# Measuring Human Rights: High School Mathematics Unit

### Learning Objectives

Students will be able to:

- Recognize that only some data are well described by a normal distribution.

- Understand how the normal distribution used to make estimates of frequencies

- Use the 68-95-99.7 rule to estimate the percent of a normal population that falls within 1, 2, or 3 standard deviations of the mean.

### Standards Addressed

**Math Content Standards**

**S.ID.4**Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate.

- Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

**Math Practice Standards **

**MP.1**Make sense of problems and persevere in solving them.

**MP.3**Construct viable arguments and critique the reasoning of others.

**MP.4**Model with mathematics.

## Instructional Approach

### Teacher Introduction

Link human rights and underweight indicators: Today we are going to explore one of the indicators that are used to determine the state of populations in regards to the human right to have adequate food. This indicator is measuring the prevalence of children <5 who are underweight. So how do we know if someone is underweight?“ Have students share some ideas. (This is a review of the homework.) Write ideas on chart paper. (Recording tip: Write legibly in dark markers and use students’ own words.)

- Project photos of children between 0-5 years of age. After each photo, pause and ask the class
*“Is this boy underweight? Is this girl underweight? How do we know?”*Collect a few answers from the students. Make sure to ask them to briefly explain how they reached their conclusion.

- Have students go to the following webpage from the WHO website: http://apps.who.int/gho/data/node.main.1098?lang=en-

- Ask students if they can explain what this data set represents
*(The percentage of underweight children < 5 according to countries)*

- Once the class agrees (with or without help) on the purpose of this data set, continue to probe by asking:
*“How do you think the people who collected this data determined which children belong to the underweight category?”*After hearing a few answers, direct students to the first row on top of the last three columns*(“Children ages <5 underweight %”)*and ask them to click on the link. This will bring them back to a WHO webpage they studied earlier in Lesson 4, where they can review the definition of the WHO indicator for underweight children used in this data set.

- Project a slide of that page with the definition highlighted

**Definition:**

Percentage of underweight (weight-for-age less than -2 standard deviations of the WHO Child Growth Standards median) among children aged 0-5 years

### Teacher Explanation

To unpack the above definition, re-introduce students to the World Health Organization and the study WHO conducted to develop the standard growth charts (Multicenter Growth Reference Study – MGRS).

The WHO Multicentre Growth Reference Study (MGRS) was undertaken between 1997 and 2003 to generate new growth curves for assessing the growth and development of infants and young children around the world. Share and explain the following information with students:

The MGRS collected primary growth data and related information from approximately 8500 healthy children from widely different ethnic backgrounds and cultural settings (Brazil, Ghana, India, Norway, Oman and the USA). The study subpopulations had socioeconomic conditions favorable to growth, and low mobility, with at least 20% of mothers following feeding recommendations and having access to breastfeeding support. The individual inclusion criteria were absence of health or environmental constraints on growth, adherence to MGRS feeding recommendations, absence of maternal smoking, single term birth, and absence of significant morbidity.

The new growth curves are expected to provide a single international standard that represents the best description of physiological growth for all children from birth to five years of age. Discuss how the idea of a world “standard” relates back to earlier reading students have done about indicators and standards elated to the UDHR.

http://www.who.int/childgrowth/standards/cht_wfa_boys_p_0_6.pdf

- Project slide showing a photo of two children who are 4 years old. One of them is very tall and appears very thin, the other is short with what appears to be standard weight. Each of them weighs xx kg.

- Ask students: Do you think any of these children are underweight? Briefly discuss with students the problems with the category weight-for-age. Weight-for-age is a composite indicator and it is useful to see trends in a large population. There are other indicators such as height-for-weight that might do a better job in describing malnutrition. However, weight-for-age is often used in population studies and it might be because the data (weight) is relatively easy to collect.

- In general the weight-for-age for children under the age of five is not normally distributed. However, according to the MGRS study, the weight of 6-month-old boys are approximately normally distributed around 7.9kg with a standard deviation σ = 0.75 kg.

**Review Normal Distribution:** Before class breaks out for individual work, review the notion of normal distribution and the 68. 95, 99.7 rule.

- Draw a line on the board. Towards the left side of the board, draw a normal curve mean=10 and standard deviation=3. Mark the number line from 1 to 30 in increments of 3 (1, 4, 7, 10….28)

- Sketch on the board a normal curve with mean =10 and standard deviation =3

- Ask a volunteer to come up to the board and draw a sketch on the same axis with mean=20 and SD remains the same 3.

- Ask the class, how does the normal curve change when the mean varies and the standard deviation stays the same?

- On another part of the board draw again a normal curve with mean 10 and SD=3

- Ask another volunteer to sketch a normal curve (on the same axis) with a mean=10 and SD=1.

- Ask: How does the normal curve change when the mean stays the same and the SD varies?

**Project** the following normal distribution slide and remind students of the 68, 95, 99.7 rule:

### Individual Work

Distribute Handout #1 and ask students to individually complete the handout.

______________________________________________________________

**Handout #1: Weight for 6-month-old Boys**

__Underweight Definition__: Percentage of underweight (weight-for-age less than -2 standard deviations of the WHO Child Growth Standards median) among children aged 0-5 years

Use the above definition of underweight and assume the weight of 6-month-old boys are normally distributed with mean μ =7.9kg, and standard deviation σ = 0.75 kg. If we study a population of 1,000 boys who are 6 months old, how many of them would you expect to be underweight?

In some instances, the category of *underweight *is divided into two subcategories:

a) Moderately Underweight: *between -2 and -3 **of the WHO Child Growth Standards median*

a) Severe Underweight: *less than **-3 standard deviations of the WHO Child Growth Standards median*

Using the definition, how many of these boys do you expect to be severely underweight?

How many of them would you expect to be moderately underweight

Notice that our definition of underweight uses the *median, *yet in our calculation of standard deviation we use the *mean. *Assuming normal distribution, does it matter? Explain your position.

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### Small Group Work

- Divide students into pairs or trios. Explain that in the group work students are going to further investigate the distribution of weight-for-age of 6-month-old boys.

- Students in small groups share and refine the work they did individually on Handout #1. Encourage students to respectfully critique each other’s work and encourage them to come to an agreement about responses wherever possible and mark areas of disagreement for further discussion.

- Distribute Handout 2. Inform students that this handout includes further data about the weight of the thousand 6-month-old boys they investigated in Handout 2.

_______________________________________________________________

**Handout #2**

The following frequency table represents the weight of a population of 1000 6-month-old boys. The mean of this data set is μ =7.9kg, and the standard deviation is σ = 0.75 kg

Frequency Table of the weight of 1,000 six-month-old boys | |||||

Bins | Midpoint | Abs. Frequency | Rel. Frequency | Cumul. Rel. Freq. | Density |

[5.5,6[ | 5.75 | 3 | 0.003 | 0.003 | 0.006 |

[6,6.5[ | 6.25 | 29 | 0.029 | 0.032 | 0.058 |

[6.5,7[ | 6.75 | 85 | 0.085 | 0.117 | 0.17 |

[7,7.5[ | 7.25 | 166 | 0.166 | 0.283 | 0.332 |

[7.5,8[ | 7.75 | 258 | 0.258 | 0.541 | 0.516 |

[8,8.5[ | 8.25 | 247 | 0.247 | 0.788 | 0.494 |

[8.5,9[ | 8.75 | 137 | 0.137 | 0.925 | 0.274 |

[9,9.5[ | 9.25 | 60 | 0.06 | 0.985 | 0.12 |

[9.5,10[ | 9.75 | 10 | 0.01 | 0.995 | 0.02 |

[10,10.5[ | 10.25 | 3 | 0.003 | 0.998 | 0.006 |

[10.5,11] | 10.75 | 2 | 0.002 | 1 | 0.004 |

The following histogram represents the above Frequency Table and the normal curve (of the weight of 1,000 six-month-old boys). Use the information you can derive from this graph to answer the following questions:

1. Does this graph support the results you got on Handout #1? If it does, explain how you reached this conclusion. If not, highlight the discrepancies and explain why you think they occurred.

2. How do you know that this data is normally distributed?

3. If a six-month-old boy weight is 14.3 lb. Approximately, what weight percentile is he in? Is he underweight? (Remember: 1 kg=2.2lb)

4. If the weight of a six-month-old boy is 5.2 kg, approximately what weight percentile is he in? Is he underweight?

5. If a six-month-old boy is in the 2^{nd} percentile weight, what is your estimate of his weight? Would he be considered underweight?

6. If a six-month-old boy is in the 70^{th} percentile weight, what is your estimate of his weight? Would he be considered underweight?

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### Classroom Discussion

- Once the class reconvenes, project the frequency table and histogram on the wall and review the answers to these questions. Make sure to rotate among the different groups while they work. Encourage students to ask clarifying questions and challenge other students.

### Homework

Child growth is the most widely used indicator for the human right to have adequate food (“nutrition status”). In addition to weight-for-age, there are several other indicators that are used to monitor children’s growth and nutrition, including height-for-age, and height-for-weight. Out of all these indicators, height-for-age for children under the age of 5, follows a normal distribution. Similar to the definition of underweight, children who are less than -2 standard deviations of the WHO Child Growth Standards median are classified as stunted.

*Stunted Definition: Percentage of stunting (height-for-age less than -2 standard deviations of the WHO Child Growth Standards median) among children aged 0-5 years*

**Answer the following questions:**

- Given that height-for-age is normally distributed, what percent of children <5 in the world would you expect to have stunted growth? Why?
- If the
*percentage*of stunted children<5 would be equally distributed among the countries in the world, what % of stunted children would you expect in each country?

The following webpage from the WHO website has data about the percentage of stunted children <5 in various countries in the world. http://apps.who.int/gho/data/node.main.1097?lang=en. Go to this webpage and explore the distribution of stunted children<5 in various countries in the world (use data from 2000 and above).

Write down at least two observations you made about the distribution of stunted children under the age of 5 in the world.

**Informational Handout: Features of the WHO Multicenter Growth Reference Study (MGRS)***

*Specific communities in the study were selected on the basis of:*

-High socioeconomic status.

-Low altitude (<1,500 m/<4,921 f).

-Low population mobility, allowing for a 2-year follow-up.

-A minimum of 20% of mothers in the community were already following the international feeding recommendations.

*Study sites had to have a research institution capable of
conducting the study.*

*Feeding criteria specified that infants had to follow
international infant feeding recommendations, including*

-Predominantly breastfeeding for at least 4 months.

-Introducing complementary foods by at least 6 months but not before 4 months.

-Continued breastfeeding for at least 12 months (no upper limit on breastfeeding duration).

*Study sites had to have a breastfeeding support system that
included lactation consultants to provide support and counseling to all mothers
to ensure that any problems with breastfeeding were addressed quickly and that
mothers received appropriate counseling on complementary feeding.*

*Exclusion criteria included:*

-Maternal smoking during pregnancy or lactation.

-Prematurity (<37 weeks gestation).

-High gestational age (≥42 weeks).

-Multiple births.

-Substantial morbidity.

-Low socioeconomic status.

-Unwillingness of the mother to follow feeding criteria.

WHO collaborated with the United Nations University Food and Nutrition Program, United Nations Children's Fund, CDC, several governments, and others to develop the WHO growth standards to replace the 1977 NCHS/WHO growth reference that had been used in the international community.

*From: http://www.cdc.gov/nccdphp/dnpao/growthcharts/who/creating/mgrs.htm

### Differentiation and Supports

**Adaptations**

Struggling students: Use a detailed print out of the normal distribution graph. Make sure the 68, 95, 99.7 are marked clearly on it, and are color-coded.

Struggling: Use the Weight-for-age chart as a visual aid to answer the questions. I

Advanced: analyze the z score WHO weight-for-age table and look for other ages that have weights that are approximately normally distributed.

Describe a way to display the distribution of weight-for-age in population that is not normally distributed.

**Supports**

Support students when using websites, providing guidance about how to navigate the sites.

While students engage in small group activity, work with a small group of students who need extra support.

Use z score weight-for-age chart to visually answer questions (without exact calculations)

### Assessment:

Student learning can be assessed through their work on Handouts #1 and 2.

The teacher can also monitor participation in small group and class discussion.