### Task 225 ... Years 4 - 8

#### Summary

Perhaps a little unusually, this task begins with the higher numbers in a pack of cards. The smallest number is 7 and the Royals are each given a numeric value in sequence beyond 10. The task is to find the total of the four suits in this special pack, but to do it in more than one way. It is an application of the mathematician's question Can I check this another way?.

The challenge in the task leads back to the usual card pack (albeit it imagined) and offers the opportunity to apply patterns discovered earlier and to know you are correct by checking another way.

#### Materials

• Playing cards 7 to King in 4 suits (28 cards)

#### Content

• algebra, concept of a variable / function
• algebra, generalisation in words & symbols
• arithmetic, multiplication / division
• consecutive numbers
• mental arithmetic
• multiplication, calculations / times tables
• multiplication, multiplicative thinking, multiplication principle of counting
• patterns, number
• patterns, visual

#### Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

Presumably the slowest way to add the pack (Question 1) would be randomly choosing cards one at a time and keeping a running total. The students might have other ideas, but the point is to set not just a baseline answer but that '...there's got to be a better way'.

It is also setting up the task so that the students never have to ask if they are correct. It is there responsibility to find a way to know they are correct. Question 2 leads into that by suggesting that grouping might help (because it helped Catharina). This leads to finding different ways of grouping such as:

• Multiplying the value of one of each card by 4, then adding the 7 results.
• Adding one suit of 7 to King, then multiplying this answer by 4.
• Using favourite facts like: 10 + 7 = 17 ... Hey so is 9 + 8 ... What can we do with the others?
• Making 21s using 8 + K, 9 + Q, 10 + J, then add 7 and multiply the total by 4.
• ...
It's most likely already been discovered in Question 2, but Question 3 makes sure that students are aware of the Rainbow Facts approach:
7, 8, 9, 10, J, Q, K
Linking 7 with K, 8 with Q, 9 with J creates three sub-totals of 20 with 10 left unpaired. (Can you see the rainbow over 10?) But there are four suits like this which explains how Antonio grouped the cards.

In the challenge the students are asked to apply the strategies they created and recorded in their maths journal to add a standard card pack. Again, they won't have to ask you if they are correct, because they are asked to check their work in three different ways.

Extensions

• Imagine five different packs of cards, each with 4 suits, that start at 1 but end at different numbers. Graph their totals, investigate and report.
• What is the total if someone designs a new pack of cards, still with 13 in each suit, but with 5 suits such as spades, clubs, diamonds, hearts and teacups?
• If I tell you any number as the last card in a 4 suit pack of cards that starts at 1, can you tell me how to add the pack?
• If I tell you any number as the last card in a pack of cards that starts at 1 and tell you any number for the number of suits, can you tell me how to add the pack?
• If I use a 4 suit pack of cards and tell you any start number and any end number can you tell me how to add the pack if the card values go up by 1?

#### Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

You only need a pack of cards for each pair, so, this one is pretty easy to turn into a whole class investigation. One reason for doing so is to highlight the mathematician's question Can I check it another way?. (See the Working Mathematically Process - you could print Page 1 for the students.) Another might be for the arithmetic practice in the task and third might be to approach the generalisations listed above.

Underlying the sequence of totals in the suggested graph above, is a quadratic function which is explored in Task 51, Staircase, and the Picture Puzzle Staircase (access code required), and extended in Task 61, Double Staircase.

To begin this lesson, you might bring the students to a central table and lay down one set of large cards (say 20cm x 20cm) numbered from 7 to 13. First ask a student to arrange them in sequence, just to be sure you are all speaking the same language. Then ask each student to '... silently add up these numbers any way you like but don't say the total'.

Now, I'll say A, B, C then you all say the total ... Good, now that's out of the way, what I am really interested in is how many different ways we can find to check that total.
Work as a community for a while to explain and list a few strategies. Then offer a pack of cards to each pair and invite them to return to their seats and sort out the cards 7 to 13. The question about '...there's no 11, 12, or 13 Miss...' will soon resolve itself. Discuss and record strategies for adding this pack, then move through the investigation above as you choose (or perhaps as the students choose).

At this stage, Add The Pack does not have a matching lesson on Maths300, however Lesson 115, Staircases, is related.

#### Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
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