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An exciting adventure awaits with the LeapLand Adventures™ learning video game. Kids go on a quest and learn about letters, numbers, shapes and colors along the way.

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The Ministry is migrating nzmaths content to Tāhurangi.             Relevant and up-to-date teaching resources are being moved to Tāhūrangi (tahurangi.education.govt.nz).  When all identified resources have been successfully moved, this website will close. We expect this to be in June 2024.  e-ako maths, e-ako Pāngarau, and e-ako PLD 360 will continue to be available. 

For more information visit https://tahurangi.education.govt.nz/updates-to-nzmaths

This seminar is based around one of the animated mathematical problems contained in the “Bright Sparks” component of the site.  These animated problems are based on a mathematical idea and are aimed at extending bright students.  In this seminar we will talk about the mathematics behind Leap Frogs games so that you can help your students if they need it.  

We suggest that you take one of two approaches here.  If you know this problem well, move straight on to ‘B.  Frogs – The Seminar’.  Otherwise you should at least skim through ‘A.  Frogs – The Game’. 

Section A. looks at the basic game and how it is played.  Then it moves into counting the number of moves that are required to mover the frogs where they want to go.  We look at some alternative ways of arriving at a formula for the general pattern and we finish by justifying this pattern. 

On the other hand, section B, lays out the steps you might take and content you might use in a seminar on frogs.  The point that we want to make here is that there is a lot more to this problem than you might expect. 

At the end of this seminar you will find Copymasters for overhead slides that you might find useful. 

A.  Frogs – The Game

1.  Introduction : Frogs is based on a game where 3 frogs on one side of a set of lily pads are trying to interchange places with three frogs on the other side.  You can see this in the picture below.  (It’s also available as Copymaster 1.)

There are essentially two key steps in the overall problem.  First, you have to master the sequence of moves that get the frogs interchanged.  Second, you have to determine the number of moves needed for any number of frogs.  This is quite enough for the average class.  However, if you have a really capable student in your class, they might ask how do you know that this is indeed the fewest number of moves?  And if they don’t ask it then you should ask them . 

2.  The sequence of moves : First you have to know what moves are possible.  Well, you must keep moving the frogs forward; you can move a frog onto an empty lily pad if there is one in front of it; and you can jump over a frog if there is an empty space on the other side of the frog that is being jumped. 

Below is a sequence of moves that will get six frogs (three on each side) where they want to go.  (GF stands for Green Frogs and BF for Brown Frogs.)

GF3   GF2   GF1             BF1   BF2   BF3

GF3   GF2   GF1   BF1             BF2   BF3

GF3   GF2              BF1   GF1 BF2   BF3

GF3              GF2   BF1   GF1 BF2   BF3

GF3   BF1   GF2              GF1 BF2   BF3

GF3   BF1   GF2   BF2   GF1            BF3

GF3   BF1   GF2   BF2   GF1  BF3            ­

GF3   BF1   GF2   BF2              BF3   GF1

GF3   BF1               BF2   GF2 BF3   GF1

            BF1    GF3   BF2   GF2 BF3   GF1

BF1               GF3   BF2   GF2 BF3   GF1

BF1   BF2    GF3              GF2 BF3   GF1

BF1   BF2    GF3   BF3   GF2            GF1

BF1   BF2    GF3   BF3             GF2   GF1

BF1   BF2               BF3   GF3 GF2   GF1

BF1   BF2    BF3              GF3 GF2   GF1

If you have four frogs on either side the sequence of moves is very similar.  The basic idea is to set up a situation where the green frogs, say, can provide a way of letting the brown frogs jump over them in succession.  There is one key idea here that has to be kept in mind.  That is, that you never want to be in a position where two frogs of the same colour are next to each other.  If you reach that state, then the whole jumping business comes to a halt.  So think ahead to make sure that you don’t get two frogs of the same colour next to each other.

3.  The number of moves : The 3-frogs-a-side problem is non-trivial.  It is not something that most students can solve immediately.  When they do manage to get the frogs interchanged it is well worth asking them to do it again to make sure that they really know the pattern of moves.  The challenge then is to predict the number of moves that it takes to complete the problem with any number of frogs on each side.  To do this it may help to construct a table such as the one below.

You might like to fill in the entries for 6, 7 and 8 frogs-a-side or, at the very least, guess what these values might be.

This will be an unusual pattern especially for primary and intermediate students.  It might help to look at it another way.  How many positions (rather than moves) are there during the jumping? If you look at the sequence of moves we gave for three frogs a side above, this will be 16.  The initial position, before any frogs have moved, will be counted as one of the positions.  This leads to a more manageable pattern.  To help to see this, construct a table similar to the one above.  Here we add one to every entry in the second row of the table above.  Is the pattern clearer?

This certainly looks as if the number of positions is always a square number but what square number? How does it relate to the number of frogs on each side? It seems to be the number of frogs plus one all squared or (n + 1) 2 .

This means that for n = 6, we should get (6 + 1) 2 = 7 2 = 49.  It wouldn’t be too difficult to check this.

So now we can go back to the number of moves.  The number of moves is one less than the number of positions so the number of moves with n frogs a side must be (n + 1) 2 - 1.  Getting this far is a huge step and is essentially the end point of the Leap Frog game.

4.  Alternatives : Students may well see different ways to get the n th term from the first (or second) table.  For instance, they may well see that to go from the first term to the second term you need to add 5; from the second term to the third term you need to add 7; from the third term to the fourth term you need to add 11; and so on.  The number that you add on each time is odd and it’s the next odd number after the last amount that you added on. 

This method is the iterative method of finding the size of a given term.  Its disadvantage is that it is not as easy to get a general (nth) term using this approach.  So if you are looking for the general term then you may need to look at another option.

Another pattern that some students will see is n 2 + 2n, despite the fact that it is not so obvious as the first one.  This seems to work just as well as (n + 1) 2 - 1.  Why is this so? Can they both be the same?

Let’s do some algebra.

(n + 1) 2 -1 = (n 2 + 2n + 1) - 1 = n 2 + 2n.

So both expressions are the same.  That’s a relief!

5.  What the Bright Sparks game does : The frogs’ game on the web site goes about the counting in an indirect way.  Every time a student successfully completes a swapping of frogs, a toad appears with a bag of money (see Copymaster 2).  This bag contains as many dollars as there are moves.  The students have to realise the significance of the dollar amount to predict what happens with different numbers of frogs.  They are then asked to enter in the general formula to complete the game.

Note that they are not asked to justify their general formula.  However, they are encouraged to tell someone, their teacher (?) how they arrived at the answer.

The rest of Section A is additional material that you may not want to cover.

6.  A Query: How do you know that there’s not a quicker way? How do you know that 3 frogs can’t be done in fewer than 15 moves?

The thing to note first is that, if nothing else, we have shown that the swapping of frogs can be done in 15 moves or less.  Now let’s see if we have to move 15.

Now no frog leaps a frog of the same colour.  To do this two frogs of the same colour would have to be next to each other.  As you have probably seen this always leads to a problem.  The running of the frogs stops when this happens.

So that means that each frog has to move 4 lily pads forward.  There are 6 frogs so altogether 24 lily pads have to be covered.  (Of course this includes repetitions.)

When two frogs of different colours meet a jump takes place.  This covers two lily pads.  There are 9 ‘meetings’ so there are 9 jumps, which means that 18 lily pads are covered by jumps.

Finally, 24 – 18 = 6.  This means that there are 6 lily pads still to be covered.  They can’t be covered by jumps because we have accounted for all of the jumps.  So they must be simple ‘move forwards one lily pad’. 

OK, then there have to be at least 9 jumps and 6 moves forward to give a total of 15 moves.  And that’s the number of moves we made. 

(If we need to we can apply the same sort of argument to 4 frogs a side or more.  So we can justify the table we produced above.)

B.  Frogs – The Seminar

You can present this in several ways.  You can use the Copymasters at the end of this seminar, you can get hold of some toy frogs so that staff can have hands on experience, you can set up a few computers so that they can work with the Leap Frogs directly, or, if you have a data projector available, you could project it on a screen so you can play it collectively.

• Play the game – 3 frogs a side: First get the staff used to the problem by actually doing it.  (If this turns out to be too much of a challenge go to the 2 frogs per side game).
• What are the things to avoid? Make sure that they see what problems are caused by having two frogs of the same colour next to each other.
• How many moves? When they have got the idea of the problem move onto the counting side.
• Play the game – 4 frogs a side: This should lead naturally on to the four frog situation.
• What are the things to avoid? Once again two frogs together is bad.
• How many moves? You will need to count the number of moves.  It’s a good idea to start to record what you are doing by setting up a table or by some other systematic means.
• Play the game – 5 frogs a side: Before they actually do this, ask them to predict how many moves are needed.
• The pattern? Discuss how you might describe the pattern. 
• Is 15 the smallest? At this point it is up to you to decide whether or not you want to go very deeply into the justification that 3 frogs a side can’t be done in fewer than 15 moves.

Copymasters

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LeapFrog PAW Patrol: To The Rescue! Learning Video Game 1.02 x 5.71 x 4.53 inches

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• Take Ryder’s Pup Pad and join him on problem-solving missions with the PAW Patrol
• Play immediately using the wireless controller and HDMI game stick; no Wi-Fi, downloads or account setup required
• Replayable games feature your kid’s favorite pups Chase, Skye, Marshall, Everest, Zuma, Rocky, Tracker and Rubble; save unique game progress so the whole family can play
• The fun isn’t done when you’ve completed all the missions, play them again from the Quick Challenge Menu
• Intended for ages 4+ years; requires 2 AA batteries; batteries not included

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Join Ryder and lead the PAW Patrol on adventures in this learning video game. Plug in and play 12 missions in Adventure Bay using a wireless controller that looks just like Ryder’s Pup Pad! No web connection, downloads or account setup required. Drive, fly and run through games like a big kid to avoid obstacles and solve problems with Chase, Skye and Marshall. Search for the right shape to repair vehicles with Rocky and Tracker and match Rubble’s dance moves with his outfits. Pick up Pup Treats in every mission to earn gold stars. Collect all the gold stars to become an honorary member of PAW Patrol! Master learning skills like how to think through problems, read a map and more by going to the Quick Challenge menu to play the games again and again. Save unique game progress with up to five player profiles so the whole family can play. Intended for ages 4+ years. Requires 2 AA batteries. Batteries not included.

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LeapFrog PAW Patrol: To the Rescue!

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• PAW Patrol: To the Rescue!, Wireless Game Controller, HDMI Game Stick, Micro USB Power Cable, Quick Start Guide

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Customer Review: Great First Video Game for Kids!

Customer Review: Decent game for little ones, easy to plug in and play

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Our kids loves this book!

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Great paw patrol game for kids!

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Customers find the controller simple to use and easy for a toddler to play. They also say the games and controls are easy to understand with clear instructions and helpful prompts. Customers also say it's easy to hook up to their TV and is a nice learning guide.

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Customers find the product to be a nice learning toy for young ones to enjoy. They say it's easy to use, has great learning tactics, and is designed to be both fun and educational. Customers also mention that it'll teach valuable lessons and is a great introduction to video games.

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Customers find the toy and game to be a great gift for the money. They also say it's fun for the price and convenience.

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"...but we tried to hook up to 3 different tv's and a laptop and it did not work . I thought it was a good value if it worked." Read more

"...It worked perfectly with our TV and was not hard to connect. I recommend this game if you have a tech savvy toddler." Read more

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Customers are mixed about the age range. Some mention it's great for children 3-5 years old, kid friendly, and designed with young children in mind. However, others say it'll be challenging for their 3 year old and get bored easily.

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Customers are dissatisfied with the glitchiness of the product. They mention that it keeps glitching and freezing during game play. Some say that the product doesn't work on all TVs and randomly goes out.

" Glitches/ freezes during game play . Could be my tv as it’s four years old? I did verify the tv has the latest updates." Read more

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LeapStart® Around the World Social Studies & Problem-Solving

Ages 3-9 yrs.

Curriculum: Various

Works With: LeapStart 3D, LeapStart Preschool Success, LeapStart Go, LeapStart, LeapStart Preschool & Pre-Kindergarten, LeapStart Kindergarten & 1st Grade

Publisher: LeapFrog

MSRP   $8.99

Join Leap, Tad and Lily as they travel to places like Egypt, France, India and Australia in the Around the World™ book for LeapStart® systems. Play problem-solving games with the frogs and learn about logic, reasoning and social studies concepts. Tap the pictures and words on every page with the LeapStart® stylus to magically bring the activity book to life. Enhance the learning with amazing animations when you play on a LeapStart® system with a screen. Visit different countries and explore exciting landmarks while learning about the culture of those countries. Play a memory game to find photographs, decipher messages that use pictures to represent words and discover musical instruments from around the world. Many activities are designed with two levels so kids can play and learn at the right level and move up when they’re ready. This book is compatible with all LeapStart® interactive learning systems. A computer with internet is required to download books onto the LeapStart® system. This book talks! Tap the pictures and words on every page with the stylus to play games and activities. Enhance learning with animations when played with a LeapStart® system with a screen. Number wheel Spin the number wheel on the dino’s tail to explore numbers. Explore different cultures Learn about the music, food, landmarks and traditions of different countries including Egypt, France, India and Australia. Play 20+ replayable activities Includes more than 20 replayable activities such as touch-and-talk games, problem-solving puzzles and more. Replay and level up Many replayable activities are designed with two levels so that kids can play and learn at their own pace and move up when they’re ready. Expand your library The LeapStart® library is a leveled series of 25+ books that grows from preschool through first grade (additional books sold separately).

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NYT Strands today — hints, answers and spangram for Wednesday, May 1 (game #59)

Our clues will help you solve the NYT's Strands today and keep that streak going

• Hint #1: Today's theme
• Hint #2: Clue words
• Hint #3: Spangram
• Hint #4: Spangram position
• Today's answers
• Yesterday's answers

New month, same old (relatively new) NYT Strands. That means a 6x8 grid of letters, hidden within which are a number of answers grouped around a theme. It's brought to you by the people behind Wordle and Connections, and it's rather difficult. Read on for some hints.

SPOILER WARNING: Information about NYT Strands today is below, so don't read on if you don't want to know the answers.

Marc is TechRadar’s Global Editor in Chief and has been obsessed with Wordle and its ilk for more than two years. He's authored dozens of articles on the game for TechRadar, including a daily today's Wordle answer column and a detailed analysis of the most common letters in Wordle in every position . He's also played every Wordle ever and only lost once and yes, he takes it all too seriously.

NYT Strands today (game #59) - hint #1 - today's theme

What is the theme of today's nyt strands.

• Today's NYT Strands theme is… A token of our appreciation

NYT Strands today (game #59) - hint #2 - clue words

What are some good clue words today.

Play any of these words to unlock the in-game hints system.

• BIRTH

• MOPE

• ARROW

• SHINE

• WIRE

• LOWER

NYT Strands today (game #59) - hint #3 - spangram

What is a hint for today's spangram.

• Pass go

NYT Strands today (game #59) - hint #4 - spangram position

Where does today's spangram start and end.

• Start: top, 4th column

• End: bottom, 6th column

Right, the answers are below, so DO NOT SCROLL ANY FURTHER IF YOU DON'T WANT TO SEE THEM.

NYT Strands today (game #59) - the answers

The answers to today's Strands, game #59, are…

• WHEELBARROW
• SPANGRAM: MONOPOLY
• My rating: Easy
• My score: Perfect

I loved MONOPOLY as a kid, although my sisters probably didn't love the zeal with which I would routinely bankrupt them. Anyway, this was a fun Strands puzzle, albeit not too taxing.

Four-letter words are the easiest to find in Strands, so it always helps when there are a couple of them among the answers. The two hidden within today's puzzle – IRON and SHIP – were enough for me to confirm what the theme here was, when added to the hint of 'A token of our appreciation'.

The only issue was that I have no idea what the tokens in MONOPOLY are these days. When did a PENGUIN get added? But finding the spangram was very easy once I knew what I was looking for, and that pointed the way to the likes of THIMBLE and RACECAR once the board had some structure.

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Yesterday's NYT Strands answers (Tuesday 30 April, game #58)

• SPANGRAM: CONDIMENT

What is NYT Strands?

Strands is the NYT's new word game, following Wordle and Connections. It's currently in Beta and can be played on the NYT Games site on desktop or mobile.

I've got a full guide to how to play NYT Strands , complete with tips for solving it, so check that out if you're struggling to beat it each day.

Marc is TechRadar’s Global Editor in Chief, the latest in a long line of senior editorial roles he’s held in a career that started the week that Google launched (nice of them to mark the occasion). Prior to joining TR, he was UK Editor in Chief on Tom’s Guide, where he oversaw all gaming, streaming, audio, TV, entertainment, how-to and cameras coverage. He's also a former editor of the tech website Stuff and spent five years at the music magazine NME, where his duties mainly involved spoiling other people’s fun. He’s based in London, and has tested and written about phones, tablets, wearables, streaming boxes, smart home devices, Bluetooth speakers, headphones, games, TVs, cameras and pretty much every other type of gadget you can think of. An avid photographer, Marc likes nothing better than taking pictures of very small things (bugs, his daughters) or very big things (distant galaxies). He also enjoys live music, gaming, cycling, and beating Wordle (he authors the daily Wordle today page).

Quordle today – hints and answers for Tuesday, April 30 (game #827)

NYT Strands today — hints, answers and spangram for Tuesday, April 30 (game #58)

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