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# Secondary Magazine - Issue 131: Building Bridges

Created on 26 February 2016 by ncetm_administrator
Updated on 16 March 2016 by ncetm_administrator

## Building Bridges

Three discussions we had in a recent department meeting began independently but soon became closely connected: a colleague asked for some suggestions about teaching his Year 7 class to simplify expression with products of algebraic terms; a different colleague was wondering why his Year 9 group of high prior attainers were finding multiplying and dividing numbers written in standard form very hard to grasp; and a third colleague’s Year 8 class were struggling to calculate areas of triangles. We explored the mathematics underpinning these seemingly different topics, and agreed that each assumed – required, even – confident understanding of, and fluency with, manipulating what we called “calculation strings”, in particular strings of multiplications and divisions. The Year 7 pupils needed to be confident that $\inline \dpi{80} \fn_jvn \small 3\times 4\times 5\times 6=3\times 5\times 4\times 6$ before they could simplify with understanding $\inline \dpi{80} \fn_jvn \small 3\times a\times 4\times b$; the Year 8 pupils needed to be sure that $\inline \dpi{80} \fn_jvn \small 0.5\times \left ( 6\times 14 \right )$ would be the same as $\inline \dpi{80} \fn_jvn \small 3\times 14$ or $\inline \dpi{80} \fn_jvn \small 6\times 7$ but not the same as $\inline \dpi{80} \fn_jvn \small 3\times 7$ – that they shouldn’t be “multiplying out the brackets”; and if the Year 9 pupils were to simplify the product and the quotient of $\inline \dpi{80} \fn_jvn \small 4\times 10^{9}$ and $\inline \dpi{80} \fn_jvn \small 8\times 10^{3}$, first they needed to simplify securely the product and the quotient of $\inline \dpi{80} \fn_jvn \small 4\times 9$ and $\inline \dpi{80} \fn_jvn \small 8\times 6$. How then, we discussed, should the depth and strength of the prerequisite knowledge be assessed – and then consolidated, if found to be lacking?

We knew that our pupils would have first experienced manipulating and simplifying calculation strings in Key Stage 2, and should have linked abstract statements such as the commutative law of multiplication $\inline \dpi{80} \fn_jvn \small a\times b=b\times a$ to concrete representations such as the area of two rectangles, each with the same dimensions but one rotated a quarter turn relative to the other. They might have seen volume of a cuboid as a good representation of the associative law of multiplication: the “base” area can be any of $\inline \dpi{80} \fn_jvn \small a\times b$, $\inline \dpi{80} \fn_jvn \small b\times c$ or $\inline \dpi{80} \fn_jvn \small c\times a$, so the volume can be any of $\inline \dpi{80} \fn_jvn \small \left ( a\times b \right )\times c$, $\inline \dpi{80} \fn_jvn \small a\times \left ( b\times c \right )$ or $\inline \dpi{80} \fn_jvn \small \left ( c\times a \right )\times b$. We had to build the bridge from there, and so we planned together activities such as

not only to assess our pupils’ familiarity with what is and isn’t permissible when manipulating multiplication strings but also to give them motivation for the formal laws of associativity and commutativity: we wanted them to realise how vexing it would be if $\inline \dpi{80} \fn_jvn \small 3\times 4\times 25$ didn't equal $\inline \dpi{80} \fn_jvn \small 3\times \left ( 4\times 25 \right )$! We were confident that once the pupils wanted to be sure that they could “start the calculating with the nice products”, then exploring

would seem to them worthwhile and relevant. We agreed in the department that we would use, and we would ensure that the pupils used, precise language: as one colleague said, if his son could reel off – and spell! –  the Latin names of 20-odd dinosaurs in Year 3, then in Year 7 he could certainly be expected to remember and use “commutative” and “associative” when simplifying multiplication strings

and when reasoning

We decided to develop our pupils’ understanding of and fluency with mixed strings similarly: beginning with “3 numbers, 2 operations”

and then going deeper with

and developing their reasoning with

We agreed not to progress immediately to “4 numbers, 3 operations” – i.e. $\inline \dpi{80} \fn_jvn \small \left ( 4\times 9 \right )\div \left ( 8\times 6 \right )$ is the same as $\inline \dpi{80} \fn_jvn \small \left ( 4\div 8 \right )\times \left ( 9\div 6 \right )$ and also $\inline \dpi{80} \fn_jvn \small \left ( 4\div 6 \right )\times \left ( 9\div 8 \right )$ - so that all the new ideas, thinking and language around “3 numbers, 2 operations” would have time to settle and be consolidated … and we rejigged all the schemes of work to postpone the topics that had prompted this discussion in the first place!

This serendipitous discussion (much more worthwhile than the admin that was on the meeting agenda!) reminded us starkly how important it is that a scheme of work is written AFTER a detailed concept map or timeline has first been written: the pupils in the Year 8 and 9 classes were struggling, and those in the Year 7 class were likely to, because the sequence of contexts wasn’t in step with the sequence of concepts. We were hoping for procedural fluency, but we hadn’t ensured first their factual knowledge or their conceptual understanding: without doing so, we were almost certainly hoping in vain.

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