This week we look at the relationship between pairs of values given in a table, and consider how the relationship changes as a result of systematic changes to the values. The new relationship can be found in a variety of ways: formally, by seeing the change in value as a transformation of one or both variables in the algebraic relation; informally, by giving meaning to the relationship and to how a set of numbers has been changed; empirically by searching for a new rule to fit the new set of values.
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MONDAY: We start with a standard linear relationship and multiply the
x- and/or the
y-values by 4.
Some students might be able to approach this formally. For example, in the case of Table B, we can think of the 'new'
x (denote it by
x', say) as replacing 4 times the old
x, ie
x' = 4
x, and so
y = 8
x + 1 = 2
×4
x + 1 becomes
y = 2
x' + 1.
Or, less formally, "the new
x-values are 4 times the old
x-values, so multiplying the old
x-values by 8 is the same as multiplying the new
x values by 2; so
y = 8
x + 1 becomes
y = 2
x + 1".
Or students might notice, by examining the numbers in Table B, that 17 = 2
×8 + 1, 41 = 2
×20 + 1, etc.
IMPORTANT NOTE: For brevity we have presented the three tables, B, C and D, on one slide. In class, it might be better to present the tables one at a time so that students can devote plenty of time to each rather than feeling they have to rush from one table to the next.
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TUESDAY: Here, we start with the same relationship as in ALG 13A, but instead of multiplying by 4, we add 4 to the
x- and/or the
y-values.
Again, students might adopt a formal, a semi-formal, or an empirical approach. In the case of Table E, the relation changes to
y = 8(
x – 4) + 1, or
y = 8
x – 31.
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WEDNESDAY: Here we start with a new linear relationship that is perhaps easier to describe succinctly.... Again, we multiply the
x- and/or the
y-values by 4.
The relationship for Table J is easy to spot or derive, the one for Table I less so.
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THURSDAY: We stick with Wednesday's relationship, and as we did earlier, we add 4 to the
x- and/or the
y-values. Finding the resulting relationships is not particularly challenging on this occasion but we include the task for completeness.
FRIDAY: We end the week with a task involving a much more challenging function. We provide four transformations for completeness, but, as suggested previously, it might be better to show the tables one at a time to prevent students feeling they have to rush from one to the next.
Note: In all of this week's task we have looked at relationships expressed symbolically and with tables of values. It is also worth drawing graphs of the relationships.
