Gruber's Essential Guide to Test Taking: Grades 6-9 - Gruber Gary R. 2019

Math refresher/review (Essential math skills)
Math strategies

Math Words, Concepts, and Rules Your Child Should Know.

The following are some basic math terms and principles that your child will need to know in order to understand many of the questions on math tests.

WORDS

TRIANGLE. Any three-sided figure.

RECTANGLE. Four-sided figure with opposite sides equal and parallel. Sides must meet at right angles.

SQUARE. Four-sided figure—all, four sides equal; opposite sides parallel. Sides meet at right angles.

PARALLELOGRAM. Four-sided figure with opposite sides equal and parallel. Sides do not have to meet at right angles as in a rectangle.

CIRCLE. A closed curve whose distance from a central point to any point on the curve is always the same. This distance is called the radius. The distance around the curve itself is known as the circumference. The diameter is twice the radius.

PERIMETER. Perimeter means the length around a figure.

EXAMPLES

AREA OF RECTANGLE. The area of a rectangle equals its length times its width.

EXAMPLE

Area = length × width = 4 × 3 = 12

AREA OF SQUARE. The area of a square equals its length times its width.

EXAMPLE

Area = length × width = 3 × 3 = 9

VOLUME OF RECTANGULAR BOX. The volume of a rectangular box equals its length times width times height.

AREA OF PARALLELOGRAM. The area of a parallelogram equals its base times its height.

EXAMPLE

Area = 5 × 3 = 15

AREA OF TRIANGLE. The area of a triangle equals one half of its base times its height.

EXAMPLE

AREA OF CIRCLE. The area of a circle equals the square of its radius times pi (πr²).

EXAMPLE

What is the area of a circle whose radius is 5?

EXAMPLE

What is the approximate area of a circle whose radius is 7?

PERIMETER OF CIRCLE (Circumference). The perimeter of a circle (its circumfereiice) equals its radius times pi times 2 (2πr).

Perimeter (circumference) = 2πr (r = radius).

EXAMPLE

What is the circumference of a circle with a radius of 5?

Solution:

C = 2πr

= 2π (5)

= 10π

EXAMPLE

What is the approximate circumference of a circle with a radius of 7? (Use for π).

NUMBER LINES. A number line is a line showing increasing or decreasing numbers.

For example, on the number line above the arrow A points to 0.5, whereas the arrow B points to 2.

EXAMPLE

What does the arrow point to on the number line below?

(A) 2.1

(B) 2.2

(C) 2.3

(D) 2.4

GREATER THAN, LESS THAN, AND EQUAL TO SYMBOLS

Greater than can be written as >.

Less than can be written as <.

Equal to can be written as = .

EXAMPLES

300 > 299 (300 is greater than 299)

299 < 300 (299 is less than 300)

300 = 300 (300 is equal to 300)

Notice that 300 > 299 can also be written as 299 < 300 (299 is less than 300) and 299 < 300 can also be written as 300 > 299 (300 is greater than 299).

From two unequal relations, you can sometimes find a third relation. Example: Mary > Sam, and Sam > John.

Then Mary > John.

What this really says is that if the first is greater than the second, and the second is greater than the third, then the first is greater than the third

FRACTIONS

A fraction has a numerator and a denominator. The numerator is on top, and the denominator is on the bottom. The numerator is divided by the denominator.

EXAMPLE

EQUALS (Rules for Adding, Subtracting Multiplying and Dividing)

1. Equals Added to Equals Are Equal.

Example:

2. Equals Subtracted from Equals Are Equal.

Example:

3. You Can Multiply Equals by Equals to Get Equals.

Example:

4. You Can Divide Equals by Equals to Get Equals.

Example:

EVEN AND ODD ESITEGERS (Rules for Adding, Subtracting Multiplying and Dividing)

An even integer is a whole number exactly divisible by 2 (2, 4, 6, 8, 10, 12, etc:.). An odd integer is a whole number not exactly divisible by 2 (1, 3, 5, 7, 9, 11, etc.).

1An even integer plus or minus another even integer always equals an even integer.

For example: 4 + 6 = 10; 6 — 4 = 2.

2An even integer plus or minus an odd integer always equals an odd integer.

For example: 4 + 1 = 5; 4 — 3 = 1.

3An odd integer plus or minus another odd integer always equals an even integer.

For example: 3 + 5 = 8; 9 — 5 = 4.

4An even integer multiplied by an even integer always equals an even integer.

For example: 2 × 4 = 8; 4 × 4 = 16.

5An even integer multiplied by an odd integer always equals an even integer.

For example: 2 × 3 = 6.

6An odd integer multiplied by an odd integer always equals an odd integer.

For example: 3 × 3 = 9; 5 × 7 = 35.

7An even integer divided by an even integer is sometimes even, sometimes odd, and sometimes not an integer.

For example: 4 ÷ 2 = 2; 12 ÷ 4 = 3; 10 ÷ 4 = 2½; 4 ÷ 8 = 1/2.

8An even integer divided by an odd integer is sometimes even, never odd, or not an integer.

For example; 2 ÷ 1 = 2; 2 ÷ 3 = 2/3; 12 ÷ 3 = 4; 12 ÷ 11 = 12/11.

9An odd integer divided by an odd integer is never even, sometimes odd, or not an integer.

For example: 9 ÷ 3 = 3; 11 ÷ 9 = 11/9.

10An odd integer divided by an even integer is never an integer.

For example: 3 ÷ 2 = 3/2; 5 ÷ 5/4.

AVERAGES

Average means the total number of one group of items divided by the number of another group of items.

Example 1:

There are 200 students in a school with 10 classes. What is the average number of students in each class?

Solution:

Example 2:

What is the average number of melons for each crate if there are 100 melons in 10 crates?

Solution:

PARTS

Example:

Solution:

Example:

What part of the rectangle is shaded?

Solution:

1/2. Since there are two parts (shaded and unshaded) and one part is shaded, so 1/2 is shaded.

NEGATIVE NUMBERS

How to Add Negative Numbers

Examples:

1.—2 —5 = —7

(Add 2 + 5, then put — in front.)

2.—2 + 5 = + 3

(This is the same as 5 — 2 = 3)

How to Subtract Negative Numbers

Examples:

1.—2 — (—5) = —2 + 5 = + 3

(The — (—5) becomes + 5.)

2.— (—2) + 3 = + 2 + 3 = 5

(The — (—2) becomes + 2.)

How to Multiply Negative Numbers

Rules:

— × — = +

— × + = —

+ × — = —

+ × + = +

Examples:

—2 × —3 = +6

—2 × +3 = —6

+ 2 × —3 = —6

+ 2 × +3 = +6

Examples:

1.3 × (—5) = —15

(Multiply 3 × 5, put — in front.)

2.—3 × —5 = + 15

(Multiply 3X5, then multiply — × —: — × — = +.)

3.—5 × + 3 = — 15

(Multiply 5 × 3, put — in front since — × + = —.

How to Divide Negative Numbers

Rules:

Examples:

FACTORING

Examples:

SQUARE ROOTS

Example:

Example:

Example:

SQUARES

Examples:

2² = 4 since 2² = 2 × 2

3² — 9 since 3² = 3 × 3

A square of a number is that number multiplied by itself.

Example:

(—2)² = —2 × —2 — +4

After you have explained to your child the words, concepts, and rules just described, have him or her try the following exercises.

PROBLEMS

1The perimeter of the triangle below is

(A)480

(B)48

(C)24

(D)12

2Which figure does not always have two opposite sides equal?

(A)a parallelogram

(B)a triangle

(C)a rectangle

(D)a square

3What is the perimeter of the 3 square below?

(A)3

(B)12

(C)9

(D)cannot tell

4What is the area of the rec tangle below?

(A)7

(B)14

(C)12

(D)10

5Which is □ true?

(A)23 > 32

(B)32 > 31

(C)33 > 35

(D)30 > 30

6Which is true?

(A)30 < 50

(B)31 < 21

(C)21 < 21

(D)15 < 14

7If 4 = 3 + , then which is true?

(A)4 — 3 =

(B)4 + 3 =

(C)4 × 3 =

(D)4 ÷ 3 =

8If 3 = , then which is true?

(A)3 × 2 = 2 ×

(B)3 × 2 = 2 +

(C)3 × 2 = 2 —

(D)3 × 2 = 3 —

9Which is an even integer?

(A)3

(B)7

(C)9

(D)12

10Which is an odd integer?

(A)2

(B)4

(C)8

(D)9

11Which is true?

12Which is true?

13What is the average number of crayons per box if there are 30 boxes of crayons and a total of 900 crayons in all the boxes?

(A)30

(B)3

(C)90

(D)9

14Which circle has 1/4 of its area shaded?

SOLUTIONS

1(C) Perimeter equals length around.

length around = 6 + 8 + 10 = 24

2(B) Choices:

3(B) Perimeter equals length around.

All sides of a square are equal.

So 3 + 3 + 3 + 3 = 12 or 3 × 4 = 12

4(C) Area of rectangle equals length times width.

length = 4, width = 3

so 4 × 3 = 12

5(B) The sign > means greater than; < means less than. Choices:

(A) 23 is not > 32; it is < 32

(B) 32 is > 31

(C) 33 is not > than 35; it is < 35

(D) 30 is not > 30; it is = 30

6(A) The sign < means less than; > means greater than.

Choices:

(A) 30 is < 50

(B) 31 is not < 21; 31 > 21

(C) 21 is not < 21; 21 = 21

(D) 15 is not < 14; 15 > 14

7(A) Equals subtracted from equals are equal.

8(A) Equals multiplied by equals are equal.

9(D) An even integer is a whole number exactly divisible by 2.

Choices;

(A) The number 3 is not exactly divisible by 2, so it is odd.

(B) The number 7 is not exactly divisible by 2, so it is odd

(C) Thc number 9 is not exactly divisible by 2, so it is odds

(D) The number 12 is exactly divisible by 2: 12 ÷ 2 = 6. So it is even.

10(D) An odd integer is a whole number not divisible by 2.

Choices:

(A) The number 2 is divisible by 2, so it is even.

(B) The number 4 is divisible by 2, so it is even.

(C) The number 8 is divisible by 2, so it is even

(D) The number 9 is not divisible by 2, so it is odd

11(C) Choices;

12(D) Multiply both numerator and denominator of Choice A, B by 3; of Choice C and D by 2. See “Fractions” section on page 175.

13(A)

14(B)

Math Shortcuts Your Child Should Know

There are many shortcuts that your child can use when working out math problems. The most important of these are discussed below.

COMPARING TWO FRACTIONS

Sometimes your child will have to find out which of two fractions is larger. Here’s a typical example:

EXAMPLE

Which is greater:

You or your child may have been taught to find a common denominator first, and then compare the fractions. There’s a much easier way that you should be aware of:

SOLUTION

Any two fractions can be compared in this way. Try it yourself:

EXAMPLE

Which is greater:

SOLUTION

ADDING FRACTIONS

EXAMPLE

SOLUTION

Here’s the quick way to add fractions:

SUBTRACTING FRACTIONS

EXAMPLE 1

EXAMPLE 2

MULTIPLYING FRACTIONS

When multiplying fractions, always try to reduce first.

EXAMPLE 1

CALCULATING PERCENTS—IT IS SOMETIMES EASIER TO MULTIPLY RATHER THAN DIVIDE

EXAMPLE 1

SUBTRACTING LARGE NUMBERS

EXAMPLE 1

11 — 98 = ?

You can do this mentally (not on paper) by saying to yourself:

112 — 100 = 12

100 — 98 = 2

Now just add 12 and 2 to get 14, and that’s the answer.

The reason this works is because 112 — 100 + 100 — 98 = 112 — 98.

EXAMPLE 2

What is 72 — 39?

Solution:

72 — 42 = 30

42 — 39 = 3

30 + 3 = 33 (answer)

Other Method:

72 — 40 = 32

40 — 39 = 1

32 + 1 = 33

Get the gist?

MULTIPLYING FRACTIONS

EXAMPLE 1

EXAMPLE 2

After you have shown your child the math shortcuts just presented, have him or her try the following exercises.

EXERCISES

Questions 1—3. Which is greater?

Questions 4—5. Add:

Questions 6—8. Subtract:

Questions 9—10. Multiply:

Questions 11—14. Find what percent these fractions are:

Questions 15—17. Subtract:

Questions 18-21. Multiply:

Questions 22-23. Divide:

2240 by .25

23700 by 50

SOLUTION

The Thirty-two Basic Math Problems for Grades 6 to 9

Here are thirty-two of the most basic math problems (for grades 6 to 9), which your child should know how to solve. After your child finishes these, check to see whether they were done correctly by comparing his or her approaches and answers with the approaches and answers given following these questions. However, don’t expect your child to have every answer, or even to be able to do all of the problems, especially the last ones, because your child may not have learned in school some of the material applicable to those questions.

Solutions to the Thirty-two Basic Math Problems