|7TH-8TH GRADE MATH LESSONS |
By Max Millard
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1. Students know the properties of, and compute with, rational numbers expressed in a variety of forms.
a. Define rational numbers. They are numbers that can be expressed as a fraction or a decimal. Examples are 2/3 and 1.75. Irrational numbers cannot be written out as exact numbers, but consist of nonrepeating decimals that go on forever, such as the square root of 2, which shows on the calculator as 1.4142136 to six decimal places but doesn't really stop there.
b. Ask the students to name some numbers that can be expressed as a fraction but not a decimal. Common ones are 1/3 and 2/3. Are these irrational numbers? No, because it's possible to measure them exactly. You can take a piece of string and measure out 1/3 quite easily, then multiply that by 3 to get the length of the whole string. But it's much harder to do with a decimal. Fractions are good for measuring things visually, and decimals are often easier when doing calculations on paper.
2. Convert fractions to decimals and percents and use these representations in estimations, computations and applications.
a. Make a numerical table, with the first column labeled "fractions," the second labeled "decimals," and the third labeled "percent." Write some fractions in the first column and let the students practice converting them into the other forms.
b. Take a deck of cards and let the class cut it approximately in half. Then estimate that it has about 26 cards. Count the cards and see how close you come. Try cutting it into thirds and fourths, and repeat the process while writing the equations to show how many cards each stack should contain. Cutting the deck is making an estimate of a fraction.
c. Weigh a bag of marbles or other small items. Write down the total weight and count the number of items. Make an estimate of how much each item should weigh. Write the estimated unit weight as a fraction of the total. Change the number of items being weighed to change the fraction, and make a chart of the results.
3. Know that every rational number is either a terminating or repeating decimal. Be able to convert terminating decimals into reduced fractions.
a. Write some fractions on the board that has repeating decimals, such as 1/6 and 1/3. Ask students to come up and do the long division, and see how the decimal keeps repeating. Talk about different ways to round them off to different decimal places. Let the students practice rounding them off to 3 places, 4 places, 5 places.
b. Do the equations in reverse, converting them into multiplication and letting the students practice them. Multiply 1.6666 x 6, .3333 x 3 and .66666 x 1.5. Do the multiplication the long way, so that students see that rounding off the decimals means that the answer is only approximate.
c. Use a calculator to show how easy it is to convert fractions to decimals and back again. Then ask how you would use the calculator to multiply a fraction by a decimal -- for example, 2/3 x .5. Answer: you can do either the fraction or the decimal first, at least with most calculators.
d. A terminating decimal is a number that has a fixed number of decimals -- for example, .125 = 1/8. Every fraction in which the denominator (bottom part of fraction) is a power of 2 (2, 4, 8, 16, 32, 64, etc.), it is equal to a terminating decimal. Let the class practice dividing 1 by these numbers to get the decimal.
e. One way to reduce fractions is to first convert them to decimals, then divide 1 by the answer. Here are some examples:
16/40 = .4
1÷.4 = 2.5
Reduced fraction: 2/5
12/60 = .2
1÷2 = 5
Reduced fraction: 1/5
13/52 = .25
1÷.25 = 4
Reduced fraction = 1/4
Make a list of other examples, and let the students use this method to practice reducing fractions.
4. Calculate the percentage of increases and decreases of a quantity.
a. First review the way to switch between fractions, decimals and percentages. The students can practice the most common fractions and their decimal and percentage equivalents. You can get some 3x5 cards and write the fraction on one side and the percentage on the other side.
b. To calculate the increase of a quantity, make a fraction in which the increase is the numerator and the original amount is the denominator. For example, if you once weighed 100 pounds and now you weigh 120 pounds:
Increase: 120 - 100 = 20
Fraction: 20/100 = .2 = 20%
c. To calculate the decrease of a quantity, make a fraction in which the decrease is the numerator and the original amount is the denominator. For example, if you once weighed 100 pounds and you now weigh 8 5pounds:
Decrease: 100 - 85 = 15
Fraction: 15/100 = .15 = 15%
d. Give some real-life examples and let the students practice them with pencil and paper. Present them as word problems.
• You start working at McDonald's at a salary of $7 per hour. You get a 50¢ raise. What's the percentage of your increase?
• You have 100 Magic cards and lose 12 of them. What's the percentage of your decrease?
e. Make the problems harder and ask the students to write down all their steps, as follows:
• You complete the Bay to Breakers race in 1 hour and 35 minutes. Last year you ran it in 1 hour and 20 minutes. How much slower are you now?
1:35 = 60 + 35 = 95 minutes
1:20 = 60 + 20 = 80 minutes
Increase in slowness = 15 minutes
15/80 = .1875 or 18.75% slower
• You earn $8 an hour working at a cafe. For the next two years, you get a 10 percent raise per year. How much are you earning after your second raise?
$8 x 10% = .80
$8 + .80 = $8.80 (pay after 1st raise)
$8.80 x 10% = .88
$8.80 + .88 = $9.68 (pay after 2nd raise)
• You're doing typing for a friend. The first month she gives you 30 pages to type. The next month she gives you 25 pages to type. The third month she gives you 20 pages to type. What's the percentage of decrease each time?
5. Solve problems that involve discounts, markups, commissions and profit. Compute simple and compound interest.
a. Discounts. Talk about the sales that are held at Macy's and other stores, and make up some problems about products that have a 25% discount, then a further 10% discount. The students should learn to calculate the discount each time by multiplication, not add the discounts together. If a $100 product has a 25% discount, then a 10% discount, the total discount is not $35. Instead, it is:
$100 -- .25(100) = $75
$75 -- .10(75) = 75 - 7.50 = $67.50
Discount = $32.50
b. Markups. One example is when a house goes on the market and the owner gets an offer that's above the selling price. Another example is when a large groups visits a restaurant where the policy is to add 20% to the bill for a service charge. Make up some problems in these contexts.
c. Commissions. This means a percentage of the selling price, which goes to the salesperson in payment for his services.
If an advertising salesman gets a 25% commission, how much advertising would he need to sell in order to earn $30,000 a year?
If a real estate agent gets a 5% commission for selling a house, how much commission would he earn from a $200,000 house?
d. Profit. This is the difference between a person's expenses and the money that comes in afterward. Start with some simple problems so the students will get the idea, and then make them more challenging.
• If a shoemaker pays $10 for leather to make shoes and sells them for $20, what is his profit?
• If a sports collector buys Barry Bond's 73rd home run ball for $1 million and sells it for $1,250,000, what is his profit?
e. Simple and compound interest. Interest is the extra money paid or charged over a certain period of time for the privilege of borrowing money. If you borrow money from the bank, you have to pay the bank interest, which means that you pay back more than you borrowed. When you put money into a bank, you are lending the bank your money to use, and the bank pays you interest. If you don't pay off your credit card bill on the month that you receive it, you are charged interest for the unpaid amount.
There are two kinds of interest: simple interest and compound interest. Simple interest is a percentage of the original amount money. So if you have a bank account that pays 10% a year in simple interest, and you start with $100, your account is worth $110 after one year, $120 after two years, and $150 after 5 years.
Compound interest means that each time you earn interest, that amount interest is added to your total, and you earn more interest on everything. Money grows much faster with compound interest. Einstein said that compound interest was the greatest invention of the 20th century. If your bank pays 10% a year with compound interest, you'll have $110 after the first year, then $121 after 2 years, $133.10 after 3 years, $146.41 after 4 years, and 161.05 after 5 years.
Give some other problems with different starting amounts and different interest rates so that the students can calculate simple and compound interest.
6. Use the inverse relationship between raising to a power and extracting the root of a perfect square integer. For an integer that is not square, determine without a calculator the two integers between which its square root lies, and explain why.
a. A Rubik's cube, which measures 3x3x3, is a good illustration of the cube of 3 and the cube root of 27. For the cube of 2, you can take 8 dice and form them into a cube, and practice counting the sides and the total number of dice.
b. Write the equations for multiplication and division, and show how they can be reversed:
2x2 = 4
4÷2 = 2
2x2x2 = 8
8÷2÷2 = 2
3x3 = 9
9÷3 = 3
3x3x3 = 27
27÷3÷3 = 27
c. Do the same for 4 and 5, and draw cube-shaped 3-D pictures on the board with the squares marked.
d. Use 5 plastic nickels and 5 quarters to demonstrate 5 cubed.
5 nickels = 1 quarter (5 squared)
5 quarters = $1.25. (5 cubed)
Show with the quarters that $1.25 divided by five equals one quarter, that one quarter equals five nickels, and that 5 nickels divided by 5 equals one nickel, or 5¢.
e. A calculator is a good way to calculate squares and square roots because if you keep pressing the = key, you won't need to retype the multiplier or divider. To calculate 6x6x6, press:
6 x 6 = =
To find the cube root of 216, press:
216 ÷ 6 = =
Try this with bigger numbers. It's especially clear with 10 and 100.
f. Determining the roots of integers that are not square. This refers to numbers that are not the square of whole numbers. 1, 4, 9, 16 and 25 are the first five squares. What about the square root of a number like 20? To find out its root, see which squares it comes between, and its root must come between the roots of those numbers. It falls between the square of 4 and the square of 5, so the square root of 20 must be between the integers 4 and 5.
Materials for 7th-8th grade math lessons:
plastic nickels and quarters
deck of cards