Mathematics Free Sample | TruePath Academics

Mixed Order Operations with Exponents

Evaluate: 18 + 72 ÷ 3².

Choose your answer

A 90

B 8

C 42

D 26

E 27

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A strategy for this skill

A zero exponent means the value is 1: a⁰ = 1 when a is not 0.
A negative exponent means reciprocal: a^-1 = 1a.
Decimal reciprocals can be quick: 0.5^-1 = 2, 0.25^-1 = 4, and 0.1^-1 = 10.
An exponent of -0.5 means reciprocal square root: a^-0.5 = 1√a for positive perfect squares.
A whole-number coefficient can sometimes be rewritten as a power of the same base, such as 81 = 3⁴.
With the same base, multiplication adds exponents and division subtracts exponents.
When two squared values are divided, compare the bases first: a² ÷ b² = (a ÷ b)².

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Evaluate: 12 + 56 ÷ 2³.

Evaluate the exponent first: 2³ = 8.
Then divide before adding: 56 ÷ 8 = 7.
Finally add 12 + 7 = 19.

Answer to this example only: 19

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Metric Speed Conversions

30 meters per second (m/s) = _____ kilometers per hour.

Choose your answer

A 60 km/h

B 30 km/h

C 108 km/h

D 8 13 km/h

E 118 km/h

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A strategy for this skill

1 kilometer = 1000 meters.
1 hour = 3600 seconds.
km/h to m/s: multiply by 518.
m/s to km/h: multiply by 185.

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40 meters per second (m/s) = _____ kilometers per hour.

To reverse the conversion, multiply meters per second by 36001000, which reduces to 185.
Then 40 × 185 = 144 km/h.

Answer to this example only: 144 km/h

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Angle Bisector Incenter Angles

Consider isosceles triangle ABC with AB = 32 and CB = 32. Segment AD bisects ∠BAC, segment CD bisects ∠BCA, and m∠ABC = 48°. What is m∠ADC?

Choose your answer

A 48°

B 90°

C 114°

D 132°

E 138°

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A strategy for this skill

An angle bisector splits an angle into two equal parts.
When two internal angle bisectors meet inside a triangle, their intersection is the incenter.
If D is the incenter, then ∠ADC = 90° + ∠B2.

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Consider isosceles triangle ABC with AB = 38 and CB = 38. Segment AD bisects ∠BAC, segment CD bisects ∠BCA, and m∠ABC = 58°. What is m∠ADC?

The two angle bisectors meet at the incenter.
For an incenter, the angle between the bisectors from A and C is 90° + half of angle B.
So m∠ADC = 90° + 582° = 119°.

Answer to this example only: 119°

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Regrouping Before Summing

Evaluate: (4 + 24 + 44 + . . . + 104) + (16 + 36 + 56 + . . . + 116).

Choose your answer

A 700

B 396

C 720

D 324

E 840

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A strategy for this skill

Changing the order of addition does not change the sum.
Parentheses can be moved around addition when every term keeps its sign.
A negative term such as -8 means add negative eight.

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Evaluate: (1 + 11 + 21 + . . . + 61) + (19 + 29 + 39 + . . . + 79).

Pair matching terms from each group first.
Each pair makes a friendlier number, then those pair sums are easier to add.

Answer to this example only: 560

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Annulus Shaded Area Percent

In the diagram, the smaller circle has radius R and the larger circle has radius 1.5R. What percent of the larger circle is shaded?

Choose your answer

A 65 59%

B 55 59%

C 60 59%

D 45 59%

E 50 59%

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A strategy for this skill

Circle area = πr².
If radius is multiplied by k, area is multiplied by k².
Shaded fraction = larger area - smaller arealarger area.

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In the diagram, the smaller circle has radius R and the larger circle has radius 2.5R. What percent of the larger circle is shaded?

Circle areas compare by the square of the radius.
The larger area is (2.5R)² = 6 14 times the smaller circle's area.
The shaded part is larger area minus smaller area, so the shaded fraction of the larger circle is larger - smallerlarger = 2125.
As a percent, that is 84%.

Answer to this example only: 84%

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Mathematics

Evaluate: 18 + 72 ÷ 3².

A strategy for this skill

30 meters per second (m/s) = _____ kilometers per hour.

A strategy for this skill

Consider isosceles triangle ABC with AB = 32 and CB = 32. Segment AD bisects ∠BAC, segment CD bisects ∠BCA, and m∠ABC = 48°. What is m∠ADC?

A strategy for this skill

Evaluate: (4 + 24 + 44 + . . . + 104) + (16 + 36 + 56 + . . . + 116).

A strategy for this skill

In the diagram, the smaller circle has radius R and the larger circle has radius 1.5R. What percent of the larger circle is shaded?

A strategy for this skill

You took the first step.

Find your child’s next step.

Start this grade’s sample?

Evaluate: 18 + 72 ÷ 32.

A strategy for this skill

30 meters per second (m/s) = _____ kilometers per hour.

A strategy for this skill

Consider isosceles triangle ABC with AB = 32 and CB = 32. Segment AD bisects ∠BAC, segment CD bisects ∠BCA, and m∠ABC = 48°. What is m∠ADC? A B C D 48°

A strategy for this skill

Evaluate: (4 + 24 + 44 + . . . + 104) + (16 + 36 + 56 + . . . + 116).

A strategy for this skill

In the diagram, the smaller circle has radius R and the larger circle has radius 1.5R. What percent of the larger circle is shaded? R 1.5R

A strategy for this skill

You took the first step.

Find your child’s next step.

Evaluate: 18 + 72 ÷ 3².

Consider isosceles triangle ABC with AB = 32 and CB = 32. Segment AD bisects ∠BAC, segment CD bisects ∠BCA, and m∠ABC = 48°. What is m∠ADC?

In the diagram, the smaller circle has radius R and the larger circle has radius 1.5R. What percent of the larger circle is shaded?