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Resources tagged with: Length/distance

There are 58 NRICH Mathematical resources connected to Length/distance , you may find related items under Measuring and calculating with units .

Car Journey

This practical activity involves measuring length/distance.

Can You Do it Too?

Try some throwing activities and see whether you can throw something as far as the Olympic hammer or discus throwers.

Olympic Measures

These Olympic quantities have been jumbled up! Can you put them back together again?

Now and Then

Look at the changes in results on some of the athletics track events at the Olympic Games in 1908 and 1948. Compare the results for 2012.

Olympic Starters

Look at some of the results from the Olympic Games in the past. How do you compare if you try some similar activities?

The Animals' Sports Day

One day five small animals in my garden were going to have a sports day. They decided to have a swimming race, a running race, a high jump and a long jump.

A group of children are discussing the height of a tall tree. How would you go about finding out its height?

Place Your Orders

Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?

Discuss and Choose

This activity challenges you to decide on the 'best' number to use in each statement. You may need to do some estimating, some calculating and some research.

Order, Order!

Can you place these quantities in order from smallest to largest?

Speed-time Problems at the Olympics

Have you ever wondered what it would be like to race against Usain Bolt?

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.

A Question of Scale

Use your skill and knowledge to place various scientific lengths in order of size. Can you judge the length of objects with sizes ranging from 1 Angstrom to 1 million km with no wrong attempts?

All in a Jumble

My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.

Sizing Them Up

Can you put these shapes in order of size? Start with the smallest.

Up and Across

Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the dot affects its vertical and horizontal movement at each stage.

How Far Does it Move?

Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the red dot affects the distance it travels at each stage.

Take Your Dog for a Walk

Use the interactivity to move Pat. Can you reproduce the graphs and tell their story?

The Man is much smaller than us. Can you use the picture of him next to a mug to estimate his height and how much tea he drinks?

Rolling Around

A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?

Four on the Road

Four vehicles travel along a road one afternoon. Can you make sense of the graphs showing their motion?

Uniform Units

Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?

Where Am I?

From the information you are asked to work out where the picture was taken. Is there too much information? How accurate can your answer be?

Lengthy Journeys

Investigate the different distances of these car journeys and find out how long they take.

Working with Dinosaurs

This article for teachers suggests ways in which dinosaurs can be a great context for discussing measurement.

Swimmers in opposite directions cross at 20m and at 30m from each end of a swimming pool. How long is the pool ?

Triangle Relations

What do these two triangles have in common? How are they related?

A Scale for the Solar System

The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?

Flight Path

Use simple trigonometry to calculate the distance along the flight path from London to Sydney.

Chippy's Journeys

Chippy the Robot goes on journeys. How far and in what direction must he travel to get back to his base?

Can you prove that the sum of the distances of any point inside a square from its sides is always equal (half the perimeter)? Can you prove it to be true for a rectangle or a hexagon?

Measure for Measure

This article, written for students, looks at how some measuring units and devices were developed.

Eclipses of the Sun

Mathematics has allowed us now to measure lots of things about eclipses and so calculate exactly when they will happen, where they can be seen from, and what they will look like.

A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.

How can the school caretaker be sure that the tree would miss the school buildings if it fell?

N Is a Number

N people visit their friends staying N kilometres along the coast. Some walk along the cliff path at N km an hour, the rest go by car. How long is the road?

The Dodecahedron Explained

What is the shortest distance through the middle of a dodecahedron between the centres of two opposite faces?

Do You Measure Up?

A game for two or more players that uses a knowledge of measuring tools. Spin the spinner and identify which jobs can be done with the measuring tool shown.

Alice's mum needs to go to each child's house just once and then back home again. How many different routes are there? Use the information to find out how long each road is on the route she took.

The Hare and the Tortoise

In this version of the story of the hare and the tortoise, the race is 10 kilometres long. Can you work out how long the hare sleeps for using the information given?

A Flying Holiday

Follow the journey taken by this bird and let us know for how long and in what direction it must fly to return to its starting point.

Watching the Wheels Go 'round and 'round

Use this information to work out whether the front or back wheel of this bicycle gets more wear and tear.

Nirmala and Riki live 9 kilometres away from the nearest market. They both want to arrive at the market at exactly noon. What time should each of them start riding their bikes?

Practice Run

Chandrika was practising a long distance run. Can you work out how long the race was from the information?

A Rod and a Pole

A lady has a steel rod and a wooden pole and she knows the length of each. How can she measure out an 8 unit piece of pole?

How many centimetres of rope will I need to make another mat just like the one I have here?

Walk and Ride

How far have these students walked by the time the teacher's car reaches them after their bus broke down?

On the Road

Four vehicles travelled on a road. What can you deduce from the times that they met?

Use your hand span to measure the distance around a tree trunk. If you ask a friend to try the same thing, how do the answers compare?

Great Squares

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

Measurement Worksheets

Welcome to the measurement worksheets page at Math-Drills.com where you can measure up, measure down or measure all around! This page includes Measurement worksheets for length, area, angles, volume, capacity, mass, time and temperature in Metric, U.S. and Imperial units.

Measurement concepts and skills give students the ability to perform tasks related to everyday life. Length, area, volume, capacity, mass, time and temperature are measurement concepts that people are exposed to everyday. Students begin using non-standard units such as their own height and progress to using standard measurement units. Being able to recognize and use for comparison, common measurement units such as the metre or foot, allows students to use their estimation skills to help them solve problems in measurement. Measurement tools enable students to learn hands-on and develop a deeper understanding of measurement concepts.

Most Popular Measurement Worksheets this Week

Reading and Converting Temperature Worksheets

Reading temperatures from a thermometer is one of those everyday skills that is transferable to other situations. Sure, students could just ask their smart device about the temperature, but that only tells them the temperature at the nearest weather station, not where they're standing. Wouldn't it be more exciting to look at a thermometer in their own yard? And what if they are working at a solar power farm (or elsewhere) in the future and have other meters and gauges to read? Their skill in reading thermometers will help introduce them to negative numbers and be useful for reading other types of meters and gauges in the future.

• Reading Temperatures From Thermometers Reading Temperatures from a Thermometer Reading Temperatures from Circular Thermometers (Fahrenheit Dominant) Reading Temperatures from Circular Thermometers (Celsius Dominant)

Millions of people live near the U.S./Canada border, so it is inevitable that once in a while, those people are going to hear or see the temperature in a less familiar unit. Perhaps traveling to another country has one baffled about the forecast. Having a knowledge of some simple equivalents (like the freezing point is 0 degrees Celsius and 32 degrees Fahrenheit) and the ability to convert between C and F will not only help in determining the temperature in a familiar form, but will also help students build math skills that are useful in other situations. The temperature conversion guide gives a summary of the formulas used and some whole number equivalents to learn. There are several practice sheets for converting from °C to °F, from °F to °C, and a couple with both conversions mixed up.

• Temperature Conversion Guide Temperature Conversion Guide
• Converting Temperatures Between °C and °F Converting °C to °F (no negatives) Converting °C to °F (with negatives) Converting °F to °C (no negatives) Converting °F to °C (with negatives) Converting Between °F and °C (no negatives) Converting Between °F and °C (with negatives)

Measuring and Converting Within Measurement Systems

Measuring length is so much more interesting if you can send students out with rulers and have them measure items in their environment. What is the width of the textbook? the classroom? the school? Have you ever met a student who didn't enjoy using a measuring wheel (you know the one that clicks every time you've traveled a yard or a meter)? How do you know they've measured things correctly though? Well, you might need something like we've provided below. You can also compare students' measurements of the same objects to see if they got the same measurement. Let's say, you had 20 students measure the height of the doorway. You should get 20 very similar answers (unless they are the sharing type then you'll get exactly the same answers) and any different answers can be quickly identified. IMPORTANT: When printing, select "Actual Size" or your measurements may be off.

• Measuring Lengths of Bars in Centimeters and Millimeters Measuring Length in Centimeters ✎ Measuring Length in Half Centimeters ✎ Measuring Length in Millimeters ✎
• Measuring Lengths of Bars in Inches Measuring Length in Half Inches ✎ Measuring Length in Quarter Inches ✎ Measuring Length in Eighth Inches ✎ Measuring Length in Sixteenth Inches ✎

Just like regrouping when adding, sometimes it is useful to regroup when measuring. If you have too many feet, it might be easier to express your number in yards or miles, for example.

• Converting Between U.S. Length Measurements Worksheets Converting Between Inches, Feet and Yards Converting Between Feet, Yards and Miles Converting Inches, Feet, Yards and Miles

Which unit to use with mass depends a lot on the quantity and what you want to communicate. Whatever your reason for converting mass measurements, these worksheets are likely able to help you take a weight off your shoulders.

• Converting Between U.S. Mass Measurements Worksheets Converting Between Ounces and Pounds Converting Between Ounces and Pounds with Fractions Converting Between Pounds and Tons Converting Between Pounds and Tons with Fractions Converting Ounces/Pounds or Pounds/Tons Converting Ounces/Pounds or Pounds/Tons with Fractions

The liquid measurement worksheets include gills because this is the key unit that results in more fluid ounces in an Imperial gallon than in a U.S. gallon. You can learn more about gills in our liquid measurement conversion guide.

• U.S. Liquid Measurements Conversion Guide Liquid Measurements Conversion Guide (U.S.)
• Converting Between U.S. Liquid Measurements Converting Liquid Measurements (U.S.) Converting Liquid Measurements (No Gills) (U.S.) Converting Between Pints, Quarts and Gallons (U.S.)

Even though Imperial and U.S. Customary units may sound the same, they aren't always the same amount. For example, there are 3.785 litres in a U.S. gallon and 4.546 litres in an Imperial gallon. Sometimes there are also different definitions for units like the gill used in liquid measurements. In the U.S., there are 4 fluid ounces in a gill and in the Imperial System, there are 5 fluid ounces in a gill.

• Imperial (U.K.) Liquid Measurements Conversion Guide Liquid Measurements Conversion Guide (Imperial)
• Converting Between Imperial (U.K.) Liquid Measurements Converting Liquid Measurements (Imperial) Converting Liquid Measurements (No Gills) (Imperial)

Converting between Metric units is really an exercise in multiplying and dividing by powers of ten. Each of the converting worksheets in this section includes a "Conversion Line" that includes the prefixes, symbols and powers. It can be used to figure out how many "steps" are required to convert from one unit to another and what operation must be used. For example, when converting from millimeters to kilometers, students would determine that it takes six steps to the left in the direction of the division sign to get from milli to kilo on the conversion line. Depending on which method they are taught, this could involve dividing by 10 six times, dividing by 10 6 or "moving the decimal" six places to the left. For squared units, each step counts as 100 or 10 2 and for cubic units, each step counts as 1000 or 10 3 . For more details, please see the Metric system conversion guide.

• Converting Within the Metric System Resources Metric System Conversion Guide (U.S. Version) Metric System Conversion Lines ✎
• Converting Between Metric Length Measurements Worksheets Converting Between Millimeters and Centimeters ✎ Converting Between Centimeters and Meters ✎ Converting Between Millimeters and Meters ✎ Converting Between Millimeters, Centimeters and Meters ✎ Converting Between Meters and Kilometers ✎ Converting Between Millimeters, Centimeters, Meters, and Kilometers ✎ Converting Between Nanometers, Micrometers, Millimeters and Centimeters ✎
• Converting Between Metric Mass Measurements Worksheets Converting Between Milligrams and Grams ✎ Converting Between Grams and Kilograms ✎ Converting Between Milligrams, Grams and Kilograms ✎ Converting Between Nanograms, Micrograms, Milligrams and Grams ✎
• Converting Between Metric Volume Measurements Worksheets Converting Between Milliliters and Liters ✎ Converting Between Microliters, Milliliters, Centiliters and Liters ✎ Converting Between Milliliters, Centiliters, Liters and Kiloliters ✎ Converting Between Milliliters, Liters, and Kiloliters ✎ Converting Between Liters, Kiloliters, Megaliters and Gigaliters ✎
• Converting Between Common Metric Measurements (Mixed) Converting Between Common Metric Length, Mass and Volume Units ✎
• Converting Between Metric Area Measurements Worksheets Converting Between Square Millimeters and Square Centimeters ✎ Converting Between Square Centimeters and Square Meters ✎ Converting Between Square Millimeters, Square Centimeters and Square Meters ✎ Converting Between Square Meters, Square Hectometers and Square Kilometers ✎

The Metric or SI system uses thin spaces for thousands separators and spells metres and litres with -re rather than -er. This section is mainly for students in English Canada, however, anyone who uses spaces for thousands separators might like these worksheets. This section is very similar to the previous section except for the differences in number formats and spelling.

• Metric System Conversion Guide (SI Format: Spelling and Space-Separated Thousands) Metric System Conversion Guide (SI Version)
• Converting Between Metric Length Measurements Worksheets (SI Format: Space-Separated Thousands) Converting Between Millimetres and Centimetres (SI number format) ✎ Converting Between Centimetres and Metres (SI number format) ✎ Converting Between Millimetres and Metres (SI number format) ✎ Converting Between Millimetres, Centimetres and Metres (SI number format) ✎ Converting Between Metres and Kilometres (SI number format) ✎ Converting Between Millimetres, Centimetres, Metres, and Kilometres (SI number format) ✎ Converting Between Nanometres, Micrometres, Millimetres and Centimetres (SI number format) ✎
• Converting Between Metric Mass Measurements Worksheets (SI Format: Space-Separated Thousands) Converting Between Milligrams and Grams (SI number format) ✎ Converting Between Grams and Kilograms (SI number format) ✎ Converting Between Milligrams, Grams and Kilograms (SI number format) ✎ Converting Between Nanograms, Micrograms, Milligrams and Grams (SI number format) ✎
• Converting Between Metric Volume Measurements Worksheets (SI Format: Space-Separated Thousands) Converting Between Millilitres and Litres (SI number format) ✎ Converting Between Microlitres, Millilitres, Centilitres and Litres (SI number format) ✎ Converting Between Millilitres, Centilitres, Litres and Kilolitres (SI number format) ✎ Converting Between Millilitres, Litres, and Kilolitres (SI number format) ✎ Converting Between Litres, Kilolitres, Megalitres and Gigalitres (SI number format) ✎
• Converting Between Common Metric Measurements (Mixed) Worksheets (SI Format: Space-Separated Thousands) Converting Between Common Metric Length, Mass and Volume Units (SI number format) ✎
• Converting Between Metric Area Measurements Worksheets (SI Format: Space-Separated Thousands) Converting Between Square Millimetres and Square Centimetres (SI number format) ✎ Converting Between Square Centimetres and Square Metres (SI number format) ✎ Converting Between Square Millimetres, Square Centimetres and Square Metres (SI number format) ✎ Converting Between Square Metres, Square Hectometres and Square Kilometres (SI number format) ✎

Similar to the previous two sections; however, these worksheets use dots for thousands separators and commas for decimals. This is often found in languages other than English. Metres and litres are spelled with -re as you would find in many countries outside of the U.S.

• Metric System Conversion Guide (European Format: Period-Separated Thousands and Comma Decimals) Metric System Conversion Guide (European Format: Period-Separated Thousands and Comma Decimals)
• Converting Between Metric Length Measurements Worksheets (European Format: Period-Separated Thousands and Comma Decimals) Converting Between Millimetres and Centimetres (Euro number format) ✎ Converting Between Centimetres and Metres (Euro number format) ✎ Converting Between Millimetres and Metres (Euro number format) ✎ Converting Between Millimetres, Centimetres and Metres (Euro number format) ✎ Converting Between Metres and Kilometres (Euro number format) ✎ Converting Between Millimetres, Centimetres, Metres, and Kilometres (Euro number format) ✎ Converting Between Nanometres, Micrometres, Millimetres and Centimetres (Euro number format) ✎
• Converting Between Metric Mass Measurements Worksheets (European Format: Period-Separated Thousands and Comma Decimals) Converting Between Milligrams and Grams (Euro number format) ✎ Converting Between Grams and Kilograms (Euro number format) ✎ Converting Between Milligrams, Grams and Kilograms (Euro number format) ✎ Converting Between Nanograms, Micrograms, Milligrams and Grams (Euro number format) ✎
• Converting Between Metric Volume Measurements Worksheets (European Format: Period-Separated Thousands and Comma Decimals) Converting Between Millilitres and Litres (Euro number format) ✎ Converting Between Microlitres, Millilitres, Centilitres and Litres (Euro number format) ✎ Converting Between Millilitres, Centilitres, Litres and Kilolitres (Euro number format) ✎
• Converting Between Common Metric Measurements (Mixed) Worksheets (European Format: Period-Separated Thousands and Comma Decimals) Converting Between Common Metric Length, Mass and Volume Units (Euro number format) ✎
• Converting Between Metric Area Measurements Worksheets (European Format: Period-Separated Thousands and Comma Decimals) Converting Between Square Millimetres and Square Centimetres (Euro number format) ✎ Converting Between Square Centimetres and Square Metres (Euro number format) ✎ Converting Between Square Millimetres, Square Centimetres and Square Metres (Euro number format) ✎ Converting Between Square Metres, Square Hectometres and Square Kilometres (Euro number format) ✎

Converting Between Measurement Systems Worksheets

Converting between Metric and U.S. customary units can be accomplished in a number of ways and usually takes a little knowledge of fractions and/or decimals. Most commonly, students will use a formula to convert and round the values. You may like our converting inches and centimeters with rulers worksheets for students who have difficulty with manipulating the numbers and formulas and need an easier method.

• Converting Between Inches and Centimeters Converting Inches to cm (whole inches) Converting Inches to cm (to 1/2 inches) Converting Inches to cm (to 1/4 inches) Converting Inches to cm (to 1/8 inches)
• Converting Between Inches and Centimeters With a Ruler Convert Inches to Centimeters with a Ruler Convert Centimeters to Inches with a Ruler Convert Between in/cm with a Ruler
• Converting Between U.S. and Metric Length Units Converting Between U.S. Inches and Centimeters Converting Between Meters and U.S. Feet and Yards Converting Between U.S. Miles and Kilometers Converting Between U.S. Feet and Kilometers and Meters and U.S. Miles Converting Between Metric and U.S. Length Units Converting Between Metric and U.S. Length Units including ft/km and m/mi
• Converting Between U.S. and Metric Mass Units Converting Between U.S. Ounces and Grams Converting Between U.S. Pounds and Kilograms Converting Between Metric and U.S. Mass Units
• Converting Between U.S. and Metric Volume Units Converting Between Milliliters and U.S. Fluid Ounces Converting Between Liters and U.S. Cups, Pints, Quarts and Gallons Converting Between Metric and U.S. Volume Units
• Converting Between Imperial (U.K.) and Metric Mass Units Converting Between Grams and Imperial Ounces Converting Between Kilograms and Imperial Pounds and Stone Converting Between Metric and Imperial Mass Units
• Converting Between Imperial (U.K.) and Metric Volume Units Converting Between Milliliters and Imperial Fluid Ounces Converting Between Liters and Imperial Cups, Pints, Quarts and Gallons Converting Between Metric and Imperial Volume Units
• Converting Worksheets For U.S. Nurses Converting mass measurements used in Nursing Converting volume measurements used in Nursing

Measuring Angles, Rectangles and Triangles

If they are available, full round protractors help students to recognize that measuring angles is the same as measuring sections of a circle. They also makes it much easier and precise to measure reflex angles.

• Measuring Angles Measuring Angles from 5° to 90° Measuring Angles from 5° to 175° Measuring Angles from 90° to 175° Measuring Angles from 185° to 355° Measuring Angles from 5° to 355°

Rectangles are fairly straight-forward polygons to measure since it is easy to find rectangles that use whole numbers. Rectangles are generally used when students first learn about perimeter and area and it is an opportune time to teach students that units are an essential part of any measurement. Without units, numbers are meaningless. Get your students into the habit of expressing all of their measurements with the correct units before they learn how to measure other polygons. Especially make sure they know that area is always expressed with squared units.

If a student is just starting to learn about perimeter and area, a few hands-on activities to learn the concepts is a good idea. Have them use square tiles to cover an area, have them paint a piece of paper and see how much paint is required. Create rectangles with straws and pipe cleaners and fill with square tiles to differentiate between perimeter and area. See if there are differently shaped rectangles that will hold the same number of square tiles.

• Perimeter and Area of Rectangles Worksheets Calculate the Area and Perimeter of Rectangles from Side Measurements (Smaller Whole Numbers) Calculate the Area and Perimeter of Rectangles from Side Measurements (Larger Whole Numbers) Calculate the Area and Perimeter of Rectangles from Side Measurements (Decimal Numbers) Area of Rectangles (grid form)
• Perimeter and Area of Rectangles Worksheets (Retro) (Retro) Rectangles (whole numbers; range 1-9) (Retro) Rectangles (whole numbers; range 5-20) (Retro) Rectangles (whole numbers; range 10-99) (Retro) Rectangles (1 decimal place; range 1-9) (Retro) Rectangles (1 decimal place; range 5-20) (Retro) Rectangles (1 decimal place; range 10-99)
• Calculating Other Rectangle Measurements Using Perimeter and Area Calculate the Side and Area Measurements of Rectangles from Perimeter and Side (Smaller Whole Numbers) Calculate the Side and Area Measurements of Rectangles from Perimeter and Side (Larger Whole Numbers) Calculate the Side and Area Measurements of Rectangles from Perimeter and Side (Decimal Numbers) Calculate the Side and Perimeter Measurements of Rectangles from Area and Side (Smaller Whole Numbers) Calculate the Side and Perimeter of Rectangles from Area and Side Measurements (Larger Whole Numbers) Calculate the Side and Perimeter of Rectangles from Area and Side Measurements (Decimal Numbers) Calculate the Side Measurements of Rectangles from Perimeter and Area (Smaller Whole Numbers) Calculate the Side Measurements of Rectangles from Perimeter and Area (Larger Whole Numbers) Calculate the Side Measurements of Rectangles from Perimeter and Area (Decimal Numbers) Calculate Various Rectangle Measurements (Smaller Whole Numbers) Calculate Various Rectangle Measurements (Larger Whole Numbers) Calculate Various Rectangle Measurements ( Decimal Numbers)

If you want students to understand the triangle area formula, you might want to study parallelograms and rectangles first. Once students get how area is calculated for rectangles and parallelograms, they simply need to cut parallelograms and rectangles in half diagonally to get related triangles. They should quickly see that the area of a triangle is simply half of the area of the related quadrilateral.

• Area and Perimeter of Triangles Worksheets Calculate the Area and Perimeter of Acute Triangles Calculate the Area and Perimeter of Acute Triangles (Rotated Triangles) Calculate the Area and Perimeter of Right Triangles Calculate the Area and Perimeter of Right Triangles (Rotated Triangles) Calculate the Area and Perimeter of Obtuse Triangles Calculate the Area and Perimeter of Obtuse Triangles (Rotated Triangles) Calculate the Area and Perimeter of Acute and Right Triangles Calculate the Area and Perimeter of Acute and Right Triangles (Rotated Triangles) Calculate the Area and Perimeter of Various Triangles Calculate the Area and Perimeter of Various Triangles (Rotated Triangles)
• Area and Perimeter of Triangles Worksheets (Retro) (Retro) Triangles (1 decimal place; range 1-5) (Retro) Triangles (1 decimal place; range 1-9) (Retro) Triangles (1 decimal place; range 5-20) (Retro) Triangles (1 decimal place; range 10-99)
• Calculating Other Triangle Measurements (Angles, Heights, Bases) Calculate Angles of a Triangle Given Other Angle(s) Calculate the Perimeter and Height Measurements of Triangles Calculate the Area and Height Measurements of Right Triangles Calculate the Base and Height Measurements of Triangles Calculate Various Measurements of Triangles Calculate the Perimeter and Area from Side Measurements of Triangles (Heron's Formula)

Measuring Other Polygons

• Area and Perimeter of Parallelograms Worksheets Parallelograms (whole number base; range 1-9) Parallelograms (1 decimal place; range 1-5) Parallelograms (1 decimal place; range 1-9) Parallelograms (1 decimal place; range 5-20) Parallelograms (1 decimal place; range 10-99)
• Area and Perimeter of Trapezoids Worksheets Calculate the Area and Perimeter of Trapezoids (Smaller Numbers) Calculate the Area and Perimeter of Trapezoids (Larger Numbers) Calculate the Area and Perimeter of Trapezoids (Even Larger Numbers) Calculate the Area and Perimeter of Trapezoids (Larger Still Numbers) Calculate the Area and Perimeter of Isosceles Trapezoids Calculate the Area and Perimeter of Right Trapezoids Calculate the Area and Perimeter of Scalene Trapezoids
• Calculating Other Trapezoid Measurements Worksheets Calculate Bases and Areas of Trapezoids Calculate Bases and the Heights of Trapezoids Calculate Bases and Perimeters of Trapezoids Calculate Bases and Sides of Trapezoids Mixed Trapezoids Questions
• Area and Perimeter of Trapeziums Worksheets (U.K. Format: Name of Shape) Calculate the Area and Perimeter of Trapeziums (Smaller Numbers) Calculate the Area and Perimeter of Trapeziums (Larger Numbers) Calculate the Area and Perimeter of Trapeziums (Even Larger Numbers) Calculate the Area and Perimeter of Trapeziums (Larger Still Numbers) Calculate the Area and Perimeter of Isosceles Trapeziums Calculate the Area and Perimeter of Right Trapeziums Calculate the Area and Perimeter of Scalene Trapeziums
• Calculating Other Trapezium Measurements Worksheets (U.K. Format: Name of Shape) Calculate Bases and Areas of Trapeziums Calculate Bases and Heights of Trapeziums Calculate Bases and Perimeters of Trapeziums Calculate Bases and Sides of Trapeziums Mixed Trapeziums Questions

The shapes are mixed up on the worksheets in this section. These area and perimeter worksheets would be best suited to students who have mastered finding the areas of triangles, rectangles, parallelograms, and trapezoids. For students who need an extra challenge, give them the compound shapes worksheet, but make sure they know how to find the area and circumference of a circle first.

• Calculating The Area And Perimeter Of Various Shapes Various Shapes (1 decimal place; range 1-9) Various Shapes (1 decimal place; range 5-20) Various Shapes (1 decimal place; range 10-99) Area and Perimeter of Measured Compound Shapes

Measuring Circles

Radius, diameter, circumference and area are all related measurements; you only need one of them to find the remaining measurements. Diameter and radius are the simplest ones because the diameter of a circle is twice the radius and, conversely, the radius is half the diameter. To calculate between radius/diameter and circumference/area, you need to use π (pi). Depending on your accessibility to calculators or computers, you may use many digits of pi in the calculation or just a few. Often, people without calculators use an estimate of pi (3 or 3.14). Just for fun we made a worksheet with pi to 100,000 decimal places. The calculations on the worksheets below use a fairly precise version of pi; you may have to adjust the answers if you use more rounded versions of pi.

• Pi to 100K Decimal Places Reference Pi to 100K Decimal Places
• Circle Measurements Worksheets (Area, Circumference, Diameter, Radius) Calculate All Circle Measurements Calculate the Area & Circumference from Radius Calculate the Area & Circumference from Diameter Calculate the Area & Circumference Calculate the Radius & Diameter from Area Calculate the Radius & Diameter from Circumference Calculate the Radius & Diameter Calculate the Area & Circumference (old)
• Calculating Arc Lengths Calculating Arc Length from Circumference Calculating Arc Length from Radius Calculating Arc Length from Diameter Calculating Arc Length from Radius or Diameter Calculating Arc Length from Circumference, Radius or Diameter
• Calculating Arc Angles Calculating Arc Angle from Circumference Calculating Arc Angle from Radius Calculating Arc Angle from Diameter Calculating Arc Angle from Radius or Diameter Calculating Arc Angle from Circumference, Radius or Diameter
• Calculating Arc Lengths and Angles Calculating Arc Length or Angle from Circumference Calculating Arc Length or Angle from Radius Calculating Arc Length or Angle from Diameter Calculating Arc Length or Angle from Radius or Diameter Calculating Arc Length or Angle from Circumference, Radius or Diameter

Measuring Three-Dimensional Forms

• Volume and Surface Area of Rectangular Prisms Worksheets Rectangular Prisms Volume and Surface Area ( Whole Numbers ) Rectangular Prisms Volume and Surface Area ( Decimal Numbers )
• Volume and Surface Area of Triangular Prisms Worksheets Triangular Prisms Volume and Surface Area ( Black and White ) Triangular Prisms Volume and Surface Area ( Color )
• Volume and Surface Area of Cylinders Worksheets Cylinders Volume and Surface Area Cylinders Volume and Surface Area ( Small Measurements )
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• Maths Questions

Measurement Questions

Measurement is the basic concept that everyone deals with in real-life situations. For instance, we generally use tonnes, kg, and g weights to purchase fruits or vegetables. When we purchase 5 kg of fruits, 5 represents the measurement and kg denotes the unit of measurement.

What is Measurement?

Measurement is the technique of finding a number that signifies the amount, quantity, and size of a thing or object. For example, the length of an object is the distance from end to end. Some basic measurements are:

• Length or distance
• Temperature

Learn more about measurement in maths here.

Measurement Questions and Answers

1. Calculate the sum of lengths: 21 m 13 cm, 33 m 55 cm and 45 m 6 cm.

21 m 13 cm + 33 m 55 cm + 45 m 6 cm

= (21 + 33 + 45) m (13 + 55 + 6) cm

= 99 m 74 cm

Therefore, the sum of given lengths = 99 m 74 cm.

2. Beena bought 3 kg 760 grams of wool to make a carpet. How much more wool does she need to make the weight 4 kg?

Given the weight of wool = 3 kg 760 grams

Let us convert this weight into grams.

3 kg 760 grams = (3 × 1000 + 760) grams

= (3000 + 760) grams

= 3760 grams

4 kg = (4 × 1000) grams = 4000 grams

Difference = (4000 – 3760) grams = 240 grams

Therefore, 240 grams of more wool is required.

3. A pile of 10 books is 10 cm high. What is the thickness of each book?

As we know,

1 cm = 10 mm

Given that the height of a pile of 10 books = 10 cm

10 books = 10 cm

= 10 x 10 mm

1 book = 100/10

Therefore, the thickness of each book = 10 mm.

4. A furlong is a unit of length used in horse racing; it equals one-eighth of a mile. To the nearest tenth, how many metres are equal to a furlong if 1.609 km equals a mile?

1.609 km = 1 mile

That means 1609 m = 1 mile

Also, one furlong = one-eighth of a mile

= (1/8) × 1.609 km

= (1/8) × 1609 m

= 201.1 m (approx)

5. The cost of 1 litre of syrup is Rs. 840.80. Find the cost of 600 ml of the syrup.

The cost of 1 litre of syrup = Rs. 840.80

1 litre = 1000 ml

The cost of 600 ml of the syrup = (600/1000) × Rs. 840.80 = Rs. 504.48

6. One inch equals 2.54 centimetres. How many centimetres tall is an 84-inch door?

1 inch = 2.54 cm

84 inch = ?

84 inch = 84 × 2.54 cm

= 213.36 cm

Thus, the door is 213.36 cm tall.

7. Vinu and Shan together weigh 72 kg 350 g. If Vinu weighs 39 kg 185 g, what is Shan’s weight?

Vinu’s weight = 39 kg 185 g

Let x be Shan’s weight.

According to the given,

39 kg 185 g + x = 72 kg 350 g

x = 72 kg 350 g – 39 kg 185 g

= (72 – 39) kg (350 – 185) g

= 33 kg 165 g

Therefore, Shan’s weight is 33 kg 165 g.

8. A tabletop measures 2 m 25 cm by 1 m 50 cm. What is the perimeter of the tabletop?

We know that,

1 m = 100 cm

1 cm = 0.01 m

Length of tabletop = 2 m 25 cm = 2.25 m

Breadth of tabletop = 1 m 50 cm = 1.50 m

Perimeter of tabletop = 2 (Length + Breadth)

= 2 (2.25 + 1.50)

Therefore, the perimeter of the tabletop is 7.5 m.

9. Two sides of a triangle are 11 cm and 15 cm. If the perimeter of the triangle is 36 cm, find its third side.

Let x cm be the third side of the triangle.

Two sides: 11 cm and 15 cm

Perimeter of triangle = 36 cm

11 + 15 + x = 36

26 + x = 36

x = 36 – 26

Hence, the third side of the triangle is 10 cm.

10. The weight of a rice packet is 40 kg 360 g. What is the total weight of 6 rice packets of the same weight?

Weight of a rice packet = 40 kg 360 g

Number of rice packets = 6

Total weight = 6 × weight of one rice packet

= 6 × (40 kg 360 g)

= (6 × 40) kg (6 × 360) g

= 240 kg 2160 g

= 240 kg (2000 + 160) g

= (240 + 2) kg 160 g

= 242 kg 160 g

Thus, the total weight of 6 rice packets = 242 kg 160 g.

Practice Questions on Measurement

• If a watermelon’s weight is 1 kg 300 g, find the weight of 12 watermelons.
• Which is the best estimation for the length of a pen?
• Convert 2550 mm to metres and cm.
• A water tank can hold 1500 l of water in it. If it has 960 l 550 ml of water; how much water can be poured into it?
• The length of a goods train is 266 m. The length of a passenger train is 113 m. Which train is longer in length? And by how much?

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6.2.3: Using Metric Conversions to Solve Problems

• Last updated
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• Page ID 62187

• The NROC Project

Learning Objectives

• Solve application problems involving metric units of length, mass, and volume.

Introduction

Learning how to solve real-world problems using metric conversions is as important as learning how to do the conversions themselves. Mathematicians, scientists, nurses, and even athletes are often confronted with situations where they are presented with information using metric measurements, and must then make informed decisions based on that data.

To solve these problems effectively, you need to understand the context of a problem, perform conversions, and then check the reasonableness of your answer. Do all three of these steps and you will succeed in whatever measurement system you find yourself using.

Understanding Context and Performing Conversions

The first step in solving any real-world problem is to understand its context. This will help you figure out what kinds of solutions are reasonable (and the problem itself may give you clues about what types of conversions are necessary). Here is an example.

In the Summer Olympic Games, athletes compete in races of the following lengths: 100 meters, 200 meters, 400 meters, 800 meters, 1500 meters, 5000 meters and 10,000 meters. If a runner were to run in all these races, how many kilometers would he run?

The runner would run 18 kilometers.

This may not be likely to happen (a runner would have to be quite an athlete to compete in all of these races) but it is an interesting question to consider. The problem required you to find the total distance that the runner would run (in kilometers). The example showed how to add the distances, in meters, and then convert that number to kilometers.

An example with a different context, but still requiring conversions, is shown below.

One bottle holds 295 deciliters while another one holds 28,000 milliliters. What is the difference in capacity between the two bottles?

There is a difference in capacity of 1.5 liters between the two bottles.

This problem asked for the difference between two quantities. The easiest way to find this is to convert one quantity so that both quantities are measured in the same unit, and then subtract one from the other.

One boxer weighs in at 85 kilograms. He is 80 dekagrams heavier than his opponent. How much does his opponent weigh?

• \(\ 5 \text { kilograms }\)
• \(\ 84.2 \text { kilograms }\)
• \(\ 84.92 \text { kilograms }\)
• \(\ 85.8 \text { kilograms }\)
• Incorrect. Look at the unit labels. The boxer is 80 dekagrams heavier, not 80 kilograms heavier. The correct answer is 84.2 kilograms.
• Correct. \(\ 80 \text { dekagrams }=0.8 \text { kilograms }\), and \(\ 85-0.8=84.2\).
• Incorrect. This would have been true if the difference in weight was 8 dekagrams, not 80 dekagrams. The correct answer is 84.2 kilograms.
• Incorrect. The first boxer is 80 dekagrams heavier , not lighter than his opponent. This question asks for the opponent’s weight. The correct answer is 84.2 kilograms.

Checking your Conversions

Sometimes it is a good idea to check your conversions using a second method. This usually helps you catch any errors that you may make, such as using the wrong unit fractions or moving the decimal point the wrong way.

A two-liter bottle contains 87 centiliters of oil and 4.1 deciliters of water. How much more liquid is needed to fill the bottle?

The amount of liquid needed to fill the bottle is 0.72 liter.

Having come up with the answer, you could also check your conversions using the quicker “move the decimal” method, shown below.

The amount of liquid needed to fill the bottle is 0.72 liters.

The initial answer checks out. 0.72 liter of liquid is needed to fill the bottle. Checking one conversion with another method is a good practice for catching any errors in scale.

Understanding the context of real-life application problems is important. Look for words within the problem that help you identify what operations are needed, and then apply the correct unit conversions. Checking your final answer by using another conversion method (such as the “move the decimal” method, if you have used the factor label method to solve the problem) can cut down on errors in your calculations.

Measurement Word Problems Worksheets

Related Pages Math Worksheets Lessons for Fourth Grade Free Printable Worksheets

Printable “Metric Measurement” Worksheets: Metric Length Conversions (km, m, cm) Metric Mass Conversions (kg, g) Metric Capacity Conversions (L, cL)

Metric Length Word Problems Metric Capacity Word Problems Metric Mass Word Problems Measurement Word Problems

Metric Measurement Word Problems Worksheets

In these free math worksheets, students practice how to solve metric measurement word problems.

How to solve measurement word problems?

Solving measurement word problems involves interpreting the information given in the problem, identifying the appropriate units of measurement, and performing the necessary mathematical operations.

• Read the problem carefully and understand the context and information presented in the word problem.
• Determine what quantities are given (knowns) and what quantity needs to be found (unknown).
• Identify the units of measurement for the known and unknown quantities. Ensure all units are consistent.
• Based on the information provided, draw a diagram and write an equation that represents the relationship between the known and unknown quantities. Use the appropriate mathematical operation.
• Use the given information and the equation to perform the necessary mathematical operations (addition, subtraction, multiplication, or division).
• Ensure that the answer makes sense in the context of the problem. Check units to ensure they are consistent.

Click on the following worksheet to get a printable pdf document. Scroll down the page for more Measurement Word Problems Worksheets .

More Measurement Word Problems Worksheets

Printable (Answers on the second page.) Measurement Word Problems Worksheet #1 Measurement Word Problems Worksheet #2 Measurement Word Problems Worksheet #3

Related Lessons & Worksheets

Adding and Subtracting Weights (kg, g, mg) Multiplying Weights (kg, g, mg) Dividing Weights (kg, g, mg)

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How to Solve Measurement Word Problems

Measurement word problems often involve understanding and using different units of measurement. These problems might ask you to convert from one unit to another or to find a total, difference, or product of given measurements.

A Step-by-step Guide to Solving Measurement Word Problems

Here’s a simple step-by-step guide for a student to solve measurement word problems:

Step 1: Understand the Problem

First, read the problem carefully. Try to understand what the problem is asking you to do. Look for keywords or phrases that indicate what operation you should use (like ‘total’ for addition or ‘difference’ for subtraction).

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Step 2: identify the units and numbers.

Next, write down all the measurements given in the problem, including their units (like meters, grams, hours, etc.). Also, note any numbers that might be important.

Step 3: Decide on the Operation(s)

Determine what operation or operations you need to perform to find the answer. It could be addition, subtraction, multiplication, or division. In some cases, you might need to convert one unit to another.

Step 4: Perform the Operation(s)

Do the math! If you need to convert units, do that first. Then perform the necessary operation or operations.

Step 5: Write Your Answer

Finally, write down your answer, making sure to include the correct unit. Make sure your answer makes sense in the context of the problem.

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Mastering grade 5 math the ultimate step by step guide to acing 5th grade math, mastering grade 3 math the ultimate step by step guide to acing 3rd grade math, mastering grade 5 math word problems the ultimate guide to tackling 5th grade math word problems, mastering grade 2 math word problems the ultimate guide to tackling 2nd grade math word problems, mastering grade 4 math word problems the ultimate guide to tackling 4th grade math word problems.

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2nd Grade Math Worksheets: Measurement

Grade 2 measurement worksheets.

Our grade 2 measurement worksheets focus on the measurement of length, weight, capacity and temperature . Measurement using non-standard units is reviewed and standard measurement units are introduced. Both the customary and metric systems are covered.

Sample Grade 2 Measurement Worksheet

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• Published: 04 May 2024

Exploring measurement estimation strategies through concept cartoons designed with Realistic Mathematics Education

• Emel Çilingir Altiner   ORCID: orcid.org/0000-0002-8085-553X 1  

Humanities and Social Sciences Communications volume  11 , Article number:  567 ( 2024 ) Cite this article

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• Development studies

Concept cartoons, in relationship with Realistic Mathematics Education (RME), not only serve as a dynamic platform for problem-solving but also intricately weave mathematical concepts into the fabric of real-world scenarios, creating a harmonious fusion of theory and practical application. By using relatable characters and presenting ideas through engaging narratives, students are encouraged to tackle problems within the given context. The aim of this study was to investigate the measurement estimation strategies used by students when exposed to concept cartoons specifically designed for RME. A qualitative research methodology was followed, involving 46 fourth-grade students from a primary school. The data collection instrument utilized was “RME-Supported Concept Cartoons.” Descriptive analysis was employed to analyze the collected data. The concept cartoon activities incorporated measurement estimation strategies such as prior knowledge, unit iteration/separation, segmentation/chunking, and comparison, which students could potentially prefer. However, it was observed that students predominantly utilized division into unit iteration/separation and segmentation/chunking strategies over other estimation strategies. Furthermore, the frequency of strategy use did not exhibit significant variations based on gender. When examining the strategies developed by the students, it is noteworthy that the presence of “another solution-oriented option” and “irrelevant answers” was prominent.

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Introduction.

Representations play a vital role in enhancing students’ understanding of mathematics by transforming complex problems into graphical, visual, symbolic, or alternative linguistic formats (National Council of Teachers of Mathematics [NCTM], 2000 ). Accomplished mathematicians have emphasized the effectiveness of visual representations in reasoning (Jitendra and Woodward, 2019 ), highlighting how they facilitate the depiction of imagination and employ diagrams and models to embody abstract information. In the field of education, visual representations are widely utilized by teachers, particularly in primary schools, as a strategy to capture students’ attention during the learning process (NCTM, 2000 ; Aldous, 2007 ).

Concept cartoons serve as one method through which teachers employ visual representations. By utilizing concept cartoons, teachers not only make abstract mathematical concepts more concrete but also engage students through the use of visually appealing illustrations and educational scenarios. Concept cartoons, in the form of illustrated arguments, facilitate active participation from all students by introducing different perspectives within familiar environments through speech bubbles (Dabell, 2008 ). These cartoons not only promote new and original learning but also foster discussion and critical thinking among students.

Cartoons play a significant role in mathematics by bridging the gap between mathematical concepts and real-world applications (Clair, 2018 ). The use of realistic characters in cartoons discussing and reviewing mathematical ideas proves valuable in capturing students’ attention. Moreover, concept cartoons align with the core principle of Realistic Mathematics Education (RME), which views mathematics as a human activity (Freudenthal, 1991 , p. 14). The introduction of RME underscores its commitment not only to connecting mathematical concepts with real-world applications but also to fostering an environment where mathematical understanding becomes genuinely real for students (Çilingir Altiner, 2021 ). This includes providing meaningful contexts and problems that resonate with students’ lived experiences, an aspect integral to RME (Fredriksen, 2021 ). The term ‘realistic’ in RME extends beyond its connection to the real-world; it involves creating contexts that are genuinely meaningful and relatable for students. Moreover, RME encompasses both horizontal and vertical mathematization (Treffers, 1993 ). Horizontal mathematization involves engaging with mathematical concepts in contexts that make sense to students, while vertical mathematization facilitates the transition from informal to more formal mathematical reasoning through dialogs and discussions. As such, RME represents an approach in which the aim is to make mathematics both accessible and comprehensible to learners (Bray and Tangney, 2015 ). Consequently, concept cartoons offer a beneficial strategy for RME, as they present problems derived from real-life situations that students need to solve. By utilizing concept cartoons, the transition from horizontal mathematization to vertical mathematization can be facilitated through dialogs and discussions.

Concept cartoons, as employed in this research, function as a method within the broader framework of RME (Gravemeijer, 1994 ). They are pedagogical tool strategically crafted to introduce problems stemming from real-life scenarios and promote active problem-solving. Featuring relatable characters and situations, concept cartoons aid in concretizing abstract mathematical concepts (Dabell, 2008 ), thus aligning with the fundamental principles of RME (reality principle of RME). By offering a visual and narrative-based approach, they enhance students’ interaction with mathematical problems in the context of their daily experiences (the activity and interaction principle of RME), embodying both a method and strategy that facilitate the implementation of RME principles in mathematics education. Despite the diverse applications of concept cartoons in mathematics education, the literature lacks a comprehensive exploration of their alignment with the core tenets of RME (Van den Heuvel-Panhuizen, 2003 ; Freudenthal, 1991 ). In the present study, the aim is to address this gap by examining the potential of concept cartoons specifically tailored to adhere to RME principles.

RME is an educational approach that individuals frequently utilize in their daily lives, often without even realizing it. RME provides contexts meaningful and relevant to students’ lived experiences. For instance, when we go grocery shopping, we engage in practical mental calculations and apply mathematical reasoning to make decisions, such as selecting a watermelon that will fit in a specific space or determining how many apples we can carry in a bag. These estimation calculations yield results that are close to actual measurements. Particularly, adults frequently rely on measurement estimation to generate quick solutions in their day-to-day activities. In a study focused on enhancing measurement estimation skills, it was found that instruction based on RME was more effective compared to traditional teaching methods (Kaba and Şengül, 2017 ). This highlights the practical relevance and effectiveness of incorporating RME principles in teaching measurement estimation skills to students.

In the present study, the measurement learning domain is addressed, which is known to be challenging for students and often involves misconceptions and difficulties related to conservation. Within this context, estimation is highlighted as a crucial aspect of learning measurement. Over the years, measurement estimation has become integral to the mathematics curriculum (Andrews et al., 2022 ). Understanding how to estimate measurements is of significant importance for two primary reasons.

Firstly, estimation is considered a fundamental arithmetic skill in its own right and is frequently employed in everyday life when precise measuring tools are unavailable or inappropriate (Levin, 1981 ; O’Daffer, 1979 ). Secondly, measurement estimation offers a practical approach to teaching physical measurement, a key topic within the elementary mathematics curriculum that lays the foundation for other mathematical concepts like fractions and ratios (Coburn and Shulte, 1986 ; Davydov and Tsvetkovich, 1991 ; NCTM, 2000 ). The NCTM has advocated measurement estimation as the central focus in preschool and elementary school mathematics curricula (NCTM, 1989 , 2000 ). There are three primary reasons supporting the inclusion of measurement estimation in the curriculum: (1) Estimation practice helps students develop a mental reference for the dimensions of measurement units. (2) Estimation is a practical and valuable skill. (3) Estimation activities allow the application of fundamental measurement concepts.

However, it has been noted that limited knowledge exists regarding children’s abilities in the area of measurement estimation, particularly at the primary school level (Ruwisch et al., 2015 ). Most research on measurement estimation has focused on upper-grade students, which prompted us in the present study to concentrate on lower-grade children (Sowder, 1992 ). The ability to estimate the length or area of an object with a reasonable degree of accuracy is a complex skill that requires a range of abilities and concepts, namely:

understanding the feature (length or area) to be measured,

understanding of the unit concept,

a mental image of the unit used in the estimation task,

ability to compare objects on the attribute to be measured,

ability to perform unit iterations,

ability to select and use appropriate strategies for estimation, and

ability to check the relevance of the estimate.

Considering the complexity of skills primary school students need to develop and their limited ability to fully articulate the measurement estimation strategies they employ, the use of supported concept cartoons is predicted to be beneficial in identifying these strategies. However, when examining the existing literature, it is evident that concept cartoons are rarely used in mathematics education compared to in other fields (Aygün et al., 2020 ; Huang and Tzu-Ying, 2020 ; Sancar and Koparan, 2019 ). Most studies on concept cartoons in mathematics have focused on teacher-oriented concept cartoons and the effectiveness of concept cartoon-supported instruction in geometry lessons for secondary school students (Samková, 2020 ; Kogler et al., 2021 ; Karaca et al., 2020 ; Aygün et al., 2020 ; Sancar and Koparan, 2019 ; Göksu and Köksal, 2016 ; Şengül and Aydın, 2013 ). However, the relationship between concept cartoons and RME has not been explicitly explored in the literature.

Therefore, the aim of the present study is to determine the measurement estimation strategies utilized by fourth-grade primary school students through concept cartoons specifically designed with realistic problems. Concept cartoons can be applied in various ways and serve different purposes. Given the identified advantages of this approach, we seek to reveal students’ existing knowledge enjoyably and ascertain the strategies they employ. The primary objective is to identify the measurement estimation strategies used by the students. Specifically, the following research questions will be addressed:

What measurement estimation strategies do fourth-grade students use in concept cartoons related to weight, length, and area problems?

Are there any differences in the measurement estimation strategies used by students in concept cartoons based on their gender?

What measurement estimation strategies do students generate for weight, length, and area problems?

The intention is to uncover the strategies they independently produce to estimate measurements. By addressing these research questions, the aim is to provide insights into the measurement estimation strategies used by fourth-grade students in concept cartoons, examine potential gender differences, and explore their ability to generate strategies for weight, length, and area problems.

The research method chosen for this study is the case study, which is a qualitative research approach. This decision was made based on several factors. Firstly, the study involves examining an existing process, namely the measurement estimation strategies of fourth-grade primary school students in concept cartoons. Secondly, real-life problems were selected, allowing an in-depth exploration of their experiences and perspectives. Thirdly, the study involves a complex and multifaceted phenomenon with potentially blurred boundaries, as it investigates measurement estimation strategies in various contexts (weight, length, and area problems) within the concept cartoons.

Participants

The study included a total of 46 fourth-grade students (23 girls and 23 boys) from a school in Adana, Turkey, during the 2021–2022 academic year. The school comprised seven classrooms for fourth graders, with a total student population of 175. To ensure the reliability of the research, purposeful random sampling was employed (Flick, 2014 ). The selection of two classes was based on the willingness of the classroom teachers to participate in the study. Fourth-grade students were chosen because they have more experience in measurement estimation compared to students in other grade levels. Additionally, students in this age group possess the ability to think more abstractly and have more developed mental estimation skills. It was believed that fourth-grade students would be capable of developing their own estimation strategies. The age range of the participating students varied between 9 and 10 years, with an average age of 9 years and 6 months. The district where the school is located has a moderate economic level, resulting in a student population with similar socioeconomic backgrounds. The socioeconomic status of the school corresponds to that of middle-class families, as reported by the Turkish Statistical Institution (TUIK, 2021 ).

The study did not include students with learning difficulties requiring special education, students with exceptional academic achievements, or students facing difficulties in accessing the learning environment. Furthermore, there were no students from different ethnic groups in the selected classes, ensuring that the language development and literacy levels of the participating students were similar. The fact that the students were in the fourth-grade indicated a good level of literacy compared to other primary school grade levels.

Data collection tool

The data collection tool used in the study was RME-Supported Concept Cartoons developed by the researcher. Unlike generic Concept Cartoons, these specialized instruments are purposefully crafted to integrate seamlessly with the RME framework, aiming to create a more profound connection between mathematical concepts and real-world applications. The differentiation lies in the meticulous alignment of content and scenarios with the core tenets of RME, ensuring that the problems presented are not only relatable but also resonate authentically with students’ lived experiences. Furthermore, the RME-Supported Concept Cartoons prioritize specific measurement-related scenarios, addressing essential learning outcomes in the primary school mathematics program, such as weight, length, and area measurement estimations. The scenarios embedded in these cartoons are deliberately designed to motivate students towards active problem-solving within the context of their daily lives. In essence, the tool serves as a targeted and tailored application of Concept Cartoons, specifically adapted to adhere to the principles of RME (for example, horizontal and vertical mathematization). This distinction ensures that the RME-Supported Concept Cartoons are not merely a generic pedagogical tool but a finely tuned and purpose-built instrument, strategically developed to enhance the integration of RME principles in the teaching and learning of mathematics. The alignment with RME goes beyond superficial visual elements, delving into the very fabric of scenario construction and problem presentation to create a more meaningful and effective learning experience for students.

After each scenario, the students were asked to identify which student in the cartoon they believed chose the most appropriate strategy for solving the problem. They were instructed to note their choice on the activity sheet provided. Furthermore, the students were asked to generate their own strategies for the given problem situation and record them. This allowed the researchers to gather insights into the measurement estimation strategies developed by the students themselves. Figure 1 provides an example of an RME-supported concept cartoon activity, illustrating the use of a realistic problem related to length measurement in the learning field.

Concept cartoon activity for the length measurement estimation problem.

Figure 1 illustrates a classroom environment with teachers and students. In the RME-supported concept cartoon activity, Elçin’s answer indicates that she relied on her own experiences, prior knowledge, and previous measurements to answer the question. This suggests that Elçin employed a strategy based on personal experiences and existing knowledge to estimate the measurement in the given problem situation.

In the RME-supported concept cartoon activity, Doruk employed the unitization strategy to estimate the size of the door. He calculated each pattern on the door as a unit and used this information to estimate the overall size. The unitization strategy involves mentally creating nonstandard units (e.g., span) and comparing them repeatedly to the object being estimated. Doruk needed to remember the image of the unit, track where the last unit ended, and determine where the next unit should begin (Joram et al., 2005 ).

On the other hand, Ceyda used the segmentation/subdividing/chunking strategy. She identified the part of the door up to the socket as a midpoint and stated that this part represented half the length of the door. In segmentation/chunking, smaller units are compared to a known length by dividing them into approximately two lengths instead of iterating them one by one. This strategy allows a reduction in the number of unit iterations required for estimation.

Ahmet employed the comparison strategy by assessing the height of the door in relation to his own height. Through this approach, he inferred that the door was taller than him and made an estimation accordingly. The comparison strategy, also known as the reference point strategy, involves mentally envisioning an object with a known measurement, such as a 3-cm-long eraser, and comparing it to the object being estimated, such as the length of a book. This strategy relies on individuals having a mental reference unit or a conceptual understanding of the size of the unit (Sowder, 1992 , as cited in Joram et al., 2005 ). For instance, a dermatologist might utilize a reference point of a “pencil eraser” when advising patients to be cautious of moles larger than 6 mm (approximately the size of a pencil eraser). This reference point is chosen because it is a familiar object. By establishing a connection between a numerical value and a known quantity, reference points enhance the meaningfulness of units of measurement for estimation purposes.

Data collection process

A 4-week study was conducted in two selected classes after approval was obtained the school administration and permission from the classroom teachers. The study adhered to the existing fourth-grade mathematics curriculum in primary schools. The weeks allocated for teaching the measurement learning area were chosen for implementing the study. During the first week of the intervention, the concept cartoons were introduced to the students by their classroom teacher. The teacher provided information about the purposes and uses of concept cartoons. Sample concept cartoon exercises were distributed and preparations were made for the actual application of the study.

The main application took place once a week during regular class hours. The students were given concept cartoon worksheets that contained realistic problems related to weight, length, and area in the measurement learning field. The teacher read the problem presented on the worksheet to the students, who were then asked to indicate their chosen answer on the worksheet. Additionally, the students were invited to share any alternative strategies they might have by noting them on the worksheet. This process was repeated for all other problem scenarios. Throughout the application, the classroom teacher acted as an observer and was present in the classroom. It is important to note that the study strictly adhered to ethical guidelines. In line with ethical guidelines, the study was designed to ensure the protection and well-being of the participants. Necessary approvals, such as parental consent forms and school board information forms, were obtained before conducting the research. The students were explicitly informed about the study’s purpose and their participation was voluntary. They were assured that their responses would be treated as confidential and solely used for research purposes. Furthermore, the students were informed that they had the right to withdraw from the study at any point if they wished to do so. These measures were put in place to uphold ethical standards and ensure the rights and privacy of the participants.

Analysis of data

The data collected were subjected to descriptive analysis, following the guidelines outlined by Yıldırım and Şimşek ( 2021 ). Descriptive analysis aims to present individuals’ opinions and experiences directly, without any added comments, by analyzing data gathered from interview transcripts, document texts, and observation notes. Direct quotations are often used to support the findings. In the present study, the strategies utilized by the students, as indicated by the names they selected from the concept cartoon worksheets, were coded. The frequency of these strategies’ use and potential differences based on gender were examined. Additionally, the students were asked if they had any strategies of their own for solving the concept cartoons, and these self-generated strategies were analyzed. All strategies were coded systematically. Throughout the analysis process, the researcher maintained objectivity and refrained from expressing personal opinions or comments. The data were presented to the reader without any additional bias or interpretation.

To enhance reliability in qualitative research, member checking is often considered a valuable method (Miles and Huberman, 2016 ). In the present study, an additional step was taken to ensure reliability by involving another researcher who is an expert in mathematics education to analyze the data and obtain the results. This approach allows independent analysis and helps mitigate potential bias or subjective interpretations. To assess the consistency between the two analyses, Miles and Huberman’s ( 2016 ) reliability formula was employed: Reliability = Consensus/(Agreement + Disagreement) × 100. The calculation resulted in a consistency value of 98%. This high consistency value indicates a strong agreement between the two researchers’ analyses, further reinforcing the reliability of the research findings. According to Miles and Huberman ( 2016 ), reliability calculations exceeding 70% are generally considered reliable for qualitative research. In this case, with a reliability value of 98%, the research findings exhibit a high level of reliability based on this criterion.

The first analysis of the research is about which of the existing estimation strategies students use. In Table 1 , the strategies chosen most by the students in the face of real-life problems through concept cartoons are given.

Upon examining Table 1 , it is evident that the students displayed varying preferences for estimation strategies when solving different measurement problems. In the weight measurement problem, the segmentation/chunking strategy was the one used most among the estimation strategies produced ( F  = 26). Following that, the comparison strategy was the second most frequently used strategy ( F  = 12). However, the pattern of students’ choices changed when it came to the length measurement problem. In this case, the unit iteration/separation strategy was predominantly favored by the students ( F  = 20). This indicates a shift in their approach to estimation when dealing with length measurements. For the area measurement problem, similar to the weight measurement problem, the segmentation/subdividing/chunking strategy emerged as the most popular strategy ( F  = 16). Additionally, the experience strategy was also notable among those chosen ( F  = 13).

These findings highlight the varying strategies employed by the students when estimating measurements in different contexts. The segmentation/chunking strategy appears to be consistently favored in weight and area measurements, while the unit iteration/separation strategy takes precedence in length measurements. The presence of the experience strategy in the area measurement problem suggests that students may draw upon their prior knowledge and familiarity with certain contexts when estimating areas. The measurement strategies used by the students according to their gender are shown in Table 2 .

According to the findings presented in Table 2 , there were no significant differences between boys and girls in terms of their use of estimation strategies across the weight, length, and area measurement problems. In the weight measurement problem, both boys and girls showed an equal inclination towards using the segmentation/subdividing/chunking strategies. Similarly, in the length measurement problem, both genders exhibited a similar preference for the unit iteration/separation strategy. Additionally, in the area measurement problem, boys and girls equally favored the segmentation/subdividing/chunking strategies. These results suggest that gender did not play a significant role in determining the measurement estimation strategies used by the students (Weight Measurement Pearson Chi-Square  =  0.912; Length Measurement Pearson Chi-Square  =  0.300; Area Measurement Pearson Chi-Square  =  0.423 ). Furthermore, the students were asked if they had any estimation strategies to suggest for the given scenarios other than the concept cartoon scenarios. The strategies proposed by the participating students are presented in Table 3 .

Upon reviewing Table 3 , it is evident that students predominantly relied on their previous experiences or prior knowledge when generating estimation strategies for the weight measurement estimation problem. Additionally, it was observed that students often opted for alternative solution-oriented approaches instead of producing direct estimations. Regarding the estimation strategies generated for the length measurement estimation problem, students showed a preference for the unit iteration/separation strategy and measuring with standard units over other options. When examining the strategies produced for the area measurement estimation problem, it is apparent that responses involving measurements in standard units were more frequently provided compared to other strategies.

These findings highlight the importance of students’ prior experiences and knowledge in their estimation strategies. Students tend to draw on their existing understanding and familiarity with weight, length, and area concepts when making estimations. This suggests that incorporating real-life examples and practical applications in mathematics education can support students in developing more accurate and meaningful estimation strategies. The prominence of alternative solution-oriented approaches in weight estimation indicates that students may employ problem-solving techniques beyond direct estimations. This highlights the need to encourage flexible thinking and diverse problem-solving strategies in mathematics instruction. The prevalence of unit iteration/separation and measuring with standard units in length estimation reflects students’ recognition of the significance of using known units and breaking down the measurement into smaller parts. It underscores the role of standard units as reference points for estimation and the value of utilizing measurement tools effectively. The emphasis on measurement in standard units in the area estimation problem suggests that students recognize the significance of using standardized measurements for accurate estimations. This underscores the importance of developing students’ understanding of measurement units and their application in different contexts. Overall, these findings contribute to our understanding of students’ estimation strategies in measurement tasks and emphasize the need for targeted instruction that builds on students’ prior knowledge, promotes problem-solving skills, and enhances their understanding of measurement concepts and units.

Discussion and conclusion

In the present study, the primary objective was to investigate the measurement estimation strategies of fourth-grade students in a classroom setting using concept cartoons. A total of 46 students participated and were presented with realistic problem situations related to weight, length, and area measurements. The students were asked to identify which answer they believed was closest to the correct estimation based on their own understanding and experiences. Furthermore, it was aimed to explore how students would respond if they were faced with the same measurement problem. The idea was to understand students’ personal approaches and thought processes when tackling measurement estimations. By engaging students in this manner, it was aimed to identify their individual perspectives and gather insights into their decision-making processes regarding measurement estimations. This approach provided valuable information on students’ reasoning abilities and problem-solving strategies and their ability to apply measurement concepts to real-life situations.

The aim of using concept cartoons in the present study was to provide students with a means to externalize their mathematical thinking processes, allowing them to express their thoughts freely and establish connections with other ideas in a familiar environment. By presenting realistic problems through concept cartoons, students were immersed in scenario-based situations that motivated them to find solutions. This approach aligns with the principles of RME and concept cartoons, as it promotes active student engagement and the application of mathematical concepts in real-life contexts. The analysis of the measurement estimation strategies provided by the students revealed that, in general, the most favored strategies in all scenarios were “Segmentation/sub-segmentation/fragmentation” and “Unit iteration/separation”. In previous studies, it was also shown that students used the “Unit iteration/separation” strategy at rates ranging from 30% to 97% (Hildreth, 1983 ; Immers, 1983 ). Additionally, Lehrer et al. ( 2003 ) emphasized the importance of students using the “Reference point/comparison” strategy before employing standard measuring instruments, as it allows them to model the process of unit iteration physically and then mentally. However, it was observed that fourth-grade students showed less interest in this strategy, which contrasts with previous findings indicating its significance. It can be inferred that comparison or reference points are important for estimation, but students tend to use this strategy less frequently (Hildreth, 1983 ). In a pilot study conducted by Joram et al. ( 2005 ) with third-year students, it was also found that the “Reference point/comparison” strategy was rarely utilized for measurement estimation. These findings suggest that there may be variations in the strategies employed by students at different grade levels, highlighting the need for further investigation into students’ understanding and utilization of estimation strategies in the context of measurement. Such variations in findings could be attributed to the mathematics teaching programs and textbooks that each country employs. The extent to which these programs and textbooks emphasize estimation strategies will influence the diversity of strategies students use. Analyzing the results of the research conducted by Bulut and colleagues ( 2017 ) on the examination of estimation skills in mathematics teaching programs between 1945 and 2015, it is noteworthy that there are a ‘limited number of studies on estimation’ and ‘insufficiency in the inclusion of estimation skills in the curriculum in Turkey.’ As of 2018, the Ministry of National Education has started to emphasize estimation more in the mathematics teaching program (MoNE, 2018 ). Consequently, the number of studies conducted on estimation in the country between 2017 and 2019 has increased compared to other years (Bağdat and Yıldız, 2023 ). Although the concept of estimation is present in the programs, it has been limited to certain grade levels. In particular, due to the necessity for students to have sufficient prior knowledge about the topic, it has been somewhat neglected at the K-4 level. Moreover, research indicates that in textbooks the only estimation strategy mentioned is usually rounding. Additionally, textbooks lack discussions about various estimation strategies and there is a lack of strategy instruction for students.

Another notable finding in the study was the similarity in measurement estimation strategies used by male and female students. While there is limited research specifically exploring gender differences in measurement strategies, Hildreth ( 1983 ) conducted a study involving students at different grade levels (fifth, seventh, and university first grade) and found no significant differences in the measurement strategies employed by students based on their gender. These results suggest that gender may not play a significant role in influencing the choice of measurement estimation strategies among students. However, further research is needed to delve deeper into this aspect and explore potential factors that might contribute to variations in strategy preferences.

In a study conducted by Ruwisch et al. ( 2015 ) involving fourth-grade students, it was discovered that these students employ a variety of strategies to estimate lengths and areas. These strategies were categorized into three main types: comparisons, division into units, and segmentation/grouping. Interestingly, the students demonstrated equal use of these strategies for both length and area measurement estimations. The students also exhibited spontaneous generation of measurement estimation strategies, including prior knowledge, another solution-oriented approach, reference point/comparison, irrelevant answers, unit iteration/separation, measuring in standard units, and segmentation/subdividing/chunking. This diversity in strategies indicates flexibility in their mathematical thinking, showcasing their ability to adapt different approaches based on the context of the measurement problem (Drijvers et al., 2019 ). When examining the strategies generated by the students in the present study, it was observed that their approaches to measuring length and area were remarkably similar. However, considering the meaningfulness of mathematical ideas when connected to other concepts and situations (Clements and McMillen, 1996 ), it was hypothesized that students would be influenced by concept cartoons and potentially develop different strategies in their original responses. Moreover, the observation of students demonstrating equal use of strategies for both length and area measurements implies a certain level of transferability in their problem-solving skills across different mathematical domains. This not only shows the versatility of students’ cognitive processes but also suggests that interventions targeting measurement estimation skills may have broad applications (Desli and Giakoumi, 2017 ). The spontaneous generation of measurement estimation strategies, including prior knowledge, another solution-oriented approach, reference point/comparison, irrelevant answers, unit iteration/separation, measuring in standard units, and segmentation/subdividing/chunking, underscores the creative and exploratory nature of students’ problem-solving processes. It indicates that students draw upon various cognitive resources, including their existing knowledge, to formulate strategies when faced with measurement estimation challenges (Tambychik and Meerah, 2010 ).

Notably, the presence of a substantial number of “other solution-oriented options” and “irrelevant answers” in the students’ responses should not be overlooked. These responses highlight the need for cultivating accuracy and relevance in their estimation strategies. As the strategies employed by children play a vital role in problem-solving, they must possess accurate and effective estimation strategies (Peeters et al., 2016 ). Hildreth ( 1983 ) emphasizes the importance of incorporating student discussions during training sessions focused on the utilization of measurement estimation strategies. Classroom examples can be utilized to illustrate each strategy and provide students with a deeper understanding. It is important to note that one limitation of the present study is the use of concept cartoons for evaluation purposes rather than instructional purposes. Consequently, the study lacked the inclusion of desired group activities, discussions, and additional examples based on similar scenarios and new problems. During the study, students were instructed to select the strategy that closely aligned with their own preferences by considering the measurement strategies presented in the speech bubbles of the concept cartoons. In this approach, the aim was to enhance their awareness of their own measurement estimation strategies and potentially inspire the development of new and spontaneous approaches, perhaps influenced by the strategies depicted in the concept cartoons.

In conclusion, in the present study, the measurement estimation strategies employed by fourth-grade students using concept cartoons and realistic problem scenarios were investigated. It was revealed that students utilize various strategies, including comparisons, division into units, and segmentation/grouping, to estimate lengths and areas. The findings emphasize the significance of connecting mathematical concepts with real-life situations. Additionally, research shows that incorporating real-life contexts into mathematical problems can significantly improve student understanding and application of concepts (Adams and Lowery, 2007 ). The aim of using concept cartoons was to influence students’ original answers and encourage the adoption of different strategies. However, the presence of a notable number of “other solution-oriented options” and “irrelevant answers” in the students’ responses highlights the need for accurate and useful estimation strategies. This suggests an area for instructional focus, emphasizing the importance of guiding students toward strategies that not only showcase creativity but also align with the requirements of the given mathematical problem (Heinze et al., 2009 ). The research underscores the importance of incorporating student discussions and examples in teaching measurement estimation strategies. By integrating these elements into instructional sessions, students can enhance their understanding and application of various strategies.

Limitations and suggestions

While the study contributes valuable insights into students’ measurement estimation strategies, it has certain limitations. Notably, the focus on evaluation rather than teaching limited the inclusion of group activities, discussions, and additional examples. Addressing these limitations and further refining the study can strengthen its contribution to the field of mathematics education. Overall, the study provides a comprehensive analysis of students’ use of measurement estimation strategies in the context of concept cartoons and realistic problems. The findings underscore the importance of promoting accurate and effective estimation strategies among students, highlighting the potential for further research and instructional improvements in this area.

As suggestions for researchers and future studies, educators, and their practices, several recommendations can be made based on the findings and limitations of the present study:

Suggestions for researchers and future studies:

Several recommendations can be proposed for researchers and future studies based on the findings and limitations of the present study. Expanding the research scope by conducting similar studies with larger sample sizes and across different grade levels can provide a more comprehensive understanding of students’ measurement estimation strategies, allowing comparisons and analysis of developmental changes in strategy choices. Longitudinal studies tracking students’ progression in measurement estimation strategies can shed light on the development and effectiveness of different strategies over time. Qualitative analysis, such as interviews or think-aloud protocols, can supplement quantitative data, providing deeper insights into students’ thought processes, reasoning, and decision-making strategies during measurement estimation tasks. Intervention studies that explicitly target measurement estimation strategies can assess their effectiveness in promoting accurate and useful estimation approaches among students. Comparing the outcomes of different instructional methods or interventions can provide evidence-based guidance for educators. Conducting studies that focus on integrating concept cartoons and realistic problem scenarios into classroom instruction rather than solely using them for evaluation purposes can offer valuable insights into the impact of such interventions on students’ strategy preferences and problem-solving abilities. Investigating potential differences in measurement estimation strategies based on various demographic factors, such as gender, cultural background, or mathematical achievement, can inform personalized instruction and support equitable mathematics education.

Implications for educators

The present study offers valuable insights that educators can use to enhance their instructional practices in the area of measurement estimation strategies. The findings underscore the significance of incorporating realistic problem scenarios, such as those presented through concept cartoons, into classroom instruction. Educators can adopt a pedagogical approach that aligns with the principles of Realistic Mathematics Education (RME) and concept cartoons, promoting active student engagement and the application of mathematical concepts in real-life contexts. By immersing students in scenario-based situations, educators can stimulate their problem-solving abilities and foster a deeper understanding of measurement estimation strategies. Moreover, the study emphasizes the importance of recognizing and addressing the diversity of strategies employed by students. The principles of RME and concept cartoons can guide educators in providing differentiated instruction that accommodates varied learning styles and preferences. The study also highlights the need for explicit instruction and discussions on measurement estimation strategies. Practically, educators can integrate the use of concept cartoons into lesson plans to facilitate discussions and activities focused on measurement estimation. Educators can incorporate classroom activities that encourage students to articulate their thought processes, share their strategies, and engage in collaborative problem-solving. By creating an environment that values diverse strategies, educators can contribute to the development of students’ reasoning abilities and problem-solving skills. Classroom activities can be designed to encourage group discussions, peer interactions, and additional examples related to the concept cartoons presented. In essence, this study opens up avenues for educators to refine their teaching methods, promote a variety of measurement estimation strategies, and create inclusive learning environments that nurture students’ mathematical reasoning.

Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author upon reasonable request.

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Teens come up with trigonometry proof for Pythagorean Theorem, a problem that stumped math world for centuries

By Bill Whitaker

May 5, 2024 / 7:00 PM EDT / CBS News

As the school year ends, many students will be only too happy to see math classes in their rearview mirrors. It may seem to some of us non-mathematicians that geometry and trigonometry were created by the Greeks as a form of torture, so imagine our amazement when we heard two high school seniors had proved a mathematical puzzle that was thought to be impossible for 2,000 years. 

We met Calcea Johnson and Ne'Kiya Jackson at their all-girls Catholic high school in New Orleans. We expected to find two mathematical prodigies.

Instead, we found at St. Mary's Academy , all students are told their possibilities are boundless.

Come Mardi Gras season, New Orleans is alive with colorful parades, replete with floats, and beads, and high school marching bands.

In a city where uniqueness is celebrated, St. Mary's stands out – with young African American women playing trombones and tubas, twirling batons and dancing - doing it all, which defines St. Mary's, students told us.

Junior Christina Blazio says the school instills in them they have the ability to accomplish anything. 

Christina Blazio: That is kinda a standard here. So we aim very high - like, our aim is excellence for all students. 

The private Catholic elementary and high school sits behind the Sisters of the Holy Family Convent in New Orleans East. The academy was started by an African American nun for young Black women just after the Civil War. The church still supports the school with the help of alumni.

In December 2022, seniors Ne'Kiya Jackson and Calcea Johnson were working on a school-wide math contest that came with a cash prize.

Ne'Kiya Jackson: I was motivated because there was a monetary incentive.

Calcea Johnson: 'Cause I was like, "$500 is a lot of money. So I-- I would like to at least try."

Both were staring down the thorny bonus question.

Bill Whitaker: So tell me, what was this bonus question?

Calcea Johnson: It was to create a new proof of the Pythagorean Theorem. And it kind of gave you a few guidelines on how would you start a proof.

The seniors were familiar with the Pythagorean Theorem, a fundamental principle of geometry. You may remember it from high school: a² + b² = c². in plain English, when you know the length of two sides of a right triangle, you can figure out the length of the third.

Both had studied geometry and some trigonometry, and both told us math was not easy. What no one told  them  was there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible, … and that was the bonus question facing them.

Bill Whitaker: When you looked at the question did you think, "Boy, this is hard"?

Ne'Kiya Jackson: Yeah. 

Bill Whitaker: What motivated you to say, "Well, I'm going to try this"?

Calcea Johnson: I think I was like, "I started something. I need to finish it." 

Bill Whitaker: So you just kept on going.

Calcea Johnson: Yeah.

For two months that winter, they spent almost all their free time working on the proof.

CeCe Johnson: She was like, "Mom, this is a little bit too much."

CeCe and Cal Johnson are Calcea's parents.

CeCe Johnson:   So then I started looking at what she really was doing. And it was pages and pages and pages of, like, over 20 or 30 pages for this one problem.

Cal Johnson: Yeah, the garbage can was full of papers, which she would, you know, work out the problems and-- if that didn't work she would ball it up, throw it in the trash. 

Bill Whitaker: Did you look at the problem? 

Neliska Jackson is Ne'Kiya's mother.

Neliska Jackson: Personally I did not. 'Cause most of the time I don't understand what she's doing (laughter).

Michelle Blouin Williams: What if we did this, what if I write this? Does this help? ax² plus ….

Their math teacher, Michelle Blouin Williams, initiated the math contest.

Bill Whitaker: And did you think anyone would solve it?

Michelle Blouin Williams: Well, I wasn't necessarily looking for a solve. So, no, I didn't—

Bill Whitaker: What were you looking for?

Michelle Blouin Williams: I was just looking for some ingenuity, you know—

Calcea and Ne'Kiya delivered on that! They tried to explain their groundbreaking work to 60 Minutes. Calcea's proof is appropriately titled the Waffle Cone.

Calcea Johnson: So to start the proof, we start with just a regular right triangle where the angle in the corner is 90°. And the two angles are alpha and beta.

Bill Whitaker: Uh-huh

Calcea Johnson: So then what we do next is we draw a second congruent, which means they're equal in size. But then we start creating similar but smaller right triangles going in a pattern like this. And then it continues for infinity. And eventually it creates this larger waffle cone shape.

Calcea Johnson: Am I going a little too—

Bill Whitaker: You've been beyond me since the beginning. (laughter) 

Bill Whitaker: So how did you figure out the proof?

Ne'Kiya Jackson: Okay. So you have a right triangle, 90° angle, alpha and beta.

Bill Whitaker: Then what did you do?

Ne'Kiya Jackson: Okay, I have a right triangle inside of the circle. And I have a perpendicular bisector at OP to divide the triangle to make that small right triangle. And that's basically what I used for the proof. That's the proof.

Bill Whitaker: That's what I call amazing.

Ne'Kiya Jackson: Well, thank you.

There had been one other documented proof of the theorem using trigonometry by mathematician Jason Zimba in 2009 – one in 2,000 years. Now it seems Ne'Kiya and Calcea have joined perhaps the most exclusive club in mathematics. 

Bill Whitaker: So you both independently came up with proof that only used trigonometry.

Ne'Kiya Jackson: Yes.

Bill Whitaker: So are you math geniuses?

Calcea Johnson: I think that's a stretch. 

Bill Whitaker: If not genius, you're really smart at math.

Ne'Kiya Jackson: Not at all. (laugh) 

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

Ne'Kiya Jackson: Well, our teacher approached us and was like, "Hey, you might be able to actually present this," I was like, "Are you joking?" But she wasn't. So we went. I got up there. We presented and it went well, and it blew up.

Bill Whitaker: It blew up.

Calcea Johnson: Yeah. 

Ne'Kiya Jackson: It blew up.

Bill Whitaker: Yeah. What was the blowup like?

Calcea Johnson: Insane, unexpected, crazy, honestly.

It took millenia to prove, but just a minute for word of their accomplishment to go around the world. They got a write-up in South Korea and a shout-out from former first lady Michelle Obama, a commendation from the governor and keys to the city of New Orleans. 

Bill Whitaker: Why do you think so many people found what you did to be so impressive?

Ne'Kiya Jackson: Probably because we're African American, one. And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part.

Bill Whitaker: So you think people were surprised that young African American women, could do such a thing?

Calcea Johnson: Yeah, definitely.

Ne'Kiya Jackson: I'd like to actually be celebrated for what it is. Like, it's a great mathematical achievement.

Achievement, that's a word you hear often around St. Mary's academy. Calcea and Ne'Kiya follow a long line of barrier-breaking graduates. 

The late queen of Creole cooking, Leah Chase , was an alum. so was the first African-American female New Orleans police chief, Michelle Woodfork …

And judge for the Fifth Circuit Court of Appeals, Dana Douglas. Math teacher Michelle Blouin Williams told us Calcea and Ne'Kiya are typical St. Mary's students.  

Bill Whitaker: They're not unicorns.

Michelle Blouin Williams: Oh, no no. If they are unicorns, then every single lady that has matriculated through this school is a beautiful, Black unicorn.

Pamela Rogers: You're good?

Pamela Rogers, St. Mary's president and interim principal, told us the students hear that message from the moment they walk in the door.

Pamela Rogers: We believe all students can succeed, all students can learn. It does not matter the environment that you live in. 

Bill Whitaker: So when word went out that two of your students had solved this almost impossible math problem, were they universally applauded?

Pamela Rogers: In this community, they were greatly applauded. Across the country, there were many naysayers.

Bill Whitaker: What were they saying?

Pamela Rogers: They were saying, "Oh, they could not have done it. African Americans don't have the brains to do it." Of course, we sheltered our girls from that. But we absolutely did not expect it to come in the volume that it came.  

Bill Whitaker: And after such a wonderful achievement.

Pamela Rogers: People-- have a vision of who can be successful. And-- to some people, it is not always an African American female. And to us, it's always an African American female.

Gloria Ladson-Billings: What we know is when teachers lay out some expectations that say, "You can do this," kids will work as hard as they can to do it.

Gloria Ladson-Billings, professor emeritus at the University of Wisconsin, has studied how best to teach African American students. She told us an encouraging teacher can change a life.

Bill Whitaker: And what's the difference, say, between having a teacher like that and a whole school dedicated to the excellence of these students?

Gloria Ladson-Billings: So a whole school is almost like being in Heaven. 

Bill Whitaker: What do you mean by that?

Gloria Ladson-Billings: Many of our young people have their ceilings lowered, that somewhere around fourth or fifth grade, their thoughts are, "I'm not going to be anything special." What I think is probably happening at St. Mary's is young women come in as, perhaps, ninth graders and are told, "Here's what we expect to happen. And here's how we're going to help you get there."

At St. Mary's, half the students get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. Here, there's no test to get in, but expectations are high and rules are strict: no cellphones, modest skirts, hair must be its natural color.

Students Rayah Siddiq, Summer Forde, Carissa Washington, Tatum Williams and Christina Blazio told us they appreciate the rules and rigor.

Rayah Siddiq: Especially the standards that they set for us. They're very high. And I don't think that's ever going to change.

Bill Whitaker: So is there a heart, a philosophy, an essence to St. Mary's?

Summer Forde: The sisterhood—

Carissa Washington: Sisterhood.

Tatum Williams: Sisterhood.

Bill Whitaker: The sisterhood?

Voices: Yes.

Bill Whitaker: And you don't mean the nuns. You mean-- (laughter)

Christina Blazio: I mean, yeah. The community—

Bill Whitaker: So when you're here, there's just no question that you're going to go on to college.

Rayah Siddiq: College is all they talk about. (laughter) 

Pamela Rogers: … and Arizona State University (Cheering)

Principal Rogers announces to her 615 students the colleges where every senior has been accepted.

Bill Whitaker: So for 17 years, you've had a 100% graduation rate—

Pamela Rogers: Yes.

Bill Whitaker: --and a 100% college acceptance rate?

Pamela Rogers: That's correct.

Last year when Ne'Kiya and Calcea graduated, all their classmates went to college and got scholarships. Ne'Kiya got a full ride to the pharmacy school at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University.

Bill Whitaker: So wait a minute. Neither one of you is going to pursue a career in math?

Both: No. (laugh)

Calcea Johnson: I may take up a minor in math. But I don't want that to be my job job.

Ne'Kiya Jackson: Yeah. People might expect too much out of me if (laugh) I become a mathematician. (laugh)

But math is not completely in their rear-view mirrors. This spring they submitted their high school proofs for final peer review and publication … and are still working on further proofs of the Pythagorean Theorem. Since their first two …

Calcea Johnson: We found five. And then we found a general format that could potentially produce at least five additional proofs.

Bill Whitaker: And you're not math geniuses?

Bill Whitaker: I'm not buying it. (laughs)

Produced by Sara Kuzmarov. Associate producer, Mariah B. Campbell. Edited by Daniel J. Glucksman.

Bill Whitaker is an award-winning journalist and 60 Minutes correspondent who has covered major news stories, domestically and across the globe, for more than four decades with CBS News.

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UW Students Marvel in Math With Participation in Wyoming Pi Days

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Institutional Communications Bureau of Mines Building, Room 137 Laramie, WY 82071 Phone: (307) 766-2929 Email:   [email protected]

Published April 30, 2024

University of Wyoming students, like many around the world, celebrated Pi Day with fun and challenging math problems.

Twenty-five UW students participated in Wyoming Pi Days, a series of three “Pizza and Problem Solving” evenings in which students got together, ate pizza and worked on a series of mathematics/logic problems.

These evenings -- March 21 and 26, and April 1 -- culminated in an examination called Pi Day Competition: a three-hour exam consisting of problems of varying difficulty that took place April 6. The problems in the exam are similar in complexity to those tackled during the “Pizza and Problem Solving” evenings. Twenty of the 25 students took the exam.

“Wyoming Pi Days is a blast, both for the students and the coordinators,” says Jorge Flores, an assistant lecturer of mathematics and one of the coordinators of Wyoming Pi Days. “We get to eat pizza, socialize and work on fun math and logic problems together. The event helps students see that math can be social and fun and helps them build a community with students of similar interests.”

Wyoming Pi Days was hosted and sponsored by the UW Department of Mathematics and Statistics. The event, for UW undergraduate students of all majors interested in mathematics, is an homage to the number pi, which is approximately 3.14.

Other event coordinators from the department were Tyrrell McAllister, an associate professor, and Christina Knox, an assistant lecturer.

The coordinators of the event discussed outstanding submissions for the Pi Day Competition and recognized the following participants, listed by hometown:

Cheyenne -- Connor Gililland, a senior majoring in statistics and chemical engineering, with a mathematics minor; and Cade Pugh, a sophomore majoring in mechanical engineering and mathematics.

Fredericktown, Ohio -- Carissa Van Slyke, a senior majoring in secondary education (mathematics) and mathematics, with minors in honors, Spanish and English as a second language.

Laramie -- Jonathan Oler, a Laramie High School junior taking classes at UW.

The competition winners each received a certificate and a grand prize -- a Möbius strip sculpture.

“We hope that Wyoming Pi Days served as a way for students to see how fun problem-solving can be and to meet other students with similar interests,” Flores says.

All registered participants who took the Pi Day Competition exam and attended at least one problem-solving session received participation prizes. The participation prizes were a math/problem-solving book and a T-shirt commemorating the event.

Eric Moorhouse, a UW professor of mathematics, created the T-shirt design. Jason Williford, department head and a UW professor of mathematics, provided financial and moral support. Organizers give a special thanks to Ping Zhong, an assistant professor of mathematics, and his National Science Foundation grant, which paid for the books given to the participants.

For more information about Wyoming Pi Days, email Flores at [email protected] .

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Intervention based on science of reading, math boosts comprehension, word problem-solving

Mon, 04/29/2024

Mike Krings

LAWRENCE — New research from the University of Kansas has found an intervention based on the science of reading and math effectively helped English learners boost their comprehension, visualize and synthesize information, and make connections that significantly improved their math performance.

The intervention, performed for 30 minutes twice a week for 10 weeks with 66 third-grade English language learners who displayed math learning difficulties, improved students’ performance when compared to students who received general instruction. That indicates emphasizing cognitive concepts involved in the science of reading and math are key to helping students improve, according to researchers.

“Word problem-solving is influenced by both the science of reading and the science of math. Key components include number sense, decoding, language comprehension and working memory. Utilizing direct and explicit teaching methods enhances understanding and enables students to effectively connect these skills to solve math problems. This integrated approach ensures that students are equipped with necessary tools to navigate both the linguistic and numerical demands of word problems,” said Michael Orosco, professor of educational psychology at KU and lead author of the study. 

The intervention incorporates comprehension strategy instruction in both reading and math, focusing and decoding, phonological awareness, vocabulary development, inferential thinking, contextualized learning and numeracy.

“It is proving to be one of the most effective evidence-based practices available for this growing population,” Orosco said.

The study, co-written with Deborah Reed of the University of Tennessee, was published in the journal Learning Disabilities Research and Practice.

For the research, trained tutors developed the intervention, developed by Orosco and colleagues based on cognitive and culturally responsive research conducted over a span of 20 years. One example of an intervention session tested in the study included a script in which a tutor examined a word problem that explained a person made a quesadilla for his friend Mario, giving him one-fourth of it, then needed to students to determine how much remained.

The tutor first asked students if they remembered a class session in which they made quesadillas, what shape they were and demonstrated concepts by drawing a circle on the board, dividing it into four equal pieces, having students repeat terms like numerator and denominator, and explaining that when a question asks how much is left, subtraction is required. The students also collaborated with peers to practice using important vocabulary in sentences. The approach both helps students learn and understand mathematical concepts while being culturally responsive.

"Word problems are complex because they require translating words into mathematical equations, and this involves integrating the science of reading and math through language concepts and differentiated instruction," Orosco said. "We have not extensively tested these approaches with this group of children. However, we are establishing an evidence-based framework that aids them in developing background knowledge and connecting it to their cultural contexts."

Orosco , director of KU’s Center for Culturally Responsive Educational Neuroscience, emphasized the critical role of language in word problems, highlighting the importance of using culturally familiar terms. For instance, substituting "pastry" for "quesadilla" could significantly affect comprehension for students from diverse backgrounds. Failure to grasp the initial scenario can impede subsequent problem-solving efforts.

The study proved effective in improving students’ problem-solving abilities, despite covariates including an individual’s basic calculation skills, fluid intelligence and reading comprehension scores. That finding is key as, while ideally all students would begin on equal footing and there were little variations in a classroom, in reality, covariates exist and are commonplace.

The study had trained tutors deliver the intervention, and its effectiveness should be further tested with working teachers, the authors wrote. Orosco said professional development to help teachers gain the skills is necessary, and it is vital for teacher preparation programs to train future teachers with such skills as well. And helping students at the elementary level is necessary to help ensure success in future higher-level math classes such as algebra.

The research builds on Orosco and colleagues’ work in understanding and improving math instruction for English learners . Future work will continue to examine the role of cognitive functions such as working memory and brain science , as well as potential integration of artificial intelligence in teaching math.

“Comprehension strategy instruction helps students make connections, ask questions, visualize, synthesize and monitor their thinking about word problems,” Orosco and Reed wrote. “Finally, applying comprehension strategy instruction supports ELs in integrating their reading, language and math cognition… Focusing on relevant language in word problems and providing collaborative support significantly improved students’ solution accuracy.”

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Mathematics > Analysis of PDEs

Title: a gauss curvature flow approach to the p-harmonic measure of minkowski problem.

Abstract: The Minkowski problem of harmonic measures was first studied by Jerison \cite{JER1991}. Recently, Akman and Mukherjee \cite{AKM2023} studied the Minkowski problem corresponding to $p$-harmonic measures on convex domains and generalized Jerison's results. In this paper, we obtain the existence of the smooth solution of the $p$-harmonic measure Minkowski problem by method of the Gauss curvature flow.

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The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Mike Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, John Ketchum, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Dan Farrell, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Summer Thomad, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Sam Dolnick, Paula Szuchman, Lisa Tobin, Larissa Anderson, Julia Simon, Sofia Milan, Mahima Chablani, Elizabeth Davis-Moorer, Jeffrey Miranda, Renan Borelli, Maddy Masiello, Isabella Anderson and Nina Lassam.

Jonathan Wolfe is a senior staff editor on the newsletters team at The Times. More about Jonathan Wolfe

Peter Baker is the chief White House correspondent for The Times. He has covered the last five presidents and sometimes writes analytical pieces that place presidents and their administrations in a larger context and historical framework. More about Peter Baker

Luke Vander Ploeg is a senior producer on “The Daily” and a reporter for the National Desk covering the Midwest. More about Luke Vander Ploeg

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