This Math Talk focuses on expressions with variables. It encourages students to think about equivalence and to rely on properties of operations to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students express answers in terms of .
To generate an equivalent expression with fewer terms, students need to look for and make use of structure (MP7).
Launch
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Give students quiet think time and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies, and record and display their responses for all to see.
Use the questions in the activity synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards. Supports accessibility for: Memory, Organization
Activity
None
For each expression, find an equivalent expression with fewer terms.
Student Response
Loading...
Building on Student Thinking
Activity Synthesis
To involve more students in the conversation, consider asking:
“Who can restate ’s reasoning in a different way?”
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”
The key takeaway is that applying properties of operations allows us to write equivalent expressions with fewer terms. This is often called "combining like terms."
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because. . . .” or “I noticed _____ so I. . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. Advances: Speaking, Representing
16.2
Activity
Standards Alignment
Building On
Addressing
7.G.B.4
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
In this activity, students calculate the area of circles given their radius or diameter. For the problem where the radius is given, students practice applying the formula . For the problem where the diameter is given, students must think about what extra steps are required. This is similar to the thinking they did previously when they were given a circle’s radius and asked to calculate its circumference.
Because of the properties of multiplication, there is some flexibility when it comes to strategies for calculating circumference given the radius, such as:
Multiplying the radius by 2 to find the diameter, then multiplying the diameter by .
Multiplying the radius by to directly calculate the circumference.
Multiplying the radius by first and then multiplying that product by 2.
However, because the formula for area of a circle involves squaring the radius, there is less flexibility possible for calculating the area given the diameter. Students are most likely to be successful using the strategy:
Dividing the diameter by 2 to find the radius, then using the formula .
During the whole-class discussion, students critique a response that is intentionally incorrect and improve it by clarifying meaning, correcting errors, and adding details (MP3).
This activity uses the Critique, Correct, Clarify math language routine to advance representing and conversing as students critique and revise mathematical arguments.
Launch
Introduce the context of this activity by asking students, “What are some special objects that you are used to seeing at a party or celebration?” Then explain that Lakota people from around South Dakota hold large celebrations called wacipis, or powwows. The wacipi (wah-CHEE-pee) involves song, dance, and colorful outfits.
Display the image of the hoop drum and beaded medallion for all to see. Explain that these are some examples of the special objects you could see at a powwow. If desired, explain that the circle shape is an important symbol in the Lakota culture.
Give students 4–5 minutes of quiet work time, followed by whole-class discussion.
Activity
None
Here are some special objects that might be seen at a Lakota powwow, or wacipi (wah-CHEE-pee).
A hoop drum has a radius of 7 inches. What is the area of the drum?
A beaded medallion has a diameter of 6 centimeters. What is the area of the medallion?
Activity Synthesis
The purpose of this discussion is to highlight students’ strategies for dealing with being given the diameter of the beaded medallion, rather than its radius.
First, invite students to share their reasoning about the area of the hoop drum. Once the class is in agreement on the answer, move the discussion on by asking “What was different about the second problem compared to the first problem?” While there are several differences students may notice, the important discussion point is that the diameter was given, rather than the radius.
Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response for finding the area of the medallion by correcting errors, clarifying meaning, and adding details.
Display this first draft:
“To find the area of the medallion, I multiplied times 62. Then I realized that 6 was the diameter and not the radius, so I divided my answer by 2.”
Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
Give students 2–4 minutes to work with a partner to revise the first draft.
Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.
The key takeaway is that it is important to divide the diameter by 2 first to find the radius before applying the formula .
16.3
Activity
Standards Alignment
Building On
Addressing
7.G.B.4
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
In this activity, students find the areas of regions involving different-sized circles and compare the strategies used. In the first set of figures, students subtract to find the area around the outside of a circle. In the second set of figures, students divide to find the area of fractions of a circle. During the whole-class discussion, students consider leaving expressions in terms of pi as a means of increasing precision in describing circles (MP6).
Monitor for students who use different strategies for comparing the area of Figures A, B, and C, such as:
Decomposing Figures B and C into smaller square sections, subtracting the area of each size of circle from its surrounding square, and then multiplying the result as needed (by 4 for Figure B and by 9 for Figure C).
Multiplying the area of the circles as needed and then subtracting the result from the area of the full-size square.
Recognizing that the areas have to be equal because Figure B is composed of 4 scaled copies of Figure A, each with as much area, and Figure C is composed of 9 scaled copies of Figure A, each with as much area.
Plan to have students present in this order to support moving them from less efficient to more efficient strategies. However, if the last strategy does not come up in students’ work, it is not necessary for the teacher to bring it up during the discussion.
As students look for ways to be more efficient when comparing the areas of the regions, they make use of structure (MP7).
Adjust the time for this activity to 20 minutes.
Launch
Arrange students in groups of 2. Display the image in the first question and ask students to make a prediction before calculating. Give 30 seconds of quiet think time before sharing with their partner. Give students partner work time followed by a whole-class discussion.
Select students who used each strategy described in the activity narrative to share later. Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven't shared recently.
Activity
None
Each square has a side length of 12 units. Compare the areas of the shaded regions in the 3 figures. Which figure has the largest shaded region? Explain or show your reasoning.
A
Figure A of three equal sized figures labeled A B and C. Figure A is a square with a circle inside of it where the circle touches the midpoint of each side of the square. The regions outside of the circle are shaded blue.
B
Figure B of three equal sized figures labeled A B and C. Figure B is a square with four identical circles inside of it that do not overlap. The circles are arranged in two rows of two circles, are tangent to each other and tangent to the sides of the square. The area outside of the circles is shaded blue.
C
Figure C of three equal sized figures labeled A B and C. Figure C is a square with nine identical circles inside of it that do not overlap. The circles are arranged in three rows of three circles, are tangent to each other and to the sides of the square. The area outside of the circles is shaded blue.
Each square in Figures D and E has a side length of 1 unit. Compare the area of the two figures. Which figure has more area? How much more? Explain or show your reasoning.
D
Figure D of two figures, labeled D and E. Figure D is composed of one square and two semi circles on top, and two bottom halves of circles and one square on the bottom. The top and bottom are divided by a horizontal dashed line.
E
Figure E of two figures, labeled D and E. Fiqure E is composed of 4 squares and 6 quarter circles. The 4 square are arranged diagonally, where the top right vertex of the first square is the same point as the bottom left vertex of the second square, the top right vertex of the second square is the same point as the bottom left vertex of the third square, and the top right vertex of the third square is the same point as the bottom left vertex of the fourth square. The corner circles align in between the sides of each square.
Activity Synthesis
There are two main goals for this discussion: for students to notice ways to be more efficient when comparing the areas of the regions and to be introduced to expressing answers in terms of pi. These two goals are related because keeping the answers in terms of pi is an additional method that can help make the comparisons more efficient.
Display Figures A, B, and C for all to see. Invite previously selected students to share how they determined the area of the figures. Sequence the discussion of the strategies in the order listed in the activity narrative. If possible, record and display their work for all to see.
Connect the different responses to the learning goals by asking questions such as:
“Why do the different approaches lead to the same outcome?”
“How does subtraction show up in each method?”
“Are there any benefits or drawbacks to one method compared to another?”
If not brought up by students, ask how we could have determined that the area of the shaded regions were equal without calculating the answer of 30.96. For example, determining that the unshaded area in each figure is 113.04 square units is enough to conclude that the shaded areas have to be equal, without completing the subtraction.
Next, focus the discussion on leaving answers in terms of for each figure. Explain to students that in Figure A, the radius of the circle is 6, so the area of the circular region is . Instead of multiplying by an approximation of , we can express this answer as . This is called answering "in terms of ." Consider writing “” inside the large circle of Figure A.
Ask students to answer these questions in terms of :
“What is the area of one of the circular regions in Figure B?” ()
“What is the area of one of the circular regions in Figure C?” ()
Consider writing “” and “” inside some of the circles in Figures B and C.
Ask students:
“What is the combined area of all four circles in Figure B?” (, or )
“What is the combined area of all nine circles in Figure C?” (, or )
“What is the area of the shaded region in each figure?” ()
“Can the expression be shortened by combining like terms?” (no)
If time permits, ask students to express their answers for the area of Figures D and E in terms of . Record and display their answers of and for all to see. Ask students to discuss how they can tell Figure E’s area is larger than Figure D’s area when they are both written in terms of . (because )
16.4
Activity
Optional
Standards Alignment
Building On
Addressing
7.G.B.4
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
In this activity, students calculate the area of a running track. Students may have investigated the perimeter of this track in a previous lesson. Here, they apply the strategies of dividing to find the area of a fraction of a circle as well as subtracting to find the area of the region around a circle. As students decide how to decompose the running track into measurable pieces and how to use the given information to calculate areas, they reason abstractly and quantitatively (MP2).
Monitor for students who use different strategies for determining the area, such as:
Decomposing the track into two straight sections and two curved sections, finding the areas, and adding all four pieces together.
Decomposing half of the track into one straight section and one curved section, finding and adding those areas, then multiplying the result by 2.
Finding the area of the larger shape (the track and the field inside) and then subtracting the area of the field inside.
Launch
Arrange students in groups of 2. Give students 5–6 minutes of partner work time followed by small-group and whole-class discussions.
MLR8 Discussion Supports. At the appropriate time, give students 2–3 minutes to make sure that both partners can explain their process. Invite groups to rehearse what they will say when they share with the larger group. Advances: Speaking, Conversing, Representing
Representation: Internalize Comprehension. Provide a blank graphic organizer for students to organize the information provided in the problem and to structure their problem-solving strategy. The graphic organizer could include the prompts: “What do I need to find out?”, “What do I need to do?”, “How I solved the problem.”, and “How I know my answer is correct.” . Supports accessibility for: Organization, Attention
Activity
None
The field inside a running track is made up of a rectangle 84.39 m long and 73 m wide, together with a half-circle at each end. The running lanes are 9.76 m wide all the way around.
A picture of a field inside a running track. The field inside the track is composed of a rectangle, indicated by two dashed vertical lines labeled 73 meters and a horizontal length labeled 84 point 3 9 meters. There is a semi circle on each vertical side of the rectangle. The running track goes completely around the field and has a width of 9 point 7 6 meters.
What is the area of the running track that goes around the field? Explain or show your reasoning.
Student Response
Loading...
Building on Student Thinking
Some students may think they can calculate the area of the running track by multiplying half of the perimeter times the radius, as if the shape were just a circle. Prompt them to see that they need to break the overall shape into rectangular and circular pieces.
Activity Synthesis
Pair the groups of 2 to create groups of 4. Have students compare answers and explain their reasoning until they reach an agreement.
Then, ask the class how finding the area of the track in this activity was similar or different from solving the two area problems in the previous activity. Make sure to highlight these points:
Like the first problem in the previous activity, finding the area of the track can be done by finding the area of the larger shape (the track and the field inside) and then taking away the area of the field inside.
Like the second problem in the previous activity, there are semicircles whose area can be found by composing them to make a full circle or by taking half the area of the corresponding full circle.
Lesson Synthesis
Share with students, “Today we applied the formula for finding the area of a circle to solve multi-step problems. We also practiced expressing answers in terms of .”
To give students extra practice expressing answers in terms of , consider asking:
“What is the area, in terms of , of a circle with a radius of 10?” (, because .)
“What is the area, in terms of , of a circle with a diameter of 10?” ( because and .)
“What is the area, in terms of , of a half-circle with a diameter of 10?” (, because .)
Student Lesson Summary
The relationship between , the area of a circle, and , its radius, is . We can use this to find the area of a circle if we know the radius. For example, if a circle has a radius of 10 cm, then the area is , or cm2. We can also use the formula to find the radius of a circle if we know the area. For example, if a circle has an area of m2 then its radius is 7 m and its diameter is 14 m.
Sometimes instead of leaving in expressions for the area, a numerical approximation can be helpful. For the examples above, a circle of radius 10 cm has an area of about 314 cm2. In a similar way, a circle with an area of 154 m2 has a radius of about 7 m.
We can also figure out the area of a fraction of a circle. For example, the figure shows a circle divided into 3 pieces of equal area. The shaded part has an area of .
A circle divided into three equal sections. From the center of the circle three line segments extend to a point on the circle. The line segment extending downward and to the right is labeled “r”. The upper left region of the circle is shaded.
Standards Alignment
Building On
Addressing
7.G.B.4
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
Some students may use 6 centimeters as if it were the radius of the medallion, rather than the diameter. Point out that the first problem gives the radius of the hoop drum, while the second question gives the diameter of the medallion. Prompt them to consider how they could determine the radius of the medallion.
Student Response
Loading...
Building on Student Thinking
In the first question, students may not know how to find the radius of the circles. Suggest having them cut off the shaded regions and rearrange them to show that the length of each side fits half way across the circle (marking the radius).
In the second question, students might benefit from cutting and rearranging the figures. Some students might assume, based on previous activities, that the areas of both figures are equal. However, Figure D has more pieces that are parts of a circle, and Figure E has more units that are a full square. Ask students whether the fourth of the circle has the same area as the square.
