Subjects › Elementary Arithmetic
Elementary Arithmetic: counting through word problems, explained step by step
This course covers the arithmetic an elementary student needs: counting and place value, the four operations on whole numbers, fractions, decimals, basic units of length/weight/capacity/time, simple 2D and 3D shapes, perimeter and area, and reading graphs. It starts from counting and number recognition and builds up, step by step, to long division with two-digit divisors, multiplying and dividing fractions and decimals, and multi-step word problems.
A lesson works like a conversation. The tutor asks a question pitched at what the student already knows, the student answers (even a partial or wrong answer gives something to work with), and the tutor responds to that specific answer before asking the next question. Instead of watching a method demonstrated, the student does the counting, the regrouping, the common-denominator finding, out loud, with the tutor checking at each step whether the idea makes sense and whether the student can actually carry it out.
Who it is for
- Elementary school students starting from counting and number recognition
- Students who need to firm up basic facts or the standard algorithms before moving on
- Parents who want to see exactly what their child can explain and what they can do
- Homeschool teachers looking for a sequenced arithmetic curriculum with built-in checking
- Students preparing for fractions, decimals, or long division who need the earlier topics solid first
What you will be able to do
- Add, subtract, multiply, and divide multi-digit whole numbers using the standard algorithms
- Compare, add, subtract, multiply, and divide fractions and decimals, and convert between them
- Do long division with one- and two-digit divisors, including interpreting remainders
- Measure length, weight, capacity, and time, and convert between units in the same system
- Find the perimeter and area of rectangles and simple composite shapes
- Read bar graphs, picture graphs, and line plots, and solve multi-step word problems involving money
A taste of a lesson on Adding and Subtracting Fractions
The tutor asks, you answer, and it works from what you actually said. No videos to sit through.
The 28 topics in Elementary Arithmetic
Each topic builds on the ones before it. The tutor checks what you already know and starts there.
Counting and Number Recognition
Counting objects one-to-one up to 100 and beyond, recognizing and writing numerals, and understanding that each number represents a quantity. Students should be able to count forward and backward and match a numeral to a set of objects.
Place Value
Understanding that digits represent different values depending on their position (ones, tens, hundreds, thousands, etc.), and being able to read, write, and decompose numbers using place value (e.g., 345 = 300+40+5). Includes comparing and ordering multi-digit numbers.
Builds on Counting and Number Recognition
Addition of Whole Numbers
Understanding addition as combining quantities, mastering basic addition facts, and learning the standard algorithm for adding multi-digit numbers including regrouping (carrying). Students should understand why carrying works using place value.
Builds on Place Value
Subtraction of Whole Numbers
Understanding subtraction as finding the difference or taking away, mastering basic subtraction facts, and learning the standard algorithm for multi-digit subtraction including regrouping (borrowing). Students should understand subtraction as the inverse of addition.
Builds on Addition of Whole Numbers
Multiplication as Repeated Addition
Understanding multiplication as repeated addition or equal groups, learning multiplication tables (facts up to 10x10 or 12x12), and recognizing the commutative property of multiplication through arrays.
Builds on Addition of Whole Numbers
Multi-Digit Multiplication
Learning the standard algorithm for multiplying multi-digit numbers by single and multi-digit numbers, understanding how place value and partial products combine to give the final answer.
Builds on Place Value, Multiplication as Repeated Addition
Division as Sharing and Grouping
Understanding division as splitting into equal groups or equal sharing, connecting division to multiplication as its inverse operation, and learning basic division facts.
Builds on Multiplication as Repeated Addition
Long Division by a One-Digit Divisor
Learning the standard long division algorithm for dividing multi-digit numbers by a single-digit divisor, including interpreting remainders in context (e.g., as leftovers, fractions, or rounding up).
Builds on Subtraction of Whole Numbers, Division as Sharing and Grouping
Long Division by a Two-Digit Divisor
Extending the long division algorithm to two-digit divisors, including estimating quotient digits, multiplying and subtracting at each step, and checking answers by multiplying back.
Builds on Multi-Digit Multiplication, Long Division by a One-Digit Divisor
Factors and Multiples
Finding all factors of a number, identifying multiples, distinguishing prime and composite numbers, and finding the greatest common factor and least common multiple of small numbers, using place value to compare and order the factors and multiples found.
Builds on Place Value, Multiplication as Repeated Addition, Division as Sharing and Grouping
Introducing Fractions
Understanding a fraction as a part of a whole and as a point on a number line, reading and writing fractions, and identifying equivalent fractions using visual models like fraction bars and pie charts.
Builds on Division as Sharing and Grouping
Comparing and Ordering Fractions
Comparing fractions with like and unlike denominators using common denominators or cross-multiplication, and ordering sets of fractions from least to greatest.
Builds on Factors and Multiples, Introducing Fractions
Adding and Subtracting Fractions
Adding and subtracting fractions with like and unlike denominators by finding common denominators, and adding/subtracting mixed numbers including regrouping when necessary.
Builds on Comparing and Ordering Fractions
Multiplying Fractions
Multiplying fractions by fractions and by whole numbers, understanding multiplication by a fraction as scaling a quantity, and simplifying products to lowest terms.
Builds on Multi-Digit Multiplication, Adding and Subtracting Fractions
Dividing Fractions
Dividing a fraction by a fraction or whole number using the reciprocal (invert and multiply) method, understanding why this method works through the relationship between multiplication and division, and simplifying results.
Builds on Long Division by a One-Digit Divisor, Multiplying Fractions
Introducing Decimals
Understanding decimal place value (tenths, hundredths, thousandths), reading and writing decimals, and connecting decimals to fractions with denominators of 10, 100, 1000.
Builds on Place Value, Introducing Fractions
Comparing, Ordering, and Rounding Decimals
Comparing and ordering decimal numbers, and rounding decimals to a given place value, using place value reasoning and number lines.
Builds on Introducing Decimals
Adding and Subtracting Decimals
Adding and subtracting decimal numbers by aligning place values, including problems with different numbers of decimal places.
Builds on Addition of Whole Numbers, Subtraction of Whole Numbers, Comparing, Ordering, and Rounding Decimals
Multiplying Decimals
Multiplying decimals by whole numbers and by other decimals, placing the decimal point correctly in the product using place value reasoning and estimation to check reasonableness.
Builds on Multi-Digit Multiplication, Adding and Subtracting Decimals
Dividing Decimals
Dividing decimals by whole numbers and by other decimals, including shifting the decimal point in both the divisor and dividend to turn a decimal divisor into a whole number before dividing.
Builds on Long Division by a Two-Digit Divisor, Multiplying Decimals
Converting Between Fractions and Decimals
Converting fractions to decimals through division and decimals to fractions using place value, recognizing common equivalents (1/2=0.5, 1/4=0.25, etc.).
Builds on Dividing Fractions, Dividing Decimals
Units of Length
Identifying and using basic units of length (inches, feet, yards, centimeters, meters), comparing lengths, and estimating and measuring the length of everyday objects with a ruler. This introduces the idea of a unit without requiring conversion across a full system.
Builds on Place Value
Units of Measurement and Conversion
Converting between units of length, weight/mass, capacity, and time within a single system (e.g., inches/feet/yards, grams/kilograms, minutes/hours), and choosing appropriate units for a given quantity, using multiplication and division of whole numbers to convert.
Builds on Multi-Digit Multiplication, Long Division by a One-Digit Divisor, Units of Length
2D and 3D Shapes and Their Properties
Identifying and naming common 2D shapes (triangle, square, rectangle, pentagon, hexagon, circle) and 3D shapes (cube, sphere, cylinder, cone, rectangular prism), and describing their properties such as number of sides, edges, vertices, and faces.
Builds on Counting and Number Recognition
Money and Making Change
Identifying coins and bills, counting money amounts, adding and subtracting money values, and solving problems involving making change, connecting to decimal place value.
Builds on Adding and Subtracting Decimals
Perimeter and Area of Basic Shapes
Calculating the perimeter of polygons by adding side lengths, and the area of rectangles and simple composite shapes by counting unit squares or multiplying length by width, using units of length and basic shape properties.
Builds on Multi-Digit Multiplication, Units of Length, 2D and 3D Shapes and Their Properties
Reading and Making Basic Graphs
Reading and interpreting data from picture graphs, bar graphs, and simple line plots, and organizing data into tables and graphs. Includes finding basic statistics like mode and range from small data sets.
Builds on Counting and Number Recognition, Addition of Whole Numbers
Solving Multi-Step Word Problems
Translating real-world situations into arithmetic operations, choosing the correct operation(s) to use, and solving multi-step word problems involving whole numbers, fractions, decimals, and money.
Builds on Dividing Fractions, Dividing Decimals, Money and Making Change
Practice questions in Elementary Arithmetic
A few of the kinds of questions you will work through, with answers.
In the number 4,costume - wait, in the number 4,528, what digit is in the hundreds place?
See the answer
5
Counting from the right, the places are ones, tens, hundreds, so the hundreds digit in 4,528 is 5.
What is 34 x 26?
See the answer
884
34 x 20 = 680 and 34 x 6 = 204; adding the partial products gives 680 + 204 = 884.
Which is greater, 3/5 or 5/8?
See the answer
5/8
Using a common denominator of 40, 3/5 = 24/40 and 5/8 = 25/40, so 5/8 is greater.
What is 7.2 divided by 0.6?
See the answer
12
Multiplying both numbers by 10 gives 72 divided by 6, which is 12.
A rectangle is 9 cm long and 4 cm wide. What is its area and its perimeter?
See the answer
Area 36 cm², perimeter 26 cm
Area is length times width (9 x 4 = 36), and perimeter is the sum of all sides (9 + 4 + 9 + 4 = 26).
A notebook costs 2.75 and a pen costs 1.20. If you pay with a $5 bill, how much change do you get?
See the answer
$1.05
The two items cost 2.75 + 1.20 = 3.95, and 5.00 minus 3.95 leaves 1.05 in change.
Questions about this subject
Does this course cover negative numbers or percents?
No. It stops before negative numbers, ratios and proportions, percents, exponents, and pre-algebra, ending with multi-step word problems using whole numbers, fractions, decimals, and money.
What math does my child need to know before starting?
None beyond recognizing numbers and counting small quantities. The course begins with counting and number recognition and builds from there.
Does it cover two-digit long division and fraction division?
Yes, those are later topics in the sequence, after long division by a one-digit divisor and multiplying fractions are solid.
How does the tutor know if my child actually understands a topic, not just the answer?
The tutor asks the student to explain their reasoning as well as carry out the steps, and comes back to earlier topics later to check the idea and the method both stuck.
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