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Build, Draw, Write Decomposing Fractions

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Build, Draw, Write Decomposing Fractions

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Represent fractions with denominators from 1 to 100 with circle and bar models.

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Number lines - same length as paper strips (8 1/2")

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Number lines - same length as paper strips (8 1/2")

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Steve Wyborney's Blog: I'm on a Learning Mission.

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Fraction Splats! have arrived! The original Splat! post rocketed up to over 35,000 hits in less than a month, which included over 19,000 lesson downloads! All 50 lessons in the Splat! post involved whole numbers. In this post, I've turned my attention to fractions, and have created 20 new lessons for you to download! This is The Fraction Splat! Series. Watch the video, download the lessons, and try them in your classroom!
Note: I've created the lessons in PowerPoint, so I recommend using PowerPoint to view them. Note: These lessons use a single splat along with a mixture of unit fractions and whole numbers. In Level 11, you control the Splat!
Instant Fraction Splat!
Note: These lessons use a single Splat! which that appears at the same moment as the whole numbers and fractions. Each question instantly presents missing information.
Multiple Splats! with Fractions
Note: These lessons include multiple Splats! and you control the timing of those Splats! The entire screen can be seen, the total can be determined, and then Splat! Remember: Each Splat! that is the same color must be covering the same number.
Multiple Splats! with Fractions
Note: These lessons use multiple Splats! which appear at the same moment as the whole numbers and fractions. Each question instantly presents missing information. Remember: Each Splat! that is the same color must be covering the same number.
Other Posts and Resources Which May Be of Interest

Splat! The original 50 downloadable Splat! lessons include 10 levels and variations.

Tiled Area Questions is a colorful, powerful, highly visual strategy for decomposing numbers - and offers many downloadable pages.

The Animated Multiplication Table remains one of the most popular downloaded resources from this blog, and this post includes several ideas about how to use it.

The Maze Hundreds Chart is another downloadable resource that can lead to some highly intriguing questions and rich discourse.

You may also be interested in Provide Massive Space to Notice.

Please take a moment to subscribe to the blog so you'll be notified each time new resources are posted!

I would treasure hearing about your experiences using The Fraction Splat! Series to promote number sense in your classroom!

All my best,
Steve

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Fraction Splats! have arrived! The original Splat! post rocketed up to over 35,000 hits in less than a month, which included over 19,000 lesson downloads! All 50 lessons in the Splat! post involved whole numbers. In this post, I've turned my attention to fractions, and have created 20 new lessons for you to download!

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Problem Solving - model and solve word problems

Common Core Connection for Grades 3, 4 and 5

Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
Join and separate parts referring to the same whole.
Solve word problems involving addition and subtraction of fractions
referring to the same whole and having like denominators.
Solve word problems involving multiplication of a fraction by a whole number.

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Problem Solving - model and solve word problems Common Core Connection for Grades 3, 4 and 5 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. Join and separate parts referring to the same whole. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators.

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This applet allows students to individually practice working with relationships among fractions and ways of combining fractions.

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Teaching to Inspire with Jennifer Findley

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Teaching students to justify their answer in math can be difficult, but thankfully it doesn't have to be. I want to share with you some ideas for teaching students how to justify answers in math, as well as some of my favorite resources for providing much needed practice with math tasks and word problems.

Any time my students are struggling with something, I like to take a close look at what they are struggling with to see if I can break it down and make it very clear for them. In regard to teaching students how to justify answers, I tried to think of all the ways students could justify their work. Here is the anchor chart that shows some of the ways my students can and have justified their answers in the past. I suggest you have in mind what you are looking for, but have a discussion from the students and get their input as well. The best scenario would be creating the chart together and having the students "help" you come up with the different ways they can justify their answers.

I know the first one on this chart (labeling your work) doesn't technically help justify their answer, but it definitely helps them stay organized and focused. We also talk about how showing all of your work is key to the justification process. For the past two years, I have had more and more students who erase their math work after finding the answer. It blew my mind when I kept seeing them do it over and over, and it took about a month for me to break them of the habit. I always tell them their work is just as important as their answer and their explanation.

Speaking of explaining, I also like to be very deliberate about teaching them how to explain their answer. As you can see from the chart, I show them how to explain how they got their answer, how they chose a particular operation, and how they know their answer is correct. I like to give students sentence stems whenever I can to help scaffold them as well. And finally, I try to push multiple representations to get the students to justify their answer in a variety of ways.

After completing this chart with my students, I like to give them a printable version that they can glue in their interactive notebooks and refer back to as they solve math problems and tasks on their own. It particularly helps my students when they have a clear list they can look over to ensure they have justified their work.

To grab the free printable shown above, click on the image below.

After you have provided clear expectations for your students to follow, the next step is to provide multiple opportunities for them to practice justifying their answers. If you are looking for resources to use to provide practice for your students, here are a few of my favorites.

Differentiated Multi Part Constructed Response Math Tasks: This resource focuses on multi part constructed math tasks that are differentiated and separated by skill. The response section of each task (not the tasks itself) is differentiated at three levels to help students organize their answers and justifications effectively. Click here to grab a freebie to try out on this post or see them in my TpT store by clicking here.

Word Problems for Interactive Math Notebooks: This next one is a student favorite, as well as teacher favorite. My students always love math interactive notebooks. Keeping this in mind, I created a set of interactive math notebook word problems for each of the common core math standards. To read a detailed post about this resource, click here. The students love these because they get to color in the graphics when they are completed, and they are half page which makes them less threatening. I love them because they are stored in their notebook for students to refer back to throughout the year. Click here to see the fourth grade version. Check out the fifth grade version by clicking here.

Constructed Response Math Tasks: I also use standards based constructed response math tasks on a regular basis throughout the year so my students have lots of opportunity to practice justifying and explaining their answers. By doing these throughout the year (about once every two weeks or so), my students become pros at this. The state test at the end of the year is just an extension of all the work they have been doing throughout the year. I typically use these as more of an assessment than the other two examples of resources I shared above. Click on the grade level you want to see in my TeachersPayTeachers store to see more. Also, there are freebies in the previews of each grade level so you can try these out with your students!
How do you help your students become proficient with solving word problems and justifying their answers?
You might also enjoy these posts:

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Teaching Students How to Justify Answers in Math - Teaching to Inspire with Jennifer Findley

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Teaching students to justify their answer in math can be difficult, but thankfully it doesn't have to be.

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Day 1: Which Shapes Show Fourths Formative Assessment

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Day 1: Which Shapes Show Fourths Formative Assessment

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100 Questions that Promote Mathematical Discourse

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100 Questions that Promote Mathematical Discourse

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Ready: 100 questions that promote Mathematical Discourse

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100 Questions to Promote Mathematical Discourse by Ready author Dr. Gladis Kersaint. A free printable with 100 questions to promote mathematical thinking and encourage classroom discourse.

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Sixteenths (same size as most fraction tile sets)

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from Math Geek Mama blog

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Robert Kaplinksy

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Beyond Common Denominators

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Strategies

Given two fractions, how do we compare them? As a young student, the way I learned to compare fractions was to use the strategy of finding common denominators. To complete the process, I simply multiplied the numerator and denominator of the first fraction by the denominator of the second fraction, and then repeated the process by multiplying the numerator and denominator of the second fraction by the denominator of the first fraction. Later, I was taught the "shortcut" of cross multiplication. Both strategies focused on procedural computation with little, if any, understanding of the process.

Fast forward to the Common Core State Standards. Does comparison of fractions look the same way it did years ago? Well sort of, but students are now presented with multiple strategies instead of just one or two, and not all of them are computational in nature. Now, comparison is about building understanding of fractions and then using that conceptual understanding to reason about the size of fractions.
Example 1:
Consider comparison of the following two fractions.

3/4 ___ 3/8

If I were to use the strategy that I was taught in elementary school, I would multiply the 3 and the 4, within 3/4, each by 8 to result in 24/32. Then I would multiply the 3 and the 8 within 3/8 each by 4 to result in 12/32.

Using cross multiplication is essentially the same process as above (in this case, making common units of thirty-seconds). The difference is that there is no mention of the unit; instead there is an emphasis on the quantity of the unit.

Now, think about the same comparison, noticing that the denominators are related denominators (i.e., 4 is a factor of 8). We rename 3/4 in terms of eighths. 3/4 is equivalent to 6/8. 6/8 is greater than 3/8, so 3/4 is greater than 3/8.

Is using one of the above strategies necessary to complete the comparison? As you have likely responded, the answer is no. Consider the following alternative strategies that illustrate a deeper understanding of fractions.

To show 3/4, a tape diagram can be drawn to model the fraction. Each fourth can be decomposed into two equal parts to show eighths. It's easy then, visually, to see that 3/4 > 3/8 (note that this may not be the best strategy to use, however, when the denominators of the fractions being compared are not related).

Similar to the fraction model, a number line can be drawn to model the comparison. Fourths are decomposed to show eighths. 3/4 and 3/8 are labeled, and we can see that the distance from 0 to 3/4 is greater than the distance from 0 to 3/8.

Now consider focusing on common numerators, a strategy that is more abstract in nature. Notice that the units (fourths and eighths) are different, but the quantity of each unit is the same. Since the unit of fourths is greater in size than the unit of eighths, and since there are 3 of each unit, 3/4 is greater than 3/8.

The same comparison can be made using benchmarks fractions. To do this, first reason about the size of 3/4 and 3/8 in comparison to the benchmark of 1/2. 2/4 is equivalent to 1/2, so 3/4 is greater than 1/2. 4/8 is equivalent to 1/2, so 3/8 is less than 1/2. We can reason, therefore (by comparing each to the familiar 1/2), that 3/4 > 3/8.
Example 2:
Now, consider the following comparison.

2/3 ___ 6/8

Again, the focus of this comparison is common numerators; this time, however, the numerators are related (i.e., 2 is a factor of 6).

In 2/3, the unit is thirds and there are two of them. To make common numerators, we need 3 times as many as 2 (resulting in 6). If there are 3 times as many, each third will be decomposed into 3 equal parts, resulting in ninths. 2/3 is equivalent to 6/9. Since ninths are a smaller unit than eighths, 6/9 < 6/8. Therefore 2/3 < 6/8.
Example 3:
Ready for another example? How about 2/3 ____ 7/8? In this example, neither the numerators nor denominators are related. Consider the benchmark of 1 and the relationship of each fraction to 1. 2/3 is 1/3 from 1. Thirds are greater than eighths, so 1/3 is a greater distance from 1 than 1/8 is from 1. Therefore, 2/3 is less than 7/8.

The aforementioned strategies may be new to you. They were certainly new to me as an adult learner and opened my eyes to a new level of thinking. Consider taking some time to think about each strategy and how it lends itself to higher-level thinking and to a deeper understanding of fractions. Then, think about how each strategy can be used by students to help reason about the size of fractions before comparing them. If you would like to see these strategies in action, refer to Lessons 12-15 of Grade 4, Module 5.

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Comparison of Fractions - Beyond Common Denominators

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Given two fractions, how do we compare them? As a young student, the way I learned to compare fractions was to use the strategy of finding common denominators. To complete the process, I simply multiplied the numerator and denominator of the first fraction by the denominator of the second fraction, and then repeated the process by multiplying the numerator and denominator of the second fraction by the denominator of the first fraction.

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Area and Linear
Model Manipulatives

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Area Model

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4th Grade Comparing Fractions task

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Alignments to Content Standards: 4.NF.A.2

Task

This activity is designed for pairs of students. ItÂ usesÂ a set of cards and a fraction mat which are supplied as an attached resource, after the commentaryÂ . The goal is to place eachÂ fraction inÂ the appropriate category on the mat.Â Â Instructions for the activity are as follows:

1. Students follow theseÂ steps with the fraction cards:

1. Each studentÂ in the pairÂ selectÂ a card.

2. Student 1 places his/herÂ card within the appropriate categoryÂ on the mat and gives an explanation for the placement.Â

3. Student 2 either agrees or disagrees with the placement.

4. Â If the partnersÂ disagree, they discuss until reaching aÂ consensus.Â

5. If the partners agree, Student 2 places his/herÂ card andÂ repeat stepsÂ 1 through 4Â with thisÂ card.

2. After 5Â rounds, each pairÂ recordÂ observations about what methods they used to categorizeÂ the fractions.

IM Commentary

The goal of this task is to determine appropriateÂ benchmarks forÂ fractions with a focus on providing explanations that demonstrate deep conceptual understanding. Â The only benchmark fractions addressed in this task are $\frac{1}{2}$ and 1. Some important ideas to look and listen for as the students work are

A fraction is bigger than one if the numerator is larger than the denominator (e.g. $\frac{7}{6}$) and a fraction is less than one if the numerator is smaller than the denominator (e.g. $\frac{2}{5}$). A fraction is equal to one if the numerator is equal to the denominator (e.g. $\frac{3}{3}$).

A fraction is bigger than $\frac{1}{2}$ if the denominator is less than two times the numerator. A fraction is equal to $\frac{1}{2}$ if the denominator is equal to two times the numerator. A fraction is less than $\frac{1}{2}$ if the denominator is more than twice the numerator.

Students can draw pictures of fractions to compare them with $\frac{1}{2}$ and $1$.

Students can plot fractions on the number line to compare them with $\frac{1}{2}$ and $1$ (3.NF.A.2).

Two different sets of cards are provided asÂ attachments, one with a picture of theÂ fractions being compared and one without. The pictures allow students to make a visual comparison of the fractions which is important. However, the teacher may wish for students to use a different method of comparison or toÂ provide these pictures as theyÂ explainÂ their decision. Some of the pictures such as those of $\frac{1}{3}$ and $\frac{2}{6}$, are not designed to visually demonstrate equivalence so the cards with pictures will still help develop non visual understanding of fractions.

Differentiated fraction mats are provided as attachmentsÂ to allow students to work with broader benchmark ranges. These mats can be used in progression based on student understandings.Â The first mat asksÂ students to identify fractions greater than or less than 1. The second mat asksÂ students to identify fractions equivalentÂ to 1/2 and 1.Â Â The third mat asksÂ students to use the prior understandings of the previousÂ mats toÂ identify fractions that fall within the benchmarks of 1/2 and 1.Â

There are four attachments below:

Fraction Cards No Picture (these have a fraction with no picture)

Fraction Cards Picture (these have a fraction with picture: if students use these the teacher should watch to make sure that students name the whole for their fraction. The intention is that the large squares are the whole but a student could think differently and then correctly classify the fraction in a different way)

Benchmark fraction mats (these mats have all possible categories for the fractions from less than $\frac{1}{2}$ through greater than $1$)

Differentiated fraction mats (described in the previous paragraph)

Solution

1. In order to place a fractionÂ in the proper category students must compare the fraction to $\frac{1}{2}$ and $1$. There are many strategiesÂ to do this and the student does not need to follow the particular order of reasoning describedÂ here:

Fractions whose value is equalÂ to 1. These are perhaps the simplest fractions to recognize:Â for example, $\frac{8}{8}$Â is equal to 1 wholeÂ because it takes 8 equal piecesÂ of 1/8 combine to createÂ aÂ whole.Â

Fractions equivalent to 1/2. For example 2/4Â is equivalent to 1/2 because the numerator is half of the denominator.Â One way to show that these fractions represent the same quantity is with a picture:

Â Â Â
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Fractions whose value isÂ less than $\frac{1}{2}$. For example, $\frac{3}{8}$Â is less than $\frac{1}{2}$Â because $\frac{4}{8}$ is equivalent to $\frac{1}{2}$and $\frac{3}{8}$ is $\frac{1}{8}$ less than $\frac{1}{2}$, therefore less than $\frac{1}{2}$.Â Â Â Â Â Â

Fractions whose value is greater than $\frac{1}{2}$, but less than 1. Â For example, $\frac{5}{8}$ is greater than $\frac{1}{2}$ because $\frac{4}{8}$ is equivalent to $\frac{1}{2}$ and $\frac{5}{8}$ has an additional fourth. $\frac{5}{8}$ is less than 1 because $\frac{8}{8}$ is equivalentÂ to 1 and $\frac{5}{8}$Â needs threeÂ more $\frac{1}{8}$'s to make a whole. Alternatively, students might draw a number line and show that $\frac{5}{8}$ lies to the right of $\frac{1}{2}$ (which is equalÂ to $\frac{4}{8}$)Â and to the left of $1$ (which is equalto $\frac{8}{8}$):

Â Â

Fractions greater than 1. For example, $\frac{6}{4}$ is greater than 1 because $\frac{4}{4}$ is aÂ wholeÂ and $\frac{6}{4}$Â has two additional fourths in addition toÂ the whole.Â

2. There are many important lessons to be learned from comparing fractions toÂ benchmarkÂ fractions, including:Â

The denominator tells me how many pieces to cut my whole into. When the whole is cut into more pieces, theÂ pieces are smaller (this is why $\frac{1}{3}$ is less than $\frac{1}{2}$).

The numerator tells me how many equal sized pieces I have. So $\frac{6}{10}$ is more than $\frac{5}{10}$ because I have one extra piece.Â

Fractions are built from the unit fractions so it is important to understand and be able to represent the unit fractions.

If using the fraction cards with pictures,Â equalÂ sizedÂ wholes areÂ important when comparing fractions.Â Â

In a picture of equivalent fractions, the whole is broken intoÂ different sized pieces, but the same total amount is shaded.Â

The relationship between the numerator and denominator can help me think about the value of the fraction.

FractionsÂ equivalent to $\frac{1}{2}$ have a numerator that is half of the denominator.

Fractions equivalent to 1 whole have the same numerator and denominator.

When the numerator is a bigger number than the denominator, the fraction is greater than one whole.Â

When doing mathematics, patterns emerge. These patterns support students in making conjectures, supporting their reasoning, and proving mathematical claims.

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Alignments to Content Standards: 4.NF.A.2 This activity is designed for pairs of students. ItÂ usesÂ a set of cards and a fraction mat which are supplied as an attached resource, after the commentaryÂ .

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5th Grade Sharing Lunch task (adding and multiplying fractions)

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Task

Alex, Bryan, and Cynthia are about to eat lunch, and they have two sandwiches to share.

1. Draw a picture to show how they could equally share the sandwiches.Â How much of a sandwich does each person get?

2. Write an equation involving addition to show how together these parts make up the 2 sandwiches. Explain how the equation you wrote represents this situation.

3. Write an equation involving multiplication to show how all the parts make up the 2 sandwiches. Explain how the equation you wrote represents this situation.

4. Write an equation using multiplication to show the fraction of a sandwich each student gets. Explain how the equation you wrote represents this situation.

5. Write an equation using division to show the fraction of a sandwich each student gets.Â Explain how the equation you wrote represents this situation.

Â

IM Commentary

This task requires students to think about how a single situation involving fractions can be accurately represented using addition or multiplication.Â

Students are first asked to draw a picture in part a showing this situation to reinforce the fact that the same situation can be modeled using three different operations. Students are asked to divide two sandwiches into three equal pieces, and must reason what that will look like.Â Students should be able to realize fairly easily that each student will get $\frac{2}{3}$ of the sandwich, and to model this using in addition in part b.Â As students' facility with operations on fractions matures, they should not be expected to always draw a picture.

In part c, students are asked to representÂ the situation using multiplication, and to explain their representation.Â Teachers may want to show multiple correct explanations, including interpretations of products as partitions of equal parts, visual displays (such as a number line or fraction diagram), or a mathematical statement such as $\frac23 + \frac23 + \frac23 = 3Â \times \frac23 = 2$ If students struggle, the teacher can tell the students to think of the problem as $2$ sandwiches $= 3Â \times$ a fraction of a sandwich.Â A teacher might also want to ask students to demonstrate multiple correct explanations, particularly if they notice students relying only on one method and being confused by others.

Parts d and e shift the focus away from using operations to represent all of the sandwiches, instead using operations to focus on each studentâ€™s share of the sandwiches. In part d, students may notice that each person will get $\frac13$ of the $2$ sandwiches, since there are $3$ people sharing them equally.Â Students may then need help realizing that since they will get $\frac13$ of each of the two sandwiches, they then will get $\frac13 \times 2$ sandwiches, which equals $\frac23$ The second diagram in the solution to part a may help, as each of the two sandwiches is divided into three equal parts (therefore,
$\frac13 \times 2$).

Finally, students are asked to consider the meaning of a fraction in part e â€" that is, the division of a numerator by a denominator.Â This task reinforces the concept that any fraction can be described as dividing a whole number into the number of pieces specified in the denominator. The explanations that the students produce for each of the three tasks may be slightly different, but will result in the same conclusion (that each student gets $\frac23$ of a sandwich).Â If students struggled with this concept, the teacher might note that if there were six sandwiches and each person got the same amount, we would use division to determine that each person gets two sandwiches $(6 \div 3 = 2)$.Â The situation here is the same, except instead of $6$ wholes, the unit is $6$ thirds.

This task was written as part of a collaborative project between Illustrative Mathematics, the Smarter Balanced Digital Library and the Teaching Channel.

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Solutions

Solution: Solution to part A

A. Students should say that each person gets $\frac23$ of one sandwich.Â The picture could be anything that shows the sandwiches being divided equally three ways.Â For example:

Â Â Â Â Â

Or

Â Â Â

The second solution has a few advantages. First, this type of picture prepares students for the tape diagrams they will see in the sixth grade when they study ratios. Secondly, this method of division into equal parts would apply if, for example, one of the sandwiches were tuna fish and the other was grilled cheese. Â Each person would receive an equal part of each type of sandwich. Third, for a more complex fraction such as $\frac{3}{5}$ the equal portions can be seen very readily (without counting) with this type of model.Â

Solution: Solution to part B

B. $\frac23 + \frac23 + \frac23 = \frac63 = 2$. Other reasonable equations, such asÂ $\frac23 +Â \frac23 + Â \frac13 +Â \frac13 Â = \frac63 = 2$, could be acceptable. Â Each person gets $\frac23$ of a sandwich and there are three people so if you add up these three pieces of size $\frac23$ you get a total of 2.

Solution: Solution to part C

C. $3 \times \frac23 = 2$. Acceptable explanations include accurate descriptions (e.g. â€œYou add two-thirds three times, which is the same as multiplying it by $3$â€), a number line model (showing $\frac23$, $\frac43$, and $2$) or a visual fraction diagram.

Solution: Solution for part D

D. $2 \times \frac13 = \frac23$. Each person gets $\frac13$ of each sandwich and there are two sandwiches so two groups of $\frac13$ is equal to $\frac23$.

Solution: Solution for part E

E. $2 \div 3 = \frac23$. Acceptable explanations include accurate descriptions (e.g. â€œYou divide two wholes into three equal parts, and each part will be $\frac23$â€), a number line model (focusing on the dividing two wholes into three parts), or a visual fraction diagram.

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Alex, Bryan, and Cynthia are about to eat lunch, and they have two sandwiches to share. Draw a picture to show how they could equally share the sandwiches.Â How much of a sandwich does each person get? Write an equation involving addition to show how together these parts make up the 2 sandwiches.

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The most popular manipulatives at your fingertips. Powerful yet easy-to-use.

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Dividing by one-half Illustrative Math Task

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Alignments to Content Standards: 5.NF.B.7

Task

Solve the four problems below. Which of the following problems can be solved by finding $3 \div \frac12$?

1. Shauna buys a three-foot-long sandwich for a party. She then cuts the sandwich into pieces, with each piece being $\frac12$ foot long. How many pieces does she get?

2. Phil makes $3$ quarts of soup for dinner. His family eats half of the soup for dinner. How many quarts of soup does Phil's family eat for dinner?

3. A pirate finds three pounds of gold. In order to protect his riches, he hides the gold in two treasure chests, with an equal amount of gold in each chest. How many pounds of gold are in each chest?

4. Leo used half of a bag of flour to make bread. If he used 3 cups of flour, how many cups were in the bag to start?

IM Commentary

This task requires students to recognize both "number of groups unknown" (part (a)) and "group size unknown" (part (d)) division problems in the context of a whole number divided by a unit fraction. It also addresses a common misconception that students have where they confuse dividing by 2 or multiplying by $\frac12$ with dividing by $\frac12$.

Solution

$$ 3 \div \frac12 = \frac31 \times \frac21 = \frac61 = 6 $$

1. If Shauna buys a 3 foot long sandwich and cuts it into pieces that are $\frac12$ foot long, her sandwich would look something like the picture below.

By looking at the picture above, we can see that there are 6 pieces that are $\frac12$ foot long, which gives the same answer as if we were to compute $3 \div \frac12$.

We can also think about this problem without the help of a picture. Shauna has a sandwich that is 3 feet long. If she slices that sandwich into pieces so that each piece is $\frac12$ foot long and wants to know how many sandwich slices she would get, Shauna is really asking, â€œHow many groups of $\frac12$ are in 3?â€ which is one of the interpretations of $3 \div \frac12$.

2. If Phil makes 3 quarts of soup for dinner, and his family eats half of the soup for dinner, his situation would look something like the picture below.

From this picture above, we can see that Philâ€™s family ate $1 \frac12$ quarts of soup. This problem cannot be modeled by $3\div \frac12$.

We can also think about this problem without the help of a picture. If Phil makes 3 quarts of soup for dinner, and his family eats $\frac12$ of the soup he made, then his family really ate $\frac12$ of the 3 quarts of soup. This is the meaning of $\frac12 \times 3=\frac12 \times \frac31= \frac32=1 \frac12$, not $3 \div \frac12=6$.

3. If a pirate finds 3 pounds of gold and hides an equal amount of gold in each of 2 chests, his situation would look something like the picture below.

From the picture above, we can see that each chest would hold half of the three pounds of gold, or $1 \frac12$ pounds of gold. This problem cannot be modeled by 3Ã·1/2.

We can also think about this problem without the help of a picture. If a pirate finds 3 pounds of gold, hides an equal amount of gold in each chest, and wants to know how much gold is in each chest, he is really asking, â€œHow much is in each group if 3 is evenly distributed amongst 2 groups?â€ This is the meaning of $3 \div 2 = 3 \div \frac21= \frac31 \times \frac12= \frac32=1 \frac12$, not $3 \div \frac12=6$.

4. From the picture above, we can see that if 3 cups of flour is $\frac12$ the bag, then 6 cups of flour were in the bag to start.

This problem is asking "3 cups is $\frac12$ of what amount?" which we can represent symbolically like this: $$\frac12 \times ? = 3$$ which is equivalent to $$3\div \frac12 = 6$$ This is the "How many in one group?" interpretation of $3\div \frac12$.

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Alignments to Content Standards: 5.NF.B.7 Solve the four problems below. Which of the following problems can be solved by finding $3 \div \frac12$? Shauna buys a three-foot-long sandwich for a party. She then cuts the sandwich into pieces, with each piece being $\frac12$ foot long. How many pieces does she get?

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5th grade worked example

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6th grade worked example

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5th grade worked example

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6th grade worked example

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Fraction Number Lines.pdf - Google Drive

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Fraction Number Lines.pdf - Google Drive

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Access Google Drive with a Google account (for personal use) or Google Workspace account (for business use).

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Jo Boaler, YouCubed

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Choose your own maths adventure with our interactive tools that allow you to build a custom playlist of inspirational activities!

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Oakdale Joint Unified School District: Math Homework Help

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Oakdale Joint Unified School District: Math Homework Help

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Engage New York (ENY) Homework provides additional practice for math that is learned in class.

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Common Core Math | Berkeley Unified School District

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Common Core Math | Berkeley Unified School District

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Here parents and teachers can find helpful resources to help students succeed.

Common Core Math Standards

Parent Resources

Twelve Steps To Increase Your Child's Math Achievement And Make Math Fun
Jo Boaler, Professor of Mathematics Education, Stanford University

Teacher Resources

Additional Teacher Resources
Scaffolding Common Core Math Instruction for English Learners

Common Core Math Family Letters (English)

Cartas para Familias (español)

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Common Core Math | Berkeley Unified School District

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Here parents and teachers can find helpful resources to help students succeed. Twelve Steps To Increase Your Child's Math Achievement And Make Math Fun Jo Boaler, Professor of Mathematics Education, Stanford University Additional Teacher Resources Scaffolding Common Core Math Instruction for English Learners

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The Teach Eureka Video Series contains 18 one-hour professional development sessions for each grade (PK-12) organized sequentially by module. In this on-demand video series, the curriculum's authors explain the mathematical concepts and instructional strategies necessary to make Eureka your own.

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Eureka Math Pacing and Preparation Guides

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Eureka Math Pacing and Preparation Guides

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Great Minds is a non-profit organization founded in 2007 by teachers and scholars who want to ensure that all students receive a content-rich education.

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SBAC 4th Grade Sample Item

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Important Thing About Fractions

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Counting Goldfish Task

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Story Contexts for Multiplying

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Fourth Grade Ice Cream Party, Multiplying Fractions

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Baking Cakes, Adding and Subtracting

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Mixed Numbers BDW

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Candy Bar Fractions, Adding and Subtracting

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Contextual Problems (Unlike Denominator)

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Decimal Squares

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Number Line Template

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Compare Rods

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Cover Up / Uncover Recording Sheet

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Build Draw Write - Converting Mixed Numbers to Fractions

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Build Draw White - Converting Fractions to Mixed Numbers

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Build Draw Write Part-Whole Meaning of Fraction

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Build Draw Write - Equivalent Fractions

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Eggsactly Eggs

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Week of Inspirational Math from Jo Boaler

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Week of Inspirational Math from Jo Boaler

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youcubed.org

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Choose your own maths adventure with our interactive tools that allow you to build a custom playlist of inspirational activities!

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Modeling Fraction Multiplication with Equal Groups: Video

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Modeling Fraction Multiplication with Equal Groups: Video

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Modeling Fraction Multiplication with Equal Groups

This video presents a visual model for solving a word problem involving the multiplication of fractions. The problem asks students to use the amount of bamboo that a panda eats in an hour to determine how much he eats in a quarter of an hour if he eats at a steady rate. The video ends with a challenge problem for students to model themselves. In the accompanying classroom activity, students use visual models to work through the challenge problem. This resource is part of the Math at the Core: Middle School Collection.

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Modeling Fraction Multiplication with Equal Groups

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This video presents a visual model for solving a word problem involving the multiplication of fractions. The problem asks students to use the amount of bamboo that a panda eats in an hour to determine how much he eats in a quarter of an hour if he eats at a steady rate.

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Fraction Multiplication with Arrays: Video

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Fraction Multiplication with Arrays: Video

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Modeling Fraction Multiplication Using Arrays

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This video illustrates a word problem involving the multiplication of fractions using an array to model the solution visually. The problem involves puffins, arctic birds that live in colonies and fish to feed their young. The accompanying classroom activity provides students with practice solving additional fraction multiplication problems using arrays.

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Fraction Multiplication with Arrays Interactive

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Fraction Multiplication with Arrays Interactive

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Practicing Fraction Multiplication with Arrays

In this interactive game, students use an array to solve word problems involving fraction multiplication. Specifically, they are asked to determine what fraction of a puffin colony is made up of males that are fishing. The game, which can be played in English or Spanish, includes a randomization feature that allows students to practice solving the same problem using different values. The accompanying classroom activity suggests ways that teachers can integrate this game into the classroom. This resource is part of the Math at the Core: Middle School Collection.

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Practicing Fraction Multiplication with Arrays

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In this interactive game, students use an array to solve word problems involving fraction multiplication. Specifically, they are asked to determine what fraction of a puffin colony is made up of males that are fishing. The game, which can be played in English or Spanish, includes a randomization feature that allows students to practice solving the same problem using different values.

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Open Middle - Challenging math problems worth solving

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Open Middle offers interactive math problems designed to foster critical thinking. Explore unique challenges that boost problem-solving skills and creativity.

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Thinking Critically With Fractions & Number Lines

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Thinking Critically with Fractions and Number Lines

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Looking for ideas to teach fractions? Check out Name That Point for a cool fraction number line estimation activity.

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Wordle word cloud example

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Directions for making a number line in Microsoft Word

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Directions for making a number line in Microsoft Word

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Decomposing with fraction tiles

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Multiple Representations Template

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Georgia Department of Education

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Mathematics Georgia Standards of Excellence (GSE) K-5

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These fraction games for kids boost understanding, critical thinking, and social interaction. Best yet, they're fun!

This activity, designed for two students, is just like the traditional card game War, but with fractions! To prepare the deck, print this free Fraction War cards printable and cut the cards out.

Students deal out the deck evenly and place their cards in a pile, face-down in front of them. Each student draws one card at a time and places it face-up in front of them. Students then compare their fractions using the colored fraction bars. The player with the highest fraction gets to keep both cards. This fraction game is great for comparing greater than, less than, and equal to.

2. Do some Newspaper Fraction Dancing.

Turn on the music and boogie down! This math activity comes from Mrs. King Rocks! Kids learn about fractions by folding and dancing on newspaper. Visit Mrs. King's Music Class to see this activity in action.

3. Take them outside and play Fraction Hopscotch.

This fraction game for kids will get your students learning outside. Kids will design their own hopscotch board- with a twist. All of the squares must be labelled with fractions. For a refresher course on how to play hopscotch, click here.

4. Go on a Fraction Picnic.

For a fun math station, provide a variety of food images cut out of magazines or printed out from the internet. Students will plan a picnic and choose what food they want to bring along. Using a ruler and scissors, they will cut portions of their food choices and glue them to a paper plate. Finally, they will label each item with the fraction name. This is a great activity for practicing fraction skills and understanding the concept of division.

5. Play Fraction Pictionary.

This fraction game for kids comes from the Oregon Department of Education.

Divide students into teams. One child draws an picture that includes a fraction on the board. Their teammates try to correctly guess what the image is as well as the fraction. If their team guesses correctly they get a point.

The student who is drawing can divide single objects to represent fractions (shown above) or they can draw multiple objects. For example for 2/5 dogs, a student could draw two dogs and three cats.

6. Play Shake and Number Bond.

All you need is a plastic cup and two-sided counters. Kids shake the cup and pour the counters on the table. Then, without flipping any of their counters over, they count how many of each color landed face up. For example, if 13 counters were rolled, six red and seven blue landed face up. Six and seven are both parts of thirteen (the whole). Have students write the number bond (see illustration above) and fractions for each color.

You could also use 10 frames with this game. After kids spill their counters on the table, they place the counters on 10 frames. Kids see that 13 equals one 10 and three ones, which helps with place value skills.

7. Play a round of Equivalent Fractions Musical Plates.

Get children up and moving with this fun, kinesthetic fraction activity. Children are given a fraction and dance around to music, trying to find its equivalent. Written and visual representations of fractions are used to assess children's understanding!

Give each student a fraction written on a sticky note. Students remember their fraction and stick it on their shirt. Next, lay out paper plates with fractions written or pasted on them. You should make a variety of equivalent fraction plates for each student and spread them far apart so that students will have to really look around to find a plate that is equivalent.

Start the music. Students dance and hunt for a fraction that is equivalent to their own. When they find it, they stand on the paper plate. If they don't find an equivalent fraction before the music stops, they're out! For each round, students keep hunting for fractions. Once there are no more fractions that are equivalent to their own, they sit out. They've found all of their fractions!

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7 Fun Fraction Games For Kids (Teacher Approved!)

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These fraction games for kids boost understanding, critical thinking, and social interaction. Best yet, they're fun! This activity, designed for two students, is just like the traditional card game War, but with fractions! To prepare the deck, print this free Fraction War cards printable and cut the cards out.

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Investigating Fractions with Pattern Blocks

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K-5 Math Teaching Resources Link - 3rd Grade

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With over 150 easy-prep, engaging centers this resource will simplify your math lesson planning and make hands-on math instruction an integral part of your classroom.

This page provides sample 3rd Grade Number tasks and games from our
3rd Grade Math Centers eBook. Try out the samples listed in blue under each Common Core State Standard or download the eBook and have all the 3rd Grade Number, Geometry, Measurement and Data Centers you'll need for the entire school year in one convenient digital file. Teaching in a state that is implementing their own specific math standards? Download our 3rd Grade Correlations document for cross-referenced tables outlining the alignment of each state's standards with the CCSS-M, as well as the page numbers in our 3rd Grade Math Centers eBook related to each standard.

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With over 150 easy-prep, engaging centers this resource will simplify your math lesson planning and make hands-on math instruction an integral part of your classroom. This page provides sample 3rd Grade Number tasks and games from our 3rd Grade Math Centers eBook.

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5th Grade

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eBook. Try out the samples listed in blue under each Common Core State Standard or download the 5th Grade Math Centers eBook and have all the 5th Grade Number, Geometry, Measurement and Data Centers you'll need for the entire school year in one convenient digital file. With over 170 easy-prep, engaging centers this resource will simplify your lesson planning and make hands-on math instruction an integral part of your classroom. that is implementing their own specific math standards? Download our
This page provides sample 5th Grade Number tasks and games from our th Grade Math Centers
Teaching in a state 5th Grade Correlations document for cross-referenced tables outlining the alignment of each state's standards with the CCSS-M, as well as the page numbers in our 5th Grade Math Centers eBook related to each standard.

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eBook. Try out the samples listed in blue under each Common Core State Standard or download the 5th Grade Math Centers eBook and have all the 5th Grade Number, Geometry, Measurement and Data Centers you'll need for the entire school year in one convenient digital file.

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K-5 Math Teaching Resources Link - 4th grade

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eBook. Try out the samples listed in blue under each Common Core State Standard or download the 4th Grade Math Centers eBook and have all the 4th Grade Number, Geometry, Measurement and Data Centers you'll need for the entire school year in one convenient digital file. With over 140 easy-prep, engaging centers this resource will simplify your lesson planning and make hands-on math instruction an integral part of your classroom. that is implementing their own specific math standards? Download our

Teaching in a state 4th Grade Correlations document for cross-referenced tables outlining the alignment of each state's standards with the CCSS-M, as well as the page numbers in our 4th Grade Math Centers eBook related to each standard.
This page provides sample 4th Grade Number tasks and games from our 4th Grade Math Centers

Generalize place value understanding for multi-digit whole numbers

Use place value understanding and properties of operations to perform multi-digit arithmetic

Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers

Understand decimal notation for fractions, and compare decimal fractions

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eBook. Try out the samples listed in blue under each Common Core State Standard or download the 4th Grade Math Centers eBook and have all the 4th Grade Number, Geometry, Measurement and Data Centers you'll need for the entire school year in one convenient digital file.

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MATH Poster

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Inside Mathematics - Problems of the Month Link

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Inside Problem Solving | Inside Mathematics

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The Inside Problem Solving problems are non-routine math problems designed to promote problem-solving in your classroom. Each problem is divided into five levels of difficulty, Level A through Level E, to allow access and scaffolding for students into different aspects of the problem and to stretch…

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NRich Math Tasks Link

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Students

In case you missed our Primary and Secondary Challenges during the summer break, we are continuing to feature them, so you can enjoy them for another month

Your Solutions

See if your solutions to our recent problems have been published
Primary
Secondary

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NRICH - Mathematics Resources for Teachers, Parents and Students to Enrich Learning

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The Nrich Maths Project Cambridge,England. Mathematics resources for children,parents and teachers to enrich learning. Problems,children's solutions,interactivities,games,articles.

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Estimation 180 Link

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ESTIMATION 180 is a place for both students and teachers to enjoy making sense of math and develop their number sense together.

Pick from over 200 visual estimation tasks that invite all your students to be part of rich math conversations and learning.

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How do I get started? 1 USE THE FREE MATH LESSONS Estimation 180 Tasks 3-Act Tasks 2 SUBSCRIBE Email Newsletter Podcast

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Tangram fractions

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Fractions Games Link

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Fractions tiles (color and b&w)

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Fractions on a Roll Game

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Illuminations Interactive Fractions Equivalents

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Fraction Scoot game

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Fractions cards

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Capture Equivalents game

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Adding like denominators (use with unifix cubes)

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Illustrative Math Link

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Illustrative Mathematics provides resources and support for giving their students an enduring understanding of mathematics.

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Mathematics Formative Assessment System Link

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mathematics formative assessment system (mfas). this system includes 230 formative assessment tasks for grades k-3

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2 professional development programs and 4 modules

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Mathematics Formative Assessment System (MFAS)

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Mathematics Formative Assessment System (MFAS). This system includes hundreds of formative assessment tasks for grades K-8, professional development programs and modules, and lesson study resource kits.