FCAT Mathematics 2010

Items on the math section of the FCAT, under the new standards, are identified by their cognitive complexity (or difficulty). They are divided into three levels: low, average and challenging. Test writers and the Item Review Committee have made the following approximate predictions about each level.

They predict that the easy level questions would be answered correctly by more than 70% of the students taking the test.

They predict that the average level questions would be answered correctly by between 40% and 70% of students taking the test.

They predict that the challenging level questions would be answered correctly by fewer than 40% of students taking the test.

The cognitive complexity is defined as the cognitive demand placed on the student in order to solve the problems.

Easy or low-complexity items might require a student to solve a one-step problem from their grade level benchmarks. These rely heavily on recall of previously learned concepts.

Sample: MA.4.A.1.2: A giant panda eats 83 pounds of bamboo per day. How many pounds of bamboo will a giant panda bear eat in 7 days?

A. 171 poundsB. 571 poundsC. 581 poundsD. 701 pounds

Average or moderate-complexity items might require a student to use multiple steps to solve a problem. These require more “flexible” thinking on the part of the student. They require an answer that goes beyond what is “habitual” or memorized, and usually requires more than one step to solve. Students have to decide how to solve these problems and use reasoning and problem-solving strategies.

Sample: (would include a picture of a graph) MA.5.S.7.1:Mrs. Reich put a basket of 20 apples in the teachers’ work room one morning. The line graph below shows the number of apples that were in the basket at several times during that morning. [ GRAPH]

According to the graph, during which of the following periods of time were the most apples taken from the basket?

A. From 7:30 a.m. to 8:30 a.m.B. From 8:30 a.m. to 9:30 a.mC. From 9:30 a.m. to 10:30 a.m.D. From 10:30 a.m. to 11:30 a.m.

Challenging or high-complexity items might require a student to analyze and synthesize information in order to solve a problem. Students must use more abstract reasoning, analysis, judgment and creative thought to solve these problems.

Sample: (would include chart)MA.3.A.6.2

Mr. Tellez is buying fish for his fish tank. The price for each fish changes according to the type and size of the fish, as shown in the chart below.[chart]

Mr. Tellez wants to buy 1 angelfish, 2 goldfish, 2 guppies and 3 mollies. Which of the following could be the total cost of the 8 fish?

A. $8.00B. $9.00C. $10.00D. $11.00

See page 15 of the new FCAT Mathematics Guide for more specific computation definitions for each of the three categories of problems.

Item Context:The manner or situation in which a test question is presented is called the item CONTEXT. FCAT math questions may be presented in either a real-world or mathematical CONTEXT; however, other variables must also be considered. These are listed below, and also described in the individual benchmark specifications. These also have more specific examples and illustrations.Sample CONTEXTS can also be found in Appendix A.

Grade 3 Content Limits

The content limits described below apply to all test items developed for Grade 3. However, the content limits defined in the individual benchmark specifications supersede these more GENERAL content limits, so pay close attention to individual benchmark specifications when designing instruction.

Whole numbers: items should not require the use of more than two operations.Place values should range from ones through hundred-thousands.

Addition: Items should not exceed three 5-digit addends or two 6-digit addends

Subtraction: Subtrahends cannot exceed 999,999.Minuends and differences should not exceed 5 digits.

Multiplication: items may include whole-number multiplication facts from 0 X 0 through 9 X 9.Multiples of 10 through 100, multiples of 100 through 1,000, and multiples of 50 through 500 may be used.

Division: items may include the related division facts for 0 X 0 through 9 X 9.

Decimals: decimal numbers are limited to amounts of money to the nearest cent.

Addition, subtraction, multiplication and division of decimals not assessed in Grade 3.

Fractions: fractions should have denominators of 1-10, 12 or 16.Items may include fractions and mixed numbers up to and including the whole number 5.

Addition, subtraction, multiplication and division of fractions not assessed in Grade 3.

Percent: Not assessed in Grade 3.

Measurement: items will not assess weight/mass, capacity or temperature in isolation.Time and linear measurement, including perimeter will be assessed.Items may use customary and/or metric units.

Grade 3 Math BenchmarksMain Objective: for students to develop an understanding of multiplication and division and strategies for solving basic multiplication fact problems and related division fact problems.

Benchmark Learning Objective ExampleMA.3.A.1.1 Students will learn to model

multiplication and division problems, using manipulatives, arrays, repeated addition, etc. They will also compare multiples (greater / less and equal), determine how many ways they can come up with the same product or quotient making “sets” or groups, determining how many sets or groups of a number they can make out of a total number or “whole set.” They will use partitioning as a strategy for solving division problems.

(e.g. Products of 16: 2 X 8, 8 X 2, 4 X 4, 16 X 1, 1 X 16)

[Given a set of math manipulatives, such as cubes, have them start with a total of ___ cubes. How many sets of ___ can you make?After several of these, then give them the algorithm that is modeled by their manipulatives.

Strategies to be taught:1)Repeated addition: 4 bags of cookies with 8 in each bag =8 + 8 + 8 + 8 = 322)Multiplicative comparisons:Joe has 4 cards; Jenn has 4 times as many ( 4 X 4 = 16)3)Arrays: draw a picture/ array to represent numbers; There are 8 students in each row and 7 rows. How many students are there?

MA.3.A.1.2 Students will learn to solve

multiplication and division problems by applying number properties to solve.

e.g. associative property:(8 X 4) X 2 = 64, 8 X (4 X 2)= 64Commutative property:The order of the multiplicands does not matter to the product

5 X 3 = 3 X 5Identity property:Any number X 1 = that numberZero property:Any number X 0 = 0Distributive property: The sum of two numbers times a third number is equal to the sum of each addend times the third number.e.g. 3( 4 + 4) = (3 X 4) + (3 X 4)

MA.3.A.1.3 Students will identify, describe and apply division and multiplication as INVERSE operations. (They will have a complete understanding of multiplication and division being opposites.)

E.g: 5 X 6 = 30 is the INVERSE or opposite of 30 ÷ 5 = 6*stress the concept that multiplication and division are the opposites of each other, as are addition and subtraction

Grade 3 Math BenchmarksMain Objective: for students to develop and understanding of fractions and fraction equivelants. (e.g. understanding fractions as FAIR SHARES, EQUAL PARTS of a whole, and that 5/10 and ½ are equivalent.)

Benchmark Learning Objective ExampleMA.3.A.2.2 Students will be able to describe how the

size of a fractional part of a whole is related to the total number of equal parts in the whole.

e.g. the NUMERATOR is the part of the whole set; the line separating the numerator and denominator is like a division symbol, that means “out of”; the DENOMINATOR is the total number of equal parts (or fair shares) that are in the whole set or group.

E.g. ⅘ means 4 parts or shares out of a total of 5 equal parts in the whole set

MA.3.A.2.1 Students will be able to represent fractions , including fractions greater than one using area models, set models and linear models.

Area model:

= 6/16Linear Model:

⅓ ⅔ 1⅓ 1⅔ 1 2Set Model: 3/5

MA.3.A.2.3 Students must be able to compare fractions and put them in proper order, including fractions greater than one, using models and strategies.

1/2

1/3

3/4

MA.3.A.2.4 Students will know how to use models to represent equivalent fractions, including fractions greater than one, and identify models that are equivalent.

Grade 3 Math BenchmarksMain Objective: students will be able to describe and analyze properties of two-dimensional shapes.

Benchmark Learning Objective ExampleMA.3.G.3.1 Students will be able to describe, analyze,

compare and classify two-dimensional shapes using sides and angles to compare and classify them. They need to be able to identify acute, obtuse and right angles and triangles, and be able to connect these identifications to defining shapes.

Acute

Obtuse

Right

Isosceles triangle

Equilateral triangle

Scalene triangle

MA.3.G.3.2 Students must be able to create or construct as well as take apart or “decompose” and “transform” or CHANGE polygons to make other polygons. These include CONCAVE and CONVEX polygons with three, four, five, six, eight or ten sides.

1.

makes

2.

1/4 1/4

1/2

Makes:

MA.3.G.3.3 Students must be able to build, draw and analyze two-dimensional shapes looking at them from different angles and orientations to identify symmetry and congruency.

Congruent triangles Cut the following rectangle into 2 congruent shapes:

Grade 3 Math BenchmarksAlgebra:

Benchmark Learning Objective ExampleMA.3.A.4.1 Students must be able to create,

analyze and represent patterns and relationships between numbers using words, variables, tables and graphs.

Find the pattern in a series of numbers:3, 7, 11, 15… [ + 4 pattern]

Relationships between numbers:X + 6 = 11 (x = 5) use counting strategies to find missing addend and to realize “X” is a symbol for the missing addend.

+ 8 = 14 (use counting strategies to find missing addend = 6) 135 > 130225 < 250175 = 175

Dogs Cats Fish BirdsIIII II III

Jenny took a survey of her friend to see which pet was the most popular. She used a table to record her data. Based on the table, which pet did her friends like the most? [dogs]Which did they like the least?[cats]

Use the chart above to answer the following:What part of the country had the hottest temperature in the third quarter of the year? [ the east ]What part of the country had the lowest

temperature during the second quarter of the year? [ the east ]

Grade 3 Math BenchmarksGeometry and Measurement:

Benchmark Learning Objective Example

MA.3.G.5.1 Students need to be able to select the appropriate units, strategies and tools to solve problems involving perimeter. (**Adding up the total of ALL the sides. This includes teaching them about the symbols for equal sides, so that if one side isn’t labeled, they can identify the size of that side based on an equivalent side. Also, this means teaching them not to forget to write the unit of measure AFTER the sum e.g. 6 cm.)

9 ft.

6 ft. 6 ft.

3 ft.Find the perimeter of the ballfield.6 + 9 = 15; 6 + 3= 9; 15 + 9= 24Or6 + 6 = 12; 9 + 3= 12; 12 + 12 = 24

6 cm

Teach students the properties of squares (all sides are equal) so they understand that whatever on e side measures, all 4 sides measure the same amount.They should be able to use repeated addition (6 + 6 + 6 + 6 = 24, doubles 12 + 12 = 24, ormultiplication 6 X 4 = 24 to find perimeter.

8 in

2 inTeach students the properties of rectangles; that they are made up of 2 sets of parallel lines; opposite sides are equal.Therefore, they would know that 2 sides of this rectangle measure 2 inches each, and the other 2 sides measure 8 inches each.Then they can use repeated addition, doubles or multiplication to find its perimeter (2 X 2=4; 8 X 2=16; 16 + 4 =20,2 + 2 + 8 + 8 = 20;

4 + 16=20)

4 ft XThe perimeter of this field is 20 ft. If the width is 4 ft, what is the length?

MA.3.G.5.2 Students must be able to measure objects using fractional parts of linear units, like ½, ¼ and 1/10.

Have students use rulers, yard sticks, tape rule, etc. to measure real life objects, such as desktops, books, classroom objects. This should be extended by then having them use these measurements to find area, perimeter, etc. and also to record their data in tables, and use their data to create graphs and charts.

MA.3.G.5.3Students must be able to tell time to the nearest minute, to the nearest QUARTER HOUR, HALF HOUR, and be able to tell the amount of ELAPSED TIME. (Elapsed time usually requires the most practice and poses the most difficulty to students at this grade level.)

Sara started breakfast at 7:29 a.m. and finished at 8:29 a.m. How long did it take her to eat breakfast?

Joey began his afternoon soccer practice at 3:00 p.m. and finished at 4:15 p.m. How long did soccer practice last?

**students sometimes have difficulty remembering to count whole hours and then minutes when figuring elapsed time.

They also will often see that the hour hand has changed, and ASSUME that an entire hour has passed:

e.g.

This is perhaps the most difficult, as the student has to recognize that the clock would have to move ALL THE WAY from 3:20 PASSED 4:20 in order for a whole hour to have passed, so even though the HOUR has changed, only MINUTES should be counted.

Grade 3 Math BenchmarksNumbers and Operations:

Benchmark Learning Objective Example

MA.3.A.6.1 Students must be able to represent, compute, estimate and solve problems using numbers through the hundred-thousands. (Place value, and lining digits up vertically are important skills for them to be successful at this!)

Compute: 14, 689 85, 782 100, 471Lining up the digits according to their place value is CRUCIAL for arriving at the correct answer.

Represent:What digit is in the ten-thousands place in 123,456(2). Students must be able to identify the DIGIT based on its place value in the number.

Estimate:Round 256,782 to the nearest hundred thousand: [300,000]Students will have to know the rules of rounding (‘five or more, up we soar’, etc.) knowing that they need to look at the DIGIT to the right of the DIGIT they are estimating, if it is 5 or larger, the digit being rounded goes up 1 and all other digits become 0)Round 256,782 to the nearest ten thousand: [260,000].To the nearest thousand:[257,000] etc.

MA.3.A.6.2 Students must be able to solve NON-ROUTINE problems by CREATING a chart or graph to represent data and to search for patterns or solutions.

Task: Students measured their desktop, the bookshelf used for portfolios, and 2 other classroom tables each. (those 2 tables were not the same for each student.) They then used the data they recorded to create a BAR GRAPH.

**See additional examples in ‘algebra’

MA.3.S.7.1 Students must be able to Frequency Table:

Fruit Number per week

construct, create and analyze frequency tables, bar graphs, pictographs and line graphs from data. This includes data collected through observations, surveys and experiments.

Peach IIIIBanana IIII IIII IIIApple IIII IIIIOrange III

Task: Students surveyed their classmates asking how often each student ate these 4 fruits on a weekly basis. They used the information to create a frequency table.

Task: 4 students each surveyed 5 other students to find out which of 4 colors was their favorite. They then compiled their data to create a bar graph.

Mon Tues Wed Thurs Fri

Task: Students were growing plants at their group table for science. They observed the size of their plant for five days and recorded the results in a pictograph.

Task: Students recorded weather for 5 days and created a line graph to display the changes .

Grade 4 General Content Limits

The content limits described below apply to all test items developed for Grade 4. However, the content limits defined in the individual benchmark specifications supersede these more GENERAL content limits, so pay close attention to individual benchmark specifications when designing instruction.

Whole Numbers: Items should not require the use of more than two operations.Place values should range from ones to hundred millions!

Addition: Items should not exceed three 7-digit addends or two 8-digit addends.

Subtraction: Subtrahends, minuends and differences should not exceed 8 digits.

Multiplication: Factors used may include up two 3-digit numbers, or, when a four-digit factor is used, the other factor may not exceed two digits.

Division: Divisors should not exceed one digit, unless it is a related division fact of 0 X 0 through 12 X 12.Dividends should not exceed three digits.Quotients may include remainders expressed only as whole numbers.

Decimals: Place values could range from tenths through thousandths with no more than five total digits.

Addition, Subtraction, Multiplication and Division of:*Not assessed at Grade 4!

Fractions: items may have denominators of 1-10, 12, or 1000, or denominators that are derived from basic multiplication facts through 12 X 12 may also be used (e.g. 24 has the factors 5 and 4; 72 has the factors 8 and 9.)

Addition, Subtraction, Multiplication and Division of:*Not assessed at Grade 4!

Percent: Percents must be equivalent only to halves, fourths, tenths or hundredths.Items dealing with percents will not involve computation using the percent.

Measurement: Items will not assess weight/mass, time, temperature, perimeter, and/or capacity in isolation.Items may use customary and/or metric units.*See Geometry & Measurement benchmarks for specifics.

Gridded Response Items: Answers may not exceed five digits.Answers may not include fractions. *See grid types for appropriate answer formats.

Grade 4 Math Benchmarks

Multiplication & Division:Main Objective: to develop quick recall of multiplication facts and related division facts and fluency with whole number multiplication

Benchmark Learning Objective Example

MA.4.A.1.1 Students will use and describe various models for multiplication in problem-solving situations, and demonstrate recall of basic multiplication and related division facts with ease.

4 step process for solving word problems: Read, Plan, Solve & Check!Students should be able to read problems and determine what strategy or strategies would be best to use to solve that particular problem:Strategies: draw a picture, make a table, find a pattern, skip count, multiply then add, make sets, make an array, etc.

MA.4.A.1.2 Students will multiply multi-digit whole numbers through four digits fluently, demonstrating understanding of the standard algorithm, and checking for reasonableness of results, including solving real-world problems.

3, 459X 62,754*Student has solved incorrectly and should recognize that this answer is impossible, since the product is smaller than the number being multiplied!Student should rework problem and arrive at 20,754.

Word problem:Charlie purchased 356 bushels of corn. Each bushel held 6 ears. How many ears of corn did Charlie purchase? 356 X 6 2,136*Students should follow regrouping steps carefully, so that errors aren’t made. They should also be able to check using repeated addition if necessary.

Grade 4 Math Benchmarks

Fractions & DecimalsMain Objective: For students to develop and understanding of decimals, including the connection between fractions and decimals.

Benchmark Learning Objective ExampleMA.4.A.2.1 Students will use decimals through

the thousandths place to name numbers between whole numbers.

Be able to represent fractions as decimals:1/10, 2 and 1/10, 4 and 5/100Are: .1 , 2.1 , 4.05*place value charts are helpful in teaching tenths and hundredths, just as they are in teaching tens and hundreds.

MA.4.A.2.2 Students will be able to describe decimals as an extension of the base-ten number system.

Students will need to understand the concept of decimals as “parts” of the number system that add up to a whole, just as fractions do. Teaching decimals as parts of a whole or fair shares by modeling the fraction AND decimal at the same time is helpful to students.

MA.4.A.2.3 Students will be able to relate equivalent fractions and decimals with and without models, including locations on a number line.

Number Line:Bobby plotted four plots on a number line. Which letter represents 2.3?

• • • • 1 2 3

What fraction would be equivalent to 2.3?

2 3/10No models:Students should have enough of a foundation to know that 2/4 = ½, 5/10=1/2, 2/3 = 4/6. Teaching the basic principal of “what you do to the numerator you must do to the denominator” for reducing fractions can help with this. E.g. 2/3…multiply both numerator and denominator by 2 and get the equivalent 4/6.Can use division to reduce or find equivalents also:6/10…divide both numerator and denominator by 2 and get the equivalent 3/5.visual model:

Write the decimal represented by the shaded area: Students should realize that the decimal(s) represented are:.03 and .3 and that they are equivalent.

MA.4.A.2.4 Students will compare and order decimals, and estimate fraction and decimal amounts in real-world problems.

Linear model:

• • 0 1

Given .2 .8 , students should be able to fill in the appropriate < , > or = sign.Estimate:About how much of the pie was eaten?

(Students draw lines to help them solve)About ¼: remind students that this is an ESTIMATE and should not necessarily be EXACT.

Now have them write the answer as a decimal: ABOUT .25 *Remind students to get in the habit of writing “about” before their answer when estimating.

Compare Decimals:Farmer Weight of

hogEthan 252.09Gina 247.99Royce 252.8Tyler 236.9

According to the table above, which farmer’s hog weighed the most?**Two important things to stress/teach here: Teach them the process of looking to the necessary DECIMAL PLACE, when the whole numbers are THE SAME (252.09 and 252.8. **Also: STRESS to them to make sure they are answering WHAT THE QUESTION IS ASKING! E.g. Instead of answering “Royce” to the problem above, the student answers 252.8.

Grade 4 Math BenchmarksMeasurementMain Objective: For students to develop and understanding of area and determine the area of two-dimensional shapes

Benchmark Learning Objective ExampleMA.4.G.3.1 Describe and determine area as the number

of same-sized units that cover a region in the plane, recognizing that a unit square is the standard unit for measuring area.

Using models to determine area:Key: Each square = 1 yd.

Above is a picture of the lake by Katie’s house. ABOUT how many square yards is Katie’s lake?(students would count the squares covered by the “lake” going across the grid, count the squares covered by the “lake” going up and down the grid, and multiply these: About 12.In this example, they are estimating, so you would remind them that their answer is not going to be EXACT.Also, note that 2 of the edges only cover half of a grid square. Teach them how to count the half on the left + the half on the right as one!

MA.4.G.3.2 Students will justify the formula for the area of the rectangle (“area = base X height)

Sample 2 step problem:

Above is a picture of Mr. Henry’s swimming pool. Find the area of the pool. For which of the following situations would Mr. Henry need to know the area of his pool?

a. To determine how much water he needs to fill his pool

b. To determine the amount of fencing he needs to go around his pool

c. To determine how much of his yard the pool is going to cover

d. To determine how much dirt that needs to be dug for the pool.

*If possible, this is a good opportunity to address that choice “a” in this scenario would be addressed by finding VOLUME, choice “b” by finding perimeter and choice “d” by capacity/volume.

MA.4.G.3.3 Students will select and use appropriate units, both customary and metric, strategies and measuring tools to estimate and solve real-world problems.

Determine appropriate unit of measure:Which would be more logical to measure your textbook in?

a. Inchesb. Feetc. Yardsd. Miles

Bobby ran for 13 hours in a local marathon. Which would be a good estimate of the distance he ran?

a. 30 cmb. 30 metersc. 30 mmd. 30 km

What would you use to measure the water in a fish tank?

a. A rulerb. A measuring cupc. A tape ruled. A yard stick

Determine the area of an irregular shape:Lana bought the patch below for her jacket. Find the area of the jacket, in centimeters.

(Students will need to know how to accurately find the area of both sections of this shape, and add them together to solve.)

Grade 4 Math BenchmarksAlgebra:

Benchmark Learning Objective ExampleMA.4.A.4.1 Students will generate algebraic

rules and use all four operations to describe patterns, including non-numeric growing or repeating patterns.

Finding a simple pattern:A number pattern is shown below. 3, 6, 9, 12, 15, 18…What algebraic rule can be used to describe the pattern?

a. Add 1 to each number b. Add 3 to each numberc. Multiply the number by 3d. Multiply the last 2 numbers

Determine how to continue a pictoral pattern:

Figure 1 Figure 2 Figure 3 Figure 4

Elijah made this pattern using squares to make pyramids. If he continues this pattern, what should be the total number of squares in Figure 5?Students will need to be VERY CAREFUL to READ THE QUESTION, so they count ALL the squares in figure 4, add on the additional squares that should be in figure 5 and give the TOTAL, (not just “five” for the number of squares that need to be ADDED TO the figure.)

MA.4.A.4.2 Students will describe mathematic relationships using expressions, equations, and visual representations.

Visual relationship:Several shapes are shown below: 1 2 3 4 If this pattern continues, which expression below can be used to find the number of sides in Shape 6?

a. 6 + 1b. 6 + 2c. 6 X 1d. 6 X 2

*The key concept here is for students to recognize that each shape has one MORE side than the shape before it; ALSO…they have to take their time so that they recognize that there are 4 shapes given, but they are asked to calculate the 6th shape, not the NEXT or fifth shape. Stress the importance of reading the problems carefully.

Understanding a variable as part of an equation:Mr. Jones is buying spoons for his ice cream party. The spoons are sold with 8 in each package. Use the expression below to calculate how many spoons Mr. Jones would have if “p” represents packages, and p=9. 8pAnswer: 72*Students will have to know that a number next to a variable means that they must multiply.

Identify a missing multiplicand represented by a variable:9n = 63N = 7

MA.4.A.4.3 Students will recognize and write algebraic expressions for functions with two operations.

Choose the appropriate expression:Alex is 4 years more than twice as old as his brother. Which expression gives Alex’s age using s to represent his brother’s age?

a. (4 + 2) X sb. (s + 4 ) X 2c. 4s + 2d. 2s + 4

*Students will have to recognize that they FIRST need to multiply (relate to order of operations if applicable) and THEN add to solve correctly.

Grade 4 Math BenchmarksGeometry

MA.4.G.5.1 Students need to classify angles of two-dimensional shapes using benchmark angles (e.g. 45, 90, 180 and 360 degrees.

Determine the closest measure of the angle in these shapes:1. Answer: 90 degrees

2.

180 degrees

3.

360 degrees

Students should know the properties of the different types of angles, also knowing that a circle is 360, semi-circle 180, etc.

MA.4.G.5.2 Students will identify and describe the results of translations, reflections and rotations of 45, 90, 180, 270 and 360 degree figures, including figures with line and rotational symmetry.

*An important pre-requisite skill for students to know are the terms “clockwise” and “counter clockwise” as these are used frequently in these types of problems.

Students should know and identify:

= slide or translation

= turn or rotation

= flip or reflection

Determine amount of rotation:

= 90 degrees clockwise

= 180 degrees Clockwise OR counter

= 270 degrees Clockwise OR 90 counter clockwise

= 360 degrees

MA.4.G.5.3 Students will identify and build a three-dimensional object from a two-dimensional representation of that object and vice-versa.

Sample:In class, Ed used wooden blocks to make this figure. Which shows what his figure would look like if viewed from the top?

Answer:

*Students should use math manipulatives to build 3-dimensional figures and then draw top-down views using grid paper to help with this concept.

Grade 4 Math BenchmarksNumbers & OperationsBenchmark Learning Objective ExampleMA.4.A.6.1 Students will use and represent

numbers through millions in various contexts, including estimation of relative sizes of amounts or distances. This includes solving real-world problems.

Real world basic estimation problem:As of 2008, Ben Hill Stadium in Jacksonville has 88,548 seats. At their recent event, 9,325 seats were empty. What is the BEST ESTIMATE of the number of seats that were NOT empty?

a. 110,000b. 100,000c. 90,000d. 80,000

*Students will need to determine that the best strategy for this type of problem is to estimate both numbers involved in the problem FIRST and then find the difference. (90,000-9,000=81,000 which is ALMOST 80,000).

Calculate using numbers to the millions:The population of Tampa this year is 1,575,204. Last year, it was 1,571,692. By how much did the population INCREASE from last year to this year? 1,575,204-1,571,692 3,512

*Students should practice lining up digits carefully according to place value when working with large numbers of digits. They should also have a review of the proper procedure for REGROUPING to avoid mistakes.** 1 in. grid paper is very useful for teaching students to line up digits by place value vertically.

MA.4.A.6.2 Students use models to represent division as:

The inverse of multiplication

As partitioning As successive

subtraction

Division as the inverse of multiplication:Using counting cubes, have students count out a set number of cubes (e.g.27). Next, have them determine how many “groups of ___”(e.g. “3”) they can make out of the whole set.Once they accurately complete this task, then have them give you the “fact family” consisting of both division algorithms and both multiplication algorithms represented:

3 X 9 = 27; 9 X 3 = 27; 27 ÷ 3 = 9; 27 ÷ 9 = 3The above problem can also be used to teach

“partitioning” by having students begin with a set number of counting cubes and “break” or “partition” the set into smaller groups of equal amounts.

Successive Subtraction:Again assign the students to count out a specific number of counters. For this example we will use 24. Next, tell them to KEEP SUBTRACTING sets of 3 from the 24 and tell you how many sets of 3 they end up with.

“subtract” groups of 3 until there are none:

(21)

(18)…

As each group of 3 is subtracted off, it is set aside, so that 8 groups of three end up subtracted successfully.

MA.4.A.6.3 Students will generate equivalent fractions and simplify fractions.

Generate Equivalent Fractions:For introducing this basic skill, fraction strips are lined up vertically. Lining up a ½ strip over 2- ¼ strips gives a visual representation of their being equivalent, etc.Simplifying fractions:Students will need to apply the rule “what you do to the numerator, you must do to the denominator” E.g. They cannot multiply a numerator by 2 and a denominator by 3 to find an equivalent fraction; Prior knowledge of inverse relationship between multiplication and division are helpful here!Start by teaching equivalent fractions by having students multiply the numerator and denominator by the same number to find an equivalent.Then,Have them do the reverse, dividing BOTH the numerator and denominator by the same number.**They must first FIND a number that divides equally into both!

MA.4.A.6.4 Students will determine factors Factors:

and multiples for specified whole numbers.

Students must understand that the multiplicands are called FACTORS and the answer to a multiplication problem is called the PRODUCT. They ALSO must understand that ALL numbers that can be divided EQUALLY into a product or number are FACTORS of that number.Sample Problem:What are the factors of 24?1, 2, 3, 4, 6, 8, 12, 24

MA.4.A.6.5 Students will relate halves, fourths, tenths and hundredths as decimals and percents.

Begin with Fractions:Students need to learn how to change halves, fourths, tenths and hundreds into decimals and then into percents.They should know that ½ = .5 or 50%. They should understand that our number system is a base ten system, and that everything relates to parts of a whole, whether it is fractions, decimals OR percents. They also have to learn that 1 whole is equal to 100%;Sample Problems:Alexis ate 50% of her sandwich at lunch. Which of the following is EQUAL to 50%?

a. ½b. 5/100c. .10d. .20

9/10 of all avocados grown in the U.S. come from California. What percent is equal to 9/10?Answer: 90%Joan wanted 75% of the money from the lemonade stand she ran with her brother last Sunday. What decimal is equal to 75%?

a. .7b. .70c. .75d. .35

MA.4.A.6.6 Students will estimate and describe the reasonableness of estimates; determine the appropriateness of an estimate versus an exact answer.

Estimate Problem:What is the BEST ESTIMATE of the RANGE of numbers for the total land area in the table below:

State AreaAlabama 50,744Georgia 57,906Florida 53,927

South Carolina 30,109*Students would estimate the lowest values for all numbers (50,000 + 50,000 + 50,000 + 30,000). This

would give them the LOWER number of the range. Then they would estimate the higher value of each number (55,000 + 60,000 +50,000 + 35,000). This would give them the larger number of the range.a.100,000 – 125,000b.125,000 – 150,000c.150,000 – 175,000d.175,000 – 200,000

Grade 5 Content LimitsThe content limits described below apply to all items developed for Grade 5. However, the content limits defined in the individual benchmark specifications can supersede these general content limits.

Whole NumbersItems should not require the use of more than three operations.Integers may range from -500 through 999,999,999.

Addition:Items should not exceed four addends.Items should not exceed four 4-digit addends, three 5-digit addends, or two 6-digit addends.

Subtraction:Subtrahends, minuends and differences should not exceed six digits.

Multiplication:Factors can have up to three digits by three digits or four digits by two digits and could include a 0 in the hundreds, tens and/or ones places.

Division:Divisors should not exceed two digits.Dividends should not exceed four digits.Quotients may be expressed as mixed numbers or include remainders.

DecimalsPlace values could range from tenths through thousandths

Addition of:Items should not require the use of more than four 4-digit addends or two 5-digit addends.

Subtraction of:Subtrahends, minuends and differences with decimals should not exceed 5 digits.

Multiplication of:Multiplication of decimals is limited to the context of money.Factors may have up to a four-digit number multiplied by a two-digit number.

Division of:Division of decimals is limited to the context of money.Divisors should not exceed two digits and must be whole numbers.Dividends should not exceed four digits.Quotients should not have remainders.

Grade 5 Math BenchmarksDivisionMain Objective: Students will develop an understanding of and fluency with division of whole numbers.

Benchmark Learning Objective ExampleMA.5.A.1.1 Students will be able to describe

the process of finding quotients involving multi-digit dividends using models, place value, properties and the relationship of division to multiplication. This includes understanding of the distributive property of division . They will be expected to solve both mathematical and real-world division problems.

Distributive Property:639 ÷ 3 can be expressed as (600 + 30 + 9) ÷ 3).Students need to be able to apply the standard algorithm to describe (and to solve) one or more of the steps of solving division problems. They also must be able to apply this algorithm in order to identify missing steps of partially completed problems.Sample Problems:1. Johnny needs to sort the base ten blocks below into 3 equal groups. Which model shows what one of the groups might look like?

There are 4 multiple choice solutions, the correct one looks like

2. Jose wants to sell roses for a fundraiser. He puts 156 roses into 12 vases. The expression below can be used to solve for the total number of roses he put in each vase. 156 ÷ 12Which of the following is equivalent to this expression?a. (156 ÷ 10) + (156 ÷ 2)b. (15 ÷ 10) + (6 ÷ 2)c. (12 ÷ 12) + (36 ÷ 12)d. (120 ÷ 12) + (36 ÷ 12)[Students should be taught to “break the whole number” into 2 numbers that have the same divisor that are easier to work with] In the example above, students know that 12 X 10=120 and 12 X 3=36, so they break the 156 down to solve more easily.

3. Jamie’s teacher gave her the following division problem. Her job is to find the missing digits.

3 5 1 5 6 1 6 4 8 8 1 8 0 1 6 1 6 0*Students will need to be very well practiced in the steps of solving long division problems in order to solve these.

MA.5.A.1.2 Students will be able to estimate quotients or calculate them mentally depending on the context and numbers involved. This will include real world problems and the ability to check for reasonableness of answer.

Estimating Quotients:Bobby learned that each year, a tornado hits Kansas nearly every 25 days. About how many tornadoes does Kansas have each year?

a. 25b. 20c. 15d. 10

*Students will have to know that there are 365 days in a year. Students will also have to be able to apply one of several strategies in order to make this estimation. They will need to be able to divide 365 by 25, which will result in 14.5, which they would round to 15; or they would divide by a closer even number, such as dividing 360 by 20 and then rounding down.

Mental Calculation Problem:If Joey earned 350.00 in allowance money this year, about how much did Joey earn per week?

a. 7.00b. 10.00c. 5.00d. 15.00

*All students will need to have a foundational concept on how to check their answer for its reasonableness. They will have to be able to not only recognize that the quotient in their solution cannot reasonably be larger than the whole number dividend. They ALSO will need to be able to check for reasonableness on estimation problems.

MA.5.A.1.3 Students will be able to interpret solutions to division situations including those with remainders depending on the context of the problem.

Sample Problems:1. A cafeteria manager baked 500 cupcakes for a school carnival and is placing them in boxes. Each box holds 24 cupcakes. What is the LEAST number of boxes the cafeteria manager will need to hold all 500 cupcakes?

a. 20b. 21c. 30d. 19

**Students will likely divide this problem and get 20 with a remainder. They will need to realize that the “remainder” represents the remaining cupcakes that won’t fit in 20 boxes. They will need to understand that because there is a remainder, they need to add one more to the number of boxes needed. This is not a typical problem with remainders because it requires more critical thinking skills.

Estimate a quotient:2. In our galaxy, a star is formed every 18 days. There are 365 days in 1 year. Which is the closest number to the total number of stars formed in our galaxy in 1 year?a. 18b. 19c. 20d. 21

MA.5.A.1.4 Students will be able to divide multi-digit whole numbers fluently, including solving real world problems, demonstrating understanding of the standard algorithm and checking the reasonableness of results.

Sample Problems:1. The fifth grade teachers at a school are renting buses to take their students on a field trip. There will be 5 teachers and 143 students going on the trip. Each bus can hold up to (or a maximum of) 35 people.

Part A: Write a division problem that can be used to determine the minimum number of buses the teachers must rent so that everyone will be able to go on the field trip.

[answer: 148 ÷ 35]

*This may be difficult for students to come up with, as it requires them to do more than one step in order to come up with the division problem. They have to recognize that they FIRST need to ADD the students and teachers together, THEN divide by the number of people each bus holds.

**They will also need to be very clear on the meaning of the terms “maximum” and “minimum”. This is ANOTHER scenario where they will end up with a remainder in the solution and will need to know that they have to add another “1” to the whole number in

the solution, recognizing that the “remainder” represents the “remaining people” that need a seat on a bus.

Part B:Solve the division problem you wrote in Part A. What is the minimum number of buses the teachers must rent for everyone to be able to go on the field trip? In the space below, show your work and EXPLAIN your answer.

**When the students solve this, they should get 4 with a remainder of 8. They should THEN recognize that if they only rent 4 buses, 8 people would not be able to go. They have to think critically to realize they need to add 1 more bus! They THEN have to explain that dividing 35 seats per bus into 148 people, they would fit on 4 buses, with 8 people left out, so they have to have 5 buses!

Grade 5 Math BenchmarksNumbers & Operations (fractions & decimals)Main Objective: Students will develop an understanding of and fluency with addition and subtraction of fractions and decimals

Benchmark Learning Objective ExampleMA.5.A.2.1 Students will represent addition and

subtraction of decimals and fractions with like AND unlike denominators using models, place value or properties. Items may include mixed numbers and/or fractions.

Sample Problems:1. Mrs. Bradford served part of a pie for dessert. The shaded parts of the pies below show how much of the pie was in the plate BEFORE and AFTER dessert.

What fraction of the whole pie, expressed in lowest terms, was eaten for dessert?

a. 1/3b. 5/12c. 7/12d. ¾

*To solve successfully, students will have to think critically. They will have to RECOGNIZE that there was NOT a whole pie available to begin with.They will also have to realize that their starting point is to count the TOTAL number of slices in the pie, to find the denominator. THEN, they have to determine that there were ONLY 9 slices in the FIRST place. So the first fraction is 9/12. Then, they see that 5/12 are remaining. They subtract these and get 4/12, THEN have to reduce to simplest form!

2. Alex and Stephanie both have some coins in their pockets. The shaded areas in the diagrams below represent the VALUE of the coins they have.Value of Alex’s Coins

Value of Stephanie’s Coins

What is the total value, in dollars, of the coins that Alex and Stephanie have?

Correct Answer: $2.55*Students MUST be careful to use the KEY to the right of the models. They ALSO must realize that each “group” has 20, not 10 in it. They will multiply 20 X .05 and get $1.00 for each Alex and Stephanie. Then, they will have to multiply .05 by the remaining shaded cubes for each and then add ALL the coins together. **They will also need to be careful about lining up decimal points when adding money.

3. Abstract problem using model:The shaded figure on the grid below represents the number 1.

Using the grid below, and the figure shown, draw a representation of 1 2/3 + ½. In the space below, show your work. Also, write your answer in SIMPLEST FORM.

Key = $.05

**Critical Thinking Issues:First, students will have to recognize that their FIRST STEP is to ADD the fractions together. This is NOT asking them to draw 2 separate pictures!

They will need to convert the 2/3 and the ½ in order to get like denominators.

Next, they will solve and find the answer = 2 1/6.

Then they will need to refer BACK to the figure given to know how to draw the picture.

MA.5.A.2.2 Students will add and subtract fractions and decimals fluently and verify reasonableness of results, including in problem situations.

Sample Problems:1. Some of the ingredients for a chocolate chip cookie recipe are shown below. 1/3 cup butter2/3 cup sugar1 1/8 cups flour

What is the minimum capacity of a bowl that can hold all of these ingredients?

**Students will have to simply add three fractions, 2 of which have unlike denominators in order to solve.**In teaching this, teach them to recognize that 1/3 + 2/3 = 1 whole. They can THEN add that 1 to the 1 in 1 1/8, and find the answer to be 2 1/8.

2. Mindy went hiking on a trail that has a total length of 3.5 miles. After she had hiked 1.98 miles, she stopped for lunch. After lunch, Mindy will continue hiking until she reaches the end of the trail. How

much further in miles will Mindy need to hike after lunch?

*since this is simple subtraction, the KEY component to stress here is the importance of lining up digits and decimal points, and pay attention to regrouping. 2 14 3.5 10 1.98 1 .5 2

3. Mr. Bruno bought two types of fruit. He bought 5 5/8 pounds of

apples and 4 ½ pounds of grapefruit. Mr. Bruno has a scale that can display a maximum weight of 10 pounds. Use mathematical terms to explain whether or not Mr. Bruno’s scale will be able to display the total weight of the fruit. Show or explain your work below.

The problem-solving work shown should include adding the fractions: 5 5/8 + 4 ½ , starting by adding the 5 + 4 = 9, then changing the denominator in ½ and adding 5/8 + 4/8 to get 9/8. They would THEN need to change 9/8 to 1 1/8, add

that to the 9 and get 10 1/8.

NEXT, their explanation needs to state that “since Mr. Bruno’s scale can only display up to 10 pounds of weight, it will NOT be able to display the total weight of the fruit, because the fruit weighs more than the maximum weight the scale can display”.**Students will have to KNOW that maximum means the most.

MA.5.A.2.3 Students will be able to make reasonable estimates of fraction and decimal sums and differences, using techniques for rounding.

Sample Problem:1. Jenny has ¾ of a bushel of apples. Sara has 2/3 of a bushel. To make the apple pies they want to make for their bake sale, they need a total of 3 bushels of apples. ABOUT how many

more apples do they need? Students should recognize that BOTH 2/3 and ¾ are ALMOST 1 whole, so the girls are very close to having 2 bushels full. So they need ABOUT 1 more bushel.

2. Frank’s truck holds 30.75 gallons of diesel fuel. His gauge tells him that he has 4.75 gallons now. ABOUT how much more fuel does he need for his tank to be full?

**Students should recognize that 30.75 rounds to 31, and that 4.75 rounds to 5. Therefore, he needs ABOUT 26 gallons.

MA.5.A.2.4 Students will be able to determine the prime factorization of numbers. They must also identify reasons why numbers are prime or composite.

Sample Problem:1. Tell whether the following numbers are PRIME or COMPOSITE.

7, 19, 23, 31, 43, 59

2. Factor the following composite number to prime factors:

330= 15 X 223 x 5 x 11 x 2

*Students will have to have a strong understanding of what makes a number PRIME and what makes it COMPOSITE. They will also need to have a strong sense of multiplication and division basic facts, in order to properly IDENTIFY a number as prime.

Grade 5 Math BenchmarksGeometry & MeasurementMain Objective: Students will be able to describe three dimensional shapes and analyze their properties, including volume and surface area.

Benchmark Learning Objective ExampleMA.5.G.3.1 Students will analyze and compare the

properties of two-dimensional figures and three-dimensional figures and three-dimensional solids including the number of edges, faces, vertices, and types of faces.

Students will need to be familiar with the following:Pyramid, prism, solid, face, edge, vertex (vertices), net, right, polyhedron, as well as previous grade vocabulary terms.

Sample Problems:1. Penelope has an aquarium in the shape of a hexagonal prism. The front view of a hexagonal prism is shown below.

Which of the following are the correct numbers of faces, edges and vertices in a hexagonal prism?

a. 4 faces, 13 edges, 10 verticesb. 8 faces, 13 edges, 10 verticesc. 6 faces, 18 edges, 12 verticesd. 8 faces, 18 edges, 12 vertices

**The key issue with this type of problem is that often students will not account for the SIDE of the figure that is NOT SHOWN in the picture! They have to remember to count the edges, faces and vertices on the back side as well.

2. Keira wants to make a triangular prism like the one shown below. 4 inches

5 inches

3 2 inchesInches

Keira will cut each face of the prism from construction paper. In the space below, draw all the shapes Keira will need to cut from the construction paper. Include the lengths of the sides, in inches, of each shape.

The response should include Five shapes shown below with measurements included. 2 inches 3 inches 4 inches

5 5 5 Inches inches in

2 3 2 3 inchesinches inches inches 4 inches 4 inches

MA.5.G.3.2 Students will describe, define and determine surface area of prisms by using appropriate units and selecting strategies and tools.

Sample Problems:1. A box of tissues is in the shape of a rectangular prism with the dimensions shown below.

6 in.

5 inches 9 inchesWhat is the VOLUME of the box of tissues?

a. 258 square inchesb. 258 cubic inchesc. 270 square inchesd. 270 cubic inches

*Students will need to remember 2 important things here:The formula for volume ( L X W X H) AND that volume has 3 dimensions so it is measured in CUBIC units.

Grade 5 Math BenchmarksAlgebra

Benchmark Learning Objective ExampleMA.5.A.4.1 Students will use the

properties of equality to solve numerical and real world situations. This can include problems with up to 2 variables..

Sample Problems:1. Mrs. Jackson purchased two identical jackets for her twin sons from an online store. The cost for shipping was $1, and the total amount Mrs. Jackson paid was $87. The equation below can be used to find j, the price of one jacket.

2j + 1 = 87What is the price, in dollars, of ONE jacket?Correct answer: $43

*When teaching these types of problems, it is important that the students understand what each number in the problem represents (e.g. 2j = jacket times 2, since there were 2 jackets purchased, and you are trying to find the cost of ONE; 1 represents the shipping cost, and 87 is the total cost for the 2 jackets AND shipping.)

2. Both Mrs. Carmen and Mr. Davis worked the same total number of hours on the weekend. Mrs. Carmen worked 5 hours on Saturday and 7 hours on Sunday. Mr. Davis worked 8 hours on Saturday.

Part A: Write an equation to represent the situation. Let d represent the number of hours Mr. Davis worked on Saturday.

Part B: Use your equation to find the number of hours Mr. Davis worked on Sunday. Show all your work.

*Critical thinking skill needed:Students will need to recognize that they are solving for a component that is NOT mentioned in the original problem. They will need to figure out what equation they can use, and how to use the variable given in that equation.They will start by adding Mrs. Carmen’s hours. If they read the problem carefully, they will realize that the answer is GIVEN in the problem! (they worked the same number of total hours) but they still must set up the equation using the variable.They will see that Mrs. Carmen worked 12 hours. The equation should look like this:

12 = 8 + jThey should find it fairly simple to solve that j = 4.

MA.5.A.4.2 Students will construct and describe a graph showing continuous data, such as a graph of a quantity that changes over time.

Terms students need to know:Coordinate, coordinate plane, ordered pairs, midpoint, x-axis, y-axis.1. Construct a line graph to show the following

data:The city of Tallahassee had the following record of registered voters :Key: (in thousands)

2000 2001 2002 2003 2004 2005 2006 2007 2008

114 118 125 136 122 140 148 163 201

*Students would be expected to create something like the graph below:

Registered Voters in Tallahasse

’00 ’01 ’02 ’03 ’04 ’05 ’06 ’07 ‘08

Key(thousands)

MA.5.A.5.1 Students will be able to identify and plot ordered pairs on the first quadrant of the coordinate plane.

Sample Problem:

A B

8

7

6

5

4

3

200

180

160

140

120

100

C D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

The letters on the coordinate grid above represent the locations of paintings hanging on a wall of an art gallery.The manager wants to hang another painting exactly halfway between points A and B. Which ordered pair best describes the location of the new painting?

a. (6, 10)b. (10,3)c. (10, 6)d. (11, 6)

MA.5.G.5.2 Students will compare, contrast, and convert units of measure within the same dimension (length, mass or time) to solve problems.

Sample Problems:1. The students in Mrs. Zavala’s fifth-grade class went on a field trip to the zoo. They arrived at the zoo in the morning and left the zoo in the afternoon. The clocks below show the time when the class arrived at the zoo and the time the class left the zoo.

Based on the times shown, for how long was Mrs. Zavala’s class at the zoo?

a. 3 hours 15 minutesb. 3 hours 45 minutesc. 4 hours 15 minutesd. 4 hours 45 minutes

2. The vending machine shown below can hold up to 24 bags of pretzels. Each bag contains 1 oz of pretzels. What is the total number of pounds of pretzels in 24 bags of this size?

**Critical Thinking Skills Needed:Students will have to read this problem CAREFULLY and recognize that it is dealing with 2 different units of measure: ounces and pounds. The amount each bag holds is given in ounces, but the question asks for a solution in pounds.It is important the students read the problems more than once so that they are aware of these things. They will need to FIRST determine that the machine holds 24 ounces of pretzels. THEN they will have to convert ounces to pounds. They will need to know that there are 16 ounces in 1 pound.

[Correct Answer: 1.5 pounds]

MA.5.G.5.3 Students will be able to solve problems requiring attention to approximation, selection of appropriate measuring tools, and precision measurement. *They will need to be familiar with both customary and metric units

Sample Problem:1. A carpenter is measuring the width of a

window in a house. Which of the following methods would provide him with the most precise measurement?

a. He should measure the width of the window to the nearest foot.

of measure. b. B. He should measure the width of the window to the nearest inch.

c. He should measure the width of the window to the nearest ¼ foot.

d. He should measure the width of the window to the nearest ½ inch.

*Students should recognize that the SMALLER the unit of measure in this problem, the more precise the measurement would be.

2. Using the information given in the previous problem, if the carpenter determined that the width of each window was 15.5 inches, approximately how many windows could he put across the front of his house if the front wall measures 18 feet across, leaving at least 5 inches between windows?

*Students AGAIN will have to read the problem very carefully and recognize that they are dealing with two different units of measure: inches and feet. Although the windows are measured in inches, the problem is requiring them to determine how many 15.5 INCH windows could go across an 18 FOOT wall. INADDITION, they must leave 5 inches between every 2 windows.

Solution:FIRST, they would multiply 18 X 12 to determine how many inches wide the wall is. They would find that they had 216 inches to deal with.

Next, they would use REPEATED SUBTRACTION to find the solution.They ALSO have to subtract 5 inches for each window placed,So, they should subtract 20.5 inches per window. OR….With a solid knowledge of basic facts, they could recognize that 20.5 is ALMOLST 21, round , and realize 21 would go into 216 10 times!

MA.5.G.5.4 Students will derive and apply formulas for area of parallelograms, triangles, and trapezoids from the area of a rectangle.

Sample Problems:1. A drama teacher drew up plans for a stage he wants to guild. A diagram of the top of the stage, which is in the shape of a trapezoid, is shown below.

Top View of Stage Design

26 feet

13 feet 12 feet 13 feet

16 feet

**It is CRITICAL that students know the formula for finding area of a TRAPEZOID, like other unconventional shapes, in order to accurately solve this problem.Solution: [ 252 square ft]

2. Diane drew a parallelogram on a grid, as shown below.

SCALE = 1 square centimeter

Part A: Draw 1 line on the parallelogram above to make 2 figures which can then be used to form a rectangle.Part B: On the grid below, draw the rectangle you can form from the 2 figures in Part A. Show or explain how you can determine the area of the parallelogram and write the area of the parallelogram on the line provided.**shown in red is where the line would be drawn.

Grade 5 Math BenchmarksNumbers & Operations

Benchmark Learning Objective Example

MA.5.A.6.1 Students will identify and relate prime and composite numbers, factors, and multiples within the context of fractions.

For samples, refer to appendix, and also to benchmarks MA.5.A.2.1, MA.5.A.2.2, and MA.5.A.2.4*Students must be able to identify and work

with PRIME and COMPOSITE whole numbers AND fractions; fractions that cannot be reduced to a lower form would be PRIME fractions.

MA.5.A.6.2 Students will use the order of operations to simplify expressions which include exponents and parentheses.

*Multiplication will be shown in parentheses, or with an X or a dot.

Sample Problems:1. Jervon needed to evaluate the expression shown below using the order of operations.

54 ÷ (9 – 3) + 1 X 6

What is the value of the above expression?a. 8b. 15c. 20d. 50

**It is critical that students are well-versed in using the order of operations in order to arrive at correct solution.

2. The values of the expression below is equal to the distance, in feet, between a ball and the ground, 2 seconds after the ball is dropped from a height of 75 feet. 2

75 – 16 X 2

What is the value of this expression?

Solution: 11MA.5.A.6.3 Students will be able to describe

real world situations using positive and negative numbers.

Items in this section may range from – 500 to 500.

Sample Problem:In one day, Sam and his family drove from Bakersfield, California to Death Valley, California. The elevation in Bakersfield is 408 feet above sea level, and the elevation in Death Valley is 282 feet below sea level. What is the difference in elevation between these two places?

a. 126 feetb. 282 feetc. 680 feetd. 690 feet

**There is a large room for error with this type

of problem. The students have to recognize the terms ABOVE and BELOW sea level, as relating to positive and negative integers. They have to be able to identify that they are not subtracting, but rather ADDING numbers in this calculation. They need to add the number of feet above sea level to the number BELOW, recognizing these as positive and negative numbers.

MA.5.A.6.4 Students will compare, order and graph integers, including integers shown on a number line.

*Items may include integers between -500 and 500, inequalities using < , > , =, and =. Items may be real world or mathematical in context.

Sample Problem:1. The table below shows the lowest recorded temperature for four states in the U.S. as of December 2007.

LOWEST RECORDED TEMPERATURESState Temperature (in F)

Delaware -17Florida -2Hawaii 12Mississippi -19

Which of these lists the temperatures shown in the table in order from lowest to highest?

A. -2, -12, -17 -19B. – 19, -17, -2, 12C. 12, -2, -17, -19D. -2, -17, -19, 12

*Students will have to have a firm grasp on the way NEGATIVE INTEGERS work; that the larger the digits, the SMALLER the unit of measure with negative integers.

MA.5.A.6.5 Students will solve non-routine problems using various strategies ding “solving a simpler problem”.”

*Prerequisite skill:Students should be adept at applying the “Four step process to solving problems”: Read, Plan, Solve and Check. They should be made to apply these steps on a regular basis on class assignments, so that they are in the habit and practice of using them.Sample Problems:1. A discount music store sells CD’s for $6 each. When a customer purchases 3 CD’s, they receive 1 free CD. Marisa went to the music store and spent $36 on CD’s. How many free CD’s did Maris receive?

a. 1b. 2c. 6d. 8*Without practice on reading and re-reading word problems, many students will answer this problem incorrectly, choosing C- 6 instead of B-2. The reason is, they read the problem and question too quickly. They ASSUME that the question is going to ask how many CD’s the person got for $36, rather than how many FREE CD’s they got for spending $36. This is a multi-step problem. They must FIRST determine that the customer purchased 6 CD’s. Since they get 1 free for each 3 they purchase, they would receive 2 free.

2. Pedro used white and grey square tiles to make models of some floor designs. The first 4 floor designs are shown below.

Floor 1 Floor 2 Floor 3 Floor 4

If Pedro continues making these floor designs, what will be the total number of grey tiles in the 100th floor design?Solution: 101*In order to solve this problem, students will have to think critically. They have to recognize the pattern is that the number of grey tiles in each design is equal to one more than the design number (e.g. Floor 1 has 2, floor 2 has 3, etc.) Using this pattern, they would conclude that the 100th floor design would have 101 grey tiles.3. Matthew has a jar of quarters. In June, Matthew added and removed quarters from the jar.

Each afternoon, from June 1 through June 30, Matthew added $1.00 (in quarters) to the jar.

On June 7, Matthew removed $1.75 from the jar.

On June 15, Matthew removed $3.25 from the jar.

One June 29, Matthew removed $2.50 from the jar.

If the value of the quarters in the jar was exactly $40.00 on the evening of June 30, find the value of the quarters in the jar on the morning of June 1. In the space below, show or explain your work.

*This requires multiple steps, and students should apply the 4-step process. As they read and reread the scenario and the question, they should determine the steps as follows:

Matthew added 30 X $1 = $30 in June.Of that money, he removed three amounts during the month of June:$1.75$3.25$2.50$7.50 total

$30.00- 7.50$22.50 was “to the good” in June.

If the value of the coins on June 30 is $40, then the student now subtracts $22.50 from $40, and finds that the value of the quarters on the morning of June 1 before Matthew started adding/removing quarters was $17.50.

MA.5.S.7.1 Students be able to construct and analyze line graphs and double bar graphs.

Prior Knowledge:Items may require the student to apply mathematical knowledge from lower grades. They have to have prior knowledge of frequency tables, single bar graphs, pictographs, and line plots from data including surveys and experiments.

Sample Problems:1. The graph below shows the number of boys and girls enrolled in three grades and Main

Street Elementary School.

Students Enrolled atMain Street Elementary School

Based on the graph, which of the following statements is true about the enrollment at Main Street Elementary School?a. The total number of 4th grade students is approximately 70.b. The total number of 5th grade students is approximately 140.c. The number of 4th grade boys is less than the number of 3rd grade boys.d. The number of 5th grade girls is greater than the number of 5th grade boys.

2. Ramon did a science experiment on evaporation. He filled a glass with water and put it on the windowsill. At the same time each day, he measured the height of the water in the glass.

Ramon made a table of his data as shown below.

Measurement of Water EvaporationDay Height of Water ( in

cm.)1 10.0

2 9.5

3 9.04

5 8.0

The height of the water is missing for Day 4 in the table. Based on the information given in the table, find the height of the water for Day 4. On

the grid below, make a line graph showing all the data in the table and the height of the water for day 4.Be sure to:

Title the graph Label the axes Accurately graph the data Use an appropriate scale

Measurement of Water Evaporation

•

•

•

•

•

1 2 3 4 5

MA.5.S.7.2 Students will be able to differentiate between continuous and discrete data and determine ways to represent those using graphs and diagrams.

Prerequisite Knowledge:Students must clearly know the difference between continuous and discrete data.

Sample Problems:1. A veterinarian measured the mass of a newborn kitten each day for 6 days. The results are shown in the table below.

Mass of Newborn KittenDay 1 2 3 4 5 6

Mass(in

grams)90 110 120 140 170 190

Which graph is the best representation of the data in the table?

CM

Day

10.0

9.5

9.0

8.5

8.0

A) Mass ofNewborn Kitten

B) Mass ofNewborn Kitten

• •

• •

• •

1 2 3 4 5 6

Correct solution: B

Day

200

160

120

80

0