Comparing and Scaling Develop students ability to make intelligent comparisons of quantitative information using ratios, fractions, decimals, rates, unit rates and percents . Students will learn different ways to reason in proportional situations. Students will use various proportional reasoning strategies they developed and apply these strategies in different context. We will also review solving 1 step equations, cross multiplying and multiplying and dividing fractions

Homework for investigation 1 All Worksheet on solving 1 and 2 step equations Worksheet on solving proportions Bookwork starting on page 19 You Pick A or B A 1,2,8,10,15,18,20 B 3,16,17,18,22,76

Ways of Comparing Ratios and Proportions Investigation 1 Ways of Comparing Ratios and Proportions

Vocabulary Needed for Investigation 1 Ratio: comparison of two quantities, lengths Proportion: setting two ratios equal to each other Part to Part Ratio: compares one part of the whole to the other part of the whole, water to concentrate Part to Whole Ratio: Compares one part to the whole, concentrate to the juice

Investigation 1.1: Analyzing Comparison Statements What do different comparisons of quantities tell you about their relationship? Students will be able to interpret mathematic statements.

Problem 1.1 (pg 8) Companies that sell soft drinks often report survey results about customers’ preferences. Here are 4 statements about the cola taste-test results.

Students at Neilson Middle School are planning an end of the year event. Of the 150 students in the school, 100 would like an athletic event and 50 would like a concert. Decide whether each statement accurately reposts the results of the survey.

What you should be able to do You should be able to read a problem based on a survey Evaluate the description of the survey and determine if they are true or false and why Are the ratios correct when they are describing the data Are they using in correctly to represent what the survey was for What do different comparisons of quantities tell you about their relationship? Which way you are comparing Which is more or less

Investigation 1.2: Comparing Ratios What strategies do you use to determine which mix is the most orangey? Students will be able to compare different mixture values to determine specific outcomes.

Problem 1.2 (pg 11) Arvin and Mariah were in charge of making orange juice for the campers. They planned to make the juice by mixing water and frozen orange juice concentrate. To find the mix that would taste the best, they decided to test some mixes. Which mix will make juice that is the most “orangey”? A – part to part or part to whole comparisons Which mix will make juice that is the least “orangey”? C – similar to above

Isabelle and Doug used fractions to express their reasoning Isabelle and Doug used fractions to express their reasoning. Do you agree or disagree? Why? Doug is correct – part to whole is needed to say the fractional part of the mix Max thinks A and C are the same. Max says “They are both the most “orangey” since the different between the number of cups of water and the number of cups of concentrate is 1.” Is Max’s thinking correct? Max is wrong – he just subtracted num from denom, he needs to look at part to whole

Assume that each camper will get ½ cup of juice. How many batches are needed to make juice for 240 campers? How much concentrate and how much water are needed to make juice for 240 campers?

What you should know Being able to compare part to part and part to whole What is the difference Which one is better Does it matter What strategies do you use to determine which mix is the most orangey?

Investigation 1.3: Scale Ratios When you scale up a recipe and change the units, like cups to ounces, what are some of the issues you have to deal with? Students will be able to set up ratios to help compare concentrates of juices.

Part to Part Ratio Scaling ratios is one comparison Strategy – compare same ratios by getting denominators and then compare numerators

Problem 1.3 (pg 13) Ratio 1:4 48oz Ratio 1:5 1/3 64 oz ½ gallon container

Problems Ratio 1:4 16 oz 16 * 4 = 64 oz Ratio is still 1:4 1 gallon is 128 oz Scale factor is 32 15 oz of concentrate Ratio 1:5 1/3 Scale up with a factor of 15 80 oz

What you should know When you scale up a recipe and change the units, like cups to ounces, what are some of the issues you have to deal with? Equivalent fractions Multiply both numerator and denominator by same value

Investigation 1.4: Scaling to Solve Proportions What strategies can you use to find a missing value in a proportion? What is your preferred strategy and why? Students will be able to set up and solve proportions.

Solving 1 and 2 step equations Solving Proportions Solving 1 step equation This means there is only one operation happening to the variable We have talked about these with rewriting using fact families Look at problem what operation is happening What is the opposite of that operation Do the opposite to both sides of the equation, this keeps the equation balance Variable should be by itself

2-step equations Look at doing PEMDAS backwards We are trying to get variable by itself Look for addition and subtraction first Undo the addition or subtraction by doing opposite to both sides Look for multiplication or division Do opposite to both sides Variable should be by itself

Examples X + 3 = 7 x – 5 = 2 3x = 12 x/5 = 3 2x – 1 = 7 x/3 + 5 = 14

Proportions We have already done but can apply solving equations Cross multiply, bring denominators up to numerator on opposite side, multiple, you may need to distribute if equation is large Solve the 1 or 2 step equation

Examples

Word Problems Be able to set up the proportion and solve for the unknown Make sure you label answer