10.1: What Do You Want to Know?
Consider the problem: A red car and a blue car enter the highway at the same time and travel at a constant speed. How far apart are they after 4 hours?
What information would you need to be able to solve the problem?
10.2: Info Gap: Hot Chocolate and Potatoes
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the problem card:
Silently read your card and think about what information you need to be able to answer the question.
Ask your partner for the specific information that you need.
Explain how you are using the information to solve the problem.
Continue to ask questions until you have enough information to solve the problem.
Share the problem card and solve the problem independently.
Read the data card and discuss your reasoning.
If your teacher gives you the data card:
Silently read your card.
Ask your partner “What specific information do you need?” and wait for them to ask for information.
If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.
Before sharing the information, ask “Why do you need that information?” Listen to your partner’s reasoning and ask clarifying questions.
Read the problem card and solve the problem independently.
Share the data card and discuss your reasoning.
Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.
10.3: Comparing Reading Rates
- Lin read the first 54 pages from a 270-page book in the last 3 days.
- Diego read the first 100 pages from a 325-page book in the last 4 days.
- Elena read the first 160 pages from a 480-page book in the last 5 days.
If they continue to read every day at these rates, who will finish first, second, and third? Explain or show your reasoning.
The ratio of cats to dogs in a room is \(2:3\). Five more cats enter the room, and then the ratio of cats to dogs is \(9:11\). How many cats and dogs were in the room to begin with?
To solve problems about something happening at the same rate, we often need:
Two pieces of information that allow us to write a ratio that describes the situation.
A third piece of information that gives us one number of an equivalent ratio. Solving the problem often involves finding the other number in the equivalent ratio.
Suppose we are making a large batch of fizzy juice and the recipe says, “Mix 5 cups of cranberry juice with 2 cups of soda water.” We know that the ratio of cranberry juice to soda water is \(5:2\), and that we need 2.5 cups of cranberry juice per cup of soda water.
We still need to know something about the size of the large batch. If we use 16 cups of soda water, what number goes with 16 to make a ratio that is equivalent to \(5:2\)?
To make this large batch taste the same as the original recipe, we would need to use 40 cups of cranberry juice.
|cranberry juice (cups)
|soda water (cups)
- double number line diagram
A double number line diagram uses a pair of parallel number lines to represent equivalent ratios. The locations of the tick marks match on both number lines. The tick marks labeled 0 line up, but the other numbers are usually different.
The word per means “for each.” For example, if the price is $5 per ticket, that means you will pay $5 for each ticket. Buying 4 tickets would cost $20, because \(4 \boldcdot 5 = 20\).
- same rate
We use the words same rate to describe two situations that have equivalent ratios.
For example, a sink is filling with water at a rate of 2 gallons per minute. If a tub is also filling with water at a rate of 2 gallons per minute, then the sink and the tub are filling at the same rate.
A table organizes information into horizontal rows and vertical columns. The first row or column usually tells what the numbers represent.
For example, here is a table showing the tail lengths of three different pets. This table has four rows and two columns.
pet tail length (inches) dog 22 cat 12 mouse 2