Sentence Frame: "These cards match because dividing by a fraction is the same as multiplying by its reciprocal..." or "Subtracting a negative has the same value as adding a positive..."
Unit 5, Lesson 13: Expressions with Rational Numbers Page 1 of 3
Instructional Delivery
Visual & Algebraic Reasoning
12 Minutes • Independent & Discussion
Students substitute rational values into 6 algebraic forms: \(-a\), \(-4b\), \(-a+b\), \(a \div -b\), \(a^2\), and \(b^3\).
| \(a\) | \(b\) | \(-a\) | \(-4b\) | \(-a+b\) | \(a \div -b\) | \(a^2\) | \(b^3\) |
|---|---|---|---|---|---|---|---|
| \(-\frac{1}{2}\) | \(6\) | \(+\frac{1}{2}\) | \(-24\) | \(6\frac{1}{2}\) | \(+\frac{1}{12}\) | \(\frac{1}{4}\) | \(216\) |
| \(\frac{1}{2}\) | \(-6\) | \(-\frac{1}{2}\) | \(+24\) | \(-6\frac{1}{2}\) | \(+\frac{1}{12}\) | \(\frac{1}{4}\) | \(-216\) |
| \(-6\) | \(-\frac{1}{2}\) | \(+6\) | \(+2\) | \(5\frac{1}{2}\) | \(-12\) | \(36\) | \(-\frac{1}{8}\) |
Magnitude vs. Signed Value:
Emphasize that "smallest value" means furthest left on the number line (\(-216\)), while "closest to 0" refers to absolute value regardless of sign.
Extension Prompt:
If \(a = 0\) and \(b = 0\), all expressions evaluate to 0 (except \(a \div -b\) which is undefined).
12 Minutes • Guided Visual Modeling
Vertical Axis Map
\(g = a - b\) (Goose: highest)
\(d = -b\) (Dragonfly: \(> 0\))
\(a\) Seagull (\(>0\))
\(e = \frac{1}{4}a\) (Eagle)
\(0\) Sea Level
\(c = \frac{-a}{2}\) (Clownfish)
\(b\) Shark (\(<0\))
\(j = 2b\) (Jellyfish: deep)
Relative positions with \(a > 0, b < 0\)
1. Dragonfly (\(d = -b\)): Since \(b < 0\), \(-b\) is positive. Dragonfly is above sea level by the exact depth of the shark.
2. Jellyfish (\(j = 2b\)): Multiplying negative depth by positive 2 yields twice the negative distance. Jellyfish is twice as deep as the shark.
3. Eagle (\(e = \frac{1}{4}a\)): One-fourth of the seagull's positive height. Flying lower than the seagull, but still above 0.
4. Clownfish (\(c = \frac{-a}{2}\)): \(-a\) is negative; dividing by 2 leaves it negative. Swimming below sea level at half the seagull's elevation.
5. Goose (\(g = a - b\)): Subtracting negative \(b\) equals adding its magnitude: \(a + |b|\). Goose is higher than the seagull!
Facilitation Move for Vulture (\(v = a + b\)):
Ask: "Does the vulture fly above sea level or swim underwater?" Guide students to see that if \(a > |b|\), the sum is positive; if the shark is deeper than the seagull is high (\(|b| > a\)), the sum would be negative!
Unit 5, Lesson 13: Expressions with Rational Numbers Page 2 of 3
Assessment & Solutions
Practice Problems & Exit Ticket
Extension / Fluency Check
1. \(24 \text{ [?]} \frac{3}{4} = 18\) \(\cdot\) (Multiply)
2. \(24 \text{ [?]} -\frac{3}{4} = -32\) \(\div\) (Divide)
3. \(12 \text{ [?]} 15 = -3\) \(-\) (Subtract)
4. \(12 \text{ [?]} -15 = 27\) \(-\) (Subtract)
5. \(-18 \text{ [?]} -\frac{3}{4} = 24\) \(\div\) (Divide)
13.1 Math Talk Solutions:
13.2 Card Sort Equivalent Clusters:
Lesson 13 Practice Problems (Attachment 2):
1. If \(x = -\frac{1}{4}\):
\(x = -0.25\); \(1 - x = 1.25\); \(x - 1 = -1.25\); \(-1 \div x = 4\).
Order: \(x - 1 < x < 1 - x < -1 \div x\)
2. Sample expressions evaluating to \(-\frac{3}{4}\):
• Sum: \(-\frac{1}{4} + (-\frac{1}{2})\) • Diff: \(\frac{1}{4} - 1\)
• Product: \((3)(-\frac{1}{4})\) • Quotient: \(-\frac{3}{2} \div 2\)
• Two ops: \(-1 + (2)(\frac{1}{8})\)
3. Values:
a. \(-22 + 5 = -17\) b. \(-22 - (-5) = -17\)
c. \((-22)(-5) = 110\) d. \(-22 \div 5 = -4.4\)
Slide 10 Exit Ticket Solutions:
Common Misconceptions & Teacher Moves
"Negative signs always mean smaller value": Students confuse distance with value. Use the horizontal or vertical axis: numbers further right/higher are greater.
"The negative symbol means negative number": \(-x\) means "the opposite of \(x\)". If \(x\) is already negative, \(-x\) is positive!
Unit 5, Lesson 13: Expressions with Rational Numbers Page 3 of 3
CARD B: \(-10 - (-7)\)
Value: -3
\(-(-7)\) becomes \(+7\), identical to Card A!
CARD C: \(-10 + (-7)\)
=
CARD D: \(-10 - 7\)
Value: -17
Both start at \(-10\) and decrease by 7!
Takeaway: Subtracting a number is always the same as adding its opposite!
Round 3
DOUBLE POINTS: 200 Pts Each
CARD A Tile #9
\(8 \div 4\)
Find the reciprocal product!
CARD B Tile #10
\(8 \div (-4)\)
Find the reciprocal product!
CARD C Tile #11
\((8)(\frac{1}{4})\)
Find the equivalent division!
CARD D Tile #12
\((8)(-\frac{1}{4})\)
Find the equivalent division!
Rule Reminder: Dividing by a number equals multiplying by its reciprocal!
Lock in your answers!
Round 3 Reveal
+400 Points Total Available
CARD A: \(8 \div 4\)
=
CARD C: \((8)(\frac{1}{4})\)
Value: +2
Dividing by 4 is multiplying by one-fourth.
CARD B: \(8 \div (-4)\)
=
CARD D: \((8)(-\frac{1}{4})\)
Value: -2
Positive divided/multiplied by negative is negative!
Also True: \((1)(4) = 4\) matches \(1 \div \frac{1}{4} = 4\), and \(-1 \cdot 4 = -4\) matches \(1 \div (-\frac{1}{4}) = -4\)!
Round 4 • Boss Level
TRIPLE POINTS: 300 Pts Each
CARD A Boss Tile #13
\(-15 \div (-6)\)
Look closely at the signs!
CARD B Boss Tile #14
\(15 \div (-6)\)
Look closely at the signs!
CARD C Boss Tile #15
\(15 \cdot \frac{1}{6}\)
Positive or negative?
CARD D Boss Tile #16
\(-15 \cdot \frac{1}{6}\)
Positive or negative?
Who will take the crown? Which pairs match?
Final guesses locked!
Round 4 Reveal
+600 Points Total Available
CARD A: \(-15 \div (-6)\)
=
CARD C: \(15 \cdot \frac{1}{6}\)
Value: +2.5
Neg \(\div\) Neg is Positive, exactly like Pos \(\times\) Pos!
CARD B: \(15 \div (-6)\)
=
CARD D: \(-15 \cdot \frac{1}{6}\)
Value: -2.5
One negative factor or divisor makes the value \(-2.5\)!
Mastery Rule: A fraction bar or reciprocal product always preserves the sign of the quotient!
Game Over
Champions of Signed Numbers
Count the negative signs. An odd number of negatives yields negative; an even number yields positive!
Dividing by a number is always identical to multiplying by its reciprocal: \(a \div b = a \cdot \frac{1}{b}\).
Subtracting a value moves left; subtracting a negative flips direction and moves right: \(a - (-b) = a + b\).
Discussion Question: Which round challenged your team the most, and what rule solved it?
Tally your team scores!
