Grade Nine
Strand: Ratio and Proportion

Instructional Notes

• Proportional thinking is a very difficult concept for students to develop. Be sure they understand the concepts of equivalent ratios and scaling up and down before using the cross multiplication algorithm. Ratios should be placed in an everyday context, whenever possible, so students develop a sense of what is and what is not reasonable.
• R-1 Provide students with a variety of strategies for solving problems involving proportional relationships. Strategies include building a table, a unit-rate approach, a factor of change approach, a fraction approach, the common multiple method, scaling up and down, and cross multiplication. Word problems involving rates can be solved in a number of different ways. Observe how students solve these problems and encourage different methods for different situations.
  When proportions are set up involving rates, the units of measure should be included.
• The newspaper or flyers are good sources of information for "best buy" problems or other word problems involving rates.
• The concept of slope is developed in Grade 10. In Grade 9, you can introduce the concept in a concrete manner as a ratio. Students can also graph the relationship as a linear function.
• R-4 Encourage students to suggest a variety of places in everyday situations where they encounter ratios and/or rates.
• R-8 Ratios can be compared by converting them to fractions, decimals, or percents and then comparing the results.
• Changing the context and numbers involved in problems changes the level of difficulty of the problem.

  R-10 Students have studied scale ratio and reading and interpreting scale drawings in previous grades. They have made scale drawings given a shape and a scale. These concepts should be reviewed before actually making scale drawings of objects. By Grade 9, students can be expected to make the necessary measurements, determine a scale, and make an accurate drawing according to that scale. A suitable group activity would be to have students construct a model or scale drawing of something they choose such as a house plan, the solar system, or a familiar object, or drawing a picture from art class or an example from drafting.

• R-15 Once students have an understanding of the proportional method, any type of percent problem can be solved. Include percents greater than 100 as well as fractional and mixed number percents.
• One way of determining whether a pair of ratios forms a proportion is to reduce each ratio to simplest form and see if they are the same. Another way is to use the product of the means equals the product of the extremes.
• R-18 The visual models used in Grade 8 for translating percent problems can also be used with the proportional method. E.g.: about 20% of the world's land area is in Africa. If the area of Africa is about 12 000 000 square units, what is the world's area?

  

• R-20 There are several strategies for solving proportions:
  • make rows of counters;
  • draw a diagram;
  • multiply or divide the terms by the same number;
  • give the two ratios a common denominator and compare their denominators;
  • use a calculator to change each ratio to an equivalent decimal; and,
  • compare the product of the means to the product of the extremes.
  Once students have had an opportunity to use a variety of methods to solve proportions, they should be permitted to use whatever method they find most appropriate. Provide a number of different questions which indicate that some problems are more easily solved by one method such as equivalence, common multiple, or algebraic equations, while others are more easily solved using cross multiplication or some other method.

  

• R-21 Remember there are several correct ways of solving a problem. Encourage a variety of approaches and have students describe their solutions to the rest of the class. Also encourage students to estimate an answer before doing the calculations so that they can judge the correctness of the answer they find. There are a variety of informal methods available to solve many problems.
• The newspaper is a good source of real-life percent problems.
• A group activity that students could do is to find the rates of interest for savings accounts and term deposits at 3 different banks, credit unions, or trust companies. Then calculate the interest earned on $1 000 in one year at each of the rates. (Assume the interest is calculated once a year -- simple interest.)
• To help students understand compound interest, do a number of exercises, using a calculator, where the student calculates the interest for each interest period. E.g.: if interest is compounded quarterly, what interest do you earn in 6 months on $500 invested at 8% annual rate of interest? Calculate the interest for the first quarter. i = prt, i = 500 x 8/100 x 1/4 = 10. The new principal is now $510. So interest for the second quarter is calculated using i = 510 x 8/100 x 1/4 = 10.20. Therefore, the interest earned over 6 months was $20.20.
• Students can gain a better understanding of the effect of compound interest by doing calculator activities such as the following. "If interest is compounded annually on an amount of money invested at 6%, how long would it take your money to double?" Enter 1.06 into the calculator and press x. Press the = sign (or x=) repeatedly until the display is at least 2. Keep track of the number of times you press the equal sign (start at 2 because the first time you press =, the answer is the product of 1.06 x 1.06). (TL) The concept could be expanded by doing the compounding semi- annually (1/2 of 6% or 1.03), quarterly (1/4 of 6% or 1.015), etc., or change the rate of interest. Remind students that each time the equal sign is pressed when interest is compounded semi-annually, it represents a six month time frame; quarterly is a 3 month time period; etc.