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Alignments to Content Standards: 7.RP.A

Task

Four different stores are having a sale. The signs below show the discounts available at each of the four stores.

1. Which of these four different offers gives the biggest price reduction? Explain your reasoning clearly.

2. Which of these four different offers gives the smallest price reduction? Explain your reasoning clearly.

IM Commentary

The Standards for Mathematical Practice focus on the nature of the learning experiences by attending to the thinking processes and habits of mind that students need to develop in order to attain a deep and flexible understanding of mathematics. Certain tasks lend themselves to the demonstration of specific practices by students. The practices that are observable during exploration of a task depend on how instruction unfolds in the classroom. While it is possible that tasks may be connected to several practices, the commentary will spotlight one practice connection in depth. Possible secondary practice connections may be discussed but not in the same degree of detail.

The purpose of this task is to engage students in Standard for Mathematical Practice 4, "Model with mathematics," and as such, the question as it is worded cannot be answered without making some assumptions. For example, if the items that are purchased do not have the same value, then the price reduction depends on the cost of the items. The answer also depends on how you interpret the meaning of "price reduction" which could be either the absolute reduction or the relative reduction. Consider the four scenarios for purchasing pairs of shoes below.

"Two for the price of one"

"Three for the price of two"

Which has the greatest price reduction? It depends, and a complete answer to this question requires a mathematical argument beyond the expectations of 7th grade. On the other hand, students need opportunities to evaluate the relative savings of advertised sales, so realizing that the best sale depends on what you are buying is a good insight to develop. The solutions below assume that you are comparing the sales for purchasing items of the same price.

It is also worth pointing out that there is a very important, although non-mathematical, issue related to whether a particular sale will save you money: you do not save money by buying things you do not need. So, for example, 3 for the price of 2 is not a better deal than buy one get the second at 25% off if you do not need three of the item.l.

The teacher might use this task after formally teaching 7.RP.1-3. Students could be given the task and asked to collaborate in small groups to solve the questions posed using all the formal instruction on ratio and proportional reasoning. The teacher might ask questions such as; "What if the price of each item is different? Does that change which discount is biggest?" "What if the price of each item is the same? Does that make a difference in which discount is biggest?" Depending on the level of the students, the teacher could direct the students by giving them a specific value or use a more abstract approach by having them solve using a variable. The students could share out their findings and compare/contrast the answers and discuss why the results vary.

Solutions

Solution: Starting with a specific value

Assume that you are comparing the sales for purchasing items of the same price (it is a much harder question to answer if you don't). Let's first look at the answer for a specific value. Suppose the regular price for all items is $60. Then the following table shows how much you will pay per item.

In general, suppose that a single item costs $x$ dollars.

So "Two for the price of one" gives the biggest price reduction per item but the total savings is the same as the "Three for the price of two" sale.

Also, "Buy one and get 25% off the second" has both the highest price per item and the lowest total savings, which means it offers the smallest price reduction.

Solution: A more abstract approach

Assume we are comparing the sales for buying identically priced items and that we are comparing the reduction in price per item (as opposed to price for the entire purchase). The sale price (per item) is the total cost divided by the number of items. The reduction in price is equal to the original price minus the sale price. We need to calculate the price reduction for every case in order to answer the two questions.

Since we don't know what the regular price per item is let's just call it p.
Two for the price of one:
The reduction is: $ p-\frac{1p}{2} = \frac{1}{2}p$
Buy one and get 25% off the second:
The reduction is: $p -\frac{(1+0.75)p}{2} = p-\frac{1.75p}{2} = p-\frac{7/4}{2}p = p -\frac{7}{4}\cdot \frac{1}{2}p = p-\frac{7}{8}p = \frac{1}{8}p$
Buy two and get 50% off the second one:
The reduction is: $ p-\frac{(1+0.50)p}{2} = p-\frac{1.5p}{2} = p -\frac{3/2}{2}p = p -\frac{3}{2}\cdot \frac{1}{2}p = p-\frac{3}{4}p =\frac{1}{4}p$
Three for the price of two:
The reduction is: $ p-\frac{2p}{3} = \frac{1}{3}p$

Now we can answer the questions

Which of these four different offers gives the biggest price reduction?

Two for the price of one with a price reduction of one half.

Which of these four different offers gives the smallest price reduction?

Buy one and get 25% off the second with a price reduction of one eighth.

# data[].attributes.attachment_link.description
Alignments to Content Standards: 7.RP.A Four different stores are having a sale. The signs below show the discounts available at each of the four stores. Which of these four different offers gives the biggest price reduction? Explain your reasoning clearly. Which of these four different offers gives the smallest price reduction?

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http://tasks.illustrativemathematics.org/content-standards/6/RP/A/tasks/2157

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http://tasks.illustrativemathematics.org/content-standards/6/RP/A/tasks/2051

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http://tasks.illustrativemathematics.org/content-standards/8/EE/C/7/tasks/999

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Alignments to Content Standards: 7.RP.A

Task

Alyssa sees a lightning bolt in the sky and counts four seconds until she hears the thunder.

1. There are 5280 feet in a mile and about 3.28 feet in a meter. Given that sound travels about 343 meters per second, is the lightning strike within one mile of Alyssa?

2. What is the speed of sound in miles per hour? <./li>

IM Commentary

The purpose of this task is to work on performing unit conversions in a real world context about the speed of sound. While the nature of the task appears to fit 6.RP.3d, the numbers involved make this a 7th grade task (6th graders work primarily with ratios of whole numbers). Seeing lightning and waiting to hear the accompanying thunder is a common experience. This task examines one side of this scenario: the time it takes for the sound from the lightning strike to reach Alyssa. The speed of sound is most often given in meters per second in a scientific context but sometimes also in miles per hour. In this task, students work first with the scientific units and then convert to the more familiar miles per hour: this provides extra practice with unit conversions and also helps students to gain an intuition for how fast sound travels. In particular, sound travels about ten times as fast as a car on the freeway and about one and a half times as fast as a passenger airplane.

A short video of a lightning strike followed by the accompanying thunder can be seen here:

The teacher may wish to show the video first and prompt the students to explain the time delay in terms of the relative speeds of light and sound.

The speed of light, a little over 186,000 miles per second, has been left out of consideration in the problem statement and solution. Because light travels so fast, even if the lightning were 100 miles away it would be visible in less than one thousandth of a second: in all cases, this amount of time is smaller than the smallest significant digit for when the thunder is audible and so does not impact the answer. Since taking the speed of light into account complicates the problem without influencing the answer, it has been left out but the teacher may wish to discuss this if it comes up in student work.

Students working on this task will need to Attend to Precision (MP6), understanding that the conversion rate of 3.28 feet per meter is only approximate as is the 343 meters per second speed of sound. This means that students must maintain 3 significant digits in their calculations and the rounding should only take place after all calculations have been completed. Interestingly, the calculated value differs from the 768 miles per hour found on wikipedia https://en.wikipedia.org/wiki/Speed_of_sound because the numbers in 343 meters per second and 3.28 feet per meter are only approximate (with three significant digits in each case). This shows that dealing accurately with significant digits requires very close attention.

Solution

1. Since the thunder from the lightning strike travels at about 343 meters per second, in 4 seconds the sound will have travelled about $$ 4 \text{ seconds} \times 343 \frac{\text{meters}}{\text{second}} = 1372 \text{ meters}. $$ Since there are about 3.28 feet in a meter this is $$ 1372 \text{ meters} \times 3.28 \frac{\text{feet}}{\text{meter}} \approx 4500 \text{ feet}. $$ This is less than a mile.

Alternatively, if Alyssa is familiar with track distances, she knows that a ''metric mile'' is 1600 meters and so 1372 meters will be less than a mile. The metric mile is a little bit shorter than a mile but since tracks are usually built with 400 meters in a lap, this is a convenient substitute for a mile and the two are very close (1600 meters is about 5249 feet).

2. To convert from meters per second to miles per hour we have to convert both the distances and the times. We will do these separately and then put all of the calculations together to find the speed of the thunder in miles per hour. First we convert 343 meters to miles:

\begin{align} 343 \text{ meters} &= 343 \text{ meters} \times 3.28 \frac{\text{feet}}{\text{meter}} \times \frac{1}{5280} \frac{\text{mile}}{ \text{feet}} \\ &= \frac{343 \times 3.28}{5280} \text{ miles} \\ &\approx 0.213 \text{ miles}. \end{align}

Next we convert seconds to hours:

\begin{align} 1 \text{ second} &= 1 \text{ second} \times \frac{1}{60} \frac{\text{minute}}{\text{seconds}} \times \frac{1}{60} \frac{\text{hour}}{ \text{minutes}} \\ &= \frac{1}{3600} \text{hours}. \end{align}

Equivalently, we have 1 hour = 3600 seconds. When we combine all of our unit conversions, we use the complete expression $\frac{343 \times 3.28}{5280}$ miles instead of the rounded $0.213$ miles since the rounding should take place after all operations have been performed. Using our calculations we find

\begin{align} 343 \frac{\text{meters}}{\text{second}} &\approx \frac{340 \times 3.28}{5280} \frac{\text{miles}}{\text{second}} \\ &= \frac{343 \times 3.28}{5280} \frac{\text{miles}}{\text{second}} \times 3600 \frac{\text{seconds}}{\text{hour}} \\ &\approx 767 \frac{\text{miles}}{\text{hour}} \end{align}

# data[].attributes.attachment_link.description
Alignments to Content Standards: 7.RP.A Alyssa sees a lightning bolt in the sky and counts four seconds until she hears the thunder. There are 5280 feet in a mile and about 3.28 feet in a meter. Given that sound travels about 343 meters per second, is the lightning strike within one mile of Alyssa?

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http://tasks.illustrativemathematics.org/content-standards/8/EE/C/8/tasks/469

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Ratios & Proportions: Baking Bread

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Alignments to Content Standards: 6.RP.A

Task

A bread recipe calls for 5 cups of flour and 2 teaspoons of yeast.

1. How much flour and how much yeast do you need for two batches of bread? Draw a picture that shows the batches as well as the total amount of flour and total amount of yeast needed.

2. How many batches can you make with 15 cups of flour and 6 teaspoons of yeast? Indicate the batches in the picture.

3. We say that the ratios 5:2, 10:4, and 15:6 are equivalent. In this case, even though the amounts of each ingredient are not the same, they would make the same kind of bread. Find another ratio equivalent to these (and that would correspond to the same kind of bread) and draw a picture to represent it.

IM Commentary

The purpose of this task is to provide students with a first informal experience with the idea of equivalent ratios. The intuitive notion that making multiple batches of the same recipe results in the same bread as a single recipe helps motivate the idea that different ratios can be "the same" in some important way. Later, students will formalize the idea that a : b is equivalent to any ratio sa : sb, where s > 0.

The task statement shows different-sized blobs, so that cups are represented with larger blobs and teaspoons are represented with smaller blobs, although the blob sizes are not proportional in any way to actual cups and teaspoons. Note that a student might produce a correct drawing where all the blobs are the same size or the relative sizes of the units are not reflected in the diagram. The important thing is that the correct values are shown for each quantity in the ratio.

A possible followup task could be:

Another recipe calls for 2 cups of flour and 1 teaspoon of yeast.

1. How much flour and how much yeast do you need for two batches of this bread?

2. Compare two batches of this bread with one batch of the first bread recipe. Do you think these ratios are equivalent?

Students might say that since they make different kinds of bread since the amount of flour is different when the amount of yeast is the same, the ratios are not equivalent.

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Alignments to Content Standards: 6.RP.A A bread recipe calls for 5 cups of flour and 2 teaspoons of yeast. How much flour and how much yeast do you need for two batches of bread? Draw a picture that shows the batches as well as the total amount of flour and total amount of yeast needed.

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Alignments to Content Standards: 6.RP.A

Task

The ratio of the number of boys to the number of girls at school is 4:5.

1. What fraction of the students are boys?

2. If there are 120 boys, how many students are there altogether?

IM Commentary

In some textbooks, a distinction is made between a ratio, which is assumed to have a common unit for both quantities, and a rate, which is defined to be a quotient of two quantities with different units (e.g. a ratio of the number of miles to the number of hours). No such distinction is made in the common core and hence, the two quantities in a ratio may or may not have a common unit. However, when there is a common unit, as in this problem, it is possible to add the two quantities and then find the ratio of each quantity with respect to the whole (often described as a part-whole relationship).

Tasks like these help build appropriate connections between ratios and fractions. Students often write ratios as fractions, but in fact we reserve fractions to represent numbers or quantities rather than relationships between quantities. For example, if we were to consider the ratio $4:5$ in this situation, then two possible ways to interpret $\frac45$ in the context are to say,

"The number of boys is $\frac45$ the number of girls,"

or to say,

"The ratio of the number of boys to the number of girls is $\frac45 : 1$."

This second interpretation reflects the fact that $\frac45$ is the unit rate (which is a number) for the ratio $4:5$.

Solutions

Solution: Using a tape diagram

For every 4 boys there are 5 girls and 9 students at the school. So that means that $\frac49$ of the students are boys. $\frac49$ of the total number of students is 120 students: $$\frac49 \times ? = 120$$ If $\frac49$ the number of students is 120, then $\frac14$ of 120 is $\frac19$ of the total number of students. In other words, $\frac14 \times 120 = 30$ is $\frac19$ the total number of students. Then 9 times this amount will give the total number of students: $$9\times 30 = 270$$ So there is a total of 270 students at the school. Note that this is equivalent to finding the answer to the division problem: $$120\div \frac49 =?$$ We can see all of this very succinctly by using a tape diagram:

There are 4 units of boys and 9 units of students. Therefore 4/9 of the students are boys.

There are 270 students altogether.

Solution: Using a table

Students can multiply the numbers in the first row by 10 to get the second row, and then double that amount to get the third row. Adding the entries in the second and third row gives the fourth row that has the solution.

Alternatively, since $120 \div 4 = 30$, students can just multiply the numbers in the first row by 30 to get the values in the fourth row.

1. In every row, we can see that the fraction of the students that are boys is $\frac{4}{9} $.

2. looking at the last row, we can see that the total number of students will be $4\times 30 + 5 \times 30 = 120 + 150 = 270$.

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Alignments to Content Standards: 6.RP.A The ratio of the number of boys to the number of girls at school is 4:5. What fraction of the students are boys? If there are 120 boys, how many students are there altogether?

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Alignments to Content Standards: 7.NS.A.1

Task

Aakash, Bao Ying, Chris and Donna all live on the same street as their school, which runs from east to west.

Aakash lives $5 \frac{1}{2}$ blocks to the west.

Bao Ying lives $4 \frac{1}{4}$ blocks to the east.

Chris lives $2 \frac{3}{4}$ blocks to the west.

Donna lives $6 \frac{1}{2}$ blocks to the east.

1. Draw a picture that represents the positions of their houses along the street.

2. Find how far is each house from every other house?

3. Represent the relative position of the houses on a number line, with the school at zero, points to the west represented by negative numbers, and points to the east represented by positive numbers.

4. How can you see the answers to part (b) on the number line? Using the numbers (some of which are positive and some negative) that label the positions of houses on the number line, represent these distances using sums or differences.

IM Commentary

The purpose of this task is for students to solve a problem involving distances between objects whose positions are defined relative to a specified location and to see how this kind of situation can be represented with signed numbers.

A full solution requires systematic listing of pairs of houses, a valid list of differences - either subtracting the smaller number from the larger, or taking absolute value of difference in any order - as well as computations which could be done by counting on the number line or subtracting fractions. Note as well the "twist" with the houses not in "alphabetical" order on the number line requires students to make sense of the problem (MP 1)attend to precision (MP 6).

This problem framework could be used with integer values (say to enhance understanding of 6.NS.5) or later for tasks involving the Pythagorean Theorem in the plane (8.G.8).

Solution

1. There are many ways to draw a picture that represents this situation. Here is one:

2. Here is a table that shows the distances between each of the student's houses.

Bao Ying

Chris

Donna

Aakash

$9\frac34$

$2\frac34$

$12$

Bao Ying

$7$

$2\frac14$

Chris

$9\frac14$

3. The colors show which point corresponds to which person in the first picture:

4. The distance between the houses is represented by the distance between the points that correspond to the houses on the number line. This can be computed by subtracting the numbers that represent the position of the house relative to the school. For example, to find the distance between Bao Ying and Chris, we subtract $-2\frac34$ from $4\frac14$: $$4\frac14-(-2\frac34)$$ We can communicate this more clearly by labeling the distance between the points with the difference of the numbers on the number line:

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Alignments to Content Standards: 7.NS.A.1 Aakash, Bao Ying, Chris and Donna all live on the same street as their school, which runs from east to west. Aakash lives $5 \frac{1}{2}$ blocks to the west. Bao Ying lives $4 \frac{1}{4}$ blocks to the east. Chris lives $2 \frac{3}{4}$ blocks to the west.

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http://tasks.illustrativemathematics.org/6

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Alignments to Content Standards: 7.G.A.3

IM Commentary

The purpose of this task is to have students explore various cross sections of a cube and use precise language to describe the shape of the resulting faces. The drawings can be either 2- or 3-dimensional; some students may not have experience in drawing 3-D projections, in which case a 2-D drawing of the shape of the slice would be appropriate. The ability to visualize and understand the relationships between 3-dimensional and 2-dimensional objects is useful in many areas. Architects need to relate 3-dimensional buildings to 2-dimensional blueprints. Physicians need to relate 2-dimensional x-rays and MRI's to 3-dimensional bodies. Engineers need to relate 2-dimensional schematics to 3-dimensional structures. Many people find such visualization challenging, often because they have had few opportunities to practice. Depending on the students, teachers may want to provide some tools to support this task. Clay can be made into cubes and sliced neatly with plastic knives, dental floss, or wire. Paper models can be cut and taped together. Isometric dot paper can also aid with drawings. There are free CAD programs such as Google sketchup that allow students to build and manipulate virtual models, but this goes beyond the intended purpose of this task. Note: We kept the language of the task simple and in line with the wording of the standard, referring to "slices" instead of "planar slices" or "planar cross-sections". Some students may ask about or want to use non-planar slices. This is an opportunity to attend to the precision of the language, and indicate that in this task "slices" means "planar slices". The use of non-planar slices changes the nature, complexity, and mathematics of this task.

The Standards for Mathematical Practice focus on the nature of the learning experiences by attending to the thinking processes and habits of mind that students need to develop in order to attain a deep and flexible understanding of mathematics. Certain tasks lend themselves to the demonstration of specific practices by students. The practices that are observable during exploration of a task depend on how instruction unfolds in the classroom. While it is possible that tasks may be connected to several practices, the commentary will spotlight one practice connection in depth. Possible secondary practice connections may be discussed but not in the same degree of detail.

Students will be engaged in Mathematical Practice Standard 5, "Use appropriate tools strategically," by considering the available tools when solving this task and making sound decisions about which and when these tools might be helpful. At the same time, students recognize both the insights to be gained and the limitations of the tool/tools selected. Being able to visualize 2-dimensional shapes within a 3-dimensional object may be very difficult for students. Making tools available will help students approach this task and give them valuable practice in visualizing these types of problems. Appropriate tools include: computer access, paper models, clay, and other tools that allow for the deconstruction of a cube. Mathematical Practice Standard 6, attend to precision, is very integral to this task also. Precision in language is required as students describe the resulting shape of the slice.

Solution

a. A rectangle.

Comment: In Grade 7, students have not had the Pythagorean theorem, so they are not expected to determine that the rectangle has sides of 1 and $\sqrt{2}$. They should, however, note that the diagonal of a face is longer than the edge, hence the cross section is not a square.

b. An equilateral triangle.

Comment: Because the three sides of the cross-section are all the diagonals of the faces, they have the same length. As noted in part A, the students are not expected to determine the length to be $\sqrt{2}$, but they are expected to note that all three lengths are the same, hence the triangle is equilateral.

c. An irregular pentagon.

Comment: This one is by far the most challenging of the three. A key insight is to figure out where the plane intersects the edges CF and BE. Many people have a great deal of difficulty visualizing the shape. However, students using clay or paper models can figure it out. Although it is not expected for 7th grade students to be able to determine the exact coordinates (it ends up that the intersections are ⅔ the way up the edge), students can estimate that it is more than halfway, and be able to make an argument that the resulting shape has five sides and not all the sides have the same length, hence it is an irregular pentagon.

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Alignments to Content Standards: 7.G.A.3 The purpose of this task is to have students explore various cross sections of a cube and use precise language to describe the shape of the resulting faces. The drawings can be either 2- or 3-dimensional; some students may not have experience in drawing 3-D projections, in which case a 2-D drawing of the shape of the slice would be appropriate.

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Ratios & Proportions: Voting for Two, Variation 4

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Alignments to Content Standards: 6.RP.A

Task

John and Will ran for 6th grade class president. John got two votes for every vote Will got. It was necessary to get more than half the votes to get elected. What fraction of the votes more than half did John get?

IM Commentary

This is the fourth in a series of tasks about ratios set in the context of a classroom election. What makes this problem interesting is that the number of voters is not given. This information isn't necessary, but at first glance some students may believe it is. On the other hand, students that solved Voting for Two, Variation #3 by recognizing it can be solved as a difference of fractions will be able to apply the same strategy here. Not knowing the number of voters makes the problem more abstract.

Note that for the first three variations in this series, students could use ratio tables instead of the more abstract approaches illustrated for each one. Here is the first time that students will have to go beyond a ratio table to solve it. Students who have had a chance to see the connection between ratio tables and other solution methods will have an easier time making this transition.

Solution

One way to think about the problem is to realize that John gets $2$ votes for every $1$ of Will's votes. Thus, John gets $2$ out of every $3$ votes or $\frac23$ of the votes. The difference between $2/3$ of the votes and $1/2$ of the votes is:

$\frac23 - \frac12 =$

$\frac46 - \frac36 = \frac16$

So, the answer is $\frac16$ of the votes.

# data[].attributes.attachment_link.description
Alignments to Content Standards: 6.RP.A John and Will ran for 6th grade class president. John got two votes for every vote Will got. It was necessary to get more than half the votes to get elected. What fraction of the votes more than half did John get?

# data[].attributes.body
http://tasks.illustrativemathematics.org/content-standards/7/RP/A/tasks/1602

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http://tasks.illustrativemathematics.org/content-standards/6/EE/A/tasks/2167

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http://tasks.illustrativemathematics.org/content-standards/8/F/A/2/tasks/641

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Ratios & Proportions: Apples to Apples

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Alignments to Content Standards: 6.RP.A.1

Task

Alice and Claire go apple picking. When they are done, Claire has 3 times as many apples in her basket as Alice has in hers. All of the apples are whole.

1. What are three different possibilities for numbers of apples that could be in the baskets?

2. What is the ratio of Alice's apples to Claire's apples?

Alice and Claire's mom measures each of their heights in inches, rounded to the nearest whole inch. She remarks, "Wow! Alice's height is exactly three fourths of Claire's height!"

1. What are three different reasonable possibilities for their heights?

2. What is the ratio of Claire's height to Alice's height?

IM Commentary

The purpose of this task is to connect students' understanding of multiplicative relationships to their understanding of equivalent ratios.

Creating tables of equivalent ratios isn't required for this task, but it is a convenient way to organize them. Students should eventually come to view ratio tables as collections of equivalent ratios, so this might be a good opportunity to suggest such a tool.

In part (b) and (d), we ask "What is the ratio?" The meaning of this question can be misinterpreted by teachers, because experienced doers-of-math usually default to lowest terms, so it would be easy to mistakenly conclude that 3:1 is the only acceptable answer for (b) and 4:3 is the only acceptable answer for (d). However, any ratio equivalent to those is acceptable.

In part (c), reasonable height values aren't determined by the task statement, but they are suggested by the context, and students should explain the thinking behind the reasonableness of their values. Paying attention to the meaning of numbers in a context and evaluating them for reasonableness is important to the standards for mathematical practice, especially MP2, MP4, and MP8.

In this task, the units for each quantity are the same and students are essentially given the values of the unit rates and asked to determine possible ratios. The task is meant to come before students have studied unit rates in order to provide a precurser to the idea that unit rates characterize sets of equivalent ratios, but without having to deal with the complexities of the units. If students have already studied unit rates, they might notice that in this case we could say, "There are 3 of Claire's apples for every 1 of Alice's apples" (so there are "apples per apple") and There are 3/4 as many inches in Alice's height to every 1 inch in Claire's height (so there are "inches per inch").

Solution

1. Here are some possibilities for the numbers of apples in each basket. (For clarity, these are organized in a table, but they don't necessarily have to be. Any response in a ratio of 3:1 is acceptable.)

2. The ratio of Alice's apples to Claire's apples is 1:3. Any equivalent statement is also acceptable, for instance, "There are 4 apples in Alice's basket for every 12 apples in Claire's."

3. Here are some possibilities for their respective heights. (For clarity, these are organized in a table, but they don't necessarily have to be.) Example reasoning about reasonable heights: the average height (usually called "length") of a newborn is 20 inches, but since Alice and Claire were presumably walking around and picking apples, we'll take 33 inches, the height of a short two-year-old, to be the least possible height. (These heights are easily googleable.) Since they are being measured by their mom, they are likely young children, so we'll say that 5 feet or 60 inches is the greatest possible height. (Students may set different bounds on the heights based on different reasoning.)

4. The ratio of Claire's height to Alice's height is 48:36 (or equivalent).

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Alignments to Content Standards: 6.RP.A.1 Alice and Claire go apple picking. When they are done, Claire has 3 times as many apples in her basket as Alice has in hers. All of the apples are whole. What are three different possibilities for numbers of apples that could be in the baskets?

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Alignments to Content Standards: 8.F.B.4

Task

Consider the relationship between the number of pounds of chicken and the number of pounds of steak that can be purchased for a barbecue on a fixed budget.

1. Write an equation for this relationship given that chicken costs \$1.29 per pound, steak costs \$3.49 per pound, and the total allotment for both is \$100.

2. Sketch a graph for the equation. Describe how the amount of steak that can be purchased depends on the amount of chicken purchased. What is the significance of the graph's $x$- and $y$-intercepts?

3. Re-write the equation to show the amount of steak purchased in terms of the amount of chicken purchased. Use this equation to describe how the amount of steak purchased changes with each pound of chicken purchased.

4. Both the equation and the graph can be used to find amounts of chicken and steak that can be purchased for the barbecue on a fixed budget of \$100. Explain how important features of the graph can be seen or found with the equation and describe what they say about the relationship between chicken and steak purchases.

IM Commentary

This task is intended strictly for instructional purposes with the goal of building understandings of linear relationships within a meaningful and, hopefully, somewhat familiar context. The expectation for part (a) is that students simply translate the situation to equation form and not necessarily express the relationship as a linear function in slope-intercept form. Parts (b) and (c) align with content standards 8.F.5 and 8.F.4 respectively. Students are asked to analyze a graph of the relationship in part (b) and construct a linear function in part (c). Part (d) was added to advance understandings of graphic and symbolic representations, specifically with regard to how they model the same situation.

Some scaffolding may be needed for eighth graders to complete the task successfully, particularly with regard to initial interpretations of the graph and transforming the equation into slope-intercept form. The use of cooperative learning structures would greatly enhance student reasoning applied to the task, as would a carefully structured class discussion of small group findings. Under these conditions the Standards for Mathematical Practice 3 (construct viable arguments and critique the reasoning of others) and 4 (model with mathematics) would most certainly come into play.

Solution

1. Let $x$ represent the number of pounds of chicken purchased and $y$ represent the number of pounds of steak purchased. Then
$$ 1.29x + 3.49y = 100 $$

2. The graph shows that the amount of steak that can be purchased on a fixed budget decreases as the amount of chicken purchased increases. The $x$-intercept shows the maximum amount of chicken that can be purchased (77.5 lbs when no steak is purchased) and the $y$-intercept shows the maximum amount of steak that can be purchased (28.7 lbs when no chicken is purchased).

3. The equation from part (a) written in slope-intercept form for linear equations is

$$ \begin{align} y &= -\left(\frac{1.29}{3.49}\right)x + \frac{100}{3.49} \\ & \text{or} \\ y &= -0.37x+28.65 \end{align} $$

with quotients rounded to the nearest hundredth. The $x$-coefficient reveals the rate of change in steak purchases per unit change in chicken purchases. Specifically, the amount of steak that can be purchased given a \$100 budget starts at 28.65 lbs and drops by 0.37 lb (or a little over one-third of a pound) with each pound of chicken purchased.

4. A robust explanation of how the graph and equation both represent the same situation is expected here. At a minimum, students should recognize (1) how the rate of change is represented by the slope of the graph and by the $x$-coefficient in the equation, (2) that the constant term in the equation represents the $y$-intercept on the graph, and (3) how the equation can be manipulated to find the graph's $x$-intercept. All explanations should be grounded in the context of the situation represented by the graph and equation.

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Alignments to Content Standards: 8.F.B.4 Consider the relationship between the number of pounds of chicken and the number of pounds of steak that can be purchased for a barbecue on a fixed budget. Write an equation for this relationship given that chicken costs \$1.29 per pound, steak costs \$3.49 per pound, and the total allotment for both is \$100.

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Task

Chichén Itzá was a Mayan city in what is now Mexico. The picture below shows El Castillo, also known as the pyramid of Kukulcán, which is a pyramid located in the ruins of Chichén Itzá.

The temple at the top of the pyramid is approximately 24 meters above the ground, and there are 91 steps leading up to the temple. How high above the ground would you be if you were standing on the 50th step?

IM Commentary

The purpose of this task is for students to solve a straight-forward problem involving a proportional relationship in a context. In order to solve the problem, students must assume that the steps are of uniform height, which looks reasonable given the picture.

Two different and important approaches to the problem are presented: calculating a unit rate (height per step) and writing an equation (the height of the 50th step is $\frac{50}{91}$ the height of the pyramid). The first of these approaches falls under the 6th grade ratio and proportion standards while the second aligns better with the 7th grade. The main goal of the third approach, using Egyptian fractions, is to provide an example of calculating fluently. The methods of the first two solutions are vital for a deep understanding of proportional reasoning.

The numbers make this an advanced 6th grade task. It could be used as a transition from 6th to 7th grade expectations for work with ratios and proportional relationships, or it could be used as an assessment task in 7th grade (either formative or summative, depending on the time and way it is used). If used in an instructional setting, the teacher could ask the students to estimate before they compute. Since there are 91 steps and we want to know about the 50th step, we know that the height should be a little more than half of the total 24 meters in height.

This task could also be used to transition to the 8th grade standard on using similar triangles to prove that the slope of a line is well-defined. Looking at the cross-section of this pyramid and the similar triangles that we are essentially asking about would be a useful extension task:

The image is courtesy Wikimedia Commons:
http://en.wikipedia.org/wiki/File:Chichen_Itza_3.jpg
Solutions

There are 91 steps needed to get you to the top, which is 24 meters high. So each step is about $24\div91$ meters tall. Since

$$24\div91=\frac{24}{91}\approx 0.264$$

we estimate that each step is about 0.264 m or 26.4 cm tall on average. Since

$$50\times \frac{24}{91}\approx 13.2 $$

we'd expect to be about 13 meters above the ground if we stood on the 50th step.

Assuming the steps are all the same size, the ratio of the height, $h$, of the 50th step to the height of the 91st step will be 50:91. So we know that

$$\frac{50}{91}=\frac{h}{24}$$

If we multiply both sides of this equation by 24, we get that

$$h=24\times\frac{50}{91}\approx 13.2$$

So the 50th step is about 13 meters off of the ground.

Solution: The solution for people who like Egyptian Fractions

Half of 91 is 45.5. One tenth of one half (or one twentieth) of 91 is 4.55. These add up to 50.05 and tell us that the 50th step is about $\frac{1}{2} + \frac{1}{20}$ of the way up the structure. The top is 24 meters. So we can estimate the height of the 50th step by adding $\frac12$ of 24 which is 12 and $\frac{1}{20}$ of 24 which is 1.2. Since $$12+1.2 = 13.2$$ we know we are about 13 meters above the ground.

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Chichén Itzá was a Mayan city in what is now Mexico. The picture below shows El Castillo, also known as the pyramid of Kukulcán, which is a pyramid located in the ruins of Chichén Itzá. The temple at the top of the pyramid is approximately 24 meters above the ground, and there are 91 steps leading up to the temple.

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6th grade: Climbing the steps of El Castillo

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6th grade: Climbing the steps of El Castillo

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http://tasks.illustrativemathematics.org/content-standards/6/RP/A/tasks/1564

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http://tasks.illustrativemathematics.org/content-standards/6/RP/A/tasks/1564