Mathematics Grade 6 - Ratios, Rates, Integers, Expressions

Welcome to Grade 6 Math!

This year, you'll learn powerful mathematical tools that have been used for thousands of years! From the ancient Hebrews to modern builders, these concepts help us understand Yahuah's ordered creation.

"Yahuah by wisdom has founded the earth; by understanding has He established the heavens." - Proverbs 3:19
Mathematics reveals the wisdom and order in Yahuah's creation!

What You'll Learn:

Unit 1: Ratios & Proportions
Unit 2: Rates & Unit Rates
Unit 3: Integers & the Number Line
Unit 4: Algebraic Expressions

1Introduction to Ratios

A RATIO compares two quantities.
We can write ratios three ways:

Using a colon: 3 : 2
Using the word "to": 3 to 2
As a fraction: 3/2

"Make thee an ark... The length of the ark shall be 300 cubits, the breadth 50 cubits, and the height 30 cubits." - Genesis 6:14-15
Noah's Ark ratio: Length : Width : Height = 300 : 50 : 30 = 6 : 1 : 0.6
This is the perfect ratio for a stable ship! Yahuah knew best!

Example: In a garden, there are 4 tomato plants and 6 pepper plants.

Ratio of tomatoes to peppers = 4 : 6 or 4 to 6 or 4/6

Simplified: 2 : 3 (divide both by 2)

Practice Problems

1. Write each ratio THREE ways:

a) 5 apples to 3 oranges:

Colon: To: Fraction:

b) 12 boys to 8 girls (simplify first):

Simplified ratio:

2. Noah's Ark:

The ratio of length to width was 300 : 50. Simplify this ratio:

3. Biblical Application:

Yahusha fed 5,000 men with 5 loaves and 2 fish (Matthew 14:17-21).
What is the ratio of loaves to fish?
Answer:

4. In a flock of 24 sheep:

If 18 are white and 6 are black, what is the ratio of white to black sheep (simplified)?

Family Activity

Find ratios in recipes (e.g., 2 cups flour : 1 cup sugar) Discuss the perfect proportions Yahuah used in creation Calculate ratios of items in your home

2Equivalent Ratios

EQUIVALENT RATIOS are ratios that have the same value.
To make equivalent ratios, multiply or divide BOTH parts by the same number.

Example: 2 : 3 is equivalent to:

2 : 3 = 4 : 6 (multiplied both by 2)
2 : 3 = 6 : 9 (multiplied both by 3)
2 : 3 = 10 : 15 (multiplied both by 5)

"A tithe of all things" - Leviticus 27:30
A tithe is a ratio of 1 : 10 (one-tenth). If you have $100, your tithe is $10.
1 : 10 = 10 : 100 = 50 : 500 (all equivalent!)

Ratio Tables

Sheep	3	6	9	12	15
Goats	2	4	6	8	10

All columns show the same ratio: 3 : 2

Practice Problems

1. Complete the equivalent ratios:

a) 3 : 5 = 6 :

b) 4 : 7 = 12 :

c) 5 : 2 = : 8

d) 6 : 9 = 2 :

2. Complete the ratio table:

Loaves	5	10	15
Fish	2	4		8

3. Tithe Problems:

If the tithe ratio is 1 : 10, fill in the amounts:
Income $50 → Tithe $
Income $200 → Tithe $
Income $750 → Tithe $

4. Are these ratios equivalent? Write YES or NO:

a) 3 : 4 and 9 : 12

b) 2 : 5 and 4 : 9

c) 6 : 8 and 3 : 4

Family Activity

Practice calculating tithes from different amounts Double a recipe using equivalent ratios Create your own ratio table

3Rates and Unit Rates

A RATE compares two quantities with different units.
A UNIT RATE has a denominator of 1.

Example: If you walk 12 miles in 3 hours, what is your rate?

Rate = 12 miles / 3 hours = 4 miles per hour (unit rate!)

The word "per" means "for each one" - 4 miles FOR EACH hour.

"Six days you shall labor and do all your work, but the seventh day is a Sabbath to Yahuah." - Exodus 20:9-10
Work rate: 6 work days per 7 days = about 0.86 work days per day
Yahuah designed rest into our work pattern!

Common Unit Rates

Rate	What It Measures
Miles per hour (mph)	Speed
Price per pound	Cost
Words per minute	Reading/Typing speed
Beats per minute	Heart rate

Practice Problems

1. Find the unit rate:

a) 150 miles in 3 hours = miles per hour

b) $24 for 6 pounds = $ per pound

c) 200 words in 4 minutes = words per minute

d) 72 beats in 1 minute = beats per minute

2. Which is the better deal?

Store A: 5 loaves of bread for $15
Store B: 3 loaves of bread for $8.40

Store A unit price: $ per loaf
Store B unit price: $ per loaf

Better deal: Store

3. Biblical Word Problem:

The Israelites walked about 15 miles in 5 hours through the wilderness.
What was their rate of travel?

miles per hour

4. Sabbath Math:

Jacob works 6 days a week and earns $480.
What is his daily rate?

$ per day

Family Activity

Compare unit prices at the grocery store Calculate your reading rate (words per minute) Find travel times using speed and distance

4Proportions

A PROPORTION is an equation showing two equivalent ratios.

a/b = c/d or a : b = c : d

Cross Products Rule: In a proportion, cross products are equal!
If a/b = c/d, then a × d = b × c

Example: Solve for x: 3/4 = x/20

Cross multiply: 3 × 20 = 4 × x
60 = 4x
x = 60 ÷ 4 = 15

Check: 3/4 = 15/20? Yes! (Both simplify to 3/4)

"And the Tabernacle was proportioned according to the pattern shown to Moses on the mountain." - Based on Exodus 25:9
Yahuah gave exact proportions for His dwelling place!

Practice Problems

1. Solve for the unknown:

a) 2/5 = x/15 x =

b) 3/4 = 12/y y =

c) 5/n = 10/6 n =

d) m/8 = 9/12 m =

2. Set up and solve:

If 3 scrolls cost 15 shekels, how much do 7 scrolls cost?

Set up: 3/ = 7/x
Cross multiply and solve: x = shekels

3. Biblical Proportion:

The ratio of clean animals to unclean animals on the ark was 7:2.
If there were 14 pairs of clean cattle, how many pairs of unclean animals of that kind?

7/2 = 14/x
x = pairs

4. Scale Drawing:

On a map, 1 inch represents 50 miles.
If two cities are 3.5 inches apart on the map, how far apart are they in real life?

1/50 = 3.5/x
x = miles

Family Activity

Use proportions to resize a recipe Measure distances on a map and calculate real distances Discuss proportions in building (blueprints, models)

5Introduction to Integers

INTEGERS are whole numbers that include positive numbers, negative numbers, and zero.

-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5

The Number Line

"Deep calls unto deep at the noise of Your waterfalls." - Psalm 42:7
Just as there are heights above sea level (+) and depths below (-), integers let us measure both directions from a starting point!

Real-World Integers

Situation	Positive (+)	Negative (-)
Temperature	Above zero	Below zero
Elevation	Above sea level	Below sea level
Money	Deposit/gain	Withdrawal/debt
Time	After event (AD)	Before event (BC)

Practice Problems

1. Write an integer for each situation:

a) 7 degrees below zero:

b) A deposit of $50:

c) 200 feet below sea level:

d) A gain of 15 points:

e) 3 years before Messiah (3 BC):

2. Compare using < or > or =:

a) -5 -3

b) 4 -7

c) -2 0

d) -8 -8

3. Order from least to greatest:

-3, 5, -7, 0, 2, -1

Answer:

4. Biblical Application:

The Dead Sea is about 1,400 feet below sea level.
Mt. Sinai is about 7,500 feet above sea level.

Dead Sea elevation: feet
Mt. Sinai elevation: feet
Difference in elevation: feet

Family Activity

Check weather temperatures for positive and negative readings Discuss Bible timeline using BC (-) and AD (+) Find elevations of places mentioned in Scripture

6Adding Integers

Rules for Adding Integers:

Same signs: Add the absolute values, keep the sign.

(+5) + (+3) = +8
(-5) + (-3) = -8

Different signs: Subtract the absolute values, use the sign of the larger number.

(+5) + (-3) = +2
(-5) + (+3) = -2

Think of a number line:
Start at the first number, then move:

RIGHT for positive (+)
LEFT for negative (-)

Example: (-3) + (+7)
Start at -3, move 7 to the right → land on +4

"Give, and it shall be given unto you... for with the same measure you use, it shall be measured to you again." - Luke 6:38
Giving (-) and receiving (+) work together in Yahuah's economy!

Practice Problems

1. Add these integers:

(+4) + (+6) =

(-5) + (-8) =

(+9) + (-4) =

(-7) + (+3) =

(-6) + (+6) =

(+8) + (-11) =

2. Word Problems:

The temperature at sunrise was -5°F. By noon, it rose 12 degrees.
What was the temperature at noon?

(-5) + (+12) = °F

Joseph's account had $75. He spent $90 on supplies.
What is his account balance now?

(+75) + (-90) = (This means he owes $)

3. Find the sum:

a) (-3) + (+8) + (-2) =

b) (+5) + (-9) + (+4) =

Family Activity

Track temperature changes over a day using integers Practice with a checking account (deposits and withdrawals) Use a number line to visualize addition

7Subtracting Integers

The Key Rule: To subtract an integer, ADD ITS OPPOSITE!

a - b = a + (-b)

Examples:

5 - 8 = 5 + (-8) = -3
-3 - 5 = -3 + (-5) = -8
-4 - (-7) = -4 + (+7) = 3

The tricky one: Subtracting a negative!
-4 - (-7) means "take away negative 7"
Taking away a negative is like ADDING a positive!
-4 - (-7) = -4 + 7 = 3

"If your brother sins against you, go and tell him his fault... If he listens, you have gained your brother." - Matthew 18:15
Removing (-) a negative (-) situation = positive outcome!

Practice Problems

1. Rewrite as addition, then solve:

a) 6 - 9 = 6 + =

b) -4 - 3 = -4 + =

c) -2 - (-8) = -2 + =

d) 7 - (-3) = 7 + =

2. Solve directly:

8 - 12 =

-5 - 7 =

-9 - (-4) =

3 - (-6) =

3. Word Problem:

The elevation of the Dead Sea is -1,400 feet.
The elevation of Jerusalem is +2,500 feet.

How much higher is Jerusalem than the Dead Sea?

(+2,500) - (-1,400) = (+2,500) + (+1,400) = feet

4. Temperature Change:

At midnight the temperature was -8°F.
By morning it was -15°F.
How much did the temperature change?

(-15) - (-8) = °F

Family Activity

Calculate elevation differences between biblical locations Practice "add the opposite" rule with flashcards Track weekly temperature highs and lows and find differences

8Multiplying & Dividing Integers

The Sign Rules (same for both operations!):

Signs	Result
(+) × (+) or (+) ÷ (+)	POSITIVE
(-) × (-) or (-) ÷ (-)	POSITIVE
(+) × (-) or (+) ÷ (-)	NEGATIVE
(-) × (+) or (-) ÷ (+)	NEGATIVE

Remember: Same signs = Positive, Different signs = Negative

Examples:

(+4) × (+3) = +12
(-4) × (-3) = +12
(+4) × (-3) = -12
(-4) × (+3) = -12

Division works the same way!

(-12) ÷ (-3) = +4
(-12) ÷ (+3) = -4

"Be fruitful and multiply." - Genesis 1:28
Multiplication is foundational to Yahuah's plan for growth and abundance!

Practice Problems

1. Multiply:

(+5) × (+7) =

(-6) × (-4) =

(+8) × (-3) =

(-9) × (+2) =

2. Divide:

(+24) ÷ (+6) =

(-36) ÷ (-9) =

(+45) ÷ (-5) =

(-56) ÷ (+8) =

3. Mixed Practice:

a) (-7) × (-8) =

b) (-48) ÷ (+6) =

c) (+9) × (-6) =

d) (-63) ÷ (-7) =

4. Word Problem:

A submarine descends 30 feet per minute for 4 minutes.
Descent = -30 feet per minute

Total change in depth: (-30) × 4 = feet

Family Activity

Create sign rule flashcards Practice integer operations with real-world examples Quiz each other on sign rules

9Introduction to Algebraic Expressions

An ALGEBRAIC EXPRESSION uses numbers, variables (letters), and operations.

Key Terms:

Variable: A letter that represents an unknown number (x, y, n)
Coefficient: The number multiplied by a variable (in 3x, the 3 is the coefficient)
Constant: A number without a variable (in 3x + 5, the 5 is a constant)
Term: Parts of an expression separated by + or - (3x + 5 has 2 terms)

Expression: 4x + 7

Variable: x
Coefficient: 4
Constant: 7
Number of terms: 2

If x = 3: 4(3) + 7 = 12 + 7 = 19

"For I know the plans I have for you, declares Yahuah, plans for welfare and not for evil, to give you a future and a hope." - Jeremiah 29:11
Yahuah knows all the "variables" in our lives. We can trust His plan even when we don't know what 'x' will be!

Writing Expressions

Words	Expression
A number plus 5	n + 5
7 times a number	7n
A number divided by 3	n ÷ 3 or n/3
4 less than a number	n - 4
Twice a number plus 6	2n + 6

Practice Problems

1. Identify the parts of 5y + 12:

Variable:

Coefficient:

Constant:

Number of terms:

2. Write an algebraic expression:

a) 8 more than a number:

b) 5 times a number:

c) A number decreased by 9:

d) A number divided by 4:

e) Triple a number plus 2:

3. Evaluate the expression when x = 4:

a) 3x + 1 =

b) 2x - 5 =

c) x² + 2 =

4. Biblical Word Problem:

Jacob worked for Laban for a number of years (y) for Leah, then the same number of years for Rachel.
Write an expression for the total years Jacob worked:

If y = 7, how many total years? years

Family Activity

Write expressions for everyday situations Practice evaluating expressions with different values Find patterns in Scripture using algebraic thinking

10Review: Putting It All Together

Key Concepts Summary

RATIOS & RATES

Ratio: compares two quantities (3:4 or 3/4)
Rate: compares quantities with different units (miles per hour)
Proportion: equation showing equal ratios (3/4 = 6/8)

INTEGERS

Adding: Same signs = add & keep sign; Different signs = subtract & use larger's sign
Subtracting: Add the opposite!
Multiplying/Dividing: Same signs = positive; Different signs = negative

ALGEBRAIC EXPRESSIONS

Variables represent unknown numbers
Coefficients are numbers multiplied by variables
Constants are numbers without variables

Comprehensive Review

1. Ratios:

Write the ratio of 15 to 25 in simplest form:

2. Unit Rate:

$36 for 4 items = $ per item

3. Proportion:

Solve: 4/5 = x/20 x =

4. Integer Operations:

a) (-8) + (+5) =

b) 6 - (-3) =

c) (-7) × (-4) =

d) (-42) ÷ (+6) =

5. Expressions:

Write: "6 less than twice a number":

Evaluate 4n - 3 when n = 5:

6. Biblical Application:

Yahusha sent out 72 disciples in pairs (Luke 10:1).
What is the ratio of disciples to pairs?
How many pairs were sent?

Celebration!

Complete the comprehensive test Review any topics that need extra practice Prepare for Grade 7 Pre-Algebra! Thank Yahuah for the gift of mathematical thinking!

Answer Key

Lesson 1: 1a) 5:3, 5 to 3, 5/3; 1b) 3:2; 2) 6:1; 3) 5:2; 4) 3:1

Lesson 2: 1a) 10; 1b) 21; 1c) 20; 1d) 3; 2) 20, 6, 8; 3) $5, $20, $75; 4) YES, NO, YES

Lesson 3: 1a) 50 mph; 1b) $4; 1c) 50 wpm; 1d) 72 bpm; 2) $3, $2.80, Store B; 3) 3 mph; 4) $80

Lesson 4: 1a) 6; 1b) 16; 1c) 3; 1d) 6; 2) 15, 35 shekels; 3) 4 pairs; 4) 175 miles

Lesson 5: 1a) -7; 1b) +50; 1c) -200; 1d) +15; 1e) -3; 2a) <; 2b) >; 2c) <; 2d) =; 3) -7, -3, -1, 0, 2, 5; 4) -1,400; +7,500; 8,900

Lesson 6: 1) +10, -13, +5, -4, 0, -3; 2) +7°F; -$15, $15; 3a) +3; 3b) 0

Lesson 7: 1a) (-9), -3; 1b) (-3), -7; 1c) (+8), +6; 1d) (+3), +10; 2) -4, -12, -5, +9; 3) 3,900 feet; 4) -7°F

Lesson 8: 1) +35, +24, -24, -18; 2) +4, +4, -9, -7; 3a) +56; 3b) -8; 3c) -54; 3d) +9; 4) -120 feet

Lesson 9: 1) y, 5, 12, 2; 2a) n+8; 2b) 5n; 2c) n-9; 2d) n/4; 2e) 3n+2; 3a) 13; 3b) 3; 3c) 18; 4) 2y or y+y, 14 years

Lesson 10: 1) 3:5; 2) $9; 3) 16; 4a) -3; 4b) 9; 4c) 28; 4d) -7; 5) 2n-6, 17; 6) 72:36 or 2:1, 36 pairs

Welcome to Grade 6 Math!

What You'll Learn:

1Introduction to Ratios

Practice Problems

Family Activity

2Equivalent Ratios

Ratio Tables

Practice Problems

Family Activity

3Rates and Unit Rates

Common Unit Rates

Practice Problems

Family Activity

4Proportions

Practice Problems

Family Activity

5Introduction to Integers

The Number Line

Real-World Integers

Practice Problems

Family Activity

6Adding Integers

Practice Problems

Family Activity

7Subtracting Integers

Practice Problems

Family Activity

8Multiplying & Dividing Integers

Practice Problems

Family Activity

9Introduction to Algebraic Expressions

Writing Expressions

Practice Problems

Family Activity

10Review: Putting It All Together

Key Concepts Summary

Comprehensive Review

Celebration!

Answer Key

Congratulations!