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Guided Practice

What is the ratio of the width to the length for the rectangle?

The width is 6 centimeters, and the length is 12 centimeters.

The ratio of width to length can be written as 6/12, 6 : 12, or "6 to 12".

Four 6th graders, two 3rd graders, and two teachers are helping at the school's Fall Festival. What is the student-to-teacher ratio?

There are 4 + 2 students, or 6 students, and 2 teachers.

The student-to-teacher ratio is 6 : 2. There are 3 times as many students as teachers helping at the Fall Festival.

Write the ratio of circles to squares as a whole number, with a colon, and in words.

With a colon:     

Whole number:

Write in words:  

Write the ratio of triangles to circles using a slash (e.g., 4/2).

Triangles to circles:

Write the ratio of triangles to circles and circles to triangles using a slash.

Triangles to circles:

Circles to triangles:

Complete each sentence with a ratio.

There are times as many triangles as circles.

There are times as many circles as triangles.

Complete the sentence with a ratio.

There are times as many triangles as circles.

Complete the sentence with a ratio.

There are times as many squares as circles.

Write the ratio of the width to the length of the square. Write the ratio using a colon (e.g., a : b) and as a whole number.

With a colon:     

Whole number:

What is the ratio of the height to the base for the triangle? Use an appropriate form for the ratio.

Height to base:

Complete the sentence with a ratio written as a whole number.

The length of the rectangle is times its width.

Complete the sentence with a ratio written as a whole number.

The length of the square is times its width.

Four 6th graders, two 3rd graders, and two teachers are helping at the school's Fall Festival. What is the ratio of 6th graders to all helpers?

6th graders to all helpers:

There are 32 students in Ms. Smith's class. Three students are absent today. Write each ratio.

Absent to present:  

Absent to total:      

of the class is absent today.

of the class is present today.

Plant B is 2 times as tall as Plant A.

Write the ratio.

B height : A height =

Ladder A is 2 steps taller than Ladder B.

Write the difference.

A height – B height = steps

There are 1/4 as many heads as legs.

Write the ratio.

heads / legs =

There is 1 fewer ticket in Pile A than in Pile B.

Write the difference.

B tickets – A tickets =

How could you compare the values for the circle and for the rectangle?

Write ratio, difference, or both:

How could you compare the values for the square and for the taller rectangle?

Write ratio, difference, or both:

Jerome has 10 toys. His brother has 3 more toys than Jerome. Write the ratio or difference which compares the number of Jerome's toys with the number his brother has.

= 3

Alice's family drove 21 miles to the concert. Corinne's family drove twice the distance Alice's family drove. Write the ratio or difference which compares the distances the two families drove.

= 2

Jasmine is 40 years old. Her son is 1/5 as old
as she is. Write the ratio or difference which

compares their ages.

= 1/5

Write the ratio as a decimal.

Mars Distance / Earth Distance =

Mars is times farther from the Sun than the Earth is.

Write the ratio as a decimal. Then complete the statement.

Venus Distance : Earth Distance =

Venus's distance from the Sun is hundredths Earth's distance.

Write the ratio as a decimal. Then complete the statement.

Uranus Distance / Earth Distance =

Uranus is times farther from the Sun than the Earth is.

Write the ratio as a decimal. Round to the nearest hundredth. Then complete the statement.

Earth Distance / Mercury Distance =

Earth is times farther from the Sun than Mercury is.

Write the ratio as a decimal. Round to the nearest hundredth. Then complete the statement.

Saturn Distance : Jupiter Distance =

Saturn is times farther from the Sun than Jupiter is.

Write the ratio as a decimal. Round to the nearest hundredth. Then complete the statement.

Uranus Distance : Mercury Distance =

Uranus is times farther from the Sun than Mercury is.

The crowd waits silently—almost silently—in the dark hall while the orchestra begins its warmup. One of the oboists plays a single note, an A note, three times. The rest of the orchestra then uses this note to tune their instruments.

Let's warm up with the orchestra. Press the Warm-Up button below. Then press the orange squares or the keys on the keyboard to hear if the orchestra is in tune.

The tuning note played by the oboe, A, is a sound wave that vibrates at a certain frequency. This frequency is given in a special unit called Hertz. The A note vibrates at 440 Hertz (Hz).

Move both sliders all the way to the right. Now the top left orange button plays A (440) and the top right button plays a higher A (880). (Try them!)

In music, the distance between an A note and the next higher A note (or a B note and the next higher B note, etc.) is called an octave (AHK-tiv). Press the button to play all the octaves.

Try it on the keyboard. When two notes are an octave apart, what do you notice about the approximate ratio of their frequencies in Hertz?

One way to remember what the interval of an octave sounds like is to think of the song "Somewhere Over the Rainbow". Press the button to listen to the song. Then press the karaoke button and try to play along with the song on the keyboard. Practice makes perfect!

Scroll down for more. Let's explore other musical intervals and the ratios of their frequencies . . .

An octave is one kind of interval. Another kind is called a major third. Move both sliders to the fourth tick mark from the left (count the leftmost tick mark as 0). Now the orange squares on the sides of each slider play a major third interval. Press the Major Thirds button to hear some intervals of a major third.

Try it on the keyboard. What is the approximate ratio of the frequencies of the notes in any major third? What is the ratio now of the frequencies directly above and below each other on the sliders?

A way to remember what the interval of a major third sounds like is to think of the first two notes of the melody in the song "When the Saints Go Marching In".

Press the button to listen to the song. Then press it again to play along with the song on the keyboard. What major thirds can you identify in the song? How can you prove that they are major thirds?

Scroll down for more.

Two notes in the interval of an octave are separated by 12 half-steps. Two notes in the interval of a major third are separated by 4 half-steps. And two notes in the interval of a major sixth are separated by 9 half-steps.

The first two notes of the song "My Bonnie Lies Over the Ocean" are in the interval of a major sixth. Press the Major Sixths button to hear some intervals of a major sixth. Then press the song button and play along with the song on the keyboard. How can you play a major sixth using the sliders? What is the ratio of the frequencies?

Here are some of the other musical intervals that musicians know:

minor second: 1 half-step
major second: 2 half-steps
minor third: 3 half-steps
perfect fourth: 5 half-steps
perfect fifth: 7 half-steps
minor sixth: 8 half-steps

Determine their frequency ratios. What patterns do you notice in the frequency ratios and on the sliders?

Can you generate any interval you learned about given any starting frequency? What about given any ending frequency?

Start freq. → ← End freq.

What is the smallest frequency ratio you can detect? What is the highest frequency you can hear? People over the age of 25 commonly have trouble hearing frequencies above 15000 Hertz. Younger people can hear higher frequencies, all the way to about 20000 Hertz.

To compare two numbers using a ratio, write one number and then the other number. You can separate the numbers by the word "to", by a colon (:), or by a fraction bar (/). For example, write the ratio of giraffes to crabs.

The order the ratio is written is important. To write the ratio of giraffes to crabs, write the number of giraffes and then the number of crabs in one of these ways:

4 to 2, 4 : 2, or 4/2

You can write a ratio comparison using the word "times." For example, compare the stars and circles using the word "times."

The ratio of stars to circles is 3 to 2, so there are 3/2 times as many stars as circles. (Three over two times as many stars as circles.)

The ratio of circles to stars is 2 to 3, so there are 2/3 times as many circles as stars. (Two thirds times as many circles as stars.)

You can write ratios using measurements too. For example, the top string is 5 inches long, and the bottom string is 6 inches long.

You can write a ratio of the lengths using the units or without using the units:

5 in. to 6 in., 5 : 6, 6 in. to 5 in., 6 : 5

You can also still say things like, "The top string is 5/6 times the length of the bottom string" and "The bottom string is 6/5 times the length of the top string."

A ratio can compare two parts or it can compare a part and a whole. For example, the ratio of the bottom length of string to the top length of string is 3 : 7. The ratio of the bottom length of string to the TOTAL length of string is 3 : 10.

The ratio of stars to circles is 3 : 2. The ratio of stars to ALL the shapes is 3 : 5.

A ratio is also a division statement. Thus, it can be a whole number. For example, the ratio of circles to triangles is 6 : 2. And 6 : 2 is equal to 6 ÷ 2 = 3.

So, 6 : 2 is the same as 3, or 3 : 1, or 3 to 1.

This means that we can say that there are 3 times as many circles as triangles. And there are 1/3 times as many triangles as circles.

Write the ratio of crabs to giraffes in each of the three ways.

to

   :

   /

Write the ratio of triangles to squares using the word "times." Then write the ratio of squares to triangles using the word "times."

There are / times as many triangles as squares.

There are / times as many squares as triangles.

The ruler shown measures the two strings in centimeters. Write the ratio of the top string length to the bottom string length.

Ratio:

The top string length is / times the length of the bottom string.

Write ratios with a colon (:) to complete each sentence.

The ratio of black circles to red triangles can be written as : .

The ratio of circles to TOTAL shapes can be written as : .

Write the ratio of rectangles to triangles as a whole number.

The ratio of rectangles to triangles is equal to .