Problem Solving: Ratios
Class Presentation Notes
LAUNCH (9 MIN) ______________________________________________________________
Before
• Who would want to know the results of this vote?
During
• What information do you know?
After
• Why do the ratios all appear different? How can they all be correct?
• If the results of the vote were changed so that three times as many votes were for comedy
as for action, what would be the new ratio of action votes to all votes, written as a fraction?
PART 1 (10 MIN) ______________________________________________________________
Before playing the solution animation
• What are the ratios of fish required for the tanks?
• Can both classes have more spotted fish than striped fish?
Jay Says (Screen 1) Use Jay Says to clarify that there are lots of ways to satisfy one ratio,
but only one way to satisfy both ratios at the same time.
After completing the solution
• How does this example help you make decisions about the fish tanks?
PART 2 (8 MIN) _______________________________________________________________
Before presenting the solution
• How is this table similar to a multiplication table?
Jay Says (Screen 1) Use the Jay Says button to reveal why maintaining the same ratios is
important.
While presenting the solution:
• What is different about equivalent ratios with mixed numbers and equivalent ratios with
whole numbers?
After completing the solution
1
• How many batches could you make if you have exactly 2 2 c of flour?
• Do you have to follow the ratios in the table to make pancakes that taste good?
PART 3 (8 MIN) _______________________________________________________________
Before presenting the solution
• What do equivalent ratios have to do with more or fewer batches of dressing?
After completing the solution
• Review your answer. Does it make sense?
• How is this example related to the first one?
CLOSE AND CHECK (5 MIN) ______________________________________________
Emphasize the connection between the Focus Question and the Essential Question.
• How has your experience with ratios in this topic prepared you to make plans and
decisions?
• What do you think you will learn next about ratios?
– grade 6 • 0-0 Teacher Guide
Problem Solving: Ratios
Lesson Preparation Notes
LESSON OBJECTIVE
1. Solve problems using ratios.
FOCUS QUESTION
In this topic you have learned different ways to work with ratios. How can you use what
you learned to make plans and decisions?
MATH BACKGROUND
Students have learned many ways to write a ratio in this topic, as well as three
different comparisons that ratios can model. Students now need practice comparing
quantities in different problem-solving situations and choosing the clearest way to
represent a ratio to others. In this lesson, the emphasis is on ratios as fractions.
Decimal representations are the emphasis in the topic “Rates.” Students should
understand that their experience with equivalent fractions is helpful when working with
equivalent ratios, but also that different real-world situations lend themselves to
different representations. The Launch shows that there are occasions when a : b or
a to b is a more useful form for a ratio than a fraction or decimal.
In the “Rates” topic, students will continue to experience problem-solving situations
involving ratios with increasing applicability to their everyday lives. They will face
problems concerning unit rates and unit prices, as well as other measurement
problems. They will need to draw on what they have learned in this topic about ways to
write ratios and methods of solving ratio problems.
LAUNCH (9 MIN) _____________________________________________________
Objective: Compare and contrast different representations of ratios.
Author Intent
Students face four different representations of the same situation in the Launch. They
need to determine in each case whether the ratio reflects the situation.
Questions for Understanding
Before
• Who would want to know the results of this vote? [Sample answer: The teacher
choosing the movie for the school party should choose a movie that the majority
of the students will enjoy. Also, other teachers can use these results to make
predictions about future students’ preferences.]
During
• What information do you know? [There are 100 students. 1/5 voted for drama.
So 4/5 voted for either comedy or action, and three times as many voted for
action as for comedy.]
After
• Why do the ratios all appear different? How can they all be correct? [The ratios
all appear different because they are written in different forms and they compare
different amounts. They are all correct because they all accurately reflect the
voting comparison, though each ratio uses different quantities from the Launch.]
• If the results of the vote were changed so that three times as many votes were
for comedy as for action, what would be the new ratio of action votes to all votes,
written as a fraction? [Accept all fractions equivalent to 1/5.]
– grade 6 • Teacher Guide
Solution Notes
Although there is only one correct answer, student explanations may vary. If students
can justify their answers using ratios, allow different methods to test that each ratio is
accurate. Explain that there are enough ways and forms to write a ratio that there are
many ways to complete this Launch. For example, students might start with 20 to 80 or
with 20 to 100.
Differentiated Instruction
You can use the Blocks and Chips tool to help students solve this example. In the
Place Value Blocks mode, choose the Explore option and create three mats, which
you can label “Drama,” “Comedy,” and “Action.” Drag 10 blocks into the workspace
and tell students that each block represents 10 votes. Allow students to drag the
blocks to the correct mats to solve the example.
Challenge: You can ask students who quickly identify that all four ratios are correct to
write each ratio in as many different forms as possible.
Connect Your Learning
Move to the Connect Your Learning screen and have students respond to what they
see. You might start by talking about how the results of the voting in the launch might
help you make plans and decisions. Carry the conversation beyond just looking at the
best type of movie choice for the school party and look at future planning for which
using a ratio is more useful. Then point to other problems in the topic in which using
ratios to make predictions or decisions was discussed.
PART 1 (10 MIN) ______________________________________________________
Objective: Divide a given number of objects into two sets given the ratios of types of
objects in each set.
Author Intent
This problem presents students with the challenge of satisfying two ratios
simultaneously. Students will use their understanding of ratios and equivalent ratios to
solve the problem using guess and check. They will guess numbers to satisfy one ratio
and will then check if the quantities remaining satisfy the second ratio.
Questions for Understanding
Before presenting the solution:
• What are the ratios of fish required for the tanks? [2 spotted fish to 1 striped fish
for Class A and 3 striped fish to 1 spotted fish for Class B.]
• Can both classes have more spotted fish than striped fish? [No, the ratio for
Class A shows they want more spotted fish, and the ratio for Class B requires
more striped fish.]
Jay Says (Screen 1) Use Jay Says to clarify that there are lots of ways to satisfy
one ratio, but only one way to satisfy both ratios at the same time.
After completing the solution:
• How does this example help you make decisions about the fish tanks? [Sample
answer: Knowing the number of spotted and striped fish in each class’s tank
helps you to plan other purchases. Class A has more fish, so they will need a
bigger tank and more money for fish food.]
Solution Notes
This problem has an animated solution designed to optimize classroom time. It
demonstrates to students how to systematically use guess and check to find the
solution. Students should use guess and check to find the correct answer. Encourage
– grade 6 • Teacher Guide
them to methodically start with 2 spotted and 1 striped and then use equivalent ratios
with higher terms until they find the solution.
Some students may want to start by using the 1 : 3 ratio for Class B and then finding
the ratio of the remaining fish in Class A’s tank. Assure students that each method is
equally correct.
Differentiated Instruction
Students may struggle to translate “twice as many” to a ratio of 2 to 1 and “three times
as many” to a ratio of 3 to 1. You may want to make visuals for what “twice as many”
or “three times as many” looks like. For example, draw 4 circles on the whiteboard and
ask students what twice as many as 4 looks like. With a visual answer of 8 circles,
manipulate the numbers into the ratio 4 : 8 and rewrite as 1 : 2 or 2 : 1.
Error Prevention
Students may accidentally reverse Class B’s ratio. Remind students that the order of
the ratio is important and make sure students understand that the question asks them
to compare spotted fish to striped fish.
You may want to have students write ratios in lowest terms whenever possible.
Students may accidentally pass over the correct answer because the ratio of fish, 2 to
6, is not identical to 1 to 3. Remind students that they should be looking for equivalent
ratios, not just identical ratios.
Got It Notes
Choice A gives the correct ratio of soccer balls and footballs, but not the right
quantities. If students chose this answer, explain that the subsequent result for the
second bin is 16 soccer balls and 11 footballs, and 16 : 11 is not equivalent to 1 : 2.
Choice B gives the correct number of soccer balls and footballs for the second bin, but
not for the first bin. Students who choose this answer may have misread the question.
Choice C ignores the ratios given in the problem and is simply half the soccer balls
and half the footballs.
PART 2 (8 MIN) _______________________________________________________
Objective: Use equivalent ratios to scale up or down a set of amounts.
Author Intent
This problem extends students’ use of ratios to mixed numbers and fractions.
Fractions are common in ingredient amounts for recipes and other problems involving
ratios. Although these problems may seem more complicated, the same principles of
equivalent ratios based on multiplication and division apply. The problem and the first
Got It involve scaling up, while the second Got It involves scaling down.
Questions for Understanding
Before presenting the solution:
• How is this table similar to a multiplication table? [Each number in this table is
found by multiplying the number at the top of the column by the number at the
far left of the row, just like a multiplication table.]
Jay Says (Screen 1) Use the Jay Says button to reveal why maintaining the
same ratios is important.
While presenting the solution:
• What is different about equivalent ratios with mixed numbers and equivalent
ratios with whole numbers? [There is no difference because the rules of
multiplication and division do not change for mixed numbers.]
– grade 6 • Teacher Guide
After completing the solution:
1
• How many batches could you make if you have exactly 2 2 c of flour? [Sample
answers: You could make one batch because you do not have enough flour to
make a second batch. You could make one batch and then a smaller batch that
uses some of the flour left over.]
• Do you have to follow the ratios in the table to make pancakes that taste good?
[No, you could vary the ingredients slightly or add and subtract ingredients to
make your own recipe. However, if, for example, you add far too much salt or do
not include enough butter, then the pancakes will be ruined. It is important to
stick to a recipe that works.]
Solution Notes
On your whiteboard, you may want to add “x 2” between the first two columns of your
table as you complete it to show students that 2 batches just requires you to multiply
every ingredient by 2. You could similarly use arrows to explain the 3-batches column.
Differentiated Instruction
If students are uncomfortable with the measurements in this example, you may want to
bring in measuring cups and spoons as visuals. You can also show that 1 tbsp = 3 tsp.
Students will be introduced to conversion factors in the topic “Rates.”
Error Prevention
Encourage students to convert mixed numbers to improper fractions if necessary. You
may want to have students come up to the whiteboard and write the improper fractions
next to the corresponding mixed numbers.
Got It Notes
Choice A is the original amount. Choice B is the amount of butter. Remind students to
reread the question before answering. If students choose choice D, ask them if it is
reasonable for 4 batches with
2
3
c of milk in each batch to use 4 c of milk in all.
Got It 2 Notes
Although this Got It does not explicitly refer to the recipe in the Dynamic Example, be
prepared for students to want to use that table to solve this problem. Encourage
students to think generally about any recipe.
Going further: Some students may point out that if you halve some recipes, you will
have a recipe that requires half an egg. Ask students if they can think of a good way to
split an egg in half. One way is to crack it in a bowl, beat the egg, and use 2 tbsp. A
typical large egg is equivalent to 4 tbsp.
PART 3 (8 MIN) _______________________________________________________
Objective: Find two quantities given the ratio between the quantities and the total
value of the quantities.
Author Intent
This example reinforces ratio understanding using part-to-part and part-to-whole
representations and equivalent ratios in a contextual example. This example also
introduces the concept of proportions, which will be studied formally later.
Questions for Understanding
Before presenting the solution:
• What do equivalent ratios have to do with more or fewer batches of dressing?
[The ratio of tablespoons of oil to tablespoons of vinegar is equivalent no matter
how many batches you are making.]
– grade 6 • Teacher Guide
After completing the solution:
• Review your answer. Does it make sense? [Yes. There is more oil than vinegar,
and the total number of tablespoons adds up to 20.]
• How is this example related to the first one? [Both examples use guess and
check as a method to find the correct equivalent ratios and identify the unknown
value(s) of the problem.]
Solution Notes
Make sure you point out how the animated solution is systematic with its guess and
check strategy.
Students may be able to tell after the first line of the table that they need to multiply the
amounts of oil and vinegar by 4, since 5 is one fourth of the total tablespoons needed.
Acknowledge any student who makes this connection immediately and allow other
students to draw similar conclusions using guess and check.
Differentiated Instruction
Challenge: Lead a discussion about whether there is a strategy to guess more
efficiently. Students may suggest that since the first guess was not close to the desired
answer, it would be more efficient to try larger numbers. Students who figured out that
you can immediately scale the ratio by 4 may express that guess and check is not the
best way to solve this example.
Error Prevention
Students may miss the first step of finding the number of total tablespoons in one
batch and instead try to divide 20 by 2 or 3. Remind students that you must find an
ratio equivalent to 3 : 2. That equivalent ratio must add up to 20, the combined total.
Got It Notes
Choice A uses the original ratio 2 : 5. The total is not 21 pounds. Choices C and D
show a total of 21 pounds, but neither ratio is 2 : 5.
CLOSE AND CHECK (5 MIN) ______________________________________
Sample Answer to the Focus Question: You can use equivalent ratios to determine
how much of a certain quantity you need. This allows you to plan for many situations
such as adjusting the ingredient amounts to make a larger batch of a recipe or
determining the number of watermelons needed for a school picnic.
Other rich responses might include using a given ratio to make predictions about
another, equivalent ratio. Student answers should lean toward using proportional
reasoning to solve problems and focus on the importance of equivalence in ratio
problems. Expressing ratios as fractions and decimals is as much a representation of
equivalence as it is of comparison.
The Focus Question mentions making plans and decisions. Similarly, the Essential
Question of the topic begins, “Comparisons are helpful for making plans, predictions,
and decisions.” Emphasize the connection between the two with the questions below.
Connection Questions
• How has your experience with ratios in this topic prepared you to make plans
and decisions? [Sample answer: Given one ratio, I can create infinitely many
equivalent ratios. I can use whatever equivalent ratio I want to help me plan or
predict. It gives me options.]
• What do you think you will learn next about ratios? [Sample answer: In this topic,
we learned about the different forms of ratios. We have not learned when it is
sensible to use one form over another. I think we will find more uses for ratios in
real-world problems and learn which ratio form works best for a given situation.]
– grade 6 • Teacher Guide