![]() ![]() Say, well, what's another way of defining Now, in both of theseĬases, I defined it as an explicit function. Term and goes to infinity, with a sub k is equaling. This sequence as a sub k, where k starts at the first Wanted to define it explicitly as a function- we could write This is a sub 2- all the way to infinity. And here, we're goingįrom our first term- so this is a sub 1, But I could use the notationī sub k or anything else. This is equal to- and people tend to use a. Now, how would I denote thisīusiness right over here? Well, I could say that The domain, is restricted to positive integers. Traditional function notation, I could have written I have essentially defined a function here. ![]() So this is one way to explicitlyĭefine our sequence with kind of this function notation. Times 3, or maybe I should write 3 times k Here, I could say a sub k is equal to some function of k. Sequence, I could define it as a sub k from k equals 1 toĤ, with- instead of explicitly writing the numbers It defining our sequence as explicitly using kind of aįunction notation or something close to function notation. Now, I could also define itīy not explicitly writing the sequence like this. So this just says, all of theĪ sub k's from k equals 1, from our first term, all This right over here is the sequence a sub kįor k is going from 1 to 4, is equal to thisĪt it this way, we can look at each of these as But I want to make usĬomfortable with how we can denote sequences andĪlso how we can define them. ![]() Of different notations that seem fancy forĭenoting sequences. So we could call thisĪn infinite sequence. Pattern going on and on and on, I'll put three dots. This is infinite, to show that we keep this Keep adding the same amount, we call these Infinite sequence- let's say we start atģ, and we keep adding 4. So this one we wouldĬall a finite sequence. And let's say I only have theseįour terms right over here. Infinite number of numbers in it- where, let's say, I Have a finite sequence- that means I don't have an And all a sequence is isĪn ordered list of numbers. Implications for teacher instruction are discussed.Video is familiarize ourselves with the notion of a sequence. We document strong relationships between children's early construction of an INS and a TNS and the likelihood of their later development of multiplicative reasoning, a measurement meaning of fractions, and general mathematics achievement, while controlling for rote computational skills. This study involved 5747 children from three cohorts surveyed at the beginning of second grade in 2013, 2014, and 2015. We report on a large-scale quantitative study of children's available number sequences and their relationships with later mathematical development, such as multiplicative reasoning and fractions knowledge. A critical benchmark in children's further numerical development is the construction of units of units, or "composite units." This development corresponds to the tacitly nested number sequence (TNS) wherein numbers are nested within other numbers. They construct other numbers as strings of 1s and can count on, by 1s, from one number to a subsequent number. Children operating with the initial number sequence (INS) construct units of 1. Number sequences are defined in terms of children's abilities to construct and transform units. ![]()
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