1. Counting & Numbers
Counting is the gateway to all of mathematics. It is the first mathematical activity children engage in and provides the conceptual underpinning for arithmetic, algebra, and beyond. The National Research Council identifies counting as one of the five foundational strands of mathematical proficiency. [1] NRC, Adding It Up, 2001, pp. 115–135
1.1 Number Sense
Number sense is an intuitive understanding of numbers, their magnitude, relationships, and how they are affected by operations. It includes the ability to compose and decompose numbers flexibly.
[2] NCTM, Principles and Standards for School Mathematics, 2000, p. 79
Number sense develops gradually. According to the NCTM, children progress through several stages:
- Rote counting — reciting number words in order (one, two, three…)
- One-to-one correspondence — matching each object to exactly one number word
- Cardinality — understanding that the last number counted represents the total quantity
- Conservation of number — recognizing that the count stays the same regardless of arrangement
- Subitizing — instantly recognizing small quantities (up to ~5) without counting
[3] Clements, D. H. & Sarama, J., Learning and Teaching Early Math: The Learning Trajectories Approach, Routledge, 2nd ed., 2014, Ch. 2
The oldest known mathematical artifact is the Ishango bone (c. 20,000 BCE), found in present-day Democratic Republic of Congo, which contains tally marks grouped in patterns suggesting an understanding of counting, doubling, and possibly prime numbers. [4] Zaslavsky, C., Africa Counts, Lawrence Hill Books, 3rd ed., 1999
Counting Principles (Gelman & Gallistel)
The foundational research by Gelman and Gallistel (1978) identified five counting principles that children must master:
| Principle | Description | Example |
|---|---|---|
| Stable-Order | Number words must be said in a fixed, repeatable order | Always: 1, 2, 3, 4… Never: 1, 3, 2, 4… |
| One-to-One | Each object gets exactly one count | Point to each apple once: “one, two, three” |
| Cardinality | The last number said is the total count | After counting “1, 2, 3, 4, 5” → “there are 5” |
| Abstraction | Any collection of objects can be counted | Count a mix of apples, books, and toys together |
| Order-Irrelevance | Objects can be counted in any order; the total stays the same | Start from left, right, or middle — still 5 objects |
[5] Gelman, R. & Gallistel, C. R., The Child’s Understanding of Number, Harvard University Press, 1978
1.2 Cardinal & Ordinal Numbers
Cardinal numbers tell “how many” (1, 2, 3, 4, 5…). Ordinal numbers tell position or order (1st, 2nd, 3rd, 4th, 5th…).
Cardinal: “There are 7 days in a week.”
Ordinal: “Thursday is the 4th day of the school week.”
The distinction between cardinal and ordinal usage is a crucial milestone in number development. [2] NCTM, 2000, p. 78
1.3 The Number Line
The number line is one of the most important representations in elementary mathematics. The Common Core State Standards introduce the number line in Grade 2 (2.MD.6) and use it increasingly through all subsequent grades. [6] CCSS, Common Core State Standards for Mathematics, 2010, 2.MD.6
The number line is not just a teaching tool — it is a mathematical model that connects counting to measurement. Each whole number represents a distance from zero, laying the groundwork for fractions, decimals, negative numbers, and eventually the real number line. [7] Wu, H.-H., Understanding Numbers in Elementary School Mathematics, AMS, 2011, Ch. 1
1.4 Comparing Numbers
Comparing numbers involves determining which is greater, which is less, or whether two numbers are equal. The symbols \(<\), \(>\), and \(=\) formalize this:
Compare 347 and 329:
- Hundreds digits: \(3 = 3\) → same, move to tens
- Tens digits: \(4 > 2\) → 347 is greater
Therefore, \(347 > 329\).
[6] CCSS, 2010, 2.NBT.4: “Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols.”
1.5 Rounding
Rounding replaces a number with a nearby number that is simpler to work with. The standard rounding rule: if the digit to the right of the rounding place is 5 or greater, round up; otherwise, round down.
Round to the nearest ten:
- \(43 \approx 40\) (3 < 5, round down)
- \(67 \approx 70\) (7 ≥ 5, round up)
- \(85 \approx 90\) (5 ≥ 5, round up)
[6] CCSS, 2010, 3.NBT.1
2. Place Value
Place value is the backbone of our Hindu-Arabic numeral system (also called the base-ten positional system). Understanding that the position of a digit determines its value is, according to the NRC, “perhaps the most important mathematical concept in elementary school.” [1] NRC, Adding It Up, 2001, p. 72
2.1 Ones, Tens, and Hundreds
In a base-ten system, each position is worth ten times the position to its right:
\(\ldots \quad \underbrace{\text{Thousands}}_{1000} \quad \underbrace{\text{Hundreds}}_{100} \quad \underbrace{\text{Tens}}_{10} \quad \underbrace{\text{Ones}}_{1}\)
\( 582 = 5 \times 100 + 8 \times 10 + 2 \times 1 \)
The digit 5 represents 500 (five hundreds), 8 represents 80 (eight tens), and 2 represents 2 (two ones).
2.2 Thousands & Beyond
The pattern extends to larger numbers following the same base-ten logic:
| Place | Value | Power of Ten |
|---|---|---|
| Ones | 1 | \(10^0\) |
| Tens | 10 | \(10^1\) |
| Hundreds | 100 | \(10^2\) |
| Thousands | 1,000 | \(10^3\) |
| Ten-thousands | 10,000 | \(10^4\) |
| Hundred-thousands | 100,000 | \(10^5\) |
| Millions | 1,000,000 | \(10^6\) |
[7] Wu, H.-H., 2011, Ch. 1–2
2.3 Expanded Form
Expanded form writes a number as the sum of the values of each digit.
Write each number in expanded form:
- \(309\) → \(300 + 0 + 9 = 3 \times 100 + 0 \times 10 + 9 \times 1\)
- \(8{,}051\) → \(8{,}000 + 0 + 50 + 1\)
- \(62{,}405\) → \(60{,}000 + 2{,}000 + 400 + 0 + 5\)
The base-ten positional system originated in India by the 6th century CE and was transmitted to Europe via Arabic mathematicians, notably al-Khwarizmi (c. 825 CE) in his book On the Calculation with Hindu Numerals. The crucial innovation was the digit zero as a positional placeholder. [8] Ifrah, G., The Universal History of Numbers, Wiley, 2000, pp. 356–398
3. Addition
Addition is the first binary operation children learn. It models the action of combining two or more quantities into a single total. [2] NCTM, 2000, p. 78
3.1 Concepts & Properties
For whole numbers \(a\) and \(b\), the sum \(a + b\) represents the total when a collection of \(a\) objects is combined with a collection of \(b\) objects.
Properties of Addition
| Property | Statement | Example |
|---|---|---|
| Commutative | \(a + b = b + a\) | \(3 + 5 = 5 + 3 = 8\) |
| Associative | \((a + b) + c = a + (b + c)\) | \((2 + 3) + 4 = 2 + (3 + 4) = 9\) |
| Identity | \(a + 0 = a\) | \(7 + 0 = 7\) |
[7] Wu, 2011, pp. 23–30
Commutativity cuts the number of addition facts to memorize nearly in half. If you know \(3 + 8 = 11\), you automatically know \(8 + 3 = 11\). This insight reduces 100 single-digit facts to 55 unique facts. [9] Baroody, A. J., Children’s Mathematical Thinking, Teachers College Press, 1987, Ch. 6
3.2 Mental Strategies
Research identifies several key strategies children develop for addition: [10] Carpenter, T. P. et al., Children’s Mathematics: Cognitively Guided Instruction, Heinemann, 2nd ed., 2015
| Strategy | How It Works | Example: \(8 + 5\) |
|---|---|---|
| Counting On | Start from the larger number and count up | Start at 8, count: 9, 10, 11, 12, 13 → 13 |
| Making Ten | Decompose to make a group of 10 | \(8 + 5 = 8 + 2 + 3 = 10 + 3 = 13\) |
| Doubles | Use known doubles as anchors | \(8 + 5 = 8 + 8 - 3 = 16 - 3 = 13\) |
| Compensation | Adjust one addend and compensate | \(8 + 5 = 10 + 5 - 2 = 13\) |
3.3 The Standard Algorithm
The standard addition algorithm works column by column from right to left, regrouping (carrying) when a column sum exceeds 9. [6] CCSS, 2010, 4.NBT.4
¹¹
4 6 7
+ 3 8 5
─────
8 5 2
Step 1: Ones: \(7 + 5 = 12\). Write 2, carry 1.
Step 2: Tens: \(6 + 8 + 1 = 15\). Write 5, carry 1.
Step 3: Hundreds: \(4 + 3 + 1 = 8\). Write 8.
What is \(256 + 178\)?
4. Subtraction
Subtraction is the inverse of addition. It has three conceptual models, each important for different types of problems. [10] Carpenter et al., 2015, Ch. 2
4.1 Three Models of Subtraction
| Model | Meaning | Example |
|---|---|---|
| Take-Away | Remove a quantity from a group | I had 9 apples. I ate 4. How many left? \(9 - 4 = 5\) |
| Comparison | Find the difference between two quantities | Sam has 12 books, Mia has 7. How many more does Sam have? \(12 - 7 = 5\) |
| Missing Addend | How much more to reach a target? | I have \$6. A toy costs \$10. How much more do I need? \(10 - 6 = 4\) |
[1] NRC, 2001, pp. 183–188
4.2 Subtraction with Regrouping (Borrowing)
4 9 13
5̶ 0̶ 3
− 2 6 7
──────
2 3 6
Step 1: Ones: Can’t do \(3 - 7\). Borrow from tens — but tens is 0, so borrow from hundreds first.
Step 2: Hundreds: \(5 \to 4\), tens: \(0 \to 10 \to 9\) (after lending 1 to ones), ones: \(3 \to 13\).
Step 3: Now subtract: \(13 - 7 = 6\), \(9 - 6 = 3\), \(4 - 2 = 2\). Answer: 236.
Children often subtract the smaller digit from the larger in each column regardless of position (e.g., computing \(52 - 37\) as \(55\) by doing \(7-2=5\) in the ones). This error reveals a lack of understanding of place value, not subtraction itself. [11] Van de Walle, J. A. et al., Elementary and Middle School Mathematics: Teaching Developmentally, Pearson, 10th ed., 2019, Ch. 12
Addition and subtraction are inverse operations. This means:
\( a + b = c \iff c - b = a \iff c - a = b \)
Understanding this relationship is critical for checking answers and for developing fact families (e.g., 3 + 5 = 8, 5 + 3 = 8, 8 − 3 = 5, 8 − 5 = 3). [6] CCSS, 2010, 1.OA.4
5. Multiplication
5.1 Multiplication as Repeated Addition
For whole numbers, \(a \times b\) means \(a\) groups of \(b\):
\( a \times b = \underbrace{b + b + b + \cdots + b}_{a \text{ times}} \)
[7] Wu, 2011, Ch. 3
Multiplication can also be understood through the array model: \(3 \times 4\) is a rectangular array with 3 rows and 4 columns, containing 12 objects total. This model naturally leads to the area model and connects multiplication to geometry. [2] NCTM, 2000, p. 151
5.2 Multiplication Tables (Times Tables)
Fluency with single-digit multiplication facts is essential. The CCSS expects fluency by end of Grade 3 (3.OA.7). [6] CCSS, 2010, 3.OA.7
| × | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| 2 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 |
| 3 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 | 33 | 36 |
| 4 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 | 44 | 48 |
| 5 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 |
| 6 | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 | 66 | 72 |
| 7 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 | 77 | 84 |
| 8 | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 | 88 | 96 |
| 9 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 | 99 | 108 |
| 10 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | 110 | 120 |
| 11 | 11 | 22 | 33 | 44 | 55 | 66 | 77 | 88 | 99 | 110 | 121 | 132 |
| 12 | 12 | 24 | 36 | 48 | 60 | 72 | 84 | 96 | 108 | 120 | 132 | 144 |
5.3 Properties of Multiplication
| Property | Statement | Example |
|---|---|---|
| Commutative | \(a \times b = b \times a\) | \(4 \times 7 = 7 \times 4 = 28\) |
| Associative | \((a \times b) \times c = a \times (b \times c)\) | \((2 \times 3) \times 5 = 2 \times (3 \times 5) = 30\) |
| Identity | \(a \times 1 = a\) | \(9 \times 1 = 9\) |
| Zero Property | \(a \times 0 = 0\) | \(6 \times 0 = 0\) |
| Distributive | \(a \times (b + c) = a \times b + a \times c\) | \(3 \times 14 = 3 \times 10 + 3 \times 4 = 42\) |
The distributive property is especially powerful: it underpins the standard multiplication algorithm and is the bridge between arithmetic and algebra. [7] Wu, 2011, pp. 51–58
5.4 Multi-Digit Multiplication
Decompose each factor: \(36 = 30 + 6\) and \(24 = 20 + 4\)
[6] CCSS, 2010, 4.NBT.5; [11] Van de Walle et al., 2019, Ch. 13
6. Division
6.1 Two Models of Division
Division is the inverse of multiplication. For whole numbers with \(b \neq 0\):
\( a \div b = c \iff b \times c = a \)
| Model | Question | Example |
|---|---|---|
| Partitive (Sharing) | If I share equally among groups, how many in each group? | 12 cookies shared among 3 friends: \(12 \div 3 = 4\) each |
| Measurement (Grouping) | How many groups of a certain size can I make? | 12 cookies, 3 per bag: \(12 \div 3 = 4\) bags |
[10] Carpenter et al., 2015, Ch. 3
6.2 Long Division
2 8 2 R1
┌──────
3 │ 8 4 7
│−6
│ 2 4
│−2 4
│ 0 7
│ −6
│ 1
\(847 \div 3 = 282\) remainder \(1\). Check: \(282 \times 3 + 1 = 846 + 1 = 847\) ✓
6.3 Understanding Remainders
For any whole numbers \(a\) (dividend) and \(b > 0\) (divisor), there exist unique whole numbers \(q\) (quotient) and \(r\) (remainder) such that:
\( a = b \times q + r \qquad \text{where } 0 \le r < b \)
[7] Wu, 2011, p. 95
The meaning of a remainder depends on the problem:
- Drop it: “How many full teams of 5 from 23 students?” → 4 teams (ignore R3)
- Round up: “How many buses for 23 students, 5 per bus?” → 5 buses (need 1 more for R3)
- The remainder IS the answer: “23 students in teams of 5; how many left over?” → 3 students
[11] Van de Walle et al., 2019, Ch. 13
7. Fractions
Fractions are widely regarded as the most challenging topic in elementary mathematics. The NRC calls them “the most important foundational skill not developed in whole-number arithmetic.” [12] NRC, How Students Learn: Mathematics in the Classroom, National Academies Press, 2005, Ch. 7
7.1 What Is a Fraction?
A fraction \(\dfrac{a}{b}\) (where \(b \neq 0\)) represents \(a\) parts out of a whole divided into \(b\) equal parts. The number \(a\) is the numerator (how many parts) and \(b\) is the denominator (how many equal parts in the whole).
[7] Wu, 2011, Ch. 12
On the number line, \(\dfrac{a}{b}\) is the point located at a distance of \(a\) copies of \(\dfrac{1}{b}\) from 0. This number line definition is mathematically precise and recommended by Wu (2011) as the foundation for all fraction work.
Types of Fractions
| Type | Description | Example |
|---|---|---|
| Proper | Numerator < denominator | \(\frac{3}{5}\) |
| Improper | Numerator ≥ denominator | \(\frac{7}{4}\) |
| Mixed Number | Whole number + proper fraction | \(1\frac{3}{4}\) |
| Unit Fraction | Numerator is 1 | \(\frac{1}{6}\) |
7.2 Equivalent Fractions
For any fraction \(\dfrac{a}{b}\) and any nonzero whole number \(n\):
\( \frac{a}{b} = \frac{a \times n}{b \times n} \)
Multiplying or dividing both numerator and denominator by the same nonzero number produces an equivalent fraction. [7] Wu, 2011, pp. 195–200
\( \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \qquad\qquad \frac{6}{8} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4} \)
7.3 Comparing Fractions
Three strategies for comparing fractions:
- Same denominator: Compare numerators. \(\frac{3}{7} < \frac{5}{7}\) because \(3 < 5\).
- Same numerator: The fraction with the smaller denominator is larger. \(\frac{3}{4} > \frac{3}{5}\) because fourths are bigger than fifths.
- Common denominator: Convert both to a common denominator, then compare. \(\frac{2}{3}\) vs. \(\frac{3}{5}\): convert to \(\frac{10}{15}\) vs. \(\frac{9}{15}\), so \(\frac{2}{3} > \frac{3}{5}\).
[6] CCSS, 2010, 3.NF.3d, 4.NF.2
7.4 Adding & Subtracting Fractions
To add or subtract fractions, they must have the same denominator:
\( \frac{a}{c} + \frac{b}{c} = \frac{a + b}{c} \qquad\qquad \frac{a}{c} - \frac{b}{c} = \frac{a - b}{c} \)
If denominators differ, find the least common denominator (LCD) first.
\( \frac{2}{3} + \frac{1}{4} \)
LCD of 3 and 4 is 12:
\( \frac{2}{3} = \frac{8}{12}, \qquad \frac{1}{4} = \frac{3}{12} \)
\( \frac{8}{12} + \frac{3}{12} = \frac{11}{12} \)
[6] CCSS, 2010, 5.NF.1–2
7.5 Multiplying Fractions
Multiply numerators and multiply denominators:
\( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \)
\( \frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15} \)
The area model gives intuition: \(\frac{2}{3} \times \frac{4}{5}\) asks “what is \(\frac{2}{3}\) of \(\frac{4}{5}\)?” Draw a rectangle, divide horizontally into 5 parts and shade 4; then divide vertically into 3 parts and shade 2. The doubly-shaded region represents \(\frac{8}{15}\) of the whole. [6] CCSS, 2010, 5.NF.4; [11] Van de Walle et al., 2019, Ch. 15
8. Decimals
8.1 Decimal Place Value
A decimal number extends the base-ten system to the right of the ones place. Each position to the right is worth one-tenth of the previous position.
| Place | Value | Fraction |
|---|---|---|
| Ones | 1 | \(10^0\) |
| Tenths | 0.1 | \(\frac{1}{10} = 10^{-1}\) |
| Hundredths | 0.01 | \(\frac{1}{100} = 10^{-2}\) |
| Thousandths | 0.001 | \(\frac{1}{1000} = 10^{-3}\) |
\(3.47 = 3 + \frac{4}{10} + \frac{7}{100} = 3 + 0.4 + 0.07\)
8.2 Operations with Decimals
Adding & Subtracting Decimals
Line up the decimal points, then add or subtract as with whole numbers: [6] CCSS, 2010, 5.NBT.7
12.50
+ 3.75
──────
16.25
Multiplying Decimals
Multiply as whole numbers, then place the decimal point by counting the total decimal places in both factors:
\(2.5 \times 0.3\): Compute \(25 \times 3 = 75\). Total decimal places: \(1 + 1 = 2\). Answer: \(0.75\).
8.3 Decimals & Fractions Connection
Every decimal is a fraction with a denominator that is a power of 10:
\( 0.7 = \frac{7}{10}, \qquad 0.25 = \frac{25}{100} = \frac{1}{4}, \qquad 0.125 = \frac{125}{1000} = \frac{1}{8} \)
To convert a fraction to a decimal, divide the numerator by the denominator.
| Fraction | Decimal | Percent |
|---|---|---|
| \(\frac{1}{2}\) | 0.5 | 50% |
| \(\frac{1}{4}\) | 0.25 | 25% |
| \(\frac{3}{4}\) | 0.75 | 75% |
| \(\frac{1}{5}\) | 0.2 | 20% |
| \(\frac{1}{3}\) | \(0.\overline{3}\) | 33.3...% |
| \(\frac{1}{8}\) | 0.125 | 12.5% |
[7] Wu, 2011, Ch. 17–18
9. Geometry
Geometry in elementary school develops spatial reasoning — the ability to visualize, reason about, and manipulate shapes. The van Hiele model describes five levels of geometric thinking, from visual recognition to rigorous proof. [13] Van Hiele, P. M., Structure and Insight: A Theory of Mathematics Education, Academic Press, 1986
9.1 Two-Dimensional Shapes
| Shape | Properties | Formula Notes |
|---|---|---|
| Triangle | 3 sides, 3 angles; angle sum = 180° | Area = \(\frac{1}{2} \times b \times h\) |
| Rectangle | 4 sides, 4 right angles (90°); opposite sides equal | Area = \(l \times w\) |
| Square | 4 equal sides, 4 right angles; special rectangle | Area = \(s^2\) |
| Parallelogram | Opposite sides parallel and equal | Area = \(b \times h\) |
| Trapezoid | Exactly one pair of parallel sides | Area = \(\frac{1}{2}(b_1+b_2) \times h\) |
| Circle | All points equidistant from center | Area = \(\pi r^2\), Circumference = \(2\pi r\) |
| Pentagon | 5 sides | — |
| Hexagon | 6 sides | — |
| Octagon | 8 sides | — |
Classifying Triangles
| By Sides | By Angles |
|---|---|
| Equilateral — 3 equal sides | Acute — all angles < 90° |
| Isosceles — 2 equal sides | Right — one angle = 90° |
| Scalene — no equal sides | Obtuse — one angle > 90° |
[6] CCSS, 2010, 3.G.1, 4.G.1–2, 5.G.3–4
9.2 Three-Dimensional Shapes
| Shape | Faces | Edges | Vertices |
|---|---|---|---|
| Cube | 6 | 12 | 8 |
| Rectangular Prism | 6 | 12 | 8 |
| Triangular Prism | 5 | 9 | 6 |
| Square Pyramid | 5 | 8 | 5 |
| Cylinder | 3 | 2 | 0 |
| Cone | 2 | 1 | 1 |
| Sphere | 1 | 0 | 0 |
For any convex polyhedron:
\( V - E + F = 2 \)
where \(V\) = vertices, \(E\) = edges, \(F\) = faces. Verify with a cube: \(8 - 12 + 6 = 2\) ✓ [14] Euler, L., “Elementa doctrinae solidorum,” Novi Commentarii Academiae Scientiarum Petropolitanae, 1758; see also Cromwell, P., Polyhedra, Cambridge University Press, 1997
9.3 Perimeter & Area
Perimeter is the total distance around a shape (measured in linear units: cm, m, in, ft).
Area is the amount of surface a shape covers (measured in square units: cm², m², in², ft²).
A rectangle is 8 cm long and 5 cm wide.
- Perimeter: \(P = 2(8) + 2(5) = 16 + 10 = 26\) cm
- Area: \(A = 8 \times 5 = 40\) cm²
Students frequently confuse perimeter and area. Research shows that hands-on activities — such as building shapes with unit tiles (area) versus surrounding shapes with string (perimeter) — help distinguish the two concepts. [11] Van de Walle et al., 2019, Ch. 19
[6] CCSS, 2010, 3.MD.5–8
9.4 Symmetry
A shape has a line of symmetry if it can be folded along that line so that the two halves match exactly (reflective symmetry).
| Shape | Lines of Symmetry |
|---|---|
| Equilateral triangle | 3 |
| Square | 4 |
| Rectangle (non-square) | 2 |
| Regular pentagon | 5 |
| Regular hexagon | 6 |
| Circle | Infinite |
[6] CCSS, 2010, 4.G.3
9.5 Angles
An angle is formed by two rays sharing a common endpoint (vertex). Angles are measured in degrees (°). A full rotation is 360°.
| Type | Measure | Example |
|---|---|---|
| Acute | \(0° < \theta < 90°\) | 45° |
| Right | \(\theta = 90°\) | Corner of a book |
| Obtuse | \(90° < \theta < 180°\) | 120° |
| Straight | \(\theta = 180°\) | A straight line |
| Reflex | \(180° < \theta < 360°\) | 270° |
[6] CCSS, 2010, 4.MD.5–7
10. Measurement
Measurement connects mathematics to the physical world. It involves assigning a numerical value to an attribute of an object using a chosen unit. [2] NCTM, 2000, pp. 170–175
10.1 Length
| System | Units | Relationships |
|---|---|---|
| Metric (SI) | mm, cm, m, km | 1 km = 1,000 m; 1 m = 100 cm; 1 cm = 10 mm |
| US Customary | in, ft, yd, mi | 1 mi = 5,280 ft; 1 yd = 3 ft; 1 ft = 12 in |
[15] Thompson, A. & Taylor, B. N., Guide for the Use of the International System of Units (SI), NIST Special Publication 811, 2008
10.2 Weight & Mass
| System | Units | Relationships |
|---|---|---|
| Metric | mg, g, kg | 1 kg = 1,000 g; 1 g = 1,000 mg |
| US Customary | oz, lb, ton | 1 ton = 2,000 lb; 1 lb = 16 oz |
10.3 Capacity & Volume
| System | Units | Relationships |
|---|---|---|
| Metric | mL, L | 1 L = 1,000 mL |
| US Customary | fl oz, cup, pt, qt, gal | 1 gal = 4 qt; 1 qt = 2 pt; 1 pt = 2 cups; 1 cup = 8 fl oz |
\( V = l \times w \times h \)
Volume is measured in cubic units (cm³, m³, in³, ft³). [6] CCSS, 2010, 5.MD.3–5
10.4 Unit Conversion
To convert between units, multiply or divide by the appropriate conversion factor:
\( 3.5 \text{ km} = 3.5 \times 1{,}000 = 3{,}500 \text{ m} \)
\( 48 \text{ in} = 48 \div 12 = 4 \text{ ft} \)
In the metric system, conversions always involve powers of 10 — this is by design. [15] NIST SP 811, 2008
[6] CCSS, 2010, 4.MD.1–2, 5.MD.1
11. Data & Graphs
Data analysis at the elementary level introduces children to the fundamental statistical question: “What does the data tell us?” [16] Franklin, C. et al., Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report, ASA, 2007
11.1 Collecting Data
The four steps of a statistical investigation:
- Formulate a question — “What is the most popular fruit in our class?”
- Collect data — Survey, observe, or measure
- Analyze data — Organize, summarize, look for patterns
- Interpret results — Answer the question and communicate findings
[16] GAISE Report, 2007, p. 11
11.2 Types of Graphs
| Graph Type | Best For | Grade Introduced (CCSS) |
|---|---|---|
| Picture Graph (Pictograph) | Categorical data with symbols | Grade 2 |
| Bar Graph | Comparing categories | Grade 2–3 |
| Line Plot (Dot Plot) | Showing frequency of measurements | Grade 2 |
| Tally Chart | Counting observations | Grade 1–2 |
| Line Graph | Showing change over time | Grade 4–5 |
| Pie Chart (Circle Graph) | Parts of a whole | Grade 5 (extension) |
11.3 Interpreting Data
Mean (Average): \(\bar{x} = \dfrac{\text{sum of all values}}{\text{number of values}}\)
Median: The middle value when data is ordered.
Mode: The value that appears most often.
Range: \(\text{Range} = \text{maximum} - \text{minimum}\)
Test scores: 78, 85, 85, 90, 92
- Mean: \(\frac{78+85+85+90+92}{5} = \frac{430}{5} = 86\)
- Median: 85 (middle value)
- Mode: 85 (appears twice)
- Range: \(92 - 78 = 14\)
[6] CCSS, 2010, 6.SP.1–5 (introduced formally in Grade 6, but conceptual groundwork is laid in Grades 3–5)
12. Patterns & Sequences
“Mathematics is the science of patterns” — this idea, articulated by mathematician Lynn Steen, captures the essence of algebraic thinking that begins in the earliest grades. [17] Steen, L. A., “The Science of Patterns,” Science, Vol. 240, 1988, pp. 611–616
12.1 Number Patterns
| Pattern | Sequence | Rule |
|---|---|---|
| Skip counting by 2 | 2, 4, 6, 8, 10, 12, … | Add 2 |
| Skip counting by 5 | 5, 10, 15, 20, 25, … | Add 5 |
| Skip counting by 10 | 10, 20, 30, 40, 50, … | Add 10 |
| Doubling | 1, 2, 4, 8, 16, 32, … | Multiply by 2 |
| Square numbers | 1, 4, 9, 16, 25, 36, … | \(n^2\) |
| Triangular numbers | 1, 3, 6, 10, 15, 21, … | \(\frac{n(n+1)}{2}\) |
| Fibonacci-like | 1, 1, 2, 3, 5, 8, 13, … | Add previous two |
12.2 Shape Patterns
Shape patterns involve repeating or growing arrangements of geometric figures:
▲ ● ■ ▲ ● ■ ▲ ● ■ …
The unit of repeat is: ▲ ● ■ (ABC pattern)
Step 1: ■ (1 square)
Step 2: ■■■ (3 squares)
Step 3: ■■■■■ (5 squares)
Rule: add 2 squares each step. The \(n\)th step has \(2n - 1\) squares.
12.3 Input–Output Rules (Early Functions)
An input–output rule (or “function machine”) takes a number in, applies a consistent rule, and produces a number out. This is the earliest encounter with the concept of a function.
| Input | Rule | Output |
|---|---|---|
| 2 | Multiply by 3, then add 1 \(f(x) = 3x + 1\) | 7 |
| 5 | 16 | |
| 10 | 31 | |
| 0 | 1 |
[6] CCSS, 2010, 4.OA.5; [2] NCTM, 2000, pp. 158–163
13. Time & Money
13.1 Telling Time
| Skill | Precision | Grade (CCSS) |
|---|---|---|
| Tell time to the hour and half hour | \(\pm 30\) min | Grade 1 |
| Tell time to the nearest 5 minutes | \(\pm 5\) min | Grade 2 |
| Tell time to the nearest minute | \(\pm 1\) min | Grade 3 |
1 day = 24 hours — 1 hour = 60 minutes — 1 minute = 60 seconds
a.m. = before noon (ante meridiem) — p.m. = after noon (post meridiem)
[6] CCSS, 2010, 1.MD.3, 2.MD.7, 3.MD.1
13.2 Elapsed Time
A movie starts at 2:45 p.m. and ends at 4:20 p.m. How long is the movie?
Method (counting up):
2:45 → 3:00 = 15 minutes
3:00 → 4:00 = 60 minutes
4:00 → 4:20 = 20 minutes
Total: \(15 + 60 + 20 = 95\) minutes = 1 hour 35 minutes
13.3 Coins & Bills
| Coin/Bill | Value | Equivalents |
|---|---|---|
| Penny | $0.01 (1¢) | — |
| Nickel | $0.05 (5¢) | 5 pennies |
| Dime | $0.10 (10¢) | 2 nickels or 10 pennies |
| Quarter | $0.25 (25¢) | 5 nickels or 25 pennies |
| Half Dollar | $0.50 (50¢) | 2 quarters |
| Dollar Bill | $1.00 | 4 quarters or 10 dimes or 100 pennies |
13.4 Making Change
An item costs $3.67. You pay with $5.00. What is the change?
Count up from $3.67:
$3.67 + 3¢ = $3.70 (3 pennies)
$3.70 + 30¢ = $4.00 (1 quarter + 1 nickel, or 3 dimes)
$4.00 + $1.00 = $5.00 (1 dollar bill)
Change: $1.33
[6] CCSS, 2010, 2.MD.8
14. Problem Solving
Problem solving is the heart of mathematics. George Pólya’s four-step framework, first published in 1945, remains the gold standard for approaching mathematical problems. [18] Pólya, G., How to Solve It, Princeton University Press, 1945 (2nd ed. 1957)
14.1 Pólya’s Four Steps
| Step | Description | Key Questions |
|---|---|---|
| 1. Understand | Read carefully; identify what is given and what is asked | What do I know? What do I need to find? Can I restate the problem? |
| 2. Plan | Choose a strategy | Have I seen a similar problem? What strategy fits? |
| 3. Execute | Carry out the plan carefully | Am I following my plan? Is each step correct? |
| 4. Reflect | Check the answer; consider other approaches | Does my answer make sense? Can I verify it? Could I solve it differently? |
Common Strategies
- Draw a picture or diagram
- Make a table or organized list
- Look for a pattern
- Guess and check (with systematic refinement)
- Work backwards
- Use simpler numbers first
- Act it out or use manipulatives
- Write an equation
[18] Pólya, 1945; [2] NCTM, 2000, pp. 116–121
14.2 Word Problem Types (CGI Framework)
Cognitively Guided Instruction (CGI) classifies addition and subtraction word problems into 11 types. Understanding the semantic structure of a problem is more important than identifying a keyword. [10] Carpenter et al., 2015, Ch. 2–3
| Category | Type | Example |
|---|---|---|
| Join | Result Unknown | Tom had 5 marbles. He found 3 more. How many now? (5 + 3 = ?) |
| Change Unknown | Tom had 5 marbles. He found some more and now has 8. How many did he find? (5 + ? = 8) | |
| Separate | Result Unknown | Tom had 8 marbles. He gave away 3. How many left? (8 − 3 = ?) |
| Start Unknown | Tom had some marbles. He gave away 3 and has 5 left. How many did he start with? (? − 3 = 5) | |
| Compare | Difference Unknown | Tom has 8 marbles. Jen has 5. How many more does Tom have? (8 − 5 = ?) |
| Part-Part-Whole | Whole Unknown | Tom has 5 red and 3 blue marbles. How many total? (5 + 3 = ?) |
Teaching students to look for “keywords” (e.g., “in all” means add, “left” means subtract) is unreliable and produces incorrect answers in many problem types. Research strongly recommends teaching problem structure instead. [19] Schoenfeld, A. H., Mathematical Problem Solving, Academic Press, 1985, pp. 24–31
14.3 Multi-Step Problems
A baker makes 4 trays of muffins. Each tray holds 12 muffins. She sells 29 muffins in the morning and 15 in the afternoon. How many muffins does she have left?
Step 1: Total muffins: \(4 \times 12 = 48\)
Step 2: Total sold: \(29 + 15 = 44\)
Step 3: Remaining: \(48 - 44 = 4\) muffins
Multi-step problems require students to determine which operations to use and in what order — a key part of mathematical modeling. [6] CCSS, 2010, 3.OA.8, 4.OA.3
A library has 3 shelves. The first shelf has 28 books, the second has twice as many as the first, and the third has 15 fewer than the second. How many books are there in total?
Solution:
Shelf 1: 28
Shelf 2: \(28 \times 2 = 56\)
Shelf 3: \(56 - 15 = 41\)
Total: \(28 + 56 + 41 = 125\) books
References & Citations
All content on this page is grounded in peer-reviewed research and authoritative educational standards. Below is the complete reference list:
- National Research Council. Adding It Up: Helping Children Learn Mathematics. Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). Washington, DC: National Academies Press, 2001. ISBN: 978-0-309-06995-3
- National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000. ISBN: 978-0-87353-480-2
- Clements, D. H. & Sarama, J. Learning and Teaching Early Math: The Learning Trajectories Approach. 2nd ed. New York: Routledge, 2014. ISBN: 978-0-415-82850-5
- Zaslavsky, C. Africa Counts: Number and Pattern in African Cultures. 3rd ed. Chicago: Lawrence Hill Books, 1999. ISBN: 978-1-55652-350-2
- Gelman, R. & Gallistel, C. R. The Child’s Understanding of Number. Cambridge, MA: Harvard University Press, 1978. ISBN: 978-0-674-11636-2
- National Governors Association Center for Best Practices & Council of Chief State School Officers. Common Core State Standards for Mathematics. Washington, DC: Authors, 2010.
- Wu, H.-H. Understanding Numbers in Elementary School Mathematics. Providence, RI: American Mathematical Society, 2011. ISBN: 978-0-8218-5260-6
- Ifrah, G. The Universal History of Numbers: From Prehistory to the Invention of the Computer. Translated by D. Bellos et al. New York: Wiley, 2000. ISBN: 978-0-471-37568-5
- Baroody, A. J. Children’s Mathematical Thinking: A Developmental Framework for Preschool, Primary, and Special Education Teachers. New York: Teachers College Press, 1987. ISBN: 978-0-8077-2837-5
- Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. Children’s Mathematics: Cognitively Guided Instruction. 2nd ed. Portsmouth, NH: Heinemann, 2015. ISBN: 978-0-325-07834-9
- Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. Elementary and Middle School Mathematics: Teaching Developmentally. 10th ed. New York: Pearson, 2019. ISBN: 978-0-13-480208-5
- National Research Council. How Students Learn: Mathematics in the Classroom. Donovan, M. S. & Bransford, J. D. (Eds.). Washington, DC: National Academies Press, 2005. ISBN: 978-0-309-08949-4
- Van Hiele, P. M. Structure and Insight: A Theory of Mathematics Education. Orlando, FL: Academic Press, 1986. ISBN: 978-0-12-714060-0
- Euler, L. “Elementa doctrinae solidorum.” Novi Commentarii Academiae Scientiarum Petropolitanae, Vol. 4, 1758, pp. 109–140. See also: Cromwell, P. R. Polyhedra. Cambridge: Cambridge University Press, 1997. ISBN: 978-0-521-66405-7
- Thompson, A. & Taylor, B. N. Guide for the Use of the International System of Units (SI). NIST Special Publication 811. Gaithersburg, MD: NIST, 2008.
- Franklin, C. et al. Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report: A Pre-K–12 Curriculum Framework. Alexandria, VA: American Statistical Association, 2007.
- Steen, L. A. “The Science of Patterns.” Science, Vol. 240, No. 4852, 1988, pp. 611–616.
- Pólya, G. How to Solve It: A New Aspect of Mathematical Method. Princeton, NJ: Princeton University Press, 1945 (2nd ed. 1957). ISBN: 978-0-691-11966-3
- Schoenfeld, A. H. Mathematical Problem Solving. Orlando, FL: Academic Press, 1985. ISBN: 978-0-12-628870-4
This page aligns with the Common Core State Standards for Mathematics (CCSSM), adopted by the majority of U.S. states, as well as the NCTM Principles and Standards (2000) and the NCTM Focal Points (2006). International standards from Singapore Math, the UK National Curriculum, and the Australian Curriculum share substantial overlap in content and sequencing at this level.