-
+
- + + + )} +
+ Result: +
+ +-
+
- + Original + + +
- + Factorise + + +
- + Cancel common factors + + +
+ Cancel only factors that appear in both numerator and denominator. +
++ Look around you. How many birds? How many cups? You can answer by + counting — pointing at one thing at a time until everything has been pointed at. + This chapter uses pictures of dots to stand for real things, like marbles or + fingers. +
+ +What is counting?
++ To count is to touch (or point at) each object once. Say a word for each touch: + “one, two, three…” The last word tells you how many there are. +
++ Try it on your fingers: one finger up for each thing you counted. If every object has a finger, + the number of fingers matches the number of objects. +
+ +Zero
++ When there is nothing to point at, we write 0 and say + zero. An empty plate, no toys in the box — zero dots: +
++ (none) +
+Zero comes before one when we count: first nothing, then one, then two, and so on.
+ +Remember
+Zero means “none here,” not “forgot to count.”
+Numbers 1 to 10
++ Each number has a digit (the symbol), a word, and a + picture of dots — one dot for each thing you counted. +
+| Digit | +Word | +Dots | +
|---|---|---|
| 1 | +one | ++ |
| 2 | +two | ++ |
| 3 | +three | ++ |
| 4 | +four | ++ |
| 5 | +five | ++ |
| 6 | +six | ++ |
| 7 | +seven | ++ |
| 8 | +eight | ++ |
| 9 | +nine | ++ |
| 10 | +ten | ++ |
Which group has more dots — the bigger picture wins:
+Patterns
++ A pattern is when something repeats or grows the same way each time. You can + often guess what comes next if you look at the pictures. +
+ +One more dot each time
+Each step adds exactly one dot:
+-
+
- + 1 + + +
- + 2 + + +
- + 3 + + +
- + 4 + + +
- + 5 + + +
Two dots at a time (even numbers)
++ Even counts split into pairs with nothing left over. We jump two dots at a time: +
+-
+
- + 0 + none + +
- + 2 + + + + +
- + 4 + + + + + +
- + 6 + + + + + + +
- + 8 + + + + + + + +
- + 10 + + + + + + + + +
One extra dot (odd numbers)
++ Odd counts also grow by two dots each step, but there is always + one dot left alone — it does not fit in a pair. Look for the single dot in orange: +
+-
+
- + 1 + + + + +
- + 3 + + + + + +
- + 5 + + + + + + +
- + 7 + + + + + + + +
- + 9 + + + + + + + + +
Circle, square, circle, square…
+Patterns can use shapes, not only dots. Watch the pictures repeat:
+-
+
- circle +
- square +
- circle +
- square +
- circle +
- square +
- + + next? + +
Addition
++ Addition means putting two groups together. Three dots and four more dots make + seven dots in one big group. We write 3 + 4 = 7. +
+ +Adding
+Start with the first group, then place the second group beside it and count all the dots.
+Multiplication
++ Multiplication is many equal groups. Three bags with two sweets each + is the same as three pairs of dots. We write 3 × 2 = 6 — “three groups of two.” +
++ You can lay the dots in a rectangle: 3 rows with 2 dots in each row fills a + block of 6 dots. +
+ +Division
++ Division is about sharing equally or making equal groups. If + 12 biscuits are shared among 4 friends, each friend gets the same number. We write + 12 ÷ 4 = 3 — “twelve split into four equal piles gives three in each pile.” +
++ Division is the partner of multiplication. If 3 × 4 = 12, then + 12 ÷ 3 = 4 and 12 ÷ 4 = 3. One picture of dots can answer both + kinds of question. +
+ +Fair shares
++ When every pile has the same number of dots, the share is fair. If dots are + left over, the total does not split evenly — those extras are the remainder. +
+Past ten: tens and hundreds
++ You can count higher than ten on your fingers — people use place value. Ten + single dots can be thought of as one group of ten. Ten groups of ten make + one hundred. +
+-
+
- + 1 one + + + +
- + 1 ten + + + +
- + 1 hundred + + +
+ The number 47 means 4 tens and 7 ones — forty-seven dots, without drawing every + dot one by one. +
+ +Big numbers in real life
++ The world is huge. Counting every grain of sand on a beach one by one would take forever, so we + use big number words to talk about amounts we cannot easily picture dot by dot. + They matter because governments, scientists, and shops all work with large totals. +
+ +| Word | +How we write it | +Why it matters | +
|---|---|---|
| hundred | +100 | +About how many people fit in a big sports field; pages in a long novel. | +
| thousand | +1,000 | +Steps you might walk in a long hike; students in a large school district. | +
| million | +1,000,000 | +People in a large city; dollars in a national budget line item. | +
| billion | +1,000,000,000 | +World population is several billion; stars in our galaxy are hundreds of billions. | +
+ Each big word is one thousand times the word before (ten hundreds make a + thousand; one thousand thousands make a million). We bundle in tens and thousands so we can + compare and add without losing track. +
+ +-
+
- Money — prices and taxes use thousands and millions. +
- Distance — kilometres between cities add up to thousands. +
- Time — minutes in a year are in the hundreds of thousands. +
- Nature — insects in a forest, cells in your body — counts go huge fast. +
Try it yourself
++ Use the tools below: count dots, add and multiply, share equally, and zoom from + one dot up to a billion to see how big numbers compare. +
+History
++ Long before writing, people kept track of herds, harvests, and trade by making marks — tallies — + one mark for each item. Archaeologists have found tally bones tens of thousands of years old. +
++ Different cultures invented symbols for amounts. The idea of zero as a number took + time: Indian mathematicians such as Brahmagupta (7th century CE) wrote rules for + arithmetic that included zero. The Maya used a shell sign for zero in their calendar. Zero let + people write “none” clearly and build place-value systems later on. +
++ Multiplication grew from repeated addition in trade and land measurement — counting equal rows of + goods or fields without adding one row at a time forever. +
++ Division appears in sharing problems as old as fair trade: split grain, land, or money into equal + parts. Large-number names (million, billion) expanded as empires, astronomy, and finance needed to + record bigger totals than everyday counting. +
+ +Derivation
+ +Why addition combines groups
++ Suppose the first group has \(a\) objects and the second has \(b\) objects, with no overlap. If + you put them in one pile and count from the start, you say \(a\) number words for the first + group, then continue with \(b\) more words. The last word is the total \(a + b\). That is why + \(3 + 4 = 7\): three objects, then four more, gives seven in all. +
+ +Multiplication as repeated addition
++ If there are \(r\) equal rows and each row has \(c\) dots, you can add + \(c + c + \cdots + c\) (\(r\) times) to get the total. Writing \(r \times c\) saves repeating + the same addition. For example, \(4 \times 3 = 3 + 3 + 3 + 3 = 12\). +
+ +Division as fair sharing
++ If \(T\) dots are split into \(g\) equal piles with nothing left over, each pile has + \(T \div g\) dots. That is the same as asking: what number times \(g\) gives \(T\)? So + \(T \div g = q\) when \(g \times q = T\). Example: \(12 \div 4 = 3\) because \(4 \times 3 = 12\). +
+ +Place value and big numbers
++ In base ten, each column is worth ten times the column to its right. So \(47 = 4 \times 10 + 7\). + Extending the pattern: \(1{,}000 = 10^3\), \(1{,}000{,}000 = 10^6\), and each “thousand step” multiplies + by \(1{,}000\) again. That is why we group digits in threes when writing millions and billions. +
+ +Checkpoints
+Pause and answer in your head or with a partner before moving on.
+ +Checkpoint
+Show the number \(7\) on your fingers. What digit symbol matches seven?
+Checkpoint
+Which is greater: \(5\) or \(9\)? How do you know?
+Checkpoint
+Is \(2 + 2 + 2\) the same total as \(3 \times 2\)? Explain with a picture of dots.
+Checkpoint
+What comes next in the pattern \(2, 4, 6, 8, \ldots\)?
+Checkpoint
+Share 15 dots into 3 equal piles. How many in each pile? What if you try 4 piles?
+Checkpoint
+Which is bigger: 1 million or 1 thousand? How do you know?
+Real-life applications
+-
+
- Sharing snacks — \(6\) biscuits shared among \(3\) friends is \(6 \div 3 = 2\) each; check with \(3 \times 2 = 6\). +
- Rows of chairs — a room with \(4\) rows of \(5\) chairs has \(4 \times 5 = 20\) seats. +
- Egg cartons — two rows of \(5\) cups hold \(2 \times 5 = 10\) eggs. +
- Counting steps — climbing \(8\) steps then \(2\) more is \(8 + 2 = 10\) steps. +
- Population — countries and cities are counted in millions or billions. +
- Packaging — eggs in cartons (\(2 \times 5\)) and sweets split among friends (\(12 \div 4\)). +
Question bank
+ +Easy
+ +How many fingers on one hand?
+Answer: \(5\).
+What digit means “none”?
+Answer: \(0\) (zero).
+Compute \(2 + 5\).
+Answer: \(7\).
+Steps: Two objects plus five more gives seven.
+Intermediate
+ +Compute \(6 + 4\).
+Answer: \(10\).
+Compute \(3 \times 4\).
+Answer: \(12\).
+Steps: \(4 + 4 + 4 = 12\), or four rows of three dots.
+Which is larger: \(8\) or \(6\)?
+Answer: \(8\) is larger.
+Difficult
+ +Compute \(5 + 3 + 2\).
+Answer: \(10\).
+Steps: \(5 + 3 = 8\), then \(8 + 2 = 10\).
+A tray has \(3\) rows of \(3\) muffins. How many muffins?
+Answer: \(9\).
+Steps: \(3 \times 3 = 9\).
+Hardcore
+ +Without counting one by one: if \(7 + 3 = 10\), what is \(3 + 7\)? Why?
+Answer: \(10\).
+Explanation: Addition of whole numbers can be done in either order (commutative property).
+Show \(5 \times 2\) as both repeated addition and as a dot rectangle.
+Answer: \(10\).
+Steps: \(2 + 2 + 2 + 2 + 2 = 10\); five rows of two dots (or two columns of five).
+Share \(12\) dots equally among \(4\) piles. How many per pile?
+Answer: \(3\) per pile.
+Steps: \(12 \div 4 = 3\) because \(4 \times 3 = 12\).
+How many zeros are in one million written in digits?
+Answer: Six zeros: 1,000,000.
+References
+-
+
- Ifrah, Georges — The Universal History of Numbers (origins of counting and zero). +
- Crossley & Henry — early Indian arithmetic including zero. +
+ Look around you. How many birds? How many cups? You can answer by + counting — pointing at one thing at a time until everything has been pointed at. + This chapter uses pictures of dots to stand for real things, like marbles or + fingers. +
+ +What is counting?
++ To count is to touch (or point at) each object once. Say a word for each touch: + “one, two, three…” The last word tells you how many there are. +
++ Try it on your fingers: one finger up for each thing you counted. If every object has a finger, + the number of fingers matches the number of objects. +
+ +Zero
++ When there is nothing to point at, we write 0 and say + zero. An empty plate, no toys in the box — zero dots: +
++ (none) +
+Zero comes before one when we count: first nothing, then one, then two, and so on.
+ +Remember
+Zero means “none here,” not “forgot to count.”
+Numbers 1 to 10
++ Each number has a digit (the symbol), a word, and a + picture of dots — one dot for each thing you counted. +
+| Digit | +Word | +Dots | +
|---|---|---|
| 1 | +one | ++ |
| 2 | +two | ++ |
| 3 | +three | ++ |
| 4 | +four | ++ |
| 5 | +five | ++ |
| 6 | +six | ++ |
| 7 | +seven | ++ |
| 8 | +eight | ++ |
| 9 | +nine | ++ |
| 10 | +ten | ++ |
Which group has more dots — the bigger picture wins:
+Patterns
++ A pattern is when something repeats or grows the same way each time. You can + often guess what comes next if you look at the pictures. +
+ +One more dot each time
+Each step adds exactly one dot:
+-
+
- + 1 + + +
- + 2 + + +
- + 3 + + +
- + 4 + + +
- + 5 + + +
Two dots at a time (even numbers)
++ Even counts split into pairs with nothing left over. We jump two dots at a time: +
+-
+
- + 0 + none + +
- + 2 + + + + +
- + 4 + + + + + +
- + 6 + + + + + + +
- + 8 + + + + + + + +
- + 10 + + + + + + + + +
One extra dot (odd numbers)
++ Odd counts also grow by two dots each step, but there is always + one dot left alone — it does not fit in a pair. Look for the single dot in orange: +
+-
+
- + 1 + + + + +
- + 3 + + + + + +
- + 5 + + + + + + +
- + 7 + + + + + + + +
- + 9 + + + + + + + + +
Circle, square, circle, square…
+Patterns can use shapes, not only dots. Watch the pictures repeat:
+-
+
- circle +
- square +
- circle +
- square +
- circle +
- square +
- + + next? + +
Addition
++ Addition means putting two groups together. Three dots and four more dots make + seven dots in one big group. We write 3 + 4 = 7. +
+ +Adding
+Start with the first group, then place the second group beside it and count all the dots.
+Multiplication
++ Multiplication is many equal groups. Three bags with two sweets each + is the same as three pairs of dots. We write 3 × 2 = 6 — “three groups of two.” +
++ You can lay the dots in a rectangle: 3 rows with 2 dots in each row fills a + block of 6 dots. +
+ +Division
++ Division is about sharing equally or making equal groups. If + 12 biscuits are shared among 4 friends, each friend gets the same number. We write + 12 ÷ 4 = 3 — “twelve split into four equal piles gives three in each pile.” +
++ Division is the partner of multiplication. If 3 × 4 = 12, then + 12 ÷ 3 = 4 and 12 ÷ 4 = 3. One picture of dots can answer both + kinds of question. +
+ +Fair shares
++ When every pile has the same number of dots, the share is fair. If dots are + left over, the total does not split evenly — those extras are the remainder. +
+Past ten: tens and hundreds
++ You can count higher than ten on your fingers — people use place value. Ten + single dots can be thought of as one group of ten. Ten groups of ten make + one hundred. +
+-
+
- + 1 one + + + +
- + 1 ten + + + +
- + 1 hundred + + +
+ The number 47 means 4 tens and 7 ones — forty-seven dots, without drawing every + dot one by one. +
+ +Big numbers in real life
++ The world is huge. Counting every grain of sand on a beach one by one would take forever, so we + use big number words to talk about amounts we cannot easily picture dot by dot. + They matter because governments, scientists, and shops all work with large totals. +
+ +| Word | +How we write it | +Why it matters | +
|---|---|---|
| hundred | +100 | +About how many people fit in a big sports field; pages in a long novel. | +
| thousand | +1,000 | +Steps you might walk in a long hike; students in a large school district. | +
| million | +1,000,000 | +People in a large city; dollars in a national budget line item. | +
| billion | +1,000,000,000 | +World population is several billion; stars in our galaxy are hundreds of billions. | +
+ Each big word is one thousand times the word before (ten hundreds make a + thousand; one thousand thousands make a million). We bundle in tens and thousands so we can + compare and add without losing track. +
+ +-
+
- Money — prices and taxes use thousands and millions. +
- Distance — kilometres between cities add up to thousands. +
- Time — minutes in a year are in the hundreds of thousands. +
- Nature — insects in a forest, cells in your body — counts go huge fast. +
Try it yourself
++ Use the tools below: count dots, add and multiply, share equally, and zoom from + one dot up to a billion to see how big numbers compare. +
+History
++ Long before writing, people kept track of herds, harvests, and trade by making marks — tallies — + one mark for each item. Archaeologists have found tally bones tens of thousands of years old. +
++ Different cultures invented symbols for amounts. The idea of zero as a number took + time: Indian mathematicians such as Brahmagupta (7th century CE) wrote rules for + arithmetic that included zero. The Maya used a shell sign for zero in their calendar. Zero let + people write “none” clearly and build place-value systems later on. +
++ Multiplication grew from repeated addition in trade and land measurement — counting equal rows of + goods or fields without adding one row at a time forever. +
++ Division appears in sharing problems as old as fair trade: split grain, land, or money into equal + parts. Large-number names (million, billion) expanded as empires, astronomy, and finance needed to + record bigger totals than everyday counting. +
+ +Derivation
+ +Why addition combines groups
++ Suppose the first group has \(a\) objects and the second has \(b\) objects, with no overlap. If + you put them in one pile and count from the start, you say \(a\) number words for the first + group, then continue with \(b\) more words. The last word is the total \(a + b\). That is why + \(3 + 4 = 7\): three objects, then four more, gives seven in all. +
+ +Multiplication as repeated addition
++ If there are \(r\) equal rows and each row has \(c\) dots, you can add + \(c + c + \cdots + c\) (\(r\) times) to get the total. Writing \(r \times c\) saves repeating + the same addition. For example, \(4 \times 3 = 3 + 3 + 3 + 3 = 12\). +
+ +Division as fair sharing
++ If \(T\) dots are split into \(g\) equal piles with nothing left over, each pile has + \(T \div g\) dots. That is the same as asking: what number times \(g\) gives \(T\)? So + \(T \div g = q\) when \(g \times q = T\). Example: \(12 \div 4 = 3\) because \(4 \times 3 = 12\). +
+ +Place value and big numbers
++ In base ten, each column is worth ten times the column to its right. So \(47 = 4 \times 10 + 7\). + Extending the pattern: \(1{,}000 = 10^3\), \(1{,}000{,}000 = 10^6\), and each “thousand step” multiplies + by \(1{,}000\) again. That is why we group digits in threes when writing millions and billions. +
+ +Checkpoints
+Pause and answer in your head or with a partner before moving on.
+ +Checkpoint
+Show the number \(7\) on your fingers. What digit symbol matches seven?
+Checkpoint
+Which is greater: \(5\) or \(9\)? How do you know?
+Checkpoint
+Is \(2 + 2 + 2\) the same total as \(3 \times 2\)? Explain with a picture of dots.
+Checkpoint
+What comes next in the pattern \(2, 4, 6, 8, \ldots\)?
+Checkpoint
+Share 15 dots into 3 equal piles. How many in each pile? What if you try 4 piles?
+Checkpoint
+Which is bigger: 1 million or 1 thousand? How do you know?
+Real-life applications
+-
+
- Sharing snacks — \(6\) biscuits shared among \(3\) friends is \(6 \div 3 = 2\) each; check with \(3 \times 2 = 6\). +
- Rows of chairs — a room with \(4\) rows of \(5\) chairs has \(4 \times 5 = 20\) seats. +
- Egg cartons — two rows of \(5\) cups hold \(2 \times 5 = 10\) eggs. +
- Counting steps — climbing \(8\) steps then \(2\) more is \(8 + 2 = 10\) steps. +
- Population — countries and cities are counted in millions or billions. +
- Packaging — eggs in cartons (\(2 \times 5\)) and sweets split among friends (\(12 \div 4\)). +
Question bank
+ +Easy
+ +How many fingers on one hand?
+Answer: \(5\).
+What digit means “none”?
+Answer: \(0\) (zero).
+Compute \(2 + 5\).
+Answer: \(7\).
+Steps: Two objects plus five more gives seven.
+Intermediate
+ +Compute \(6 + 4\).
+Answer: \(10\).
+Compute \(3 \times 4\).
+Answer: \(12\).
+Steps: \(4 + 4 + 4 = 12\), or four rows of three dots.
+Which is larger: \(8\) or \(6\)?
+Answer: \(8\) is larger.
+Difficult
+ +Compute \(5 + 3 + 2\).
+Answer: \(10\).
+Steps: \(5 + 3 = 8\), then \(8 + 2 = 10\).
+A tray has \(3\) rows of \(3\) muffins. How many muffins?
+Answer: \(9\).
+Steps: \(3 \times 3 = 9\).
+Hardcore
+ +Without counting one by one: if \(7 + 3 = 10\), what is \(3 + 7\)? Why?
+Answer: \(10\).
+Explanation: Addition of whole numbers can be done in either order (commutative property).
+Show \(5 \times 2\) as both repeated addition and as a dot rectangle.
+Answer: \(10\).
+Steps: \(2 + 2 + 2 + 2 + 2 = 10\); five rows of two dots (or two columns of five).
+Share \(12\) dots equally among \(4\) piles. How many per pile?
+Answer: \(3\) per pile.
+Steps: \(12 \div 4 = 3\) because \(4 \times 3 = 12\).
+How many zeros are in one million written in digits?
+Answer: Six zeros: 1,000,000.
+References
+-
+
- Ifrah, Georges — The Universal History of Numbers (origins of counting and zero). +
- Crossley & Henry — early Indian arithmetic including zero. +
+ 100 squares × 10,000 each = 1 million +
++ Each orange square is a whole million. This board shows 100 of them — imagine + 10 identical boards stacked (10 × 100 = 1,000 + millions) to reach one billion. +
+- Numbers on a line can lie to the right of zero (positive) or to the - left (negative). Temperatures below freezing, bank overdrafts, and positions - below sea level are everyday examples of directed numbers. + You already know how to count and work with whole numbers from 0 to + 10. On a number line, those values lie on or to the right of zero. + Numbers can also lie to the left of zero — we call them negative. + Temperatures below freezing, bank overdrafts, and positions below sea level are everyday examples + of directed numbers.
The number line
diff --git a/content/math/directed-numbers-rates/core.html b/content/math/directed-numbers-rates/core.html index a942c92..94da8b4 100644 --- a/content/math/directed-numbers-rates/core.html +++ b/content/math/directed-numbers-rates/core.html @@ -1,8 +1,10 @@- Numbers on a line can lie to the right of zero (positive) or to the - left (negative). Temperatures below freezing, bank overdrafts, and positions - below sea level are everyday examples of directed numbers. + You already know how to count and work with whole numbers from 0 to + 10. On a number line, those values lie on or to the right of zero. + Numbers can also lie to the left of zero — we call them negative. + Temperatures below freezing, bank overdrafts, and positions below sea level are everyday examples + of directed numbers.
The number line
diff --git a/content/math/polynomials-expansion-factorisation/_index.md b/content/math/polynomials-expansion-factorisation/_index.md new file mode 100644 index 0000000..24097dd --- /dev/null +++ b/content/math/polynomials-expansion-factorisation/_index.md @@ -0,0 +1,374 @@ ++++ +title = "Polynomials: expansion & factorisation" +description = "Expand products, collect like terms, and factorise expressions using common factors and identities." +template = "chapter.html" +weight = 1 +[extra] +subject = "math" +chapter_id = "polynomials-expansion-factorisation" +strand = "Number & Algebra" ++++ + ++ An algebraic expression combines numbers, variables, and operations. A + polynomial in one variable \(x\) is a sum of terms of the form + \(a x^n\) where \(a\) is a constant and \(n\) is a non-negative integer. Examples: + \(3x^2 - 5x + 1\), \(x^3 + 2x\), and \(7\). +
+ +Terms, coefficients, and degree
++ In \(5x^2 - 3x + 4\), the terms are \(5x^2\), \(-3x\), and \(4\). The + coefficient of \(x^2\) is \(5\); the coefficient of \(x\) is \(-3\). The + degree of the polynomial is the highest power of \(x\) — here, \(2\). + Like terms have the same variable part (same power of \(x\)); only their + coefficients may differ. +
+ +Adding and subtracting polynomials
++ Line up like terms and add or subtract their coefficients: + \[ (3x^2 + 2x - 1) + (x^2 - 5x + 4) = 4x^2 - 3x + 3. \] + When subtracting, distribute the minus sign to every term in the bracket: + \[ (2x^2 + x) - (x^2 - 3x + 2) = 2x^2 + x - x^2 + 3x - 2 = x^2 + 4x - 2. \] +
+ +Expanding products
++ Use the distributive law: each term in the first bracket multiplies each term in + the second. For two binomials, + \[ (a+b)(c+d) = ac + ad + bc + bd. \] + Example: + \[ (x+3)(x-2) = x^2 - 2x + 3x - 6 = x^2 + x - 6. \] +
++ A binomial times a trinomial works the same way — multiply each of the two terms by each of the + three: + \[ (x+1)(x^2 - x + 2) = x^3 - x^2 + 2x + x^2 - x + 2 = x^3 + x + 2. \] +
+ +Collecting like terms
++ After expanding, combine like terms so the answer is simplified. From above, + \(x^2 - 2x + 3x - 6\) becomes \(x^2 + x - 6\). Constants, \(x\)-terms, and \(x^2\)-terms are + collected separately. +
+ +Factorising
++ Factorisation is the reverse of expansion: write an expression as a product of + simpler factors. Choose a method by inspecting the shape of the expression: +
+| Situation | +Method | +
|---|---|
| Every term shares a common factor | +Highest common factor (HCF) | +
| Four terms, pairs share factors | +Grouping | +
| Matches \(a^2-b^2\) or \((a\pm b)^2\) | +Standard identities | +
| Quadratic in \(x\): \(ax^2+bx+c\) | +Two-number search or ac-method | +
Highest common factor
++ Factor out the largest factor common to every term: + \[ 6x^2 + 9x = 3x(2x + 3), \qquad 4x^3 - 8x^2 = 4x^2(x - 2). \] + Always check whether the bracket can be factorised further. +
+ +Factorising by grouping
++ With four terms, split into two pairs and factor each pair: + \[ ax + ay + bx + by = a(x+y) + b(x+y) = (a+b)(x+y). \] + Example: + \[ 2x^2 + 6x - x - 3 = 2x(x+3) - 1(x+3) = (2x-1)(x+3). \] +
+ +Standard identities
+Identities
++ \[ a^2 - b^2 = (a+b)(a-b) \] + \[ (a+b)^2 = a^2 + 2ab + b^2 \] + \[ (a-b)^2 = a^2 - 2ab + b^2 \] +
++ Recognising these forms speeds up factorisation. For example, + \(x^2 - 9 = x^2 - 3^2 = (x+3)(x-3)\) and + \(x^2 + 6x + 9 = (x+3)^2\). +
+ +Quadratic trinomials
++ To factorise a monic quadratic \(x^2 + bx + c\), find two numbers that + multiply to \(c\) and add to \(b\): + \[ x^2 + 5x + 6 = (x+2)(x+3) \quad\text{since } 2\times 3 = 6,\; 2+3 = 5. \] +
++ For a non-monic quadratic \(ax^2 + bx + c\) with \(a \neq 1\), use the + ac-method: find two numbers that multiply to \(ac\) and add to \(b\), split the + middle term, then factor by grouping: + \[ 2x^2 + 7x + 3 = 2x^2 + 6x + x + 3 = 2x(x+3) + 1(x+3) = (2x+1)(x+3). \] +
++ This chapter stops at factorising expressions. Solving equations of the form \(ax^2+bx+c=0\) is + covered in the chapter on quadratic equations. +
+ +Try it yourself
++ Use the area model to see how \((a+b)(c+d)\) splits into four partial products. Then practise + choosing a factorisation method and checking your factors in the explorer below. +
+History
++ Long before modern symbols, people solved problems that required multiplying lengths and + collecting areas — the same ideas behind expanding and factorising. Babylonian scribes (around + 1800 BCE) recorded rectangle problems on clay tablets: given area and side relationships, + find unknown lengths. Their recipes are early cousins of “expand, then simplify.” +
+ +
+ + In the 9th century, al-Khwarizmi organised methods for equations and products + in Arabic texts that later spread to Europe. Renaissance mathematicians such as + Viète introduced letters for unknowns and constants, making it easier to see + patterns like \(a^2 - b^2\). By the 17th–18th centuries, polynomial algebra was central to + calculus and science; factorisation became a routine step before solving equations or + simplifying formulas. +
+ +Derivation
+ +Area proof of \((a+b)(c+d)\)
++ Draw a rectangle with width \(a+b\) and height \(c+d\). The total area is \((a+b)(c+d)\). Split + the rectangle into four smaller rectangles with areas \(ac\), \(ad\), \(bc\), and \(bd\). The + sum of partial areas equals the whole: + \[ (a+b)(c+d) = ac + ad + bc + bd. \] + This is the distributive law in two dimensions — the same picture as in the area-model widget + above. +
+ +Deriving standard identities
++ Set \(c = a\) and \(d = b\) in the area formula for a square of side \(a+b\): + \[ (a+b)^2 = a^2 + ab + ab + b^2 = a^2 + 2ab + b^2. \] + For a square of side \(a-b\) (with \(a > b\)), remove a strip of width \(b\) from a square of + side \(a\); the remaining L-shape rearranges to show + \[ (a-b)^2 = a^2 - 2ab + b^2. \] + Difference of squares: a large square of side \(a\) minus a small square of side \(b\) leaves two + congruent rectangles, which rearrange to a rectangle of dimensions \((a+b)\) by \((a-b)\): + \[ a^2 - b^2 = (a+b)(a-b). \] +
+ +Grouping walkthrough
++ Factorise \(3x^2 + 12x - 2x - 8\). +
+-
+
- Group: \((3x^2 + 12x) + (-2x - 8)\). +
- Factor each pair: \(3x(x+4) - 2(x+4)\). +
- Common bracket \((x+4)\): \((3x-2)(x+4)\). +
Checkpoints
++ Pause and reason before continuing. Discuss with a partner or write your work in a notebook. +
+ +Checkpoint
++ In \(4x^3 - 2x^2 + 5\), what is the degree? Which terms are like terms with \(4x^3\)? +
+Checkpoint
++ Expand \((2x-1)(x+4)\) without skipping steps. How many terms do you get before collecting? +
+Checkpoint
++ How would you factorise \(x^2 - 16\)? Which identity applies? +
+Checkpoint
++ Expand \(x(x+1) - (x+1)\). What common factor appears? What is the factorised form? +
+Applications
+ +Geometry: area and perimeter
++ A rectangle has sides \((x+3)\) cm and \((2x-1)\) cm. Its area is + \[ (x+3)(2x-1) = 2x^2 + 5x - 3 \quad\text{cm}^2. \] + The perimeter is \(2\bigl((x+3)+(2x-1)\bigr) = 2(3x+2) = 6x+4\) cm. Expansion links algebraic + descriptions of sides to measurable quantities. +
+ +Revenue and pricing
++ If a shop sells \(n\) items at price \((50 - 0.5n)\) dollars each, revenue is + \(n(50 - 0.5n) = 50n - 0.5n^2\). Expanding a product of two linear factors often appears before + finding maximum revenue (a quadratic in \(n\)). +
+ +Number patterns
++ Notice that \(11^2 = 121\), \(111^2 = 12321\). In general, expanding \((10^k + 10^{k-1} + + \cdots + 1)^2\) explains palindromic squares — a playful link between identities and arithmetic + patterns. +
+ +Question bank
++ Work each question, then open the solution to check your reasoning. Difficulty increases + through the sets. +
+ +Easy
+ +Expand \(3(x+2)\).
+Answer: \(3x + 6\).
+Steps: \(3 \cdot x + 3 \cdot 2 = 3x + 6\).
+Simplify \(2x + 5x - 1\).
+Answer: \(7x - 1\).
+Explanation: Combine like terms \(2x + 5x = 7x\).
+Factorise \(6x + 9\).
+Answer: \(3(2x + 3)\).
+Steps: HCF is \(3\); \(6x \div 3 = 2x\), \(9 \div 3 = 3\).
+Intermediate
+ +Expand \((x+4)(x-2)\).
+Answer: \(x^2 + 2x - 8\).
+Steps: \(x^2 - 2x + 4x - 8 = x^2 + 2x - 8\).
+Factorise \(x^2 - 9\).
+Answer: \((x+3)(x-3)\).
+Explanation: Difference of squares: \(x^2 - 3^2\).
+Factorise \(x^2 + 6x + 9\).
+Answer: \((x+3)^2\).
+Explanation: Perfect square: \(3 \times 3 = 9\), \(3 + 3 = 6\).
+Difficult
+ +Factorise \(2x^2 + 6x - x - 3\) by grouping.
+Answer: \((2x-1)(x+3)\).
+Steps: \(2x(x+3) - 1(x+3) = (2x-1)(x+3)\).
+Factorise \(2x^2 + 7x + 3\).
+Answer: \((2x+1)(x+3)\).
+Steps: \(ac = 6\); split \(7x = 6x + x\); \(2x(x+3) + 1(x+3) = (2x+1)(x+3)\).
+Hardcore
+ +Expand and then factorise \(x(x+2) - (x+2)\).
+Answer: \((x+2)(x-1)\) after factorising (expansion gives \(x^2 + x - 2\)).
+Steps: Factor common \((x+2)\): \((x+2)(x-1)\). Check: \((x+2)(x-1) = x^2 + x - 2\).
+A rectangle has sides \((x+3)\) cm and \((2x-1)\) cm. Find its area in expanded form.
+Answer: \((2x^2 + 5x - 3)\) cm².
+Calculation: \((x+3)(2x-1) = 2x^2 - x + 6x - 3 = 2x^2 + 5x - 3\).
+References
+-
+
- al-Khwarizmi, Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala (c. 820 CE) — systematic treatment of algebra. +
- HKDSE Compulsory Part curriculum — Number and Algebra strand (polynomials, expansion, factorisation). +
+ An algebraic expression combines numbers, variables, and operations. A + polynomial in one variable \(x\) is a sum of terms of the form + \(a x^n\) where \(a\) is a constant and \(n\) is a non-negative integer. Examples: + \(3x^2 - 5x + 1\), \(x^3 + 2x\), and \(7\). +
+ +Terms, coefficients, and degree
++ In \(5x^2 - 3x + 4\), the terms are \(5x^2\), \(-3x\), and \(4\). The + coefficient of \(x^2\) is \(5\); the coefficient of \(x\) is \(-3\). The + degree of the polynomial is the highest power of \(x\) — here, \(2\). + Like terms have the same variable part (same power of \(x\)); only their + coefficients may differ. +
+ +Adding and subtracting polynomials
++ Line up like terms and add or subtract their coefficients: + \[ (3x^2 + 2x - 1) + (x^2 - 5x + 4) = 4x^2 - 3x + 3. \] + When subtracting, distribute the minus sign to every term in the bracket: + \[ (2x^2 + x) - (x^2 - 3x + 2) = 2x^2 + x - x^2 + 3x - 2 = x^2 + 4x - 2. \] +
+ +Expanding products
++ Use the distributive law: each term in the first bracket multiplies each term in + the second. For two binomials, + \[ (a+b)(c+d) = ac + ad + bc + bd. \] + Example: + \[ (x+3)(x-2) = x^2 - 2x + 3x - 6 = x^2 + x - 6. \] +
++ A binomial times a trinomial works the same way — multiply each of the two terms by each of the + three: + \[ (x+1)(x^2 - x + 2) = x^3 - x^2 + 2x + x^2 - x + 2 = x^3 + x + 2. \] +
+ +Collecting like terms
++ After expanding, combine like terms so the answer is simplified. From above, + \(x^2 - 2x + 3x - 6\) becomes \(x^2 + x - 6\). Constants, \(x\)-terms, and \(x^2\)-terms are + collected separately. +
+ +Factorising
++ Factorisation is the reverse of expansion: write an expression as a product of + simpler factors. Choose a method by inspecting the shape of the expression: +
+| Situation | +Method | +
|---|---|
| Every term shares a common factor | +Highest common factor (HCF) | +
| Four terms, pairs share factors | +Grouping | +
| Matches \(a^2-b^2\) or \((a\pm b)^2\) | +Standard identities | +
| Quadratic in \(x\): \(ax^2+bx+c\) | +Two-number search or ac-method | +
Highest common factor
++ Factor out the largest factor common to every term: + \[ 6x^2 + 9x = 3x(2x + 3), \qquad 4x^3 - 8x^2 = 4x^2(x - 2). \] + Always check whether the bracket can be factorised further. +
+ +Factorising by grouping
++ With four terms, split into two pairs and factor each pair: + \[ ax + ay + bx + by = a(x+y) + b(x+y) = (a+b)(x+y). \] + Example: + \[ 2x^2 + 6x - x - 3 = 2x(x+3) - 1(x+3) = (2x-1)(x+3). \] +
+ +Standard identities
+Identities
++ \[ a^2 - b^2 = (a+b)(a-b) \] + \[ (a+b)^2 = a^2 + 2ab + b^2 \] + \[ (a-b)^2 = a^2 - 2ab + b^2 \] +
++ Recognising these forms speeds up factorisation. For example, + \(x^2 - 9 = x^2 - 3^2 = (x+3)(x-3)\) and + \(x^2 + 6x + 9 = (x+3)^2\). +
+ +Quadratic trinomials
++ To factorise a monic quadratic \(x^2 + bx + c\), find two numbers that + multiply to \(c\) and add to \(b\): + \[ x^2 + 5x + 6 = (x+2)(x+3) \quad\text{since } 2\times 3 = 6,\; 2+3 = 5. \] +
++ For a non-monic quadratic \(ax^2 + bx + c\) with \(a \neq 1\), use the + ac-method: find two numbers that multiply to \(ac\) and add to \(b\), split the + middle term, then factor by grouping: + \[ 2x^2 + 7x + 3 = 2x^2 + 6x + x + 3 = 2x(x+3) + 1(x+3) = (2x+1)(x+3). \] +
++ This chapter stops at factorising expressions. Solving equations of the form \(ax^2+bx+c=0\) is + covered in the chapter on quadratic equations. +
+ +Try it yourself
++ Use the area model to see how \((a+b)(c+d)\) splits into four partial products. Then practise + choosing a factorisation method and checking your factors in the explorer below. +
+History
++ Long before modern symbols, people solved problems that required multiplying lengths and + collecting areas — the same ideas behind expanding and factorising. Babylonian scribes (around + 1800 BCE) recorded rectangle problems on clay tablets: given area and side relationships, + find unknown lengths. Their recipes are early cousins of “expand, then simplify.” +
+ +
+ + In the 9th century, al-Khwarizmi organised methods for equations and products + in Arabic texts that later spread to Europe. Renaissance mathematicians such as + Viète introduced letters for unknowns and constants, making it easier to see + patterns like \(a^2 - b^2\). By the 17th–18th centuries, polynomial algebra was central to + calculus and science; factorisation became a routine step before solving equations or + simplifying formulas. +
+ +Derivation
+ +Area proof of \((a+b)(c+d)\)
++ Draw a rectangle with width \(a+b\) and height \(c+d\). The total area is \((a+b)(c+d)\). Split + the rectangle into four smaller rectangles with areas \(ac\), \(ad\), \(bc\), and \(bd\). The + sum of partial areas equals the whole: + \[ (a+b)(c+d) = ac + ad + bc + bd. \] + This is the distributive law in two dimensions — the same picture as in the area-model widget + above. +
+ +Deriving standard identities
++ Set \(c = a\) and \(d = b\) in the area formula for a square of side \(a+b\): + \[ (a+b)^2 = a^2 + ab + ab + b^2 = a^2 + 2ab + b^2. \] + For a square of side \(a-b\) (with \(a > b\)), remove a strip of width \(b\) from a square of + side \(a\); the remaining L-shape rearranges to show + \[ (a-b)^2 = a^2 - 2ab + b^2. \] + Difference of squares: a large square of side \(a\) minus a small square of side \(b\) leaves two + congruent rectangles, which rearrange to a rectangle of dimensions \((a+b)\) by \((a-b)\): + \[ a^2 - b^2 = (a+b)(a-b). \] +
+ +Grouping walkthrough
++ Factorise \(3x^2 + 12x - 2x - 8\). +
+-
+
- Group: \((3x^2 + 12x) + (-2x - 8)\). +
- Factor each pair: \(3x(x+4) - 2(x+4)\). +
- Common bracket \((x+4)\): \((3x-2)(x+4)\). +
Checkpoints
++ Pause and reason before continuing. Discuss with a partner or write your work in a notebook. +
+ +Checkpoint
++ In \(4x^3 - 2x^2 + 5\), what is the degree? Which terms are like terms with \(4x^3\)? +
+Checkpoint
++ Expand \((2x-1)(x+4)\) without skipping steps. How many terms do you get before collecting? +
+Checkpoint
++ How would you factorise \(x^2 - 16\)? Which identity applies? +
+Checkpoint
++ Expand \(x(x+1) - (x+1)\). What common factor appears? What is the factorised form? +
+Applications
+ +Geometry: area and perimeter
++ A rectangle has sides \((x+3)\) cm and \((2x-1)\) cm. Its area is + \[ (x+3)(2x-1) = 2x^2 + 5x - 3 \quad\text{cm}^2. \] + The perimeter is \(2\bigl((x+3)+(2x-1)\bigr) = 2(3x+2) = 6x+4\) cm. Expansion links algebraic + descriptions of sides to measurable quantities. +
+ +Revenue and pricing
++ If a shop sells \(n\) items at price \((50 - 0.5n)\) dollars each, revenue is + \(n(50 - 0.5n) = 50n - 0.5n^2\). Expanding a product of two linear factors often appears before + finding maximum revenue (a quadratic in \(n\)). +
+ +Number patterns
++ Notice that \(11^2 = 121\), \(111^2 = 12321\). In general, expanding \((10^k + 10^{k-1} + + \cdots + 1)^2\) explains palindromic squares — a playful link between identities and arithmetic + patterns. +
+ +Question bank
++ Work each question, then open the solution to check your reasoning. Difficulty increases + through the sets. +
+ +Easy
+ +Expand \(3(x+2)\).
+Answer: \(3x + 6\).
+Steps: \(3 \cdot x + 3 \cdot 2 = 3x + 6\).
+Simplify \(2x + 5x - 1\).
+Answer: \(7x - 1\).
+Explanation: Combine like terms \(2x + 5x = 7x\).
+Factorise \(6x + 9\).
+Answer: \(3(2x + 3)\).
+Steps: HCF is \(3\); \(6x \div 3 = 2x\), \(9 \div 3 = 3\).
+Intermediate
+ +Expand \((x+4)(x-2)\).
+Answer: \(x^2 + 2x - 8\).
+Steps: \(x^2 - 2x + 4x - 8 = x^2 + 2x - 8\).
+Factorise \(x^2 - 9\).
+Answer: \((x+3)(x-3)\).
+Explanation: Difference of squares: \(x^2 - 3^2\).
+Factorise \(x^2 + 6x + 9\).
+Answer: \((x+3)^2\).
+Explanation: Perfect square: \(3 \times 3 = 9\), \(3 + 3 = 6\).
+Difficult
+ +Factorise \(2x^2 + 6x - x - 3\) by grouping.
+Answer: \((2x-1)(x+3)\).
+Steps: \(2x(x+3) - 1(x+3) = (2x-1)(x+3)\).
+Factorise \(2x^2 + 7x + 3\).
+Answer: \((2x+1)(x+3)\).
+Steps: \(ac = 6\); split \(7x = 6x + x\); \(2x(x+3) + 1(x+3) = (2x+1)(x+3)\).
+Hardcore
+ +Expand and then factorise \(x(x+2) - (x+2)\).
+Answer: \((x+2)(x-1)\) after factorising (expansion gives \(x^2 + x - 2\)).
+Steps: Factor common \((x+2)\): \((x+2)(x-1)\). Check: \((x+2)(x-1) = x^2 + x - 2\).
+A rectangle has sides \((x+3)\) cm and \((2x-1)\) cm. Find its area in expanded form.
+Answer: \((2x^2 + 5x - 3)\) cm².
+Calculation: \((x+3)(2x-1) = 2x^2 - x + 6x - 3 = 2x^2 + 5x - 3\).
+References
+-
+
- al-Khwarizmi, Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala (c. 820 CE) — systematic treatment of algebra. +
- HKDSE Compulsory Part curriculum — Number and Algebra strand (polynomials, expansion, factorisation). +
-
+ const NavList = (props: { listRef?: (el: HTMLUListElement) => void; class?: string }) => (
+
- jump(item.id)}
>
{item.label}
@@ -191,12 +271,12 @@ export function ChapterHeadingNav() {
setRailOpen(true)} - onMouseLeave={() => setRailOpen(false)} - onFocusIn={() => setRailOpen(true)} + onMouseEnter={openRail} + onMouseLeave={scheduleRailClose} + onFocusIn={openRail} onFocusOut={(e) => { if (!e.currentTarget.contains(e.relatedTarget as Node)) { - setRailOpen(false); + scheduleRailClose(); } }} > @@ -218,9 +298,13 @@ export function ChapterHeadingNav() {
- +@@ -232,7 +316,7 @@ export function ChapterHeadingNav() { class="chapter-heading-nav__fab" aria-expanded={sheetOpen()} aria-controls="chapter-heading-sheet" - onClick={() => setSheetOpen((open) => !open)} + onClick={toggleSheet} > On this page @@ -245,6 +329,7 @@ export function ChapterHeadingNav() { />On this page
-+ = { + Foundation: "catalog-badge--foundation", DSE: "catalog-badge--dse", IB: "catalog-badge--ib", "A-Level": "catalog-badge--a-level", diff --git a/scripts/sync-chapters.ts b/scripts/sync-chapters.ts index 9384c66..502fc21 100644 --- a/scripts/sync-chapters.ts +++ b/scripts/sync-chapters.ts @@ -6,6 +6,7 @@ import { $ } from "bun"; const ROOT = join(import.meta.dir, ".."); const CONTENT = join(ROOT, "content"); const GENERATED_META = join(ROOT, "data/_generated/chapters-meta.json"); +const GENERATED_NAV = join(ROOT, "data/_generated/chapter-nav.json"); const GENERATED_WIDGETS = join(ROOT, "frontend/src/generated/chapter-widgets.ts"); const STATIC_ROOT = join(ROOT, "static"); const CHAPTER_CSS_DIR = join(STATIC_ROOT, "css/chapters"); @@ -19,6 +20,30 @@ type ChaptersMetaFile = { chapters: Recorddiff --git a/themes/openfreebooks/templates/partials/chapter-footer-nav.html b/themes/openfreebooks/templates/partials/chapter-footer-nav.html new file mode 100644 index 0000000..61ce18c --- /dev/null +++ b/themes/openfreebooks/templates/partials/chapter-footer-nav.html @@ -0,0 +1,38 @@ +{% set nav_data = load_data(path="data/_generated/chapter-nav.json", format="json") %} +{% set nav_entry = nav_data.byKey[chapter_key] | default(value=false) %} + +{ + const discoveredSet = new Set(discovered.map((c) => `${c.subject}/${c.slug}`)); + const byKey: Record