Open Middle Problems
Problems with a closed beginning, a closed end, but an open middle. They encourage deeper thinking, multiple strategies and mathematical reasoning.
How Open Middle Problems work
Everyone starts with the same problem and aims for the same goal, but there are lots of ways to get there.
Each problem ends with a challenge: can you find the biggest, smallest or closest answer? There is always a better answer to hunt for.
I first heard about Open Middle Problems from Robert Kaplinsky on my podcast. The original collection is at openmiddle.com.
Four ways to make them work in class
The problem does most of the work. These four routines do the rest.
- 1
Try one together
Choose some numbers as a class and test them, so everyone understands the rules and structure of the problem.
- 2
Use mini-whiteboards
These problems get messy. Students will try numbers, rub them out and try again, which is much harder in an exercise book.
- 3
Use the 4-2 approach
Four minutes of silent, independent work to find a “best” answer, then two minutes sharing strategies with a partner.
- 4
Track the best answer
Stop regularly, ask who has the best answer, get everyone to check it, write it on the board, then challenge the class to beat it.
No problem sets match
Try fewer words, or a different word for the same thing.
Number
- Place valueOrdering integersKS1KS2
6 problems, from “Two-digit neighbours” up to “Multiply to match”.
- Mental arithmeticNumber bondsKS1KS2
6 problems, from “Two numbers make a total” up to “Into the hundreds”.
- Mental arithmeticTimes tablesKS1KS2
6 problems, from “Single digit product” up to “Two-digit times one-digit”.
- Written methodsWritten additionKS1KS2
6 problems, from “Two-digit plus single digit” up to “The classic”.
- Written methodsWritten subtractionKS1KS2
6 problems, from “Two-digit minus single digit” up to “Vanishing difference”.
- Written methods
6 problems, from “Two-digit times single digit (small)” up to “The classic”.
- Factors, multiples and primesFactorsKS2KS3
6 problems, from “Two factors” up to “Two two-digit factors”.
- Factors, multiples and primesMultiplesKS2KS3
6 problems, from “Two multiples” up to “All nine digits”.
- Factors, multiples and primesProduct of prime factorsKS2KS3
6 problems, from “Product of 2 primes” up to “Three-digit, 7 primes”.
- Factors, multiples and primes
6 problems, from “Two two-digit numbers” up to “All nine, three-digit HCF”.
- Factors, multiples and primes
6 problems, from “Two single digits” up to “Three-digit LCM”.
- Negative numbers
6 problems, from “Two negative, two positive” up to “All two-digit negative”.
- Negative numbers
6 problems, from “Negative + negative” up to “Three-digit negative result”.
- Negative numbers
6 problems, from “Subtracting a negative” up to “Three-digit result”.
- Negative numbers
6 problems, from “Positive × negative” up to “Three-digit positive”.
- Negative numbers
6 problems, from “Positive ÷ negative” up to “Three-digit, neg ÷ neg”.
- Rounding and estimation
6 problems, from “Round to the nearest 10” up to “Triple constraint”.
- Rounding and estimation
6 problems, from “Round to 1 decimal place” up to “Triple constraint”.
- Rounding and estimation
6 problems, from “Round to 1 significant figure” up to “Triple constraint”.
- Fractions
6 problems, from “Equivalent pair” up to “Three equivalent fractions”.
- Fractions
6 problems, from “Two fractions make one” up to “Three fractions make two”.
- Fractions
6 problems, from “Integer times a fraction” up to “Three fractions multiply to one”.
- FractionsDividing with fractionsKS2KS3
6 problems, from “Integer divided by a fraction” up to “Chain division equals one”.
- DecimalsDecimal multiplicationKS2KS3
6 problems, from “Integer × one decimal place” up to “All nine digits”.
- DecimalsDecimal divisionKS2KS3
6 problems, from “Integer ÷ integer = decimal” up to “All seven digits”.
- Decimals
6 problems, from “Fraction to decimal” up to “Simplify to a decimal”.
- PercentagesPercentage of an amountKS2KS3
6 problems, from “Single-digit percentage” up to “All nine digits”.
- Percentages
6 problems, from “Percentage increase” up to “The multiplier connection”.
- PercentagesReverse percentagesKS3GCSE
6 problems, from “Reverse an increase” up to “Three-digit reverse decrease”.
- Ratio
6 problems, from “Equivalent single-digit ratios” up to “All nine digits”.
- Indices, roots and surds
6 problems, from “Multiply: add exponents” up to “Power of a power + multiply”.
- Indices, roots and surds
6 problems, from “Negative from division” up to “Combine two power-of-powers”.
- Indices, roots and surds
6 problems, from “Simplify a two-digit surd” up to “Write back under one root”.
- Indices, roots and surds
6 problems, from “Multiply like surds” up to “Multiply unlike surds”.
- Indices, roots and surds
6 problems, from “Two-digit coefficient” up to “Bigger calculation”.
- Indices, roots and surds
6 problems, from “Add (same power)” up to “Subtract with exchange”.
- Indices, roots and surds
6 problems, from “Multiply (simple)” up to “Divide with decimals”.
Algebra
- Introduction to algebra
6 problems, from “One variable” up to “Three x-terms”.
- Introduction to algebra
6 problems, from “Linear expression” up to “Full quadratic”.
- SequencesSimple number sequencesKS2KS3
6 problems, from “Going up by the same amount” up to “Add the last two”.
- Sequences
6 problems, from “Finding a term” up to “Which position?”.
- Linear equations
6 problems, from “Addition equation” up to “Two-digit divisor”.
- Linear equations
6 problems, from “Coefficient and addition” up to “Fraction with expression”.
- Linear equations
6 problems, from “Both sides add” up to “Mixed signs, two-digit”.
- Quadratic equations
6 problems, from “Positive adjustment” up to “Two-digit remainder”.
- Quadratic equations
6 problems, from “Rearrange: simple” up to “The trap”.
- Other equations
6 problems, from “Add the equations” up to “Same coefficient”.
- Coordinates
6 problems, from “Translate right” up to “Rectangle perimeter”.
Also in: Perimeter and area
- CoordinatesMidpoint of two pointsKS3GCSE
6 problems, from “Same x-coordinate” up to “Ratio splitting”.
- Straight-line graphs
6 problems, from “Rise and run” up to “Three collinear points”.
- Straight-line graphs
6 problems, from “Point on a line” up to “Intersecting lines”.
Also in: Other equations
- Straight-line graphs
6 problems, from “Both intercepts (integer)” up to “Perpendicular (fractional)”.
- Quadratic and other graphsPoints on quadratic curvesKS3GCSE
6 problems, from “Point on a quadratic” up to “Leading coefficient”.
- Quadratic and other graphs
6 problems, from “Roots from factorised form” up to “TP from factorised form”.
Geometry
- Perimeter and area
6 problems, from “Perimeter” up to “The ultimate rectangle”.
- Perimeter and area
6 problems, from “L-shape area” up to “The full l-shape”.
- Perimeter and area
6 problems, from “Area of a circle” up to “Semicircle perimeter”.
- Perimeter and areaArea of a triangleKS2KS3
6 problems, from “Single-digit area” up to “The ultimate triangle”.
- Volume and surface areaVolume of a prismKS2KS3
6 problems, from “Cuboid” up to “The ultimate cuboid”.
Statistics & Probability
- Averages and spreadMean from a list of dataKS2KS3
6 problems, from “Mean of two values” up to “Two-digit data, two-digit mean”.
- Averages and spread
6 problems, from “Median of four values” up to “Fixed median”.
- Averages and spread
6 problems, from “Range of three values” up to “Two-digit value”.
- Averages and spread
6 problems, from “Mean and range” up to “Mode, mean and range”.
