What we need to be able to do

We are not redoing all of 6th grade. We are rebuilding the few skills we need right now.

1

Fractions & Mixed Numbers

Convert forms and know when denominators must match.

2

Decimals

Align decimal points and use place value correctly.

3

Operations

Add, subtract, multiply, and divide fractions.

4

Positive & Negative

Apply the sign rules after the fraction skill is set up correctly.

Teacher use: move quickly. Stop only where students show a real prerequisite gap.

Fraction basics

Numerator = TOP

3 / 5

The numerator tells how many parts we have.

Denominator = BOTTOM

3 / 5

The denominator tells how many equal parts make one whole.

Whole number as a fraction

Any whole number can be written over 1.

4 = 4/1 -3 = -3/1

Mixed number → improper fraction

Classroom nickname: the “Texas Method.” Technical skill: convert a mixed number to an improper fraction.

The rule

BOTTOM × WHOLE + TOP
Keep the BOTTOM

Example

2 1/3

1. 3 × 2 = 6
2. 6 + 1 = 7
3. Keep denominator 3

2 1/3 = 7/3

Work these together

3 1/4 = ?

13/4

5 2/3 = ?

17/3

-2 3/5 = ?

-13/5

Improper fraction → mixed number

We mainly need this when an answer is bigger than 1.

The rule

TOP ÷ BOTTOM

Quotient = whole number. Remainder = new numerator. Keep the same denominator.

Example

11/4

11 ÷ 4 = 2 remainder 3

11/4 = 2 3/4

7/3 = ?

2 1/3

9/4 = ?

2 1/4

14/5 = ?

2 4/5

WORKSHEET PAUSE 1: Students complete Fraction Foundations, then stop.

Fractions and decimals are connected

They are two ways to name the same rational number.

Fraction → decimal

TOP ÷ BOTTOM

3/4 = 3 ÷ 4 = 0.75

2/3 = 0.666...

A decimal either terminates or repeats.

Decimal → fraction

Use place value, then simplify.

0.6 = 6/10 = 3/5

1.25 = 125/100 = 5/4

1/2 = ? decimal

0.5

-7/8 = ? decimal

-0.875

0.35 = ? fraction

35/100 = 7/20

Add & subtract: SAME denominator

Same bottoms? Work with the TOPS.
Keep the denominator.

Add

2/7 + 3/7 = 5/7

Subtract

6/9 - 2/9 = 4/9

Work these together

1/5 + 3/5

4/5

7/8 - 2/8

5/8

-2/3 + 5/3

3/3 = 1

Add & subtract: DIFFERENT denominators

Different bottoms? STOP.
Make the denominators match first.

Example

1/2 + 1/4

1/2 = 2/4
2/4 + 1/4 = 3/4

Equivalent fraction rule

Whatever you multiply the bottom by, multiply the top by the same number.

1/3 × 2/2 = 2/6

1/2 + 1/3

3/6 + 2/6 = 5/6

3/4 - 1/2

3/4 - 2/4 = 1/4

2/3 + 1/6

4/6 + 1/6 = 5/6

WORKSHEET PAUSE 2: Students complete Add & Subtract Fractions, then stop.

Add & subtract decimals

Rewrite subtraction if needed.
LINE UP the decimal points.
Add zeros as placeholders.

Different signs

(-8.1) - (-8.9)

Add the opposite: -8.1 + 8.9
Subtract magnitudes: 8.9 - 8.1 = 0.8
The larger magnitude was positive.

Answer: 0.8

Unequal decimal places

3.305 + 1.7

Rewrite 1.7 as 1.700.
Line up the decimal points.
3.305 + 1.700 = 5.005

2.9 - 9.4

-6.5

-4.3 + 14.5

10.2

6 - 2.75

6.00 - 2.75 = 3.25

Positive & negative rational numbers

First do the fraction skill correctly. Then apply the sign rule.

Situation	What to do	Example
Same signs	Add. Keep the sign.	-2 + -4 = -6
Different signs	Subtract. Keep the sign of the number farther from 0.	-7 + 3 = -4
Subtracting	Change subtraction to adding the opposite.	6 - (-4) = 6 + 4

Fraction example

2 1/3 + (-1 2/3)

7/3 + (-5/3) = 2/3

Multiply fractions

Multiply TOP × TOP.
Multiply BOTTOM × BOTTOM.

Example

2/3 × 4/5

= 8/15

If there is a mixed number

1 1/2 × 2/3

Convert first: 1 1/2 = 3/2

3/2 × 2/3 = 6/6 = 1

3/4 × 2/5

6/20 = 3/10

-2/3 × 3/5

-6/15 = -2/5

2 1/3 × 3/7

7/3 × 3/7 = 1

Multiply decimals

1. Determine the sign.
2. Multiply as whole numbers.
3. Count the TOTAL decimal places.

Example 1

-5.5 × -4.87

Same signs → positive
55 × 487 = 26,785
Three total decimal places

26.785

Example 2

1.7 × -2.1

Different signs → negative
17 × 21 = 357
Two total decimal places

-3.57

-3.2 × 0.5

-1.6

2.4 × 1.25

3.00 = 3

-0.6 × -0.08

0.048

WORKSHEET PAUSE 3: Students complete Signs & Multiplication, then stop.

Divide fractions

Keep – Change – Flip

Keep the first fraction. Change ÷ to ×. Flip the second fraction.

Example

2/3 ÷ 4/5

2/3 × 5/4 = 10/12 = 5/6

Whole number?

3 = 3/1

Write the whole number over 1 before using the fraction process.

1/2 ÷ 3/4

1/2 × 4/3 = 2/3

3 ÷ 1/2

3/1 × 2/1 = 6

-2/5 ÷ 4/5

-2/5 × 5/4 = -1/2

Divide decimals

1. Determine the sign.
2. Make the DIVISOR a whole number.
3. Move both decimal points the same distance.

Example 1

13.96 ÷ 0.4

Move both decimals 1 place:
139.6 ÷ 4

34.9

Example 2

10.6 ÷ 0.2

Move both decimals 1 place:
106 ÷ 2

53

-8.4 ÷ 0.7

-84 ÷ 7 = -12

6.75 ÷ 1.5

67.5 ÷ 15 = 4.5

-4.2 ÷ -0.06

-420 ÷ -6 = 70

WORKSHEET PAUSE 4: Students complete Division & Cumulative Review, then stop.

Teacher-led practice: add & subtract

These mirror the completed assignment: integers, decimals, then fractions.

1) 10 + 3 - (-8)

10 + 3 + 8 = 21

2) (-8.1) - (-8.9)

-8.1 + 8.9 = 0.8

3) 2.9 - 9.4

-6.5

4) 3.305 + 1.7

3.305 + 1.700 = 5.005

5) -3/4 + 1/4

-2/4 = -1/2

6) 2 - 13/8

16/8 - 13/8 = 3/8

7) 1/2 + 1/6

3/6 + 1/6 = 4/6 = 2/3

8) 2 1/3 + (-1 2/3)

7/3 + (-5/3) = 2/3

Teacher-led practice: multiply & divide

Ask students to identify the sign and number form before calculating.

1) -5 × -4 × -10

-200

2) -5.5 × -4.87

26.785

3) 1.7 × -2.1

-3.57

4) 13.96 ÷ 0.4

139.6 ÷ 4 = 34.9

5) -2/3 × 5/4

-10/12 = -5/6

6) -1/2 ÷ 5/4

-1/2 × 4/5 = -2/5

7) -2 ÷ -3 4/5

-2/1 ÷ -19/5 = 10/19

8) 10.6 ÷ 0.2

106 ÷ 2 = 53

Exit ticket

Students should identify the operation, number form, and correct first move before solving.

A) Convert 3 2/5 to an improper fraction.

17/5

B) 2/3 + 1/4

8/12 + 3/12 = 11/12

C) -4.3 + 14.5

10.2

D) -1/2 × 3/4

-3/8

E) 1.7 × -2.1

-3.57

F) 10.6 ÷ 0.2

53

Decision rule: If most students can do A-F with coaching, return to the 7th-grade lesson. If one skill collapses, reteach only that skill for 10 minutes.