Fourth Grade. Order the following numbers from smallest to largest: 576, 543, 562, A: 461.5, 543, 562, 576

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1 Standard for Excellence: Number Sense Fourth Grade Mathematics Content Standard Performance Standard SAT 9 Number Sense 1.0 Students understand the place value of whole numbers and decimals to two decimal places and how whole numbers and decimals relate to simple fractions. Students use the concepts of negative numbers: 1.1 Read and write whole numbers in the millions. Write the following number in words: 7,035,926 A: seven million, thirty-five thousand, nine hundred twenty-six Order and compare whole numbers and decimals to two decimal places. Order the following numbers from smallest to largest: 576, 543, 562, A: 461.5, 543, 562, Round whole numbers through the millions to the nearest ten, hundred, thousand, ten thousand, or hundred thousand. Round the following numbers to the nearest hundred thousand: a. 5,567,981; b. 1,338,467 A: a. 5,600,000; b. 1,300, Denotes Key Standard Standards for Excellence: Mathematics Grade 4 1

2 Mathematics Content Standard Performance Standard SAT Decide when a rounded solution is called for and explain why such a solution may be appropriate. In which instance would an estimated answer be appropriate? Which would not? a. Five students had 237 straws for a science project. How many would each have to use? b. $24 was to be divided evenly between 15 students. How much would each student receive? A: Problem a. could be answered by using an estimated answer, while problem b. would be better answered by an exact amount Explain different interpretations of fractions, for example, parts of a whole, parts of a set, and division of whole numbers by whole numbers; explain equivalents of fractions (see Standard 4.0). Which number represents the shaded portion of the figure? a. 1/7 b. 3/7 c. 4/7 d. 1 3/7 A: 3/7 1.6 Write tenths and hundredths in decimal and fraction notations and know the fraction and decimal equivalents for halves and fourths (e.g., ½ = 0.5 or.50; 7 / 4 =1¾ = 1.75). See pages Standards for Excellence: Mathematics Grade 4 2

3 Mathematics Content Standard Performance Standard SAT Write the fraction represented by a drawing of parts of a figure; represent a given fraction by using drawings; and relate a fraction to a simple decimal on a number line. Find the following fraction on the number line: A: Standards for Excellence: Mathematics Grade 4 3

4 Mathematics Content Standard Performance Standard SAT Use concepts of negative numbers (e.g., on a number line, in counting, in temperature, in owing ). Find the following temperature on the thermometer: - 12º 5º A: º -5º -10º -15º Standards for Excellence: Mathematics Grade 4 4

5 Mathematics Content Standard Performance Standard SAT Identify on a number line the relative position of positive fractions, positive mixed numbers, and positive decimals to two decimal places. Locate 1 ¼ on the number line: A: Number Sense 2.0 Students extend their use and understanding of whole numbers to the addition and subtraction of simple decimals: 2.1 Estimate and compute the sum or difference of whole numbers and positive decimals to two places. Solve the following: = A: Estimate = 80; Answer = Round two-place decimals to one decimal or the nearest whole number and judge the reasonableness of the rounded answer. Round to the nearest tenth: 34.67, 12.49, A: 34.7, 12.5, Standards for Excellence: Mathematics Grade 4 5

6 Mathematics Content Standard Performance Standard SAT 9 Number Sense 3.0 Students solve problems involving addition, subtraction, multiplication, and division of whole numbers and understand the relationships among the operations: 3.1 Demonstrate an understanding of, and the ability to use, standard algorithms for the addition and subtraction of multidigit numbers. Find: a. 632, ,921 b. 74, ,246 A: a. 188,231 b. 210,079 1, Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multidigit number by a two-digit number and for dividing a multidigit number by a one-digit number; use relationships between them to simplify computations and to check results. Solve the following problems: a. 252 X 47 =? b =? a. 83 X 97 =? A: a. 11,844; b. 235; c. 8, Solve problems involving multiplication of multidigit numbers by two-digit numbers. During the festival this year, Harvest Acres made $65 on candy-apple sales. It made 16 times that amount selling apple cider. How much did Harvest Acres make on apple-cider sales? A: $1040 2, Standards for Excellence: Mathematics Grade 4 6

7 Mathematics Content Standard Performance Standard SAT Solve problems involving division of multidigit numbers by one-digit numbers. An insect zoo has 365 insects displayed in 5 equal groups. How many insects are in each group? Explain your answer. A: 73 insects; divide the number of insects by the number of groups: = Number Sense 4.0 Students know how to factor small whole numbers: 4.1 Understand that many whole numbers break down in different ways (e.g., 12 = 4 x 3 = 2 x 6 = 2 x 2 x 3). Write 90 as the product of two numbers in as many ways as you can. A: 1 X 90, 2 X 45, 3 X 30, 5 X 18, 6 X 15, 9 X Know that numbers such as 2, 3, 5, 7, and 11 do not have any factors except 1 and themselves and that such numbers are called prime numbers. Which numbers between 10 and 20 are prime numbers? A: 11, 13, 17, 19 2, Standards for Excellence: Mathematics Grade 4 7

8 Standard for Excellence: Algebra and Functions Mathematics Content Standard Performance Standard SAT 9 Algebra and Functions 1.0 Students use and interpret variables, mathematical symbols, and properties to write and simplify expressions and sentences: 1.1 Use letters, boxes, or other symbols to stand for any number in simple expressions or equations (e.g., demonstrate an understanding and the use of the concept of a variable). Solve the following and select the expression that best represents the problem and answer: A math book has 60 multiplication and division problems in it. If 23 of the problems are division, how many of the problems are multiplication? a N = 23 b = N c = N, N = 37 A: c. 1, 2, Interpret and evaluate mathematical expressions that now use parentheses. Solve the problem: ( ) 12 =? A: Use parentheses to indicate which operation to perform first when writing expressions containing more than two terms and different operations. Insert parentheses to make the equation true: = A: 65 (3 + 2) = Standards for Excellence: Mathematics Grade 4 8

9 Mathematics Content Standard Performance Standard SAT Use and interpret formulas (e.g., area = length x width or A = lw) to answer questions about quantities and their relationships. In the formula A = lw, A stands for area. What do the l and w stand for? How would you use this formula for a rectangle that has sides of 5 cm and 9 cm, and what unit would you use to express the answer? A: l stands for length (the longest side) and w stands for the width (the shortest side). The answer would be expressed on square centimeters or cm², with the answer being 45 cm² Understand that an equation such as y = 3x + 5 is a prescription for determining a second number when a first number is given. Solve the equation: y = 4x + 8, if x = 6; y =? A: y = Algebra and Functions 2.0 Students know how to manipulate equations: 2.1 Know and understand that equals added to equals are equal = ; (41 X 15) + 5 = (15 X 41) Know and understand that equals multiplied by equals are equal. (42 X 16) + 5 = (16 X 42) + 5; 123 X 44 = 44 X Standards for Excellence: Mathematics Grade 4 9

10 Standard for Excellence: Measurement and Geometry Mathematics Content Standard Performance Standard SAT 9 Measurement and Geometry 1.0 Students understand perimeter and area: 1.1 Measure the area of rectangular shapes by using appropriate units, such as square centimeter (cm 2 ), square meter (m 2 ), square kilometer (km 2 ), square inch (in 2 ), square yard (yd 2 ), or square mile (mi 2 ). Determine if the area of a 7 cm square is larger or smaller than the area of a rectangle with a perimeter of 30 cm and one side that is 5 cm. A: smaller: 49 < Recognize that rectangles that have the same area can have different perimeters. Rectangles a and b both have the same area (8 square units). Do they have the same perimeter? a b A: No: 12 < 18 Standards for Excellence: Mathematics Grade 4 10

11 Mathematics Content Standard Performance Standard SAT Understand that rectangles that have the same perimeter can have different areas. Rectangles a and b both have the same perimeter (10 units). Do they have the same area? a units A: No: units 1.4 Understand and use formulas to solve problems involving perimeters and areas of rectangles and squares. Use those formulas to find the areas of more complex figures by dividing the figures into basic shapes. The length of a rectangle is 12 cm and the perimeter is 34 cm. What is the area of the rectangle in square centimeters? A: 60 square centimeters (cm²) Teaching ideas/extensions/assessment/planning: Standards for Excellence: Mathematics Grade 4 11

12 Mathematics Content Standard Performance Standard SAT 9 Referenc e Measurement and Geometry 2.0 Students use two-dimensional coordinate grids to represent points and graph lines and simple figures: 2.1 Draw the points corresponding to linear relationships on graph paper (e.g., draw 10 points on the graph of the equation y = 3x and connect them by using a straight line). Draw three points for the equation y = 2x and connect them by using a straight line Understand that the length of a horizontal line segment equals the difference of the x- coordinates. What is the length of the line segment joining the points (2, 1) and (2, 7)? A: 6 units 2 4 Standards for Excellence: Mathematics Grade 4 12

13 Mathematics Content Standard Performance Standard SAT Understand that the length of a vertical line segment equals the difference of the y- coordinates. What is the length of the line segment joining the points (2, 3) and (7, 3)? A: 5 units Measurement and Geometry 3.0 Students demonstrate an understanding of plane and solid geometric objects and use this knowledge to show relationships and solve problems: 3.1 Identify lines that are parallel and perpendicular. Name the parallel lines and name two perpendicular lines. A B C D E F K H I G A: Parallel lines are D E and F G Perpendicular lines are D E and A K, or F G and A K Standards for Excellence: Mathematics Grade 4 13

14 Mathematics Content Standard Performance Standard SAT Identify the radius and diameter of a circle. Name the radius and the diameter of the circle A B X C D A: Diameter is CB, radius is DX 3.3 Identify congruent figures. Name the two congruent figures A: The two congruent figures are # 1 and # 4. Standards for Excellence: Mathematics Grade 4 14

15 Mathematics Content Standard Performance Standard SAT 9 Standards Benchmark 3.4 Identify figures that have line (or bilateral) and rotational symmetry. Which figure has line ( or bilateral) symmetry, and which one has rotational symmetry? A: Know the definitions of a right angle, an acute angle, and an obtuse angle. Understand that 90º, 180º, 270º, and 360º are associated, respectively, with ¼, ½, ¾, and full turns. Which of the following angles is obtuse? a. b. c A: b Between noon and 12:30, the minute hand on a clock moves from 12 to 6. Has the minute hand made a quarter turn, a half turn, or a threequarter turn? What time will it be when the minute hand has made a full turn? A: Half turn; 1 o clock Standards for Excellence: Mathematics Grade 4 15

16 Mathematics Content Standard Performance Standard SAT Visualize, describe, and make models of geometric solids (e.g., prisms, pyramids) in terms of the number and shape of faces, edges, and vertices; interpret two-dimensional representations of three-dimensional objects; and draw patterns (of faces) for a solid that, when cut and folded, will make a model of the solid. Which net can be folded to make a cube? a. b. c. d. A: d Which of the following is a cone? a. b. c A: c Standards for Excellence: Mathematics Grade 4 16

17 Mathematics Content Standard Performance Standard SAT Know the definitions of different triangles (e.g., equilateral, isosceles, scalene) and identify their attributes. Which of the following is a right triangle? a. b. c A: c Which of the following is an acute triangle? a. b. c. A: b 3.8 Know the definition of different quadrilaterals (e.g., rhombus, square, rectangle, parallelogram, trapezoid). List three different kinds of quadrilaterals and their definitions Standards for Excellence: Mathematics Grade 4 17

18 Standard for Excellence: Statistics, Data Analysis, and Probability Mathematics Content Standard Performance Standard SAT 9 Standards Benchmark Statistics, Data Analysis, and Probability 1.0 Students organize, represent, and interpret numerical and categorical data and clearly communicate their findings: 1.1 Formulate survey questions; systematically collect and represent data on a number line; and coordinate graphs, tables, and charts. List five questions to ask students in your classroom concerning food preferences (e.g. favorite ice cream flavor), conduct a survey of at least 25 people, and graph the results. Write a conclusion based on the information collected Identify the mode(s) for sets of categorical data and the mode(s), median, and any apparent outliers for numerical data sets. Have each member of the class shoot 10 free throws with a basketball. Record the number of baskets for each person. Identify the mode(s). 2, Standards for Excellence: Mathematics Grade 4 18

19 Mathematics Content Standard Performance Standard SAT 9 Standards Benchmark 1.3 Interpret one- and two- variable data graphs to answer questions about a situation. Which of these bar graphs show the same information? 3 Graph A Graph B Graph C Graph D A: A and B Standards for Excellence: Mathematics Grade 4 19

20 Mathematics Content Standard Performance Standard SAT 9 Standards Benchmark Statistics, Data Analysis, and Probability 2.0 Students make predictions for simple probability situations: 2.1 Represent all possible outcomes for a simple probability situation in an organized way (e.g., tables, grids, tree diagrams). See pages , Express outcomes of experimental probability situations verbally and numerically (e.g., 3 out of 4; ¾). A coin is tossed once. What is the probability it will land on heads? 1 A: or 1 out of Standards for Excellence: Mathematics Grade 4 20

21 Standard for Excellence: Mathematical Reasoning Mathematics Content Standard Performance Standard SAT 9 Mathematical Reasoning 1.0 Students make decisions about how to approach problems: 1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and prioritizing information, and observing patterns. In the following problem, underline any information that is relevant to finding a solution. A group of five students were working on a project to graph the percentage of times a coin landed on heads. They flipped the coin 100 times. The coin landed on tails 47 times. 1, 2, 3, Determine when and how to break a problem into simpler parts. Clown noses are $4 a bag. There are 3 noses in a bag. How many clown noses can you buy for $12? A: $12 $4 = 3 bags 3 bags X 3 noses per bag = 9 noses Standards for Excellence: Mathematics Grade 4 21

22 Mathematics Content Standard Performance Standard SAT 9 Mathematical Reasoning 2.0 Students use strategies, skills, and concepts in finding solutions: 2.1 Use estimation to verify the reasonableness of calculated results. Estimate the answer, then calculate. Compare the estimated answer to the calculated answer to see if it is reasonable ; Estimate 474 (480) 6 = 80; Exact calculation = 79; Compare calculated answer of 79, to estimate of 80. A: They are close, so the calculated answer is reasonable Apply strategies and results from simpler problems to more complex problems. In a bicycle race, there are judges posted at the beginning and end of each mile. If the race is 20 miles, how many judges are needed? miles judges A: 21 judges Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. Write the steps needed to explain how to work a division problem. Make a chart showing all of the steps in the correct order Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work. Write the steps needed to explain how to work an addition of fractions problem. Make a chart showing all of the steps in the correct order. 2, 3, Standards for Excellence: Mathematics Grade 4 22

23 Mathematics Content Standard Performance Standard SAT Indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy. See pages Make precise calculations and check the validity of the results from the context of the problem. See page Standards for Excellence: Mathematics Grade 4 23

24 Mathematics Content Standard Performance Standard SAT 9 Mathematical Reasoning 3.0 Students move beyond a particular problem by generalizing to other situations: 3.1 Evaluate the reasonableness of the solution in the context of the original situation. Teachers and parents helped out backstage. Half as many teachers as parents were backstage. If 5 teachers helped out backstage, how many parents were backstage? A: 10 parents Is 5 half of 10? Yes Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems. In a bicycle race, there are judges posted at the beginning and end of each mile. Can you predict how many judges would be needed for a 30 mile race? (See chart in Mathematical Reasoning, 2.2) A: 31 judges Develop generalizations of the results obtained and apply them in other circumstances. Which of the following numbers are divisible by 5? a. 10 b. 12 c. 25 d. 17 A: a and c Is 115 divisible by 5? A: yes: it ends in 0 or Teaching ideas/extensions/assessment/planning: Standards for Excellence: Mathematics Grade 4 24

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