Models You Can Count On RATIO TABLES A ratio table is a useful tool to organize and solve problems To set up a ratio table, label each row and set up the first column ratio Different operations (addition, subtraction, multiplication, division) may be used within the same ratio table Just remember, whatever you do to the top, you must also do to the bottom! For the example on the next page, one movie ticket costs $9, and we want to know the price for 3, 6, and 9 tickets.

Ratio Table x3 x2 Movie Tickets 1 3 6 9 Price (Dollars) $9 $27 $54 $81 To get from the first column to the second column, we multiplied by three. Then to get from the second column to the third column, we multiplied by two. Notice that we did the same thing to both the top and bottom of the column. Lastly, to get the value of the last column, we add the second and third columns together (3+6=9), so we do the same for the bottom ($27+$54 = $81).

Ratio Table There are other combinations that can be used to find the values of the last column. We could have multiplied the second column by three. 3 tickets x 3 = 9 tickets, so $27 x 3 = $81. So, as you can see, there is more than one way to create a ratio table. How you do it will be up to you and it will depend on what the question is asking you.

Ratio Table - Recipes Most recipes need to be changed depending on the number of people you are cooking for. You will need more ingredients for a recipe for 10 people, than you would if you were only cooking for two people. So how do know how many ingredients to use? A ratio table is always helpful for this type of problem.

Ratio Table - Recipe X 5 Servings 2 10 Apples 6 cups Cups Flour 3 cups Butter 1 tbsp 5 tbsp Cinnamon 1 cup 5 cups

Ratio Table - Recipe As you can see in the recipe on the last page, the recipe calls for a certain amount of ingredients for two servings. However, you are cooking for 10 people (10 servings) but the recipe doesn’t tell us how much we need. To do this, all we do is multiply 2 servings by five, to get 10 servings. Then we multiply all of the ingredients by five as well. This will give us the correct amount of ingredients for 10 servings.

Bar Model Fraction Bars Fraction bars can be used to represent how something is divided into pieces. For example, if four friends needed to share a plot of land to plant their favorite garden, we good create a fraction bar to represent each friend’s portion of the land. Friend 1 Friend 2 Friend 3 Friend 4

Bar Model We can also include fractions to represent each friend’s portion of the land. Friend 1 Friend 2 Friend 3 Friend 4 ¼ ¼ ¼ ¼

Bar Model If the number of friends changes, so does the size of each portion of land. The more friends that share, the smaller each portion becomes. Friend 1 Friend 2 Friend 3 Friend 4 Friend 5 Friend 6 Friend 7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1 2 3 4 5 6 7 8 9 10 1/10 1/10 1/10 1/10 1/10 1/10 1/10 1/10 1/10 1/10

Bar Model We can also add percents to a fraction bar. Of course if you use percents, you may want to change the name to a percent bar. Sometimes, you may want both percents and fractions. Friend 1 Friend 2 Friend 3 Friend 4 ¼ ¼ ¼ ¼ 25% 25% 25% 25%

Bar Model Sometimes our fraction or percent bars will be horizontal (up and down) instead of vertical (left to right). Friend 1 Friend 2 Friend 3 Friend 4 ¼ or 25% ¼ or 25% ¼ or 25% ¼ or 25%

Bar Model Some bar models are not divided into fractions. They contain overall numbers, which you must use to determine the fraction or percent. For example, if a water tank holds a maximum of 300 liters, but the water gauge is not filled to the top, how do you know how many liters of water are in the tank? With a little bit of thought and work, a correct answer can be determined.

Bar Model 300 L Here we have a water tank that holds 300 liters (L), but the gauge is not full to the top, so there is less than 300 L in the tank. But how can we determine the amount of water? Knowing your fractions is the best tool to begin solving this problem. As you can see, there are five sections in the bar, therefore we can divide this into fifths. Our next step is to determine the amount of water that is in 1/5 of a tank that holds 300 L. 300 L divided by 5 = 60 L Therefore, each 1/5 section contains 60 L of water.

Bar Model Now we can see how each 1/5 section is 60 L. 1/5 or 60 L Now we can see how each 1/5 section is 60 L. Here is the last step to solving this problem. We need to determine how many total L are in the tank. There are two 1/5 sections that are full of water, and each section has 60 L of water. To find our final answer, we multiply 60 L by two and we determine that there are 120 L of water in this tank.

300 L 1/5 or 60 L If each section is 1/5 as a fraction, then it must be 20% as a percent. Remember, this is one of our benchmarks. If we want to find 20% of the 300L, then we have to change 20% to a decimal, then multiply it to 300L. 300 x .20 = 60, therefore, there are 60 L in each 20% section of the bar graph. Since there are two sections, multiply 60 L by two to get a total of 120L in the tank – the same as the answer we found using the fractions.

Tips You just received your bill at a local restaurant. Your bill is $40.00 How much of a tip do you live for your waitress? If the service was great, you leave 20%, if it wasn’t real good, maybe 10%, and if the service was just good, 15% would do it. So how do we find out how much each tip is worth?

Tips To find a 10% tip, we can just move the decimal one place to the left of the bill. So, if we move the decimal in our $40.00 bill, our tip would be $4.00 Our other option is to simply change the tip percentage to a decimal then multiply to the bill. The answer will be the tip amount. Example - $40 x .10 = 4.00, so a 10% tip for this bill is $4.00

Tips For a 20% tip, we can find 10%, then multiply by two. Our other choice is to change 20% to a decimal (.20) then multiply by the bill. $40 x .20 = 8.00, so your tip for the bill would be $8.00 For the 15% bill, we can change the 15% to a decimal and multiply by the bill. $40 x .15 = 6.00, so $6.00 is the tip for the $40 bill.

Tips Bill $40.00 10% tip: $4.00 15% tip: $6.00 20% tip: $8.00

Number Lines Number lines can be divided into sections using fractions or percents. 0% 25% 50% 75% 100% 1/4 1/2 3/4 1 whole

Number Lines Fractions with different denominators can be used on the same number line. The denominator tells you how many sections the number line should be divided into, even if you are not dividing the entire number line. For example, we can label 1/3, 1/4, and 1/5 on the same number line. But we only label these three fractions. We do not label 2/3, 3/4, 4/5, etc on the number line, so the line should be divided into all of the thirds, fourths, and fifths. See next picture………

Number Lines Fractions with different denominators on the same number line: 1/5 1/4 1/3 1

Number Lines Number lines can also be used to organize decimals. 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 1

Double Number Lines Double number lines can be used to compare and label different numbers – much like a ratio table. For example, if it takes 10 minutes to walk 1/2 mile, we can label that on a double number line. We can then use that information to find out how long it takes to go, 1, 2, or even ten miles. We could also use it to find out how far we can go in 20, 30, or even 90 minutes.

Double Number Lines It takes 10 minutes to walk 1/2 mile. It would be labeled on a double number line like this: Minutes 10 Miles 1/2

Double Number Lines Weights and prices can also be labeled on double number lines. Weight (lbs) 1/4 1/2 3/4 1 1 1/4 1 1/2 .50 $1.00 $1.50 $2.00 $2.50 $3.00 Price ($)