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STANDARDS OF LEARNING
CONTENT REVIEW NOTES
Grade 6 Mathematics
3rd Nine Weeks, 2018-2019
Content Review: Standards of Learning in Detail Grade 6 Mathematics: Third Nine Weeks
2018-2019 This resource is intended to be a guide for parents and students to improve content knowledge and understanding. The information below is detailed information about the Standards of Learning taught during the 3rd grading period and comes from the Mathematics Standards of Learning Curriculum Framework, Grade 6 issued by the Virginia Department of Education. The Curriculum Framework in its entirety can be found at the following website.
http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/2016/cf/grade6math-cf.pdf
SOL 6.8 The student will
(a) identify the components of the coordinate plane; (b) identify the coordinates of a point and graph ordered pairs in a coordinate
plane.
(a) Identifying the coordinates of a point in a coordinate plane
A Coordinate plane is formed by the intersection of a horizontal number line and
a vertical number line. The number lines intersect at the origin and separate the coordinate plane into four regents called quadrants.
The coordinates of a point are represented by the ordered pair (x,y), where x is the first coordinate and y is the second coordinate. The order of the coordinates matter.
Example: Think of an airplane. An airplane runs horizontal (x axis) first and then flies vertical (y axis). Coordinates are plotted X axis first, Y axis second.
The coordinates of the origin are (0,0) Quadrants are named in a counterclockwise order. The signs for the quadrants are
as follows: Quadrant I (+,+) Quadrant II (-,+) Quadrant III (-, -) Quadrant IV (+,-)
Coordinates are written (x,y)
The Coordinate Plane
Graphing ordered pairs in a coordinate plane.
To graph the ordered pair (4, 2), start at the origin. Move four units to the right and 2 units up. Then plot the point.
Practice graphing the following points on the coordinate plane below: (-3,-2) (-5, 2) (0,-4) (5, 0)
SOL Practice Items provided by the VDOE, http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/index.shtml
Answers are located on the last page of the booklet.
SOL 6.8 (Coordinate Plane)
1. On a regular coordinate grid, the point (-7, 10) is in which quadrant?
a) I b) II c) III d) IV
2. Which ordered pair describes a point with a location in the second quadrant on
a coordinate grid?
a) (-6, -5) b) (-6, 5) c) (6, -5) d) (6, 5)
3. Look at the coordinate grid.
Which point is located in Quadrant III?
a) M b) P c) L d) N
P
M
N
L
4. On a regular coordinate grid, the point (-7, -10) is in which quadrant? a) I b) II c) III d) IV
5. Which coordinate grid has the quadrant correctly labeled?
a)
b)
c)
d)
I
I
I
I
II
II
II
II
III
III
III
III
IV
IV
IV
IV
6. In which quadrant is the point (17, -18) located?
a) Quadrant I b) Quadrant II c) Quadrant III d) Quadrant IV
7. Use the given numbers to create an ordered pair representing a point located on the x-axis.
SOL 6.9 The student will determine congruence of segments, angles and polygons. Regular Polygons
equal sides
equal angles
Congruent figures have exactly the same size and the same shape.
Non-congruent figures may have the same shape but not the same size.
The symbol for congruency is .
Example:
Congruent Figures Non-congruent Figures
The matching or corresponding angles of congruent polygons have the same measure, and the matching or corresponding sides of congruent polygons have the same measure.
The direct comparison of congruent or non-congruent figures can be accomplished by placing one figure on top of the other or by measuring all sides and angles.
Construction of congruent line segments, angles, and polygons helps students understand congruency.
Congruent Line Segments Non-congruent Line Segments Congruent Angles Non-congruent Angles
SOL Practice Items provided by the VDOE, http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/index.shtml
Answers are located on the last page of the booklet. SOL 6.9 (Congruence)
5
SOL 6.7 The student will
(a) derive ๐ ;
(b) solve problems, including practical problems, involving circumference and area of a circle;
(c) solve problems, including practical problems, involving area and perimeter of triangles and rectangles.
Derive ๐
๐ is the ratio of the circumference to the diameter.
Circumference and Area of Circles
A diameter is a chord that goes through the center of a circle.
o When using formulas, sometimes you are given the diameter and must find the
radius. To do this, divide the given diameter by 2 to find the radius.
Example: If the diameter of a circle is 10 and you need the radius, divide by 2. 10 รท 2 = 5 so, the radius = 5
A radius is a segment from the center of a circle to any point on the circle. Two radii end-to-end form a diameter of a circle.
o When using formulas, sometimes you are given the radius and must find the diameter. To do this, multiply the given radius by 2 to find the diameter.
Example: If the radius of a circle is 6, and you need the diameter, multiply by 2.
6 ร 2 = 12 so, the diameter = 12
The value of pi (๐) is the ratio of the circumference of a circle to its diameter.
The ratio of the circumference to the diameter of a circle is a constant value, pi (๐), which
can be approximated by measuring various sizes of circles.
The fractional approximation of pi generally used is 22
7.
The decimal approximation of pi generally used is 3.14. Circumference is the distance around or perimeter of a circle.
Formulas:
Finding the circumference of a circle: Use the formula ๐ถ = ๐๐ ๐๐ ๐ถ = 2๐๐
Example: If the diameter of a circle is 6.5 inches, which is closest to the circumference of the circle?
Step 1: Choose the correct formula and write it down. There are 2 different formulas that can be used for finding the circumference of a circle.
๐ถ = ๐๐ ๐๐ ๐ถ = 2๐๐
Step 2: Fill in the value of pi.
๐ถ = 3.14 ร ๐ ๐๐ ๐ถ = 2 ร 3.14 ร ๐
Step 3: Replace the variable with the given value. If the diameter is given, use the formula on the left in the above step. If the radius is given, use the formula on the right in the above step.
๐ถ = 3.14 ร 6.5 ๐๐ ๐ถ = 2 ร 3.14 ร 3.25
Step 4: Complete the calculations.
๐ถ = 20.4 (rounded to the nearest tenth)
Finding the Area of a Circle: Use the formula ๐ด = ๐๐2
Example: Which is the closest to the area of a circle if the diameter is 9 inches? Step 1: Start by writing the correct formula. Make sure you select the formula for area, not circumference.
๐ด = ๐๐2
Step 2: Fill in the value of pi.
๐ด = 3.14 ร ๐2
Step 3: Replace the variable with the value of the radius. If the diameter is given, divide the diameter by 2; this will give you the radius. Remember, the radius is half the diameter. So in this example, the diameter given is 9. Divide 9 รท 2 = 4.5
๐ด = 3.14 ร 4.52
Step 4: Calculate the radius squared. This is done by multiplying the radius times the radius. Remember, ๐2 = ๐ ร ๐
๐ด = 3.14 ร 4.5 ร 4.5 ๐ด = 3.14 ร 20.25
Step 5: Complete the calculations. Remember that area is calculated in square units, so include that in your answer.
๐ด = 63.585 ๐๐2
The area of a polygon is the number of non-overlapping square units required to fill the region enclosed by the curve.
Applications of area include:
Covering the table with newspaper A bucket of paint covers 220 square feet Painting the door red The glass in the window The tarp that covers the baseball field when it is raining The tiles on the kitchen floor The entire city of Suffolk Public Schools
The perimeter of a polygon is the measure of the distance around the polygon.
Applications of perimeter include: Painting a gold trim around the edge of the coffee table A blue border on the carpet A fence around the playground The border around the bulletin board The trim around the window The baseboard trim around the room The border of the state of Virginia
Area of Triangles Formulas:
๐ด =1
2๐โ
(A is area, b is base, h is height)
Finding the area of a triangle: use the formula ๐ด = 1
2๐โ
Example:
What is the area of the triangle?
๐ด = 1
2๐โ ๐ด = ๐โ รท 2
๐ด = 1
2(10)(6) ๐ด = (10)(6) รท 2
๐ด = 1
2(60) ๐ด = (60) รท 2
๐ด = 30 square meters ๐ด = 30 meters2
OR
Perimeter and Area of Squares and Rectangles
Formulas: (s is side of a square, p is perimeter, A is area, l is length, w is width, h is height, and b is base.)
Finding the perimeter of a square: Use the formula ๐ = 4๐
Example: A picture measures 5 inches by 5 inces. How much wood is needed to frame the picture?
๐ = 4๐ ๐ = 4(5)
๐ = ๐๐ ๐ข๐ง๐๐ก๐๐ฌ
Finding the perimeter of a rectangle: Use the formula ๐ = 2๐ + 2๐ค
Example: A picture measures 8 inches by 10 inches. How much wood is need to frame the picture?
๐ = 2๐ + 2๐ค ๐ = 2(8) + 2(10)
๐ = 16 + 20 ๐ = ๐๐ ๐ข๐ง๐๐ก๐๐ฌ
Finding the area of a square: use this formula ๐ด = ๐ 2
Example: A rug measures 4 feet by 4 feet. What is the area of this rug?
๐ด = ๐ 2 ๐ด = 42
๐ด = ๐๐ ๐ข๐ง๐๐ก๐๐ฌ
Finding the area of a rectangle: Use the formula ๐ด = ๐ ร ๐ค
Example: A table measures 3 feet by 6 feet. How much paper is needed to cover the table?
๐ด = ๐ ร ๐ค ๐ด = 3 ร 6
๐ด = ๐๐ ๐ฌ๐ช๐ฎ๐๐ซ๐ ๐๐๐๐ญ ๐จ๐ซ ๐๐ ๐๐๐๐ญ๐
1. Logan needs to order a cover for his
swimming pool. The circular swimming pool has
a radius of 17 feet.
Which is closest to the number of square feet
needed to completely cover the pool?
A 106.76 B 289 C 907.46 D 1,156
2. In the circle shown, Point T is the center of
the circle and Point R is on the circle.
Which is closest to circumference of circle T ?
A 15.70 cm
B 17.27 cm
C 31.40 cm
D 34.54 cm
3. Lou is making a pizza that has a radius of 9
inches. Which is closest to the area of the pizza?
F. 1,017.36 sq in.
G. 254.34 sq in
H. 56.52 sq in
J. 25.434 sq in
4. Which is closest to the circumference of
circle O shown?
F 113.04 cm
G 75.36 cm
H 37.68 cm
J 18.84 cm
5. Point M is the center of the circle shown. Point N lies on circle M. Which is closest to the area of the circle? F 66๐๐2
G 132๐๐2
H 1,386๐๐2
J 5,544 ๐๐2
6. If the diameter of a circle is 7 inches, which is closest to the circumference? F. 21.98 in G 38.47 in H 43.96. in J 153.96 in
6.7
7. A Wheel has a radius of 67.5 meters. Which is closest to the circumference of this wheel? A. 211.95 m B. 423.9 m C. 4,556.25 m D. 14,306.62 m 8. The diameter of the circular base of a storage container is 18.8 meters. The circumference of the base is approximate 59 meters. Which of these could be used to estimate the value of ะป?
A. 9.4
59 B.
59
9.4 C.
18.8
59 D.
59
18.8
9. The circular campfire site at Camp Willow has a diameter of 5 yards. Which is closest to the area of the campfire site? A. 15.7 sq yd B. 19.6 sq yd C. 31.4 sq yd D. 78.5 sq yd
SOL 6.7 (Area and Perimeter)
1. What Is the area of the triangle shown?
A. 32 cm2 B. 40 cm2 C. 48 cm2 D. 96 cm2
2. Andrea is buying a rectangular rug that is 3 feet wide and 4 feet long. What is the total area that the rug will cover?
A. 12 square feet B. 14 square feet C. 24 square feet D. 28 square feet
3.. Susan has a rectangular garden that measures 20 feet by 10 feet. What is the least amount of fencing that she needs to buy in order to enclose the garden?
A. 30 feet B 60 feet C. 80 feet D. 200 feet
4. Casey and her friends went to the library to work on their social studies project. They pushed the two tables pictured together. What will be the total area of the top of the two tables when they are pushed together? F. 18 square feet G. 24 square feet H. 36 square feet J. 48 square feet 5. Rory and Curtis are on the stage crew for the school play. The rectangular stage measures 20 feet by 40 feet. What is the minimum amount of tape they will need to outline the stage? A. 60 feet B. 120 feet C. 400 feet D. 800 feet
6. What is the area of the large rectangle shown if each small square is 4 inches wide and 4 inches long?
A. 480 sq in. B. 129 sq in. C. 80 sq in. D. 30 sq in.
7. Mr. Miller is putting a border around the edges of a rectangle ceiling. The perimeter of the ceiling is 18 meters. Identify the measurements that could be the two dimensions of the ceiling.
2
meter
s
3
meter
s
4
meter
s
5
meter
s
8
meter
s
9
meter
s
SOL 6.11 The student will
a) represent the mean of a data set graphically as the balance point; and b) determine the effect on measures of center when a single value of a data set is
added, removed, or changed.
Mean as a balance point โ the mean of a set of data can be described as the balanced point based on the distribution of the data. If you think of scores or the average as a weight distribution, the mean is the number that balances the data. The other measures surround the mean.
Measures of central tendencies are types of averages for a data set. They represent numbers that best describe a data set. Mean, median, and mode are measures of central tendency that are useful for describing the average for different situations.
o Mean works well for sets of data with no very high or low numbers. o Median is a good choice when data sets have a couple of values much higher or
lower than most of the others. o Mode is a good descriptor to use when the set of data has some identical values.
The mean is the numerical average of the data set and is found by adding the numbers in the data set together and dividing the sum by the number of data pieces in the set.
Example: 15, 10, 5, 25, 30
For the set of data shown above, the mean is 17. This can be found by adding all the items in the set (which equals 85) and then dividing the number (85) by the number of items in the set (5).
15 + 10 + 5 + 25 + 30 = 85; then, 85 รท 5 = 17
The median is the middle value of a data set in ranked order. If there are an odd number of pieces of data, the median is the middle value in ranked order. If there is an even number of pieces of data, the median is the numerical average of the two middle values.
Example: 87, 85, 90, 100, 78
For the set of data shown above, the median is 87. This can be found by putting the numbers in order from least to greatest and finding the middle number.
Step 1: Order the numbers from least to greatest: 78, 85, 87, 90, 100 Step 2: Find the middle number. The number that is in the middle is 87.
Sometimes there are two middle numbers. 4,7,10,12,14,17 In this case, find the mean of the two middle numbers. 11 would be the median.
The mode is the piece of data that occurs most frequently. If no value occurs more often than any other, there is no mode. If there is more than one value that occurs most often, all these most-frequently-occurring values are modes. When there are exactly two modes, the data set is bimodal.
Examples:
- For 2, 3, 4, 5, 5, 6, 7, 8, 8, 8, 9, 11, the mode is 8.
- For 2, 3, 4, 5, 5, 5, 7, 8, 8, 8, 9, 11, the modes are 5 and 8 (bimodal).
- For 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 17, there is no mode.
The Outlier is a number that is numerically distant from the rest of the data. If students scored 100, 89, 92, 79, and 23 on the exam, then the outlier is 23 because it is distant from the other scores.
Balance point can be determined with a number line. The Xโs can โhopโ to the balance point, or you can solve mathematically. List the numbers 0, 1, 2, 2, 3, 3, 3, 10. Add all the numbers and divide by 8, this is the mean. The mean is the balance point. The balance point is 24. 0 + 1+ 2+ 2+ 3+ 3+ 3+ 10 =24
X
X X X X X X X 0 1 2 3 4 5 6 7 8 9 10 11
24
SOL Practice Items provided by the VDOE,
http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/index.shtml
Answers are located on the last page of the booklet.
6.11 (mean)
1
2
3
4
1 2
25
5
6
7
26
8 8
9
27
SOL 6.10 The student, given a problem situation, will
a) represent data in a circle graph; b) make observations and inferences about data represented in a circle
graph; and c) compare circle graphs with the same data represented in bar graphs,
pictographs, and line plots. General information about data collection and graphic representations of data:
To collect data for any problem situation, an experiment can be designed, a survey can be conducted, or other data-gathering strategies can be used. The data can be organized, displayed, analyzed, and interpreted to answer the problem.
Data can be discrete or continuous.
Different types of graphs are used to display different types of data. โ Bar graphs use categorical (discrete) data (e.g., months or eye color). โ Line graphs use continuous data (e.g., temperature and time). โ Circle graphs show a relationship of the parts to a whole.
All graphs include a title and data categories should have labels.
A scale should be chosen that is appropriate for the data.
A key is essential to explain how to read the graph.
A title is essential to explain what the graph represents.
Bar Graphs
Data are analyzed by describing the various features and elements of a graph.
Inferences and convincing arguments are based on data analysis.
Bar graphs are used to compare counts of different categories (categorical or discrete data). โ A bar graph uses either horizontal or vertical parallel bars to represent counts for several
categories. One bar is used for each category, with the length of the bar representing the count for that category.
โ There is space before, between, and after the bars. โ The axis displaying the scale representing the count for the categories should extend one
increment above the greatest recorded piece of data. The values should represent equal increments.
โ Each axis should be labeled, and the graph should have a title.
โ Statements representing an analysis and interpretation of the characteristics of the data in the graph (e.g., similarities and differences, mode, least and greatest) should be included.
Here is an example of a vertical bar graph.
28
Line Graphs
Line graphs should be utilized to show how one variable changes over time. By looking at a single-line graph, it can be determined whether the variable is increasing, decreasing, or staying the same with the passage of time.
The values along the horizontal axis represent continuous data on a given variable, usually some measure of time (e.g., time in years, months, or days). The data presented on a line graph is referred to as continuous data because it represents data collected over a continuous period of time.
The values along the vertical axis represent the frequency with which those values occur in the data set. The values should represent equal increments of multiples of whole numbers, fractions, or decimals, depending upon the data being collected. The scale should extend one increment above the greatest recorded piece of data.
Each axis should be labeled, and the graph should have a title.
Statements representing an analysis and interpretation of the characteristics of the data in the graph (e.g., trends of increase and/or decrease, least and greatest) should be included.
Here is an example of a line graph.
29
Circle Graphs
Circle graphs show relationships of the parts to a whole. The pie-shaped sections show groups.
Circle graphs show percent (out of 100). The percents add up to 100%. The sum of the angle measures in a circle graph are 3600.
Example:
Apples40%
Oranges25%
Bananas35%
0%
Favorite Fruit
Math48%
Social Studies
17%Music
1%
Reading 26%
Science 8%
Sam's Homework
Example: 1. The circle graph shows the subjects Sam studies
during homework time. Which subject does Sam spend most of his time studying?
The largest section of the graph is the section representing math. So, he spends most of his time studying math.
2. How does the time spent studying social studies compare to the time spent studying science?
The section representing social studies is about twice the size of the section representing science. So, twice as much time is spent on social studies as on science.
The percents total 100%.
Each section represents part of the data.
The circle represents all of the data.
30
Example:
A school survey asked students to choose their favorite color. The results are shown in the two data displays below. You want to compare the portion of students who chose red to all the studentsโ responses. Which data display should you use?
To use the bar graph, you have to estimate the number of students surveyed and the number of students that chose red. The circle graph shows exact percents, so it is the better choice.
Orange5%
Blue 25%
White8%
Red16%
Green12%
Yellow4%
Black 30%
Favorite Color
0
50
100
150
200
250
Blue White Red Green Yellow Black Orange
Nu
mb
er o
f St
ud
ents
Color
Favorite Color
31
SOL Practice Items provided by the VDOE, http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/index.shtml
Answers are located on the last page of the booklet.
SOL 6.10 (graphs)
1. The first 200 visitors to a state park asked about their favorite park activity. The results are shown in this circle graph. Which of the following is closest to the number of these 200 visitors who said hiking was their favorite activity?
2.
32
3. Look at the graph.
Which type of clothing appears to have had 5 times the sales of dresses?
4. Mr. Walker surveyed 24 students. He asked each student to rate a television show. The results are shown in this circle graph.
33
5. Twelve students answered a question that had answer choices labeled as A, B, C, and D. This circle graph represents the answer choices selected by the 12 students.
6. Bob asked a group of people to identify their favorite vegetable. The circle graph shows the results. Which graph on the next slide could represent the same data?
34
Math Smarts! Math + Smart Phone = Math Smarts!
Need help with your homework? Wish that your teacher could explain the math concept to you one more time? This resource is for you! Use your smart phone and scan the QR code and instantly watch a 3 to 5 minute video clip to get that extra help. (These videos can also be viewed without the use of a smart phone. Click on the links included in this document.) Directions: Using your Android-based phone/tablet or iPhone/iPad, download any QR barcode scanner. How do I do that?
1. Open Google Play (for Android devices) or iTunes (for Apple devices). 2. Search for โQR Scanner.โ 3. Download the app.
After downloading, use the app to scan the QR code associated with the topic you need help with. You will be directed to a short video related to that specific topic!
Itโs mobile math help when you need it! So next time you hear, โYouโre always on that
phoneโ or โPut that phone away!โ you can say โItโs homework!!!โ
PLEASE READ THE FOLLOWING: This resource is provided as a refresher for lessons learned in class. Each link will connect to a YouTube or TeacherTube video related to the specific skill noted under โConcept.โ Please be aware that advertisements may exist at the beginning of each video.
35
Math 6 SOL Concept Link QR Code
6.8 Coordinate Plane: naming quadrants, plotting points, etc.
https://www.youtube.com/watch?v=r16I6LB2YbQ
6.7 Area and Circumference of Circles
https://www.khanacademy.org/math/basic-geo/basic-geo-area-and-perimeter/area-circumference-circle/v/circles-radius-diameter-and-circumference
6.7 Perimeter and Area of Squares and Rectangles
https://www.khanacademy.org/math/geometry/hs-geo-foundations/hs-geo-area/v/intuition-for-area-of-a-triangle
6.11 Mean โ Balance Point https://www.youtube.com/watch?v=9of5gI9Yg00
6.11 Mean โ Balance Point
https://www.youtube.com/watch?v=mIAJggg25k4
36
6.11 Mean โ Increasing or decreasing an outlier
https://www.khanacademy.org/math/ap-statistics/summarizing-quantitative-data-ap/mean-median-more/v/impact-on-median-and-mean-when-increasing-highest-value
6.10 Creating Circle/Pie Graphs https://www.youtube.com/
watch?v=y3EEoI5wngE
6.10 Understanding Data Presented in
Circle/Pie Graphs
https://www.youtube.com/watch?v=KzXZfv9anpU
6.10 Finding the percent of a number
from a circle graph
https://www.youtube.com/watch?v=G5vqSr0YRZ8
37
Released Test Answers (3rd Nine Weeks)
SOL 6.8 (Coordinate Planes)
1. b 2. b 3. a 4. c 5. a 6. d 7. Any combination of (x,y) with 0 as the
y.
SOL 6.9 (Congruence) 1. f 2. b 3. j 4. c 5. ๐๐ฬ ฬ ฬ ฬ
SOL 6.7 (Pi, Circumference, Area)
1. c 2. b 3. g 4. j 5. h 6. f 7. b 8. d 9. b
SOL 6.7 (Area, Perimeter) 1. c 2. f 3. b 4. h 5. b 6. a 7. 4 meters by 5 meters
SOL 6.11 (Mean)
1. c 2. b 3. a 4. c 5. g 6. a 7. c 8. c 9. c
SOL 6.10 (Circle Graphs) 1. H 2. Action - 15 Comedy - 30 Other - 5 Musical - 10
3. F 4. C 5. B 6.