Solving Ratios with a Tape Diagram

Once students understand how the tape diagram works, they are ready to start solving ratio problems using this visual tool.  You can print and laminate ratio cards, or have students draw boxes.

We are going to look at 2 examples on how to use the tape diagram to solve ratio problems.  The first example is when we are given one of the parts and the second example is when we are given the total.

Example 1:  The ratio of boys to girls is 2:3.  If there are 10 boys, how many girls are there? 

Have students represent the ratio with their tape diagram cards and label boys and girls.

 

If there are 10 boys, that means each of the two cards must be worth 5.  Since all of the ratio cards have to be the same in order to have an equivalent ratio, put a 5 on every card.  This means that there must be 15 girls.

Example 2:  The ratio of cats to dogs in a shelter is 5:1.  If there are 24 animals, how many cats are there?   

Have students represent the ratio with their tape diagram cards and label cats and dogs.

This time, we were given the total number of animals.  That means that the 6 total cards is representing 24.  Therefore, each card must be worth 4, because 24 divided by 6 is 4.  Our final answer is 20 cats.

I hope this visual model is an effective strategy for your students!
~MN

Intro to Using a Tape Diagram for Ratios

Tape diagrams are a great visual tool for understanding ratios.  Students can easily draw tape diagrams or you can print and laminate tape diagram cards and have students write on them with dry erase markers.  I like using the tape diagram cards because sometimes students have a hard time making equal size boxes.

Example 1

Have students represent the ratio 2:3 using the tape diagram cards or by drawing 2 boxes and underneath, 3 boxes.

When we write on the tape diagram cards, we must write the same number in each card in order to keep our ratios equivalent.  This is super important!  Let’s write 2 in each box.  That would give us 4 in the top ratio and 6 in the bottom ratio.  Therefore, a ratio of 4:6.  That means that the ratio of 2:3 equals the ratio of 4:6.

If we write a 3 in each box, that would give us a ratio of 6 to 9.  If we write a 4 in each box, that would give us a ratio of 8:12.  All of the ratios when simplified equal 2:3.

Example 2

Let’s look at another example.  Represent the ratio 6:5.

Have students write a number of their choice in the boxes to represent equivalent ratios.  Have students share their results.  Some possible answers would be 12:10, 18:15, 24:20.

Discuss with students how it is important to keep the order of the ratio the same.  If the first number is larger than the second, it should stay that way when you find your equivalent ratios.  Also, we do not convert ratios into mixed numbers.

Once students have a strong understanding of how the tape diagram works, they are ready to start solving ratio problems using the tape diagram.

~MN

 

Finding the Missing Whole using Bar Models

After students are able to find the percent of a number, they can explore what it means to find the missing whole.  We will look at 3 examples of how to teach this with percent bars.

Example 1

6 is 20% of what ?  First, we need to break our model into 10% sections.  6 is the 20%, so above 20% put the 6.

Now we have to determine what the whole bar is worth.  If 20% is worth 6, then 10% is worth 3.  If we continue to count up by 3, we can see that the whole bar is worth 30.  Therefore, 6 is 20% of 30.

Example 2

12 is 75% of what?  Since this problem is asking for 75%, we can break our model into 25% sections.

12 is the 75% so above 75% we are going to write a 12.  If 75% is worth 12, then every 25% is worth 4.  if we count up by 4 we see that the whole bar is worth 16.

Example 3

12 is 15% of what?  If we use our 10% section model, we can determine that 15% is right between 10% and 20%.

There are 3, 5% sections in 15% so if we divide 12 by 3 we can determine that each 5% section is worth 4.  If we continue counting up, we see that the whole bar is worth 80.  Therefore, 12 is 15% of 80.

Using percent bar models has been monumental for my students.  This visual gives them what they need to understand what it means to find the missing whole.  I hope your students find success with this!

 

 

 

Finding the Percent of a Number with Bar Models

Percent bars are a great way to get students to understand what it means to find the percent of a number.  Before students start finding the percent of a number, they should have some understanding of equivalent ratios and should understand that percent is a special type of ratio that always compares number to 100.

When we are finding the percent of a number, we are using proportional reasoning to create two equal ratios.  This why percent bars are such a great visual tool.  On the bottom of the bar we are looking at the percent’s and at the top of the bar we are looking at the number we are finding the percent of.

Example 1

What is 80% of 20?”  First, we have to break our bar model into 10% sections.  20 is the whole amount that we are trying to find the percent of.  Therefore, we want to line it up with 100%.  Explain to students that if we had 20 out of 20, that would be 100%.

Have students ignore the percent’s for a minute and just look at the 20.  If 20 represents the whole bar, what is each box worth?  Students should see that since there are 10 boxes, each box is worth 2 because 20 divided by 10 is 2.

Since the question is asking us to find 80% of the number, we are going to shade in up to the 80%.  Now we can easily see that 80% of 20 is 16.  To relate this to a real-life example, we could say if you received an 80% on a test out of 20 questions (all worth the same amount), that means you got 16 questions right.

Example 2:

“What is 15% of 40?”  We are going to start by placing 40 at the end, above 100%.

Since 40 divided by 10 if 4, we can fill in the top number line by counting by 4’s.

If we want 15%, that would be a full box and half of a box.  A full box is worth 4, half a box is worth 2, so 15% would be 6.  15% of 40 is 6.

Example 3

What happens when out last number is not divisible by 10 like in the problem “What is 20% of 36?”  Start by placing 36 above 100%.  36 divided by 10 is 3.6, so we can fill in the top number line by counting by 3.6.  Shade in 2 boxes for 20% and see that 20% of 36 is 7.2.

Example 4

You can even use this visual for smaller percents.  For instance, what is 2% of 30?  We would place 30 above 100%.  Since 30 divided by 10 is 3, we would place a 3 in each box.  If 10% is equal to 3, then 1% would be one-tenth of 3 or 0.3.  If 1% is 0.3, then 2% would be 0.6.  2% of 30 is 0.6.

Using bar models will be monumental for your students understanding of what it means to take the percent of a number!  Once students practice solving percent problems this way, they will start to gain a deeper understanding of what it means to find the percent of a number.  Once that deeper understanding starts to happen, students can move to a more abstract way to solve percents problems such as the percent proportion.

~MN

Converting Fractions to Percents

10×10 grids are a great tool when introducing percents because they provide a visual for fractions that are out of 100.  Before starting the lesson, discuss with students that the word percent means,  “out of 100.”  As soon as we have a number out of 100, we have a percent.  Percents are a useful tool for comparing because they make every number out of the same thing, 100.  It is much easier to compare 80% and 90% than to compare 4/5 and 9/10.

Have students represent 3/100 by coloring in 3 squares out of 100.  “What percent is this?”   Since 3 are shaded in out of 100, this is 3%.

Have students represent 3/10.  Discuss how the fraction means 3 out of 10, therefore we need to start by breaking our whole into 10 equal groups.  Students may do this differently and that is fine, as long as they have 10 equal groups.

Once students have 10 equal groups, have them shade in 3 of the 10.  Then ask, “what percent is this?”  Since percent is out of 100, we have to think, how many are shaded in out of 100, or how many of the tiny squares are shaded in.  Students should see that 30 out of 100 are shaded and therefore 3/10 = 30/100 or 30%.

Have students represent 3/4.  We need to start by breaking the whole into 4 equal groups.  Again, students may break the whole into fourths differently which is fine, as long as there are 4 equal groups.

Then have students shade in 3 of the 4 groups.  Ask, “what percent is this?”  Since there are 75 shaded out of 100, 3/4 = 75/100 = 75%.

Have students try several examples like this until they have a strong understanding of how fractions and percents relate.  It is so important that students get a visual when making sense of fractions and percents.

~MN

Fraction Divided by a Fraction

Using visuals is essential for deep mathematical understanding.  Today we are going to explore what happens when two fractions are divided.

First, we are going to look at 1/3 ÷ 1/4.  Have students represent both fraction tiles.  “How many times can 1/4 fit in 1/3?”  Have students ponder that questions and discuss with a partner an estimate.

Students should have a sense that since 1/3 is larger than 1/4, it can fit in 1 time with a little bit left over.  How can we find out what fraction is left over?  Let’s explore!  If you line up the twelfth piece, you can see that there is 1/12 left over.  But how does this relate to what we are dividing by, the 1/4 piece.  If you convert fourths into twelfths, you can see that 1/12 is 1/3 of 1/4.  Therefore, 1/3 ÷ 1/4 is 1 and 1/3.

Let’s look at the reverse, 1/4 ÷ 1/3.  “How many times can 1/3 fit in 1/4?”

This time, the divisor is larger than the dividend.  Students should see that 1/3 can’t fit into 1/4, because 1/3 is larger then 1/4.  Therefore, our answer must be less than 1.  We have to determine what fraction 1/4 is of 1/3.  If we use our twelfth pieces again, we can see that 1/4 = 3/12 and 1/3 = 4/12.  “What fraction of 4/12 is 3/12?” It is 3/4 of 4/12.  Therefore, 1/4 ÷ 1/3 is 3/4.

Try some more similar examples with your students using the fraction tiles.  You will see that they start to really develop that deep mathematical understanding that will stay with them forever.

~MN

CCSS:  6.NS.A.1  and 7.NS.A.2

Check for understanding:  When you divided a smaller fraction by a bigger fraction, does your answer get bigger or smaller?  Explain.  When you divided a bigger fraction by a smaller fraction, does your answer get bigger or smaller?  Explain.

Fraction divided by a Whole Number

Today we will be focusing on a fraction being divided by a whole number.  I will be using Fraction Tiles which can be purchased here, or you can print and laminate a set of fractions tiles for free here.

Let’s look at the problem 6/8 ÷ 2.  Have students represent 6/8 with their fraction tiles.

We could say, “How many times can 2 fit in to 6/8?”  Have students ponder that question and have a discussion about it.  Students may be puzzled because it can’t fit in at all.  Therefore, it makes sense that our answer is going to be a fraction.  Have students line up 16/8 to represent 2 wholes.  We could say we have 6/8 out of 16/8.  Therefore, 6/16 which simplifies to 3/8.  So, 6/8 divided by 2 = 3/8.

Another way to look at the same problem is with partitive division.  If we have 6/8 ÷ 2, we could say, “What is 6/8 divided into two groups?”  Students can easily split their pieces into to groups to see that there are 3/8 in each group.

Let’s look at another problem, 1/3 ÷ 2.  Have students represent one-third.  We could say, “How many times can 2 fit into 1/3?”  Have students line up 6/3 to represent 2.  We have 1/3 out of 6/3.  Therefore, we have 1/6.  1/3 ÷ 2 = 1/6.

If you want to see the same problem using partitive division, have students represent 1/3.  We can’t break 1/3 into two pieces, but if we look at our fraction tiles, 2/6 = 1/3.  Therefore, 1/3 ÷ 2 = 1/6.

Let’s try one more, 3/5 ÷ 2.  “How many times can we fit 2 into 3/5.”  Have students line up 10/5 to represent 2.  We have 3/5 out of 10/5.  Therefore we have 3/10.  3/5 ÷ 2 = 3/10.

Using partitive division, have students represent 3/5.  We can’t break 3/5 into two pieces, but if we use our tenth pieces, we can see that 3/5 = 6/10 and 6/10 divided by 2 = 3/10.

Students may start to see that you can just multiply the whole number by the denominator.  That is a beautiful thing, because they are starting to see why multiplying by the reciprocal works!!  Don’t let them take short cuts yet!  We are building their fraction sense so when you finally teach the rule, they will have that deep understanding!!

~MN

 

Whole Number divided by a Fraction

Over the next few weeks we are going to dive deep into dividing fractions!  I can honestly say I have been teaching fractions for years and while I did understand why the multiplicative inverse worked, I never fully understood what happened when two fractions got divided.  Now, after looking at fraction division using manipulatives and visuals, I can finally say I get it!  It is amazing how much visuals really help in deep understanding.  Today we will be focusing on a whole number divided by a fraction.  I will be using Fraction Tiles which can be purchased here, or you can print and laminate a set of fractions tiles for free here.  

The first problem we will look at is 1 ÷ 1/8.  Or we could say, “How many one-eighth pieces are in 1 whole?”  Have students use their fraction tiles to justify their answer.  Students will see that there are 8 one-eighths pieces in 1 whole.

Try several similar problems using the fraction tiles and have students justify their answer.  For example, 2 ÷ 1/4, 1 ÷ 1/10.

What about 2 ÷ 2/3?  “How many 2/3 pieces are in 2?”  Have students represent 2 and then line up their third pieces.  Students should see how it takes 3 two-third pieces to make 2.

Then ask, “Why is our answer getting so much bigger?”  Have a class discussion on the fact that since the piece we are dividing by is smaller that the dividend, it can fit inside many times.  It is essential that students understand that when you divide a whole number by a fraction, your answer gets bigger.

What happens if the division doesn’t work out evenly?  For example, 2 ÷ 3/4.  In this case we have to look at the number of pieces that are left out.  We are able to make two whole groups with two-fourth pieces left out.  The two-fourths left out is two thirds of what we are dividing by.  Therefore, our answer would be 2 and 2/3.

Once students have a strong understanding, they can solve problems on a worksheet by drawing a model to get their answer.  I would not rush to multiplying by the reciprocal until a deep understanding is formed.

I hope you enjoy this lesson with your students.  I would love your feedback.

~MN

Comparing Fractions and Decimals

Students sometimes have a difficult time understanding how fractions and decimals are related and how they can be compared.  As always, providing a visual is essential for student understanding.  Once students have a strong understanding on how to compare fractions, you can add decimals to the mix.

Begin by having students put together their fraction tiles.  Give students 2 fractions to compare using one of the 5 strategies for Comparing Fractions.  For example, 2/5 and 2/8.  Have a discussion on how 2/5 is larger because both fractions have 2 pieces and fifths are larger than eighths.

Ask, how would the decimal 0.3 compare to those two fractions?  Students should realize that 0.3 is 3/10 and therefore would fall between the two fractions.

Finally, ask students “where would the decimal 0.35 fall?”  Discuss with students who this decimal is half way between 0.3 and 0.4.

This is a common misconception for students and providing that visual is a great way to strengthen student understanding of fractions and decimals.  Have students try several similar problems to strengthen their understanding of how decimals and fractions compare.

What strategies do you have for comparing fractions and decimals?

5 strategies for Comparing Fractions

Today we are going to talk about 5 strategies for comparing fractions.  Discussing and exploring each of these strategies with your students provides them with some great tools for comparing fractions as well as builds their number sense and understanding of fractions.  The visual is key to students understanding.  I believe using actual manipulatives that students can touch and move is ideal.  If you prefer, you can print and laminate a set of fractions tiles for free. 

Begin by having students line up all of the fraction tiles.  Have students write down everything they notice.  

• As the denominator gets bigger, the size of the pieces gets smaller
• The colors are all different
• There are no 7ths, 9ths or 11ths
• The number in the denominator is the same number of pieces it takes to make a whole

Have a class discussion about everything that students came up with and write all responses on the board. 

On the board write:

Number if pieces (numerator)

Size of the pieces (denominator)

Have a discussion with students that the numerator tells us how many pieces we have and the denominator tells us how big the pieces are.  For example, if we have the fraction ⅔, the two tell us that we have 2 pieces that are the size of 3rds.  

Strategy 1:  Common Denominator 

If the denominator is the same, you can compare the numerator which tells us the number of pieces.  If we are comparing 3/12 and 7/12, we can see that 7/12 is bigger because the pieces are the same size and 7 is more than 3.  Have students compare several fractions where the denominators are the same.  For example, 4/5 and 2/5, 3/10 and 4/10.   

Strategy 2:  Common Numerator

If the numerators are the same, you can compare the denominators (the size of the pieces).  The smaller the denominator, the larger the piece.  For example, if we compare 2/3 and 2/5, we can see that both fractions have 2 pieces but since thirds are larger pieces, 2/3 is greater than 2/5.  Have students compare several fractions where the numerators are the same.  For example, 4/10 and 4/8, 1/6 and 1/8.

Strategy 3:  Compliments 

Line up the fractions bars and remove one fraction tile from each whole.  Have students look at the fraction bars and discuss what they notice.  Guide students to notice that the fractions at the bottom are larger because they are missing a smaller piece.  Have a discussion with students about this.  Have students compare 7/8 and 11/12.  11/12 is bigger because it is missing a smaller piece and therefore leaving more.

Strategy 4: Benchmark to 1/2 

For this strategy, we will be compare our given fractions to 1/2.  Have students take the 1/2 fraction bar out and we will put one fraction above and one fraction below.  First we will compare 3/8 and 6/10.  Discuss how 3/8 is less than 1/2 (4/8) and 6/10 is greater than 1/2 (5/10).  Therefore, 6/10 is larger.

It gets a little trickier when both fractions are greater than or less than 1/2.  For example, let’s compare 5/8 and 6/10.  Both fractions are greater than 1/2.  Therefore, we have to look at how much greater they are to 1/2.  5/8 is 1/8 greater and 6/10 is 1/10 greater.  Since 1/8 is larger than 1/10, 5/8 is greater than 6/10.

When both fractions are less than 1/2 we have to think carefully.  Let’s compare 4/10 and 5/12.  Both fractions are less than 1/2.  4/10 is 1/10 less than 1/2 and 5/12 is 1/12 less than 1/2.  Since 1/10 is “more less than” 1/2, it is smaller.  Therefore, 4/10 is less than 5/12.

Strategy 5:  Common Sense

Sometimes when we compare fractions, common sense tells us which fraction is larger.  For example, if we look at 1/8 and 3/4 we could say that 3/4 is larger because it is close to 1 whole and 1/8 is closer to 0.  Give students some examples to try and have them discuss their reasoning on why one fraction is larger than the other.

 

I hope you enjoy all of the great math discuss that take place with exploring these strategies with your students.  I would love your feedback!  Please comment below with any questions or comments.

~MN