girls or number of red, blue, white, and mixed shirts. For example: 12B, 14G, 8r, 4b, 42, 10m. Write fractions (in lowest terms) to show comparisons such as: 14 7 the fractional part of the class that is girls. = 26 13 4 2 the fractional part of the class wearing a blue shirt. = 26 13 45) Adds, subtracts, multiplies, and divides fractions and mixed numbers Have students roll fraction number cubes twice. Have the students performs all four operations using the two fractions rolled. Have the students work with a partner. Provide each group with a baggie and grease pencil. The groups are to draw a picture on the baggie with the grease pencil. Have each group individually place its baggie on the overhead projector. Have the class to write an addition, subtraction, multiplication, and division word problem that fits the picture. Have the students share the problem with the class. Have the students work with a partner to construct a spinner containing the digits 1-9. Have a student write on a sheet of paper the following: (215) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Have the same student spin the spinner four times. (The first number will be placed in the numerator spot of the first fraction. The second number will be placed in the denominator spot of the second fraction. The third number will be placed in the numerator spot of the second fraction, and the fourth number will be placed in the denominator spot of the second fraction.) Have another student spin four times and do the same thing as the other student. Have the students then add their fractions. The student with the larger fraction is the winner. This activity can also be done with subtraction, multiplication, and division. In subtraction, if the first fraction is smaller than the second fraction, the student must swap them. 46) Uses estimation to add, subtract, multiply, and divide fractions Have the student use Cuisennaire Rods to estimate adding, subtracting, multiplying and dividing fractions. Have the students locate fractions on a ruler and round to the nearest whole number. Example: 0 x 1 x 2 3 13 1 4 16 1 + 2 = 3 estimation Have the students use the stock quotes in a newspaper to estimate gains or losses for a stock on a particular day. Add or subtract to compare with their estimates. 47) Explores equivalent ratios and expresses them in simplest form Write a ratio on one side of fifteen index cards and its equivalent (simplest form) on the other side. Cut the cards in various ways in the middle. Give each student 1 of a card. Do not let the students see each other’s cards. 2 One student stands and calls out his ratio and the student who thinks he has a match stands. If they are correct, their cards will fit together. Give students several pairs of equivalent ratios. Have students draw a rectangle to show the ratio draw a rectangle the same size showing the ratio with the smaller numbers. Compare the shaded part of the two rectangles. It 1 2 should be the same. Example: , 2 4 (216) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 48) Solves problems involving proportions Model the procedure for solving a proportion using cross products. Point out to the students that the position of the terms in each ratio MUST correspond. Have the students set up a proportion. Example: ―Sheila wants to buy a CD player that costs $240. She earns $25 in 5 hours babysitting. How many hours will Sheila have to work in order to be able to buy the CD player? (hours worked)5 n(hours worked) (earnings) 25 (earnings) Have the students draw a rectangle that measures 1 cm by 3 cm on graph paper. Have the students draw another rectangle whose sides are 3 times the length of the first rectangle. Have the students label the lengths. Have the students write a proportion whose ratios are the widths over the length of each rectangle. Provide each student with centimeter squares. Have students construct rectangles with one side twice as long as the other. Have students create their own proportions using the pieces and share these with the class. Utilize the Russian dolls that come in proportional sizes for students to actually see how the dolls are proportionally made. Sand is a good material for students to experiment with. 49) Determines unit rates Set up a class store. Have at least a produce section, a meat section, and a canned good section. Have students bring pictures of items that are found in these sections. The students will cut out the picture and glue them to construction paper. Provide prices by each item, such as, green beans- 4/$1.00, apples-3/$0.89, etc. Set up the items in centers. Have the students rotate from one item to the next to determine the cost of one of each item. Have the students then work with a partner to see if they have the same answers. Have the student bring in grocery ads from newspapers. Have the students figure and compare unit prices on given items at two different stores. 50) Uses models to illustrate the meaning of percent Allow the student time to experiment with place value mats and blocks to build decimal numbers in relation to percents. Write percent and decimal numbers on index cards. Give each student an index card that has a decimal number on it. Have the students model their number as a percent on the place value chart. Have the students play the ―Memory Game‖. Make index cards for students to match. Have the students work with a partner or individually. Give each group a deck of 16 cards. Write a percent on eight of the cards. Provide a picture of a 10-by-10 cm grid that has been shaded on the other eight cards. (See example below.) The 16 cards are placed face down in rows of four. One player turns up 2 cards trying to and get a match, such as 16% is the (217) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 same as a 10-by-10 cm grid that has 16 squares colored. If a correct match is not made, the other player turns up 2 cards. If a correct match is made, that player continues until a match is not made. The player with the most matches at the end of the game is the winner. 16% Provide each student with a laminated 10-by10 cm grid and an index card that has a percent written on it. Have the students use a watercolor marker to model their number on the grid. 51) Converts among decimals, fractions, mixed numbers, and percents Place the students in groups of 2-4. Provide each group with a deck of 43 cards. Write a decimal number on fifteen of the cards, a fraction or mixed number on fourteen cards, and a percent on the remaining fourteen. Deal each student 7 cards. Place the remainder of the cards in the center. Students will play according to the rules of ―Gin Rummy‖. Model for students that division is another way to convert fractions to percents, 1 1 1 such as: 1 4 25% and that 1 8 0.125 12.5% 12 % . 4 8 2 Have the students work in groups of 4. Give each group 4 index cards. Have the students write their first name on their index card and determine the part of their first name that is vowels. These will be written in fraction, decimal, and percent form. Students will then do the same for their last name. These cards can be ordered from least to greatest within the individual groups or for the whole class. The fraction forms can be hung on a number line. Students are asked to look in any of their textbooks and locate 3 lines. They can do any of the following to determine fraction, decimal, and percent form. Make a line plot depicting the letters of the alphabet. For instance: A B C D E F G H I J K L M X X X X X X X X X X X X X X X Students can then determine fraction, decimal, and percent forms for what part of the 3 lines of print each letter represents. Use the same 3 lines and compare the number of 3-letter words vs. the 4-letter words. These will be written in fraction, decimal, and percent form. Using a paragraph from the newspaper, students will determine the fraction, decimal, and percent form of common nouns vs. adjectives. Graphs can be constructed to display the data in these activities. (218) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 52) Determines the percent of a number Students work with a partner. Percent problems will be written on the back of index cards such as: 25% of 200 37.5% of 95 19% of 125 50% of 315 The cards will be divided evenly between the two. Each student will place a card face up. The student with the greater card takes both cards. This will continue until all cards are played. The student with the most cards is the winner. Have students practice using mental math with easier numbers, such as 1 1 1 of 16, of 10, of a dozen. 8 2 3 Have the students work with a partner. Provide each group with a newspaper that shows regular price and rate of discount of items, scissors, glue, and construction paper. Have students cut and paste ads on the construction paper. Have each group compute the sale prices of the items. Use 10-by-10 cm grids for students to model such problems as: 10% of 100 30% of 60 50% of 75 53) Estimates decimals, fractions, and percents Give students various decimal, fraction, and percent problems. Ask leading 3 1 questions to help determine the estimate. Example: 7 + 2 4 3 ―Is the sum more or less than ?‖ (Fill in the blank with various numbers.) Have the student use calculators to check for exact answer after estimation has been done. 3 1 Hold up cards that have problems on them, such as 46.68 – 14.66, 2 +7 , 4 8 or 28% 54%. Have the students write the estimated answer. The first student to calculate the correct answer will hold up the next card. 54) Uses proportions and equations to solve problems with rate, base, and part with and without calculators Provide each student with a triangular card. *See below. Display the following problem: 15 is what percent of 60? Identify P = 15 b = 60 (219) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Have the students cover the unknown part to determine the appropriate operation. (If ―r‖ or ―b‖ is the missing part, the student is to divide; if ―p‖ is the missing part, the student is to multiply. p r x b Use the proportion to set up and solve percent problems. For example: 7% of 7 n 80 is _______. ; Cross multiply and divide. (7x80) 100 = 5.60 100 80 60 2 2 1 Example: 60 is _____% of 90 (Simplify first ); ; Cross multiply 90 3 3 100 2 and divide. (2 x 100) 3 = 66 % . Use a calculator for multiplication and 3 division. 56) Finds the percent of increase and decrease Have students use base ten blocks to model percent increase. For example, students put a hundred square on their desk and identify it as 1 whole. Ask the students what fraction is 1 one-square (1/100). Then ask, ―What percent is this?‖ (1%). Next have the students put 1-square next to the hundred squares. By doing this hundred square is increased by 1%. Have the students use the blocks to increase the hundred-square by 10%. Students will add a ten-strip. Ask what fraction is 1 ten-strip. (10/100) and what percent is this? (10%). Inform students that by adding a ten-strip they increase the hundred-square by 10%. Have students use the blocks to increase the hundred-square by 20% (2 ten-strips added), 6% (6 one-squares added), and 14% (1 ten-strip and 4 one-squares added). In finding percent decrease, students use their hands to cover the amount decreased. Having worked students through this strategy, provide them with this information. Added or Unknown x Original = Subtracted Percent Amount Amount (220) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Provide the students with given data from line graphs to plot. Have the students determine the percent of increase and decrease based on the data. For example: Unemployment Rate 7.5 7.0 6.5 6.0 5.5 5.0 Jan. Feb. Mar. Apr. Ask the question, ―What was the percent increase from January to February?‖ 57) Solves problems involving sales tax, discount, and simple interest with and without calculators Have student get a sales advertisement from a department store. Pretend the student has $100 to spend. Have the student make a purchase and figure the total cost including a 7% sales tax without overspending. Have the student find the amount of discount, by purchasing the item or items on sale. Have the students look through the newspaper car sales. The students should select a car, and then figure the total cost with the given rate of interest and the selected time period. Students work in groups of 2-4. Provide each group with two triangular cards* as shown below. Provide one problem at a time for the groups to determine the missing part. Have the students cover the missing part to determine the appropriate operation. If ―r‖ or the other item in the lower right hand corner is the missing part; the student will divide to find the answer. If the C and D located at the top of the triangle are missing; the student will multiply to find the answer. * C D original r x amount r x price Have the students make up their own menu or provide one. Have each student order a meal from the menu. Have the students determine the total cost of the meal. Have the students calculate the amount of sales tax with and without calculators. (221) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Have the students determine the gratuity (tip) of 15% and add to the final cost. *The triangular cards can be used to assist the student. Calculators can be used. Have the students work with a partner. Provide each group newspaper ads showing regular price and sale price, a pair of scissors, glue, and a sheet of construction paper. Have the students cut out the ads, glue them to the sheet of construction paper, and calculate the discount and rate of discount. (222) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 MATHEMATICS BENCHMARKS Eighth Grade O – means teacher should be able to observe throughout the day – possibly use anecdotal records. I – Informal Assessment—those marked ―I‖ have an assessment task attached. Uses properties to create and simplify algebraic expressions and solves linear equations and inequalities 1) I - Identifies and applies the commutative, associative, and distributive properties 2) I - Distinguishes between numerical and algebraic expressions, equations, and inequalities 3) I - Converts between word phrases or sentences and algebraic expressions, equations, or inequalities 4) I - Simplifies and evaluates numerical and algebraic expressions 5) I - Solves and checks one and two-step linear equations and inequalities 6) I - Solves and checks multi-step linear equations using the distributive property 7) I - Graphs solutions to inequalities on a number line 8) I - Writes a corresponding real life situation from an algebraic expression Interprets, organizes, and makes predictions using appropriate probability and statistics techniques 9) I - Interprets and constructs frequency tables and charts 10) I - Finds the mean, median, mode, and range of a given set of data 11) I - Interprets and constructs bar graphs, line graphs, circle graphs, and pictographs from given data 12) I Interprets and constructs stem and leaf, box and whisker, and scatter plots from given data 13) I - Predicts patterns or trends based on given data 14) I - Uses combinations and permutations in application problems 15) I - Calculates and applies basic probability Writes and solves problems involving standard units of measurement 16) I - Converts, performs basic operations, and solves word problems using standard measurements 17) I - Measures line segments and finds dimensions of given figures using standard measurements 18) I - Writes and solves real life problems involving standard measurements 19) I - Selects appropriate units of measurement for real life problems (223) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Determines the relationships and properties of two and three-dimensional geometric figures and the application of properties and formulas of coordinate geometry 20) I - Identifies parallel, perpendicular, intersecting, and skew lines 21) I - Identifies and describes characteristics of polygons 22) I - Finds the perimeter and area of polygons and circumference and area of circles 23) I - Classifies, draws, and measures acute, obtuse, right, and straight angles 24) I - Identifies and finds the missing angle measure for adjacent, vertical, complementary and supplementary angles 25) I - Locates and identifies angles formed by parallel lines cut by a transversal (e.g., corresponding, alternate interior, and alternate exterior) 26) I - Classifies triangles by sides and angles and finds the missing angle measure 27) I - Identifies three-dimensional figures and describes their faces, vertices, and edges 28) I - Uses the Pythagorean Theorem to solve problems with and without a calculator 29) I - Identifies the x and y-axis, the origin, and the quadrants of a coordinate plane 30) I - Plots ordered pairs 31) I - Labels the x and y coordinates for a given point 32) I - Uses tables and graphs simple linear equations Uses basic concepts of number sense and performs operations involving exponents, scientific notation, and order of operations 33) I - Simplifies expressions using order of operations 34) I - Uses the rules of exponents when multiplying or dividing like bases and when raising a power to a power 35) I - Multiplies and divides numbers by powers of ten 36) I - Converts between standard form and scientific notation 37) I - Multiplies and divides numbers written in scientific notation 38) I - Evaluates and estimates powers, squares, and square roots with and without calculators Determines relationships among real numbers to include fractions, decimals, percents, ratios, and proportions in real life problems 39) I - Classifies and gives examples of real numbers such as natural, whole, integers, rational, and irrational 40) I - Identifies, compares, and orders fractions and decimals 41) I - Rounds and estimates using fractions and decimals 42) I - Solves real life problems involving addition, subtraction, multiplication, and division of fractions, decimals, and mixed numbers 43) I - Determines the absolute value and additive inverse of real numbers (224) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 44) I - Classifies, compares, and orders integers and rational numbers 45) I - Adds, subtracts, multiplies, and divides integers and rational numbers with and without calculators 46) I - Writes ratios comparing given data 47) I - Converts among ratios, decimals, and percents 48) I - Solves proportions 49) I - Solves for part, rate, or base 50) I - Finds commissions and rates of commission, discounts, sale prices, sales tax, and simple interest 51) I - Finds percent of increase and decrease 52) I - Writes and solves real life word problems using percents with and without calculators (225) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Informal Assessment for Eighth Grade Guiding Questions Task Sample 1) Can the student identify and apply Provide the student with the following the commutative, associative, and statements: distributive properties? (2 + 5) + 3 = 2 + (5 + 3) (6a)b = 6(ab) 7 x 32 = 32 x 7 8(6 + 7) = 8(6) + 8(7) Have the student label each as the commutative, associative, or distributive property. 2) Can the student distinguish between Provide the student with cards containing numerical and algebraic expressions, the following 1, 2, 3, 4, 5, 6, 7, 8, 9, x, y, z, equations, and inequalities? +, -, =, <, and > written on them. Using the cards have the student create numerical and algebraic expressions, equations, and inequalities using the cards. 3) Can the student convert between Provide the student with a word phrase or word phrases or sentences and sentence similar to the following: algebraic expressions, equations, or 1) Three times a number less than inequalities? sixteen 2) Twenty more than twice is a negative thirty. 3) Three times a number increased by 4 is at least 16. Have the student write expressions, equations, or inequalities for each. Provide the student with the following expressions, equations, and inequalities: 1) 10x – 7 2) 6x + 2 = 20 3) 4x – 7 < 17 Have the student write a word phrase for each. (226) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 4) Can the student simplify and Provide the student with algebraic evaluate numerical and algebraic expressions. Have the student roll a expressions? number cube. Have the student evaluate the expression by substituting the number rolled for the variable. 5) Can the student solve and check one Provide the student with two-step linear and two- step linear equations and equations and inequalities. For example: inequalities? a) 2x + 5 = 9 b) ‾5a – 2 < 8 Have the student solve the equations and inequalities with manipulatives. Have the student check the solution of the equation by substituting it for the variable. 6) Can the student solve and check Provide the student with a set of multi-step multi-step linear equations using the linear equations. Have the student solve distributive property? and check the equation with or without the use of manipulatives. For example: 2(3x – 4) = 10 7) Can the student graph solutions to Provide the student with inequalities and inequalities on a number line? number lines. Have the student solve and graph the inequality on the number line. Example: 2x – 5 > 15 8) Can the student write a real-life Provide the student with an equation such situation from an algebraic as 3x – 5 = 15. Have the student write a expression? real-life situation that is represented by this equation. 9) Can the student interpret and Have the student gather responses to a construct frequency tables and survey question and construct a frequency charts? table and/or chart. Have the student answer leading questions by the teacher concerning the table and/or chart. (227) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 10) Can the student find the mean, Provide the student with a given set of median, mode, and range of a given data such as grades. set of data? Have the students find the mean. Have the students find the mode. Have the students find the median. Have the students find the range. 11) Can the student interpret and Divide the class into groups. Have each construct bar and line graphs, circle group take a survey of favorite television graphs, and pictographs from given programs. Assign each group a graph to data? construct. Have each group interpret a different group’s graph. 12) Can the student interpret and Provide the student with a set of data construct stem-and-leaf, box-and- gathered by measuring the circumference whisker, and scatterplots from of the students’ heads. Have the student given data? construct stem-and-leaf and box-and- whisker graphs. 13) Can the student predict patterns or Provide the student with a graph showing trends based on given data? minimum wages for a 10-year period. Have the student predict what the minimum wage will be in five years. 14) Can the student use combinations Provide the student with the following and permutations in application examples: problems? 654 Combinations C (6, 3) = 20 3 2 1 Permutations P(6, 3) = 6 5 4 =20 Have students find how many ways a first, second, and third place winner can be selected from 20 people. (228) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 15) Can the student calculate and apply Have the student find the number of basic probability? phone numbers that can be assigned if the three-digit prefix remains the same and the last four numbers can be rearranged. Example : 736 – N N N N 10 · 10 · 10 · 10 10,000 16a) Can the student convert from one Provide the student with a chart of equal unit to another using standard measures. Have the student convert the measurements? following measurements showing his/her work. 16b) Can the student perform basic 1. 10 quarts to gallons. operations using standard 2. 4 feet 7 inches to inches measurements and solve word 3. 34 ounces to pounds and ounces problems using standard measurements? Provide the student with the following line chart. k h da meter d c m Have the student convert the following conversions: 1. 9 g = ______ g 2. 14.5 ml = ______ L 3. 25 m = _____ km Provide the student with a chart of equal measures. Have the student perform the following. 3 qt 2 pt 12 lb + 4 qt 3 pt - 3 lb 4 oz 5 ft 3 in x 5 in (229) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample Provide the student with standard English and metric units of measure. Have the student answer the following questions and show all work. 1. Janet’s room is 12 feet by 11 feet. How many square feet of carpet will she need? 2. If Mary rode her bike 3.2 miles, how many yards did she ride the bike? 3. Tyler has a board 24 cm long. He needs 3 boards, 7 cm long. Can he get all three from the board? 4. If a nickel weighs 5 grams, how many nickels weigh 1 kilogram? 17) Can the student measure line Provide the student with some pre-drawn segments and find dimensions of line segments. Have the student measure given figures using standard 1 these to the nearest of an inch or to measurements? 16 the nearest cm. Provide the student with an English and metric ruler. Have the student measure a side of a square on the floor, the diameter of a penny, the length of book, etc., in English and metric measures. 18) Can the student write and solve real- Provide the student with several vegetable life problems involving standard can labels. Ask the student to find the measurements? number of cans needed to feed 3, 10, or 25 people. Provide the student with a grocery ad and ask the student to find the cost of a certain number of pounds of some food. Use a grocery ad and have the student write a problem involving the purchase of 2 or 3 items such as fruit or deli (per pound) items. Provide the student with information about races or some event involving metric measurement. Have the student write a problem using this information. (230) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 19) Can the student select appropriate Present the student with a list of objects units of measurement for real-life such as length of a pencil, volume of a problems? tank, or area of a room, and have the student tell if it should be measured in inches, feet, square inches, square feet, cubic inches, or cubic feet. Provide the student with the following items to match with the appropriate measurement. a) Distance from Memphis to Columbus 1. 270 km b) Length of shoe 2. 270 m c) Diameter of quarter 3. 19 cm 4. 19 mm 5. 3 cm 6. 3 mm 20) Can the student identify parallel, Ask the student to use the shape below to perpendicular, intersecting, and skew identify a pair of parallel, perpendicular, lines? intersecting, and skew lines. A B E F D C G H 21) Can the student identify and describe Have the student draw and name a characteristics of polygons? polygon. Have them describe the characteristics of the polygon drawn. 22) Can the student find the perimeter Using a map of Mississippi, have the and area of polygons and student find the perimeter of the polygon circumference and area of formed by joining Jackson, Meridian, circles? Laurel, and Collins. Ask the student to build a five-sided, four- sided, and six-sided polygon on the geoboard and find the area of each. Ask the student to find the circumference of a circle with a diameter of 8 cm using the formula c= d. (231) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 23) Can the student classify, draw, and Ask the student to name an acute, obtuse, measure acute, obtuse, right, and right, and straight angle using the figure straight angles? below: A C D B E F G Have the student use a protractor to find the actual measure of ABC, BFG, and EFG in the figure above. 24) Can the student identify and find the Ask the student to draw and label a missing angle measure for adjacent, diagram to show each of the following: vertical, complementary, and 1) Angles ABC and CBD are adjacent supplementary angles? angles 2) Angles XYZ and AYC are vertical angles 3) Angles ABC and CBD are complementary Have the student find the measure for the missing angle in each figure. a) b) 75 ? 110 ? c) 45 ? 25) Can the student locate and identify Provide the students with two parallel lines angles formed by parallel lines cut by cut by a transversal. Have student identify a transversal (e.g., corresponding, the corresponding, alternate interior, and alternate interior and alternate alternate exterior angles. exterior)? 1 2 3 4 5 6 7 8 (232) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 26a) Can the student classify triangles Have the student draw and label an by sides and angles? isosceles triangle, a scalene triangle, and an equilateral triangle. Have the student draw a sketch of an acute, obtuse, and right triangle. 26b) Can the student find a missing Have the student find the value of the angle measure? missing angle(s). Then classify each triangle as acute, right, and obtuse. a) b) 45 60 60 c) 120 30 27) Can the student identify three- Provide the students the following figures: dimensional figures and describe their faces, vertices, and edges? a) b) c) Have the student name the type of polygon that forms the faces and edges. Have the student list the number of faces, vertices and edges of each solid. Can the student use the Pythagorean Have the student use the Pythagorean Theorem to solve problems with and Theorem (a2 + b2 = c2) to find the length of without a calculator? the missing leg. 5 3 (233) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 29) Can the student identify the x-axis, Provide the student a coordinate plane the y-axis, the origin, and the and ask the student to label the x-axis, y- quadrants of a coordinate plane? axis, the origin, and the quadrants. 30) Can the student plot ordered pairs? Ask the student to plot these points on the coordinate plane: (0,0) (-2,3) (4, -2) (-3, -3) (0, 4) (-5, 0). Provide the student a negative x coordinate of an ordered pair. Ask the student ―In which quadrants will the point be plotted?‖ Have the student give two examples of an ordered point using the given x coordinate. Provide the student a positive y-coordinate of an ordered pair. Ask the student ―In which quadrants will the point be plotted?‖ Have the student give two examples of an ordered pair using the given y coordinate. 31) Can the student label the x and y Provide the student with a coordinate coordinates for a given point? plane on which several points have been drawn. Have the student name the x- coordinate and the y-coordinate of each. Have the student tell which quadrant contains each ordered pair. 32) Can the student use tables to graph Provide the student with a pegboard, golf simple linear equations? tees, and a simple linear equation. Have the student make a table of values. Have the student graph the equation by placing the pegs in the correct holes. Have the student complete the table and graph the solution to the equation: y = 2x + 1. x 2x + 1 y x, y (234) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 33) Can the student simplify expressions Provide the student with several using order of operations? expressions containing more than one operation. Have the student simplify each expression by using order of operations. Example: 4 + 3 5 = 4 + 15 = 19 34) Can the student use the rules of Provide the student with multiplication and exponents to multiply or divide like division problems with like bases and bases and to raise a power to a exponents. Provide the student with power? problems raising a power to a power. Have the student use the rule of exponents to solve the problems. Example: 35 32 = 37, 98 9 = 97, (24)5 = 220 35) Can the student multiply and divide Provide the student with a set of problems numbers by powers of ten? requiring multiplication and division by powers of ten. Example: 2.1 x 104 = 21,000 2.1 ÷ 104 = 0.00021 36) Can the student convert between Provide the student with a set of numbers standard form and scientific notation? in standard form. Have the student convert to scientific notation. Example: 3,500= 3. 5 x 103 Provide the student with a set of numbers written in scientific notation. The student must convert to standard form. Example: 7.4 x 10-2 = 0.074 37) Can the student multiply and divide Provide the student with a set of problems numbers written in scientific in scientific notation. Have the student notation? multiply and divide the numbers. Example: 2.4 x 105 = 1.2 x 10 2 2 x 10 3 (235) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 38) Can the student evaluate and Provide the student with a set of numbers estimate powers, squares, and written as powers, squares and square square roots with and without roots. Have the student estimate and calculators? evaluate the numbers with calculators? Have the student estimate and evaluate the numbers without calculators? Example: Estimate then evaluate. 38 21 2 39) Can the student classify and give Provide the student with a set of numbers. examples of real numbers (natural, Have the student classify each number as whole, integers, rational and natural, whole, integer, rational or irrational)? 1 7 irrational. Ex: ( -2, , 0, 5, 0.6, ) 3 2 Have the student give examples of natural numbers, whole numbers, integers, rational numbers and irrational numbers. 40) Can the student identify, compare Provide the student with a group of and order fractions and decimals? fractions and decimals. Have the student compare fractions and decimals 3 3 Example: ( , 0.6) ( 0.6) 4 4 Order the group of numbers from least to 1 1 2 greatest. (0.7 , , , , 0.5) 3 9 5 1 1 2 ( , , , 0.5, 0.7) 9 3 5 41) Can the student round and estimate Give the student a variety of fractions and fractions and decimals? decimals. Have the student round each fraction to the nearest whole number. Ex: 3 1 = 2. Round each decimal to a given 4 place value. Ex: Round 7.0236 to the nearest thousandth. ( Answer 7.024) (236) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample Have student answer questions, such as 3 7 ―If I have 2 pizzas, and Mary has 1 4 8 pizzas, do we have as much as 4 whole pizzas together?‖ 43) Can the student determine absolute Ask the student to state the absolute value value and additive inverse of real for each of the following: numbers? 3, 0, 2 , 4 .5 3 Ask the students to state the additive 1 inverse of 6, -5, 0, 2 , 0.45. 44) Can the student classify, compare, Given the following numbers: and order integers and rational 2 ˉ5, , 7 , 0, ˉ2.7, 45, 0.4 numbers? 3 Ask the student to tell whether the number is an integer, a rational number, neither, or both. Ask the student to arrange the following numbers in order from least to greatest. 5 2 , -4, 0, -1, , 2 4 3 45) Can the student add, subtract, Provide the student with a variety of math multiply, and divide integers and problems using the four basic operations. rational numbers with and without Include integers and rational numbers in calculators? the problems. Have the student solve some of the problems without calculators. Have the student solve some of the problems using a calculator. Ex: -3 + 24 = 21 (237) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 47) Can the student convert among Have the student perform the following ratios, decimals, and percents? tasks: Convert a ratio to a decimal and to a percent. Example: 3:5 = 0.6 = 60% Convert a decimal to a ratio and a percent. 6 3 Example: 0.6 = 60% 10 5 Convert a percent to a ratio and to a decimal. 60 3 Example: 60% = 0 .6 100 5 48) Can the student solve proportions? Ask the student to explain how to solve a proportion. Provide the student with a set of dominoes. Have the student draw two dominoes and determine if they form a proportion. (If a blank domino is drawn, the student must determine the number needed to form a proportion.) 49) Can the student solve for part, rate or Have the student count the number of girls base? and boys in the classroom. Have the student use this information to predict how many boys and how many girls are in the eighth grade class with a total of 1000 students. 50) Can the student find commissions Provide the student with a list or pictures and rates of commission, discounts, of prices of specific items (e.g., homes, sale prices, sales tax, and simple CD players, food, cars). Provide the interest? student with interest rates or sales tax rates. Have them determine discount, sale price, sales tax, and/or simple interest. (238) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Guiding Questions Task Sample 51) Can the student find percent of Provide the student with several increase and decrease? basketball players shooting averages from last year. Have them find the percent of increase or decrease as compared to this year’s averages. 52) Can the student write and solve real- Write several percents on the overhead life word problems using percents board for the student to select at least two. with and without calculators? Have the student write and solve a problem involving the two percents chosen. Example: 60% 40% 33 1 % 3 25% 17% Janie got 40% of her questions correct on a reading test that had 100 questions. Maurice got 25% of his questions correct on a spelling test that had 80 questions. How many questions did each student miss? (239) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Suggested Teaching Strategies for EIGHTH GRADE Uses the properties to create and simplify algebraic expressions and solves linear equations and inequalities 1) Identifies and applies the commutative, associative, and distributive properties. Make a deck of cards with the name, the algebraic notation, and a numerical example of the commutative, associative, and distributive properties. Include cards with incorrect properties written on them. Have the students play Rummy. An example of a set would be Distributive a(b + c) = ab + ac 6(2 + 3)= 6(2) + 6(3) Give the student expressions already simplified. Have the students determine which expressions are equivalent and which ones are not. Have the students correct the mistakes made and identify the name of the property. Examples: ˉ2(x -5) = ˉ2x – 5 incorrect ˉ2(x – 5) = ˉ2x + 10 Distributive 3 + (x + 5) = (3 + x) + 5 correct Associative for addition Use boxes to help students remember to distribute the number on the outside of the parentheses correctly. Examples: ˉ6(x – 8) ˉ6x 48 = ˉ6x + 48 2) Distinguishes between numerical and algebraic expressions, equations, and inequalities Divide the class into three teams. Hold up a card with an algebraic expression, equation, or inequality written on it. Give one point to the team who can correctly identify the card as an expression, equation, or an inequality. Give bonus points to the team that can simplify the expression or solve the equation or inequality. Provide bonus points or a small prize to the team who earns the most points. Discuss the difference between operation and relation signs. Provide students with various examples of numerical and algebraic expressions, equations and inequalities. Have students identify the type example it is. They must explain ―why‖ in order to receive full credit. Example: 4a – 2 is an algebraic expression - Why: variable, no relation sign Example: y < 6 inequality - Why: relation sign, less than (240) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Put several examples of numerical and algebraic expressions, equations and inequalities on the board. Have the student classify each example. k 3 + a; 5 x 7 + 12 ÷ 4; a +16 = 72; 5b = 75; <5 13 3) Converts between word phrases or sentences and algebraic expressions, equations, or inequalities Divide the students into teams. Play Jeopardy with the categories being expressions, equations, and inequalities. Have the team choose the category and the point value for the question. If the team can simplify the expression or solve the equation or inequality, they earn an additional 100 points. Example: Team A picks equations for 200 points. The teacher would read an equation problem such as three times a number increased by 7 is 52. Team A would respond, ―What is 3x + 7 = 52?‖ Team A would earn 200 points for the correct equation and 100 points for the correct solution. Provide students with algebraic phrases or sentences written on one side and word phrases or sentences on the other side. Orally discuss the odds on each side together. Students complete the evens individually or in groups. Have students come to the overhead to work and explain their answers. Discuss multiple ways to explain the algebraic phrases or sentences. Example: 2 + 5a, two plus five times ―a,‖ the sum of 2 and five times ―a.‖ 4) Simplifies and evaluates numerical and algebraic expressions Give students an expression such as 3 52 2 4 . Have students write the problem several times on a piece of paper and insert parentheses to create as many expressions as possible. Have the student evaluate each expression. Have the class make a list of all the possible answers. Give the class four numbers and the desired answer. Have the students add, subtract, multiply, or divide the four numbers to get the desired answer. Example: 4, 6, 5, 2 desired answer is 9 (6 – 4) 2 + 5 6–4=2x2=4+5=9 Give each group a deck of cards with algebraic expressions written on them and a number cube. Give each person in the group one card. Have a student roll the number cube and have the other students evaluate the expressions with the number generated. (The person with the highest answer earns a point. After using all cards, the person with the most points is the winner.) Another option for scoring is to keep a sum of all possible answers. The person with the largest sum is the winner. (241) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 5) Solves and checks one and two-step linear equations and inequalities Have the student solve equations and inequalities using a flow chart. Example: ˉ5x + 7 > ˉ38 Start: X ˉ5 ˉ5x +7 ˉ5x + 7 Answer < 9 ÷-5 ˉ45 ˉ7 ˉ38 > x<9 When solving inequalities using a flow chart, remind students to look at the number with the variable. If the number is negative the inequality must reverse. Discuss opposites, zero values, equations, and inequalities. Demonstrate how to use a balance to solve equations. Have students come to the overhead to solve, then check equations. Example: 3a – 2 = 10 aaa –2 = 10. Replace each original with 4. Extend this to solving inequalities. +2 +2 aaa 2 10 aaa 12 a 4 Give the student one and two-step equations and inequalities. Have the student solve. Let the student use a calculator to check the solution. 4a + 2 = 22 4a + 2 – 2 = 22 – 2 subtract 2 from both sides 4a = 20 divide both sides by 4 4 4 a=5 (242) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 6) Solves and checks multi-step linear equations using the distributive property Have the students use the flow chart, algebra tiles, etc., to solve the equation. Example: -4(2x + 11) = 92 2x – 4 -8x + 11 -4 -44 -8x + -44 = 92 x -8 -8x – 44 – 8x – 44 = -17 -8 136 + 44 92 Answer: x = 17 Use parentheses and arrows to model the solution of equations and inequalities using the distributive property. Example: -2(4a – 5) = -22 -8a + 10 = -22 -10 = -10 8a 32 = 8 8 a =4 Have the student use the distributive property to solve the equation. Use a calculator to check solution. 5(2 x + 3) = 65 (5 2x) + (5 3) = 65 10 x + 15 = 65 10 x +15 – 15 = 65 - 15 10 x = 50 10 10 x =5 7) Graphs solutions to inequalities on a number line Create a number line on the floor using masking tape. Give each student a card with a number written on it. Write this number on the number line. Provide an example such as x < 2. Have any student who has a number less than two stand on the number line. Have a class discussion about the spaces in between the numbers now marked on the number line. Have the student on the number line hold hands to represent that the correct graph would include all numbers less than 2, even the fractions and decimals in between each number. (243) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 8) Writes a corresponding real life situation from an algebraic expression Have students create a small business that will sell a product or service. Have the students create an advertisement for the product or service that states what they are selling and the cost. Have the students write an expression showing the relationship. Divide the class into 4 or 5 groups. Have each group write an algebraic expression. Groups then pass their expression to the next group who will write a real-life situation from the expression. Continue passing around until each group receives their original expression back. Each group will present to the class all the real life situations given for their expression. Example: 2a – 5. My sister is 5 years less than twice my age. Have student write an expression for each of the following: a. Tom has two times as many marbles as John. Let m stand for John’s marbles. Write an expression for Tom’s marbles, such as 2m. b. Denise ran 5 more laps today than she did yesterday. Let y stand for the laps run yesterday and write an expression for the number of laps she ran today. Example: 5 + y Interprets, organizes, and makes predictions using appropriate probability and statistics techniques 9) Interprets and constructs frequency tables and charts Have students make paper airplanes and go outside to fly them. Have the students record the distance in feet and make a frequency table using the distances the airplanes flew. Punch out holes from several colors of construction paper. Provide each student with a small amount of different color holes. Have the students glue dots on graph paper to create a frequency table. Have the student use this information to create a story. 10) Finds the mean, median, mode, and range of a given set of data Have the student count the number of each color of M&M’s. Have the student find the mean, median, mode, and range for each color of M&M. Discuss the results. Have students measure their height in cm. Make a frequency table. Then find the mean, median, mode and range of their heights. Give the student the following test scores: 100, 80, 93, 85, 80, 90, 65, 50, 93, 98, 57. Have the student place the test scores in order from least to greatest and find the mean, mode, median and range. Solution: 50, 57, 65, 80, 80, 85, 90, 93, 93, 98, 100 Mode: 80 and 93 Median: 85 Range: 100 – 50 = 50 Mean: 891 ÷ 11 = 81 (244) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 11) Interprets and constructs bar graphs, line graphs, circle graphs, and pictographs from given data Construct a frequency table from the class’ favorite football team. Divide the class into 3 groups. Assign each group a different type of graph to construct from the data (circle, bar, pictograph). Have the groups present their graphs to the class. Conduct a class discussion to help decide which type of graph best represents the data. Provide the class with several examples of bar graphs, circle graphs, and pictographs. Working in pairs or small groups, have the students select a graph. The students will interpret the data given in their graphs. They should be able to discuss the key, the intervals used, the type of graph used and what the graph represents. 12) Interprets and constructs stem-and-leaf, box and whisker, and scatter plots from given data Have the students measure their height and armspan to the nearest centimeter. Construct two frequency tables. Divide the class into 3 groups. Have one group construct a stem-and-leaf plot, another group a box and whisker plot and the last group a scatterplot from both sets of data. Have each group present and discuss the graphs. Use the list function on a graphing calculator to create the above graphs. 13) Predicts patterns or trends based on given data Provide a chart showing the number of cell phones per family since 1990. Have the students predict the number of cell phones per family in 2025. Use the Internet to track the progress of a given stock for the past year. Predict patterns or trends for the next month. 14) Uses combinations and permutations in application problems Introduce the factorial key on the calculator. Provide students with permutation and combination problems to solve using the calculator. Have the student determine how many ways a president, vice president, and secretary can be selected from a group of 10 people. Example: C (10, 3) = 10 9 8 3 2 1 15) Calculates and applies basic probability Have the students find the number of possible car tags that can be assigned in a state if the tag must have 3 letters and 3 numbers. Discuss why a town of 20,000 households with one phone each must have multiple prefixes in addition to the 4 digit phone numbers. Example: 10 x 10 x 10 x 10 10,000 possible numbers = 10,000 possible numbers < 20,000 households (245) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Have students show how prefixes could determine 400,000 different phone numbers. Use only the face cards from a deck of cards. Lay the cards face-up in a random order. State this rule: A card is chosen at random. What is the probability of each of the following events: 4 1 a. Choosing a queen? or 12 3 6 1 b. Choosing a red card or 12 2 8 2 c. Choosing jack or king? or 12 3 Writes and solves problems involving standard units of measurement 16) Converts, performs basic operations, and solves word problems using standard measurements Provide students with calculators, standard English and metric units of measures, and a series of questions to be answered in class Examples: 1. Your new hot water heater is calibrated in degrees Celsius. What setting should you use if you want a hot water temperature of 149˚ F? 2. How many pounds of ground beef should you buy to make 140 hamburgers, if each hamburger patty weighs 6 ounces before cooking? 1 3. You want to increase a recipe by . Your measuring cup is marked with 4 thirds of a cup and milliliters. The recipe calls for 2 cups of sugar. a. If each cup is equivalent to 8 fl. oz., how many ounces of sugar do you need? b. If a cup is about 250 milliliters, how many milliliters do you need? c. Can you express the amount of sugar in ―thirds of a cup‖? If so, how many thirds do you need? Give the student the following problems to solve. 1. A carpenter uses 4 pieces of wood to make a frame. The pieces of wood total 8ft 2in. What is the average length of each piece of wood? Simplify the answer. 17) Measures line segments and finds dimensions of given figures using standard measurements Let the students make figures with tangrams, trace around their figures, and write measurements in English and metric measures above each line. Have a scavenger hunt for two or three days. Ask students to bring in objects that are specified lengths. (246) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Measure temperature, length, liquid measure, and weight/mass of given objects. Measure line segments and find dimensions of given figures using standard measurements. 18) Writes and solves real life problems involving standard measurements Provide the students a recipe of something they can make in class like ―nuts and bolts‖ or Cheks Party mix. Have them double, triple, half, 1 1 times the 2 recipe. Have the students make the recipe and enjoy. Use an easy cookie recipe and do the same thing. They love to cook. Let them bring cookies or verify that they made the recipe for extra credit. Divide class into groups. Have each group build a swing out of straws, string, cardboard and glue. Final presentation must list lengths and heights of each object like swings, play area, and seesaw. 19) Selects appropriate units of measurements for real life problems Have students take 4 index cards each. Write an object to be measured on two cards. Write the appropriate measure of the objects on the other two cards. Have the students gather the information from local newspapers, magazines, etc. Take up cards every day for several days. Display about 4 or 5 objects and the correct answers in random order. Have the students match the object with the correct measure. Have the student choose the most appropriate unit of measurement: a. The length of a football field. b. The volume of a fish tank. c. The height of a door. d. The weight of a book. e. The amount of water a glass can hold. f. The distance from Jackson to the Gulf Coast. Determines the relationships and properties of two and three-dimensional geometric figures and the application of properties and formulas of coordinate geometry 20) Identifies parallel, perpendicular, intersecting, and skew lines Use the classroom as the model and have students identify parallel, perpendicular, intersecting and skew lines. Use a city street map. Have students name streets that are parallel, streets that intersect, and streets that are perpendicular. Use given examples and have the students identify skew lines, parallel lines, and perpendicular lines. 21) Identifies and describes characteristics of polygons Arrange the students in groups of four. Ask the students to discuss the set of shapes, then begin sharing sentences that involve ALL of the shapes, (247) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 SOME of the shapes, and NONE of the shapes. The recorder will post on chart paper and groups will report out. ALL of the shapes SOME of the shapes NONE of the shapes Give students sets of pictures of polygons with several characteristics in common. Have students give at least one similarity and one difference in each set. Example: Rectangle Square Same: Right angle Different: Length of sides (Note: A square is a rectangle, but not all rectangles are squares.) 22) Finds the perimeter and area of polygons and circumference and area of circles Have students make shapes of various polygons on the geoboard. Count the number of squares to find the area. Count the border edges to find the perimeter. Give a shape and ask students to add tiles until the perimeter of the figure is 16. (Squares that are added must meet so that they are touching on at least one side) What is the area of the original figure? What is the area since the addition of the tiles? Where would you place a tile to increase the perimeter by 1? By 2? By 3? What happened to the area as the perimeter increased? How could you increase the area by 3 and not increase the perimeter? What is the fewest number of tiles that can be added to increase perimeter to 16 units? The greatest number? Describe this new shape. What is the area of each shape? FINDING THE AREA OF A CIRCLE Method 1: Counting Squares 1. Draw a circle on the grid paper. Count all the whole centimeter squares that lie completely inside the circle. (This underestimates the area of the circle). (248) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 2. Now make an overestimate of the area of the circle by taking the number of whole squares that lie inside the circle (the same number you got for step 1 above) and add to it the number of squares that touch the circle and lie partly inside and partly outside the circle. 3. Find the average of your two counts. Method 2: Inscribing and Circumscribing Squares 1. Circumscribe a square about the circle. Find its area. 2. Inscribe a square inside the circle. Find its area. 3. Find the average of the two areas. Method 3: The “Curvy Parallelogram” Method 1. The circle has been divided into 8 congruent sectors. 2. Cut out the sectors and arrange them to form a curvy parallelogram. 3. Approximate the area of the curvy parallelogram. Recording the Results Record the information from methods 1, 2, & 3 in the table below. Method No. Title Area 1 Counting Squares 2 Inscribing and Circumscribing Squares 3 The ―Curvy Parallelogram‖ ______________________________________________________________ When you and your partner have completed all the methods, answer the following question in writing: Which method do you ―trust‖ the most? Why? 23) Classifies, draws, and measures acute, obtuse right, and straight angles Use a circular geoboard, template and protractor. Using a rubber band connect P1 to P19. Then use another rubberband to connect P19 to P17 What is the vertex of the angle? Estimate the measure of the angle. Use a protractor to measure the angle. What type of angle is formed? Construct any angle on geoboard; transfer to template. Name, measure, and classify the angle according to its measure. Construct polygons on the circular template. Measure and classify each angle of the polygon. (249) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Enrichment: Construct these 3 polygons on the same circle using colored pencils and color after completion. Note: Circle may be divided into units other than 24. Students may color in the designs made and have beautiful art work on display. 24) Identifies and finds the missing angle measure for adjacent, vertical, complementary, and supplementary angles Draw 90 angles. Draw another ray from the vertex to create two adjacent angles. Measure the two angles created. Ask students to write a statement regarding these angles. The following are 180 angles. Draw another ray from the vertex to create adjacent angles. Measure the angels created. Write a statement regarding these angles? Use dowel rods to model two intersecting lines. Have the students name the angles formed and note their findings about the relationships of the angles. Identify and find the missing angle measure for adjacent, vertical and complementary angles. Find the measure of angle X, Y and Z. Y X Z 60 25) Locates and identifies angles formed by parallel lines cut by a transversal (e.g., corresponding, alternate interior, and alternate exterior) Give students these materials: grid sheet, colored pencils, and tracing paper. This grid is formed by sets of parallel lines. Example: Use one color and color in one set of corresponding angles. Use a different color and color in a different set of corresponding angles. (250) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Use a third color and color in a set of alternate interior angles. Using a fourth color and color in a set of alternate exterior angles. Using the tracing paper, trace over the grid marking the corresponding angles on top of the other. What observations can be made? Do the same with alternate interior and alternate exterior. Try several to validate the conjecture. Use a subway map to locate and identify parallel lines, transversals, interior and exterior angles. 26) Classifies triangles by sides and angles and finds the missing angle measure Have the class construct triangles on the geoboard with bands. Students will sort triangles according to their sides and angles and name triangles. On geoboard construct a triangle. Working in pairs, do not show it to your partner, but give partner careful instructions on how to create triangle on his geoboard. When finished, compare your triangles. Are they congruent? If not why? Reverse role. Use a circular geoboard to connect. 1) P1 and P7 What type of triangle do you have? P7 and P15 P15 and P1 Measure each angle to verify the type of triangle. 2) P10 and P14 P14 and P19 P19 and P10 3) P1 and P5 Measure the sides to determine the type of P5 and P13 triangle. P13 and P1 27) Identifies three-dimensional figures and describes their faces, vertices, and edges Use materials such as straws, bobby pins or pipe cleaners to: 1. Make a square with straws 2. At each corner, attach a straw (Straws must be the same length) 3. Join these four straws at a point. 4. What shape was constructed? How many faces? edges? vertices? Make a triangular pyramid. Count its vertices, edges and faces. Complete the chart Vertices Edges Faces Square pyramid Triangular pyramid (251) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Use three-dimensional figures or pictures to compare faces, vertices, and edges of these figures. Example: Rectangular prism and rectangular pyramid Example: Cube and rectangular prism Example: Triangular prism and triangular pyramid 28) Uses the Pythagorean Theorem to solve problems with and without a calculator Have student use the Pythagorean Theorem to determine whether the triangles are right triangles. 3 5 5 15 4 12 Find the unknown length of given right triangles 25 x x 8 24 15 When a TV is advertised a having a 19-inch screen, it means that the diagonal is 19 inches long. If a 19 inch TV screen has a height of 12 inches, what is the width? 29) Identifies the x -and y-axis, the origin, and the quadrants of a coordinate plane Identify a point on the x-axis, y-axis and identify the origin when given a coordinate plane with several points on it. Tell in which quadrants the following points are located. (-3,2) (0, 0) (6,1) (5,-4) (-2,-2) 30) Plots ordered pairs Trace a picture of Mississippi on a coordinate plane. Write the ordered pairs that form the state. Give the set of ordered pairs to a friend to plot. See if they get Mississippi. (252) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Play a game of ―Battleship‖. Try to sink your opponent’s ship by giving the correct coordinates. 31) Labels the x and y coordinates for a given point The grid shows the location of three sailboats, A, B, and C. Give an ordered pair for each sailboat. Use the game Battleship to locate coordinates for a given point. 32) Uses tables and graphs simple linear equations Write and graph an equation that shows the relationship between the amount of time Susan studies and the amount of time she practices the piano. Susan spends four more hours each week studying than she does practicing the piano. Solution: Let x = time she practices piano Let y = time she studies y=x+4 Make a table of values and graph the ordered pairs on a coordinate plane. x x +4 y x, y 0 0+4 4 0, 4 1 1+4 5 1, 5 2 2+ 4 6 2, 6 Use a graphing calculator to solve linear equations for given values of x. Then graph. Uses basic concepts of number sense and performs operations involving exponents, scientific notation and order of operations 33) Simplifies expressions using order of operations Give students a Bingo Card with expressions written in each space. Allow the students time to simplify the expressions. Call out numbers, which are solutions to various expressions. Students mark their cards according to their solutions. (The winner must show how they simplified each expression used.) Give one group of students a scientific calculator and another group a regular calculator. Give all students 10 expressions to simplify. After they have worked them with the given calculator, compare answers. Discuss which they have alike and which are different and why. (253) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 34) Uses the rules of exponents when multiplying or dividing like bases and when raising a power to a power Have students model/demonstrate the meaning of multiplying the bases. For example: 35 · 32 =( 3 · 3 · 3 · 3 · 3) · (3 · 3) = 37 Discuss the shortcut to use when multiplying like bases. (Adding exponents). Work examples by using the shortcut. Have students demonstrate the meaning of dividing like bases. Ex. 35 32 3 3 3 3 3 33 3 3 Discuss the short cut for dividing like bases – (subtracting exponents). Work examples using the shortcut. Have the students demonstrate the meaning of (82)3=(8 8) (8 8) (8 8) = 86 Discuss the shortcut of raising powers to powers (multiplying exponents). Work examples. 35) Multiplies and divides numbers by powers of ten Have the student multiply or divide by powers of ten by using exponents. 3.5 103 (3 places) = 3500 3.5 104 (4 places) = .00035 Have the student count the number of zeroes in order to know how many places to move. 7.2 x 10000 (4 zeroes) (4 places) 72,000 7.22 ÷ 10000 (4 zeroes) (4 places) 0.000722 Compare multiplying by powers of ten with negative exponents (6.8 x 10 -3) and dividing by powers of ten with positive exponents (6.8 ÷ 10 3). Work several example. Generalize a rule. 6.8 10-3 = 0.0068 6.8 103 = 0.0068 Multiplying by a negative power of ten and dividing by the same positive power of ten will give the same answer. 36) Converts between standard form and scientific notation Find examples of large whole numbers or very small decimal numbers in magazines or newspapers and convert to scientific notation. Give each group a chemistry book. Have them write 5 numbers they found written in scientific notation and what they were describing. Convert to standard form. Have each group demonstrate the numbers and explain ways they were used. (254) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 37) Multiplies and divides numbers written in scientific notation Use properties and powers of ten to regroup and multiply. (7.1 103) (0.14 104) (7.1 0.14) (103 x 104) 0.994 107 9,940,000 Have the student use properties and powers of ten to regroup and divide. 6.8 10 3 (6.8 103) (0.4 102) 17 101 170 0.4 10 2 38) Evaluates and estimates powers, squares, and square roots with and without calculators Find the missing standard numerals 32 = , 33 = , 105 = , 9 squared = , 8 cubed = , 26 = Have the student write the expanded form and standard form of each 42 = 108 = Have the student estimate the square root by determining which perfect squares the number is between. 40 Estimate between 6 and 7, because 40 is between 36 and 49. Estimate which numbers the square root of 85 is closer to and why. 85 Is closer to 9 or 10? (9, because 85 is closer to 81 than 100) Use the x2 , x4 , and keys on a calculator to simplify the given expression. 192 = 37= 22500 = Determines relationships among real numbers to include fractions, decimals, percents, ratios and proportions in real life problems 39) Classifies and gives examples of real numbers such as natural, whole, integers, rational, and irrational Give each student an index card with natural, whole, integer, rational, or irrational written on it. Tell the students to write on their card any number that called out that is an example of the word on their card. Call out various numbers. (The student with the most correct numbers written on their card for each category wins). Let the class help decide if the numbers are correct and why. (255) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Use Venn Diagrams to show relationships among natural, whole, integers, rational, and irrational numbers. R I W N IR 40) Identifies, compares, and orders fractions and decimals Have students match the fraction or decimal form and give a word name. Seven and 8 tenths 7.8 Nine hundredths 9 100 Have the students cross multiply to compare fractions. 3 5 < 8 9 27<40 Find common denominators to compare fractions. 3 27 3 5 < 8 72 8 9 5 40 9 72 Change fractions to decimals, and then compare. 3 0.375 8 0.375 < . 5 5 0.5 9 Use baseball cards to record batting averages. Order from least to greatest. Discuss what the average means. (256) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 41) Rounds and estimates fractions and decimals Have the student round decimals to a given place. Example: Round 7.176 to the nearest tenth. 7 Estimate to which whole number of a given fraction is closer. 2 is closer to 8 which whole number ( 2 or 3)? 42) Solves real life problems involving addition, subtraction, multiplication, and division of fractions, decimals, and mixed numbers Have students select two stocks they would be interested in buying. Students will work individually or in a group. Provide each student/group with a set amount of money. Students will use the stock market page to determine increase and/or decrease of their stock for at least a week. Graphs can be drawn to model how their stocks did for that period of time. Percentages can be calculated with this activity also. Use the sports page to determine shooting percentages during basketball season and batting averages during baseball season. Students can also determine these with their own basketball and baseball teams. Have students work with a partner. Each group will be given a catalog, such as J.C. Penny, Sears, etc. Each group will choose 10 items and record each item and its price on a sheet of paper. Have the students write a word problem for each of the operations. 43) Determines the absolute value and additive inverse of real numbers Draw a number line. Have students locate points on the number line such as 6 and -6 Discuss that they are both 6 units from zero. Give students examples of integers and ask them to give the absolute value and the additive inverse of each. Compare the additive inverse of each. Compare answers. Example: 3 absolute value = 3 additive inverse = -3 -2 absolute value = 2 additive inverse = 2 44) Classifies, compares, and orders integers and rational numbers 1 1 1 Give students a set of rational numbers such as ( , 0, 0.5, 2 , -3, ) 3 2 4 Have students identify the integers, place all numbers in ascending order, and compare each consecutive pair of numbers. On a poster board write 10 integers. Write 2 using blue numbers, 2 using red numbers, 2 using green numbers, 2 using yellow numbers and 2 using orange numbers. a. Compare the 2 green integers. b. Order the green, blue and red integers from least to greatest. (257) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 45) Adds, subtracts, multiplies and divides integers and rational numbers with and without calculators Pair students and play a game using dominos. The students can use addition and subtraction. Example: start finish game 0 board 0 _ (+) (-) + • • • • • • -2 + 4 = 2 move forward 2 places. The students will pull a domino and place on the card. - + • • • -3 + 2 = ˉ1 move backward 1 place. • • The student that reaches the finish line 1st wins. When students multiply and divide they use the hand-flipping rule to remind them of the appropriate rules. Have the student put a positive sign on the top of one hand and a negative sign on the palm of the other hand. Begin with the positive sign facing up for each problem. Each time you have a negative sign appear, turn your hand over. (Which ever sign is facing up at the end of the problem is the sign of the number in the answer). Example: 4 x ˉ3 = ˉ12 *Begin with + up *Flip hand because of the ˉ3. *Negative ends facing up so the answer is negative ˉ8 x ˉ6 = 48 *Begin with + up *Flip hand because of the ˉ8 *Flip again because of the ˉ6 *Positive ends facing up so the answer is positive 46) Writes ratios comparing given data Have the students work in groups of 4. Provide each group a small package of M&M’s or baggie that contains different colored squares. Students will determine what part of the package/baggie contains each color. These will be written in ratio form. Students can also change these ratios to percent form. Data will be shared with other groups. Students could then graph the data gathered from the entire class. (258) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Provide each student an index card that has a ratio or a fraction on it. Instruct the students to locate their match. (For instance, if a student is holding the 3 fraction , he/she will look for the ratio that matches his. These can also be 4 placed in order from least to greatest with the entire class and placed on a number line that has been hung across the classroom.) Have the students work with a partner. Provide each group with a measuring tape that will be used to determine the measure* of each of the following for each partner. Write in ratio form: -arm to foot -foot to height -around the neck to arm -foot to arm -head to foot Example: A person has a wrist measurement of 5 in. and a height of 50 in. The ratio of wrist to height is 5:50 or 1:10 in simplest form. Measurements should be given to the nearest inch and in simplest form. 47) Converts among ratios, decimals, and percents Provide students with fictitious sports data or data that has been collected from your school’s teams. Students can determine shooting percentages during basketball season and batting average during baseball season. Play the ―Memory Game‖. Students work with a partner. Each group is given a stack of 24 cards. Eight of the cards will have a ratio form written on them; eight will have the decimal form and the remaining eight will have the percent form. *See example below. Shuffle the cards and place face down in rows of four. One player turns up 3 cards to see if there is a 3-way match of a ratio, decimal, and percent. The game continues until a 3-way match is not made. When this happens, the other player plays. The student with the most matches at the end of the game is the winner. * 33 1 % 1 .3333 3 3 Students work in groups of 2-4. Each group receives a deck of 42 digit cards. These cards have fraction, decimals, and/or percents written on them. Each student is dealt 7 cards. The remainder of the cards go in the center of the table. Students play according to ―Gin Rummy‖ rules. Provide the student an index card that has a ratio, decimal, or percent form on it. Students will be instructed to get out of their seats and find their match. 3 (For instance, if a student is holding the fraction , find the student that has 4 0.75 and the 75%.) (259) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Students can bring in grocery store receipts. Have the student get receipts from at least two different grocery stores. Collect these for at least two weeks before this activity is done. Divide the class into two groups. Provide one group the receipts from one store, and the other gets the receipts from the store. Subdivide these two groups when the activity begins. Students will predict which digit they think will appear most frequently as the last digit in a price from the receipts. Have each group construct a line plot for each digit 0- 9 using the data from the receipts. See example below. 0 1 2 3 4 5 6 7 8 9 x x x x x x x x x x Have the students express the results in decimal, fraction, or percent form. Have them construct a graph to display the data. The results can be recorded as to which grocery store had the number _____ to show up most often. Students work with a partner. Each group will be given a page from the local newspaper and a transparent centimeter grid sheet to be used as an overlay. Students will estimate the total area of the newspaper page, excluding the margins, and determine the area of each category. The categories could consist of local news, national news, advertisements, entertainment, sports, photographs, weather, obituaries, etc. Students then express the area of the article to the area of the page as a fraction, decimal, and percent. The class then records all findings. The total area of each of the categories is calculated. These totals are compared to the total number of pages of the newspaper. Using the data, students decide how much of the newspaper is really news. 48) Solves proportions Have students conduct a survey of 5 blocks in the area in which they live and count the number of cats and dogs seen in a given period of time. Students can set up a proportion and make an estimate of the total number of cats and dogs in their neighborhood. Provide each student with several 2 colored cubes or counters, for instance, yellow and red. Have the students model a ratio of 1 to 3 using the cubes/counters. Have them set up a proportion to determine the number of yellow or red cubes/counters it will take to solve this proportion. 1 n 3 12 The student will also model this proportion using the cubes or counters. (260) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 Utilize the Russian dolls that come in proportional sizes for students to actually see how the dolls are proportionally made. Sand is a good material for students to experiment with. Show the class small and large photographs of the same picture to review the concept of ratio. Have students investigate the relationship between their arm span and their height or the measurement of the length of their foot to the measurement of the distance around their fist. The ratio is about 1 to 1. Show a picture of the statue of Liberty to the students. Inform students that the Statue of Liberty’s nose measures 4 feet 6 inches from the bridge to the tip. Having this information, groups are to try and determine the length of the Statue of Liberty’s right arm, the one holding the torch. The actual length of her arm is 42 feet. Have students then compare the ratios of the measurements of the lengths of their noses to the lengths of their arms. 49) Solves for part, rate, or base Provide student with a triangular card. *See below. The teacher will display the following problem: 15 is what percent of 80? Identify p = 15 b = 80 Students cover the unknown part, which will determine the appropriate operation. (If ―r‖ or ―b‖ is the missing part, the student is to divide; if ―p‖ is the missing part, the students is to multiply.) * p r x b Conduct a class survey of favorite things (ice cream, songs, TV shows, etc.). Have students find the percent of their favorite compared to the whole class. Example: Ice Cream 10 – chocolate 5 – vanilla 4- strawberry 1- banana Student A likes vanilla. 5 1 25% 20 4 50) Finds commissions and rates of commission, discounts, sale prices, sales tax, and simple interest Work in groups of 2-4. Each group is provided 3 triangular cards as shown below. The teacher will provide one problem at a time for the groups to determine the missing part. Students are to cover the missing part, which will (261) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 determine the appropriate operation. If r and the other item in the lower right hand angle is the missing part, the student will divide to find the answer. If the C, T, or D located at the top of the triangle is missing, the student will multiply to find the answer. COMMISSION TAX DISCOUNT C T D * • • • • • • original r x sales r x amount r x price Make up a menu or provide one. Each student is to order a meal from the menu. The total cost of the meal is to be determined. They are to find the amount of sales tax. Gratuity (tip) of 15% can be determined and added to the final cost. *The triangular cards can be used to assist the student. Work with a partner. Each group is to be given newspaper ads showing regular price and sale price, a pair of scissors, glue, and a sheet of construction paper. Students cut out the ads, glue them to the sheet of construction paper, and calculate the discount and rate of discount. Find the commission a real estate agent makes on the sale of a $65,000 home. Use a variety of rates. Example: $65,000 x 3%; $65,000 x .035. 51) Finds the percent of increase and decrease Students will be given data from line graph to plot. They will determine the percent of increase and decrease based on the data. For example: Unemployment Rate Rate of unemployment 7.5 7.0 6.5 6.0 • 5.5 5.0 Jan. Feb. Mar. April What was the percent of increase from Jan. to Feb.? Have students use base ten blocks to model percent of increase. For example, students put a hundred squares on their desks and identify this as 1 1 whole. Ask the students what fraction is 1 one-square. ( ). Then ask 100 (262) Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies) 2000 what percent is this (1%). Next have the students put a 1-square next to the hundred squares. By doing this, the hundred square is increased by 1%. Have students use the blocks to increase the hundred-square by 10%. They should add a ten-square ten-strip. Ask what fraction is 1 ten-strip. (10/100) and what percent is this. (10%). Inform students that by adding a ten-strip they increase the hundred-square by 10%. Have students use the blocks to increase the hundred-square by 20% (2-ten strips added), 6% (6 one-squares added), and 14% (1 ten-strip and 4 one-squares added). Having worked students through this strategy, provide them with this information: Unknown x Original = Added Percent Amount Amount p x 100 = 10 = 10/100 = 1/10 = 10% In finding percent of decrease, students use their hands to cover the amount decrease. 52) Writes and solves real life word problems using percents with and without calculators Allow students each day for a 2-week (ten-day) period to shoot a wad of paper from a designated spot into the garbage can. Each student is to keep up with his or her data (number of hits and misses). This information will be given to the teacher at the end of the 2-week period. Students will be given the data, such as, ―Michael shot at the garbage can 10 times. The paper went into the garbage can 60% of the time. How many times did the paper go into the can?‖ The teacher will supply the data of several more students. Students determine the percent of common nouns vs. the percent of adjectives in a paragraph from a reading, math, or social studies book. Newspapers can also be used. Conduct a class survey of favorite things, such as, ice cream, song, brand shoe, TV show, radio station, etc. Students will work in-groups of 2-4 and create a circle graph displaying the data in percents. Two students are given a deck of index cards that contain percents. Each student draws from the deck and writes a word problem incorporating their percent. The two students will swap cards* and work the problem. A calculator will be used to check the answers. *These cards can be put in a file for students to use throughout the year as enrichment. (263)