Open-ended Maths: a selection of 101qs favourites

The 101qs website was created in 2012 by Dan Meyer, a great Maths teacher and dedicated apostle of open-ended Maths.

‘Open-ended Maths’ is an expression that sounds pretty scary to many Maths teachers. A lot worse than ‘out of the textbook’, which is rather trendy and flattering. Dan Meyer’s first paragraph on his About page does not seek to cajole them back to their comfort zone either:

We don’t care how well you lecture. We don’t care how well you engage us. We aren’t impressed by your fancy slide transitions or your interactive whiteboard. We care how well you perplex us.

This sets the tone perfectly: open-ended Maths is about stimulating students by presenting them with a perplexing picture or video — perplexing in the sense that it raises questions. Students have to formulate the questions themselves and subsequently solve them, which often means developing appropriate resources on the way.

If you are a regular reader of my posts, you know by now that this blog is not about being ‘for’ or ‘against’, but understanding precisely WHY and HOW an idea is potentially useful for teaching Maths. In this instance, it so happens that open-ended Maths is not only cool, it is also an essential teaching tool — for reasons I shall explain in a later post — on the condition that it is not used systematically (for reasons I shall also explain).

I will not write about how 101qs material should be used in the classroom, as this has already been done very clearly by Dan Meyer himself, for example in his post entitled The Three Acts Of A Mathematical Story.

Nor will I make a list of topics covered by 101qs activities: there is an excellent search engine on the 101qs website, with keywords and grades.

What I would like to do is to start from this idea of perplexing students and propose a selection that will highlight the 7 main forms of this perplexing power. (Please note: all links open a new window of the related page on Dan Meyer’s website).

 

Perplexing power 1: the power of Maths to prove something incredible

Domino Skyscaper

This video-based resource is a fabulous introduction to ascending geometric sequences, as well as a great activity on powers or fraction multiplication.

 

 

Shrinking dollar

This video-based activity plays with the same idea, but with a multiplier lower than 1 (descending geometric sequences).

Students will be surprised to prove that it doesn’t take that many reductions for a dollar bill to become microscopic.

 

 

Perplexing power 2: the magic of records and ‘big things’

Hot coffee

 

This activity can be used when working on volume, unit conversions, speed or ratios.

 

 

In the same category, see also Super Bear.

 

Dollar wall

This activity is about areas and various calculations based on ratios.

Students also get the extra pleasure of working on a real net of this incredible dollar-carpeted room.

 

172-wikibook.jpg

 

With its triggering question (How tall would all of wikipedia be ?), this activity is purely about the joy of big numbers.

 

 

 

Printer Paper.jpg

Unlike the previous activity, where the idea is to produce a plausible ballpark number, in this instance it is possible to calculate (among other things) a precise number of pages.

But for this, students have to exercise their 3D vision and be quite selective as to which information is really relevant.

 

33-gumballs.jpg

 

This situation is also about ‘big things’, but it induces a different discussion about the nature of space. Is that a sphere filled with spheres ? what about the space between gumballs ? etc.

Many possible sequels with colours, estimated revenue, etc.

 

 

 

 

 

 

112-viva-las-colas         434-please-take-a-seat-gulliver

These are 2 examples from which students can work on proportionality, scale, ratios, etc.

 

 

 

Perplexing power 3: the curious Maths of everyday life situations

Graduation

 

This hilarious video is virtually limitless in its applications on speed, line equations, linear and non-linear functions, etc.

 

 

 

Paint mixup

 

This is a classic problem involving ratios, but the video and humour add an extra dimension. Lots of possible open-ended extensions as well.

 

On the same idea (but with a mix of continuous and discontinuous variables), see also Nana’s Lemonade.

 

Lucky cow

This is an example of many possible problems involcing cuts, and therefore fractional areas of various shapes.

It is also illustrates the versatility of open-ended resources: for example, this video can be used either as such for A-level students on sector areas, or with a triangle approximation for younger students.

 

Double sunglasses

 

This very simple everyday life situation raises an important mathematical question about fraction multiplication  (versus addition).

 

 

Sugar packets

 

There again, this activity is about various ratios, with a highly educational dimension on the absurdity of soft drinks.

 

On the same idea, see also Soda.

 

Leaky faucet

 

A volume estimate problem, with an interesting question about the size of a drop of water, as well as many possible cross-topic sequels.

 

 

Dandy candiesCoffee traveller

 

These 2 activities are  interesting for students to relate 1D, 2D and 3D concepts such as length, area and volume.

The first activity (Dandy candies) is probably more suitable for younger students, while the second (Coffee traveller) is a beauty, as it raises questions on graphs, discontinuities induced by hypotheses, etc. without going into too complex Maths.

 

Gas Light Act1

Unlike previously selected activities, this is just one picture, but it’s absolutely packed with information and potential calculations: the situation is easy to understand and students can formulate their own questions.

 

66-the-ticket-roll.jpg

As per the previous one, this picture is a concentrate of fascinating Maths question: is about length ? perimeter ? radius ? area ? cross-section ?

It also introduces students to the idea of modelling reality, as they will probably have to conceptually approximate the idea of roll to find an estimate.

On the same idea, see also Toilet paper roll.

 

64-transcontinental-license-plates.png

Registration plates have long been a subject of interest for kids, and this example can be the starting point to a lesson on combinatorics.

Interesting implications on standard form as well.

 

 

 

 

Perplexing power 4: how beautiful visuals raise Maths questions

39-car-caravan.jpg

 

Much as the Ticket Roll activity, with a different feel (more accessible to younger students, because cars are definitely 3D objects with a width)… and a more artistic touch.

 

 

 

 

140-ball-of-mackerel

 

Same idea, but even more beautiful… and more complex too.

 

 

 

 

 

37-penny-pyramid

This goes well beyond applying the formula for the volume of a pyramid. It questions how you measure volume, as this pyramid is actually made of cylinders ? Or is it cuboids ?

On close analysis, this is well worth a discussion, isn’t it ?

Can also be used for sequences (going from one layer to the next).

 

 

775-circle-or-polygon

There is much to understand from this picture, which can be an opportunity for estimating the number of trolleys, but also for modelling a polygon into a circle.

Perhaps a circle is a polygon with an infinite number of sides ?

 

 

Neptune.JPG

Apart from the sheer perplexing poetry of the picture, this activity introduces students to the idea of scale and how large distances can be modelled into smaller ones.

Just like the Maine solar system project, students can then choose a suitable scale and position planets in one of the school’s open spaces.

 

 

Perplexing power 5: how a little Maths helps you to outplay rip-offs

2125-negative-sale

 

An interesting intellectual shift on negative numbers…

 

 

 

 

 

 

Money duck

This activity is potentially limitless on probability, probability distributions, expected value and optimisation.

 

 

 

 

221-great-grocery-sale

zH6jQlp.jpg

These are 2 complementary activities on equivalent fractions.

 

 

Perplexing power 6: how Maths helps students to become active scientists

Volcano

This simulation-based activity is how about a volcano and how long will it take the lava to reach a village called Tarata.

It is about speed and circles, but with potentially many implications on graphs, imagining and modelling the impact of slope on speed, etc.

 

Falling rocks

Beyond the teen story, this video raises the interesting question of how to measure the depth of a hole in the dark and with no tool.

Much to discuss about the relevant information on speed, weight, gravity, acceleration, etc.

 

 

Perplexing power 7: how Maths can be used to analyze the imperfections of this world

Best circle

In Maths, and particularly geometry, students are accustomed to working with theoretically perfect objects, like circles for instance. But what about imperfect objects ?

This activity introduces students to error percentages, and more importantly to the idea of distance criteria.

There are also similar activities for best midpoint, best square and best triangle.

 


 

Conclusion

There are dozens more resources on the 101qs website, which also features a search engine with keywords and grades. Here is also a link to Dan Meyer’s own bank of Three-Act Maths tasks.

It should be noted that, among other advantages, many 101qs activities (and particularly the most visual ones)  provide excellent practice for estimating (cf. feel for numbers in Maths ability pyramid). Open-ended Maths also reduces the fear of numbers (cf. post on Three things your Maths students should not be afraid of).

Beyond that, 101qs resources and investigations often defy both topical classification and level classification. They are so rich that most of them can be used at different levels with different objectives.

However, there are several areas of the curriculum where 101qs is definitely a goldmine, for example:

 

My daughter’s top 5:

 

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