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Prealgebra / LEVEL 1 · DIFFICULTY 1/5

Arithmetic & Number Structure

Use signed numbers, operation structure and reciprocals to explain calculations.

3 stages · 24 practice problems · two 6-question assessment forms

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MINI QUEST 1.1Signed Numbers & Order of OperationsRead the lesson
MINI QUEST 1.2Structure & Efficient CalculationRead the lesson
MINI QUEST 1.3Division, Reciprocals & ZeroRead the lesson
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STAGE 1.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Signed Numbers & Order of Operations

Goal: Track direction and group operations before calculating.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

Numbers describe position and change

A negative position is below the chosen zero. A negative change moves toward smaller numbers. Subtracting a change reverses that change.

\displaystyle a - \left(-b\right) = a + b

Signs belong to numbers

Multiplication by a negative reverses direction. Two reversals restore the original direction; a product with an odd number of negative factors is negative.

\displaystyle \left(-a\right)\, \left(-b\right) = ab

Read the structure first

Resolve grouping, then powers, then multiplication and division from left to right, then addition and subtraction from left to right. Equal-priority operations do not jump the queue.

\displaystyle 18 \div 3 \times 2 = 12

WORKED EXAMPLE 1

Evaluate $\displaystyle 7 - 2\, \left(5 - 9\right)$ .

Inside parentheses: $\displaystyle 5 - 9 = -4$ .
Multiply: $\displaystyle 2\, \left(-4\right) = -8$ .
Subtracting −8 adds 8.

WORKED EXAMPLE 2

A sensor reads $\displaystyle -11^{\circ}\, C$ and warms by $\displaystyle 6^{\circ}\, C$ , then cools by $\displaystyle 4^{\circ}\, C$ . What does it read?

Start at −11.
Warming gives −5; cooling gives −9.

$\displaystyle -9^{\circ}\, C$

Common pitfalls

Possible mix-up: Subtracting always makes a number smaller.

Subtracting a negative makes it larger.

Possible mix-up: Multiply before dividing, wherever multiplication appears.

Multiplication and division have equal priority; read left to right.

Explain it to yourself

Explain why $\displaystyle -5 - \left(-8\right)$ and $\displaystyle -5 + \left(-8\right)$ have different signs.

Preview the eight practice prompts

Problem 1
Compute $\displaystyle -14 + 9$ .
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Problem 2
Compute $\displaystyle 6 - \left(-13\right)$ .
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Problem 3
Compute $\displaystyle \left(-7\right)\, \left(-4\right) - 10$ .
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Problem 4
Evaluate $\displaystyle 42 \div 7 \times 3 - 5$ .
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Problem 5
Evaluate $\displaystyle 5 - 3\, \left(2 - 6\right)$ .
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Problem 6
What number n makes $\displaystyle -8 - n = 5$ ?
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Problem 7New context
An elevator starts on floor 4, moves down 9 floors, then up 6. Basement floors are negative and ground is 0. Which floor is it on?
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Problem 8New context
A game adds −6 points for each of three penalties, then removes one of those penalties. Starting from 20 points, what is the final score?
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Open stage 1.1 in the student workspace →

STAGE 1.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Structure & Efficient Calculation

Useful preparation: Signed Numbers & Order of Operations

Goal: Use regrouping and distribution without changing the value.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

Swap or regroup only where justified

Addition and multiplication allow rearrangement and regrouping. Subtraction and division do not generally allow these changes.

\displaystyle a + b = b + a

;

\displaystyle \left(ab\right)\, c = a\, \left(bc\right)

Distribute across every term

A common factor can be spread across a sum, or collected from separate products. The same structure works with subtraction.

\displaystyle a\, \left(b + c\right) = ab + ac

Use a nearby convenient number

Rewrite an awkward factor as a friendly number plus or minus a small correction. The correction must also be multiplied.

\displaystyle 39 \times 12 = \left(40 - 1\right) \times 12

WORKED EXAMPLE 1

Compute $\displaystyle 24 \times 49$ mentally.

Rewrite 49 as $\displaystyle 50 - 1$ .
$\displaystyle 24 \times 50 - 24 = 1200 - 24$ .

1176

WORKED EXAMPLE 2

Find $\displaystyle 17 \times 8 + 17 \times 12$ .

Both terms contain a factor of 17.
Factor to get $\displaystyle 17\, \left(8 + 12\right) = 17 \times 20$ .

340

Common pitfalls

Possible mix-up: $\displaystyle 8 - 3$ equals $\displaystyle 3 - 8$ .

Subtraction is not commutative.

Possible mix-up: $\displaystyle 4\, \left(x + 3\right)$ equals $\displaystyle 4\, x + 3$ .

The factor 4 multiplies both x and 3.

Explain it to yourself

Give two different efficient ways to compute $\displaystyle 25 \times 36$ .

Preview the eight practice prompts

Problem 1
Compute $\displaystyle 25 \times 16 \times 4$ using a useful regrouping.
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Problem 2
Compute $\displaystyle 13 \times 7 + 13 \times 3$ .
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Problem 3
Compute $\displaystyle 62 \times 99$ .
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Problem 4
What number fills the blank: $\displaystyle 8 \times 37 = 8 \times 30 + 8$ × ___?
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Problem 5
Is $\displaystyle \left(24 \div 6\right) \div 2$ equal to $\displaystyle 24 \div \left(6 \div 2\right)$ ? Enter yes or no.
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Problem 6
Compute $\displaystyle 103 \times 18 - 3 \times 18$ .
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Problem 7New context
A makerspace orders 14 packs with 9 red clips and 11 blue clips in each pack. How many clips are ordered?
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Problem 8New context
A warehouse ships 48 boxes. Each should contain 25 tiles, but every box has 2 fewer. How many tiles are shipped?
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Open stage 1.2 in the student workspace →

STAGE 1.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Division, Reciprocals & Zero

Useful preparation: Structure & Efficient Calculation

Goal: Interpret division and identify operations that are undefined.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

MOVE FIRST. WRITE THE MATH AFTER.

Share 12 counters into 3 equal groups

Drag a counter into a group. Or tap a counter, then “Put selected here.” Everyone should get the same number.

One small move: Move one counter from the tray into group 1. It has 0 right now.

Counter tray · 12 left

Group $\displaystyle 1 \cdot 0$

Group $\displaystyle 2 \cdot 0$

Group $\displaystyle 3 \cdot 0$

Move one counter from the tray into group 1. It has 0 right now.

Division asks for a missing factor

The quotient $\displaystyle a \div b$ is the number q satisfying bq = a. This definition requires b to be nonzero.

bq = a ⇔

\displaystyle q = a \div b

\displaystyle b \ne 0

A reciprocal undoes multiplication

A nonzero number and its reciprocal multiply to one. To divide by a nonzero number, multiply by its reciprocal.

\displaystyle a \div b = a \times \left(\frac{1}{b}\right)

Zero has a special boundary

Zero divided by a nonzero number is zero. Division by zero has no unique quotient: no product 0q can equal a nonzero number, and every q gives 0 when the numerator is also zero.

\displaystyle 0 \div 7 = 0

;

\displaystyle 7 \div 0

is undefined

WORKED EXAMPLE 1

Find the reciprocal of $\displaystyle \frac{-5}{8}$ .

Swap the roles of numerator and denominator, keeping the sign.
Check: $\displaystyle \left(\frac{-5}{8}\right)\, \left(\frac{-8}{5}\right) = 1$ .

$\displaystyle \frac{-8}{5}$

WORKED EXAMPLE 2

A robot covers 18 meters in equal moves of $\displaystyle \frac{3}{2}$ meters. How many moves?

Number of moves = total distance ÷ distance per move.
$\displaystyle 18 \div \left(\frac{3}{2}\right) = 18 \times \left(\frac{2}{3}\right)$ .

Common pitfalls

Possible mix-up: The reciprocal of 0 is 0.

No number multiplied by 0 gives 1.

Possible mix-up: Dividing by a number always makes the result smaller.

Dividing a positive number by a positive number less than 1 makes it larger.

Explain it to yourself

Why is $\displaystyle 0 \div 5$ defined but $\displaystyle 5 \div 0$ undefined?

Preview the eight practice prompts

Problem 1
What is the reciprocal of 7? Enter a fraction.
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Problem 2
What is the reciprocal of $\displaystyle \frac{-3}{8}$ ?
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Problem 3
Compute $\displaystyle 15 \div \left(\frac{1}{5}\right)$ .
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Problem 4
Compute $\displaystyle 0 \div \left(-9\right)$ .
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Problem 5
Is $\displaystyle 12 \div 0$ defined? Enter yes or no.
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Problem 6
Find x if $\displaystyle \left(\frac{-4}{7}\right)\, x = 1$ .
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Problem 7New context
A 9-meter roll is cut into pieces of length $\displaystyle \frac{3}{4}$ meter with no waste. How many pieces are made?
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Problem 8New context
A machine multiplies an input by $\displaystyle \frac{2}{7}$ and produces 6. What was the input?
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Open stage 1.3 in the student workspace →

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Arithmetic & Number Structure

Signed Numbers & Order of Operations

Numbers describe position and change

Signs belong to numbers

Read the structure first

Evaluate 7−2 (5−9)\displaystyle 7 - 2\, \left(5 - 9\right)7−2(5−9).

A sensor reads −11∘ C\displaystyle -11^{\circ}\, C−11∘C and warms by 6∘ C\displaystyle 6^{\circ}\, C6∘C, then cools by 4∘ C\displaystyle 4^{\circ}\, C4∘C. What does it read?

Common pitfalls

Explain it to yourself

Structure & Efficient Calculation

Swap or regroup only where justified

Distribute across every term

Use a nearby convenient number

Compute 24×49\displaystyle 24 \times 4924×49 mentally.

Find 17×8+17×12\displaystyle 17 \times 8 + 17 \times 1217×8+17×12.

Common pitfalls

Explain it to yourself

Division, Reciprocals & Zero

Share 12 counters into 3 equal groups

Division asks for a missing factor

A reciprocal undoes multiplication

Zero has a special boundary

Find the reciprocal of −58\displaystyle \frac{-5}{8}8−5​.

A robot covers 18 meters in equal moves of 32\displaystyle \frac{3}{2}23​ meters. How many moves?

Common pitfalls

Explain it to yourself

Evaluate $\displaystyle 7 - 2\, \left(5 - 9\right)$ .

A sensor reads $\displaystyle -11^{\circ}\, C$ and warms by $\displaystyle 6^{\circ}\, C$ , then cools by $\displaystyle 4^{\circ}\, C$ . What does it read?

Compute $\displaystyle 24 \times 49$ mentally.

Find $\displaystyle 17 \times 8 + 17 \times 12$ .

Find the reciprocal of $\displaystyle \frac{-5}{8}$ .

A robot covers 18 meters in equal moves of $\displaystyle \frac{3}{2}$ meters. How many moves?