This one’s tricky to me. It’s pretty straightforward that the correct math class answer is 185, and that, in math class, we would count the single block as step 1. However, I think there is an argument to be made that the problem establishes a step as the process of undergoing change, but doesn’t establish that change has occurred with the lone block — that would make the set of 5 blocks the result of the first step. I would be willing to mark correct either answer if a student explained their reasoning.
I don’t like the unnecessary dissonance between the math class usage of terms and the real world usage of terms. It’s unavoidable sometimes, but it seems like more of an accident in this problem than an intentional lesson. (The “47st” typo, to me, supports the idea that this might not be a well thought out problem.)
It’s unavoidable sometimes, but it seems like more of an accident in this problem than an intentional lesson.
How can you tell whether something like that is accident or intentional?
Hi Michael. Monday’s lesson would start with, “Draw what you think the next step looks like. Then the one after that.” Rarely does a student not see how the pattern grows. It’s important to capture this initial success and honor that. The challenge, and hopefully there is one, is for the student to come up with the rule, for them to make a leap in their understanding to the 47th step. And if they have the patience (and persistence) to do this recursively, applaud them for that, but then ask for the 841st step.
I’m really puzzled why there’s a statement right underneath the pattern, “In the pattern above, 4 blocks are added at every step. (Meaning: Next Step = Current + 4)”
This seems to immediately have done the thinking/investigating for the students. When we do visual patterns, it’s always about how THEY SEE the patterns change from step to step. We get to talk about what part of the pattern they see stays the same/constant (if any) and what part changes.
I’d love to hear/read a kid’s explanation for his/her answer of 4x – 3. Where did the “minus 3” come from?
I tell my kids that I want them to write the equation the same way as they SEE the pattern, meaning I don’t want them to simplify it just yet. My equation for how I see it would be 4(x – 1) + 1.
If kids learned to solve these patterns by only finding the difference between each term (some of my current 8th graders came to me having learned it this way from their last year’s teacher or someone else), then I think the beauty of the pattern itself is lost because the focus is on the numbers and formula rather than the HOW-does-it-grow.
Thanks, Michael!
I have little in the way of defense to this excellent criticism, but I can offer an explanation. (This is from my classroom.)
1. I’ll start with the weirdest thing, the “Next Step = Current + 4” line. The reason why I included it was because many of my students repeatedly wrote recursive formulas as equations and called it a day. On many assessments I got answers such as “n + 4 = c” to a question like this one. Now, in theory the prodding to provide the 47th step should take care of this, but it wasn’t, and I wanted my students to dig for an explicit formula. But it wasn’t working. So I tried to provide them with a recursive rule, to cue them that I wasn’t interested in their observation that the pattern increases by 4 each time.
But I have no defense for this. It’s a weird attempt to prime my students to do the hard work of finding an explicit formula. It probably should have been left on the cutting room floor.
2. In instruction I gave no procedures to students on how to find rules. So I don’t think that I’m guilty of teaching this stuff proceduraly. But, through the course of many examples, most students developed a sort of procedure of their own for these. First, they figure out the pattern. Then, if the pattern is “+4” they take a 4x, and then they tinker with one of the “points” until they jigger an equation out of it. Some students count the first step as the “zero” step. Others count it as the “first”. That’s why there are both 4x – 3 and 4x + 1. We’ll sort out what counts as the “first step” once we explicitly connect this to graphing.
3. I didn’t teach this to properly represent the pattern in algebra in the way that you do, Fawn. I should have, but didn’t. It was a missed opportunity. Instead, I shifted between visual patterns, “missing step” patterns, cell phone rate problems, plumber problem, lemonade stand problems and finding equations for two points or finding equations from graphs. The instructional idea was that I’d show a bunch of problems with common underlying math, and that I’d let the kids make the connections between the problems on their own. My focus was on linear equations, though, not on representing patterns algebraically. I think I limited what these problems are capable of in my focus.
I think the students do 4x-3 because step 1 is 4x + 1. But you need step 0, which is 4x-3.
I think these block patterns are insanely annoying used as a serious method of teaching math. Fun enough as a game.
I have found that when working with visual patterns I have adjusted the way I ask the questions and that has helped.
I realized that there was no point in having students look at and build visual patterns and then ask them “how many” blocks are there? By asking “how many” I found students were focused on the number of blocks and the question just became the same as asking what is the pattern in the numbers 1, 5, 9, …. which defeats the purpose of giving them a visual – if I just wanted them to look at the number of blocks, why bother giving them a visual representation.
Instead I now ask students questions such as “what does the 10th step look like? what does the 47th step look like?”, etc”. Students might then describe the 10th one as 1 block in the middle and 4 arms that are 9 blocks long and the 47th one as 1 block in the middle and 4 arms that are 46 blocks long. Students who see it this way might come up with a pattern rule similar to the one Fawn mentioned 1 + 4(x – 1).
I find asking students what does the …….th one look like leads them to think more functionally rather than recursively.
It is a small shift in wording that has made a big difference in the work I have done with students.
Liisa, that’s a lovely approach! Thanks. I think it’ll be particularly helpful with one group of students I work with, who need more help articulating their thinking than they do solving problems.
7 replies on “Visual Patterns”
This one’s tricky to me. It’s pretty straightforward that the correct math class answer is 185, and that, in math class, we would count the single block as step 1. However, I think there is an argument to be made that the problem establishes a step as the process of undergoing change, but doesn’t establish that change has occurred with the lone block — that would make the set of 5 blocks the result of the first step. I would be willing to mark correct either answer if a student explained their reasoning.
I don’t like the unnecessary dissonance between the math class usage of terms and the real world usage of terms. It’s unavoidable sometimes, but it seems like more of an accident in this problem than an intentional lesson. (The “47st” typo, to me, supports the idea that this might not be a well thought out problem.)
How can you tell whether something like that is accident or intentional?
Hi Michael. Monday’s lesson would start with, “Draw what you think the next step looks like. Then the one after that.” Rarely does a student not see how the pattern grows. It’s important to capture this initial success and honor that. The challenge, and hopefully there is one, is for the student to come up with the rule, for them to make a leap in their understanding to the 47th step. And if they have the patience (and persistence) to do this recursively, applaud them for that, but then ask for the 841st step.
I’m really puzzled why there’s a statement right underneath the pattern, “In the pattern above, 4 blocks are added at every step. (Meaning: Next Step = Current + 4)”
This seems to immediately have done the thinking/investigating for the students. When we do visual patterns, it’s always about how THEY SEE the patterns change from step to step. We get to talk about what part of the pattern they see stays the same/constant (if any) and what part changes.
I’d love to hear/read a kid’s explanation for his/her answer of 4x – 3. Where did the “minus 3” come from?
I tell my kids that I want them to write the equation the same way as they SEE the pattern, meaning I don’t want them to simplify it just yet. My equation for how I see it would be 4(x – 1) + 1.
If kids learned to solve these patterns by only finding the difference between each term (some of my current 8th graders came to me having learned it this way from their last year’s teacher or someone else), then I think the beauty of the pattern itself is lost because the focus is on the numbers and formula rather than the HOW-does-it-grow.
Thanks, Michael!
I have little in the way of defense to this excellent criticism, but I can offer an explanation. (This is from my classroom.)
1. I’ll start with the weirdest thing, the “Next Step = Current + 4” line. The reason why I included it was because many of my students repeatedly wrote recursive formulas as equations and called it a day. On many assessments I got answers such as “n + 4 = c” to a question like this one. Now, in theory the prodding to provide the 47th step should take care of this, but it wasn’t, and I wanted my students to dig for an explicit formula. But it wasn’t working. So I tried to provide them with a recursive rule, to cue them that I wasn’t interested in their observation that the pattern increases by 4 each time.
But I have no defense for this. It’s a weird attempt to prime my students to do the hard work of finding an explicit formula. It probably should have been left on the cutting room floor.
2. In instruction I gave no procedures to students on how to find rules. So I don’t think that I’m guilty of teaching this stuff proceduraly. But, through the course of many examples, most students developed a sort of procedure of their own for these. First, they figure out the pattern. Then, if the pattern is “+4” they take a 4x, and then they tinker with one of the “points” until they jigger an equation out of it. Some students count the first step as the “zero” step. Others count it as the “first”. That’s why there are both 4x – 3 and 4x + 1. We’ll sort out what counts as the “first step” once we explicitly connect this to graphing.
3. I didn’t teach this to properly represent the pattern in algebra in the way that you do, Fawn. I should have, but didn’t. It was a missed opportunity. Instead, I shifted between visual patterns, “missing step” patterns, cell phone rate problems, plumber problem, lemonade stand problems and finding equations for two points or finding equations from graphs. The instructional idea was that I’d show a bunch of problems with common underlying math, and that I’d let the kids make the connections between the problems on their own. My focus was on linear equations, though, not on representing patterns algebraically. I think I limited what these problems are capable of in my focus.
I think the students do 4x-3 because step 1 is 4x + 1. But you need step 0, which is 4x-3.
I think these block patterns are insanely annoying used as a serious method of teaching math. Fun enough as a game.
I have found that when working with visual patterns I have adjusted the way I ask the questions and that has helped.
I realized that there was no point in having students look at and build visual patterns and then ask them “how many” blocks are there? By asking “how many” I found students were focused on the number of blocks and the question just became the same as asking what is the pattern in the numbers 1, 5, 9, …. which defeats the purpose of giving them a visual – if I just wanted them to look at the number of blocks, why bother giving them a visual representation.
Instead I now ask students questions such as “what does the 10th step look like? what does the 47th step look like?”, etc”. Students might then describe the 10th one as 1 block in the middle and 4 arms that are 9 blocks long and the 47th one as 1 block in the middle and 4 arms that are 46 blocks long. Students who see it this way might come up with a pattern rule similar to the one Fawn mentioned 1 + 4(x – 1).
I find asking students what does the …….th one look like leads them to think more functionally rather than recursively.
It is a small shift in wording that has made a big difference in the work I have done with students.
Liisa, that’s a lovely approach! Thanks. I think it’ll be particularly helpful with one group of students I work with, who need more help articulating their thinking than they do solving problems.