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In this unit, your student will be learning about analyzing data. Statistics can help us recognize trends in what is typical, as well as how far from typical we can expect to go.

For example, imagine you discover that one of your student's childhood toys is actually worth a lot of money from toy collectors. What is a good price for the toy, and where should you sell it?

You find two sites where you could auction the toy: one is a site for toy collectors and one is a general online site for selling items. After doing some research, you come across two histograms showing the price when other people sold this toy, as well as some statistics about the prices.

General site:

Collector site:

The first thing you may notice is that one of the toys on the collector site sold for between $250 and $260. An extremely high value like this can be called an "outlier" and should be investigated to understand why it is so different. In this case, the toy was in its original box and signed by the toy maker. Although we don't usually like to throw out data, it doesn't make sense to include that value when comparing to our toy, so you might take out that value to get new statistics. This changes the average price at the collector site to \(\) $150.51 and the standard deviation to $18.65.

Next, you might notice that the prices for the general site are more spread out than at the collector site. This spread can be represented with a value called the "standard deviation." The general site has a much larger standard deviation, indicating its prices are more spread out. For you, that means you are more likely to sell our toy near the average price on the collector site while the general site is more likely to sell significantly higher or lower than average.
Here is a task to try with your student:
A football coach is considering adding one of two running backs to the team. Some statistics for the number of yards gained on each run for each player are given. Which player should the coach choose? Explain your reasoning.

Running back A:

Running back B:
Solution:
Either running back can be a good choice for the team depending on what the coach is looking for.

Running back A gets more yards per run on average, but is a lot more variable (based on the standard deviation). This means Player A sometimes has really long runs and sometimes very short (or even negative) runs. Running back A will likely be more exciting to watch, but could also be frustrating when they don't get the needed yards.

Running back B gets fewer yards per run on average, but is a lot less variable (based on the standard deviation). This means that Player B is more consistent and gets close to 4 yards every play they are allowed to run. Running back B will likely be less exciting to watch, but can be relied on to get consistent gains.

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Illustrative Mathematics Algebra 1, Unit 1 - Families | Kendall Hunt

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In this unit, your student will be learning about analyzing data. Statistics can help us recognize trends in what is typical, as well as how far from typical we can expect to go. For example, imagine you discover that one of your student's childhood toys is actually worth a lot of money from toy collectors.

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In this unit, your student will analyze constraints on different quantities. For example, the amount you spend on a bicycle may be limited by how much you have saved. To qualify for a sports team, you may need to practice at least a certain number of hours, or lift at least a certain number of pounds.

Here are some ways to write constraints using mathematical notation:

\(w < 20\). An apartment building only allows dogs that weigh less than 20 pounds.

\(m + g + b = 4\). A casserole recipe calls for four cups of vegetables. You have mushrooms, green beans, and broccoli.

\(12.5c + 15a \geq 1,\!000\). In order for a concert to be performed, the artists need to be sure of $1,000 in ticket sales. Tickets for children under 18 are $12.50, and tickets for adults are $15.

\(5n+10d=150\). You need $1.50 in coins for a parking meter. You have a bunch of nickels and dimes in your pocket.

Each point on the graph represents a combination of nickels and dimes that totals $1.50. For example, if you use 8 nickels, you will need 11 dimes.
Here is a task for you to try with your student:
Priya is saving money to go on an overnight school trip. The cost of the trip is $360. She has a job at a convenience store, which pays $9 per hour, and sometimes babysits for a family in her neighborhood, which pays $12 per hour.

The equation \(9x+12y=360\) represents all the combinations of hours Priya could work at each job and earn a total of $360. Here is a graph showing those combinations:

Solution:

1. Priya works 20 hours at the convenience store and 15 hours babysitting.

2. Point \(B\): \((32,6)\). Priya works 32 hours at the convenience store and 6 hours babysitting. Point \(C\): \((40,0)\). Priya works 40 hours at the convenience store and does not babysit at all.

3. Priya does not make enough money. She works 24 hours at the convenience store and 8 hours babysitting. She makes only $312, since \(24 \cdot 9 + 8 \cdot 12 = 312\).

4. Priya makes more than enough money: $438. She works 30 hours at the convenience store and 14 hours babysitting. \(30 \cdot 9 + 14 \cdot 12 = 438\).

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Illustrative Mathematics Algebra 1, Unit 2 - Families | Kendall Hunt

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In this unit, your student will analyze constraints on different quantities. For example, the amount you spend on a bicycle may be limited by how much you have saved. To qualify for a sports team, you may need to practice at least a certain number of hours, or lift at least a certain number of pounds.

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In this unit, students learn about two-way tables and use them to determine if two categories have an association. For example, would you predict that a forecast of rain over a school is associated with the number of students wearing rain boots? What association do you think the rain has with students' eye colors?

It is possible for two things to have no association, as you may have guessed for rain and students' eye colors. With your student, make predictions about associations found in daily life. For example, do these pairs share an association?

length of time a plant spends in sunlight and its growth

the size of a car and the amount of gas it takes to be full

number of open apps on your mobile phone and battery percentage

What other associations can you think of together?

Students can use tables and collected data to determine if two things are associated. One type of table is a two-way table, which organizes two categorical variables. A categorical variable is a variable that takes on values which can be divided into groups or categories. For example, color is a categorical variable which can take on values like red, blue, or green. In the table, you may notice that it has a total of four categories, but only two categorical variables (hand dominance and fruit preference).

With your student, examine the data collected from 1,914 other students.

1. Predict if there is an association between hand dominance and fruit preference.

2. Complete the table with the totals for each column and row.

3. Among the students who are left-handed, the proportion who prefer pineapples is about 0.57, since \(66 \div 116=0.57\). This means that about 57% of students in this group who are left-handed prefer pineapples over mangoes. What proportion of those who are left-handed prefer mangoes?

4. What proportion of those who are right-handed prefer mangoes?

5. Is there a significant difference between the proportion of left-handed students who prefer mangoes and the proportion of right-handed students who prefer mangoes?

6. Was your prediction accurate? Is there an association between hand dominance and fruit preference?

Solution

1. Sample response: I think there shouldn't be any association between hand dominance and fruit preference, since neither should influence the other.

2. 0.43, because \(50 \div 116=0.43\) or \(1 - 0.57 = 0.43\)

3. 0.46, because \(826 \div1,798=0.46\)

4. No, there is no significant difference between left-handed students who prefer mangoes and right-handed students who prefer mangoes, because 0.43 and 0.46 are close in value.

5. Sample response: I predicted there would be no association, and I think that the math supports my prediction. No, there is no association between hand dominance and fruit preference.

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Illustrative Mathematics Algebra 1, Unit 3 - Families | Kendall Hunt

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In this unit, students learn about two-way tables and use them to determine if two categories have an association. For example, would you predict that a forecast of rain over a school is associated with the number of students wearing rain boots? What association do you think the rain has with students' eye colors?

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In this unit, students learn about functions, building on their work in middle school. A function is a relationship between an input and an output, where for every input there is exactly one output. Here are some examples of functions:

The relationship between a name (input) and the number of letters in it (output). If the input is "Maya Angelou," the only possible output is 11.

The relationship between the number of seconds since an oven was turned on (input) and the temperature in the oven (output). For example, 50 seconds after the oven was turned on, the temperature in the oven was 124 degrees Fahrenheit.

We often use the phrase "(output) is a function of (input)" to express how the input and output sets are related. For example, "the number of letters in a name is a function of the name," or "the temperature in the oven is a function of time since it was turned on."

To make it easier to talk about and work with functions, we often use letters to name them, and we use function notation to represent their input and output.

Suppose \(f\) is a function that tells us the distance, in feet, that a child ran over time, \(t\), in seconds. So: \(f\) is the name of the function, time is the input, and distance is the output.

Here is how we represent this information in function notation:

​​​​​​

A function can also be represented with a graph. Here is a graph of function \(f\).

Sometimes a rule tells us what to do to the input of a function to get the output.

Suppose function \(g\) gives the dollar cost of buying \(x\) burritos at $5 each. To get the output (the cost), we multiply the input (the number of burritos) by 5. We can write: \(g(x) = 5x\).
Here is a task to try with your student:
The height of a plant in centimeters is a function of its height in inches, \(h\).

1. Let \(p\) represent this function. Name the input and output of \(p\).

2. What does \(p(10)\) mean in this situation? What about \(p(h)= 50.8\)?

3. Because there are 2.54 centimeters in 1 inch, a rule that defines \(p\) is \(p(h)=2.54h\). What is the value of \(p(10)\)?

4. What is the value of \(h\) when \(p(h)\) is 50.8?

Solution:

1. The input is height in inches.The output is height in centimeters.

2. \(p(10)\) represents the plant's height in centimeters when its height is 10 inches. \(p(h)= 50.8\) tells us that the plant is 50.8 centimeters tall when its height is \(h\) inches.

3. 25.4 centimeters. (\(p(10)=2.54(10)=25.4\))

4. 20 inches

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Illustrative Mathematics Algebra 1, Unit 4 - Families | Kendall Hunt

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In this unit, students learn about functions, building on their work in middle school. A function is a relationship between an input and an output, where for every input there is exactly one output. Here are some examples of functions: The relationship between a name (input) and the number of letters in it (output).

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In this unit, your student is introduced to exponential relationships. Earlier, your student studied what mathematicians call linear relationships, where they start with a quantity and add or subtract the same amount repeatedly. In an exponential relationship, they start with a quantity and multiply by the same amount repeatedly.

Exponential relationships are represented by equations in the form \(y=a\boldcdot b^x\), where \(a\) is the quantity you start with, \(b\) is the growth factor that you are going to multiply it by, and \(x\) is how many times you are going to multiply by \(b\). If \(b\) is bigger than 1, the amount is growing and if \(b\) is less than 1, the amount is shrinking. When \(b\) is equal to 1, the amount is staying the same.

If you start with 50 bees in your apiary (bee garden), and the number of bees doubles each year, how many bees would you have in 5 years? Let \(y\) represent the number of bees and \(x\) represent time in years. The starting amount is 50 bees and the multiplier is 2.

\(\begin{align} y &= a\boldcdot b^x \\ &= 50\boldcdot2^5 \\ &= 50\boldcdot2\boldcdot2\boldcdot2\boldcdot2\boldcdot2\\ &= 1,\!600 \text{ bees after 5 years} \end{align}\)

While multiplication works well for a situation like this, where we are multiplying by 2 five times, a graph can be a useful tool. If you wanted to know how many bees you would have in 10 years, you could graph \(y = 50\boldcdot 2^x\) and see how many bees there are in 10 years.

Graphing is especially helpful when looking a long way into the future or when you want to know when something will happen, like when the bee population will reach 1 million.

Here is a task to try with your student: Florida is having a problem with a toxic green algae that is floating on their waterways, contaminating the water and killing the marine life. Kiran lives on a small lake in south Florida. One day he noticed the algae floating on a 3 square meter area of the lake. A month later, the algae had doubled in size, growing to 6 square meters.

1. If the doubling pattern continues, how many square meters of the lake will be covered in algae in 4 months?

2. If the surface area of the lake is about 1,500 square meters, after how many months will the entire lake be covered?

Solution: 1. This can be solved using a variety of strategies. You could use a table, an equation, or a graph.

Let \(x\) represent the time in months and \(y\) represent the area in square meters. \(y=3\boldcdot2^x\). Substitute 4 for \(x\) and solve for \(y\), which yields 48.

2. Just as with part A, there are several strategies for finding when the algae will cover the whole lake. Extending the graph, adding the graph of \(y = 1,\!500\), and finding where they intersect is a nice way to find the month. In just under 9 months, the algae will cover 1,500 square meters of the lake.

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Illustrative Mathematics Algebra 1, Unit 5 - Families | Kendall Hunt

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In this unit, your student is introduced to exponential relationships. Earlier, your student studied what mathematicians call linear relationships, where they start with a quantity and add or subtract the same amount repeatedly. In an exponential relationship, they start with a quantity and multiply by the same amount repeatedly.

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In this unit, students learn about quadratic functions. Earlier, they learned about linear functions that grow by repeatedly adding or subtracting the same amount and exponential functions that grow by repeatedly multiplying by the same amount.

Quadratic functions also change in a predictable way. Here, the number of small squares in each step is increasing by 3, then 5, then 7, and so on. How many squares are in Step 10? How many in Step \(n\)?

Here is a table that shows the pattern.

​​​​​​

Both of the following expressions give the ball's distance above the ground: \(5x(2-x)\) and \(10x-5x^2\), where \(x\) represents the number of seconds since it was thrown. Quadratic expressions are most recognizable when you can see the "squared term," \(\text-5x^2\), as shown in \(10x-5x^2\).

Your student will learn even more about quadratics in the next unit.

Here is a task to try with your student:
The equation \(h=1+25t^2-5t^2\) models the height in meters of a model rocket \(t\) seconds after it is launched in the air. Here is a graph representing the equation.

1. What was the height of the rocket above the ground at the time it was launched?

2. How high did it go up in the air?

3. When did the rocket land back on the ground?

Solution:

1. 1 meter

2. about 32 meters

3. a little more than 5 seconds after launch

Video Lesson Summaries

Here are the video lesson summaries for Algebra 1, Unit 6: Introduction to Quadratic Functions. Each video highlights key concepts and vocabulary that students learn across one or more lessons in the unit. The content of these video lesson summaries is based on the written Lesson Summaries found at the end of lessons in the curriculum. The goal of these videos is to support students in reviewing and checking their understanding of important concepts and vocabulary. Here are some possible ways families can use these videos:

Keep informed on concepts and vocabulary students are learning about in class.

Watch with their student and pause at key points to predict what comes next or think up other examples of vocabulary terms (the bolded words).

Consider following the Connecting to Other Units links to review the math concepts that led up to this unit or to preview where the concepts in this unit lead to in future units.

Video 1 Video 2 Video 3 Video 4 Video 5 Video 6 Connecting to Other Units

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Illustrative Mathematics Algebra 1, Unit 6 - Families | Kendall Hunt

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In this unit, students learn about quadratic functions. Earlier, they learned about linear functions that grow by repeatedly adding or subtracting the same amount and exponential functions that grow by repeatedly multiplying by the same amount. Quadratic functions also change in a predictable way.

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In this unit, your student will be learning how to solve quadratic equations using several methods. In the previous unit, students saw how quadratic functions can represent a variety of situations such as the height of a ball thrown into the air over time.

The graph shows that the ball is 10 feet above the ground at about 0.28 seconds, and again at about 2.22 seconds after being thrown.

To learn how to solve these more complicated equations, students first reason about solving equations like \(x^2=9\) or \((x-1)^2=9\). Can you figure out the solutions to these equations?

You probably noticed that one solution to \(x^2=9\) is 3 because \(3^2=9\). Also, -3 is a solution because \((\text-3)^2\) is also equal to 9. By similar reasoning, the solutions to \((x-1)^2=9\) are 4 and -2. You can check those solutions because \(4-1=3\) and \(\text-2-1=\text-3\).

Later in the unit, your student will learn to rewrite expressions to quickly find the values that make an equation equal to 0. A diagram can be useful. Here is a diagram showing \(x^2+3x\) is equal to \(x(x+3)\).

That means solutions to the equation \(x^2+3x=0\) are the same as the solutions to the equation \(x(x+3)=0\). Can you "see" from the second equation that the solutions are 0 and -3?

By the end of the unit, students have learned that the quadratic formula that can be used to find the exact solutions to any quadratic equation.
Here is a task to try with your student:
Solve the equation \(x^2-4x+3=0\) in two ways.

\(\displaystyle \begin {align} x^2 - 4x +3&=0\\x^2-4x+4 &=1\\(x-2)^2 &= 1\end {align}\)
Solution:

1. \((x-1)(x-3)=0\) and the solutions are \(x=1\) and \(x=3\).

2. One solution is \(x=1\) because \((1-2)^2=(\text-1)^2\), which equals 1. The other solution is \(x=3\) because \((3-1)^2=(1)^2\), which also equals 1.

Video Lesson Summaries

Here are the video lesson summaries for Algebra 1, Unit 7: Quadratic Equations. Each video highlights key concepts and vocabulary that students learn across one or more lessons in the unit. The content of these video lesson summaries is based on the written Lesson Summaries found at the end of lessons in the curriculum. The goal of these videos is to support students in reviewing and checking their understanding of important concepts and vocabulary. Here are some possible ways families can use these videos:

Keep informed on concepts and vocabulary students are learning about in class.

Watch with their student and pause at key points to predict what comes next or think up other examples of vocabulary terms (the bolded words).

Consider following the Connecting to Other Units links to review the math concepts that led up to this unit or to preview where the concepts in this unit lead to in future units.

Video 1 Video 2 Video 3 Video 4 Video 5 Video 6 Connecting to Other Units

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Illustrative Mathematics Algebra 1, Unit 7 - Families | Kendall Hunt

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In this unit, your student will be learning how to solve quadratic equations using several methods. In the previous unit, students saw how quadratic functions can represent a variety of situations such as the height of a ball thrown into the air over time.