Picture Problems

Problems to solve from viewing a picture

Real Functions HSF.IF.A

jessicacole Algebra Teaching Issues, HS Algebra, HS Functions, Picture Problems

 

 

These three pictures show three steps of the recycling process of plastic containers.  This process begins with many different containers being stored together because they are made out of plastic.  Then the containers are loaded into a machine that compresses them together so that when they come out of the machine they hold a uniform shape.  I would use this process to help students conceptualize functions.  The storage of the many different containers represents a function’s

domain, the compacting machine represents the function itself, and the compressed and uniform cubes of plastic containers represent the function’s range and the organization of the ordered pairs that are produced by the domain and range.  By including this analogy in your lesson, students will all have at least one real world occurrence that they can use as they reason through the concept of a function.

 

One problem that I would give students to think about after introducing functions and talking about the recycling process would be:

If f(x) = 7x-10, what is f (5)?

 

If f(x) =, what is f (3)?

 

If f(x-1) = 3x +2, what is f (8)?

 

 

CCSS.MATH.CONTENT.HSF.IF.A.1
Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

CCSS.MATH.CONTENT.HSF.IF.A.2
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Shape Sorter Activity CCSS.Math.Content.6.SP.B.5.b

nicholasspencer Common Core State Standards Mathematics, Geometry Teaching Issues, ML Modeling, Picture Problems

By: Nick Spencer, Sam Marcoe, Grayson Windle, Elizabeth Englehart

With the Shape Sorter activity, which can be located at this website, we enable our students to work with several different concepts such as venn diagrams and geometric patterns.  The students work with a venn diagram in which they can set up two “rules” for each side of the diagram, and then are tasked with organizing the shapes accordingly to the rules.  If a shape meets both standards set by the rules, the student can place the shape in the center of the venn diagram, and if the shape doesn’t meet the criteria for either rule the student can leave it outside of the venn diagram.

In the figure above, my group of teacher candidates created our own venn diagram with these two rules:

  1. The figure has at least one line of symmetry (left side)
  2. The figure has rotational symmetry (right side)

In the beginning of this activity, we have several different shapes and begin organizing them in the matching part of the venn diagram based on their geometric characteristics.  The figures on the left side of the diagram have at least one line of symmetry while the figures on the right have rotational symmetry.  The figures in the center have both characteristics set by our rules, while the figures on the outside have neither lines of symmetry nor rotational symmetry.

This online educational activity allows the students to further develop conceptual understanding, and practice procedural fluency for geometric shapes and laws, as well as develop understanding for venn diagrams.  By allowing students to combine geometry with venn diagrams, students can open their perspective that venn diagrams can be utilized in various and unique situations.

 

Geometric Unicorn 7.G.A

munsonbill Picture Problems

The problem aligns with the following Standards:

CCSS.MATH.CONTENT.7.G.A.1

CCSS.MATH.CONTENT.6.G.A.1

ART GLE: 1.1.2

ART GLE: 4.2.1

The question will be how can we scale up this animal using what we have learned about scaling geometric shapes. The students will be given a sheet with an animal made of geometric shapes on it. I would let the class decide what animal will be used in the lesson. The unicorn and the bear are examples I found.  Each student will be assigned a piece of the animal. They will all need to find the area of their piece by measuring it, and for most pieces they need to split them into shapes for which they already know how to find the area. Then the students scale the figure up with a scale factor of 5, and find the area again. The students will have to use poster paper for the bigger pieces, and they will color and cut out the pieces when finished. Once every student has finished they will get to put them all together to make a big version of the animal to put on the wall in the room.

After it’s all put together, as a class you can talk about how the area of each shape changed and how the area of the bear as a whole changed. Also you can talk about how making this shows that math can be used in the shape and form of visual artwork to link the lesson to the art standards. The math standards are about scaling and finding the area of geometric shapes. The art standards are about learning how shape and form affect visual art, and how other content areas are connected to art.

Pizza Fractions 5.NF.1

jaimiebarry ML Number Systems, Picture Problems

This is a fifth grade mathematics lesson integrated with technology. The focus of this lesson is adding fractions with unlike denominators. This activity would take place when students are beginning to understand how to add fractions with unlike denominators. The pizza problem is a classic example used when it comes to fractions, however this activity is going to have a little bit of a twist.

Normally, math is in the morning but for the day of this lesson, math will be moved to after lunch. During lunch students would be able to eat pizza that would be provided. After lunch, students will need to add up how many pieces of pizza are left, the top of the boxes will say how many slices each pizza was cut into. This should be fairly easy because the pizzas would most likely be cut into the same amount of pieces.

After the students have added up how much pizza is left, they will figure out how much pizza the class ate as a whole. Again, should be fairly simple for fifth grade students.

This is when the lesson will take advantage of the technology provided. You can either do this activity as a class or, depending on how many computers you have, you can have them work individually or in small groups.

On VisualFractions.com, there is an adding unlike fraction circles activity that students can do. The website will give the students the first addend and they will have to input the fraction that is shaded. Then, they will be provided with the second addend, again, having to provide a fraction for the shaded amount. Finally, they will have to add the two unlike fractions by finding equivalent fractions. If students get the answer wrong, it will tell them if the solution is greater than or less than the answer they gave.

Website: http://www.visualfractions.com/AddUnlikeCircle/

Education Technology EALR 1.1.2 Use models and simulations to explore systems, identify trends and forecast possibilities.

CCSS-Math 5.NF.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.

Art from a Different Angle 7.G.B.5

nicholasspencer Geometry Curriculum Aligned with CCSS-Math, Picture Problems


This math problem is one in which students are using their knowledge of different kinds of angles (supplementary, complimentary, vertical, adjacent) to find as many angles as they can within some sort of abstract art.

The main idea is to locate as many angles of each time as they can within the art-piece, as well as give reasoning in written form of how they know they are correct, based on the different laws of angles and not just a protractor.  Students can also be expected to identify patterns during discussions as well as in written forms, and can present their arguments to the class.  Students can be given various samples of art to work with, and be separated into groups, so that at the end of the lesson the groups can present their art and which angles they found and where.

This lesson is an integration of math into art, and a great way for students to explore how math really is a part of more than most would believe.  This lesson could also lead into the idea that there are different kinds of math within all art, and students could begin trying to identify the various mathematics involved in art and widen their understanding of mathematics in the real world.

CCSS.MATH.CONTENT.7.G.B.5
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.

CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.

CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.

CCSS.MATH.PRACTICE.MP6 Attend to precision.

CCSS.ELA-LITERACY.W.7.1.B
Support claim(s) with logical reasoning and relevant evidence, using accurate, credible sources and demonstrating an understanding of the topic or text.

Math in Minecraft? 6.G.A.2

pelhamt Picture Problems

Talking about Minecraft for math? How exciting for middle school students!

The focus of this lesson would be on the concept of volume while using the popular game of Minecraft. Students will be given the task of designing a classroom Minecraft city. Students would be presented with buildings that need to have their volume identified in order to figure out how many blocks it will take to recreate the building. For one building, students will be given the dimensions with which they can use the algorithm to solve. For another building, students will be given the volume and need to identify the dimensions. The last building students will identify is one they would create on their own as a group. For this building, students will design it, figure out what dimensions they would like it to be, calculate the building’s volume, and lastly think about what the building would be used for in the city.

Students would be provided with pictures of the buildings to base their calculations on as well as inspiration for how they would like their final building to look. Students will also be provided with manipulatives which they can use to construct the buildings to better help them visualize the dimensions and volume. After students have designed and calculated their final building, their task as a group would be to draw their building so it can be presented to the class. The students would give a short explanation of how they designed their building, why they chose the dimensions they did, how they made their calculations, and what their building would be used for in the city.

Standards addressed include:

CCSS.MATH.CONTENT.6.G.A.2: Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism.

Grade 6 GLE: 1.1.2: Applies, analyzes, and creates the elements shape and form when producing a work of art.

Theme Park Expressions 6.EE.B.6

heathercarlton ML Expressions & Equations, Picture Problems

This activity is a real-world problem that uses a context that student will enjoy. They will be using variables to create and analyze a theme parks attendance. Students will be given a theme park scenario with variables for such things as the employees, visitors to the park, admissions cost and days of operation.  For example, F could be the # of female employees and E for the # of male employees. The students will collect and analyze the data to determine how the park is doing financially and figure out which rides are the most efficient. They must develop math expressions to save a lot of time and energy. They will use the variables to write expressions for: 1.) The total number of people visiting the park on a given day. 2.) The total number of people in the park on a given day. 3.) The amount of money the amusement park collects from tickets on a given day if all visitors pay for a single-day pass for their respective age groups. 4.) The number of people who ride roller coasters if 3/5 of all visitors ride roller coasters on a given day. 5.) The number of visitors over age 18 who ride roller coasters during the park’s season if 1/4 of all visitors ride roller coasters on a given day.

To extend this activity students could write up a proposal about building an additional roller coaster. Students will need to use the data to select the most important variables and expressions the owners should consider to make their decision.

CCSS.MATH.CONTENT.6.EE.B.6
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

CCSS.ELA-LITERACY.W.6.2.D
Use precise language and domain-specific vocabulary to inform about or explain the topic.

CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.

CCSS.Math.Practice.MP2 Reason abstractly and quantitatively.

Finding Abstract Shape’s Area 6.G.A.1

goldingj Picture Problems

This whole lesson is about how to find the area of abstract shapes. Students would work on breaking the shapes down into simpler ones that they know and then find the area from there. They would start out with finding the area of simple polygon shapes to refresh their memory on how to find the area. Then as the lesson continues, the shapes would get bigger and look more abstract to continue with the idea of finding the area by breaking it down into other shapes.

The main problem the students would be working to solve is: The neighborhood playground is in need of new wood chips to cover the ground beneath the playground. One bag of wood chips covers approximately 25 square feet. How many bags of wood chips will be needed to cover the area the playground sits upon? After determining the answer to this problem, students would have the opportunity to try and find something that was abstract in the classroom and then find the area of that shape. The students would then be asked to write down an explanation for how they found their answer when looking at the abstract shape.

In the classroom, I would use this picture and problem to show students how to find the area of strangely shaped objects. It would also be used to show that even though something may seem like it is going to be hard to accomplish at first, there is always some way to make it easier, which in this case is by breaking the huge shape into smaller shapes.

CCSS.MATH.CONTENT.6.G.A.1

Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

CCSS.ELA-LITERACY.W.6.2

Write informative/explanatory texts to examine a topic and convey ideas, concepts, and information through the selection, organization, and analysis of relevant content.

CCSS.MATH.PRACTICE.MP1: Make sense of problems and persevere in solving them.

CCSS.MATH.PRACTICE.MP4: Model with mathematics.

CSS.MATH.PRACTICE.MP6: Attend to precision.

Student Handouts: Handout

Valentines Day Candy! CCSS. Math.Content.7.SP.C

rodriguees CCSS-Math Learning Progressions, ML Statistics & Probability, Picture Problems

Photo Credit

Learning Objective: I can collect the probability of sweetheart candies; colors or sayings by making collecting data and determine the frequency.

With Valentines Day fast approaching, students can take advantage and do math with candies. In this activity students can use regular 1 oz boxes to determine the frequency of each colored heart, or of each saying. Students can connect math with candy, of course all students enjoy candies! When students are making connections with what is relevant to them, than they are able to retain information better. Students can start out by predicting the number of candies in each box. After students predict they can have a small classroom discussion by sharing their predictions. Students can also share which color or saying in the box is their favorite. When students are are finished having their discussion they can individually count and graph data. To extend this activity students will use their data that they found with their class and determine the average in terms of color or saying. Students can also write down fractions to describe their candy color or saying as a fraction of the total.

To extend this activity Sweetheart candies can be bought in different languages. For example the picture above says “cutie pie”, but instead candies can be purchased to have Spanish sayings instead. Students can become familiar with a different language and incorporate what other countries do to celebrate this holiday.

CCSS.Math.Content.7.SP.C.6

Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.

CCSS.MATH.PRACTICE.MP4 Model with mathematics.

Constructing Shape Art 7.G.A.2

hibbards Picture Problems

CCSS.MATH.CONTENT.7.G.A.2
Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

Art Standard: Students use aesthetic criteria to present and reflect upon artwork.

 

This lesson will focus on both art and math standards. Much of art that you see is based on different shapes and the interaction of different shapes together. The students will learn about the connection between the construction of different shapes and art in this lesson.

The students would have learned how to make constructions with a compass and a protractor before this lesson. This lesson will give the students free reign to create their own artistic shape composition.

Once the art is created the students can use another classmate’s art to find different shapes inside of their artwork. This activity will lead to more assignments about finding shapes in different pieces of art.