Mathematics

Numbers and algebra, measurement and data — the section where method beats memory, because the exam changes the numbers and keeps the method.

Introduction

Of the four sections on this exam, Mathematics is the one that most rewards deliberate practice, and it is the one most people study wrong. They study it by reading. They read a worked solution, follow every line, nod, and conclude that they understand it. Then they sit a timed set and discover that following a solution and producing one are not the same skill at all. Reading a solution exercises recognition. Producing one exercises retrieval, and retrieval is what the exam measures.

So this chapter is built to be worked, not read. Every method carries a worked example set out line by line, and every major method also carries a wrong turn — the specific mistake that the method invites, spelled out with the wrong number it produces, so that when you produce that wrong number yourself you will recognize it as an old acquaintance instead of trusting it.

The second thing to understand about this section is that it is not a memory test. There is very little to memorize here. A short list of conversion factors, a short list of area and volume formulas, the order of operations, and the definition of the mean: that is close to the whole of it. Everything else is method. You are not asked whether you remember that 3/8 + 1/6 = 13/24; you are asked whether you can find a common denominator when the two numbers are ones you have never seen together before. The exam changes the numbers every time. It does not change the method. That is very good news, because methods are learnable in a way that long memorized lists are not.

The third thing is that this section is heavily weighted toward two clusters: numbers and algebra on one side, and measurement and data on the other. Numbers and algebra is the larger of the two. If your study time is limited, the parts of this chapter on fractions, decimals, percent, ratio and basic equation-solving will move your score further than anything else, because those four topics feed into almost every other question on the section. A measurement question often reduces to a proportion. A data question often reduces to a percent. A geometry question often reduces to substituting into a formula and doing decimal arithmetic without slipping a place value. Master the core arithmetic and you have partly mastered everything else.

Finally: a basic four-function on-screen calculator is available on the Mathematics items. Use it. But understand exactly what it protects you from and what it does not. It protects you from arithmetic slips. It does nothing whatever about a wrong setup. A calculator will divide 45 by 63 as cheerfully and as accurately as it will divide 63 by 45, and only one of those is the answer to the question you were asked. The whole of this chapter's discipline is aimed at the setup, because the setup is where the marks are lost.

What the Mathematics section asks you to do

Strip away the wording and the Mathematics items ask a small number of things over and over.

Compute cleanly with the four number forms. Whole numbers, fractions, decimals and percents are four ways of writing the same kind of quantity, and a great many items are simply asking whether you can move between them and operate on them without error. Items in this family look like bare arithmetic, or like a short word problem dressed around bare arithmetic.

Compare and order quantities. Given a mixed list — a fraction, a decimal, a percent, perhaps a negative number — put them in order. This looks trivial and is a reliable source of lost marks, because the comparison must happen in a single common form and people try to do it by eye.

Set up and solve a proportion. A very large number of items reduce to "these two things stay in the same relationship; one of the four numbers is missing." Recipes, dosages, map scales, unit prices, fuel economy, staffing ratios, and most unit conversions all live here.

Translate words into algebra and solve. One unknown, one equation, occasionally an inequality. The algebra itself is modest — linear equations, sometimes with fractions. The translation is the hard part, and it is a learnable skill with its own vocabulary.

Convert units, within a system and across systems. Metric to metric, customary to customary, and the handful of bridges between them. Rates and squared units make the same task harder, and both appear.

Measure figures. Perimeter, circumference, area, volume, surface area, and composite shapes built from those. The formulas you need are few, and the errors are almost never in the formula.

Read and interpret data. Mean, median, mode, range, and what a bar graph, a line graph, a circle graph or a scatterplot actually claims. Also — and this matters more than students expect — what a graph does not claim, and how a graph misleads.

Judge whether an answer is reasonable. This is not a named topic but it is present in every item. An estimate made before you compute is the single most efficient error-catcher available to you, and it costs a few seconds.

Learning objectives

By the end of this chapter you should be able to:

• Apply the order of operations correctly, including to expressions containing exponents, nested grouping symbols, and negative numbers.
• Add, subtract, multiply and divide fractions and mixed numbers, and explain why a common denominator is required for two of those operations and not the other two.
• Convert freely among fractions, decimals and percents, and order a mixed list of all three.
• Solve all three forms of percent question from a single setup, and handle percent increase, percent decrease, successive changes, and working backwards to an original amount.
• Build a proportion whose units line up, solve it by cross-multiplication, and check the result against the original relationship.
• Compute and compare unit rates and unit prices.
• Evaluate expressions, combine like terms, distribute, and solve one-step, two-step, two-sided, fractional and literal equations.
• Solve a linear inequality and state the one operation that reverses the inequality sign.
• Translate an English sentence into an equation, identifying what the variable stands for before writing anything.
• Convert within the metric system and within US customary units, and across the two systems using standard factors.
• Convert a rate and convert squared and cubed units, without treating them as ordinary units.
• Convert between Fahrenheit and Celsius in both directions.
• Compute perimeter, circumference, area, volume and surface area for the standard figures, and decompose a composite figure into figures you know.
• Compute mean, median, mode, range and a weighted mean, and choose the measure of center that a given data set deserves.
• Read a bar graph, line graph, circle graph and scatterplot, and identify the ways each can mislead.
• Estimate an answer before computing it and use that estimate to reject impossible answer choices.

Key concepts at a glance

How to read the worked examples in this chapter

Each worked example is set out in numbered steps, and each step does exactly one thing. That is deliberate. When you make an arithmetic error you almost never make it in the step you were thinking about; you make it in the step you performed automatically while thinking about the next one. Writing one action per line makes the automatic step visible.

After most worked examples you will find a paragraph headed The wrong turn. This is not padding, and it is not a joke at anyone's expense. It is the most useful part of the page. Every method in arithmetic has a characteristic failure — a plausible-looking move that produces a specific wrong number. Those wrong numbers are not random: they are the numbers that appear in the answer choices. A distractor is usually not a random near-miss; it is the result you get if you make the standard mistake. When you learn the standard mistake alongside the method, two things happen. You stop making it, and you start recognizing the choice that was placed there to catch it.

Read the wrong turn and ask yourself honestly whether it is the move you would have made. If it is, that subsection is one you should work rather than read.

You will also find callout boxes — blockquotes whose first line is a heading. They hold the things worth returning to: conversion factors, formula lists, and the short rules that are easier to look up than to reconstruct. If you are reviewing this chapter in a hurry, read the callouts and work the self-check problems at the end.

The four-step habit this chapter is built on

Every worked example in this chapter, whatever the topic, follows the same underlying shape. It is worth naming it, because the shape is transferable and the individual methods are not.

One: say what is being asked, in your own words, before you touch a number. "They want the percent, not the part." "They want how much is left, not how much was used." An astonishing share of lost marks are correct arithmetic answering the wrong question.

Two: estimate. One sentence, no pencil. "A bit less than half of forty, so call it eighteen." The estimate has no authority over the exact answer, but it has absolute authority over an exact answer that is wildly different from it.

Three: set up, then compute. Write the equation, the proportion or the expression completely before evaluating any part of it. Setting up and computing are different mental activities and interleaving them is where setups go wrong.

Four: check by a second route. Reverse the operation. Substitute the answer back in. Compare against the estimate. Convert to a different form and see whether it still looks right. A check that repeats the original method catches nothing, because it makes the same mistake again; a check that takes a different route catches almost everything.

### Remember this — the four-step habit | Step | Question you are answering | Time it costs | |---|---|---| | 1. Read | What exactly is being asked for? | A few seconds | | 2. Estimate | Roughly how big should the answer be? | A few seconds | | 3. Set up, then compute | What is the full expression before I evaluate anything? | Most of the item | | 4. Check a second way | Does it survive a route I have not already used? | A few seconds |

Part A — Number sense and whole numbers

Place value, and reading a number out loud

Place value is the reason our arithmetic works at all, and the reason a decimal point matters. Each column in a written number is worth ten times the column to its right. In 4,7 0 6, the 4 sits in the thousands column and is worth 4,000; the 7 sits in the hundreds column and is worth 700; the 0 holds the tens column open so that the 6 lands in the ones column and not the tens. That zero is doing real work. Remove it and you have 476, a different number entirely.

Reading a number out loud, properly, is a habit worth building, because the spoken form carries the place value with it. "Four thousand, seven hundred six" contains the information that the 7 is hundreds. "Four seven zero six" does not, and people who read numbers that way are the people who lose a place value when they write them down again.

Worked example. In the number 38,472, what is the value of the digit 8, and what is the value of the digit 4?

1. Label the columns from the right: 2 is ones, 7 is tens, 4 is hundreds, 8 is thousands, 3 is ten-thousands.
2. The 8 sits in the thousands column, so its value is 8 × 1,000 = 8,000.
3. The 4 sits in the hundreds column, so its value is 4 × 100 = 400.
4. Answer: the 8 is worth 8,000 and the 4 is worth 400.
5. Check by rebuilding the number: 30,000 + 8,000 + 400 + 70 + 2 = 38,472. It reassembles, so the labeling was right.

The wrong turn. A student counts the columns from the left instead of the right and reports the 8 as being in the hundreds place, worth 800. Counting from the left cannot work, because the leftmost column's value depends on how long the number is, while the rightmost column is always ones. Always count from the ones column outward — and when a decimal point is present, count outward from the point in both directions.

The order of operations, and the two places people break it

An expression like 8 + 3 × 4 has two possible readings, and mathematics settles the dispute by convention: multiplication binds more tightly than addition. The conventional order is grouping symbols first, then exponents and roots, then multiplication and division, then addition and subtraction.

The two words that do the damage are "then". Multiplication and division are not two ranks; they are one rank, performed left to right in the order they appear. The same is true of addition and subtraction. Almost every order-of-operations error on this exam comes from treating them as two ranks — doing all the multiplying before any dividing, or all the adding before any subtracting.

### Remember this — order of operations | Rank | Operations | Direction | |---|---|---| | 1 | Grouping: parentheses, brackets, the bar of a fraction | Innermost first | | 2 | Exponents and roots | As written | | 3 | Multiplication and division together | Left to right | | 4 | Addition and subtraction together | Left to right |

Worked example. Evaluate 8 + 3 × (10 − 6)² ÷ 4 − 5.

1. Grouping first: 10 − 6 = 4, so the expression is 8 + 3 × 4² ÷ 4 − 5.
2. Exponent next: 4² = 16, giving 8 + 3 × 16 ÷ 4 − 5.
3. Multiplication and division, left to right. The multiplication comes first as written: 3 × 16 = 48. Now 8 + 48 ÷ 4 − 5.
4. Continue left to right at the same rank: 48 ÷ 4 = 12. Now 8 + 12 − 5.
5. Addition and subtraction, left to right: 8 + 12 = 20, then 20 − 5 = 15.
6. Answer: 15.

The wrong turn. A student who reaches step 3 and decides that multiplication outranks division computes 3 × 16 = 48 and then, seeing 8 + 48 ÷ 4 − 5, reaches the same place — this particular expression forgives that error. Change it to 3 ÷ 4 × 16 and it does not: left to right gives 0.75 × 16 = 12, while multiplying first gives 4 × 16 = 64, then 3 ÷ 64, which is about 0.047. The more common break is at the start: a student adds 8 + 3 = 11 first, then 11 × 16 = 176, ÷ 4 = 44, − 5 = 39. That 39 is a number you should learn to distrust on sight; it is the classic left-to-right-regardless answer.

Grouping symbols inside grouping symbols

When parentheses are nested, work from the innermost outward. A fraction bar is itself a grouping symbol: everything above the bar is grouped and everything below the bar is grouped, even though no parentheses are printed.

Worked example. Evaluate (24 − 4 × 3) ÷ (2 + 1)².

1. Inside the first parentheses, multiplication before subtraction: 4 × 3 = 12, so 24 − 12 = 12.
2. Inside the second parentheses: 2 + 1 = 3.
3. Now the expression reads 12 ÷ 3². Exponent before division: 3² = 9.
4. 12 ÷ 9 = 4/3, which is 1 1/3, or about 1.33.
5. Answer: 4/3 (about 1.33).
6. Check: 4/3 × 9 = 12, and 12 was the numerator. Reversing the division returns the numerator, so the division was performed correctly.

The wrong turn. A student squares before adding inside the parentheses, reading (2 + 1)² as 2 + 1², which is 2 + 1 = 3. The answer then comes out as 12 ÷ 3 = 4 instead of 4/3 — three times too large. The exponent applies to the whole grouped quantity, because the parentheses were closed before the exponent was written. Anything inside a grouping symbol is finished before the grouping symbol's own exponent is applied.

Adding and subtracting negative numbers

Think of a number line with zero in the middle, positives to the right and negatives to the left. Adding a positive moves you right; adding a negative moves you left. Subtracting is the same as adding the opposite, which is why subtracting a negative moves you right.