If you’re new to our blog, take a look at our earlier post, Why Guided Solutions for the Problems of the Week?. Here’s the solution for last week’s problem. Share other approaches.

1. Look at the square on the upper left. How wide is it across the middle? How wide is it across the top? How wide is it across the bottom? What is the length of the left side? What is the length of the right side? What is the perimeter (the distance around) of the square?

2. A comparison of two amounts is called a ratio. Tell this ratio: the perimeter of the square compared to its width. Write the ratio so that it looks like a fraction. Now reduce the fraction. What is the reduced ratio?

3. What is the width (diameter) of the circle on the upper right? Is it wider than the square that it is in? Is it wider than the square on the upper left?

4. Trace your finger around the circumference of the circle. Trace your finger around the perimeter of the square that contains the circle. Which distance is longer— the circumference of the circle, or the perimeter of the square? If you are not sure, think about walking around a corner on a sidewalk: if you are in a hurry, would you go straight and then turn right, or would you “cut the corner” by leaving the sidewalk and taking a curving path?

5. Think about the ratio of perimeter of the square to its width: the width times _____ (what number) = the perimeter? Think about the ratio of circumference of the circle to its width: the diameter times _____ (what number) = the circumference? Do we know? If you multiplied the width of the square times 4, would you get its perimeter? If you multiplied the diameter of the circle times 4, would you get its circumference? Or would you have multiply by some number smaller than 4?

6. Look at the hexagon on the middle left. How many equilateral triangles does it contain? What exactly is an equilateral triangle? How many sides does it have? Are all the sides of equal length? The hexagon is 7 cm wide. What is half of that distance? How wide is the base of the upper left triangle? How wide is the base of the upper right triangle? Is every side of every triangle that same length? What is the perimeter (the distance around) of the hexagon?

7. A ratio is a comparison of two amounts. Tell this ratio: the perimeter of the hexagon compared to its width. Write the ratio so that it looks like a fraction. Now reduce the fraction. What is the reduced ratio?

8. What is the width (diameter) of the circle on the middle right? Is it wider than the hexagon that is in it? Is it wider than the hexagon on the middle left?

9. Trace your finger around the perimeter of the hexagon contained in the circle. Trace your finger around the circumference of the circle. Which distance is longer—the circumference of the circle, or the perimeter of the hexagon?

10. Think about the ratio of perimeter of the hexagon to its width: the width times _____ (what number) = the perimeter? Think about the ratio of circumference of the circle to its width: the diameter times _____ (what number) = the circumference? Do we know? If you multiplied the width of the hexagon times 3, would you get its perimeter? If you multiplied the diameter of the circle times 3, would you get its circumference? Or would you have multiply by some number larger than 3?

11. Look at the figure on the bottom: a square with a circle inside it, and a hexagon inside the circle.

Which is longer, the perimeter of the square, or the circumference of the circle?

Which is longer, the perimeter of the hexagon, or the circumference of the circle?

What is the ratio of the perimeter of the square, compared to its width?

What is the ratio of the perimeter of the hexagon, compared to its width?

What is the ratio of the circumference of the circle, compared to its diameter?

Is it more than? Is it less than? Is it more than? Is it less than?