Counting Blocks Problem

Could be used as an introduction to solving equations of the form:

 

x² + bx + c = 0.

 

Could be a lead into factoring.

If students have already done some factoring ( c > 0 type), could be a lead in to more advanced factoring techniques ( c < 0 type).

 

Will typical take at least 2 periods of 45 minutes.

Introduction: Hello class. Today we are going to work on the following problem (see next page): Have them do the first page of the problem and discuss it. Then give them the second page of problem.

 

For student teachers, before you read the rest of the lesson plan, workout the problem yourself and then discuss among yourselves who you did the problem

A landscape artist wanted to build a decorative wall of cinder blocks that had the shape of a triangle

How many blocks total would she use for four layers?     How many blocks total would she use for six layers?

Can you find a formula giving the number of blocks she would use for n layers?

 

 

 

 

 

 

She recalls that the other day, she built a triangular wall for someone and it looked great. She can’t recall how many layers it was, but does recall that she needed 55 blocks. Can you figure out for her the number of layers?

 

 

Lesson Structure

 

-           first have students work on their own for about 5-10 minutes and see how far they can get.

-           Then have students work in their groups of four using standard Cooperative Learning techniques: roles for students: leader, recorder, questioner, presenter. Teacher circulates around the class watching what the students are doing, responds to any questions posed by the team questioner, gives hints and suggestions. Be prepared to encourage students to use their problem solving techniques.

-           Students will typically find it quite challenging to find a formula with n layers.

Scaffolding:

o        ask students what problem solving techniques they might use:

§          table (is often used)

§          prod them to think about trying to graph the data.

§          Does the data fit the types of graphs they know about? What types of graphs do they know (linear, quadratic)? How can they tell if the data has a linear fit? If it is not linear, what other type of function might they try? What does shape look like? Do they now any methods to decide if a function is quadratic?

§          If they try quadratic, how would they solve for coefficients of the quadratic?

Opportunity to assess students understanding of several important ideas from earlier in the course.

 

 

Student Presentations: Stop work part way through class for student presentations.

-           use observations of how the groups are doing to decide whether to have presentations of both the four and six layer problem or just one of them (then check that students have correct answer to the other one).

o        Sometimes level four is not enough for students to see the pattern; this will only emerge as they examine the level six problem.

-           Have one team give their answer to four (six) layer, being sure to explain how they got their answer. Use questioning technique to ask another group what they thought of this group’s answer.

o        Invite several other teams to present their solutions; choose teams that have used different approaches. Select teams by looking at their solutions techniques when walking around class. In most years, students have at least four different ways of solving problem:

§          Drawing picture and counting (without noticing any pattern)

§          # bricks in 6^th layer = 1 + 2 + 3 + 4 + 5 + 6; can be seen from drawing but now student notices the pattern.

§          Noticing that the 6^th level problem is related to the 5^th level problem: to make 6 levels, take the 5 level stack and then add one more layer on the bottom with 6 bricks:

# bricks in 6^th layer = # bricks in five layers + 6

Noticing this pattern leads to a very efficient calculation system with tables.

§          When we have multiple presentations, at the end ask the students

·          if they can see connections between the different ways the problem is solved

·          which way they prefer and why

 

After presentations on level 4 and level 6, have the students go back to working on level n. Point out the difference between finding a value for a specific case versus finding a formula that works in general (example of importance of variables).

 

Have students do presentations on what they tried to do for level n (very rare that anyone has gotten the problem on the first go around). Have students explicitly describe their problem solving strategies and their partial solutions.

-           often students can get as far as: 1 + 2 + 3 + … + n = # bricks in n levels but can not find a formula for this.

Then do a group brainstorm about other possible problem solving strategies to try (see above).

 

Before having someone solve the general formula for the whole class, let the students go on to the 2^nd page and solve the 55 brick problem:

-           enumeration; with enough effort, students can find the answer just by continuing to figure out how many bricks it would take for successive layers.

-           at this point. Some of the top students will have figured out that they need to solve a quadratic equation.

-           If this is continuation of factoring, then some students will be able to solve it even though they have not seen this type of problem before.

 

Presentations:

                  Have several presentations using different techniques from easiest to more sophisticated.

                  Have final presentation show that one gets a quadratic equation. Opportunity to talk about the meaning of negative solutions that are not physically meaningful.

 

 

Finale: Either :

-now we are going to learn how to solve such equations in a more efficient way than just enumeration (if this is first introduction to factoring)

Or

-now we need to learn how to factor equations when c < 0.