Gallery Problems Exercise

Gallery Problems Exercise

Building Bridges

Opening

Work Time

Building Bridges

The diagrams in this table represent a series of bridges. The bars are the segments that make up a bridge. n represents the number of triangles. b represents the number of bars.

The following algebraic expressions show different ways to calculate the number of bars based on the number of triangles.

b = 3 + 4(n – 1)

b = 3n + (n – 1)

b = 4n – 1

Each expression was found based on a different method for constructing the bridges. These methods are explained and illustrated here.

Method A: At each stage add a triangle at the end and then add a bar across the top.

Method B: At each stage (after the first stage) add a triangle with a bar at the top that points to the left.

Method C: At each stage add a triangle with a bar at the top that points to the right. At the end subtract the last bar pointing to the right.

 

  1. Determine which expression matches which method/diagram.


     
  2. Evaluate each expression for n = 10. 

Patterns in a Table

Work Time

Patterns in a Table

  • Complete the table. In the top row of each column write an expression that relates the values for n. For example, the numbers in the second column are all two times the numbers in the first column, so the heading for the second column is 2n
  • Then evaluate the expression you wrote in the top row, using the values in the first column, to fill in any empty cells in that column.

Expressions for Perimeter and Area

Work Time

Expressions for Perimeter and Area

For each of the figures:

  • Write two or more equivalent expressions for the perimeter, and two or more equivalent expressions for the area of each figure.
  • Then express the perimeter and area in the simplest form that you can.

Figure 1:

Figure 2:

Figure 3:

Multiplication Table

Work Time

Multiplication Table

This multiplication table has been started for you.

  • Copy and complete the table.

Garden Beds

Work Time

Garden Beds

A garden nursery sells square tiles for making borders around groups of plants (planting beds). The border tiles go only on the outside of all the plants—no tiles go between the plants. In the diagram, the square tiles are represented by tan squares and each plant is represented by 1 green square.

  1. How many square tiles are needed for 1 row of 100 plants?
  2. Write an algebraic expression that represents the number of square tiles needed for any number of plants planted in a single row.
  3. Find how many square tiles are needed for the following planting beds:
    • 2 rows of 50 plants
    • 4 rows of 25 plants
    • 5 rows of 20 plants
    • 10 rows of 10 plants
  4. Use a double number line or look at the relationship between the number of plants and the number of square tiles.

Telephone Tree

Work Time

Telephone Tree

A club set up a telephone tree so that all club members can receive important news quickly. The club secretary calls 3 people, and each of those people call 3 people, and so on.

  1. Finish the telephone tree to show all the calls that will be completed by the third round.
  2. What expression can be used to show the number of calls made during the fifth round? Evaluate this expression.
  3. What expression can be used to show the number of calls completed during the nth round?

 

Stacks of DVDs

Work Time

Stacks of DVDs

There are DVD cases and boxed sets of DVDs stacked on a shelf. The DVD cases are 14 millimeters thick and the boxed sets are 42 millimeters thick.

  1. Write an expression for the width of the stack on the shelf if the numbers of DVD cases and sets of DVDs are unknown.
  2. What is the width of a stack if there are 10 DVD cases and 5 boxed sets?
  3. What is the width of a stack if there are 32 DVD cases and 15 boxed sets?

Exponent Card Sort

Work Time

Exponent Card Sort

Work alone or with a partner on the Exponent Card Sort.

  • Match a number and a factor form from the right to each expression form to complete the table.
  • Some sets will have a missing form. Use a blank card to write the missing form for that set.

HANDOUT: Exponent Card Sort

Matching Words and Expressions

Matching Words and Expressions

  • Sort the Expression Cards to match the Word Cards.
  • When you find a match, explain how you know the cards match.

HANDOUT: Matching Words and Expressions

Investigating Factors and Multiples

Work Time

Investigating Factors and Multiples

  1. Choose any pair of numbers and follow these steps:
    • Find the greatest common factor (GCF) and the lowest common multiple (LCM).
    • Find the product of your two original numbers.
    • Find the product of the GCF and LCM.
    • Compare the two products.
  2. Choose other pairs of numbers and repeat the steps above.
  3. What do you notice about the products in each case? How do you explain your results?

Fourth Rock

Work Time

Fourth Rock

The planet Zantron is an imaginary planet that is a lot like Earth. It is the third planet from its sun. Its year—the time it takes the planet to go around the sun—is exactly 12 months.

Zort—the fourth planet in that solar system—takes exactly 20 months.

Scientists on Zantron want to communicate with the colony on Zort, but their radio only works when the planets are aligned so that they are at the closest distance possible from each other.

  1. If the two planets are at their closest point now, how many months will it be before they are at their closest point again?
  2. Explain the math you used to get your answer.

Factors of a Number

Work Time

Factors of a Number

Read the following mathematical claim:

For any whole number n greater than 6, if 6 and 2 are both factors of n, then 12 is also a factor of n.

  1. Decide whether the claim:

a.  Is true for every value of n.

b.  Is true for only some values of n.

c.  Is not true for any values of n.

     2.  Defend your decision mathematically.

Common Factors

Work Time

Common Factors

Suppose that the greatest common factor of two positive integers and is 20.

  • List all the other factors that must be common to and q. How do you know that these numbers must be common factors?

History of Variables

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History of Variables

Research the use of variables in the history of mathematics throughout the world.

  • What country or countries were variables first used in?

    • When were they used there?
    • What were they used for?
    • Create a timeline showing the use of variables in one of the countries you read about, or create a map showing the time each country you read about started using variables.
  • In what ways has the use of variables changed over time? In what ways has it stayed the same?

Create a Video

Work Time

Create a Video

Create a video about expressions.

  • Think about the key concepts and terminology involved with expressions.
  • Develop a story line that encompasses those concepts.
  • Shoot and edit the video.