Proportional Problems: Teacher Notes

Overview

In this activity students examine and reflect on how multiple strategies can be used for solving proportion problems.

Important Mathematical Ideas

  • Problems involving proportions can be solved with a variety of strategies: (e.g., using equivalent ratios, ratio tables, percents, or rates).

Prior Knowledge

  • Solving proportion problems using a variety of methods: (e.g., percent, ratio table, unit rate).

Common Misconceptions

  • There is only one way to solve each proportion problem.
  • Lack of understanding that solving a ratio, rate, or percent problem involves determining an equivalent ratio in an appropriate form for the context.

Information to Support/ Enhance/ Extend Learning

  • Students are asked to keep a journal for each unit in the course. It should contain notes of important mathematical ideas with examples and new vocabulary.
  • ePortfolio may be used for these journal entries.
  • Students can make individual choices whether this is a paper or digital personal resource.
  • Consider a variety of formats as alternatives to journal entries (e.g., student note, pair/share, group discussion, exit card, poster, electronic posting).

Minds On

Task 1: Blue Paint Problem

  • Students solve the problem using equivalent ratios, percents, a ratio table and unit rates.
  • sample solutions are provided
  • Can be set up as aFour Cornersactivity. The corners reflect the strategies used: percent, unit rates, ratio table and equivalent ratios. Students move to the corner that matches the strategy they think they'd like to use. In the corner, students solve the problem, discuss their solutions and prepare to present their method to the rest of the class. After presentations, as a class, discuss the connections and merits of the various strategies.

Action

Task 2: Bicycle Sale Problem and Sample Solution

The price of a bicycle is $200. It is reduced by 25%. Use two different strategies(e.g. equivalent ratios, percents, ratio table, unit rates) determine the new price of the bicycle.

The bicycle is reduced by $50. The new price of the bicycle is $200 - $50 = $150.

Discussion Prompt and Sample Response

The new price was reduced by a further 20%. This means a total discount of 45%. Do you agree or disagree? Explain your reasoning.

Further reduced by 20%

The bicycle is further reduced by $30. The new price of the bicycle is $150 – $30 = $120.

Discount of 45%

The bicycle is reduced by $90. The new price for the bicycle is $200 - $90 = $110.

I disagree with the statement. The 25% discount, followed by a further 20% discount saves $80. The 45% discount saves $90.

Responses should include:

  • an explanation of thinking in written or video form
  • a comparison of the two solutions
  • a contrast of the two solutions
  • selection of a solution and an explanation why

Common Error:

  • an unclear understanding of a further discount

Task 3: Basketball Season Problem and Sample Solution

The Hagersville Hoyas basketball team plays 80 games in a season. So far, Jackson played 27 games and scored 450 points. After an injury, Gerdon played 18 games and scored 320 points. If these players continue playing the way they have been, who will have the better season?

  • Solution using unit rates:

Jackson played 27 games and scored 450 points. His unit rate is 450 ÷ 27 = 16.67 points/game.

Gerdon played 18 games and scored 320 points. His unit rate is 320 ÷ 18 = 17.78 points/game.

Gerdon has a better points per game ratio and will have the better season.

  • Solution using ratio tables:

Jacksonwill score 150 points in 9 games.

Gerdon will score 160 points in 9 games.

Gerdon has a better points game ratio, so will have a better season if they continue playing this way.

Responses should include:

  • an explanation of thinking in written or video form
  • a comparison of the two solutions
  • a contrast of the two solutions
  • selection of a solution and an explanation why

Common Errors:

  • difficulty using more than one strategy
  • calculation errors, especially if not using friendly numbers

Consolidation

Task 4:Journal Prompts and Sample Responses

It may be helpful to have a discussion about the variety of solutions before students complete the journal.

1)Reflect on theBicycle Sale and theBasketball Seasonproblems.

Which strategy(s) are challenging to use.

Answers will vary.

Students might say ratio tables if they are not paying attention to friendly numbers.

Students may say percent or equivalent ratios because sometimes they involve non-friendly numbers

2)Why it is it important to have more than one strategy to solve a problem?

Some strategies are easier to use with non-friendly numbers. I need to pay attention to the numbers to help me decide which strategy to use.

Task 5: Assignment 1Frayer Model for Proportional Reasoning

  • Students revisit and update their Frayer Model for proportional reasoning. They add new and edit existing information.
  • Sample solution provided in Teacher notes posted on vLE.

Task 6:Assignment 2 Gasoline Costs

  • Posted with unit.
  • See sample solution in the Teacher Notes posted on the vLE.

Task 7: Student Reflection

  • Students are asked to reflect on their understanding of this topic.
  • These reflections can be used as assessment for learning to help determine next steps for individual students.

Grade 9 Applied Blended Learning: Unit 4Activity 7 Page 1 of 4