Proposed ALF Math Book 6 Corrections

hfriedman

28 Proposed ALF Math Book 6 Corrections

*Note: For this to chapter to display properly, QuickLaTeX plugin must be deactivated.

This chapter details the issues and proposed changes from the Adult Literacy Fundamental Mathematics – Book 6.

The following solutions work with Apple’s screen reading software. Other screen readers have not been tested.

Operands within fractions

There are several examples as well as Exercises that use the format $\dfrac{4}{5}\left(\dfrac{\times2}{\times2}\right)$ . The two multiplication signs within the parenthesis are not read out correctly. The proposed change is to move the multiplication (or division sign in some cases) outside the parentheses.

Example A: Original

Express $4:5$ in higher terms.

$4:5=\dfrac{4}{5}\longrightarrow\dfrac{4}{5}\times\left(\dfrac{\times2}{\times2}\right)\longrightarrow\dfrac{8}{10}$

$4:5$ is equivalent to $8:10$

Example A: Proposed solution

Express $4:5$ in higher terms.

$4:5=\dfrac{4}{5}\longrightarrow\dfrac{4}{5}\times\left(\dfrac{2}{2}\right)\longrightarrow\dfrac{8}{10}$

$4:5$ is equivalent to $8:10$

Example A: Edited proposed solution

Express $4:5$ in higher terms.

$4:5=\dfrac{4}{5}\longrightarrow\dfrac{4}{5}\times\left(\dfrac{2}{2}\right)\longrightarrow \dfrac{4\times2}{5\times2}\longrightarrow\dfrac{8}{10}$

$4:5$ is equivalent to $8:10$

This issue is seen in Unit 1 (Topic A, B, C).

Crossed-off numbers when simplifying equations

Frequently, the book shows numbers or variables being struck through as a visual shorthand to show the number being reduced through simplification of fractions, such as

(1) $\begin{equation*}\begin{split} \dfrac{\cancel{6}\textit{N}}{\cancel{6}}\end{split}\end{equation*}$

. The screenreader doesn’t acknowledge this strike through (\cancel command) and simply reads the number as normal.

Adding an extra equation

In most cases, the surrounding text provides enough context that we can simply add an extra equation that shows the equation before and after simplification, like the example below.

Example B: Original

$\dfrac{6}{7}=\dfrac{24}{\textit{N}}$

Cross multiply:

(2) $\begin{equation*}\begin{split} 6\times\textit{N} &= 7\times24 \\ 6\textit{N} &= 168 \end{split}\end{equation*}$

Divide both sides by 6. The 6’s with the N will cancel (reduce), and the N will be alone.

(3) $\begin{equation*}\begin{split} \dfrac{\cancel{6}\textit{N}}{\cancel{6}} &= \dfrac{168}{6} \\ \\ \textit{N} &= 168\div6 \\ \\ \textit{N} &= 28 \\ \\ \dfrac{6}{7} &= \dfrac{24}{28} \end{split}\end{equation*}$

Check by cross-multiplying:

(4) $\begin{equation*}\begin{split} \text{Is }6\times28 &= 7\times24? \\ 6\times28 &= 168 \\ 7\times24 &= 168 \\ \text{the cross-product }168 &= \text{the cross-product }168 \\ \text{Yes - }6:7 &= 24:28 \end{split}\end{equation*}$

Example B: Proposed solution

$\dfrac{6}{7}=\dfrac{24}{\textit{N}}$

Cross multiply:

(5) $\begin{equation*}\begin{split} 6\times\textit{N} &= 7\times24 \\ 6\textit{N} &= 168 \end{split}\end{equation*}$

Divide both sides by 6. The 6’s with the N will cancel (reduce), and the N will be alone.

(6) $\begin{equation*}\begin{split} \dfrac{{6}\textit{N}}{{6}} &= \dfrac{168}{6} \\ \\ N &=\dfrac{168}{6} \\ \\ \textit{N} &= 168\div6 \\ \\ \textit{N} &= 28 \\ \\ \dfrac{6}{7} &= \dfrac{24}{28} \end{split}\end{equation*}$

Check by cross-multiplying:

(7) $\begin{equation*}\begin{split} \text{Is }6\times28 &= 7\times24? \\ 6\times28 &= 168 \\ 7\times24 &= 168 \\ \text{the cross-product }168 &= \text{the cross-product }168 \\ \text{Yes - }6:7 &= 24:28 \end{split}\end{equation*}$

This solution can be used in Units 1 Topic C, Unit 2 Topic A, Unit 3 Topic A, Unit 4 Topic A.

Adding an additional text explanation

In other cases, more explanation may be necessary. In this case, we’ll opt to put the equation into a table so we can provide more of an explanation when we remove the strike-throughs.

Example A: Original

*** QuickLaTeX cannot compile formula:
\begin{equation*}\begin{split}
\frac{3}{4} &= \text{_____}\% \ \ \ \ \ \ \ \ \ \ \ &\frac{3}{4} &\times 100\% \ &= \frac{3}{\cancel{4} 1} &\times \frac{\cancel{100} 25}{1} \ &= 75\% \\
1\frac{1}{5} &= \text{_____}\% \ \ \ \ \ \ \ \ \ \ \ 1&\frac{1}{5} &\times 100\% \ &= \frac{6}{\cancel{5} 1} &\times \frac{\cancel{120} 20}{1} \ &= 120\%
\end{split}\end{equation*}

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Example A: Proposed solution

$\frac{3}{4} = \text{_____}\%$

$\frac{3}{4}$

$\frac{3}{4} \times 100\%$

Multiply by 100%

$\frac{3}{4} \times \frac{100}{1}\%$

Convert 100% to a fraction.

$\frac{3}{1} \times \frac{25}{1}\%$

Divide 100 by 4. The 4 cancels and 100 is reduced to 25.

$3 \times 25\%$

Multiply 3 by 25%.

$75\%$

$1\frac{1}{5} = \text{_____}\%$

$1\frac{1}{5}$

$\frac{6}{5}$

Convert number to a fraction.

$\frac{6}{5} \times 100\%$

Multiply by 100%.

$\frac{6}{5} \times \frac{100}{1}\%$

Convert 100% to a fraction.

$\frac{6}{1} \times \frac{20}{1}\%$

Divide 100 by 5. The 5 cancels and 100 is reduced to 20.

$6 \times 20\%$

Multiply 6 by 20%.

$120\%$

This solution can be used in Unit 2 Topic A.

Arrows

This author uses arrows to indicate when the student should move the decimal point to the left or right while converting between decimals and percentages, which looks like

(9) $\begin{equation*}\begin{split} &\curvearrowright \\ 0&.125 \end{split}\end{equation*}$

. This symbol isn’t accessible as a screen reader does not pronounce it.

Removing the arrows and showing moved decimal point

In some cases, the surrounding text gives us adequate instructions or explanations so that we are able to simply remove the number with the arrow.

In this particular example, the following textbox is directly before the example:

To change a decimal to a percent, move the decimal point two places to the right and then write the percent sign after the number.

Example B: Original

Change each decimal to a percent.

(10) $\begin{equation*}\begin{split} &\curvearrowright \\ 0.125=0&.125=12.5\% \end{split}\end{equation*}$

(11) $\begin{equation*}\begin{split} &\curvearrowright \\ 1.375=1&.375=137.5\% \end{split}\end{equation*}$

Example B: Proposed solution

Change each decimal to a percent.

(12) $\begin{equation*}\begin{split} 0.125=12.5\% \end{split}\end{equation*}$

(13) $\begin{equation*}\begin{split} 1.375=137.5\% \end{split}\end{equation*}$

This solution can be used in Unit 2 Topic A.

Removing the arrows and multiplying/dividing by 100%

In some cases, it is simply enough to remove the arrows and show that the number or percentage is being multiplied or divided by 100%, respectively, since the surrounding text is enough to explain what is happening.

Exercise 4: Original

Change each percent to its decimal equivalent.

	Percent	÷ 100% Move decimal 2 places to left	= Decimal
A.	$23\%$	(14) $\begin{equation}\begin{split} &\curvearrowleft \\ & \ 23. \end{split}\end{equation}$	$=0.23$
B.	$1\%$
C.	$112\%$
D.	$10.3\%$
E.	$36\%$
F.	$147\%$

Answers to Exercise 4

$0.01$
$1.12$
$0.103$
$0.36$
$1.47$

Exercise 4: Proposed solution

Change each percent to its decimal equivalent.

	Percent	÷ 100% Move decimal 2 places to left	= Decimal
A.	$23\%$	$23. \div 100\%$	$=0.23$
B.	$1\%$
C.	$112\%$
D.	$10.3\%$
E.	$36\%$
F.	$147\%$

Answers to Exercise 4

$0.01$
$1.12$
$0.103$
$0.36$
$1.47$

This solution can be used in Unit 2 Topic A.

Specific changes

Unit 1 Topic C

The author uses $\dfrac{2}{5}\rlap{\nearrow}{\searrow}\dfrac{4}{10}$ to show which numbers are multiplied together when cross-multiplying. These arrows are only useful visually. I would propose to change the arrows to an equal sign as the rest of the text explains what numbers are being cross-multiplied.

Original

Review cross products:

$\dfrac{2}{5}\rlap{\nearrow}{\searrow}\dfrac{4}{10}$

(15) $\begin{equation*} \begin{split} 2\times10 & =5\times4 \\ 20 & =20 \end{split} \end{equation*}$

$2\times10=20$ and $5\times4=20$

Therefore: $\dfrac{2}{5}=\dfrac{4}{10}$

Remember that when the cross products are the same, the fractions are equivalent.

When finding the missing terms in a proportion, cross-multiplication can be used. Follow the examples carefully.

Proposed solution

Review cross products:

$\dfrac{2}{5} = \dfrac{4}{10}$ ?

(16) $\begin{equation*} \begin{split} 2\times10 & =5\times4 \\ 20 & =20 \end{split} \end{equation*}$

$2\times10=20$ and $5\times4=20$

Therefore: $\dfrac{2}{5}=\dfrac{4}{10}$

Remember that when the cross products are the same, the fractions are equivalent.

When finding the missing terms in a proportion, cross-multiplication can be used. Follow the examples carefully.

Unit 2, Topic A – 1

The author uses downward arrows that line up with the words fractions, decimals, and percents to show examples of what equivalent fractions, decimals, and percents look like. These are not read out correctly and is a purely visual shorthand. My proposed solution is to place it in a table after the text.

Original

Writing equivalent fractions is an important math skill.

(17) $\begin{equation*}\begin{split} \text{Equivalent common } &\text{fractions, } &\text{decimals, and } &\text{percents all represent the same amount} \\ &\downarrow &\downarrow &\downarrow \\ &\tfrac{1}{2} = &0.5 = &50\% \\ &\tfrac{3}{10} = &0.3 = &30\% \end{split}\end{equation*}$

Proposed solution

Writing equivalent fractions is an important math skill.

Equivalent common fractions, decimals, and percents all represent the same amount.

Equivalent fractions, decimals, and percentages
Fractions	Decimals	Percentages
$\frac{1}{2}$	$0.5$	$50\%$
$\frac{3}{10}$	$0.3$	$30\%$

Unit 2 Topic A – 2

In this case, both arrows and crosses are used. In this case, I would put the equations into a table along with text to better explain solution.

Original

If the decimal point moves to the end of the number it is not necessary to write the decimal point. Remember that zeros at the beginning of a number are also not necessary.

(18) $\begin{equation*}\begin{split} &\curvearrowright \\ 0.24=0&.24=\cancel{0}24\%=24\% \end{split}\end{equation*}$

(19) $\begin{equation*}\begin{split} &\curvearrowright \\ 0.05=0&.05=\cancel{00}5\%=5\% \end{split}\end{equation*}$

Proposed solution

If the decimal point moves to the end of the number it is not necessary to write the decimal point. Remember that zeros at the beginning of a number are also not necessary.

Convert 0.24 to a percentage
$0.24$
$24.00\%$	Move decimal point two places to the right and add percent sign.
$24\%$	Remove unnecessary zeros.

Convert 0.05 to a percentage
$0.05$
$5.00\%$	Move decimal point two places to the right and add percent sign.
$5\%$	Remove unnecessary zeros.

Unit 2, Topic A – 3

The long division in this example does not show up in either the PDF version correctly nor is it accessible for screen readers. I am unsure of alternatives other than reinserting it as an image with alt text.

As for the rest of the equation, I would put everything into a table, adding text explanation where necessary. I’d also remove the arrow when moving the decimal point.

Original

*** QuickLaTeX cannot compile formula:
\begin{equation*}\begin{split}
&  & \ \ 0.375 &\hspace{3.9cm}\curvearrowright \\
\tfrac{3}{8} &= \text{_____}\% \ \ \ \ \ \ \ \ \ \ \ 8&\overline{\smash{)}3.000} \ \ \ \ \ \ \ \ \ \ \ 0&\tfrac{3}{8}=0.375\hspace{1.5cm}0.375=37.5\% \\
&  & \ \ 2 \ 4 \smash{\downarrow}\smash{\downarrow} \\
&  &  \ \ \ \ \ 60 \smash{\downarrow} \\
&  &  \ \ \ \ \ \underline{56} \smash{\downarrow} \\
&  &  \ \ \ \ \ \ \ 40
\end{split}\end{equation*}

*** Error message:
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*** QuickLaTeX cannot compile formula:
\begin{equation*}\begin{split}
&  & \ \ 0.33\overline{3} &\curvearrowright \\
\tfrac{1}{3} &= \text{_____}\% \ \ \ \ \ \ \ \ \ \ \ 3&\overline{\smash{)}1.000} \ \ \ \ \ \ \ \ \ \ \ 0&.33\overline{3}=33.\overline{3}\% \\
&  &  &\text{also written as }33\tfrac{1}{3}\%
\end{split}\end{equation*}

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*** QuickLaTeX cannot compile formula:
\begin{equation*}\begin{split}
&  &  \ \ \ \ 0.916\overline{6} &\curvearrowright \\
\tfrac{11}{12} &= \text{_____}\% \ \ \ \ \ \ \ \ \ \ \ 12&\overline{\smash{)}11.000} \ \ \ \ \ \ \ \ \ \ \ \ 0&.91\overline{6}=91.\overline{6}\% \\
&  & \ \ 10 \ 8 \smash{\downarrow}\smash{\downarrow} \\
&  &  \ \ \ \ \ \ \ 20 \smash{\downarrow} \\
&  &  \ \ \ \ \ \ \ 12 \smash{\downarrow} \\
&  &  \ \ \ \ \ \ \ \ \ 80 \\
&  &  \ \ \ \ \ \ \ \ \ \underline{72} \\
&  &  \ \ \ \ \ \ \ \ \ \ \ 8
\end{split}\end{equation*}

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Proposed solution

Convert $\frac{3}{8}$ to a percentage
$\frac{3}{8} = \text{_____}\%$
	Use long division to write fraction as a decimal.
$\frac{3}{8} = 0.375$
$0.375 = 37.5\%$	Move the decimal point two places to the right and add the percent sign.

Convert $\frac{1}{3}$ to a percentage
$\frac{1}{3} = \text{_____}\%$
	Use long division to write fraction as a decimal.
$\frac{1}{3} = 0.33\overline{3}$
$0.33\overline{3}=33.\overline{3}\% \text{ also written as } 33\frac{1}{3}\%$	Move the decimal point two places to the right and add the percent sign.

Convert $\frac{11}{12}$ to a percentage
$\frac{11}{12} = \text{_____}\%$
	Use long division to write fraction as a decimal.
$\frac{11}{12} = 0.91\overline{6}$
$0.916\overline{6} = 91.\overline{6}\%$	Move the decimal point two places to the right and add the percent sign.

Unit 2 Topic A – 4

The long division in this example does not show up in either the PDF version correctly nor is it accessible for screen readers. I am unsure of alternatives other than reinserting it as an image with alt text.

As for the rest of the equation, I would simply remove it from the {equation} and simply have it in simple LaTeX.

Original

(23) $\begin{equation*}\begin{split} & \ \ \ \ \ \ 1 \\ 110\%=\tfrac{110}{100}=1\tfrac{10}{100}=1\tfrac{1}{10} \ \ \ \ \ \ \ \ \ \ \ 100&\overline{\smash{)}110} \\ & \ \ \underline{100} \\ & \ \ \ \ 10 \end{split}\end{equation*}$

(24) $\begin{equation*}\begin{split} & \ \ \ \ \ \ 1 \\ 120\%=\tfrac{120}{100}=1\tfrac{20}{100}=1\tfrac{1}{5} \ \ \ \ \ \ \ \ \ \ \ 100&\overline{\smash{)}120} \\ & \ \ \underline{100} \\ & \ \ \ \ 20 \end{split}\end{equation*}$

Remember 100% is the whole thing. $100\%=1$ .

Proposed Solution

$110\%=\tfrac{110}{100}=1\tfrac{10}{100}=1\tfrac{1}{10}$ equation alt text

$120\%=\tfrac{120}{100}=1\tfrac{20}{100}=1\tfrac{1}{5}$ equation alt text

Remember 100% is the whole thing. $100\%=1$ .

License

Icon for the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License

Operands within fractions

Crossed-off numbers when simplifying equations

Adding an extra equation

Adding an additional text explanation

Arrows

Removing the arrows and showing moved decimal point

Removing the arrows and multiplying/dividing by 100%

Specific changes

Unit 1 Topic C

Original

Proposed solution

Unit 2, Topic A – 1

Original

Proposed solution

Unit 2 Topic A – 2

Original

Proposed solution

Unit 2, Topic A – 3

Original

Proposed solution

Unit 2 Topic A – 4

Original

Proposed Solution

License

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