rates patterns and problem solving

Problem Solving with Rates & Patterns

Algebra is a part of mathematics in which letters and other general symbols are used as a language to represent numbers and quantities in formulae and equations. Examples of Algebra can be as simple as “how many burgers do you need to make for five people, if everyone will eat two burgers” to advanced scientific mechanics to designing your smartphone. Algebra is the formula we use to build and discover everything around us.

For the following rate and ratio questions, use multiplication and division. As always, show your work and make sure you consistently include your units.

The solutions in the above animation are repeated with a more detailed explanation below.

Question 1.1

If there are 6 apples per packet, how many apples do we have if we have 3 packets of apples?

What do we know from the question? There are 6 apples PER packet. For these types of questions, it helps to think of that phrase as “6 apples per 1 packet”. A reminder, “per” also can be thought of as division or a “ / “

We want to know how many apples are in 3 packets, so we will use multiplication.

Note how the "packets" cancel each other out.

Question 1.2

If a bus is travelling at 45 miles per hour, how far will it travel in 4 hours?

Again we use multiplication. The bus is travelling at 45 miles per 1 hour or (45 Miles) ÷ (1 Hour)

Note again how the "hours" cancel each other out.

Question 1.3

A pumpkin pie has 12 slices and 4 people want to split it evenly. How many slices are there per person?

In this one we will have to use division. A hint to knowing this is in the question. It wants to know how many slices are there per person or slices/person.

We have 12 slices and we want to split (or divide) it amongst 4 people.

Question 1.4

If a runner travels 10 miles in one hour, how many minutes does its take per mile travelled?

We know, one hour = 60 minutes, and we want to know minutes per mile travelled or minutes/mile.

Question 2.1

A tap pours out 60 milliliters (or 60 ml) per second.

Create an equation that gives the volume, V (ml) , the tap has poured where the variable that you know is time, t (seconds).

For this equation we do not need to include units as the symbols in the question have been described with units already.

An equation that GIVES the volume, V . That means an equation for V or an equation that starts like:

We know that after one second we have the volume = 60 x 1 after two seconds the volume = 60 x 2 and so on. So for the equation we write

Question 2.2

How long does it take for the hose to pour a volume of 900 ml?

For this question, we know the volume, V , that we want and its now asking us for the time, t .

We use the equation above

Then substitute what we have been told from the question ( V=900 )

And then start to re-arrange to have an equation that gives us the value of t.

Divide both sides by 60

We will now get more into the fundamentals of Algebra as we start to increase the complexity of the relationship between sentences and words and symbolist that in equations.

A motorbike and a car are driving towards each other down a single track road that is 150 miles long. They start at each end of the road, 150 miles a part. The motorbike is travelling at 3 miles per minute and the car is travelling at 2 miles per minute.

We need to multiply the speed (in miles per minute) by the number of minutes. So:

i) After 1 minute: Motorbike's distance = 3 x 1 = 3 and car's distance = 2 x 1 = 2

ii) 8 minutes: Motorbike's distance = 3 x 8 = 24 and car's distance = 2 x 8 = 16

iii) 15 minutes: Motorbike's distance = 3 x 15 = 45 and car's distance = 2 x 15 = 30

iv) 20 minutes: Motorbike's distance = 3 x 20 = 60 and car's distance = 2 x 20 = 40

v) What is the total distance of the car and motorbike after 20 minutes?

Simply add the distance of the car and motorbike from iv)

v) 60 + 40 = 100

Let us develop these questions further:

Create equations for each distance in miles, d , that the motorbike and car have travelled over time in minutes, t and use them to calculate at what time they will meet.

First, find equations that gives us the value of distance, d , depending on the time passed, t .

Note: This is similar to question 2.1.

Truck: d = 3t

Car: d = 2t

Secondly, what must be true about the total distance travelled by the motorbike and car for them to meet?

Try thinking of the question as, “What must be true about the total distance travelled by the motorbike and car for them to meet in the 150 mile long road?”

If they have each started at each end of the road, 150 miles apart and meet at some point, the total they have both travelled can't be any more or any less than 150 miles.

The question isn't looking for how far the car and motorbike have travelled individually, it just needs to know the total distance travelled.

Another way of thinking about it would be to imagine the car and motorbike have met, and you knew the distance that the car had travelled and the motorbike had travelled. If you added the two distances together, what would it be? Yes - it would be 150 miles. We know that when the car and motorbike meet, the total distance travelled is equal to 150 miles.

How do we find out the total distance travelled by the car and motorbike? Add the two distances together. What's another way we can represent the distance the car and motorbike travel individually?

Truck, d = 3t and car, d = 2t .

Adding these two equations for distances together will give us

2t + 3t = total distance travelled (which we know also) = 150

2t + 3t = 150

So we know that the car and motorbike will meet at t = 30, or 30 minutes.

Fluency and Language

In particular, at this stage, fluency means understanding what is being asked and identifying whether to use multiplication or division.

Question 4.1

Tim is running at 120 steps per minute. If Tim runs for 6 minutes, how many steps will he have taken?

Start with writing what we know from the question.

Tim is running at 120 steps per minute. Remember “per” also means divide and it also helps to write minute as 1 minute, so we can write that statement as:

120steps / 1 minute

So in one minute, Tim runs 120 steps. The question asks us how many steps Tim takes in 6 minutes. Think of as Tim running 1 minute, 6 times. 6 times one minute. We now start to see that we are going to use MULTIPLICATION.

Writing out our work will look like:

Tim runs 120 steps / 1 minute for 6 minutes

With the minutes cancelling out (remember, anything divided by itself is equal to 1) we are left with,

120 steps x 6 = 720 steps

If you write down what you know from the question including all the units, and cancel out units through your working, you will always be left with the right unit.

Another way to figure out if we need to use multiplication or division is by looking ahead at the unit that the ANSWER must be in. Our answer is going to be a certain amount of steps, say that the number of steps we want is represented by the letter x .

So are answer will look like,

From what the questions gives us, we know we will be working with. i.e. 120 steps per minute or 120 steps / 1 minute and also 6 minutes.

Think about what we will have to do with this to end up with an answer that just has steps as the units. The minutes will have to cancel out, so we will have to multiply.

Question 4.2

David bought 6 bags of candy. Each bag contains 35 pieces of candy per bag. How many pieces of candy in total does he have?

Start with writing what the question has given us:

And each bag has

35 pieces per 1 bag, or 35 pieces / 1 bag or 35 pieces / bag

He has 1 bag, 6 times.

Times that by 6 bags, will look like

(35 pieces / 1 bag) x 6 bags =

Remember, the unit of our answer will have to be in pieces. The answer gives us 35 pieces/bag and 6 bags, to have the unit pieces remaining, the bags must cancel out.

Question 5.1

Sam has 28 treats that he wants to divide evenly for his 4 cats. He distributes the treats evenly amongst 4 bowls. How many treats are there per bowl?

Write what we know from the question.

There are 28 treats.

There are 4 bowls.

The question says Sam wants to DIVIDE the treats evenly, which gives us a hint that we will use division.

We can also use the extra Tip regarding units. We want the answer in treats per bowl or treats / bowl. The question gives us 28 treats and 4 bowls, to get the unit of treats per bowl (treats/ bowl) we will have to divide the amount of treats by the amount of bowls.

Question 5.2

Oleg is selling paintings for $100 each. If he has sold 7 paintings, how much money has he made?

This time the units given are not as obvious.

Oleg is selling paintings for $100 each it may help to rewrite this as:

$100 per painting

$100/ painting

If it helps, you can also write it as 100 dollars / painting as long as you stay consistent with that unit and then change the “dollars” back to $ at the end. For now we will stick with the $.

The question also gives us that Oleg sold 7 paintings

The questions asks us how much he makes from selling 7 paintings, so the unit the answer will be in will be $ or dollars. Again, the units we want are not as obvious from the question.

Oleg sold 7 paintings at $100 / painting

Oleg made $700

If there are 6 buildings in a block, and 10 apartments in a building, and 4 rooms in an apartment, then how many rooms are in a block?

This question is worded slightly differently, but we still use the same method we've been using. This is where we learn to change the language we speak into the language of algebra.

Write what we know from the question

6 buildings in a block

10 apartments in a building

4 rooms in an apartment

And we want to know how many rooms are in a block.

If we first start by changing “in a” to “per” and “per to “/” that will help visualize the math that we have to do.

6 buildings in a block → 6 buildings per 1 block → 6 buildings / 1 block

10 apartments in a building → 10 apartments per 1 building → 10 apartments / 1 building

4 rooms in an apartment → 4rooms per 1 apartment → 4 rooms / 1 apartment

We want to know how many rooms are in a block, so rewrite as:

→ rooms per 1 block → rooms / 1 block

First, work out how many rooms are in a building

4 rooms per apartment and 10 apartments per building,

So one building has 10 apartments with 4 rooms per apartment.

(apartment/ apartment cancelling out)

(10 x 4rooms) / building

40 rooms / building

40 rooms per building,

40 rooms in a building.

Now, use the same method for amount of rooms per block

6 buildings / block and 40 rooms / building,

(buildings/ building cancelling out)

= (6 x 40 rooms) / block =

240 rooms / block or,

240 rooms per block or,

240 rooms in a block

If we have a strong understanding of this and once we are familiar with the methods, we can do it all at once

(6 buildings / block) x (10 apartments / building) x (4 rooms / apartment) =

With buildings x apartments on top cancelling out with building x apartment below.

(6 x 10 x 4 x rooms) / block =

240 rooms / block

Also, if we use our extra Tip method, we will have to approach this one differently. Looking at the units of the answer we want, we have rooms / block, so we will want rooms as the numerator and block as the denominator.

Just because the unit of the answer is rooms/block does not necessarily mean we will use division. Looking at what the question gives us, we have the units buildings/block, apartments/building, and rooms/apartment.

Notice, we already have here rooms as the numerator and blocks as the dominator, and if we were to use division, it would switch this around. Notice also that if all of those units were multiplied together, apartments and buildings would be cancelled out, leaving us with rooms/ block. This hinting indicates the method that we should use.

Another way to help is to use a visual:

Question 7.1

Tim is running at a speed of 9 miles per hour. In 2.5 hours how far will he have run?

Write what we know from the question:

Tim running at 9 miles per hour or, 9 miles / hour.

Tim runs for 2.5 hours. So,

For 2.5 hours, Tim runs at 9 miles / hour

(hours/hour cancels out)

2.5 x 9 miles = 22.5 miles

Question 7.2

Mr. Lahey is a teacher who wants to know how many students he will have at each of his classroom tables in his class of 35 students. He has 7 tables. How many students will be at each table, evenly distributed?

Write what we know,

35 students

We want to know how many students will be at each table. Again this is where we learn to turn our spoken language into the language of algebra. Students at each table means students per table, which means students/table. So divide:

35 students / 7 tables =

5 students / table or 5 students per table.

Question 7.3

Lucy can do 60 push ups in 4 minutes. If she has been doing her push ups at a constant rate, how many push ups will she have done in 1 minute?

So, we know Lucy can do push ups at a rate of 60 push in 4 minutes.

This is the same as 60 push ups per 4 minutes, or 60 push ups / 4 minutes.

We want to know how many push ups she will have done in 1 minute.

Although the rate is per 4 minutes and not what you are probably used to i.e. per 1 minute, we still follow the same methods.

Take the rate and then multiply by the time,

The minutes on top cancels the minutes on the bottom, we are left with

60 push ups / 4 which is:

15 push ups

Another way to think of it is, if we know she does 60 push ups in 4 minutes, and we want to know how many push ups she can do in 1 minute, we can divide 4 minutes by 4, to give us 1 minute. If we do this we must also divide the 60 push ups by 4. Which gives us 15 push ups.

How To Solve Rate Problems: A Step-By-Step Guide For Math Students

How do you solve a problem involving rates.

To solve a problem involving rates, follow these steps:

1. Identify what you are trying to find: Before you begin solving a problem involving rates, you need to identify what you are trying to find. This will help guide you in setting up the problem.

2. Write down the formula: Once you have identified what you are trying to find, write down the formula for the rate you need to calculate. This formula may be given to you in the problem or you may need to derive it from the information given.

3. Plug in the values: Now that you have the formula, plug in the values given in the problem. Make sure to match units (e.g., time, distance, weight) so that they cancel out appropriately.

4. Solve for the unknown: Once you have plugged in all the values, solve for the unknown variable (the one you’re trying to find).

5. Check your answer: Always double-check your work to ensure that your answer makes sense and is reasonable given the context of the problem.

6. Convert units: Finally, convert your answer to the required units if necessary.

More Answers:

Recent posts, ramses ii a prominent pharaoh and legacy of ancient egypt.

Ramses II (c. 1279–1213 BCE) Ramses II, also known as Ramses the Great, was one of the most prominent and powerful pharaohs of ancient Egypt.

Formula for cyclic adenosine monophosphate & Its Significance

Is the formula of cyclic adenosine monophosphate (cAMP) $ce{C_{10}H_{11}N_{5}O_{6}P}$ or $ce{C_{10}H_{12}N_{5}O_{6}P}$? Does it matter? The correct formula for cyclic adenosine monophosphate (cAMP) is $ce{C_{10}H_{11}N_{5}O_{6}P}$. The

Development of a Turtle Inside its Egg

How does a turtle develop inside its egg? The development of a turtle inside its egg is a fascinating process that involves several stages and

The Essential Molecule in Photosynthesis for Energy and Biomass

Why does photosynthesis specifically produce glucose? Photosynthesis is the biological process by which plants, algae, and some bacteria convert sunlight, carbon dioxide (CO2), and water

How the Human Body Recycles its Energy Currency

Source for “The human body recycles its body weight of ATP each day”? The statement that “the human body recycles its body weight of ATP

Don't Miss Out! Sign Up Now!

Sign up now to get started for free!

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

1.3: Problem Solving Strategies

• Last updated
• Save as PDF
• Page ID 9823

• Michelle Manes
• University of Hawaii

Think back to the first problem in this chapter, the ABC Problem. What did you do to solve it? Even if you did not figure it out completely by yourself, you probably worked towards a solution and figured out some things that did not work.

Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.

How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985. [1]

George Pólya, circa 1973

• Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY 2.0 ( http://creativecommons.org/licenses/by/2.0 )], via Wikimedia Commons ↵

In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

• First, you have to understand the problem.
• After understanding, then make a plan.
• Carry out the plan.
• Look back on your work. How could it be better?

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

We have already seen one problem solving strategy, which we call “Wishful Thinking.” Do not be afraid to change the problem! Ask yourself “what if” questions:

• What if the picture was different?
• What if the numbers were simpler?
• What if I just made up some numbers?

You need to be sure to go back to the original problem at the end, but wishful thinking can be a powerful strategy for getting started.

This brings us to the most important problem solving strategy of all:

Problem Solving Strategy 2 (Try Something!).

If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate. You need to just try something! Put pencil to paper (or stylus to screen or chalk to board or whatever!) and try something. This is often an important step in understanding the problem; just mess around with it a bit to understand the situation and figure out what is going on.

And equally important: If what you tried first does not work, try something else! Play around with the problem until you have a feel for what is going on.

Last week, Alex borrowed money from several of his friends. He finally got paid at work, so he brought cash to school to pay back his debts. First he saw Brianna, and he gave her 1/4 of the money he had brought to school. Then Alex saw Chris and gave him 1/3 of what he had left after paying Brianna. Finally, Alex saw David and gave him 1/2 of what he had remaining. Who got the most money from Alex?

Think/Pair/Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem? This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.

Problem Solving Strategy 3 (Draw a Picture).

Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric, like this one, thinking visually can help! Can you represent something in the situation by a picture?

Draw a square to represent all of Alex’s money. Then shade 1/4 of the square — that’s what he gave away to Brianna. How can the picture help you finish the problem?

After you have worked on the problem yourself using this strategy (or if you are completely stuck), you can watch someone else’s solution.

Problem Solving Strategy 4 (Make Up Numbers).

Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!

You can work forwards: Assume Alex had some specific amount of money when he showed up at school, say $100. Then figure out how much he gives to each person. Or you can work backwards: suppose he has some specific amount left at the end, like $10. Since he gave Chris half of what he had left, that means he had $20 before running into Chris. Now, work backwards and figure out how much each person got.

Watch the solution only after you tried this strategy for yourself.

If you use the “Make Up Numbers” strategy, it is really important to remember what the original problem was asking! You do not want to answer something like “Everyone got $10.” That is not true in the original problem; that is an artifact of the numbers you made up. So after you work everything out, be sure to re-read the problem and answer what was asked!

(Squares on a Chess Board)

How many squares, of any possible size, are on a 8 × 8 chess board? (The answer is not 64... It’s a lot bigger!)

Remember Pólya’s first step is to understand the problem. If you are not sure what is being asked, or why the answer is not just 64, be sure to ask someone!

Think / Pair / Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem, even if you have not solved it completely?

It is clear that you want to draw a picture for this problem, but even with the picture it can be hard to know if you have found the correct answer. The numbers get big, and it can be hard to keep track of your work. Your goal at the end is to be absolutely positive that you found the right answer. You should never ask the teacher, “Is this right?” Instead, you should declare, “Here’s my answer, and here is why I know it is correct!”

Problem Solving Strategy 5 (Try a Simpler Problem).

Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said: “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3?

Of course the ultimate goal is to solve the original problem. But working with smaller boards might give you some insight and help you devise your plan (that is Pólya’s step (2)).

Problem Solving Strategy 6 (Work Systematically).

If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.

For example, in this problem you might keep track of how many 1 × 1 squares are on each board, how many 2 × 2 squares on are each board, how many 3 × 3 squares are on each board, and so on. You could keep track of the information in a table:

Problem Solving Strategy 7 (Use Manipulatives to Help You Investigate).

Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!

For example, in this problem it can be difficult to keep track of which squares you have already counted. You might want to cut out 1 × 1 squares, 2 × 2 squares, 3 × 3 squares, and so on. You can actually move the smaller squares across the chess board in a systematic way, making sure that you count everything once and do not count anything twice.

Problem Solving Strategy 8 (Look for and Explain Patterns).

Sometimes the numbers in a problem are so big, there is no way you will actually count everything up by hand. For example, if the problem in this section were about a 100 × 100 chess board, you would not want to go through counting all the squares by hand! It would be much more appealing to find a pattern in the smaller boards and then extend that pattern to solve the problem for a 100 × 100 chess board just with a calculation.

If you have not done so already, extend the table above all the way to an 8 × 8 chess board, filling in all the rows and columns. Use your table to find the total number of squares in an 8 × 8 chess board. Then:

• Describe all of the patterns you see in the table.
• Can you explain and justify any of the patterns you see? How can you be sure they will continue?
• What calculation would you do to find the total number of squares on a 100 × 100 chess board?

(We will come back to this question soon. So if you are not sure right now how to explain and justify the patterns you found, that is OK.)

(Broken Clock)

This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. ( Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15.)

Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2.)

Remember that your first step is to understand the problem. Work out what is going on here. What are the sums of the numbers on each piece? Are they consecutive?

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What progress have you made?

Problem Solving Strategy 9 (Find the Math, Remove the Context).

Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.

In this case, worrying about the clock and exactly how the pieces break is less important than worrying about finding consecutive numbers that sum to the correct total. Ask yourself:

• What is the sum of all the numbers on the clock’s face?
• Can I find two consecutive numbers that give the correct sum? Or four consecutive numbers? Or some other amount?
• How do I know when I am done? When should I stop looking?

Of course, solving the question about consecutive numbers is not the same as solving the original problem. You have to go back and see if the clock can actually break apart so that each piece gives you one of those consecutive numbers. Maybe you can solve the math problem, but it does not translate into solving the clock problem.

Problem Solving Strategy 10 (Check Your Assumptions).

When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?

In the clock problem, because the first solution has the clock broken radially (all three pieces meet at the center, so it looks like slicing a pie), many people assume that is how the clock must break. But the problem does not require the clock to break radially. It might break into pieces like this:

Were you assuming the clock would break in a specific way? Try to solve the problem now, if you have not already.

Fun teaching resources & tips to help you teach math with confidence

Math Strategies: Problem Solving by Finding a Pattern

One important math concept that children begin to learn and apply in elementary school is reading and using a table. This is essential knowledge, because we encounter tables of data all the time in our everyday lives! But it’s not just important that kids can read and answer questions based on information in a table, it’s also important that they know how to create their own table and then use it to solve problems, find patterns, graph equations, and so on. And while some may think of these as two different things, I think problem solving by making a table and finding a pattern go hand in hand!

–>Pssst! Do your kids need help making sense of and solving word problems? You might like this set of editable word problem solving templates ! Use these with any grade level, for any type of word problem :

Finding Patterns in Math Problems: 

So when should kids use problem solving by finding a pattern ? Well, when the problem gives a set of data, or a pattern that is continuing and can be arranged in a table, it’s good to consider looking for the pattern and determining the “rule” of the pattern.

As I mentioned when I discussed problem solving by making a list , finding a pattern can be immensely helpful and save a lot of time when working on a word problem. Sometimes, however, a student may not recognize the pattern right away, or may get bogged down with all the details of the question.

Setting up a table and filling in the information given in the question is a great way to organize things and provide a visual so that the “rule” of the pattern can be determined. The “rule” can then be used to find the answer to the question. This removes the tedious work of completing a table, which is especially nice if a lot of computation is involved.

But a table is also great for kids who struggle with math, because it gives them a way to get to the solution even if they have a hard time finding the pattern, or aren’t confident that they are using the “rule” correctly.

Because even though using a known pattern can save you time, and eliminate the need to fill out the entire table, it’s not necessary. A student who is unsure could simply continue filling out their table until they reach the solution they’re looking for.

Helping students learn how to set up a table is also helpful because they can use it to organize information (much like making a list) even if there isn’t a pattern to be found, because it can be done in a systematic way, ensuring that nothing is left out.

If your students are just learning how to read and create tables, I would suggest having them circle their answer in the table to show that they understood the question and knew where in the table to find the answer.

If you have older students, encourage them to find a pattern in the table and explain it in words , and then also with mathematical symbols and/or an equation. This will help them form connections and increase number sense. It will also help them see how to use their “rule” or equation to solve the given question as well as make predictions about the data.

It’s also important for students to consider whether or not their pattern will continue predictably . In some instances, the pattern may look one way for the first few entries, then change, so this is important to consider as the problems get more challenging.

There are tons of examples of problems where creating a table and finding a pattern is a useful strategy, but here’s just one example for you:

Ben decides to prepare for a marathon by running ten minutes a day, six days a week. Each week, he increases his time running by two minutes per day. How many minutes will he run in week 8?

Included in the table is the week number (we’re looking at weeks 1-8), as well as the number of minutes per day and the total minutes for the week. The first step is to fill in the first couple of weeks by calculating the total time.

Once you’ve found weeks 1-3, you may see a pattern and be able to calculate the total minutes for week 8. For example, in this case, the total number of minutes increases by 12 each week, meaning in week 8 he will run for 144 minutes.

If not, however, simply continue with the table until you get to week 8, and then you will have your answer.

I think it is especially important to make it clear to students that it is perfectly acceptable to complete the entire table (or continue a given table) if they don’t see or don’t know how to use the pattern to solve the problem.

I was working with a student once and she was given a table, but was then asked a question about information not included in that table . She was able to tell me the pattern she saw, but wasn’t able to correctly use the “rule” to find the answer. I insisted that she simply extend the table until she found what she needed. Then I showed her how to use the “rule” of the pattern to get the same answer.

I hope you find this helpful! Looking for and finding patterns is such an essential part of mathematics education! If you’re looking for more ideas for exploring patterns with younger kids, check out this post for making patterns with Skittles candy .

And of course, don’t miss the other posts in this Math Problem Solving Series:

• Problem Solving by Solving an Easier Problem
• Problem Solving by Drawing a Picture
• Problem Solving by Working Backwards
• Problem Solving by Making a List

One Comment

I had so much trouble spotting patterns when I was in school. Fortunately for her, my daughter rocks at it! This technique will be helpful for her when she’s a bit older! #ThoughtfulSpot

Comments are closed.

Similar Posts

Quadratic equations project (with free printables).

Summer Math Camp: Multiplication!

3 Simple Ideas for Using Math Games In the Classroom

Investigating Exponent Properties {FREE Lesson!}

Pattern Puzzles to Build Algebraic Thinking | DIGITAL Puzzles

“Build a Turkey” Number Sense Activity {FREE Printable}

Find more resources to help make math engaging, join 165k+ parents & teachers.

Who learn new tips and strategies, as well as receive engaging resources to make math fun!

• Privacy Policy

Math Time Doesn't Have to End in Tears

Join 165,000+ parents and teachers who learn new tips and strategies, as well as receive engaging resources to make math fun. Plus, receive my guide, "5 Games You Can Play Today to Make Math Fun," as my free gift to get you started!

How to master the seven-step problem-solving process

In this episode of the McKinsey Podcast , Simon London speaks with Charles Conn, CEO of venture-capital firm Oxford Sciences Innovation, and McKinsey senior partner Hugo Sarrazin about the complexities of different problem-solving strategies.

Podcast transcript

Simon London: Hello, and welcome to this episode of the McKinsey Podcast , with me, Simon London. What’s the number-one skill you need to succeed professionally? Salesmanship, perhaps? Or a facility with statistics? Or maybe the ability to communicate crisply and clearly? Many would argue that at the very top of the list comes problem solving: that is, the ability to think through and come up with an optimal course of action to address any complex challenge—in business, in public policy, or indeed in life.

Looked at this way, it’s no surprise that McKinsey takes problem solving very seriously, testing for it during the recruiting process and then honing it, in McKinsey consultants, through immersion in a structured seven-step method. To discuss the art of problem solving, I sat down in California with McKinsey senior partner Hugo Sarrazin and also with Charles Conn. Charles is a former McKinsey partner, entrepreneur, executive, and coauthor of the book Bulletproof Problem Solving: The One Skill That Changes Everything [John Wiley & Sons, 2018].

Charles and Hugo, welcome to the podcast. Thank you for being here.

Hugo Sarrazin: Our pleasure.

Charles Conn: It’s terrific to be here.

Simon London: Problem solving is a really interesting piece of terminology. It could mean so many different things. I have a son who’s a teenage climber. They talk about solving problems. Climbing is problem solving. Charles, when you talk about problem solving, what are you talking about?

Charles Conn: For me, problem solving is the answer to the question “What should I do?” It’s interesting when there’s uncertainty and complexity, and when it’s meaningful because there are consequences. Your son’s climbing is a perfect example. There are consequences, and it’s complicated, and there’s uncertainty—can he make that grab? I think we can apply that same frame almost at any level. You can think about questions like “What town would I like to live in?” or “Should I put solar panels on my roof?”

You might think that’s a funny thing to apply problem solving to, but in my mind it’s not fundamentally different from business problem solving, which answers the question “What should my strategy be?” Or problem solving at the policy level: “How do we combat climate change?” “Should I support the local school bond?” I think these are all part and parcel of the same type of question, “What should I do?”

I’m a big fan of structured problem solving. By following steps, we can more clearly understand what problem it is we’re solving, what are the components of the problem that we’re solving, which components are the most important ones for us to pay attention to, which analytic techniques we should apply to those, and how we can synthesize what we’ve learned back into a compelling story. That’s all it is, at its heart.

I think sometimes when people think about seven steps, they assume that there’s a rigidity to this. That’s not it at all. It’s actually to give you the scope for creativity, which often doesn’t exist when your problem solving is muddled.

Simon London: You were just talking about the seven-step process. That’s what’s written down in the book, but it’s a very McKinsey process as well. Without getting too deep into the weeds, let’s go through the steps, one by one. You were just talking about problem definition as being a particularly important thing to get right first. That’s the first step. Hugo, tell us about that.

Hugo Sarrazin: It is surprising how often people jump past this step and make a bunch of assumptions. The most powerful thing is to step back and ask the basic questions—“What are we trying to solve? What are the constraints that exist? What are the dependencies?” Let’s make those explicit and really push the thinking and defining. At McKinsey, we spend an enormous amount of time in writing that little statement, and the statement, if you’re a logic purist, is great. You debate. “Is it an ‘or’? Is it an ‘and’? What’s the action verb?” Because all these specific words help you get to the heart of what matters.

Want to subscribe to The McKinsey Podcast ?

Simon London: So this is a concise problem statement.

Hugo Sarrazin: Yeah. It’s not like “Can we grow in Japan?” That’s interesting, but it is “What, specifically, are we trying to uncover in the growth of a product in Japan? Or a segment in Japan? Or a channel in Japan?” When you spend an enormous amount of time, in the first meeting of the different stakeholders, debating this and having different people put forward what they think the problem definition is, you realize that people have completely different views of why they’re here. That, to me, is the most important step.

Charles Conn: I would agree with that. For me, the problem context is critical. When we understand “What are the forces acting upon your decision maker? How quickly is the answer needed? With what precision is the answer needed? Are there areas that are off limits or areas where we would particularly like to find our solution? Is the decision maker open to exploring other areas?” then you not only become more efficient, and move toward what we call the critical path in problem solving, but you also make it so much more likely that you’re not going to waste your time or your decision maker’s time.

How often do especially bright young people run off with half of the idea about what the problem is and start collecting data and start building models—only to discover that they’ve really gone off half-cocked.

Hugo Sarrazin: Yeah.

Charles Conn: And in the wrong direction.

Simon London: OK. So step one—and there is a real art and a structure to it—is define the problem. Step two, Charles?

Charles Conn: My favorite step is step two, which is to use logic trees to disaggregate the problem. Every problem we’re solving has some complexity and some uncertainty in it. The only way that we can really get our team working on the problem is to take the problem apart into logical pieces.

What we find, of course, is that the way to disaggregate the problem often gives you an insight into the answer to the problem quite quickly. I love to do two or three different cuts at it, each one giving a bit of a different insight into what might be going wrong. By doing sensible disaggregations, using logic trees, we can figure out which parts of the problem we should be looking at, and we can assign those different parts to team members.

Simon London: What’s a good example of a logic tree on a sort of ratable problem?

Charles Conn: Maybe the easiest one is the classic profit tree. Almost in every business that I would take a look at, I would start with a profit or return-on-assets tree. In its simplest form, you have the components of revenue, which are price and quantity, and the components of cost, which are cost and quantity. Each of those can be broken out. Cost can be broken into variable cost and fixed cost. The components of price can be broken into what your pricing scheme is. That simple tree often provides insight into what’s going on in a business or what the difference is between that business and the competitors.

If we add the leg, which is “What’s the asset base or investment element?”—so profit divided by assets—then we can ask the question “Is the business using its investments sensibly?” whether that’s in stores or in manufacturing or in transportation assets. I hope we can see just how simple this is, even though we’re describing it in words.

When I went to work with Gordon Moore at the Moore Foundation, the problem that he asked us to look at was “How can we save Pacific salmon?” Now, that sounds like an impossible question, but it was amenable to precisely the same type of disaggregation and allowed us to organize what became a 15-year effort to improve the likelihood of good outcomes for Pacific salmon.

Simon London: Now, is there a danger that your logic tree can be impossibly large? This, I think, brings us onto the third step in the process, which is that you have to prioritize.

Charles Conn: Absolutely. The third step, which we also emphasize, along with good problem definition, is rigorous prioritization—we ask the questions “How important is this lever or this branch of the tree in the overall outcome that we seek to achieve? How much can I move that lever?” Obviously, we try and focus our efforts on ones that have a big impact on the problem and the ones that we have the ability to change. With salmon, ocean conditions turned out to be a big lever, but not one that we could adjust. We focused our attention on fish habitats and fish-harvesting practices, which were big levers that we could affect.

People spend a lot of time arguing about branches that are either not important or that none of us can change. We see it in the public square. When we deal with questions at the policy level—“Should you support the death penalty?” “How do we affect climate change?” “How can we uncover the causes and address homelessness?”—it’s even more important that we’re focusing on levers that are big and movable.

Would you like to learn more about our Strategy & Corporate Finance Practice ?

Simon London: Let’s move swiftly on to step four. You’ve defined your problem, you disaggregate it, you prioritize where you want to analyze—what you want to really look at hard. Then you got to the work plan. Now, what does that mean in practice?

Hugo Sarrazin: Depending on what you’ve prioritized, there are many things you could do. It could be breaking the work among the team members so that people have a clear piece of the work to do. It could be defining the specific analyses that need to get done and executed, and being clear on time lines. There’s always a level-one answer, there’s a level-two answer, there’s a level-three answer. Without being too flippant, I can solve any problem during a good dinner with wine. It won’t have a whole lot of backing.

Simon London: Not going to have a lot of depth to it.

Hugo Sarrazin: No, but it may be useful as a starting point. If the stakes are not that high, that could be OK. If it’s really high stakes, you may need level three and have the whole model validated in three different ways. You need to find a work plan that reflects the level of precision, the time frame you have, and the stakeholders you need to bring along in the exercise.

Charles Conn: I love the way you’ve described that, because, again, some people think of problem solving as a linear thing, but of course what’s critical is that it’s iterative. As you say, you can solve the problem in one day or even one hour.

Charles Conn: We encourage our teams everywhere to do that. We call it the one-day answer or the one-hour answer. In work planning, we’re always iterating. Every time you see a 50-page work plan that stretches out to three months, you know it’s wrong. It will be outmoded very quickly by that learning process that you described. Iterative problem solving is a critical part of this. Sometimes, people think work planning sounds dull, but it isn’t. It’s how we know what’s expected of us and when we need to deliver it and how we’re progressing toward the answer. It’s also the place where we can deal with biases. Bias is a feature of every human decision-making process. If we design our team interactions intelligently, we can avoid the worst sort of biases.

Simon London: Here we’re talking about cognitive biases primarily, right? It’s not that I’m biased against you because of your accent or something. These are the cognitive biases that behavioral sciences have shown we all carry around, things like anchoring, overoptimism—these kinds of things.

Both: Yeah.

Charles Conn: Availability bias is the one that I’m always alert to. You think you’ve seen the problem before, and therefore what’s available is your previous conception of it—and we have to be most careful about that. In any human setting, we also have to be careful about biases that are based on hierarchies, sometimes called sunflower bias. I’m sure, Hugo, with your teams, you make sure that the youngest team members speak first. Not the oldest team members, because it’s easy for people to look at who’s senior and alter their own creative approaches.

Hugo Sarrazin: It’s helpful, at that moment—if someone is asserting a point of view—to ask the question “This was true in what context?” You’re trying to apply something that worked in one context to a different one. That can be deadly if the context has changed, and that’s why organizations struggle to change. You promote all these people because they did something that worked well in the past, and then there’s a disruption in the industry, and they keep doing what got them promoted even though the context has changed.

Simon London: Right. Right.

Hugo Sarrazin: So it’s the same thing in problem solving.

Charles Conn: And it’s why diversity in our teams is so important. It’s one of the best things about the world that we’re in now. We’re likely to have people from different socioeconomic, ethnic, and national backgrounds, each of whom sees problems from a slightly different perspective. It is therefore much more likely that the team will uncover a truly creative and clever approach to problem solving.

Simon London: Let’s move on to step five. You’ve done your work plan. Now you’ve actually got to do the analysis. The thing that strikes me here is that the range of tools that we have at our disposal now, of course, is just huge, particularly with advances in computation, advanced analytics. There’s so many things that you can apply here. Just talk about the analysis stage. How do you pick the right tools?

Charles Conn: For me, the most important thing is that we start with simple heuristics and explanatory statistics before we go off and use the big-gun tools. We need to understand the shape and scope of our problem before we start applying these massive and complex analytical approaches.

Simon London: Would you agree with that?

Hugo Sarrazin: I agree. I think there are so many wonderful heuristics. You need to start there before you go deep into the modeling exercise. There’s an interesting dynamic that’s happening, though. In some cases, for some types of problems, it is even better to set yourself up to maximize your learning. Your problem-solving methodology is test and learn, test and learn, test and learn, and iterate. That is a heuristic in itself, the A/B testing that is used in many parts of the world. So that’s a problem-solving methodology. It’s nothing different. It just uses technology and feedback loops in a fast way. The other one is exploratory data analysis. When you’re dealing with a large-scale problem, and there’s so much data, I can get to the heuristics that Charles was talking about through very clever visualization of data.

You test with your data. You need to set up an environment to do so, but don’t get caught up in neural-network modeling immediately. You’re testing, you’re checking—“Is the data right? Is it sound? Does it make sense?”—before you launch too far.

Simon London: You do hear these ideas—that if you have a big enough data set and enough algorithms, they’re going to find things that you just wouldn’t have spotted, find solutions that maybe you wouldn’t have thought of. Does machine learning sort of revolutionize the problem-solving process? Or are these actually just other tools in the toolbox for structured problem solving?

Charles Conn: It can be revolutionary. There are some areas in which the pattern recognition of large data sets and good algorithms can help us see things that we otherwise couldn’t see. But I do think it’s terribly important we don’t think that this particular technique is a substitute for superb problem solving, starting with good problem definition. Many people use machine learning without understanding algorithms that themselves can have biases built into them. Just as 20 years ago, when we were doing statistical analysis, we knew that we needed good model definition, we still need a good understanding of our algorithms and really good problem definition before we launch off into big data sets and unknown algorithms.

Simon London: Step six. You’ve done your analysis.

Charles Conn: I take six and seven together, and this is the place where young problem solvers often make a mistake. They’ve got their analysis, and they assume that’s the answer, and of course it isn’t the answer. The ability to synthesize the pieces that came out of the analysis and begin to weave those into a story that helps people answer the question “What should I do?” This is back to where we started. If we can’t synthesize, and we can’t tell a story, then our decision maker can’t find the answer to “What should I do?”

Simon London: But, again, these final steps are about motivating people to action, right?

Charles Conn: Yeah.

Simon London: I am slightly torn about the nomenclature of problem solving because it’s on paper, right? Until you motivate people to action, you actually haven’t solved anything.

Charles Conn: I love this question because I think decision-making theory, without a bias to action, is a waste of time. Everything in how I approach this is to help people take action that makes the world better.

Simon London: Hence, these are absolutely critical steps. If you don’t do this well, you’ve just got a bunch of analysis.

Charles Conn: We end up in exactly the same place where we started, which is people speaking across each other, past each other in the public square, rather than actually working together, shoulder to shoulder, to crack these important problems.

Simon London: In the real world, we have a lot of uncertainty—arguably, increasing uncertainty. How do good problem solvers deal with that?

Hugo Sarrazin: At every step of the process. In the problem definition, when you’re defining the context, you need to understand those sources of uncertainty and whether they’re important or not important. It becomes important in the definition of the tree.

You need to think carefully about the branches of the tree that are more certain and less certain as you define them. They don’t have equal weight just because they’ve got equal space on the page. Then, when you’re prioritizing, your prioritization approach may put more emphasis on things that have low probability but huge impact—or, vice versa, may put a lot of priority on things that are very likely and, hopefully, have a reasonable impact. You can introduce that along the way. When you come back to the synthesis, you just need to be nuanced about what you’re understanding, the likelihood.

Often, people lack humility in the way they make their recommendations: “This is the answer.” They’re very precise, and I think we would all be well-served to say, “This is a likely answer under the following sets of conditions” and then make the level of uncertainty clearer, if that is appropriate. It doesn’t mean you’re always in the gray zone; it doesn’t mean you don’t have a point of view. It just means that you can be explicit about the certainty of your answer when you make that recommendation.

Simon London: So it sounds like there is an underlying principle: “Acknowledge and embrace the uncertainty. Don’t pretend that it isn’t there. Be very clear about what the uncertainties are up front, and then build that into every step of the process.”

Hugo Sarrazin: Every step of the process.

Simon London: Yeah. We have just walked through a particular structured methodology for problem solving. But, of course, this is not the only structured methodology for problem solving. One that is also very well-known is design thinking, which comes at things very differently. So, Hugo, I know you have worked with a lot of designers. Just give us a very quick summary. Design thinking—what is it, and how does it relate?

Hugo Sarrazin: It starts with an incredible amount of empathy for the user and uses that to define the problem. It does pause and go out in the wild and spend an enormous amount of time seeing how people interact with objects, seeing the experience they’re getting, seeing the pain points or joy—and uses that to infer and define the problem.

Simon London: Problem definition, but out in the world.

Hugo Sarrazin: With an enormous amount of empathy. There’s a huge emphasis on empathy. Traditional, more classic problem solving is you define the problem based on an understanding of the situation. This one almost presupposes that we don’t know the problem until we go see it. The second thing is you need to come up with multiple scenarios or answers or ideas or concepts, and there’s a lot of divergent thinking initially. That’s slightly different, versus the prioritization, but not for long. Eventually, you need to kind of say, “OK, I’m going to converge again.” Then you go and you bring things back to the customer and get feedback and iterate. Then you rinse and repeat, rinse and repeat. There’s a lot of tactile building, along the way, of prototypes and things like that. It’s very iterative.

Simon London: So, Charles, are these complements or are these alternatives?

Charles Conn: I think they’re entirely complementary, and I think Hugo’s description is perfect. When we do problem definition well in classic problem solving, we are demonstrating the kind of empathy, at the very beginning of our problem, that design thinking asks us to approach. When we ideate—and that’s very similar to the disaggregation, prioritization, and work-planning steps—we do precisely the same thing, and often we use contrasting teams, so that we do have divergent thinking. The best teams allow divergent thinking to bump them off whatever their initial biases in problem solving are. For me, design thinking gives us a constant reminder of creativity, empathy, and the tactile nature of problem solving, but it’s absolutely complementary, not alternative.

Simon London: I think, in a world of cross-functional teams, an interesting question is do people with design-thinking backgrounds really work well together with classical problem solvers? How do you make that chemistry happen?

Hugo Sarrazin: Yeah, it is not easy when people have spent an enormous amount of time seeped in design thinking or user-centric design, whichever word you want to use. If the person who’s applying classic problem-solving methodology is very rigid and mechanical in the way they’re doing it, there could be an enormous amount of tension. If there’s not clarity in the role and not clarity in the process, I think having the two together can be, sometimes, problematic.

The second thing that happens often is that the artifacts the two methodologies try to gravitate toward can be different. Classic problem solving often gravitates toward a model; design thinking migrates toward a prototype. Rather than writing a big deck with all my supporting evidence, they’ll bring an example, a thing, and that feels different. Then you spend your time differently to achieve those two end products, so that’s another source of friction.

Now, I still think it can be an incredibly powerful thing to have the two—if there are the right people with the right mind-set, if there is a team that is explicit about the roles, if we’re clear about the kind of outcomes we are attempting to bring forward. There’s an enormous amount of collaborativeness and respect.

Simon London: But they have to respect each other’s methodology and be prepared to flex, maybe, a little bit, in how this process is going to work.

Hugo Sarrazin: Absolutely.

Simon London: The other area where, it strikes me, there could be a little bit of a different sort of friction is this whole concept of the day-one answer, which is what we were just talking about in classical problem solving. Now, you know that this is probably not going to be your final answer, but that’s how you begin to structure the problem. Whereas I would imagine your design thinkers—no, they’re going off to do their ethnographic research and get out into the field, potentially for a long time, before they come back with at least an initial hypothesis.

Want better strategies? Become a bulletproof problem solver

Hugo Sarrazin: That is a great callout, and that’s another difference. Designers typically will like to soak into the situation and avoid converging too quickly. There’s optionality and exploring different options. There’s a strong belief that keeps the solution space wide enough that you can come up with more radical ideas. If there’s a large design team or many designers on the team, and you come on Friday and say, “What’s our week-one answer?” they’re going to struggle. They’re not going to be comfortable, naturally, to give that answer. It doesn’t mean they don’t have an answer; it’s just not where they are in their thinking process.

Simon London: I think we are, sadly, out of time for today. But Charles and Hugo, thank you so much.

Charles Conn: It was a pleasure to be here, Simon.

Hugo Sarrazin: It was a pleasure. Thank you.

Simon London: And thanks, as always, to you, our listeners, for tuning into this episode of the McKinsey Podcast . If you want to learn more about problem solving, you can find the book, Bulletproof Problem Solving: The One Skill That Changes Everything , online or order it through your local bookstore. To learn more about McKinsey, you can of course find us at McKinsey.com.

Charles Conn is CEO of Oxford Sciences Innovation and an alumnus of McKinsey’s Sydney office. Hugo Sarrazin is a senior partner in the Silicon Valley office, where Simon London, a member of McKinsey Publishing, is also based.

Explore a career with us

Related articles.

Strategy to beat the odds

Five routes to more innovative problem solving

In mathematics, a pattern is a sequence of numbers , shapes , or objects that follow a certain rule or repetition. Patterns can be found in many different areas of math, including arithmetic , geometry , and algebra.

Number Patterns

Number patterns are sequences of numbers that follow a specific rule or operation. For example, in the sequence 2, 4, 6, 8, 10, the pattern is adding 2 to the previous number. Recognizing number patterns can help in understanding basic arithmetic operations and in solving more complex mathematical problems.

Shape Patterns

In geometry , patterns can also be found in shapes and figures. For example, a pattern of triangles , squares , and circles can repeat in a sequence. Recognizing and extending shape patterns can help in developing spatial reasoning and understanding geometric properties.

Identifying and Extending Patterns

Identifying and extending patterns is an important skill in mathematics. Students are often asked to identify the core of a pattern and extend it by predicting the next numbers or shapes in the sequence. This helps in developing critical thinking and problem-solving skills.

Using Patterns in Problem Solving

Patterns can also be used to solve problems in mathematics. By recognizing and understanding patterns, students can make predictions and test their hypotheses, leading to a deeper understanding of mathematical concepts and relationships.

Overall, patterns play a significant role in mathematics, helping students develop logical thinking, problem-solving skills, and a deeper understanding of mathematical concepts.

[Patterns] Related Worksheets and Study Guides:

• Syllable Patterns/Word Families English Language Arts • Third Grade
• Skip Counting Mathematics • Third Grade
• Weather Science • Third Grade
• Number Words to 1,000 Mathematics • Third Grade
• R Controlled Vowels English Language Arts • Third Grade
• Sequencing Mathematics • First Grade

Read More...

◂ Math Worksheets and Study Guides Third Grade. Patterns

The resources above cover the following skills:

• Download and Print thousands of standards-based ELA, Social Study, Science and Math Worksheets and Study Guides!
• Terms of Use
• Privacy Policy
• Membership Benefits
• Completing Worksheets Online
• Share to Google Classroom
• NewPathLearning

Kaizen is about changing the way things are. If you assume that things are all right the way they are, you can’t do kaizen. So change something! —Taiichi Ohno

Inspect and Adapt

Inspect & adapt: overview.

The Inspect and Adapt (I&A) is a significant event held at the end of each PI, where the current state of the Solution is demonstrated and evaluated. Teams then reflect and identify improvement backlog items via a structured problem-solving workshop.

The Agile Manifesto emphasizes the importance of continuous improvement through the following principle: “At regular intervals, the team reflects on how to become more effective, then tunes and adjusts its behavior accordingly.”

In addition, SAFe includes ‘relentless improvement’ as one of the four SAFe Core Values as well as a dimension of the Continuous Learning Culture core competency. While opportunities to improve can and should occur continuously throughout the PI (e.g., Iteration Retrospectives ), applying some structure, cadence, and synchronization helps ensure that there is also time set aside to identify improvements across multiple teams and Agile Release Trains .

All ART stakeholders participate along with the Agile Teams in the I&A event. The result is a set of improvement backlog items that go into the ART Backlog for the next PI Planning event. In this way, every ART improves every PI. A similar I&A event is held by Solution Trains .

The I&A event consists of three parts:

PI System Demo

• Quantitative and qualitative measurement
• Retrospective and problem-solving workshop

Participants in the I&A should be, wherever possible, all the people involved in building the solution. For an ART, this includes:

• The Agile teams
• Release Train Engineer (RTE)
• System and Solution Architects
• Product Management ,  Business Owners , and other stakeholders

Additionally, Solution Train stakeholders may also attend this event.

The PI System Demo is the first part of the I&A, and it’s a little different from the regular system demos after every iteration. This demo shows all the Features the ART has developed during the PI. Typically the audience is broader; for example, Customers or Portfolio representatives are more likely to attend this demo. Therefore, the PI system demo tends to be a little more formal, and extra preparation and setup are usually required. But like any other system demo, it should be timeboxed to an hour or less, with the level of abstraction high enough to keep stakeholders actively engaged and providing feedback.

Before or as part of the PI system demo, Business Owners collaborate with each Agile Team to score the actual business value achieved for each of their Team PI Objectives , as illustrated in Figure 1.

The achievement score is calculated by separately totaling the business value for the plan and actual columns. The uncommitted objectives are not included in the total plan. However, they are part of the total actual. Then divide the actual total by the planned total to calculate the achievement score illustrated in Figure 1.

Quantitative and Qualitative Measurement

In the second part of the I&A event, teams collectively review any quantitative and qualitative metrics they have agreed to collect, then discuss the data and trends. In preparation for this, the RTE and the Solution Train Engineer are often responsible for gathering the information, analyzing it to identify potential issues, and facilitating the presentation of the findings to the ART.

Each team’s planned vs. actual business value is rolled up to create the ART predictability measure, as shown in Figure 2.

Reliable trains should operate in the 80–100 percent range; this allows the business and its external stakeholders to plan effectively. (Note: Uncommitted objectives are excluded from the planned commitment. However, they are included in the actual business value achievement, as can also be seen in Figure 1.)

Retrospective

The teams then run a brief (30 minutes or less) retrospective to identify a few significant issues they would like to address during the problem-solving workshop . There is no one way to do this; several different Agile retrospective formats can be used [3].

Based on the retrospective and the nature of the problems identified, the facilitator helps the group decide which issues they want to tackle. Each team may work on a problem, or, more typically, new groups are formed from individuals across different teams who wish to work on the same issue. This self-selection helps provide cross-functional and differing views of the problem and brings together those impacted and those best motivated to address the issue.

Key ART stakeholders—including Business Owners, customers, and management—join the retrospective and problem-solving workshop teams. The Business Owners can often unblock the impediments outside the team’s control.

Problem-Solving Workshop

The ART holds a structured, root-cause problem-solving workshop to address systemic problems. Root cause analysis provides a set of problem-solving tools used to identify the actual causes of a problem rather than just fixing the symptoms. The RTE typically facilitates the session in a timebox of two hours or less.

Figure 3 illustrates the steps in the problem-solving workshop.

The following sections describe each step of the process.

Agree on the Problem(s) to Solve

American inventor Charles Kettering is credited with saying that “a problem well stated is a problem half solved.” At this point, the teams have self-selected the problem they want to address. But do they agree on the details of the problem, or is it more likely that they have differing perspectives? To this end, the teams should spend a few minutes clearly stating the problem, highlighting the ‘what,’ ‘where,’ ‘when,’ and ‘impact’ as concisely as possible. Figure 4 illustrates a well-written problem statement.

Perform Root Cause Analysis

Effective problem-solving tools include the fishbone diagram and the ‘5 Whys.’ Also known as an Ishikawa Diagram , a fishbone diagram is a visual tool to explore the causes of specific events or sources of variation in a process. Figure 5 illustrates the fishbone diagram with a summary of the previous problem statement written at the head of the ‘fish.’

For our problem-solving workshop, the main bones often start with the default categories of people, processes, tools, program, and environment. However, these categories should be adapted as appropriate.

Team members then brainstorm causes that they think contribute to solving the problem and group them into these categories. Once a potential cause is identified, its root cause is explored with the 5 Whys technique. By asking ‘why’ five times, the cause of the previous cause is uncovered and added to the diagram. The process stops once a suitable root cause has been identified, and the same process is then applied to the next cause.

Identify the Biggest Root Cause

Pareto Analysis, also known as the 80/20 rule, is used to narrow down the number of actions that produce the most significant overall effect. It uses the principle that 20 percent of the causes are responsible for 80 percent of the problem. It’s beneficial when many possible courses of action compete for attention, which is almost always the case with complex, systemic issues.

Once all the possible causes-of-causes are identified, team members then cumulatively vote on the item they think is the most significant factor contributing to the original problem. They can do this by dot voting. For example, each person gets five votes to choose one or more causes they think are most problematic. The team then summarizes the votes in a Pareto chart, such as the example in Figure 6, which illustrates their collective consensus on the most significant root cause.

Restate the New Problem

The next step is to pick the cause with the most votes and restate it clearly as a problem. Restating it should take only a few minutes, as the teams clearly understand the root cause.

Brainstorm Solutions

At this point, the restated problem will start to imply some potential solutions. The team brainstorms as many possible corrective actions as possible within a fixed timebox (about 15–30 minutes). The rules of brainstorming apply here:

• Generate as many ideas as possible
• Do not allow criticism or debate
• Let the imagination soar
• Explore and combine ideas

Create Improvement Backlog Items

The team then cumulatively votes on up to three most viable solutions. These potential solutions are written as improvement stories and features, planned in the following PI Planning event. During that event, the RTE helps ensure that the relevant work needed to deliver the identified improvements is planned. This approach closes the loop, thus ensuring that action will be taken and that people and resources are dedicated as necessary to improve the current state.

Following this practice, problem-solving becomes routine and systematic, and team members and ART stakeholders can ensure that the train is solidly on its journey of relentless improvement.

Inspect and Adapt for Solution Trains

The above describes a rigorous approach to problem-solving in the context of a single ART. If the ART is part of a Solution Train, the I&A event will often include key stakeholders from the Solution Train. In larger value streams, however, an additional Solution Train I&A event may be required, following the same format.

Due to the number of people in a Solution Train, attendees at the large solution I&A event cannot include everyone, so stakeholders are selected that are best suited to address the problems. This subset of people consists of the Solution Train’s primary stakeholders and representatives from the various ARTs and Suppliers .

Last update: 22 January 2023

Privacy Overview

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Praxis Core Math

Course: praxis core math   >   unit 1.

• Rational number operations | Lesson
• Rational number operations | Worked example

Ratios and proportions | Lesson

• Ratios and proportions | Worked example
• Percentages | Lesson
• Percentages | Worked example
• Rates | Lesson
• Rates | Worked example
• Naming and ordering numbers | Lesson
• Naming and ordering numbers | Worked example
• Number concepts | Lesson
• Number concepts | Worked example
• Counterexamples | Lesson
• Counterexamples | Worked example
• Pre-algebra word problems | Lesson
• Pre-algebra word problems | Worked example
• Unit reasoning | Lesson
• Unit reasoning | Worked example

What are ratios and proportions?

What skills are tested.

• Identifying and writing equivalent ratios
• Solving word problems involving ratios
• Solving word problems using proportions

How do we write ratios?

• The ratio of lemon juice to sugar is a part-to-part ratio. It compares the amount of two ingredients.
• The ratio of lemon juice to lemonade is a part-to-whole ratio. It compares the amount of one ingredient to the sum of all ingredients.
• Determine whether the ratio is part to part or part to whole.
• Calculate the parts and the whole if needed.
• Plug values into the ratio.
• Simplify the ratio if needed. Integer-to-integer ratios are preferred.
• 1 5 ‍   of the students on the varsity soccer team are lower-level students.
• 1 ‍   in 5 ‍   students on the varsity soccer team are lower-level students.

How do we use proportions?

• Write an equation using equivalent ratios.
• Plug in known values and use a variable to represent the unknown quantity.
• If the numeric part of one ratio is a multiple of the corresponding part of the other ratio, we can calculate the unknown quantity by multiplying the other part of the given ratio by the same number.
• If the relationship between the two ratios is not obvious, solve for the unknown quantity by isolating the variable representing it.
• (Choice A)   1 : 4 ‍   A 1 : 4 ‍  
• (Choice B)   1 : 2 ‍   B 1 : 2 ‍  
• (Choice C)   1 : 1 ‍   C 1 : 1 ‍  
• (Choice D)   2 : 1 ‍   D 2 : 1 ‍  
• (Choice E)   4 : 1 ‍   E 4 : 1 ‍  
• (Choice A)   1 6 ‍   A 1 6 ‍  
• (Choice B)   1 3 ‍   B 1 3 ‍  
• (Choice C)   2 5 ‍   C 2 5 ‍  
• (Choice D)   1 2 ‍   D 1 2 ‍  
• (Choice E)   2 3 ‍   E 2 3 ‍  
• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  

Things to remember

Want to join the conversation.

• Upvote Button navigates to signup page
• Downvote Button navigates to signup page
• Flag Button navigates to signup page