Homework Headaches

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Homework Headache: Solving Fractions Using Bar Graphs

Hi Homework Unlocked, 

My whole family (husband, daughter and even our cat and dog 😉 are having trouble solving this problem.  Could you give us a hand?

HH_fractions

Thanks!

The Harrod Family

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Not to worry, Homework Unlocked is here to help. We’ll walk you through this problem step by step so you can explain it to your daughter (plus your dog and cat!). This is a great example of a homework headache because it uses a bar diagram to solve a word problem with some algebra mixed in …phew!

Since we are given a partially completed bar diagram some of our work is already done for us, but we’ll start from the beginning so you can explain each step of the problem to your kids.

Step 1. When we start working with word problems read through them carefully and underline the relevant information.   

Problem states: Of the beads in a box, ¼ are red. There are 24 more yellow beads than red beads. The remaining 76 beads are blue. How many beads are there altogether?

Step 2. Let’s work with each piece of underlined information to recreate the diagram from scratch. 

A) ¼ of the beads are red: Simply draw a bar and shade a quarter of it in red.

hh_fractions_step1a

B) There are 24 more yellow beads than red beads.  We know that ¼ of the bead are red so color another ¼ in yellow and then mark 24 additional beads.

hh_fractions_step1b

C) The remaining beads are blue- this is the easiest step, color all the remaining beads in blue.

hh_fractions_step1c

D) Now we need to solve for the total number of beads

hh_fractions_step1d

Step 3. Now that we can visualize the problem we can work on solving it.  Looking at the problem we can easily see that ½ of the total beads equals 24 + 76, let’s write this out as an equation- we’ll represent the total number of beads with an X.

hh_fractions_step3_1

Now let’s solve for X to figure out how many beads we had total. 

hh_fractions_step3_2  

Now we need to isolate the X. To do this let’s get rid of the ½ by multiplying both sides of the equation by 2.

hh_fractions_step3_3

2/2 = 1 and we are left with    X = 200

Step 4. Let’s check our work by solving for the number of beads in each color category. 

We’re told ¼ of the beads are red.  We now also know that the total number of beads = 200, therefore, there are 50 red beads. There are 24 more yellow beads than red beads, therefore, there are 50 + 24 = 74 yellow beads and 76 blue beads.  If our work is correct these should add to make 200. 

50 + 74 + 76 = 200        Our work checks!

Sincerely,

Homework Unlocked

Do you have a homework problem you can’t explain to your kids? Don’t give up!

Email it to homeworkheadaches@homeworkunlocked.com

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Thank you to Anne for contributing our first Homework Headache problem!

Anne (like so many parents!) is struggling to help her 6th grade student complete his math homework. Her son brought home a homework assignment that asked him to find the greatest common factor of two numbers using three different methods. Let’s solve Anne’s and her son’s problem by walking through each method! (Hint: If we are right, our answer will be the same for each method!)

  • examples
  • homework worksheet

Method 1: Find all the factors

method1

 

You’re given the following

factors

 

 

 

 

method1_text

Method 2: Prime Factorization

method2

You’re given following:

method2_text

 

To find the greatest common factor of two numbers through prime factorization, first find the greatest common prime factor of the two numbers and then multiply them together. The product is the greatest common factor.

Let’s start with 20

Step 1. Let’s create a prime factor pyramid for the number 20. Look at our list of factors. We see that the prime pyramid20factors of 20 are 5 and 2.

 

20 = 2 ⋅ 5 · 2

 

 

Next let’s create a prime factor pyramid for 32

pyramid32

Step 2. The only prime factor of 32 is 2.

 

32 = 2 ⋅ 2 · 2 ⋅ 2 ⋅ 2

 

 

 

 

 

 

Step 3. The common prime factor of 20 and 32 is 2. We can find our greatest common factor by multiplying common prime factors from each number.

Greatest common factor

= 2 ⋅ 2

= 4

Thus, the greatest common factor of 20 and 32 is 4.

Method 3: Using a Factor Chart to Find the Greatest Common Factor

method3

 

You’re given following:

 

method3_text

 

Method 3 is another alternative way of finding the largest common factor of 20 and 32 through prime factorization. T0 review this method lets first refer back to the example we were given:

method3_ex

Tip:
In this example we see a factor chart where the numbers we start with, in this example 45 and 75 are inside the chart on the top layer. If we divide through by 3 (one of our prime common factors, placed outside the chart on the upper left side) we get the numbers on the inside lower layer, 15 and 25. Next, multiply through by the other prime factor, 5 to get 3 and 5 written below the chart.

Step 1. Use the prime factor that is given to complete the inside of the factor chart. If we look at our factor chart, we see you’re given 2. Complete the inside of the chart by dividing both 20 and 32 by 2.

method3_step1

 

Step 2. Find the second prime factor: If we look at our factor chart, we still don’t know the second prime factor, however we do know the result of dividing 10 and 16 by the missing factor, so let’s work backwards

10 ÷ ? = 5, and 16 ÷ ? = 8

10 ÷ 5 = 2, and 16 ÷ 8 = 2,

Thus, we know the missing prime factor is 2 and we can complete our factor chart!

method3step2

 

Step 3. Multiply the common prime factors to find the greatest common factor

2 x 2 = 4

Therefore, the greatest common factor of 20 and 32 is 4 .

For more information on Factors, please see the Homework Unlocked video “Finding Factors.”

 

Phew. We got the same answer (4) using these three methods to find the greatest common factor of 20 and 32.

Anne, we hope that our solution helps you as well as other parents with similar problems. Let us know! And parents, keep on sending us your homework and we will try to answer as many as possible.

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Thank you Michelle for contributing our newest homework headache.

Michelle is having trouble helping her daughter Tanya complete her math homework that asks Tanya to find the area of an odd shape.  Here is the problem that Michelle and Tanya are struggling to solve:

hh2_question

Let’s solve this problem step by step!

When you approach a problem like this one, first encourage your kids to list all of the relevant information they already know as well as the information they are given in the problem.

Based on prior knowledge, we know:

1: How to find the area of a triangle:

area

 

 

2: How to find the Area of a rectangle:

area_rectangle

 

 

3) To find the area of the remaining piece of paper, we need to know the area of the entire rectangle and then subtract the area of each triangle.  Since there are five equal triangles:

 Area of the remaining piece of paper = area of rectangle – 5 x area of triangle

 Next, let’s look at what information we are directly given:

 

  • If we look at the shape, we see that the height of the whole shape is 30 cm and base is 60 cm.
  • We’re also told that there are 5 identical triangles in the 60 cm page.
  • We are told that the height of each triangle is exactly half of the height of the whole shape.

Now, ask your child what other information we need to know?  Remember, problems like this are usually solved by finding the area of the larger shape and then subtracting the area of the smaller shapes.  Therefore we need to know the area of the larger shape, the rectangle, and the area of the smaller shape, the triangles.

Now, let’s solve it:

hh2_areaStep 1:  Find the area of the larger shape, the rectangle. We can find the area of the rectangle simply by multiplying base x height.

Area of a Rectangle = Length of the Base x Length of the Height

Area of the Rectangle =  60 cm x 30 cm

Area of the Rectangle = 1,800 cm2

hh2_area_step2Step 2:  Now let’s find the area of all 5 triangles. Michelle, at this point, it is helpful to remind Tanya to go back to the original problem.  Even though we aren’t directly told the base and height of each triangle, the answer is there!

 

Remember, from reading the problem, we know that the height of each triangle is half of 30 cm, therefore:

htequals

We also know that “Sharon” cut five identical triangles.  Since the base of the entire shape is 60 cm,

btequals

Step 3:  Now subtract the area of all five triangles from the area of the entire rectangle since

Area of the Rectangle –  Area of 5 Triangles = Area of the Grey Shape

1,800 cm2 – 450 cm2 = 1,350 cm2

Therefore, the Area of the Grey Shape is 1,350 cm2

owl _headache_93Thank you Lynsey for contributing this Homework Headache.

Lynsey is having trouble helping her son, Max, with his math homework that asks to find the quotient of two decimals using a number line.   Here is the problem that Max is struggling to solve:

hh4-problem

Lynsey, like so many of us, learned division using more traditional methods and is struggling with the ‘new methods’.  Don’t worry Lynsey, Homework Unlocked is here to help.  We’re going to help you explain this problem step by step.

How to Find a Quotient Using A Number Line

The problem states:

Find 0.9 ÷ 0.3

The division expression 0.9 ÷ 0.3 means “How many __?__s are in __?__?”


Step 1: Rewrite the expression

When the problem states a number ‘a’ is being divided by a number ‘b’ it’s really asking how many ‘b’s’ can fit into number ‘a’. Therefore, another way of writing 0.9 ÷ 0.3 that might make more sense to your child is,

How many 0.3s are in 0.9?

Step 2. Fill in all the intervals of our number line:

In this problem we are given a number line with the following information

hh4-numline-blanks

Since we are trying to answer the question, “how many 0.3s are in 0.9,” our number line interval in this example is 0.3.  Instruct your child to fill in each blank on our number line by adding 0.3 to the number to the left of the blank.  Since we are adding, move from left to right.

Looking at the number line, we see that our first number is 0.

0 + 0.3 = 0.3, now we can fill in our fist blank with 0.3.

hh4-numline-oneblank

Now let’s move one step to the right to the next blank.

0.3 + 0.3 = 0.6, now we can fill in our second blank with 0.6.

hh4-numline-noblanks

Before we move on, it’s helpful to remind your child to check his or her work by adding another interval.  If we did this correctly our second blank (0.6) plus our interval (0.3) will equal 0.9.

0.6 + 0.3 = 0.9 Great!

Step 3 Solve:

At this point, go back through the original problem with your child.  Remember, the problem asks:

  1. What value is each small interval?
  2. How many 0.3’s are in 0.9?
  3. So 0.9 ÷ 0.3 = ?

Show your child that they solved part 1—each small interval represents 0.3!

Next, look at the number line to answer part 2.

Since our interval is 0.3 we can simply count how many intervals are in our number line to answer this question. 

hh4-numline-wjumps

Counting from left to right we see that it takes 3 intervals of 0.3 to get from 0 to 0.9.  Therefore, there are 3, 0.3s in 0.9.

Finally, we need to solve for 0.9 ÷0.3.

0.9 ÷ 0.3 =  number of intervals = 3

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The last few Homework Headaches we’ve chosen were from the parents of older students, so we decided to mix it up with a headache from a parent of a 3rd grader.

Thanks Dinah for the submission!

hh_cubes

The problem above depicts a 3-step staircase made up of 6 cubes. Using this as a guide, the problem asks how many cubes we would need to build a staircase with 11 steps. 

When a problem gives you an example and then asks a question, like this one, make sure you go over the example carefully first. Guide your child through what we already know.

1) In our staircase, each step has one less cube than the step below it. 

2) The number of cubes at the base of our staircase must equal the total number of steps in the staircase.

hh_cubes_1

In this example, we are given both the total number of cubes and the number of steps in the staircase but let’s see if we can solve for the number of cubes using the number of steps. That way we can test the method that we will use to calculate the number of cubes in our 11-step staircase.

We know that the base of the staircase is made of three cubes and each step has one less cube than the step below it- so we can just add the cubes up step by step, starting with the bottom and working our way to the top:

hh_cubes_2

3 + 2 + 1  =  total number of cubes in the staircase = 6

Great! This agrees with what we already knew! And it shows us that we can solve for the number of cubes knowing only how many steps we started with. 

Let’s apply that same method to a staircase with 11-steps. We know we have 11 cubes in the base and we know each step will have one less cube than the step below it- so let’s just add the cubes up step by step!

11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 66

A staircase with 11-steps would have 66 cubes.

hh_cubes_3

All done! Often with a problem like this our kids just want to count all the boxes but it’s important kids learn the skills to calculate rather than count!

Dear Homework Unlocked,

My 4th grade son is learning several different methods of long division and came home today with an assignment to work on ‘Division without Dividing’.  His math notes show 1 example. I understand long division but neither of us can work out what this example shows.  Any insight?

Here’s the example:

hh_5

Thanks,

Jerry

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Hi Jerry,

Thanks for our newest Homework Headache.  The example your son brought home is one of the new methods of long division that our kids are learning in school.   This method uses the product of multiples of 10 and the divisor to break up the dividend into smaller parts- in effect, it allows you to divide without using division. This particular method works really well for students who find multiplication much easier than division. 

Your son’s example shows the 423 ÷ 3.  We’ll go through the same problem step by step to show you how this method works.hh_5_setup  We’ve added some color-coding so you can follow more easily.

First, set up the problem in a division bracket just like you normally would but place a vertical line to the right of the bracket.

hh5_steps

  423 ÷ 3 = 141! hh5_solution

All done! 

owl _headache_93

Thanks Howard for our latest Homework Headache.  Howard has a question about converting decimals into fractions, he wrote:

Dear Homework Unlocked,

I’m on Homework Help duty tonight and I can’t for the life of me remember how to go from a decimal to a fraction.  I know 0.5 equals ½ but we’re trying to find out what fraction 0.04 equals and we are having trouble- can you help?

Best,

Howard

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Dear Howard,

Most of us don’t convert decimals to fractions every day (or even once a decade!) so there’s no surprise that you don’t remember.  We’ll walk you through it step by step. We just posted some new videos on decimals and fractions so for more examples and information see our lesson library here: “Fractions Lessons”.

Let’s get started!

Problem: Convert 0.04 into a fraction

To Change a Decimal into a Fraction we need to follow the steps below:

Rewrite the decimal as a fraction

Determine what multiple of 10 will make the decimal a whole number.

Multiply the numerator and denominator by the same multiple of 10 to get rid of the decimal. 

Step 1. Rewrite the decimal as a fraction

Any number can be rewritten as a fraction; that number over 1. 

For example, 7 is equal to hh_7_fraction

This is true for decimals as well as whole numbers. 

So for our current example let’s rewrite 0.04 as hh_04_fraction 

Step 2. Determine what multiple of 10 will make the decimal a whole number.

Now that we have our fraction hh_04_fraction  we need to remove the decimal place.  To do this we have to figure out what multiple of 10 to multiply the decimal by to make it a whole number.  In this case, we have to shift the decimal point two places to the right to make 0.04 a whole number, 4.

hh_convert_04

To shift a decimal point two places to the right we have to multiply by 100

Step 3. Multiply the denominator and numerator by the same multiplier to remove the decimal point in your fraction. 

Since we had to multiply the numerator by 100 to make 0.04 a whole number we also have to multiply the denominator by the same value. Remember, if you only multiplied the numerator we would change the value of our fraction, which we don’t want to do! 

hh_04_fractions_convert

0.04 is equal to hh_04_as_fraction

We hope this helps, Howard!  Remember, if you want a video version of how to convert a decimal into a fraction, watch this lesson.

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Homework Headache: Using Number Bonds

For this week’s homework headache, we chose an example that has recently made the rounds as a common source of common core math confusion.  The problem asks the student to use “Number Bonds” to add.

hh7_bond

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Since it is easier to add a 1-digit number to 10 than two 1-digit numbers whose sum is greater than 10, this problem asks you to break up numbers into smaller parts so you can create 10 and then add what is left over.  For most of us, this might seem a bit silly; particularly since as adults we can pretty quickly figure out that 8 + 7 is 15 in our heads.

For many of our kids, however, this mental calculation does not come as naturally.  Therefore, if our kids can learn to break down smaller numbers into something that equals 10 their Mental Math may improve. 

Now let’s get started.  In the first example, 8 + 7 you are given two circles below the 7. This means you need to break up the 7 into two smaller numbers.  Since we are trying to make the 8 into a 10 in order to do the mental calculation, the first number should be the difference between 8 and 10. In other words, ask yourself, what do you need to add to 8 to get to 10.  The answer is 2 so in the first circle under the 7 write a 2.

hh7_step1

7 – 2 equal 5 so place a 5 in the second circle below the 7.  Now when you add the numbers up you can just add 10 to the number in the second circle.  10 + 5 = 15.

Now let’s go through one more example on the upper row together. 

Example 2.

hh7_step2

In this example, first ask yourself how do you turn 6 into a 10.  Since 6 + 4 =10, write a 4 in the first circle.  Next, complete the second circle by subtracting 4 from 5 to get 1.  6 + 5 is equivalent to 10 + 1, but conceptually it is much easier to see that 10 + 1 = 11 rather than 5 + 6.

Here are the answers to the other examples on the top row:

Example 3.

hh7_ex3

Ask your child what number plus 7 equals 10.  Since 7 + 3 =10, record that 3.  Next subtract 3 from 6.  6 – 3 = 3. Record a 3 in the second circle.

Example 4.

hh7_ex4

9 plus what number equals 10?  Since 9 + 1 =10, record that 1 in the first circle.  Next subtract 1 from 5.  5 – 1 = 4. Record a 4 in the second circle.

Need more help? For more complicated problems using Mental Math see our Mental Math Videos in the Homework Unlocked lesson library. 

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You may link to and publish this content provided you acknowledge HomeworkUnlocked as the original source.  www.homeworkunlocked.com

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