2018 Summer Review for Students Entering Pre-Algebra … · 2018 Summer Review for Students Entering Pre-Algebra 7 & Pre-Algebra 8 1. Area and Perimeter of Polygons 2. Multiple Representations

2018 Summer Review for Students Entering Pre-Algebra 7 & Pre-Algebra 8

1. Area and Perimeter of Polygons

2. Multiple Representations of Portions

3. Multiplying Fractions and Decimals

4. Order of Operations

5. Writing and Evaluating Algebraic Expressions

6. Simplifying Expressions

7. Data Displays: Histograms and Box Plots

8. Solving Multi-Step Equations

9. Unit Rates and Proportions


Area and Perimeter of Polygons

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.

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:


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)

Writing and Evaluating Algebraic Expressions

There are some vocabulary words that are frequently used to represent arithmetic operations.

Addition is often suggested by: sum, increased, more than, greater than, total

Subtraction is often suggested by: difference, decreased by, less than, smaller than

Multiplication is often suggested by: product, times, twice, double

Division is often suggested by: quotient, divided by, shared evenly

Examples

Five more than m: Five more than a number, increases a number by 5 so it would be m + 5 .

Three less than x: Three less than a number makes the number smaller by 3 so it would

be x − 3.

Triple m: Tripling a number is the same as multiplying the number by 3 so it would be 3m.


Order of Operations

The acronym PEMDAS may help: (Please Excuse My Dear Aunt Sally)

First evaluate expressions in parentheses

Evaluate each exponential (for example, 52 = 5 · 5 = 25).

Multiply and divide each term from left to right.

Combine like terms by adding and subtracting from left to right.

Numbers above or below a “fraction bar” are considered grouped. A good way to remember is to circle the

terms like in the following example. Remember that terms are separated by + and – signs.

Example 1: Simplify 12 ÷ 22 − 4 + 3(1 + 2)

3

Simplify within the circled terms: Be sure to perform the exponent operations before

dividing.

12 ÷ 22 = 12 ÷ 2 · 2 = 3

Then perform the exponent operation: 33 = 3 · 3 · 3 = 27

Next, multiply and divide left to right: 3(27) = 81

Finally, add and subtract left to right: 3 − 4 = −1

Example 2: Simplify −32 − + 8 ÷

Simplify within the circled terms: −32 = −3 · 3 = −9

= = 3 8 ÷ = 8 · = 16

Then add and subtract, left to right.

Simplifying Expressions Like terms are two or more terms that are exactly the same except for their coefficients. That is, they have the

same variable(s), with corresponding variable(s) raised to the same power. Like terms can be combined into one

quantity by adding and/or subtracting the coefficients of the terms. Terms are usually listed in the order of

decreasing powers of the variable. Combining like terms, one way of simplifying expressions, using algebra

tiles is shown in the first two examples.

Example 1: Example 2:

Simplify (2x2 + 4x + 5) + (x

2 + x + 3) means Simplify x

2 + 3x − 4 + 2(2x

2 − x) + 3

combine 2x2 + 4x + 5 with x

2 + x + 3.


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.

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.


Solving Multi-Step Equations

A general strategy for solving equations is to first simplify each side of the equation. Next isolate the variable on one side

and the constants on the other by adding equal values on both sides of the equation or removing balanced sets or zeros.

Finally determine the value of the variable–usually by division.

Note: When the process of solving an equation ends with different numbers on each side of the equal sign (for example, 2

= 4), there is no solution to the problem. When the result is the same expression or number on each side of the equation

(for example, x + 2 = x + 2 ) it means that all numbers are solutions.

Example 1: Solve 3x + 3x − 1 = 4x + 9

Example 2: Solve −2x + 1 + 3(x −1) = −4 + −x − 2

Unit Rates and Proportions

A rate is a ratio comparing two quantities and a unit rate has a denominator of one after simplifying. Unit rates or

proportions may be used to solve ratio problems. Solutions may also be approximated by looking at graphs of lines

passing through the origin and the given information.

Example 1: Judy’s grape vine grew 15 inches in 6 weeks. What is the unit growth rate (inches per week)?

Solution: The growth rate is . To create a unit rate we need a denominator of

"one." = . Solve by using a Giant One: = · ⇒ 2.5

Example 2: Bob’s favorite oatmeal raisin cookie recipe use 3 cups of raisins for 5 dozen cookies. How

many cups are needed for 40 dozen cookies?

Solution: The rate is so the problem may be written as this proportion: .

One method of solving the proportion is to use a Giant One:


1. One way to find the height of a tall tree is to measure its shadow and compare that with the shadow of an

object whose height is known. Why does this work? Explain completely and use this idea to write and

solve a proportion based on this picture, to calculate the height of the tree.

2. Compute. Show your work. (No Calculator)

a. 17.6 + 3.4 b. 12.0 – 3.6

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

3. Compute:

a. b. c. d.

4. Calculate the area of the shape. Are there any rectangles or other shapes you see in the figure that will

help you calculate the total area? Show these shapes as you show your work. All sides are either

vertical or horizontal.

( 14)3 ( 3

7)2 (6

3)2 (2

3)4

1 m 0.75 m 18 m

16

12

4

8

3 15


5. Morticia’s exam paper is shown. Why did she get this problem wrong? Explain to Morticia what she

did incorrectly.

6. Write an equation based on the diagram below. What is the value of x?

7. The area of a right triangle is 36 square feet. The base is twice the height.

a. Draw and label the triangle.

b. Write an equation and solve it to find base and height.

8. 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.

Evaluate: .

x

15

30 10 20 40 50 0


9. Simplify by combining like terms.

a. 2a – 8a2 + a – 3a

2 + 5 – 2

b. 25 + 5b + 6b2 – 8b – (–5) + 13b

2

c. 2(x + 5) + 3x2 – 7x

2 – 6x – 9

10. Simplify by combining like terms.

a. –9x2 – 4 – 5x + 8x

2 – 7 + 15x

b. –6x2 – (–6) + (–9x) – 8 – 7x + 3x

2

11. Simplify.

a. 24 – (6 + 9) 3 + 42 2

b. 30 + 10 5 – 45 5

c. 25 5 (4 + 5)

12. Simplify.

a.

b.

c.

15 10 2.5 3 65

5


13. Trish collected the following information about inches of snowfall in her town:

a. What is the mean snowfall for February and March?

b. What is the mean snowfall for all 5 months on the graph?

c. If June has 0 inches of snow, explain how to find the mean for all six months.

14. Explain how to find the area of the rectangle, using square centimeters as your units. Twelve square

centimeters are shown in the shape.

15. Think about tiles and simplify each expression.

a. (2x2 + 4x + 1) + (x

2 + 2x + 5) b. (3x + 7) + (x + 4)

c. (5x2 + 9x + 6) – (2x

2 + 2x + 5) d. (7x + 14) – (3x + 10)

Ja Fe M Ap M

29.

16. 16.

8.3 0.8

Snowfall (inches) in Mt.

Shasta, CA


16. A recipe calls for 2 cups of flour and 3 eggs and it serves 8 people. If we need to serve 24 people, how

much flour and how many eggs should we use? Why?

17. Fill in the table below with the missing form of the number.

Decimal Percent

0.45

0.675

8%

37.5%

0.005

18. Convert the following:

a. 1.074 to a percent.

b. 534.3% to a decimal.

19. Complete each of these Diamond Problems:

a. b. c.

20. Simplify without a calculator:

a. 6 + 3 (7 – 3 2)(7 – 3 2) b.

3 −7

−10

24

−5

-3


21. What is the area of the shape shown? Clearly show how the area of each triangle and rectangle was

calculated. All units are in feet.

22. Consider the parallelogram at right.

a. What is the length of the base of the parallelogram?

b. What is the length of the height of the parallelogram?

c. What is the area of the parallelogram?

d. Draw a rectangle that would have the same area as this parallelogram. Include the dimensions

on the rectangle!

23. What is the area and perimeter of a rectangle with a width of 7 meters and a length of twice that amount?

24. Simplify the expression without a calculator:

7

6

9

8

11

3

6


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

a. a fraction.

b. a decimal.

c. a percent.

26. Consider the representation below. Write the portion shaded as

a. a fraction.

b. a decimal.

c. a percent.

27. Using any variable you wish to represent the unknown number, write each of the following as an

algebraic expression.

a. The product of a number and 6.

b. The sum of a number and 12.

c. The difference of a number and 8.

d. The quotient of a number and 15.


ANSWERS

1.) Explanations will vary -If we connect the top

of the meter stick to the end of the shadow, and

similarly connect the top of the tree with the

end of its shadow, we form two similar

triangles. From that we can set up a

proportion. 24 m

2.) a: 21, b: 8.4, c: 5.12, d: 3.85

3.) a: ; b: ; c: simplifies to 2, 23 = 4; d:

4.) A = 224 sq. units

5.) Explanations will vary -Morticia: when you are

left with just multiplication and division, don’t

do the multiplication first! Go left to right!

The answer is 16.

6.) , x = 25

7.) a: drawing will have triangle with height marked x and the base marked 2x; b:

, h = 6 ft, b = 12 ft

8.) a: 14, b: 7.5, c: 34 Min: 2 Lower Quartile: 7.5

Upper Quartile: 33.25 Max: 43

9.) a: –11a2 + 3a + 3 , b: 19b

2 – 3b + 30,

c: –4x2 – 4x + 1

10.) a: –x2 + 10x – 11, b: –3x

2 – 16x – 2

11.) a: 40, b: 71, c: 45

12.) a: -1, b: 2, c:–7

13.) a: 16, b: 14.02, c: Divide sum of 70.1 by 6

months resulting in a mean of 11.68.

14.) 40 square cm.

15.) a: 3x2 + 6x + 6, b: 4x + 11, c: 3x

2 + 7x + 1,

d: 4x + 4

16.) Explanations will vary - It would be 3 times as

many so 6 cups of flour and 9 eggs.

17.) 45%, 67.5%, 0.08, 0.375, 0.5%

18.) a: 107.4%, b: 5.343

19.) a: xy = −21, x + y = −4; b: y = −5, x + y =

−3; c: x and y are −4 and −6

20.) a: 9, b: 2

21.) (8∙11) + (6∙6/8) + (7∙6) + (6∙2/2) =

88+24+42+27 = 181 sq. feet

22.) a: 6, b: 3, c: 18 square units, d: rectangle

with same base and height as parallelogram

23.) area 98 sq. meters, perimeter: 42 meters

24.) 1

25.) a: or ; b: 0.24; c: 24%

26.) a: 9/100, b: .09, c: 9%

27.) a: 6x, b: x + 12, c: x – 8, d:

164

949

63

1681

24100

625

x15

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2018 Summer Review for Students Entering Pre-Algebra … · 2018 Summer Review for Students Entering Pre-Algebra 7 & Pre-Algebra 8 1. Area and Perimeter of Polygons 2. Multiple Representations