2017 summer review for students entering concepts of math ... · pdf file2017 summer review...

2017 Summer Review for Students Entering Concepts of Math 7 or Mathematics 7

1. Using Place Value to Round and Compare Decimals

2. Addition and Subtraction of Decimals

3. Multiple Representations of Portions

4. Addition and Subtraction of Fractions

5. Locating Points on a Number Line and Coordinate Graph

6. Multiplication and Division of Fractions

7. Multiplication and Division of Decimals

8. Area and Perimeter of Quadrilaterals and Triangles

9. Rewriting and Evaluating Variable Expressions

10. Data Displays: Box Plots and Histograms

11. Solving One-Step Equations

12. Greatest Common Factor (GCF)/Least Common Multiple (LCM)

13. Measures of Central Tendency

(Suggested Timeline on Reverse Side)


June 2017

Sun Mon Tues Wed Thur Fri Sat

19

Complete

Problems 1-5

20 21 22 23 24

25 26

Complete

Problems 6-10

27 28 29 30 1

July 2017

2 3

Complete

Problems 11-15

4 5 6 7 8

9 10

Complete

Problems 16-20

11 12 13 14 15

16 17

Complete

Problems 21-25

18 19 20 21 22

23 24

Complete

Problems 26-30

25 26 27 28 29

August 2017

30 31

Complete

Problems 31-34

1 2 3 4 5

6 7

Complete

Problems 35-39

8 9 10 11 12

13 14

Complete

Problems 40-42

15 16 17 18 19

20 21

Complete

Problems 43-45

22 23 24 25 26


Using Place Value to Round and Compare Decimals

Example 1: Round 17.23579 to the nearest hundredth.

Solution: We start by identifying the digit in the hundredths place—the 3. The digit to the right of it is 5

or more so hundredths place is increased by one. 17.24

Example 2: Round 8.039 to the nearest tenth.

Solution: Identify the digit in the tenths place– the 0. The digit to the right of it is less than 5 so the tenths

place remains the same. 8.0 (the zero must be included)

Addition and Subtraction of Decimals

To add or subtract decimals, write the problem in column form with the decimal points in a vertical column so that digits

with the same place value are kept together. Include zeros so that all decimal parts of the number have the same number

of digits. Add or subtract as with whole numbers. Place the decimal point in the answer aligned with those in the

problem.

Example 1: Add: 37.68 + 5.2 + 125 Example 2: Subtract: 17 − 8.297

Solution:

Solution:

Multiple Representations of Portions

Portions of a whole may be represented in various ways as represented by this web. Percent means ―per

hundred‖ and the place value of a decimal will determine its name. Change a fraction in an equivalent fraction

with 100 parts to name it as a percent.

Example 1: Write the given portion as a fraction and as a percent. 0.3

Solution: The digit 3 is in the tenths place so, . On a diagram or a hundreds grid, 3

parts out of 10 is equivalent to 30 parts out of 100 so .

Example 2: Write the given portion as a fraction and as a decimal. 35% Solution:


Addition and Subtraction of Fractions To add or subtract two fractions that are written with the same denominator, simply add or subtract

the numerators and then simplify if possible. For example: .

If the fractions have different denominators, a common denominator must be found. One way to

find the lowest common denominator (or least common multiple) is to use a table as shown below.

The multiples of 3 and 5 are shown in the table at right. 15 is the least common multiple and a

lowest common denominator for fractions with denominators of 3 and 5.

After a common denominator is found, rewrite the fractions with the same denominator (using the

Giant One, for example).

Example 1: Solution:

Example 2: Solution:

To add or subtract two mixed numbers, you can either add or subtract their parts, or you can change the

mixed numbers into fractions greater than one.

Example 3: Compute the sum:

Solution: This addition example shows adding the whole number parts and the fraction parts

separately. The answer is adjusted because the fraction part is greater than one.

Example 4: Compute the difference:

Solution: This subtraction example shows changing the mixed numbers to fractions greater than one and then

computing in the usual way.


Locating Points on a Number Line and on a Coordinate Graph

Points on a number line represent the locations of numbers. Numbers to the right of 0 are positive;

to the left of 0, they are negative. For vertical lines, normally the top is positive.

Point a at right approximates the location of 2

.

Two perpendicular intersecting number lines (or axes) such as the ones below create a coordinate

system for locating points on a graph. Points are located using a pair of numbers , or coordinates,

where x represents the horizontal direction and y represent the vertical direction. In this case ―A‖

represents the point (2, −3).

Area and Perimeter of Quadrilaterals and Triangles

Area is the number of square units in a flat region. The formulas to calculate the areas of several kinds of

quadrilaterals or triangles are:

Perimeter is the number of units needed to surround a region. To calculate the perimeter of a quadrilateral or

triangle, add the lengths of the sides.


Multiplication of Fractions

To multiply fractions, multiply the numerators and then multiply the denominators. To multiply mixed numbers, change

each mixed number to a fraction greater than one before multiplying. In both cases, simplify by looking for factors than

make ―one.‖

Example 1: Multiply Example 2: Multiply

Solution: Solution:

Note that we are simplifying using Giant Ones but no longer drawing the Giant One.

Multiplication of Decimals

There are at least two ways to multiply decimals. One way is to use the method that you have used to multiply

integers; the only difference is that you need to keep track of where the decimal point is (place value) as you

record each line of your work. The other way is to use a generic rectangle.

Example 1: Multiply 12.5 · 0.36 Example 2: Multiply 1.4(2.35) Solution:

(2 + 0.3 + 0.05 + 0.8 + 0.12 + 0.020= 3.29)

Division of Decimals

To divide decimals, change the divisor to a whole number by multiplying by a power of 10. Multiply the

dividend by the same power of 10 and place the decimal directly above in the answer. Divide as you would with

whole numbers. Sometimes extra zeros may be necessary for the number being divided.

Example 1: Find 53.6 ÷ 0.004.


Rewriting and Evaluating Variable Expressions

Expressions may be rewritten by using the Distributive Property: .

This equation demonstrates how expressions with parenthesis may be rewritten without parenthesis. Often this

is called multiplying. If there is a common factor, expressions without parenthesis may be rewritten with

parenthesis. This is often called factoring.

To evaluate a variable expression for particular values of the variables, replace the variables in the expression

with their known numerical values (this process is called substitution) and simplify using the rules for order of

operations.

Example 1: Multiply and then simplify .

Solution:

First rewrite using the Distributive Property and then combine like terms.

Example 2: Evaluate for .

Solution:

Solving One-Step Equations

To solve an equation (find the value of the variable which makes the equation true) we want the variable by

itself. To undo something that has been done to the variable, do the opposite arithmetical operation.

Example 1: Solve: x − 17 = 49

Solution: 17 is subtracted from the variable. To undo subtraction of 17, add 17.

x = 49 + 17 ⇒ x = 66

Greatest Common Factor

The greatest common factor of two or more integers is the greatest positive integer that is a factor of both (or all) of the

integers.

For example, the factors of 18 are 1, 2, 3, 6, and 18 and the factors of 12 are 1, 2, 3, 4, 6, and 12, so the greatest common

factor of 12 and 18 is 6.


Division of Fractions

Division of fractions can be shown using an area model or a Giant One. Division using the invert and multiply

method is based on a Giant One.

Example 1: Use an area model to find .

Example 2: Use a Giant One to find .

Histograms

A histogram is a method of showing data. It uses a bar to show the frequency (the number of times something occurs).

The frequency measures something that changes numerically. (In a bar graph the frequency measures something that

changes by category.) The intervals (called bins) for the data are shown on the horizontal axis and the frequency is

represented by the height of a rectangle above the interval. The labels on the horizontal

axis represent the lower end of each interval or bin.

Example: Sam and her friends weighed themselves and here is their weight in

pounds: 110, 120, 131, 112, 125, 135, 118, 127, 135, and 125. Make a histogram to

display the information. Use intervals of 10 pounds.

Solution: See histogram at right. Note that the person weighing 120 pounds is counted

in the next higher bin. Solution: See histogram at right. Note that the person weighing

120 pounds is counted in the next higher bin.

Least Common Multiple

The least common multiple (LCM) of two or more positive or negative whole numbers is the lowest positive whole

number that is divisible by both (or all) of the numbers.

For example, the multiples of 4 and 6 are shown in the table at right. 12 is the least common multiple, because it is the

lowest positive whole number divisible by both 4 and 6.

4 8 12 16 20 24 28 32

6 12 18 24 30 36 42 48


Box Plots

A box plot displays a summary of data using the median,

quartiles, and extremes of the data. The box contains the

―middle half‖ of the data. The right segment represents

the top 25% of the data and the left segment represent the

bottom 25% of the data.

Example: Create a box plot for the set of data given in the

previous example.

Solution:

Place the data in order to find the median (middle

number) and the quartiles (middle numbers of the upper

half and the lower half.)

Based on the extremes, first quartile, third quartile, and

median, the box plot is drawn. The interquartile range

IQR = 131–118 = 13.

Measures of Central Tendency

Numbers that locate or approximate the ―center‖ of a set of data are called the measures of central tendency. The mean

and the median are measures of central tendency.

The mean is the arithmetic average of the data set. One way to compute the mean is to add the data elements and then to

divide the sum by the number of items of data. The mean is generally the best measure of central tendency to use when

the set of data does not contain outliers (numbers that are much larger or smaller than most of the others). This means

that the data is symmetric and not skewed.

The median is the middle number in a set of data arranged numerically. If there is an even number of values, the median

is the average (mean) of the two middle numbers. The median is more accurate than the mean as a measure of central

tendency when there are outliers in the data set or when the data is either not symmetric or skewed.

When dealing with measures of central tendency, it is often useful to consider the distribution of the data. For symmetric

distributions with no outliers, the mean can represent the middle, or ―typical‖ value, of the data well. However, in the

presence of outliers or non-symmetrical distributions, the median may be a better measure.

Examples: Suppose the following data set represents the number of home runs hit by the best seven players on a Major

League Baseball team:

16, 26, 21, 9, 13, 15, and 9.

The mean is .

The median is 15, since, when arranged in order (9, 9, 13, 15, 16, 21, 26), the middle number is 15


1. Add or subtract as indicated. NO calculators!

a. 1039.9 + 93.07 b. 398.32 – 129

2. Multiply or divide as indicated. NO calculators!

a. 8 325 b. 827 ×14

c. 12.5 ×3.4 d. 12.62÷ 0.4

3. Find the area of the rectangle at right.

4. Round each number to the given place.

a. 23.679 b. 55.55 c. 2,840.12

(hundredths) (ones) (tenths)

5. Consider the list of numbers below.

0.34 0.4 0.034 0.304 0.314

a. Which number is the smallest? How do you know?

b. Which number is the greatest? How do you know?

inch

inch


6. Fill in the blank with either < , > , or = .

a. 91.01 _______ 91.10 b. 0.123 _______ 0.0123

7. If you have 8 pieces of licorice to share among 5 people, how much licorice will each person get? Show your

thinking.

8. Calculate.

a.

110

+ 35

b. 5/8 – 3/16

9. The two spinners below are incomplete. What fraction is missing in each spinner?

a. b.

10. Evaluate the expression x2 y2 for x 5 and y 2 .

11. What number is being described in each of these puzzles?

a. When I multiply my number by 12 I get 60.

b. When I divide my number by 14 I get 4.

c. When I subtract 19 from my number I get 22.

d. When I add 24 to my number I get 20.

12. Fill in the missing number to make each equation true.

a. 2(____) + 4 = 11 b. 30 – 2 (___) = 16 c. 28 + 9 (___) = 55

?

?


13. Mr. Jones feeds his big cat of a can of cat food and he feeds his small cat half that amount.

a. How much of a can does he feed his small cat?

b. How much of a can does he feed both cats?

14. Simplify each statement fully:

a. b. c.

15. Complete the generic rectangle below by filling in any missing dimensions or area. Write a mathematical sentence

showing the multiplication problem and the product.

16. What is the greatest common factor of 6, 9 and 12?

17. On the number line below, make a box-and-whisker plot for this data.

2, 7, 9, 12, 14, 22, 32, 36, 43

a. What is the median for this data? Label it on the graph.

b. What is the lower quartile? Label it on the graph.

c. What is the upper quartile? Label it on the graph.

13

32 ( 3 9 )

4 6 15 10

2.5 3 65 5

3 ( 32 5 ) 100 211

70

20

350

1400

05

30 10 20 40 50 0


18. Add or subtract as indicated. NO calculators!

a. 1039.9 + 93.07 c. 67.234 – 23.05

b. 0.827 + 432.1 d. 398.32 – 129

19. At right is a graph. What are the coordinates of each point, A through G?

20. Place the following numbers in the approximate locations on the number

line:

4.5, –2.5, , 3.1, , .

21. Represent the portion in three different ways.

Of 25 students interviewed, 21 said they believe in the Loch Ness monster.

Fraction: _____

Decimal: _____

Percent: _____

22. Label the following numbers at their approximate place on the number line.

a. b. –0.1 c. d.

e. 210% f. 1.75 g. 1.9 h. 1.99

23

154

15

89

98

–5 0 5

0 1 2


23. Calculate the area of the shape below. Do you see rectangles or other shapes within this shape? If so, what?

Does that help you? Explain.

24. Solve.

a. x + 14 = –4 b. y – 14 = –4 c. 14 z = 42

25. Consider the representation at right. Write the portion shaded as

a. a fraction.

b. a decimal.

c. a percent.

26. What value of n makes the following equation true? n 6 = 18

27. Kris read each day of her vacation. The following is a list of how many pages she read each day:

105, 39, 83, 54, 84, 75, 52, 96

a. Create a step-and-leaf plot to represent this data.

b. Create a box-and-whisker plot to represent this data.

c. What are the upper and lower quartiles?

d. Which representation gives you more information about Kris’ data? Why?

15

10

45


28. Represent the portion in three different ways.

Of 50 students surveyed, 19 said they had soda for breakfast.

Fraction: _____

Decimal: _____

Percent: _____

29. Multiply. Do not use a calculator.

a. b. c.

30. Sandra works at a flower store. The store charges $1.60 for each rose and $1.15 for each carnation. Write a

numerical or variable expression for each of the following:

a. the cost of y roses.

b. the cost of 12 carnations.

c. the cost of m roses and f carnations.

31. Explain what you know about each of the following measures of central tendency:

a. The mode

b. The median

c. The mean

32. A farmer has fenced in a square piece of land that measures 209 feet by 209 feet. While this might seem like a

strange length and width, this fence encloses exactly one acre of land. We still measure the lengths of the sides in

feet, but when we are talking about the area enclosed, we can say 43,681 square feet (209×209), but usually we

say one acre.

a. What is the perimeter of this piece of land?

b. If the farmer doubles the length and the width of his land, what is the new perimeter? What is the

new area in acres?

c. If he triples the length and width of the fence, will the area be three acres? Explain.

13× 1

9(1 1

4) × 4

535

× 56


33. The perimeter of the rectangle is 18 inches. How long is m?

34. Denise has 12 yards of fabric. How many ties can be made from the fabric if each tie uses yard of fabric?

35. Consider the graph shown here.

a. What is the coordinate point for A?

b. What is your estimate for coordinate point for D?

c. Which two points have the same y value?

36. At right are the results of a recent survey. What portion of the students walked to school?

Fraction: _____

Decimal: _____

Percent: _____

37. Compute. Show your work.

a. 17.6 + 3.4 b. 12.0 – 3.6

c. (3.2)(1.6) d. (3.5)(1.1)

34

A

C

D

EB

How Do You Get To School?

Walk Bus Car

5.5 inches

m


38. In their last game of the season, Jessica has made 6 out of 10 shots so far, and Linda has made 12 out of 15 shots.

As the coach, you must decide which player will shoot the final deciding shot of the game. Who should do it and

why? Be clear and convincing! One of them is going to upset she isn’t chosen, so your reasoning needs to be

sound.

39. The following wingspan measurements of butterflies (in mm) have been recorded:

42, 38, 20, 45, and 38.

Find the mean, median, and mode of the data.

40. Simplify.

a. b. c.

d. e. f.

g. h. i.

3 8 1

4 5 9 1

3 1 2 1

4

3 8

1 5 10 1

2 1 9 1

6

0.2 1 10

1 9

1 6 0.5 1

4


41. Logan bought 5 books that all cost the same amount. His bill was $78.25. How much did each book cost?

42. Manuel’s class created a bar graph for the data collected from the question ―How many brothers and sisters do

you have?‖

a. How many students said they had 3 siblings?

b. How many more students said they had 1 sibling than said they had none?

c. How many students are represented in the histogram?

d. What is the mean, median, and mode number of the siblings represented? Are any of those measures a

central tendency impossible to find? Why?

43. Solve.

a. x + 14 = –4 b. y – 14 = –4 c.

44. When asked how many minutes it takes to get ready for school, the following responses were given:

22, 5, 7, 16, 18, 28, 29, 28, 32, and 45.

a. Find the mean, median, and mode of the data.

b. Create a stem-and-leaf plot for the data.

45. Draw a picture for each of the following situations and determine the fractional product.

a. of b. of c. of

z14

= 3

12

14

13

12

23

14

0 1 2 3 4 or

more

1

2

3

4

5

6

7


ANSWERS

1.) a. 1132.97 b. 269.32

2.) a. 40.625 b. 11578 c. 42.5 d.

31.55

3.) 1/6 in2

4.) a. 23.68 b. 56 c. 2840.1

5.) a. 0.034 (explanations will vary) b.

0.4 (explanations will vary)

6.) a. < b. ˃

7.) 1.6

8.) a. 7/10 b. 7/16

9.) a. 5/18 b. 1/24

10.) 29

11.) a. 5 b. 56 c. 41 d. -

4

12.) a. 3 ½ b. -7 c. 3

13.) a. 1/6 b. ½

14.) a. 2 b. -7 c. -1

15.) 25 x 74 = 1850

16.) 3

17.) a. 14 b. 8 c. 34

18.) a. 1132.97 b. 432.927 c. 44.184 d.

269.32

19.) A(6, 4), B(3, 6), C(4.5, 4.5), D(6, 5), E(2, 0), F(–1,

–2), G(–3, 1)

20.)

21.) 21/25, 0.84, 84%

22.) From least to greatest: –0.1, , , , 1.75, 1.9,

1.99, 210%. Look for reasonableness on placement.

23.) 300 units squared

24.) a. -18 b. 10 c. 3

25.) 8/25, 0.32, 32%

26.) 108

27.) c: lower is 53, upper is 90 d: Answers will

vary. We can read off more information from the box-

and-whisker plot, namely the median, upper and lower

quartile, so some may argue that it shows more

information.

28.) 19/50, 0.38, 38%

29.) a. 1/27 b. 1 c. ½

30.) a. 1.60y b. 12 x $1.15 c. 1.60m + 1.15 f

31.) Answers will vary.

32.) a. 836 feet b. 1672 feet (perimeter),

174,724 ft2 (area) c. No, explanations will vary

33.) 3.5 inches

34.) 16

35.) A(2,3), B(3.5,8), C(4,1.5), D(5.5, 4.5), E (8,8)

36.) 11/45, 0.24, 24%

37.) a. 21 b. 8.4 c. 5.12 d. 3.85

38.) Linda (Reasoning will vary)

39.) 36.6, 38, 38

40.) a. 1/8 b. 8/9 c. -1/4 d. 3/40 e. 5 f. 5/18 g. 0.1 h. 1/45 i. 0.75

41.) $15.65

42.) a. 4 b. 3 c. 22 d. Mode: 1, Median:

1.5, not possible to find mean because the two people

with 4 or more could have any number of siblings that

are > 3

43.) a. -18 b. 10 c. 42

44.) a: mean: 23, median: 25, mode: 28 b:

45.) a. 1/8 b. 1/6 c. 1/6

15

89

98

2017 summer review for students entering concepts of math ... · pdf file2017 summer review...

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