Readiness Diagnostic and Targeted Repair

Measure prerequisite skills by subdomain, diagnose the cause of errors, and complete a targeted correction and reassessment loop.

Why this matters

Advanced problems can fail for a basic reason. A correct plan for a parameter problem may collapse because of a sign error; a calculus derivation may fail because fractions were combined incorrectly; a model may be numerically correct but meaningless because units were ignored. The remedy is not to repeat an entire elementary course. It is to identify the exact weak link, repair it, and verify the repair on a parallel problem.

This chapter uses a diagnostic cycle:

\[ \text{attempt}\longrightarrow\text{classify}\longrightarrow \text{repair}\longrightarrow\text{reattempt}\longrightarrow\text{reflect}. \]

A diagnostic is not an entrance barrier and its total score is not a measure of mathematical potential. It is evidence about present fluency. Honest evidence lets you spend time where it produces the largest gain.

What you should be able to do

After completing the chapter, you should be able to:

  • calculate exactly with signed integers and rational numbers;
  • use factors, powers, roots, and scientific notation without false laws;
  • solve proportions and interpret rates, percentages, and dimensional units;
  • read coordinates, slope, scale, and qualitative behavior from a graph;
  • estimate sign, order of magnitude, and plausible bounds before computing;
  • classify an error by its first cause rather than its last symptom;
  • build and complete a targeted repair route from subskill evidence;
  • use a calculator or SymPy to verify work without surrendering method choice;
  • demonstrate repair on a second attempt and document the change in reasoning.

Prerequisites

Complete How to read and write an algebraic argument first. In particular, use \(=\) only for equality, preserve exact values, and mark steps that produce candidates rather than equivalent equations.

Five-minute retrieval warm-up

  1. Which is more informative after a mistake: “careless error” or “I distributed the negative sign to the first term but not the second”? Why?
  2. If a calculator reports 0.3333333333, what exact number might have produced it, and what can the display alone prove?
  3. Estimate \(49.8\times0.203\) before multiplying.
  1. The second description identifies a reproducible operation and therefore a repair. “Careless” does not specify what to practice.
  2. \(1/3\) is one possibility, but many real numbers round to that display. The display proves only a finite-precision approximation to the calculator’s internal result.
  3. \(50\times0.2=10\), so the product should be close to \(10\) (in fact \(10.1094\)). An output near \(100\) or \(1\) should be rejected immediately.

1. How to take the baseline diagnostic

Set aside 45–60 uninterrupted minutes. Use paper, but do not use notes, calculator, search, or computer algebra on the first attempt. Write enough work to reveal your method. If you are unsure, record a best attempt and mark your confidence as high, medium, or low.

The structured diagnostic near the end contains 24 questions in six strands:

Questions Strand Evidence sought
1–4 exact rational arithmetic grouping, signs, common denominators, exact form
5–8 powers, roots, factors laws with conditions, square roots, divisibility
9–12 ratios, percentages, units multiplicative comparison and dimensional meaning
13–16 elementary equations and order inverse operations, inequalities, checks
17–20 coordinates and graphs scale, slope, intercepts, qualitative reading
21–24 estimation and interpretation magnitude, plausibility, exact versus approximate

After submitting, score each strand separately. A result of 3 or 4 correct means “continue, then retrieve again this week.” A result of 2 means “complete the strand repair below.” A result of 0 or 1 means “complete the repair slowly, practice five additional examples, and reassess before Unit 1.” Confidence also matters: a correct low-confidence answer belongs in retrieval practice.

2. Diagnose the first cause

Use one of these categories for every missed or guessed problem.

  • Concept error: a definition or law is misunderstood, such as believing \(a^{-2}\) is negative.
  • Representation error: a correct idea is translated incorrectly among words, symbols, tables, graphs, or units.
  • Strategy error: the chosen method is valid in principle but poorly suited, or no plan was selected.
  • Validity error: a condition is lost, an implication reversed, or an operation applied outside its domain.
  • Algebra error: a symbolic law is misapplied, such as distributing a power over addition.
  • Arithmetic error: the symbolic plan is correct but a numeric operation is wrong.
  • Notation error: the work says something different from the intention, such as using \(=\) for approximation.
  • Interpretation error: the computed number is not connected back to the question, scale, or units.

Do not classify from the final wrong line. Mark the first line that would need to change for the remainder to become valid.

Example 1: error classification

A learner computes

\[ \frac23-\frac58=\frac{-3}{5}. \]

This is not merely an arithmetic slip. The learner subtracted numerators and denominators as if fraction bars were independent components. That is an algebra/representation error: unlike units were combined. The repair is to rename both fractions in fifteenths:

\[ \frac23-\frac58=\frac{16}{24}-\frac{15}{24}=\frac1{24}. \]

A useful reattempt changes the surface form, for example \(5/6-7/10\), while testing the same common-unit idea.

3. Repair strand A: exact rational arithmetic

A fraction \(a/b\) with \(b\ne0\) is a number, not a pair of independent digits. Addition compares common-sized parts:

\[ \frac ab+\frac cd=\frac{ad+bc}{bd},\qquad b,d\ne0. \]

Division asks how many copies of one number fit in another, so multiplying by a reciprocal is derived from the inverse property:

\[ \frac ab\div\frac cd=\frac ab\cdot\frac dc, \qquad b,c,d\ne0. \]

Example 2: a signed fraction calculation

Evaluate

\[ -\frac34+\frac56\div\left(-\frac{10}{9}\right). \]

Division comes before addition:

\[ \frac56\div\left(-\frac{10}{9}\right) =\frac56\left(-\frac9{10}\right) =-\frac{45}{60}=-\frac34. \]

Therefore the total is \(-3/4-3/4=-3/2\). A decimal check gives \(-1.5\), consistent with adding two negative numbers of magnitude \(0.75\).

Guided repair A

Compute \(\frac7{12}-\frac58\) exactly, then estimate to check the sign.

The least common denominator is \(24\): \(14/24-15/24=-1/24\). Since \(7/12\approx0.583\) and \(5/8=0.625\), a small negative answer is plausible.

Mastery signal: four mixed fraction problems correct in succession, with unsimplified denominators and signs checked rather than guessed.

4. Repair strand B: signs, order, and absolute size

A leading negative sign means multiplication by \(-1\). Thus \(-(a-b)=-a+b\), not \(-a-b\). On a number line, \(-7<-2\) because \(-7\) lies farther left, even though its absolute value is larger.

Multiplying an inequality by a negative reverses order. If \(a<b\), then \(b-a>0\); multiplying by \(-1\) gives \(a-b<0\). This is the reason, not a memorized symbol trick.

Example 3: compound signs and order

Simplify \(-2[3-(x-4)]\).

First preserve the inner grouping:

\[ 3-(x-4)=3-x+4=7-x. \]

Then distribute \(-2\) to the entire bracket:

\[ -2(7-x)=-14+2x. \]

At \(x=0\), both the original and final expressions equal \(-14\), a quick check that catches the common answer \(-14-2x\).

Guided repair B

Solve \(-3(2x-1)\le9\) and test one allowed and one excluded value.

Expand: \(-6x+3\le9\), so \(-6x\le6\). Division by \(-6\) reverses order, giving \(x\ge-1\). The boundary \(x=-1\) yields \(9\le9\); the excluded value \(x=-2\) yields \(15\le9\), false.

5. Repair strand C: factors, powers, roots, and scientific notation

For nonzero \(a\) and integers \(m,n\),

\[ a^ma^n=a^{m+n},\qquad \frac{a^m}{a^n}=a^{m-n},\qquad a^{-n}=\frac1{a^n}. \]

The product and quotient laws require a common base. Powers do not distribute over addition: \((a+b)^2=a^2+2ab+b^2\). Over the reals, \(\sqrt{u}\) denotes the nonnegative number whose square is \(u\), so \(u\ge0\) and \(\sqrt{x^2}=|x|\).

Scientific notation writes a nonzero number as \(c\times10^n\) with \(1\le|c|<10\). It separates significant size from decimal placement.

Example 4: exact powers and scale

Compute

\[ \frac{(3\times10^{-4})(8\times10^7)}{6\times10^2}. \]

Separate coefficients and powers:

\[ \frac{3\cdot8}{6}\times10^{-4+7-2} =4\times10^1=40. \]

The exponent estimate \(-4+7-2=1\) predicts tens, so \(40\) has a plausible scale.

Guided repair C

Simplify \((12x^5y^{-2})/(18x^{-1}y)\) for nonzero \(x,y\).

Reduce coefficients and subtract exponents:

\[ \frac{12}{18}x^{5-(-1)}y^{-2-1} =\frac23x^6y^{-3} =\frac{2x^6}{3y^3}. \]

The conditions \(x,y\ne0\) justify the negative exponents and original quotient.

6. Repair strand D: ratios, percentages, and units

A ratio compares quantities multiplicatively. A rate carries units, such as kilometers per hour. A percentage change is measured relative to the original value:

\[ \text{relative change}=\frac{\text{new}-\text{old}}{\text{old}}. \]

An increase of \(20\%\) multiplies by \(1.20\); a later decrease of \(20\%\) multiplies by \(0.80\). The combined factor is \(0.96\), not \(1\).

Dimensional analysis treats units as algebraic factors. It catches impossible additions and missing conversions.

Example 5: unit-aware rate

A vehicle travels \(180\) kilometers in \(2.5\) hours. Its average speed is

\[ \frac{180\ \mathrm{km}}{2.5\ \mathrm h}=72\ \mathrm{km/h}. \]

Converting to meters per second,

\[ 72\frac{\mathrm{km}}{\mathrm h} \cdot\frac{1000\ \mathrm m}{1\ \mathrm{km}} \cdot\frac{1\ \mathrm h}{3600\ \mathrm s} =20\ \mathrm{m/s}. \]

The cancelled units are part of the reasoning.

Guided repair D

A price rises from \(80\) to \(92\), then receives a \(10\%\) discount. Find the final price and the overall percentage change from the original.

The discount multiplier is \(0.90\), so the final price is \(92(0.90)=82.80\). The overall change is \(2.80/80=0.035=3.5\%\) increase. Adding \(15\%-10\%=5\%\) would be wrong because the changes use different bases.

7. Repair strand E: equations and order

Solving means describing every value in the stated universe that makes the original statement true. Reversible operations preserve the solution set; nonreversible operations create candidates that need checks.

For a proportion \(a/b=c/d\), cross multiplication is shorthand for multiplying by \(bd\). It requires \(b,d\ne0\). For a linear equation with a parameter, isolate the coefficient before dividing and split the zero case.

Example 6: a parameter case

Solve \(k(x-2)=3k\) for real \(x\).

If \(k\ne0\), division by \(k\) gives \(x-2=3\), so \(x=5\). If \(k=0\), the equation is \(0=0\) for every real \(x\). Thus

\[ S_k=\begin{cases} \mathbb R,&k=0,\\ \{5\},&k\ne0. \end{cases} \]

The classification is more complete than the unqualified answer \(x=5\).

Guided repair E

Solve \(\frac{x-1}{3}-\frac{x+2}{4}=2\).

Multiply both sides by \(12\): \(4(x-1)-3(x+2)=24\). Expanding gives \(4x-4-3x-6=24\), so \(x=34\). Check: \(33/3-36/4=11-9=2\).

8. Repair strand F: coordinates, graphs, and estimation

A point \((x,y)\) records an input and output. The slope between distinct points \((x_1,y_1)\) and \((x_2,y_2)\) is

\[ m=\frac{y_2-y_1}{x_2-x_1}. \]

Read axis labels and scale before estimating. A steep-looking line can have a small numerical slope if the axes use different units. A graph suggests behavior on its window; it does not prove exact roots or global behavior.

Example 7: interpret slope with units

A tank contains \(120\) liters at minute \(2\) and \(84\) liters at minute \(8\). The average rate of change is

\[ \frac{84-120}{8-2}=-6\ \mathrm{liters/minute}. \]

The negative sign means volume decreased as time increased. A linear extrapolation would predict \(132\) liters at minute \(0\), but that conclusion is a model assumption, not a fact supplied by two data points.

Guided repair F

A line passes through \((-2,5)\) and \((4,-7)\). Find its slope and equation, then check both points.

The slope is \((-7-5)/(4-(-2))=-12/6=-2\). Using \(y=mx+b\) with \((-2,5)\) gives \(5=4+b\), so \(b=1\) and \(y=-2x+1\). Substitution gives \(5\) and \(-7\) at the stated inputs.

9. Common mistakes and limitations

Mistake: using the total score as the diagnosis

Two learners with 16 correct answers may need entirely different repairs. One may miss all unit questions while another misses sign and order questions. Record strand scores and first causes; the total is only a summary.

Mistake: studying the displayed answer instead of the failed operation

Rereading -13/36 does not repair fraction addition. Close the solution and perform the same operation on a new pair of fractions. A transfer problem—not recognition of the old answer—is the evidence of repair.

Mistake: taking the reassessment immediately

An immediate parallel problem can measure short-term imitation. Insert another activity or a delay, then retrieve the method without the model in view. Repeat again later in mixed practice.

Mistake: using a calculator to hide representation gaps

A decimal output may conceal a wrong denominator, a lost exact form, or an incorrect unit. Write the mathematical operation first, estimate its scale, and then use a tool for independent verification.

Mistake: treating speed as mastery

Early repair should prioritize a correct, explainable method. Add time pressure only after several accurate attempts. A fast invalid rule becomes harder—not easier—to unlearn through repetition.

This diagnostic samples prerequisites; it cannot prove mastery of every possible school-algebra task. Low confidence, a long pause, or an unexplained guess is useful evidence even when the selected answer is correct.

10. Topic checkpoint: baseline diagnostic

Record each answer and confidence before submitting. If website controls are not available, the generated notebook provides the same questions and collapsed solutions.

11. Build a correction portfolio

For every missed or low-confidence item, use this six-column record:

Field Required entry
Problem ID stable question identifier
First attempt enough work to reveal the original method
First invalid step exact line or decision where validity was lost
Error class one category from Section 2
Corrected argument complete work, not only the corrected answer
Transfer check a new problem testing the same idea

A repair is complete only when the transfer check is solved without copying the worked example. Schedule another one-question retrieval after two or three days. This delay tests learning rather than short-term imitation.

Example 8: correction record

Problem. Decrease \(200\) by \(15\%\).

First attempt. \(200-15=185\).

First invalid step. The percent sign was treated as 15 units rather than \(15/100\) of the original quantity.

Class. Representation error.

Correction. \(200(1-0.15)=200(0.85)=170\).

Transfer check. Increase \(240\) by \(12\%\): \(240(1.12)=268.8\).

Reflection. Translate a percentage to a multiplier before performing the arithmetic.

12. Second-attempt mastery check

Complete these after the targeted repairs, preferably on a later day.

  1. Compute \(-\frac79+\frac5{12}\) exactly.

  2. Simplify \(-[2x-(3-x)]\).

  3. Simplify \((6a^{-2}b^4)(3a^5b^{-1})\) for nonzero \(a,b\).

  4. Convert \(3.6\times10^5\) millimeters to kilometers.

  5. A quantity falls from \(250\) to \(215\). Find its percentage decrease.

  6. Solve \(-4<2x+6\le12\) and write interval notation.

  7. Solve \(m(x+1)=m^2\) for \(x\), classifying all real \(m\).

  8. A line through \((1,8)\) has slope \(-3\). Find where it crosses each axis and interpret the sign of the slope.

  9. Without a calculator, place \(\sqrt{70}\) between consecutive tenths.

  10. A value is increased by \(p\%\) and then decreased by \(p\%\). Prove that the result is below the original for every \(0<p<100\), and find the percentage loss in terms of \(p\).

  1. Common denominator \(36\) gives \(-28/36+15/36=-13/36\).
  2. \(-[2x-3+x]=-(3x-3)=-3x+3\).
  3. \(18a^3b^3\).
  4. \(3.6\times10^5\text{ mm}\cdot10^{-6}\text{ km/mm}=0.36\text{ km}\).
  5. \((250-215)/250=35/250=0.14\), so \(14\%\).
  6. Subtract \(6\) and divide by \(2\): \(-5<x\le3\), or \((-5,3]\).
  7. If \(m=0\), every real \(x\) works. If \(m\ne0\), \(x+1=m\), so \(x=m-1\).
  8. \(y-8=-3(x-1)\), hence \(y=-3x+11\). The \(y\)-intercept is \((0,11)\) and the \(x\)-intercept is \((11/3,0)\). Output decreases three units per one-unit input increase.
  9. \(8.3^2=68.89<70\) and \(8.4^2=70.56>70\), so \(8.3<\sqrt{70}<8.4\).
  10. See the full solution below.

Let the original positive value be \(V\). The increase multiplies it by \(1+p/100\) and the decrease multiplies the result by \(1-p/100\). Thus

\[ V\left(1+\frac p{100}\right)\left(1-\frac p{100}\right) =V\left(1-\frac{p^2}{10000}\right). \]

For \(0<p<100\), we have \(0<p^2/10000<1\), so the multiplier is positive and strictly below \(1\). The loss is \(V p^2/10000\), which is \(p^2/100\) percent of the original. The proof explains why equal percentage changes do not cancel: the second percentage uses a different base.

13. Examination strategy clinic

Problem: reason before computing

Positive numbers \(a\) and \(b\) satisfy \(a/b=3/5\). Without assigning arbitrary values, determine

\[ \frac{2a+3b}{4a-b} \]

and state why the denominator is nonzero.

Plan. A ratio determines a common scale. Write \(a=3t\) and \(b=5t\) for some \(t>0\).

Execution. Substitution gives

\[ \frac{2(3t)+3(5t)}{4(3t)-5t} =\frac{21t}{7t}=3. \]

Because \(t>0\), the denominator \(7t\) is nonzero and cancellation is valid.

Why this method. Choosing \(a=3,b=5\) would produce the right number but would not explain why every positive pair with the ratio works. Introducing the scale \(t\) converts an example into a general argument.

14. Exercises by purpose and difficulty

Use these if a strand score remains below three. Select at least two from the weak strand and one mixed problem.

  1. Evaluate \(5-2[3-(4-9)]\).

  2. Compute \(7/15\div(-14/25)\).

  3. Write \(0.000072\) in scientific notation.

  4. Explain why \((-3)^4\) and \(-3^4\) differ.

  5. A recipe ratio is \(3\) cups flour to \(2\) cups liquid. Scale it to \(7.5\) cups flour.

  6. Convert \(54\) kilometers per hour to meters per second.

  7. Solve \(3(x-4)-2(x+1)=5\) and check.

  8. Find the slope through \((-3,-2)\) and \((5,10)\).

  9. Find the number that becomes \(126\) after a \(12\%\) increase.

  10. Determine all real \(q\) for which \(q/(q-2)>0\) using sign reasoning.

  11. A graph’s horizontal axis is seconds and vertical axis is meters. Explain the units and meaning of a secant slope of \(-4.2\).

  12. Prove that if \(0<a<b\), then \(1/a>1/b\), without relying on decimal examples.

  1. \(-11\). 2. \(-5/6\). 3. \(7.2\times10^{-5}\). 4. Parentheses make the first base \(-3\), giving \(81\); exponent precedence makes the second \(-(3^4)=-81\).
  2. The scale factor is \(2.5\), so liquid is \(5\) cups. 6. \(15\) m/s. 7. \(x=19\).
  3. \(12/8=3/2\). 9. \(126/1.12=112.5\). 10. Critical points are \(0\) and \(2\); the quotient is positive on \((-\infty,0)\cup(2,\infty)\). 11. Average position changes by \(-4.2\) meters per second; the negative sign indicates decreasing position over the interval. 12. Since \(a,b>0\), \(ab>0\). Multiplying \(a<b\) by the positive number \(1/(ab)\) preserves order and gives \(1/b<1/a\).

15. Cumulative retrieval

Return to argument language. For each claim below, state whether the displayed relation should be \(\Rightarrow\) or \(\Leftrightarrow\), and name conditions.

  1. \(x/5=3\) ___ \(x=15\).
  2. \(x=-2\) ___ \(x^2=4\).
  3. \(ax=ay\) ___ \(x=y\).
  4. \(1/x<0\) ___ \(x<0\).
  1. \(\Leftrightarrow\) over the reals because multiplication by nonzero \(5\) is reversible. 2. Only \(\Rightarrow\). 3. \(\Leftrightarrow\) if \(a\ne0\); only the forward implication from \(x=y\) is unconditional. 4. \(\Leftrightarrow\) on the domain \(x\ne0\): a reciprocal has the same sign as its nonzero input.

16. Summary and next step

  • Score diagnostics by subskill, not only by total.
  • Preserve first attempts and classify the first cause of each error.
  • Repair with a worked model, a new transfer problem, delayed retrieval, and a second attempt.
  • Keep exact arithmetic and units visible; estimate before accepting output.
  • Use tools only after a manual attempt, and interpret their precision and assumptions.
  • Advance when the reassessment shows reliable reasoning, not when the examples merely look familiar.

Proceed to Sets, intervals, statements, and quantifiers. If one strand remains below mastery, interleave its repair exercises with the next chapter rather than postponing all progress.

References and further study

  • OpenStax, Prealgebra 2e, for open prerequisite practice with complete examples.
  • OpenStax, Algebra and Trigonometry 2e, for algebraic review in a later-course context.
  • George Pólya, How to Solve It, for planning and retrospective checking.
  • The Learning Scientists, Retrieval Practice, for an accessible research-informed explanation of delayed self-testing.
  • SymPy documentation, Gotchas and Pitfalls, for exact arithmetic and symbol assumptions.

Using this lesson with fcmath and SymPy

Use Python to verify a completed correction, not to skip the first attempt. The first cell preserves exact fractions; the second compares an exact value with a floating approximation.

from fractions import Fraction

result = -Fraction(3, 4) + Fraction(5, 6) / Fraction(-10, 9)
result
Fraction(-3, 2)

The output Fraction(-3, 2) is exact. A binary floating-point result would be an approximation even if its printed digits looked simple.

import sympy as sp

x = sp.symbols("x", real=True)
exact = sp.sqrt(70)
approximation = sp.N(exact, 12)
bounds = (sp.Rational(83, 10) ** 2, sp.Rational(84, 10) ** 2)
exact, approximation, bounds
(sqrt(70), 8.36660026534, (6889/100, 1764/25))

The squared endpoints \(68.89\) and \(70.56\) prove \(8.3<\sqrt{70}<8.4\); the decimal is a convenient report after the bound has already explained its plausibility.

For a portfolio, store only problem IDs, classifications, and your own written corrections. Do not build code that exposes solutions before an honest attempt or uploads raw answers without informed consent.

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