Year 2 Lesson Plan 22 - Rates and Proportions

1. (5 min) Mental Math
1. What is 1/5 of 35? [7]
2. What is 1/20 of 100? [5]
3. What is 1/9 of 36? [4]
4. What is 23% of 200? [46]
5. What is 12% of 300? [36]

2. (5 min) RATES
Rates involve numbers and units expressed as fractions. They are expressed with a "/" or "____" which is read as "per". Rates usually measure speed, with time as the denominator, but sometimes there are other units instead.

```   Sally types 45 words          45 words/min          45 words
per minute                                          -----
min

I drove 30 miles              30 miles/hr           30 miles
per hour                                            -----
hour

Pistachios cost five          \$5/lb.                \$5
dollars per pound                                --
lb
```

Both numbers and units can be multiplied and canceled when using rates. (All problems in this section assume a constant rate.) I drove 30 miles per hour. How far did I drive in 4 hours?

```          30 miles x 4 hours = 120 miles
-----
hour
```

I drove 30 miles per hour. How fast did I drive per minute?

```          30 miles x 1 hour      = 1 mile
-----   ----------    - -------
hour    60 minutes    2 minutes
```

I drove 1/2 mile in 1 minute! You know you have it set up right if the hours cancel. This is wrong:

```          30 miles x 60 minutes      180 mile-minutes
-----   ----------  =       ------------
hour    1 hour              hour2
```

3. PROPORTIONS
Now lets set up some equations:

If 6 bananas cost \$1.20, how much will 4 bananas cost? Set it up as two equivalent fractions. Make sure that the thing you want to find (x) is in the numerator and solve for x. Make sure that you have the same units on top and the same units on the bottom. In this problem we will put money on top and bananas on the bottom:

```        \$1.20      =     X
--------      ---------
6 bananas     4 bananas

Solve:

4 bananas x \$1.20   =  X
-----------------
6 bananas

X = 80 cents
```

Here is another way to do the same problem using the idea from algebra that we do the same thing to both sides of an equivalence. Here we setup the bananas on the left and the money on the right.
```        6 bananas   are   \$1.20
|               |
|               |
V               V
(take 2/3)       (take 2/3)
|               |
|               |
V               V
4 bananas   are      ???     (80 cents)
```
4. (Remainder of class) In-class exercise

5. Hand out homework as students successfully complete the in-class exercise.
6. Competitions
The first competition (Functions and Relations) stresses setting up equations from word problems. Each equation is worth 2 points, and each right answer is worth 1 point.

The second competition is an organized approach to guess and check. The first one has a table to work in and the first guess made, just to get the kids started. The last one needs the kids to set up their own table.