![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEindGkNwGOO4hHwjL7SIfOB7BYpmmo-u6NXQ6ZdysdT2B4TKv-N1Mtne7I0_J14um_ihywyUqi4dURE3IDhArw0E7ux9pJFaBaz9p1ZFkuY3w-Rd42yQsbL6ZIj_ln71099tYuChtSEqL5r/s1600/image009.png)
and one like this
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXJ36oAlypDfaLMD7Oue2_8EeR1dKXOY0xfy3oUL6UoPPON4niCrr8VPjvTFbXl7ekBqAz6yinyefeCriL2ws3y3pMrY6yT0cl1qTmHplz92HWbR87jseWiYMI2nc3t3LpU-uXBwj33Gq1/s1600/image014.png)
In the first gift box there are five boxes of three apples
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEQi1GAK75q24ngkdJMhF0HLSHSPpgqqcixlcqTugNSwpNim15FCrQaQya6Jd7lZ8Pjwa8LxLbBR0U-fikOY_muKUR-l140y6VU4eMfsNjJ6Vzy0hEPrPUJPRZ_r5Bwtf4j6KMhyphenhyphenkKZnYq/s1600/image016.png)
And five boxes of four pears
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgL5AyIdTpK_rJYb8f9pb5rsI7_-eahWrUtPMJ72tUdBg9t9q1dUwNWfePKpc8smUQzc6XlvKvjIsNTSj4k6NjdgVgWZou0pcrPusLG_inJfGhLxyBmND9xyEPeYSGgYhRd1i5e5Jz_OOKA/s1600/image018.png)
In the second gift box there five boxes containing three apples and four pears
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrYuf2LHq18c6oeV9AODQ2hJuUN_duZTqqomjBk2DDnPjSmfZlIVEWx-wSV9446i_SEQ5FbgfiPdRetyKVuCgcRzcNFfBzfhkSolhbx9SX9uSQVB0SYEsEb2lvPQFh-MHLfXcqanpzfKxb/s1600/image006.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrYuf2LHq18c6oeV9AODQ2hJuUN_duZTqqomjBk2DDnPjSmfZlIVEWx-wSV9446i_SEQ5FbgfiPdRetyKVuCgcRzcNFfBzfhkSolhbx9SX9uSQVB0SYEsEb2lvPQFh-MHLfXcqanpzfKxb/s1600/image006.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrYuf2LHq18c6oeV9AODQ2hJuUN_duZTqqomjBk2DDnPjSmfZlIVEWx-wSV9446i_SEQ5FbgfiPdRetyKVuCgcRzcNFfBzfhkSolhbx9SX9uSQVB0SYEsEb2lvPQFh-MHLfXcqanpzfKxb/s1600/image006.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrYuf2LHq18c6oeV9AODQ2hJuUN_duZTqqomjBk2DDnPjSmfZlIVEWx-wSV9446i_SEQ5FbgfiPdRetyKVuCgcRzcNFfBzfhkSolhbx9SX9uSQVB0SYEsEb2lvPQFh-MHLfXcqanpzfKxb/s1600/image006.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrYuf2LHq18c6oeV9AODQ2hJuUN_duZTqqomjBk2DDnPjSmfZlIVEWx-wSV9446i_SEQ5FbgfiPdRetyKVuCgcRzcNFfBzfhkSolhbx9SX9uSQVB0SYEsEb2lvPQFh-MHLfXcqanpzfKxb/s1600/image006.jpg)
In both cases there are the same number of apples and pears in the gift box.
Instead of apples and pears we can do the same thing with Threes and Fours.
The Game of Maths
This is the same as arranging a hand of cards in two different ways.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgErIVNwtU9koatrtJYT39K1lwHnwC3sb4UK7kDBb2Sbw4tYxObjPWHXWK_LABFezG8mDrlDqW0dR1n1PYVVCcJHCsmgDDvgQqzubckiykd6rW6JHTAfYSDxFqCUzwasgOxsswpX_kMCLf2/s1600/image022.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxLIbix8XMRCzJsp6lFlW-8OF_PWHA0G03A46Ldb2jX-bg8KFlKD4t-L1pecuuMy1Ndvp8rRKTD6JjANfyG_scbnoxKb8UUU19f0jnsClbWgNYt7ceDdpxSMNrkldHJh-rjpuWTX3CJQuk/s1600/image024.png)
and using holders
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFbnc0-Pf7HM52xYyt2TmkZoun9LWi6KUsTyFZd8cZ-oXqTFCuwGLj5_5gDhKVqBeXqPsjlE4tGAG4lW5BsvWfMuJjbPG8Wn0DO78C4vTzIR85MYj4xy6UJW_2nMA5SR-n7uKv6_czHFj5/s1600/image026.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOXGH4uy0oJ1tZXzMKL3RDW97_1y41yVsHkW2kC2lCfVXE8b0fWOCNOfMDFpi8a-M2Qoen280Cx6M0rYAJx5IsQzYn4B8Dz1gxmJXnwot_ppgXl0F8rkYwAL04BPOcRkEnvErpTDXUcsgm/s1600/image028.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6_Lxessr-7qNQG7S7-UW1uPqMJeJuhjTXbLPh4WtPUxmPXhhXt0fdfmYVxYe2DC-wg9051OiEXvi4DFgBoa9GmwZK2tZRBHCwvn-U8RLQ6sZyNhQxhPKV2e7dIz5wtLhpzUzLJebHZiNT/s1600/image030.png)
In the recording we show that two or more cards held in one holder using brackets.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFbnc0-Pf7HM52xYyt2TmkZoun9LWi6KUsTyFZd8cZ-oXqTFCuwGLj5_5gDhKVqBeXqPsjlE4tGAG4lW5BsvWfMuJjbPG8Wn0DO78C4vTzIR85MYj4xy6UJW_2nMA5SR-n7uKv6_czHFj5/s1600/image026.png)
Record | |
List 5 x 3 = 15 5 x 4= 20 | Maths 5 x 3 + 5 x 4 =15 + 20 = 35 |
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOXGH4uy0oJ1tZXzMKL3RDW97_1y41yVsHkW2kC2lCfVXE8b0fWOCNOfMDFpi8a-M2Qoen280Cx6M0rYAJx5IsQzYn4B8Dz1gxmJXnwot_ppgXl0F8rkYwAL04BPOcRkEnvErpTDXUcsgm/s1600/image028.png)
Record | |
List 5 x (3 + 4) = 5 x 7 = 35 35 | Maths 5 x (3 + 4) =5 x 7 = 35 |
We can see that 5 x (3 + 4) = 5 x 3 + 5 x 4
More generally using wildcards
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqLrUkoTGXonzPLi191zeULlnUL7KcJL1BfKJF9Mssyc64LVJ2W_5bF5aqGzLyYSQlboQTt7T2lvNBXm982G4qjrwq_qAAawXyK9AZhElPqLn1Z472q6O9u9m3nS-YD30VWTPR_IkMZJnJ/s1600/image034.png)
Record | |
List![]() ![]() ![]() | Maths ![]() |
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjee965FitlIKVFP6Rm8IRcE5pZcQOOCAIqIMs8GTkpI9jHh5yLl_g3PKonXX0EFX0CkkwpDg6mEMJGg9RV1j7RnWEgvGR21yChUUPlyH0tDBLwWouZRmkzfLIo6rD4fCkKVtVp_0Ah4Zny/s1600/image043.png)
Record | |
List![]() ------------- ![]() | Maths ![]() |
We cannot work out a value without being given the declared values for
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjw2Wh60EUfB0U6AYFwxwNHaNr_SfzRKdlStrdOBlJ0lPipSNjhRacaUfhPLUzIIBBL-Y0Q6Aj5PpEdFemMNyWtWslxlsebjEXClN0bSBtWs_BfO56bwDN8aKSbW2qrjWkjuoPKVYFkUy6-/s1600/image049.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqblNKPat8Onv7cRGULc2VjL9UYq73BpowG6xJEZV1wCnaX-Ik3hel9ZFub_yjf8mVZeVd4JUH4873F3d1R1_jM9Re24c4eQ8Zx18n_adlStAnNfkRjV11RZcbf4fuD5u5FMP6Enj99bPq/s1600/image051.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMJy3L0qyfdWBqV5_xrp95o3X7AqU6N3wVYxo2gmSqVIvjMmHP2cygGN01jSpqInJ_akpntVFx79zyIT_atN3RCKLrnPs-vN4physUvkIjJFx0up_UVCvt3JfH0wCq-wFTxV6JDPBYtFC2/s1600/image054.png)
We have found that
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhh2LKXJdoO3tIfLoTpKitQVE0nGW6apTBQTNoKqMcdXnzWjgC98M_4YgxHNfLQY__taXl9a5_HzWlN4_7GgPZOaoMWi3Ck7uH8WFsQkpoFBeTiNkcMWi-VBodfWYG7TkVqPW47qyr6NVEH/s1600/image039.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbNOhr_vfEu2vD4z08ROvkad24zlo3MogmBS5EESlG2p5Fyv5WzoPHyOO7TsFtRyBj3884CRa3GK_mlGn4__B6V1B000DD8-2_mqp0GVybuc1Trdtc-hF38zg-HBlNu2Iza3tqMMgxZ5vl/s1600/image055.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiek3Gz6AtGJ-cMt4JB7Kaw4snYuElMDY8jdtu8NpZcYJ8irtjuqrBAqpbfI85nBXzs8ZHpOU55hsAsh-i-tqJkey2Wx9-Coio4ndZ5Vq82Bttv9XjBkHg2BStsi-mR-XEUaD4Bm9sJBug4/s1600/image057.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhh2LKXJdoO3tIfLoTpKitQVE0nGW6apTBQTNoKqMcdXnzWjgC98M_4YgxHNfLQY__taXl9a5_HzWlN4_7GgPZOaoMWi3Ck7uH8WFsQkpoFBeTiNkcMWi-VBodfWYG7TkVqPW47qyr6NVEH/s1600/image039.png)
When dealing with different wildcards think back to The Importance of Counting. Here we saw that all we could do was list ten yachts and twenty crew and could not add the ten and the twenty in any meaningful way.
In the same way because we do not know anything about
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjw2Wh60EUfB0U6AYFwxwNHaNr_SfzRKdlStrdOBlJ0lPipSNjhRacaUfhPLUzIIBBL-Y0Q6Aj5PpEdFemMNyWtWslxlsebjEXClN0bSBtWs_BfO56bwDN8aKSbW2qrjWkjuoPKVYFkUy6-/s1600/image049.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqblNKPat8Onv7cRGULc2VjL9UYq73BpowG6xJEZV1wCnaX-Ik3hel9ZFub_yjf8mVZeVd4JUH4873F3d1R1_jM9Re24c4eQ8Zx18n_adlStAnNfkRjV11RZcbf4fuD5u5FMP6Enj99bPq/s1600/image051.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiek3Gz6AtGJ-cMt4JB7Kaw4snYuElMDY8jdtu8NpZcYJ8irtjuqrBAqpbfI85nBXzs8ZHpOU55hsAsh-i-tqJkey2Wx9-Coio4ndZ5Vq82Bttv9XjBkHg2BStsi-mR-XEUaD4Bm9sJBug4/s1600/image057.png)
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