We are beginning our unit on patterns.  If your child has not yet showed you their trick (they can read your mind!), ask them about it… it’s fun.  We will be posting some pattern work soon.

Today’s question (Sept 15th):

The students were asked to look at the pattern.  Then to try to figure out the number of tiles (red, blue, total) in figure 5 (grade 4’s) and figure 10 (grade 5’s).

Here’s what the students came up with:

Some students used drawings to expand the pattern:

Some students drew the first three figures, but noticed and were able to explain the pattern (that first you add 3 red, then you add 3 blue) to figure out the answer:

The most efficient strategy was the use of a table:

With the tables, the students noticed the obvious patterns going down (0,3,3,6,6,9,9…).  Some students also (brilliantly) began to focus on a pattern across to figure out the total number of tiles:

In figure 2 the total number of tiles was (2×2) 4,
In figure 3 the total number of tiles was (3×2+1) 7,
In figure 4 the total number of tiles was (4×2+2) 10
In figure 5 the total number of tiles was (5×2+3) 13

It was pointed out that you take the figure number, multiply it by 2 and then add 2 less than the figure number to get the answer!
It would look something like this:
total tiles = f (figure number) x 2 + (f-2)

So I gave them a random number (27) and asked them to tell me how many tiles would figure 27 have.

And they figured it out!

It was a great start to the BANSHO math programme…

Today’s question (Sept 16th):

Some information the students wanted clarified:

Some students used drawings/designs to solve the problem:

Some students used a chart and pictures to solve the problem, realizing that the pattern was to increase by 6 each time:

Some students used only a chart to solve the problem:

Some students found a very interesting pattern, and were able to predict the answer without having to complete the entire chart.  The pattern they noticed was that at the first table there were 10 people.  If you added 5 tables, there were 40 people.  If you added 5 more tables, there were 70 people.  They realized that every time you add 5 tables, you add 30 people.  Therefore they were able to predict that at 16 tables there would be 100 people!  Awesome thinking.

From this the students were able to see that since every 5 tables there were 30 more people (raising by increments of 6) therefore they could use multiplication to solve the problem.

Some students found a great solution (using multiples of 6, they were able to divide the number of people by (the multiple of) 6 and find the answer).  Awesome thinking!

Consolidation:  We explored the idea of using multiples of 6 to find a pattern.  In this pattern we realized that the total number of people were always the number of tables multiplied by 6, and then add on the 2 people at each end (4).

The students were able to come up with an algorithm (total # of people = # tables x 6 + 4 or P = T x 6 + 4).

Using the formula can you figure out how many people can sit at 25 tables?  How about 53 tables?

Today’s problem (Sept 19):
Today’s problem did not go as well as we would have liked.  We got several different answers (which is fine) but NO explanations of the student’s thinking!  We can learn a lot about math patterns, even if the answer is wrong.  So the students have been asked to do the question again for homework (in their notebooks), but to be sure to include an explanation.

(fig. 5 for grade 4, fig.10 for grade 5, figure 50 for a challenge)

Here were some of the answers (which have very interesting ideas and inventive patterns, but no explanation which makes it difficult to move forward)

Sept 20: Here were some of the correct answers (I did not want to post them before they had a chance to try it for homework)…

Today’s problem (Sept 20):

Some students used a table to find the answer…

Some students saw a multiplication pattern, and did not create a table…

Consolidation: We focused our discussion on how we could create a number sentence (algorithm) that could represent the pattern that we found.  We discussed the pattern for sugar first (increasing by 25 ml each time).  The students used P to represent the number of people (or desserts needed), and S to represent the total amount of sugar needed. 

The algorithm they came up with was P x 25 = S.  We also discussed why it might be a good idea to come with some kind of math sentence to solve the problem, and the consensus was that it was simply more efficient.  Excellent work!

Today’s problem (Sept 27):

Today’s focus was two-fold, first to try a question that could be solved with a pattern (chart), and second, for the students to show their thinking.

The criteria we came up with together (showing your thinking) was:

Today’s question was…

Some students focused on the pattern by creating a table…

Some students did a great job including an explanation:

Some students showed also showed their thinking by including (labeling) the pattern on the chart:

Some students also found new patterns.  This group noticed that every 2 teams the number jumps by 30, and 30 is a friendly number (it’s easy to do mental math with), so they focused on this pattern…

Some groups made the connection between the table and multiplication.  Upon finding this connection they realized that they had found the ‘algorithm’:

Consolidation:  We discussed how patterns are found in all sorts of math situations and questions.  We reviewed the idea that using a table is very efficient, and that finding the secret code (the algorithm) is the most efficient way to solve the problem.


7 Responses to Patterns

  1. I.S. says:

    Hi Mr. W., I don’t understand how the children came up the this numbers. Ex.fig.1 has 1, fig.2 has 5 fig.3 has 14 fig.4 has27 and fig.6 has44. I don’t know how to explain this.

    • mrwendler says:

      This was my point to the students too. If they don’t explain where they are getting their answers, how can we understand. The students did orally explain their thinking in class, but if it is not on the paper we all have trouble following. So the discussion we had in class centred on the importance of explaining where the math ideas come from and the use of math language when doing ‘bansho’ math problems. We also reviewed what the correct answer is (I did not post the correct answer because I wanted the students to try it at home). I hope that helps.

  2. I.S. says:

    We actually figured it out and she finished the table by herself

  3. N-mom says:

    Before I came home, he had finished his work. I think he was able to explain it for the 10. Wonder if there is a trick to get the answer for 50 without actually working on an entire table……that’s my coffee-break challenge now:-)

  4. Mrs. L says:

    Unfortunately, my little one wrote the figures incorrectly….this is why she had difficulty! We have asked her to sit down and show us everything that was explained in class. Thank you 🙂

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