Tier 1: Foundation Absolute Value & Plotted Distance on Number Line
10 Problems • 1 pt each
-8 -6 -4 -2 0 2 4 6 8 P (-7) Q (-2) R (3) S (7)
1. \(|-19| =\)
2. \(|34| =\)
3. Distance from \(0\) to Point P (\(-7\)):
4. Distance from Point Q (\(-2\)) to R (\(3\)):
5. Distance from Point P (\(-7\)) to Q (\(-2\)):
6. Distance from Point R (\(3\)) to S (\(7\)):
7. Express dist P to R as \(|p - q|\):
8. Evaluate \(-|-15| =\)
9. Value of \(|0| =\)
10. Can \(|a - b|\) ever be negative?
Tier 2: Approaching Absolute Value of Difference: \(|p - q|\) Model
10 Problems • 1 pt each
Distance Formula: \(|(-4) - 5| = |-9| = 9\text{ units}\)
-4 -2 0 2 5
11. Distance from visual \(|-4 - 5| =\)
12. \(8 - 19 =\)
13. \(-14 - 9 =\)
14. \(11 - (-16) =\)
15. \(-21 - (-8) =\)
16. \(6.4 - 10.1 =\)
17. \(-4.5 - 7.9 =\)
18. \(5.8 - (-6.4) =\)
19. Temp: \(-8^\circ\) to \(15^\circ\). Diff:
20. Distance \(-6\) to \(5\): \(|-6-5|\) or \(-6-5\)?
Tier 3: At Standard Rational Differences & Coordinate Distance
10 Problems • 1 pt each
21. \(-\dfrac{7}{8} - \dfrac{1}{4} =\)
22. \(\dfrac{5}{6} - \left(-\dfrac{2}{3}\right) =\)
23. \(-3\dfrac{1}{2} - \left(-1\dfrac{3}{4}\right) =\)
24. \(-\dfrac{3}{5} - \dfrac{9}{10} - \left(-\dfrac{1}{2}\right) =\)
25. Distance between \(-5.6\) and \(4.25\):
26. Elevation \(-92\text{m}\) to \(1,650\text{m}\). Dist:
27. Diver \(-55\text{ft}\) to sub \(-210\text{ft}\). Dist:
28. Find \(|a - b|\) for \(a=-4.6, b=-9.1\):
29. True/False: \(|a - b| = |b - a|\) for all rationals:
30. Dist from \(x\) to \(5\) is \(11\) (\(x < 0\)). \(x=\):
Score Tracker: Tier 1: ___/10 • Tier 2: ___/10 • Tier 3: ___/10 • Total: ___/30 Page 2 of 6
Standard 7.NS.3 Form B
Apply multiplication properties, signed rules (\((-1)(-1)=1\)), and distributive property models.
Name:
Date: Period:
Tier 1: Foundation Number Line Hop Model & Integer Products
10 Problems • 1 pt each
Repeated Jump Model: \(3 \times (-4) = -12\)
-12 -8 -4 0
1. Multiplication modeled by the 3 hops:
2. \(5 \times (-7) =\)
3. \((-6) \times (-8) =\)
4. \((-9) \times 7 =\)
5. \((-1) \times (-1) \times (-1) =\)
6. Sign of \((-2) \times 5 \times (-7)\):
7. Sign of \((-4) \times (-3) \times (-6)\):
8. \((-14) \times 0 =\)
9. \((-3)^3 =\)
10. \((-12) \times (-5) =\)
Tier 2: Approaching Multi-Factor Products & Distribution
10 Problems • 1 pt each
11. \((-2) \times (-5) \times (-6) =\)
12. \((-4) \times 3 \times (-7) =\)
13. \((-2.5) \times (-4) =\)
14. \(6 \times (-3.2) =\)
15. Distribute: \(4(y - 6) =\)
16. Distribute: \(-3(3x + 7) =\)
17. Distribute: \(-6(2m - 5) =\)
18. Distribute: \(-5(8 + (-4)) =\)
19. Evaluate: \((-3)^4 =\)
20. Lose \(\$55\)/day for \(4\) days. Net:
Tier 3: At Standard Rational Products & Rational Distribution
10 Problems • 1 pt each
21. \(\left(-\dfrac{4}{7}\right) \times \left(-\dfrac{14}{16}\right) =\)
22. \(\left(-\dfrac{3}{5}\right) \times \dfrac{10}{21} =\)
23. \(-1\dfrac{2}{3} \times 2\dfrac{2}{5} =\)
24. \((-0.5) \times 3.2 \times (-1.5) =\)
25. Distribute: \(-\dfrac{3}{4}(8x - 20) =\)
26. Distribute: \(-\dfrac{2}{5}(-15y + 35) =\)
27. Distribute: \(-1.6(2k - 4) =\)
28. \(-\dfrac{1}{3}(27 - (-18)) =\)
29. If \(a < 0\) and \(b < 0\), sign of \(a \times b\):
30. Stock drops \(\$3.25\)/hr for \(3.5\) hrs. Net:
Score Tracker: Tier 1: ___/10 • Tier 2: ___/10 • Tier 3: ___/10 • Total: ___/30 Page 3 of 6
Standard 7.NS.4 Form B
Understand \(-\frac{p}{q} = \frac{-p}{q} = \frac{p}{-q}\); compute quotients with number line models.
Name:
Date: Period:
Tier 1: Foundation Fraction Equivalences & Basic Quotients
10 Problems • 1 pt each
1. \(-32 \div 8 =\)
2. \(-54 \div (-6) =\)
3. \(48 \div (-6) =\)
4. \(-72 \div (-8) =\)
5. Equal to \(-\dfrac{5}{9}\): \(\dfrac{-5}{9}\) or \(\dfrac{-5}{-9}\)?
6. Is \(\dfrac{7}{-10}\) equal to \(-\dfrac{7}{10}\)?
7. \(0 \div (-15) =\)
8. Sign of \((-27) \div (-9)\):
9. Sign of \(63 \div (-7)\):
10. Is \(\dfrac{-8}{-13}\) Pos or Neg?
Tier 2: Approaching Partition Model & Signed Decimals
10 Problems • 1 pt each
Quotient Partition Model: \((-16) \div 4 = -4\) (4 equal intervals of \(-4\))
-16 -12 -8 -4 0
11. Division modeled above:
12. \(-8.1 \div 9 =\)
13. \(16.8 \div (-1.4) =\)
14. \(-8.4 \div (-0.4) =\)
15. Equiv form of \(-\dfrac{4}{7}\) (denom 21):
16. Is \(-\dfrac{3}{5} = \dfrac{-3}{-5}\)? (Yes/No):
17. Reciprocal of \(-\dfrac{5}{9}\) is:
18. \(-96 \div 16 =\)
19. Descend \(540\text{ft}\) in \(9\text{min}\). Rate/min:
20. \(-4.8 \div (-0.08) =\)
Tier 3: At Standard Rational Quotients & Algebraic Equivalence
10 Problems • 1 pt each
21. \(-\dfrac{5}{8} \div \dfrac{15}{16} =\)
22. \(\dfrac{8}{15} \div \left(-\dfrac{16}{25}\right) =\)
23. \(-2\dfrac{1}{4} \div \left(-1\dfrac{7}{8}\right) =\)
24. \(-4\dfrac{1}{2} \div 6 =\)
25. \(-18.75 \div (-1.5) =\)
26. Simplify to lowest terms: \(\dfrac{-24}{36} =\)
27. True/False: \(\dfrac{-a}{b} = -\dfrac{a}{b} = \dfrac{a}{-b}\):
28. Find \(\dfrac{p}{q}\) for \(p=-\frac{3}{4}, q=-\frac{9}{8}\):
29. Leak \(22.5\text{gal}\) in \(1.5\text{hr}\). Rate/hr:
30. Why is \((-a) \div (-b)\) always positive?
Score Tracker: Tier 1: ___/10 • Tier 2: ___/10 • Tier 3: ___/10 • Total: ___/30 Page 4 of 6
Standard 7.NS.5 Form B
Decompose whole numbers into prime factors and write expressions using exponential notation.
Name:
Date: Period:
Tier 1: Foundation Prime vs. Composite & Basic Factoring
10 Problems • 1 pt each
1. Classify \(19\): (Prime or Composite)
2. Classify \(33\): (Prime or Composite)
3. Classify \(37\): (Prime or Composite)
4. Classify \(51\): (Prime or Composite)
5. Only even prime number:
6. Prime factors of \(16\):
7. Prime factors of \(24\):
8. Prime factors of \(28\):
9. Evaluate: \(3^2 \times 5 =\)
10. \(42 = 2 \times 3 \times\)
Tier 2: Approaching Exponents & Factoring < 150
10 Problems • 1 pt each
11. Prime factorization of \(54\):
12. Prime factorization of \(64\):
13. Prime factorization of \(84\):
14. Prime factorization of \(120\):
15. If \(98 = 2 \times 7^x\), find \(x =\)
16. If \(140 = 2^a \times 5 \times 7\), find \(a =\)
17. Evaluate: \(2^3 \times 3 \times 5 =\)
18. Evaluate: \(2^2 \times 7^2 =\)
19. Is \(4 \times 15\) a prime factorization?
20. If \(112 = 2^m \times 7\), find \(m =\)
Tier 3: At Standard Factoring < 225 & Variable Exponents
10 Problems • 1 pt each
21. Prime factorization of \(150\):
22. Prime factorization of \(168\):
23. Prime factorization of \(192\):
24. Prime factorization of \(200\):
25. Prime factorization of \(220\):
26. \(450 = 2^a 3^b 5^c\). Find \(a+b+c\):
27. If \(3^x \times 2^3 = 72\), find \(x =\)
28. Evaluate: \(2^4 \times 3^2 =\)
29. Is \(2^4 \times 3^2\) a perfect square?
30. GCF of \(60\) and \(90\) via factors:
Score Tracker: Tier 1: ___/10 • Tier 2: ___/10 • Tier 3: ___/10 • Total: ___/30 Page 5 of 6
Standard 7.NS.6 Form B
Apply the inverse relationship between squaring and finding square roots of perfect squares.
Name:
Date: Period:
Tier 1: Foundation Dual Number Line: \(\sqrt{x}\) and \(x^2\) Inverse Scale
10 Problems • 1 pt each
Square Root: \(\sqrt{x}\) Inverse Operation Relationship Square: \(x^2\)
0 0 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144
1. \(\sqrt{36} =\)
2. \(\sqrt{16} =\)
3. \(\sqrt{4} =\)
4. \(\sqrt{81} =\)
5. \(\sqrt{144} =\)
6. \(\sqrt{1} =\)
7. \(\sqrt{100} =\)
8. \(8^2 =\)
9. \(\sqrt{11^2} =\)
10. \((\sqrt{49})^2 =\)
Tier 2: Approaching Roots 144 to 400 & Equations
10 Problems • 1 pt each
11. \(\sqrt{196} =\)
12. \(\sqrt{169} =\)
13. \(\sqrt{256} =\)
14. \(\sqrt{225} =\)
15. \(\sqrt{324} =\)
16. \(\sqrt{289} =\)
17. \(\sqrt{400} =\)
18. If \(y^2 = 196\) and \(y > 0\), \(y =\)
19. If \(\sqrt{w} = 16\), find \(w =\)
20. Write \(17\) using a square root:
Tier 3: At Standard Negative Roots & Radical Equality Statements
10 Problems • 1 pt each
21. \(-\sqrt{100} =\)
22. \(-\sqrt{196} =\)
23. \(-\sqrt{256} =\)
24. Solve: \(\sqrt{p} = 17 \implies p =\)
25. Solve: \(z^2 = 256 \implies \sqrt{z^2} =\)
26. Solve: \(5 + \sqrt{y} = 22 \implies y =\)
27. Solve: \(3\sqrt{m} = 42 \implies m =\)
28. Value of \(\sqrt{169 - 25} =\)
29. True/False: \(\sqrt{25 \times 4} = \sqrt{25} \times \sqrt{4}\):
30. Is \(\sqrt{64} = 8\) or \(\pm 8\)? (Principal root):
Score Tracker: Tier 1: ___/10 • Tier 2: ___/10 • Tier 3: ___/10 • Total: ___/30 Page 6 of 6
10 Problems • 1 pt each
-8 -6 -4 -2 0 2 4 6 8 P (-8) Q (-3) R (2) S (6)
1. \(|-22| =\)
2. \(|41| =\)
3. Distance from \(0\) to Point P (\(-8\)):
4. Distance from Point Q (\(-3\)) to R (\(2\)):
5. Distance from Point P (\(-8\)) to Q (\(-3\)):
6. Distance from Point R (\(2\)) to S (\(6\)):
7. Express dist Q to S as \(|p - q|\):
8. Evaluate \(-|18| =\)
9. Is \(|x|\) always greater than or equal to 0?
10. Value of \(|-9.4| =\)
Tier 2: Approaching Absolute Value of Difference: \(|p - q|\) Model
10 Problems • 1 pt each
Distance Formula: \(|(-6) - 2| = |-8| = 8\text{ units}\)
-6 -3 0 2 6
11. Distance from visual \(|-6 - 2| =\)
12. \(6 - 18 =\)
13. \(-16 - 7 =\)
14. \(13 - (-15) =\)
15. \(-19 - (-5) =\)
16. \(7.1 - 12.4 =\)
17. \(-5.2 - 8.6 =\)
18. \(3.9 - (-8.3) =\)
19. Temp: \(-12^\circ\) to \(9^\circ\). Diff:
20. Distance \(-7\) to \(8\): \(|-7-8|\) or \(-7-8\)?
Tier 3: At Standard Rational Differences & Coordinate Distance
10 Problems • 1 pt each
21. \(-\dfrac{4}{5} - \dfrac{3}{10} =\)
22. \(\dfrac{7}{8} - \left(-\dfrac{1}{2}\right) =\)
23. \(-2\dfrac{2}{3} - \left(-1\dfrac{1}{6}\right) =\)
24. \(-\dfrac{5}{6} - \dfrac{2}{3} - \left(-\dfrac{1}{2}\right) =\)
25. Distance between \(-6.4\) and \(5.15\):
26. Elevation \(-75\text{m}\) to \(1,820\text{m}\). Dist:
27. Diver \(-65\text{ft}\) to sub \(-240\text{ft}\). Dist:
28. Find \(|p - q|\) for \(p=-5.4, q=-8.9\):
29. If \(|x - 3| = 7\) and \(x < 0\), find \(x =\):
30. Does \(|x - y| = |-(y - x)|\)? (Yes/No):
Score Tracker: Tier 1: ___/10 • Tier 2: ___/10 • Tier 3: ___/10 • Total: ___/30 Page 2 of 6
Standard 7.NS.3 Form C
Apply multiplication properties, signed rules (\((-1)(-1)=1\)), and distributive property models.
Name:
Date: Period:
Tier 1: Foundation Number Line Hop Model & Integer Products
10 Problems • 1 pt each
Repeated Jump Model: \(4 \times (-2) = -8\)
-8 -6 -4 -2 0
1. Multiplication modeled by the 4 hops:
2. \(6 \times (-5) =\)
3. \((-8) \times (-4) =\)
4. \((-7) \times 8 =\)
5. \((-1)^5 =\)
6. Sign of \((-3) \times (-6) \times (-4)\):
7. Sign of \((-8) \times 2 \times (-5)\):
8. \(0 \times (-28) =\)
9. \((-4)^2 =\)
10. \((-15) \times (-4) =\)
Tier 2: Approaching Multi-Factor Products & Distribution
10 Problems • 1 pt each
11. \((-4) \times (-2) \times (-7) =\)
12. \((-3) \times 8 \times (-4) =\)
13. \((-3.5) \times (-6) =\)
14. \(5 \times (-4.4) =\)
15. Distribute: \(5(a - 8) =\)
16. Distribute: \(-5(2y + 6) =\)
17. Distribute: \(-4(4p - 3) =\)
18. Distribute: \(-8(6 + (-2)) =\)
19. Evaluate: \((-4)^3 =\)
20. Lose \(\$60\)/day for \(5\) days. Net:
Tier 3: At Standard Rational Products & Rational Distribution
10 Problems • 1 pt each
21. \(\left(-\dfrac{5}{9}\right) \times \left(-\dfrac{18}{25}\right) =\)
22. \(\left(-\dfrac{4}{7}\right) \times \dfrac{21}{32} =\)
23. \(-2\dfrac{1}{4} \times 1\dfrac{1}{3} =\)
24. \((-0.6) \times 4.5 \times (-1.4) =\)
25. Distribute: \(-\dfrac{4}{5}(10x - 25) =\)
26. Distribute: \(-\dfrac{3}{8}(-16a + 32) =\)
27. Distribute: \(-1.8(3y - 5) =\)
28. \(-\dfrac{1}{5}(35 - (-20)) =\)
29. If \(a > 0, b < 0, c < 0\), sign of \(a \cdot b \cdot c\):
30. Stock drops \(\$4.20\)/hr for \(2.5\) hrs. Net:
Score Tracker: Tier 1: ___/10 • Tier 2: ___/10 • Tier 3: ___/10 • Total: ___/30 Page 3 of 6
Standard 7.NS.4 Form C
Understand \(-\frac{p}{q} = \frac{-p}{q} = \frac{p}{-q}\); compute quotients with number line models.
Name:
Date: Period:
Tier 1: Foundation Fraction Equivalences & Basic Quotients
10 Problems • 1 pt each
1. \(-36 \div 9 =\)
2. \(-48 \div (-8) =\)
3. \(42 \div (-7) =\)
4. \(-81 \div (-9) =\)
5. Equal to \(-\dfrac{4}{11}\): \(\dfrac{-4}{11}\) or \(\dfrac{-4}{-11}\)?
6. Is \(\dfrac{8}{-15}\) equal to \(-\dfrac{8}{15}\)?
7. \(0 \div (-18) =\)
8. Sign of \((-35) \div (-7)\):
9. Sign of \(72 \div (-8)\):
10. Is \(\dfrac{-9}{-14}\) Pos or Neg?
Tier 2: Approaching Partition Model & Signed Decimals
10 Problems • 1 pt each
Quotient Partition Model: \((-18) \div 3 = -6\) (3 equal intervals of \(-6\))
-18 -12 -6 0 -6 -6 -6
11. Division modeled above:
12. \(-6.3 \div 7 =\)
13. \(19.5 \div (-1.5) =\)
14. \(-7.5 \div (-0.5) =\)
15. Equiv form of \(-\dfrac{6}{11}\) (denom 22):
16. Is \(-\dfrac{5}{8} = \dfrac{-5}{-8}\)? (Yes/No):
17. Reciprocal of \(-\dfrac{7}{12}\) is:
18. \(-108 \div 12 =\)
19. Descend \(480\text{ft}\) in \(8\text{min}\). Rate/min:
20. \(-5.4 \div (-0.09) =\)
Tier 3: At Standard Rational Quotients & Algebraic Equivalence
10 Problems • 1 pt each
21. \(-\dfrac{6}{7} \div \dfrac{12}{21} =\)
22. \(\dfrac{9}{14} \div \left(-\dfrac{18}{35}\right) =\)
23. \(-3\dfrac{1}{3} \div \left(-2\dfrac{2}{9}\right) =\)
24. \(-5\dfrac{1}{4} \div 7 =\)
25. \(-21.25 \div (-2.5) =\)
26. Simplify to lowest terms: \(\dfrac{-28}{42} =\)
27. True/False: \(\dfrac{-x}{-y} = \dfrac{x}{y}\) for \(y \neq 0\):
28. Find \(\dfrac{u}{v}\) for \(u=-\frac{7}{10}, v=-\frac{14}{5}\):
29. Leak \(31.5\text{gal}\) in \(3.5\text{hr}\). Rate/hr:
30. Can a quotient of nonzeros equal 0?
Score Tracker: Tier 1: ___/10 • Tier 2: ___/10 • Tier 3: ___/10 • Total: ___/30 Page 4 of 6
Standard 7.NS.5 Form C
Decompose whole numbers into prime factors and write expressions using exponential notation.
Name:
Date: Period:
Tier 1: Foundation Prime vs. Composite & Basic Factoring
10 Problems • 1 pt each
1. Classify \(23\): (Prime or Composite)
2. Classify \(39\): (Prime or Composite)
3. Classify \(41\): (Prime or Composite)
4. Classify \(57\): (Prime or Composite)
5. Prime number between 24 and 30:
6. Prime factors of \(15\):
7. Prime factors of \(32\):
8. Prime factors of \(50\):
9. Evaluate: \(2 \times 3^3 =\)
10. \(70 = 2 \times 5 \times\)
Tier 2: Approaching Exponents & Factoring < 150
10 Problems • 1 pt each
11. Prime factorization of \(60\):
12. Prime factorization of \(96\):
13. Prime factorization of \(105\):
14. Prime factorization of \(128\):
15. If \(135 = 5 \times 3^x\), find \(x =\)
16. If \(150 = 2 \times 3 \times 5^a\), find \(a =\)
17. Evaluate: \(2^2 \times 5^2 \times 3 =\)
18. Evaluate: \(2^3 \times 11 =\)
19. Is \(6 \times 10\) a prime factorization?
20. If \(144 = 2^m \times 3^2\), find \(m =\)
Tier 3: At Standard Factoring < 225 & Variable Exponents
10 Problems • 1 pt each
21. Prime factorization of \(175\):
22. Prime factorization of \(189\):
23. Prime factorization of \(208\):
24. Prime factorization of \(210\):
25. Prime factorization of \(224\):
26. \(504 = 2^a 3^b 7^c\). Find \(a+b+c\):
27. If \(5^x \times 2^3 = 200\), find \(x =\)
28. Evaluate: \(3^3 \times 2^2 \times 5 =\)
29. Prime factor condition for perfect cubes:
30. GCF of \(48\) and \(72\) via factors:
Score Tracker: Tier 1: ___/10 • Tier 2: ___/10 • Tier 3: ___/10 • Total: ___/30 Page 5 of 6
Standard 7.NS.6 Form C
Apply the inverse relationship between squaring and finding square roots of perfect squares.
Name:
Date: Period:
Tier 1: Foundation Dual Number Line: \(\sqrt{x}\) and \(x^2\) Inverse Scale
10 Problems • 1 pt each
Square Root: \(\sqrt{x}\) Inverse Operation Relationship Square: \(x^2\)
0 0 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144
1. \(\sqrt{49} =\)
2. \(\sqrt{9} =\)
3. \(\sqrt{64} =\)
4. \(\sqrt{121} =\)
5. \(\sqrt{25} =\)
6. \(\sqrt{100} =\)
7. \(\sqrt{144} =\)
8. \(7^2 =\)
9. \(\sqrt{12^2} =\)
10. \((\sqrt{81})^2 =\)
Tier 2: Approaching Roots 144 to 400 & Equations
10 Problems • 1 pt each
11. \(\sqrt{225} =\)
12. \(\sqrt{169} =\)
13. \(\sqrt{289} =\)
14. \(\sqrt{196} =\)
15. \(\sqrt{256} =\)
16. \(\sqrt{361} =\)
17. \(\sqrt{400} =\)
18. If \(a^2 = 225\) and \(a > 0\), \(a =\)
19. If \(\sqrt{c} = 19\), find \(c =\)
20. Write \(18\) using a square root:
Tier 3: At Standard Negative Roots & Radical Equality Statements
10 Problems • 1 pt each
21. \(-\sqrt{121} =\)
22. \(-\sqrt{289} =\)
23. \(-\sqrt{324} =\)
24. Solve: \(\sqrt{r} = 20 \implies r =\)
25. Solve: \(q^2 = 289 \implies \sqrt{q^2} =\)
26. Solve: \(6 + \sqrt{k} = 21 \implies k =\)
27. Solve: \(4\sqrt{x} = 48 \implies x =\)
28. Value of \(\sqrt{225 - 81} =\)
29. True/False: \(\sqrt{\dfrac{100}{25}} = \dfrac{\sqrt{100}}{\sqrt{25}}\):
30. Can \(\sqrt{x}\) produce a negative principal root?
Score Tracker: Tier 1: ___/10 • Tier 2: ___/10 • Tier 3: ___/10 • Total: ___/30 Page 6 of 6
18. -7
19. -60 ft/m
20. 60
T3: Target
21. -1 1/2
22. -3/8
23. 2/3
24. -2/3
25. 6.3
26. -2/3
27. True
28. 1/4
29. -7.5 gal
30. Mult
7.NS.2 • Distance & Absolute Value 30 Pts
T1: Found.
1. 16
2. 23
3. 6
4. 5
5. 5
6. 4
7. |-6-4|=10
8. No
9. 0
10. -12
T2: Appr.
11. 8
12. -8
13. -20
14. 23
15. -12
16. -3.5
17. -10.2
18. 12.5
19. 23°F
20. |-4-7|
T3: Target
21. -1 1/6
22. 1 1/8
23. -5/8
24. -3/5
25. 8.45
26. 1,506 m
27. 143 ft
28. 4.3
29. True
30. -5
7.NS.5 • Prime Factors & Exponents 30 Pts
T1: Found.
1. Prime
2. Comp
3. Prime
4. Comp
5. 2
6. 2² × 3
7. 2 × 3²
8. 2² × 5
9. 24
10. 5
T2: Appr.
11. 2⁴ × 3
12. 2³ × 3²
13. 2×3²×5
14. 2² × 5²
15. x = 2
16. a = 2
17. 180
18. 189
19. No
20. m = 4
T3: Target
21. 2⁴ × 3²
22. 2²×3²×5
23. 2² × 7²
24. 2³ × 3³
25. 3² × 5²
26. 6
27. x = 3
28. 168
29. Even
30. 36
7.NS.3 • Multiplication & Distributive 30 Pts
T1: Found.
1. 4×(-3)=-12
2. -24
3. 35
4. -72
5. 1
6. Neg
7. Pos
8. 0
9. -8
10. 100
T2: Appr.
11. -60
12. 60
13. 9
14. -11.2
15. 3x - 21
16. -8a - 20
17. -15y + 20
18. -42
19. 16
20. -$270
T3: Target
21. 2/3
22. -3/4
23. -4
24. 1.2
25. -4x + 10
26. 6a - 15
27. -4.2m + 7
28. -11
29. 1
30. -$12.38
7.NS.6 • Squares & Square Roots 30 Pts
T1: Found.
1. 5
2. 7
3. 8
4. 9
5. 10
6. 11
7. 12
8. 81
9. 7
10. 36
T2: Appr.
11. 13
12. 14
13. 15
14. 16
15. 17
16. 18
17. 20
18. x = 12
19. y = 225
20. √196
T3: Target
21. -9
22. -15
23. -19
24. k = 324
25. 18
26. n = 225
27. x = 169
28. 8
29. False
30. No
Grade 7 Number Sense Progression • Form A Master Solutions (180 Total Problems) Page 1 of 4
Answer Key Form B
Page 2 of 4
7.NS.1 • Additive Inverses & Sums 30 Pts
T1: Found.
1. 4
2. Pt A (-4)
3. 0
4. 9
5. 6
6. 0
7. 0
8. +24 m
9. 8+(-8)
10. 0°
T2: Appr.
11. -5+12=7
12. 8
13. -9
14. -33
15. 5.8
16. -7.5
17. -14.1
18. 0
19. $23.00
20. -15.8
T3: Target
21. 3/10
22. -1/8
23. -4 1/6
24. -3/8
25. -11.45
26. $53.00
27. 1.2°C
28. 7/12
29. Neg
30. True
7.NS.4 • Negative Fractions & Div 30 Pts
T1: Found.
1. -4
2. 9
3. -8
4. 9
5. (-5)/9
6. Yes
7. 0
8. Pos
9. Neg
10. Pos
T2: Appr.
11. -16÷4=-4
12. -0.9
13. -12
14. 21
15. -12/21
16. No
17. -9/5
18. -6
19. -60 ft/m
20. 60
T3: Target
21. -2/3
22. -5/6
23. 1 1/5
24. -3/4
25. 12.5
26. -2/3
27. True
28. 2/3
29. -15 gal
30. Pos
7.NS.2 • Distance & Absolute Value 30 Pts
T1: Found.
1. 19
2. 34
3. 7
4. 5
5. 5
6. 4
7. |-7-3|=10
8. -15
9. 0
10. No
T2: Appr.
11. 9
12. -11
13. -23
14. 27
15. -13
16. -3.7
17. -12.4
18. 12.2
19. 23°F
20. |-6-5|
T3: Target
21. -1 1/8
22. 1 1/2
23. -1 3/4
24. -1
25. 9.85
26. 1,742 m
27. 155 ft
28. 4.5
29. True
30. -6
7.NS.5 • Prime Factors & Exponents 30 Pts
T1: Found.
1. Prime
2. Comp
3. Prime
4. Comp
5. 2
6. 2⁴
7. 2³ × 3
8. 2² × 7
9. 45
10. 7
T2: Appr.
11. 2 × 3³
12. 2⁶
13. 2²×3×7
14. 2³×3×5
15. x = 2
16. a = 2
17. 120
18. 196
19. No
20. m = 4
T3: Target
21. 2×3×5²
22. 2³×3×7
23. 2⁴ × 13
24. 2³ × 5²
25. 2²×5×11
26. 5
27. x = 2
28. 144
29. Yes
30. 30
7.NS.3 • Multiplication & Distributive 30 Pts
T1: Found.
1. 3×(-4)=-12
2. -35
3. 48
4. -63
5. -1
6. Pos
7. Neg
8. 0
9. -27
10. 60
T2: Appr.
11. -60
12. 84
13. 10
14. -19.2
15. 4y - 24
16. -9x - 21
17. -12m+30
18. -20
19. 81
20. -$220
T3: Target
21. 1/2
22. -2/7
23. -4
24. 2.4
25. -6x + 15
26. 6y - 14
27. -3.2k+6.4
28. -15
29. Pos
30. -$11.38
7.NS.6 • Squares & Square Roots 30 Pts
T1: Found.
1. 6
2. 4
3. 2
4. 9
5. 12
6. 1
7. 10
8. 64
9. 11
10. 49
T2: Appr.
11. 14
12. 13
13. 16
14. 15
15. 18
16. 17
17. 20
18. y = 14
19. w = 256
20. √289
T3: Target
21. -10
22. -14
23. -16
24. p = 289
25. 17
26. y = 289
27. m = 196
28. 12
29. True
30. 8
Grade 7 Number Sense Progression • Form B Master Solutions (180 Total Problems) Page 2 of 4
Answer Key Form C
Page 3 of 4
7.NS.1 • Additive Inverses & Sums 30 Pts
T1: Found.
1. 7
2. Pt A (-7)
3. 0
4. 2
5. 7
6. 0
7. 0
8. +30 m
9. -11+11
10. 0°
T2: Appr.
11. -6+11=5
12. 8
13. -12
14. -36
15. 7.7
16. -7.5
17. -13.4
18. -4
19. $32.00
20. -18.4
T3: Target
21. -1/4
22. -1/10
23. -4 3/8
24. -5/8
25. -13
26. $26.25
27. -0.1°C
28. 8/15
29. 0
30. True
7.NS.4 • Negative Fractions & Div 30 Pts
T1: Found.
1. -4
2. 6
3. -6
4. 9
5. (-4)/11
6. Yes
7. 0
8. Pos
9. Neg
10. Pos
T2: Appr.
11. -18÷3=-6
12. -0.9
13. -13
14. 15
15. -12/22
16. No
17. -12/7
18. -9
19. -60 ft/m
20. 60
T3: Target
21. -1 1/2
22. -1 1/4
23. 1 1/2
24. -3/4
25. 8.5
26. -2/3
27. True
28. 1/4
29. -9 gal
30. No
7.NS.2 • Distance & Absolute Value 30 Pts
T1: Found.
1. 22
2. 41
3. 8
4. 5
5. 5
6. 4
7. |-3-6|=9
8. -18
9. Yes
10. 9.4
T2: Appr.
11. 8
12. -12
13. -23
14. 28
15. -14
16. -5.3
17. -13.8
18. 12.2
19. 21°F
20. |-7-8|
T3: Target
21. -1 1/10
22. 1 3/8
23. -1 1/2
24. -1
25. 11.55
26. 1,895 m
27. 175 ft
28. 3.5
29. -4
30. Yes
7.NS.5 • Prime Factors & Exponents 30 Pts
T1: Found.
1. Prime
2. Comp
3. Prime
4. Comp
5. 29
6. 3 × 5
7. 2⁵
8. 2 × 5²
9. 54
10. 7
T2: Appr.
11. 2²×3×5
12. 2⁵ × 3
13. 3 × 5 × 7
14. 2⁷
15. x = 3
16. a = 2
17. 300
18. 88
19. No
20. m = 4
T3: Target
21. 5² × 7
22. 3³ × 7
23. 2⁴ × 13
24. 2×3×5×7
25. 2⁵ × 7
26. 6
27. x = 2
28. 540
29. Div by 3
30. 24
7.NS.3 • Multiplication & Distributive 30 Pts
T1: Found.
1. 4×(-2)=-8
2. -30
3. 32
4. -56
5. -1
6. Neg
7. Pos
8. 0
9. 16
10. 60
T2: Appr.
11. -56
12. 96
13. 21
14. -22
15. 5a - 40
16. -10y - 30
17. -16p+12
18. -32
19. -64
20. -$300
T3: Target
21. 2/5
22. -3/8
23. -3
24. 3.78
25. -8x + 20
26. 6a - 12
27. -5.4y + 9
28. -11
29. Pos
30. -$10.50
7.NS.6 • Squares & Square Roots 30 Pts
T1: Found.
1. 7
2. 3
3. 8
4. 11
5. 5
6. 10
7. 12
8. 49
9. 12
10. 81
T2: Appr.
11. 15
12. 13
13. 17
14. 14
15. 16
16. 19
17. 20
18. a = 15
19. c = 361
20. √324
T3: Target
21. -11
22. -17
23. -18
24. r = 400
25. 17
26. k = 225
27. x = 144
28. 12
29. True
30. No
Grade 7 Number Sense Progression • Form C Master Solutions (180 Total Problems) Page 3 of 4
Answer Key Form D • 4-Tier Edition
Page 4 of 4
7.NS.1 • Additive Inverses & Sums 40 Pts
T1: Below
1. 8
2. -15
3. 0
4. 0
5. 12
6. 13
7. -31
8. +22 ft
9. -9+9
10. 0°
T2: Appr.
11. 8
12. -13
13. -42
14. 6.5
15. -6.3
16. -16.2
17. 0
18. 8
19. $28
20. -21.5
T3: At
21. 1/8
22. 1/4
23. -4 1/4
24. -3/5
25. -11
26. $40.25
27. 0.2°C
28. 9/14
29. Neg
30. True
T4: Above
31. < 0
32. x = -y
33. Pos
34. k+(-|m|)
35. True
36. |b| > a
37. -7/36
38. 0
39. $0.00
40. x = y
7.NS.4 • Negative Fractions & Div 40 Pts
T1: Below
1. -4
2. 8
3. -6
4. 7
5. (-2)/7
6. Yes
7. 0
8. Pos
9. Neg
10. Pos
T2: Appr.
11. -0.8
12. -13
13. 22
14. -9/24
15. No
16. -7/3
17. -6
18. 9
19. -70 ft/m
20. 60
T3: At
21. -1 1/3
22. -1
23. 1 1/2
24. -5/6
25. 6.5
26. -3/5
27. True
28. 3/4
29. -8.5 gal
30. Yes
T4: Above
31. -5x/2y
32. k = 21
33. -1 1/3
34. m < 0
35. 2
36. True
37. w = 3/4
38. Undef
39. Equal
40. -3 1/2
7.NS.2 • Distance & Absolute Value 40 Pts
T1: Below
1. 18
2. 29
3. 8
4. 7
5. 9
6. 5
7. |-5-4|=9
8. -14
9. 0
10. No
T2: Appr.
11. -8
12. -23
13. 28
14. -13
15. -3.5
16. -13.2
17. 13.2
18. 15
19. 25°
20. |-5-8|
T3: At
21. -1 1/10
22. 1 3/8
23. -1 5/8
24. -7/8
25. 9.95
26. 1,875 m
27. 165 ft
28. 4.5
29. True
30. -4
T4: Above
31. 0
32. True
33. Neg
34. |x| - |y|
35. 4, -10
36. -9, 5
37. 1 17/24
38. p
39. -43, -127
40. True
7.NS.5 • Prime Factors & Exponents 40 Pts
T1: Below
1. Prime
2. Comp
3. Prime
4. Comp
5. 37
6. 2 × 7
7. 5²
8. 3³
9. 28
10. 5
T2: Appr.
11. 2³ × 7
12. 2⁴ × 5
13. 2² × 3³
14. 2×3²×7
15. x = 4
16. a = 2
17. 360
18. 99
19. No
20. m = 4
T3: At
21. 2⁵ × 5
22. 2⁴ × 11
23. 2×3²×11
24. 2×3×5×7
25. 2⁵ × 7
26. 6
27. x = 3
28. 240
29. Even
30. 42
T4: Above
31. 2²×3²×7
32. 2²×3×5²
33. 3² × 7²
34. 2²×3³×5
35. 8
36. 150
37. 432
38. Comp
39. 12
40. p² q³
7.NS.3 • Multiplication & Distributive 40 Pts
T1: Below
1. -40
2. 54
3. -63
4. 1
5. Neg
6. Pos
7. 0
8. -20
9. -27
10. 144
T2: Appr.
11. -60
12. 84
13. 9
14. -18
15. 4x - 36
16. -15a-20
17. -12y+30
18. -35
19. -32
20. -$250
T3: At
21. 2/5
22. -3/8
23. -3
24. 2.7
25. -6x + 15
26. 8a - 16
27. -6m + 9
28. -17
29. 1
30. -$11.90
T4: Above
31. > 0
32. a⁴
33. -9x+12y-6
34. Neg
35. k < 0
36. -6a(3x+4)
37. |x| > |x²|
38. -3/2
39. Dist.
40. Pos
7.NS.6 • Squares & Square Roots 40 Pts
T1: Below
1. 4
2. 5
3. 7
4. 8
5. 9
6. 10
7. 11
8. 36
9. 10
10. 25
T2: Appr.
11. 12
12. 13
13. 14
14. 15
15. 16
16. 18
17. 20
18. x = 13
19. y = 289
20. √169
T3: At
21. -12
22. -16
23. -18
24. k = 361
25. 17
26. n = 324
27. x = 169
28. 12
29. False
30. No
T4: Above
31. 17
32. x = 33
33. y = 12
34. -15
35. 12, 13
36. <
37. m = 81
38. 2
39. 72 cm
40. |x| ≥ 0
Grade 7 Number Sense Progression • Form D 4-Tier Master Solutions (240 Total Problems) Page 4 of 4
| Tier 2 ≥ 8, Tier 3 ≥ 8 |
| Demonstrates rigorous rational computation, algebraic verification, and word problem modeling. |
| Peer tutoring roles, student-created application problems, and Indiana Above-Standard extensions. |
Grade 7 Number Sense Mastery Series • Teacher Facilitation Guide Page 1 of 2
Instructional Scripts Standards 7.NS.1–6
High-frequency student errors with scripted educator verbal corrections and pacing routines.
Indiana Checkpoint 1
Teacher Notes
Standards 7.NS.1 & 7.NS.2: "Subtracting a negative always stays negative" Sign Errors
Student Error: Calculating \(7 - (-15) = -8\) or \(-12 - 8 = -4\) by combining signs indiscriminately.
Teacher Script: "Remember: subtraction is adding the opposite. Replace \(a - b\) with \(a + (-b)\). For \(7 - (-15)\), flipping the operation and the sign gives \(7 + (+15) = 22\). On our number line, distance between coordinates is always positive: \(|p - q|\)."
Standard 7.NS.3: "Distributing a negative only changes the first term" Algebraic Distribution
Student Error: Writing \(-4(2a + 5) = -8a + 5\) or \(-5(3y - 4) = -15y - 20\).
Teacher Script: "The coefficient outside must multiply every term inside the parentheses. Draw arrows to both terms. \((-4) \times (2a) = -8a\) and \((-4) \times (+5) = -20\). Multiplying a negative by a negative inside turns positive: \((-5) \times (-4) = +20\)."
Standard 7.NS.4: "Putting negatives in numerator and denominator keeps it negative" Fraction Equivalence
Student Error: Writing \(-\dfrac{2}{3} = \dfrac{-2}{-3}\) thinking both positions require a negative sign.
Teacher Script: "A negative divided by a negative equals a positive! So \(\frac{-2}{-3} = +\frac{2}{3}\). A single negative fraction has exactly ONE negative sign: either in front \(-\frac{2}{3}\), on top \(\frac{-2}{3}\), or on the bottom \(\frac{2}{-3}\)."
Standards 7.NS.5 & 7.NS.6: Incomplete Factorization & Distributing Radicals Exponents & Radicals
Student Error: Stating \(8 \times 9\) is the prime factorization of 72, or believing \(\sqrt{64 + 36} = \sqrt{64} + \sqrt{36}\).
Teacher Script: "8 and 9 are composite numbers, not primes. Decompose until all leaves are prime: \(2^3 \times 3^2\). For square roots, you must evaluate the sum inside first: \(\sqrt{64 + 36} = \sqrt{100} = 10\). Notice that \(\sqrt{64} + \sqrt{36} = 8 + 6 = 14 \neq 10\)."
Model A: 10-Minute Daily Warm-Up (Sprint Pacing)
• Mon–Wed: Complete 1 tier per day on current standard.
• Thursday: Review Tier 3 error analysis and explanations.
• Friday: Weekly 15-minute diagnostic re-check with Form B.
Model B: Station Rotation / Small-Group Block
• Station 1 (Independent): Form A Tier 1 & 2 skill builders.
• Station 2 (Teacher Table): Tier 3 multi-step rational problems.
• Station 3 (Assessment): Form B exit ticket verification.
Grade 7 Number Sense Mastery Series • Misconceptions & Facilitation Scripts Page 2 of 2
Tier 3: Fraction Products
How to Solve: Cross-simplify first. Distribute fractional coefficients to both terms.
\(-\frac{2}{3}(6x - 15)\)
\(= -4x + 10\)
Grade 7 Number Sense Mastery Series • Quick Solving Notes Page 1 of 2
Quick Notes Standards 7.NS.4–6
Core rules, step-by-step procedures, and worked examples for Below, Approaching, and At Standard tiers.
Study • Review Guide
Page 2 of 2
Standard 7.NS.4 • Equivalent Fractions & Signed Division Rule: \(-\frac{p}{q} = \frac{-p}{q} = \frac{p}{-q}\)
Tier 1: Sign Placement
How to Solve: Negative fraction has EXACTLY ONE negative sign (front, numerator, OR denominator). Two negatives equal positive!
\(-\frac{3}{7} = \frac{-3}{7} = \frac{3}{-7}\)
\(\frac{-6}{-11} = +\frac{6}{11}\)
Tier 2: Decimal Quotients
How to Solve: Count negative signs: opposite signs give negative quotient. For reciprocals, flip fraction but KEEP the negative sign.
\(-14.4 \div 1.2 = -12\)
Recip of \(-\frac{4}{7} \to -\frac{7}{4}\)
Tier 3: Fraction Division
How to Solve: Keep-Change-Flip: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\). Convert mixed numbers to improper fractions first!
\(-\frac{4}{5} \div \frac{8}{15} = -\frac{4}{5} \times \frac{15}{8}\)
\(= -\frac{3}{2} = -1\frac{1}{2}\)
Standard 7.NS.5 • Prime Factorization & Exponents Rule: Decompose to prime leaves
Tier 1: Prime vs. Comp
How to Solve: Primes have exactly 2 factors (1 & itself). 2 is the ONLY even prime. 0 and 1 are neither.
Primes: \(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31\dots\)
Tier 2: Exponential Form
How to Solve: Use factor tree. Group repeated prime factors using exponent powers: \(a \times a \times a = a^3\).
\(72 = 8 \times 9 \to \text{not prime!}\)
\(72 = 2^3 \times 3^2\)
Tier 3: Equations & GCF
How to Solve: Divide out known factors to find variable exponent. Perfect squares have ALL EVEN exponents.
\(360 = 2^3 \times 3^2 \times 5^1\)
\(a+b+c = 3+2+1 = 6\)
Standard 7.NS.6 • Squares, Roots & Inverse Operations Rule: \((\sqrt{x})^2 = x\) • \(\sqrt{x^2} = x\)
Tier 1: Roots ≤ 144
How to Solve: \(\sqrt{n}\) asks: "What number multiplied by itself equals \(n\)?" Memorize perfect squares 1 to 144.
\(\sqrt{64} = 8\), \(\sqrt{121} = 11\)
\((\sqrt{36})^2 = 36\)
Tier 2: Roots 144–400
How to Solve: Memorize benchmarks: \(13^2=169, 14^2=196, 15^2=225, 16^2=256, 17^2=289, 18^2=324, 19^2=361, 20^2=400\).
\(x^2 = 144 \implies x = 12\)
\(\sqrt{y} = 15 \implies y = 225\)
Tier 3: Radicals & Equations
How to Solve:
• Neg outside: \(-\sqrt{a}\) means \(-(\sqrt{a})\). (Note: \(\sqrt{-a}\) is NOT a real number).
• Solve \(\sqrt{x}\): Isolate radical, then square both sides.
• Operations: Simplify inside the radical first!
\(2\sqrt{x} = 26 \implies \sqrt{x} = 13 \implies x = 169\)
\(-\sqrt{81} = -9\) • \(\sqrt{100 - 36} = \sqrt{64} = 8\)
Grade 7 Number Sense Mastery Series • Quick Solving Notes Page 2 of 2
39. Net of \(-\$4.35, +\$6.10, -\$8.25, +\$6.50\):
40. If \(x + (-y) = 0\), relation of \(x\) and \(y\):
Score Tracker: T1: ___/10 • T2: ___/10 • T3: ___/10 • T4: ___/10 • Total: ___/40 Page 1 of 6
Standard 7.NS.2 Form D • 4-Tier Edition
Distance is \(|p - q|\); subtraction as adding opposites; algebraic coordinate generalizations.
Name:
Date: Period:
Tier 1: Foundation Absolute Value & Plotted Distance on Number Line
10 Problems • 1 pt each
1. \(|-18| =\)
2. \(|29| =\)
3. Distance from \(0\) to \(-8\):
4. Distance between \(2\) and \(9\):
5. Distance between \(-3\) and \(6\):
6. Distance between \(-7\) and \(-2\):
7. Express dist \(-5\) to \(4\) as \(|p - q|\):
8. Evaluate \(-|14| =\)
9. Value of \(|0| =\)
10. Can distance on number line be negative?
Tier 2: Approaching Subtraction as Adding Opposites & Decimals
10 Problems • 1 pt each
11. \(9 - 17 =\)
12. \(-15 - 8 =\)
13. \(12 - (-16) =\)
14. \(-22 - (-9) =\)
15. \(6.3 - 9.8 =\)
16. \(-4.7 - 8.5 =\)
17. \(5.4 - (-7.8) =\)
18. Distance \(|-9 - 6| =\)
19. Temp: \(-9^\circ\) to \(16^\circ\). Diff:
20. Distance \(-5\) to \(8\): \(|-5-8|\) or \(-5-8\)?
Tier 3: At Standard Rational Differences & Coordinate Distance
10 Problems • 1 pt each
21. \(-\dfrac{7}{10} - \dfrac{2}{5} =\)
22. \(\dfrac{5}{8} - \left(-\dfrac{3}{4}\right) =\)
23. \(-3\dfrac{1}{4} - \left(-1\dfrac{5}{8}\right) =\)
24. \(-\dfrac{3}{4} - \dfrac{5}{8} - \left(-\dfrac{1}{2}\right) =\)
25. Distance between \(-5.2\) and \(4.75\):
26. Elevation \(-95\text{m}\) to \(1,780\text{m}\). Dist:
27. Diver \(-60\text{ft}\) to sub \(-225\text{ft}\). Dist:
28. Find \(|a - b|\) for \(a=-4.2, b=-8.7\):
29. True/False: \(|p - q| = |q - p|\) always:
30. Dist from \(x\) to \(6\) is \(10\) (\(x < 0\)). \(x=\):
Tier 4: Above Standard Generalizations of Differences & Coordinate Proofs
10 Problems • 1 pt each
31. If \(a < b\), evaluate \(|a - b| - (b - a)\):
32. Prove/Disprove: \(|p - q| = |q - p|\) for all numbers:
33. If \(x < 0\) and \(y > 0\), sign of \(x - y\) is:
34. If \(x, y < 0\) with \(|x| > |y|\), simplify \(|x - y|\):
35. Solve: \(|k - (-3)| = 7 \implies k =\)
36. Midpoint is \(-2\), dist is \(14\). Endpoints:
37. Evaluate: \(\left|-\dfrac{7}{8} - \dfrac{5}{6}\right| =\)
38. If \(p - q > 0\), which is further right on line?
39. Diver at \(-85\text{ft}\) moves \(42\text{ft}\). New depths:
40. True/False: For nonzeros, \(|p - q| < |p| + |q|\):
Score Tracker: T1: ___/10 • T2: ___/10 • T3: ___/10 • T4: ___/10 • Total: ___/40 Page 2 of 6
Standard 7.NS.3 Form D • 4-Tier Edition
Properties of signed multiplication, algebraic distributive expansions, and inequality generalizations.
Name:
Date: Period:
Tier 1: Foundation Basic Sign Rules & Integer Products
10 Problems • 1 pt each
1. \(5 \times (-8) =\)
2. \((-9) \times (-6) =\)
3. \((-7) \times 9 =\)
4. \((-1) \times (-1) =\)
5. Sign of \((-2) \times (-3) \times (-4)\):
6. Sign of \((-6) \times 4 \times (-3)\):
7. \(0 \times (-35) =\)
8. Model: \(5\) groups of \(-4\) equals:
9. \((-3)^3 =\)
10. \((-12) \times (-12) =\)
Tier 2: Approaching Multi-Factor Products & Integer Distribution
10 Problems • 1 pt each
11. \((-4) \times (-3) \times (-5) =\)
12. \((-2) \times 7 \times (-6) =\)
13. \((-1.8) \times (-5) =\)
14. \(5 \times (-3.6) =\)
15. Distribute: \(4(x - 9) =\)
16. Distribute: \(-5(3a + 4) =\)
17. Distribute: \(-6(2y - 5) =\)
18. Distribute: \(-7(8 + (-3)) =\)
19. Evaluate: \((-2)^5 =\)
20. Lose \(\$50\)/day for \(5\) days. Net:
Tier 3: At Standard Rational Signed Products & Distribution
10 Problems • 1 pt each
21. \(\left(-\dfrac{5}{8}\right) \times \left(-\dfrac{16}{25}\right) =\)
22. \(\left(-\dfrac{4}{9}\right) \times \dfrac{27}{32} =\)
23. \(-2\dfrac{1}{3} \times 1\dfrac{2}{7} =\)
24. \((-0.5) \times 3.6 \times (-1.5) =\)
25. Distribute: \(-\dfrac{3}{5}(10x - 25) =\)
26. Distribute: \(-\dfrac{4}{7}(-14a + 28) =\)
27. Distribute: \(-1.5(4m - 6) =\)
28. \(-\dfrac{1}{3}(36 - (-15)) =\)
29. If \(-1 + (-1)(-1) = 0\), \((-1)(-1)=\)
30. Stock drops \(\$3.40\)/hr for \(3.5\) hrs. Net:
Tier 4: Above Standard Magnitude, Variable Inequations & Equivalence Proofs
10 Problems • 1 pt each
31. If \(x < 0, y < 0\), compare \(x \cdot y\) __ \(0\):
32. If \(a < 0\), which is larger: \(a^3\) or \(a^4\)?
33. Expand: \(-\dfrac{3}{4}(12x - 16y + 8) =\)
34. If \(m < 0, n > 0\), sign of \((-m)^2 \cdot (-n)\):
35. For what values of \(k\) is \(-5k > 0\)?
36. Factor GCF: \(-18ax - 24a =\)
37. If \(-1 < x < 0\), compare \(|x|\) and \(|x^2|\):
38. \(\left(-\dfrac{2}{3}\right)^3 \times \left(-\dfrac{9}{4}\right)^2 =\)
39. Evaluate \(a + (-1)a\) to prove \((-1)a = -a\):
40. If \(p \cdot q < 0\) and \(p + q > 0\), larger \(|\cdot|\):
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Standard 7.NS.4 Form D • 4-Tier Edition
Rules for rational quotients, variable fraction equivalences, and multi-step rate contexts.
Name:
Date: Period:
Tier 1: Foundation Fraction Equivalences & Basic Quotients
10 Problems • 1 pt each
1. \(-28 \div 7 =\)
2. \(-48 \div (-6) =\)
3. \(54 \div (-9) =\)
4. \(-56 \div (-8) =\)
5. Equal to \(-\dfrac{2}{7}\): \(\dfrac{-2}{7}\) or \(\dfrac{-2}{-7}\)?
6. Is \(\dfrac{5}{-8}\) equal to \(-\dfrac{5}{8}\)?
7. \(0 \div (-14) =\)
8. Sign of \((-36) \div (-4)\):
9. Sign of \(45 \div (-5)\):
10. Is \(\dfrac{-7}{-12}\) Pos or Neg?
Tier 2: Approaching Signed Decimal Division & Reciprocals
10 Problems • 1 pt each
11. \(-6.4 \div 8 =\)
12. \(15.6 \div (-1.2) =\)
13. \(-8.8 \div (-0.4) =\)
14. Equiv form of \(-\dfrac{3}{8}\) (denom 24):
15. Is \(-\dfrac{4}{5} = \dfrac{-4}{-5}\)? (Yes/No):
16. Reciprocal of \(-\dfrac{3}{7}\) is:
17. \(-72 \div 12 =\)
18. \(-144 \div (-16) =\)
19. Descend \(490\text{ft}\) in \(7\text{min}\). Rate/min:
20. \(-4.2 \div (-0.07) =\)
Tier 3: At Standard Rational Signed Quotients & Fractions
10 Problems • 1 pt each
21. \(-\dfrac{3}{5} \div \dfrac{9}{20} =\)
22. \(\dfrac{6}{11} \div \left(-\dfrac{18}{33}\right) =\)
23. \(-2\dfrac{2}{3} \div \left(-1\dfrac{7}{9}\right) =\)
24. \(-4\dfrac{1}{6} \div 5 =\)
25. \(-16.25 \div (-2.5) =\)
26. Simplify to lowest terms: \(\dfrac{-21}{35} =\)
27. True/False: \(-\dfrac{a}{b} = \dfrac{-a}{b} = \dfrac{a}{-b}\):
28. Find \(\dfrac{x}{y}\) for \(x=-\frac{2}{3}, y=-\frac{8}{9}\):
29. Leak \(25.5\text{gal}\) in \(3\text{hr}\). Rate/hr:
30. Does \(p \div q = p \cdot \frac{1}{q}\) hold for \(q \neq 0\)?
Tier 4: Above Standard Variable Equivalences & Rational Equivalence Proofs
10 Problems • 1 pt each
31. Write 2 forms of \(-\dfrac{5x}{2y}\) with 1 sign:
32. Solve: \(\dfrac{-18}{k} = \dfrac{6}{-7} \implies k =\)
33. Simplify complex fraction: \(\dfrac{-\frac{3}{5}}{\frac{9}{20}} =\)
34. For what values of \(m\) is \(\dfrac{m}{-8} > 0\)?
35. If \(\dfrac{-p}{q} = 4\), evaluate \(\dfrac{p}{-2q}\):
36. True/False: \(\dfrac{-x-y}{z} = -\dfrac{x+y}{z}\):
37. Solve: \(w \div \left(-\dfrac{4}{9}\right) = -\dfrac{27}{16} \implies w =\)
38. Why is \(a \div 0\) undefined for all \(a\)?
39. If \(a, b < 0\), compare \(\dfrac{a}{b}\) to \(\dfrac{-a}{-b}\):
40. Rate decreases \(-\frac{7}{8}\) each \(\frac{1}{4}\text{hr}\). Hourly:
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Standard 7.NS.5 Form D • 4-Tier Edition
Decomposition into prime powers, multi-step factoring ≥ 225, and variable exponent equations.
Name:
Date: Period:
Tier 1: Foundation Prime vs. Composite & Basic Factors
10 Problems • 1 pt each
1. Classify \(13\): (Prime or Composite)
2. Classify \(21\): (Prime or Composite)
3. Classify \(29\): (Prime or Composite)
4. Classify \(35\): (Prime or Composite)
5. Prime number between 31 and 37:
6. Prime factors of \(14\):
7. Prime factors of \(25\):
8. Prime factors of \(27\):
9. Evaluate: \(2^2 \times 7 =\)
10. \(45 = 3 \times 3 \times\)
Tier 2: Approaching Exponents & Factoring < 150
10 Problems • 1 pt each
11. Prime factorization of \(56\):
12. Prime factorization of \(80\):
13. Prime factorization of \(108\):
14. Prime factorization of \(126\):
15. If \(112 = 7 \times 2^x\), find \(x =\)
16. If \(180 = 2^a \times 3^2 \times 5\), find \(a =\)
17. Evaluate: \(2^3 \times 3^2 \times 5 =\)
18. Evaluate: \(3^2 \times 11 =\)
19. Is \(9 \times 8\) a prime factorization?
20. If \(162 = 2 \times 3^m\), find \(m =\)
Tier 3: At Standard Factoring Numbers < 225 & Exponent Equations
10 Problems • 1 pt each
21. Prime factorization of \(160\):
22. Prime factorization of \(176\):
23. Prime factorization of \(198\):
24. Prime factorization of \(210\):
25. Prime factorization of \(224\):
26. \(400 = 2^a \times 5^b\). Find \(a + b\):
27. If \(2^x \times 3^3 = 216\), find \(x =\)
28. Evaluate: \(2^4 \times 3 \times 5 =\)
29. Perfect square exponents condition:
30. GCF of \(84\) and \(126\) via factors:
Tier 4: Above Standard Factorization ≥ 225 & Divisibility Analysis
10 Problems • 1 pt each
31. Prime factorization of \(252\):
32. Prime factorization of \(300\):
33. Prime factorization of \(441\):
34. Prime factorization of \(540\):
35. \(1,008 = 2^a 3^b 7^c\). Find \(a \times b \times c\):
36. Min factor to multiply \(180\) to make cube:
37. LCM of \(144\) and \(216\) via factors:
38. Why is \(504 = 4 \times 9 \times 14\) invalid?
39. Total factors of \(n = 2^3 \times 3^2\):
40. If \(p, q\) primes, prime factors of \(p^2 q^3\):
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Standard 7.NS.6 Form D • 4-Tier Edition
Inverse relationships, radical equations, negative roots, and geometric area applications.
Name:
Date: Period:
Tier 1: Foundation Perfect Squares & Roots ≤ 144
10 Problems • 1 pt each
1. \(\sqrt{16} =\)
2. \(\sqrt{25} =\)
3. \(\sqrt{49} =\)
4. \(\sqrt{64} =\)
5. \(\sqrt{81} =\)
6. \(\sqrt{100} =\)
7. \(\sqrt{121} =\)
8. \(6^2 =\)
9. \(\sqrt{10^2} =\)
10. \((\sqrt{25})^2 =\)
Tier 2: Approaching Roots 144 to 400 & Radical Equations
10 Problems • 1 pt each
11. \(\sqrt{144} =\)
12. \(\sqrt{169} =\)
13. \(\sqrt{196} =\)
14. \(\sqrt{225} =\)
15. \(\sqrt{256} =\)
16. \(\sqrt{324} =\)
17. \(\sqrt{400} =\)
18. If \(x^2 = 169\) and \(x > 0\), \(x =\)
19. If \(\sqrt{y} = 17\), find \(y =\)
20. Write \(13\) using a square root:
Tier 3: At Standard Negative Roots & Radical Equations
10 Problems • 1 pt each
21. \(-\sqrt{144} =\)
22. \(-\sqrt{256} =\)
23. \(-\sqrt{324} =\)
24. Solve: \(\sqrt{k} = 19 \implies k =\)
25. Solve: \(m^2 = 289 \implies \sqrt{m^2} =\)
26. Solve: \(7 + \sqrt{n} = 25 \implies n =\)
27. Solve: \(3\sqrt{x} = 39 \implies x =\)
28. Value of \(\sqrt{225 - 81} =\)
29. True/False: \(\sqrt{16+9} = \sqrt{16}+\sqrt{9}\):
30. Can a real number have \(\sqrt{-64}\)?
Tier 4: Above Standard Complex Radicals, Geometric Proofs & Equations
10 Problems • 1 pt each
31. \(\sqrt{400} - \sqrt{225} + \sqrt{144} =\)
32. Solve: \(\sqrt{3x + 1} = 10 \implies x =\)
33. Solve for \(y > 0\): \((2y)^2 = 576 \implies y =\)
34. Evaluate: \(-\sqrt{(-15)^2} =\)
35. Consecutive integers enclosing \(\sqrt{150}\):
36. Compare: \(\sqrt{25+144}\) __ \(\sqrt{25}+\sqrt{144}\):
37. Solve: \(5\sqrt{m} - 8 = 37 \implies m =\)
38. Simplify: \(\sqrt{\dfrac{196}{49}} =\)
39. Square area is \(324\text{ cm}^2\). Perimeter:
40. Explain why \(\sqrt{x^2} = |x|\) for all real \(x\):
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